Galileo: The Starry Messenger, Letter to the Grand Duchess Christina, and Two New Sciences (First Day)

Source

Source: The Sidereal Messenger, trans. Edward Stafford Carlos (London: Rivingtons, 1880), Project Gutenberg #46036; Letter to the Grand Duchess Christina, trans. Thomas Salusbury, Mathematical Collections and Translations (London, 1661), EEBO-TCP A61244 (CC0); Dialogues Concerning Two New Sciences, trans. Henry Crew and Alfonso de Salvio (New York: Macmillan, 1914), via the Internet Archive (dialoguesconcern00galiuoft) Source URL: https://www.gutenberg.org/ebooks/46036 Translator: Edward Stafford Carlos; Thomas Salusbury; Henry Crew and Alfonso de Salvio Public domain basis: Public-domain translations. Selection: Sidereus Nuncius complete (1610); the Letter to Christina complete (1615) in Salusbury's 1661 spelling; Two New Sciences, the First Day only (1638; OCR, diagrams omitted). Letter: https://github.com/textcreationpartnership/A61244 ; Two New Sciences: https://archive.org/details/dialoguesconcern00galiuoft

The Starry Messenger (1610)

THE SIDEREAL MESSENGER

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THE SIDEREAL MESSENGER

RIVINGTONS

London Waterloo Place

Oxford Magdalen Street

Cambridge Trinity Street

[Illustration: GALILEO’S BROKEN LENS.

EXHIBITED IN THE LOAN COLLECTION OF SCIENTIFIC APPARATUS AT THE SOUTH KENSINGTON MUSEUM, 1876.

From a photograph of the Science and Art Department, South Kensington.]

THE SIDEREAL MESSENGER

OF

GALILEO GALILEI

AND A PART OF THE PREFACE TO KEPLER’S DIOPTRICS

CONTAINING THE ORIGINAL ACCOUNT OF GALILEO’S ASTRONOMICAL DISCOVERIES.

A Translation with Introduction and Notes

BY

EDWARD STAFFORD CARLOS, M.A.

HEAD MATHEMATICAL MASTER IN CHRIST’S HOSPITAL.

RIVINGTONS WATERLOO PLACE, LONDON Oxford and Cambridge MDCCCLXXX

PREFATORY NOTE.

About five years ago I was engaged in preparing a catalogue of the ancient books which belong to Christ’s Hospital. One portion of these books consisted of a collection of ancient mathematical works presented at various times for the use of that part of the school which is known as the Royal Mathematical Foundation of King Charles II. Amongst them were some well known by name to every mathematical student, but which few have ever seen. Perhaps the most interesting of them all was a little volume, printed in London in 1653, containing Gassendi’s Explanation of the Ptolemaic and Copernican Systems of Astronomy, as well as that of Tycho Brahe, Galileo’s Sidereus Nuncius, and Kepler’s Dioptrics. I found Galileo’s account of his astronomical discoveries so interesting, both in matter and in style, that I translated it as a recreation from school-work. I venture to think that others also will be interested in following Galileo through the apprehension of his famous discoveries, and in reading the language in which he announced them.

INTRODUCTION.

In 1609, Galileo, then Professor of Mathematics at Padua, in the service of the Venetian Republic, heard from a correspondent at Paris of the invention of a telescope, and set to work to consider how such an instrument could be made. The result was his invention of the telescope known by his name, and identical in principle with the modern opera-glass. In a maritime and warlike State, the advantages to be expected from such an invention were immediately recognised, and Galileo was rewarded with a confirmation of his Professorship for life, and a handsome stipend, in recognition of his invention and construction of the first telescope seen at Venice. In his pamphlet, The Sidereal Messenger, here translated, Galileo relates how he came to learn the value of the telescope for astronomical research; and how his observations were rewarded by numerous discoveries in rapid succession, and at length by that of Jupiter’s satellites. Galileo at once saw the value of this discovery as bearing upon the establishment of the Copernican system of astronomy, which had met with slight acceptance, and indeed as yet had hardly any recommendation except that of greater simplicity. Kepler had just published at Prague his work on the planet Mars (Commentaria de motibus Stellæ Martis), on which he had been engaged apparently for eight years; there he heard of Galileo’s discoveries, and at length was invited by Galileo himself, through a common friend, Giuliano de’ Medici, ambassador of the Grand-Duke of Tuscany, Cosmo de’ Medici II., to the Emperor Rudolph II., to correspond with Galileo on the subject of these discoveries. The Emperor also requested his opinion, and Kepler accordingly examined Galileo’s Sidereal Messenger in a pamphlet, entitled A Discussion with the Sidereal Messenger (Florence, 1610).

In this Discussion Kepler gives reasons for accepting Galileo’s observations—although he was not able to verify them from want of a telescope—and entirely supports Galileo’s views and conclusions, adducing his own previous speculations, or pointing out, as in the case of Galileo’s idea of earth-light on the moon, the previous conception of the same explanation of the phenomenon. He rejects, however, Galileo’s explanation of the copper colour of the moon in eclipses. Kepler ends by expressing unbounded enthusiasm at the discovery of Jupiter’s satellites, and the argument it furnishes in support of the Copernican theory.

Soon after, in 1611, Kepler published another pamphlet, his Narrative, giving an account of actual observations made in verification of Galileo’s discoveries by himself and several friends, whose names he gives, with a telescope made by Galileo, and belonging to Ernest, Elector and Archbishop of Cologne. Kepler and his friends saw the lunar mountains and three of the satellites of Jupiter, but failed to make out any signs of the ring of Saturn corresponding to the imperfect description of Galileo.

Kepler had previously published a treatise on Optics (Frankfort, 1604). He now extended it to the consideration of the theory of the telescope, and explained the principle of Galileo’s telescope; he also showed another combination of lenses which would produce a similar effect. This was the principle of the common astronomical telescope, often called, from this circumstance, Kepler’s telescope, though he did not construct it. The account of Galileo’s later astronomical discoveries of Saturn’s ring and the phases of Venus is taken from the preface of this work.—(Kepler’s Dioptrics; Augsburg, 1611.)

In 1612 Galileo published a series of observations of solar spots, and in 1618 some observations of three comets. There exist also long series of minute observations of Jupiter and his satellites, continued to November 1619.—(Galileo’s Works; Florence, 1845.)

Further astronomical researches may have been hindered by failing sight. One more astronomical discovery, however, that of the moon’s librations, was made as late as 1637, and the announcement of it is dated “dalla mia carcere di Arcetri.” Galileo died January 8, 1642.

The following editions have been used for the translation:—

Galileo’s Works.

1. Florence, 1718. 2. Padua, 1744. 3. Florence, 1842-56.

Sidereus Nuncius.

1. Venice, 1610. 2. London, 1653.

Kepler’s Works, ed. C. Frisch. Frankfurt a. M., 1858-71.

Prodromus dissertationum mathematicarum continens Mysterium Cosmographicum de admirabili proportione orbium cœlestium. Tübingen, 1596.

Astronomia nova αἰτιολογητός (Commentaria de motibus stellæ Martis). [Prague,] 1609.

THE

SIDEREAL MESSENGER

OF

GALILEO GALILEI

THE

SIDEREAL MESSENGER

UNFOLDING GREAT AND MARVELLOUS SIGHTS, AND PROPOSING THEM TO THE ATTENTION OF EVERY ONE, BUT ESPECIALLY PHILOSOPHERS AND ASTRONOMERS,

BEING SUCH AS HAVE BEEN OBSERVED BY

GALILEO GALILEI

A GENTLEMAN OF FLORENCE, PROFESSOR OF MATHEMATICS IN THE UNIVERSITY OF PADUA,

WITH THE AID OF A

TELESCOPE

lately invented by him,

_Respecting the Moon’s Surface, an innumerable number of Fixed Stars, the Milky Way, and Nebulous Stars, but especially respecting Four Planets which revolve round the Planet Jupiter at different distances and in different periodic times, with amazing velocity, and which, after remaining unknown to every one up to this day, the Author recently discovered, and determined to name the_

MEDICEAN STARS.

Venice 1610.

TO THE MOST SERENE

COSMO DE’ MEDICI, THE SECOND,

FOURTH GRAND-DUKE OF TUSCANY.

There is certainly something very noble and large-minded in the intention of those who have endeavoured to protect from envy the noble achievements of distinguished men, and to rescue their names, worthy of immortality, from oblivion and decay. This desire has given us the lineaments of famous men, sculptured in marble, or fashioned in bronze, as a memorial of them to future ages; to the same feeling we owe the erection of statues, both ordinary and equestrian; hence, as the poet[1] says, has originated expenditure, mounting to the stars, upon columns and pyramids; with this desire, lastly, cities have been built, and distinguished by the names of those men, whom the gratitude of posterity thought worthy of being handed down to all ages. For the state of the human mind is such, that unless it be continually stirred by the counterparts[2] of matters, obtruding themselves upon it from without, all recollection of the matters easily passes away from it.

[1] Propertius, iii. 2. 17-22.

[2] Compare Lucretius iv. 881:

Dico animo nostro primum simulacra meandi Accidere, atque animum pulsare.

But others, having regard for more stable and more lasting monuments, secured the eternity of the fame of great men by placing it under the protection, not of marble or bronze, but of the Muses’ guardianship and the imperishable monuments of literature. But why do I mention these things, as if human wit, content with these regions, did not dare to advance further; whereas, since she well understood that all human monuments do perish at last by violence, by weather, or by age, she took a wider view, and invented more imperishable signs, over which destroying Time and envious Age could claim no rights; so, betaking herself to the sky, she inscribed on the well-known orbs of the brightest stars—those everlasting orbs—the names of those who, for eminent and god-like deeds, were accounted worthy to enjoy an eternity in company with the stars. Wherefore the fame of Jupiter, Mars, Mercury, Hercules, and the rest of the heroes by whose names the stars are called, will not fade until the extinction of the splendour of the constellations themselves.

But this invention of human shrewdness, so particularly noble and admirable, has gone out of date ages ago, inasmuch as primeval heroes are in possession of those bright abodes, and keep them by a sort of right; into whose company the affection of Augustus in vain attempted to introduce Julius Cæsar; for when he wished that the name of the Julian constellation should be given to a star, which appeared in his time, one of those which the Greeks and the Latins alike name, from their hair-like tails, comets, it vanished in a short time and mocked his too eager hope. But we are able to read the heavens for your highness, most Serene Prince, far more truly and more happily, for scarcely have the immortal graces of your mind begun to shine on earth, when bright stars present themselves in the heavens, like tongues to tell and celebrate your most surpassing virtues to all time. Behold therefore, four stars reserved for your famous name, and those not belonging to the common and less conspicuous multitude of fixed stars, but in the bright ranks of the planets—four stars which, moving differently from each other, round the planet Jupiter, the most glorious of all the planets, as if they were his own children, accomplish the courses of their orbits with marvellous velocity, while all the while with one accord they complete all together mighty revolutions every ten years round the centre of the universe, that is, round the Sun.

But the Maker of the Stars himself seemed to direct me by clear reasons to assign these new planets to the famous name of your highness in preference to all others. For just as these stars, like children worthy of their sire, never leave the side of Jupiter by any appreciable distance, so who does not know that clemency, kindness of heart, gentleness of manners, splendour of royal blood, nobleness in public functions, wide extent of influence and power over others, all of which have fixed their common abode and seat in your highness,—who, I say, does not know that all these qualities, according to the providence of God, from whom all good things do come, emanate from the benign star of Jupiter? Jupiter, Jupiter, I maintain, at the instant of the birth of your highness having at length emerged from the turbid mists of the horizon, and being in possession of the middle quarter of the heavens, and illuminating the eastern angle, from his own royal house, from that exalted throne, looked out upon your most happy birth, and poured forth into a most pure atmosphere all the brightness of his majesty, in order that your tender body and your mind—though that was already adorned by God with still more splendid graces—might imbibe with your first breath the whole of that influence and power. But why should I use only plausible arguments when I can almost absolutely demonstrate my conclusion? It was the will of Almighty God that I should be judged by your most serene parents not unworthy to be employed in teaching your highness mathematics, which duty I discharged, during the four years just passed, at that time of the year when it is customary to take a relaxation from severer studies. Wherefore, since it evidently fell to my lot by God’s will, to serve your highness, and so to receive the rays of your surpassing clemency and beneficence in a position near your person, what wonder is it if you have so warmed my heart that it thinks about scarcely anything else day and night, but how I, who am indeed your subject not only by inclination, but also by my very birth and lineage, may be known to be most anxious for your glory, and most grateful to you? And so, inasmuch as under your patronage, most serene Cosmo, I have discovered these stars, which were unknown to all astronomers before me, I have, with very good right, determined to designate them with the most august name of your family. And as I was the first to investigate them, who can rightly blame me if I give them a name, and call them the Medicean Stars, hoping that as much consideration may accrue to these stars from this title, as other stars have brought to other heroes? For not to speak of your most serene ancestors, to whose everlasting glory the monuments of all history bear witness, your virtue alone, most mighty sire, can confer on those stars an immortal name; for who can doubt that you will not only maintain and preserve the expectations, high though they be, about yourself, which you have aroused by the very happy beginning of your government, but that you will also far surpass them, so that when you have conquered others like yourself, you may still vie with yourself, and become day by day greater than yourself and your greatness?

Accept, then, most clement Prince, this addition to the glory of your family, reserved by the stars for you; and may you enjoy for many years those good blessings, which are sent to you not so much from the stars as from God, the Maker and Governor of the stars.

Your Highness’s most devoted servant,

Galileo Galilei.

Padua, March 12, 1610.

THE ASTRONOMICAL MESSENGER

Containing and setting forth Observations lately made with the aid of a newly invented Telescope respecting the Moon’s Surface, the Milky Way, Nebulous Stars, an innumerable multitude of Fixed Stars, and also respecting Four Planets never before seen, which have been named

THE COSMIAN STARS.[3]

[3] The satellites of Jupiter are here called “the Cosmian Stars” in honour of Cosmo de’ Medici, but elsewhere Galileo calls them “the Medicean Stars.” Kepler sometimes calls them “the Medicean Stars,” but more often “satellites.”

[Sidenote: Introduction.]

IN the present small treatise I set forth some matters of great interest for all observers of natural phenomena to look at and consider. They are of great interest, I think, first, from their intrinsic excellence; secondly, from their absolute novelty; and lastly, also on account of the instrument by the aid of which they have been presented to my apprehension.

The number of the Fixed Stars which observers have been able to see without artificial powers of sight up to this day can be counted. It is therefore decidedly a great feat to add to their number, and to set distinctly before the eyes other stars in myriads, which have never been seen before, and which surpass the old, previously known, stars in number more than ten times.

Again, it is a most beautiful and delightful sight to behold the body of the Moon, which is distant from us nearly sixty semi-diameters[4] of the Earth, as near as if it was at a distance of only two of the same measures; so that the diameter of this same Moon appears about thirty times larger, its surface about nine hundred times, and its solid mass nearly 27,000 times larger than when it is viewed only with the naked eye; and consequently any one may know with the certainty that is due to the use of our senses, that the Moon certainly does not possess a smooth and polished surface, but one rough and uneven, and, just like the face of the Earth itself, is everywhere full of vast protuberances, deep chasms, and sinuosities.

[4] Galileo says, “per sex denas fere terrestres diametros a nobis remotum” by mistake for semi-diametros, and the same mistake occurs in p. 11.

Then to have got rid of disputes about the Galaxy or Milky Way, and to have made its nature clear to the very senses, not to say to the understanding, seems by no means a matter which ought to be considered of slight importance. In addition to this, to point out, as with one’s finger, the nature of those stars which every one of the astronomers up to this time has called nebulous, and to demonstrate that it is very different from what has hitherto been believed, will be pleasant, and very fine. But that which will excite the greatest astonishment by far, and which indeed especially moved me to call the attention of all astronomers and philosophers, is this, namely, that I have discovered four planets, neither known nor observed by any one of the astronomers before my time, which have their orbits round a certain bright star, one of those previously known, like Venus and Mercury round the Sun, and are sometimes in front of it, sometimes behind it, though they never depart from it beyond certain limits. All which facts were discovered and observed a few days ago by the help of a telescope[5] devised by me, through God’s grace first enlightening my mind.

[5] The words used by Galileo for “telescope” are perspicillum, specillum instrumentum, organum, and occhiale (Ital.). Kepler uses also oculare tubus, arundo dioptrica. The word “telescopium” is used by Gassendi, 1647.

Perchance other discoveries still more excellent will be made from time to time by me or by other observers, with the assistance of a similar instrument, so I will first briefly record its shape and preparation, as well as the occasion of its being devised, and then I will give an account of the observations made by me.

[Sidenote: Galileo’s account of the invention of his telescope.]

About ten months ago a report reached my ears that a Dutchman had constructed a telescope, by the aid of which visible objects, although at a great distance from the eye of the observer, were seen distinctly as if near; and some proofs of its most wonderful performances were reported, which some gave credence to, but others contradicted. A few days after, I received confirmation of the report in a letter written from Paris by a noble Frenchman, Jaques Badovere, which finally determined me to give myself up first to inquire into the principle of the telescope, and then to consider the means by which I might compass the invention of a similar instrument, which a little while after I succeeded in doing, through deep study of the theory of Refraction; and I prepared a tube, at first of lead, in the ends of which I fitted two glass lenses, both plane on one side, but on the other side one spherically convex, and the other concave. Then bringing my eye to the concave lens I saw objects satisfactorily large and near, for they appeared one-third of the distance off and nine times larger than when they are seen with the natural eye alone. I shortly afterwards constructed another telescope with more nicety, which magnified objects more than sixty times. At length, by sparing neither labour nor expense, I succeeded in constructing for myself an instrument so superior that objects seen through it appear magnified nearly a thousand times, and more than thirty times nearer than if viewed by the natural powers of sight alone.

[Sidenote: Galileo’s first observations with his telescope.]

It would be altogether a waste of time to enumerate the number and importance of the benefits which this instrument may be expected to confer, when used by land or sea. But without paying attention to its use for terrestrial objects, I betook myself to observations of the heavenly bodies; and first of all, I viewed the Moon as near as if it was scarcely two semi-diameters[6] of the Earth distant. After the Moon, I frequently observed other heavenly bodies, both fixed stars and planets, with incredible delight; and, when I saw their very great number, I began to consider about a method by which I might be able to measure their distances apart, and at length I found one. And here it is fitting that all who intend to turn their attention to observations of this kind should receive certain cautions. For, in the first place, it is absolutely necessary for them to prepare a most perfect telescope, one which will show very bright objects distinct and free from any mistiness, and will magnify them at least 400 times, for then it will show them as if only one-twentieth of their distance off. For unless the instrument be of such power, it will be in vain to attempt to view all the things which have been seen by me in the heavens, or which will be enumerated hereafter.

[6] “Vix per duas Telluris diametros,” by mistake for “semi-diametros.”

[Sidenote: Method of determining the magnifying power of the telescope.]

But in order that any one may be a little more certain about the magnifying power of his instrument, he shall fashion two circles, or two square pieces of paper, one of which is 400 times greater than the other, but that will be when the diameter of the greater is twenty times the length of the diameter of the other. Then he shall view from a distance simultaneously both surfaces, fixed on the same wall, the smaller with one eye applied to the telescope, and the larger with the other eye unassisted; for that may be done without inconvenience at one and the same instant with both eyes open. Then both figures will appear of the same size, if the instrument magnifies objects in the desired proportion.

[Sidenote: Method of measuring small angular distances between heavenly bodies by the size of the aperture of the telescope.]

After such an instrument has been prepared, the method of measuring distances remains for inquiry, and this we shall accomplish by the following contrivance:—

[Illustration]

For the sake of being more easily understood, I will suppose a tube A B C D.[7] Let E be the eye of the observer; then, when there are no lenses in the tube rays from the eye to the object F G would be drawn in the straight lines E C F, E D G, but when the lenses have been inserted, let the rays go in the bent lines E C H, E D I,—for they are contracted, and those which originally, when unaffected by the lenses, were directed to the object F G, will include only the part H I. Hence the ratio of the distance E H to the line H I being known, we shall be able to find, by means of a table of sines, the magnitude of the angle subtended at the eye by the object H I, which we shall find to contain only some minutes. But if we fit on the lens C D thin plates of metal, pierced, some with larger, others with smaller apertures, by putting on over the lens sometimes one plate, sometimes another, as may be necessary, we shall construct at our pleasure different subtending angles of more or fewer minutes, by the help of which we shall be able to measure conveniently the intervals between stars separated by an angular distance of some minutes, within an error of one or two minutes. But let it suffice for the present to have thus slightly touched, and as it were just put our lips to these matters, for on some other opportunity I will publish the theory of this instrument in completeness.

[7] [Illustration]

The line C H in Galileo’s figure represents the small pencil of rays from H which, after refraction through the telescope, reach the eye E. The enlarged figure shows that if O P be the radius of the aperture employed, the point H of the object would be just outside the field of view. The method, however, is at best only a very rough one, as the boundary of the field of view in this telescope is unavoidably indistinct.

Now let me review the observations made by me during the two months just past, again inviting the attention of all who are eager for true philosophy to the beginnings which led to the sight of most important phenomena.

[Sidenote: The Moon. Ruggedness of its surface. Existence of lunar mountains and valleys.]

Let me speak first of the surface of the Moon, which is turned towards us. For the sake of being understood more easily, I distinguish two parts in it, which I call respectively the brighter and the darker. The brighter part seems to surround and pervade the whole hemisphere; but the darker part, like a sort of cloud, discolours the Moon’s surface and makes it appear covered with spots. Now these spots, as they are somewhat dark and of considerable size, are plain to every one, and every age has seen them, wherefore I shall call them great or ancient spots, to distinguish them from other spots, smaller in size, but so thickly scattered that they sprinkle the whole surface of the Moon, but especially the brighter portion of it. These spots have never been observed by any one before me; and from my observations of them, often repeated, I have been led to that opinion which I have expressed, namely, that I feel sure that the surface of the Moon is not perfectly smooth, free from inequalities and exactly spherical, as a large school of philosophers considers with regard to the Moon and the other heavenly bodies, but that, on the contrary, it is full of inequalities, uneven, full of hollows and protuberances, just like the surface of the Earth itself, which is varied everywhere by lofty mountains and deep valleys.

Sketches by Galileo to shew:—

[Illustration: the indentation of the terminator and illuminated summits of mountains in the dark part of the moon;]

[Illustration: the shape of a lunar mountain and of a walled plain. Galileo: ‘Sidereus Nuncius,’ Venice 1610.]

The appearances from which we may gather these conclusions are of the following nature:—On the fourth or fifth day after new-moon, when the Moon presents itself to us with bright horns, the boundary which divides the part in shadow from the enlightened part does not extend continuously in an ellipse, as would happen in the case of a perfectly spherical body, but it is marked out by an irregular, uneven, and very wavy line, as represented in the figure given, for several bright excrescences, as they may be called, extend beyond the boundary of light and shadow into the dark part, and on the other hand pieces of shadow encroach upon the light:—nay, even a great quantity of small blackish spots, altogether separated from the dark part, sprinkle everywhere almost the whole space which is at the time flooded with the Sun’s light, with the exception of that part alone which is occupied by the great and ancient spots. I have noticed that the small spots just mentioned have this common characteristic always and in every case, that they have the dark part towards the Sun’s position, and on the side away from the Sun they have brighter boundaries, as if they were crowned with shining summits. Now we have an appearance quite similar on the Earth about sunrise, when we behold the valleys, not yet flooded with light, but the mountains surrounding them on the side opposite to the Sun already ablaze with the splendour of his beams; and just as the shadows in the hollows of the Earth diminish in size as the Sun rises higher, so also these spots on the Moon lose their blackness as the illuminated part grows larger and larger. Again, not only are the boundaries of light and shadow in the Moon seen to be uneven and sinuous, but—and this produces still greater astonishment—there appear very many bright points within the darkened portion of the Moon, altogether divided and broken off from the illuminated tract, and separated from it by no inconsiderable interval, which, after a little while, gradually increase in size and brightness, and after an hour or two become joined on to the rest of the bright portion, now become somewhat larger; but in the meantime others, one here and another there, shooting up as if growing, are lighted up within the shaded portion, increase in size, and at last are linked on to the same luminous surface, now still more extended. An example of this is given in the same figure. Now, is it not the case on the Earth before sunrise, that while the level plain is still in shadow, the peaks of the most lofty mountains are illuminated by the Sun’s rays? After a little while does not the light spread further, while the middle and larger parts of those mountains are becoming illuminated; and at length, when the Sun has risen, do not the illuminated parts of the plains and hills join together? The grandeur, however, of such prominences and depressions in the Moon seems to surpass both in magnitude and extent the ruggedness of the Earth’s surface, as I shall hereafter show. And here I cannot refrain from mentioning what a remarkable spectacle I observed while the Moon was rapidly approaching her first quarter, a representation of which is given in the same illustration, placed opposite page 16. A protuberance of the shadow, of great size, indented the illuminated part in the neighbourhood of the lower cusp; and when I had observed this indentation longer, and had seen that it was dark throughout, at length, after about two hours, a bright peak began to arise a little below the middle of the depression; this by degrees increased, and presented a triangular shape, but was as yet quite detached and separated from the illuminated surface. Soon around it three other small points began to shine, until, when the Moon was just about to set, that triangular figure, having now extended and widened, began to be connected with the rest of the illuminated part, and, still girt with the three bright peaks already mentioned, suddenly burst into the indentation of shadow like a vast promontory of light.

At the ends of the upper and lower cusps also certain bright points, quite away from the rest of the bright part, began to rise out of the shadow, as is seen depicted in the same illustration.

[Sidenote: The lunar spots are suggested to be possibly seas bordered by ranges of mountains.]

In both horns also, but especially in the lower one, there was a great quantity of dark spots, of which those which are nearer the boundary of light and shadow appear larger and darker, but those which are more remote less dark and more indistinct. In all cases, however, just as I have mentioned before, the dark portion of the spot faces the position of the Sun’s illumination, and a brighter edge surrounds the darkened spot on the side away from the Sun, and towards the region of the Moon in shadow. This part of the surface of the Moon, where it is marked with spots like a peacock’s tail with its azure eyes, is rendered like those glass vases which, through being plunged while still hot from the kiln into cold water, acquire a crackled and wavy surface, from which circumstance they are commonly called frosted glasses.[8] Now the great spots of the Moon observed at the same time are not seen to be at all similarly broken, or full of depressions and prominences, but rather to be even and uniform; for only here and there some spaces, rather brighter than the rest, crop up; so that if any one wishes to revive the old opinion of the Pythagoreans, that the Moon is another Earth, so to say, the brighter portion may very fitly represent the surface of the land, and the darker the expanse of water. Indeed, I have never doubted that if the sphere of the Earth were seen from a distance, when flooded with the Sun’s rays, that part of the surface which is land would present itself to view as brighter, and that which is water as darker in comparison. Moreover, the great spots in the Moon are seen to be more depressed than the brighter tracts; for in the Moon, both when crescent and when waning, on the boundary between the light and shadow, which projects in some places round the great spots, the adjacent regions are always brighter, as I have noticed in drawing my illustrations, and the edges of the spots referred to are not only more depressed than the brighter parts, but are more even, and are not broken by ridges or ruggednesses. But the brighter part stands out most near the spots, so that both before the first quarter and about the third quarter also, around a certain spot in the upper part of the figure, that is, occupying the northern region of the Moon, some vast prominences on the upper and lower sides of it rise to an enormous elevation, as the illustrations show. This same spot before the third quarter is seen to be walled round with boundaries of a deeper shade, which just like very lofty mountain summits appear darker on the side away from the Sun, and brighter on the side where they face the Sun; but in the case of the cavities the opposite happens, for the part of them away from the Sun appears brilliant, and that part which lies nearer to the Sun dark and in shadow. After a time, when the enlightened portion of the Moon’s surface has diminished in size, as soon as the whole or nearly so of the spot already mentioned is covered with shadow, the brighter ridges of the mountains mount high above the shade. These two appearances are shown in the illustrations which are given.

[8] Specimens of frosted or crackled Venetian glass are to be seen in the Slade Collection, British Museum, and fully justify Galileo’s comparison.

[Sidenote: Description of a lunar crater, perhaps Tycho.][9]

[9] Webb, Celestial Objects for Common Telescopes, p. 104, suggests this identification.

There is one other point which I must on no account forget, which I have noticed and rather wondered at. It is this:—The middle of the Moon, as it seems, is occupied by a certain cavity larger than all the rest, and in shape perfectly round. I have looked at this depression near both the first and third quarters, and I have represented it as well as I can in the second illustration already given. It produces the same appearance as to effects of light and shade as a tract like Bohemia would produce on the Earth, if it were shut in on all sides by very lofty mountains arranged on the circumference of a perfect circle; for the tract in the Moon is walled in with peaks of such enormous height that the furthest side adjacent to the dark portion of the Moon is seen bathed in sunlight before the boundary between light and shade reaches half-way across the circular space. But according to the characteristic property of the rest of the spots, the shaded portion of this too faces the Sun, and the bright part is towards the dark side of the Moon, which for the third time I advise to be carefully noticed as a most solid proof of the ruggednesses and unevennesses spread over the whole of the bright region of the Moon. Of these spots, moreover, the darkest are always those which are near to the boundary-line between the light and the shadow, but those further off appear both smaller in size and less decidedly dark; so that at length, when the Moon at opposition becomes full, the darkness of the cavities differs from the brightness of the prominences with a subdued and very slight difference.

[Sidenote: Reasons for believing that there is a difference of constitution in various parts of the Moon’s surface.]

These phenomena which we have reviewed are observed in the bright tracts of the Moon. In the great spots we do not see such differences of depressions and prominences as we are compelled to recognise in the brighter parts, owing to the change of their shapes under different degrees of illumination by the Sun’s rays according to the manifold variety of the Sun’s position with regard to the Moon. Still, in the great spots there do exist some spaces rather less dark than the rest, as I have noted in the illustrations, but these spaces always have the same appearance, and the depth of their shadow is neither intensified nor diminished; they do appear indeed sometimes a little more shaded, sometimes a little less, but the change of colour is very slight, according as the Sun’s rays fall upon them more or less obliquely; and besides, they are joined to the adjacent parts of the spots with a very gradual connection, so that their boundaries mingle and melt into the surrounding region. But it is quite different with the spots which occupy the brighter parts of the Moon’s surface, for, just as if they were precipitous crags with numerous rugged and jagged peaks, they have well-defined boundaries through the sharp contrast of light and shade. Moreover, inside those great spots certain other tracts are seen brighter than the surrounding region, and some of them very bright indeed, but the appearance of these, as well as of the darker parts, is always the same; there is no change of shape or brightness or depth of shadow, so that it becomes a matter of certainty and beyond doubt that their appearance is owing to real dissimilarity of parts, and not to unevennesses only in their configuration, changing in different ways the shadows of the same parts according to the variations of their illumination by the Sun, which really happens in the case of the other smaller spots occupying the brighter portion of the Moon, for day by day they change, increase, decrease, or disappear, inasmuch as they derive their origin only from the shadows of prominences.

[Sidenote: Explanation of the evenness of the illuminated part of the circumference of the Moon’s orb by the analogy of terrestrial phenomena, or by a possible lunar atmosphere.]

But here I feel that some people may be troubled with grave doubt, and perhaps seized with a difficulty so serious as to compel them to feel uncertain about the conclusion just explained and supported by so many phenomena. For if that part of the Moon’s surface which reflects the Sun’s rays most brightly is full of sinuosities, protuberances, and cavities innumerable, why, when the Moon is increasing, does the outer edge which looks toward the west, when the Moon is waning, the other half-circumference towards the east, and at full-moon the whole circle, appear not uneven, rugged, and irregular, but perfectly round and circular, as sharply defined as if marked out with a pair of compasses, and without the indentations of any protuberances or cavities? And most remarkably so, because the whole unbroken edge belongs to that part of the Moon’s surface which possesses the property of appearing brighter than the rest, which I have said to be throughout full of protuberances and cavities. For not one of the Great Spots extends quite to the circumference, but all of them are seen to be together away from the edge. Of this phenomenon, which affords a handle for such serious doubt, I produce two causes, and so two solutions of the difficulty.

The first solution which I offer is this:—If the protuberances and cavities in the body of the Moon existed only on the edge of the circle that bounds the hemisphere which we see, then the Moon might, or rather must, show itself to us with the appearance of a toothed wheel, being bounded with an irregular and uneven circumference; but if, instead of a single set of prominences arranged along the actual circumference only, very many ranges of mountains with their cavities and ruggednesses are set one behind the other along the extreme edge of the Moon, and that too not only in the hemisphere which we see, but also in that which is turned away from us, but still near the boundary of the hemisphere, then the eye, viewing them afar off, will not at all be able to detect the differences of prominences and cavities, for the intervals between the mountains situated in the same circle, or in the same chain, are hidden by the jutting forward of other prominences situated in other ranges, and especially if the eye of the observer is placed in the same line with the tops of the prominences mentioned. So on the Earth, the summits of a number of mountains close together appear situated in one plane, if the spectator is a long way off and standing at the same elevation. So when the sea is rough, the tops of the waves seem to form one plane, although between the billows there is many a gulf and chasm, so deep that not only the hulls, but even the bulwarks, masts, and sails of stately ships are hidden amongst them. Therefore, as within the Moon, as well as round her circumference, there is a manifold arrangement of prominences and cavities, and the eye, regarding them from a great distance, is placed in nearly the same plane with their summits, no one need think it strange that they present themselves to the visual ray which just grazes them as an unbroken line quite free from unevennesses. To this explanation may be added another, namely, that there is round the body of the Moon, just as round the Earth, an envelope of some substance denser than the rest of the ether, which is sufficient to receive and reflect the Sun’s rays, although it does not possess so much opaqueness as to be able to prevent our seeing through it—especially when it is not illuminated. That envelope, when illuminated by the Sun’s rays, renders the body of the Moon apparently larger than it really is, and would be able to stop our sight from penetrating to the solid body of the Moon, if its thickness were greater; now, it is of greater thickness about the circumference of the Moon, greater, I mean, not in actual thickness, but with reference to our sight-rays, which cut it obliquely; and so it may stop our vision, especially when it is in a state of brightness, and may conceal the true circumference of the Moon on the side towards the Sun.

[Illustration]

This may be understood more clearly from the adjoining figure, in which the body of the Moon, A B C, is surrounded by an enveloping atmosphere, D E G. An eye at F penetrates to the middle parts of the Moon, as at A, through a thickness, D A, of the atmosphere; but towards the extreme parts a mass of atmosphere of greater depth, E B, shuts out its boundary from our sight. An argument in favour of this is, that the illuminated portion of the Moon appears of larger circumference than the rest of the orb which is in shadow.

Perhaps also some will think that this same cause affords a very reasonable explanation why the greater spots on the Moon are not seen to reach to the edge of the circumference on any side, although it might be expected that some would be found about the edge as well as elsewhere; and it seems credible that there are spots there, but that they cannot be seen because they are hidden by a mass of atmosphere too thick and too bright for the sight to penetrate.

[Illustration]

[Sidenote: Calculation to show that the height of some lunar mountains exceeds four Italian miles[10] (22,000 British feet).]

[10] In the list of the heights of lunar mountains determined by Beer and Maedler, given in their work Der Mond (Berlin, 1837), there are six which exceed 3000 toises, or 19,000 British feet.

I think that it has been sufficiently made clear, from the explanation of phenomena which have been given, that the brighter part of the Moon’s surface is dotted everywhere with protuberances and cavities; it only remains for me to speak about their size, and to show that the ruggednesses of the Earth’s surface are far smaller than those of the Moon’s; smaller, I mean, absolutely, so to say, and not only smaller in proportion to the size of the orbs on which they are. And this is plainly shown thus:—As I often observed in various positions of the Moon with reference to the Sun, that some summits within the portion of the Moon in shadow appeared illumined, although at some distance from the boundary of the light (the terminator), by comparing their distance with the complete diameter of the Moon, I learnt that it sometimes exceeded the one-twentieth (1/20th) part of the diameter. Suppose the distance to be exactly 1/20th part of the diameter, and let the diagram represent the Moon’s orb, of which C A F is a great circle, E its centre, and C F a diameter, which consequently bears to the diameter of the Earth the ratio 2:7; and since the diameter of the Earth, according to the most exact observations, contains 7000 Italian miles, C F will be 2000, and C E 1000, and the 1/20th part of the whole, C F, 100 miles. Also let C F be a diameter of the great circle which divides the bright part of the Moon from the dark part (for, owing to the very great distance of the Sun from the Moon this circle does not differ sensibly from a great one), and let the distance of A from the point C be 1/20th part of that diameter; let the radius E A be drawn, and let it be produced to cut the tangent line G C D, which represents the ray that illumines the summit, in the point D. Then the arc C A or the straight line C D will be 100 of such units, as C E contains 1000. The sum of the squares of D C, C E is therefore 1,010,000, and the square of D E is equal to this; therefore the whole E D will be more than 1004; and A D will be more than 4 of such units, as C E contained 1000. Therefore the height of A D in the Moon, which represents a summit reaching up to the Sun’s ray, G C D, and separated from the extremity C by the distance C D, is more than 4 Italian miles; but in the Earth there are no mountains which reach to the perpendicular height even of one mile. We are therefore left to conclude that it is clear that the prominences of the Moon are loftier than those of the Earth.

[Sidenote: The faint illumination of the Moon’s disc about new-moon explained to be due to earth-light.]

I wish in this place to assign the cause of another lunar phenomenon well worthy of notice, and although this phenomenon was observed by me not lately, but many years ago, and has been pointed out to some of my intimate friends and pupils, explained, and assigned to its true cause, yet as the observation of it is rendered easier and more vivid by the help of a telescope, I have considered that it would not be unsuitably introduced in this place, but I wish to introduce it chiefly in order that the connection and resemblance between the Moon and the Earth may appear more plainly.

When the Moon, both before and after conjunction, is found not far from the Sun, not only does its orb show itself to our sight on the side where it is furnished with shining horns, but a slight and faint circumference is also seen to mark out the circle of the dark part, that part, namely, which is turned away from the Sun, and to separate it from the darker background of the sky. But if we examine the matter more closely, we shall see that not only is the extreme edge of the part in shadow shining with a faint brightness, but that the entire face of the Moon, that side, that is, which does not feel the Sun’s glare, is illuminated with a pale light of considerable brightness. At the first glance only a fine circumference appears shining, on account of the darker part of the sky adjacent to it; whereas, on the contrary, the rest of the surface appears dark, on account of the contiguity of the shining horns, which destroys the clearness of our sight. But if any one chooses such a position for himself, that by the interposition of a roof, or a chimney, or some other object between his sight and the Moon, but at a considerable distance from his eye, the shining horns are hidden, and the rest of the Moon’s orb is left exposed to his view, then he will find that this tract of the Moon also, although deprived of sunlight, gleams with considerable light, and particularly so if the gloom of the night has already deepened through the absence of the Sun; for with a darker background the same light appears brighter. Moreover, it is found that this secondary brightness of the Moon, as I may call it, is greater in proportion as the Moon is less distant from the Sun; for it abates more and more in proportion to the Moon’s distance from that body, so much so that after the first quarter, and before the end of the second, it is found to be weak and very faint, although it be observed in a darker sky; whereas, at an angular distance of 60° or less, even during twilight, it is wonderfully bright, so bright indeed that, with the help of a good telescope, the great spots may be distinguished in it.

This strange brightness has afforded no small perplexity to philosophical minds; and some have published one thing, some another, as the cause to be alleged for it. Some have said that it is the inherent and natural brightness of the Moon; some that it is imparted to that body by the planet Venus; or, as others maintain, by all the stars; while some have said that it comes from the Sun, whose rays, they say, find a way through the solid mass of the Moon. But statements of this kind are disproved without much difficulty, and convincingly demonstrated to be false. For if this kind of light were the Moon’s own, or were contributed by the stars, the Moon would retain it, particularly in eclipses, and would show it then, when left in an unusually dark sky, but this is contrary to experience. For the brightness which is seen on the Moon in eclipses is far less intense, being somewhat reddish, and almost copper-coloured, whereas this is brighter and whiter; besides, the brightness seen during an eclipse is changeable and shifting, for it wanders over the face of the Moon, so that that part which is near the circumference of the circle of shadow thrown by the Earth is bright, but the rest of the Moon is always seen to be dark. From which circumstance we understand without hesitation that this brightness is due to the proximity of the Sun’s rays coming into contact with some denser region which surrounds the Moon as an envelope; owing to which contact a sort of dawn-light is diffused over the neighbouring regions of the Moon, just as the twilight spreads in the morning and evening on the Earth:[11] but I will treat more fully of this matter in my book upon the System of the Universe.[12]

[11] The illumination of the Moon in eclipses, noticed by Galileo, is now referred to the refraction of the sunlight by the earth’s atmosphere, and the reddish colour of the light is explained by Herschel (Outlines of Astronomy, ch. vii.) to be due to the absorption of the violet and blue rays by the aqueous vapour of the Earth’s atmosphere. The idea of a sensible lunar atmosphere is not in accordance with the observed phenomena of the occultations of stars.

[12] Galileo’s Systema Mundi. Owing to the violent opposition provoked by the discussion of the discoveries of Galileo, and their bearing on the Copernican system of astronomy, Galileo seems to have found it necessary to delay the publication of this work until 1632, when, believing himself safe under the friendship and patronage of Pope Urban VIII. and others in power at Rome, he at length published it. Urban, however, now turned against him, denounced the book and its author, and summoned him to Rome, where the well-known incidents of his trial and condemnation took place.

Again, to assert that this sort of light is imparted to the Moon by the planet Venus is so childish as to be undeserving of an answer; for who is so ignorant as not to understand that at conjunction and within an angular distance of 60° it is quite impossible for the part of the Moon turned away from the Sun to be seen by the planet Venus?

But that this light is derived from the Sun penetrating with its light the solid mass of the Moon, and rendering it luminous, is equally untenable. For then this light would never lessen, since the hemisphere of the Moon is always illumined by the Sun, except at the moment of a lunar eclipse, yet really it quickly decreases while the Moon is drawing near to the end of her first quarter, and when she has passed her first quarter it becomes quite dull. Since, therefore, this kind of secondary brightness is not inherent and the Moon’s own, nor borrowed from any of the stars, nor from the Sun, and since there now remains in the whole universe no other body whatever except the Earth, what, pray, must we conclude? What must we assert? Shall we assert that the body of the Moon, or some other dark and sunless orb, receives light from the Earth? Why should it not be the Moon? And most certainly it is. The Earth, with fair and grateful exchange, pays back to the Moon an illumination like that which it receives from the Moon nearly the whole time during the darkest gloom of night. Let me explain the matter more clearly. At conjunction, when the Moon occupies a position between the Sun and the Earth, the Moon is illuminated by the Sun’s rays on her half towards the Sun which is turned away from the Earth, and the other half, with which she regards the Earth, is covered with darkness, and so in no degree illumines the Earth’s surface. When the Moon has slightly separated from the Sun, straightway she is partly illumined on the half directed towards us; she turns towards us a slender silvery crescent, and slightly illumines the Earth; the Sun’s illumination increases upon the Moon as she approaches her first quarter, and the reflexion of that light increases on the Earth; the brightness in the Moon next extends beyond the semicircle, and our nights grow brighter; at length the entire face of the Moon looking towards the Earth is irradiated with the most intense brightness by the Sun, which happens when the Sun and Moon are on opposite sides of the Earth; then far and wide the surface of the Earth shines with the flood of moonlight; after this the Moon, now waning, sends out less powerful beams, and the Earth is illumined less powerfully; at length the Moon draws near her first position of conjunction with the Sun, and forthwith black night invades the Earth. In such a cycle the moonlight gives us each month alternations of brighter and fainter illumination. But the benefit of her light to the Earth is balanced and repaid by the benefit of the light of the Earth to her; for while the Moon is found near the Sun about the time of conjunction, she has in front of her the entire surface of that hemisphere of the Earth which is exposed to the Sun, and vividly illumined with his beams, and so receives light reflected from the Earth. Owing to such reflexion, the hemisphere of the Moon nearer to us, though deprived of sunlight, appears of considerable brightness. Again, when removed from the Sun through a quadrant, the Moon sees only one-half of the Earth’s hemisphere illuminated, namely the western half, for the other, the eastern, is covered with the shades of night; the Moon is, therefore, less brightly enlightened by the Earth, and accordingly that secondary light appears fainter to us. But if you imagine the Moon to be set on the opposite side of the Earth to the Sun, she will see the hemisphere of the Earth, now between the Moon and the Sun, quite dark, and steeped in the gloom of night; if, therefore, an eclipse should accompany such a position of the Moon, she will receive no light at all, being deprived of the illumination of the Sun and Earth together. In any other position, with regard to the Earth and the Sun, the Moon receives more or less light by reflexion from the Earth, according as she sees a greater or smaller portion of the hemisphere of the Earth illuminated by the Sun; for such a law is observed between these two orbs, that at whatever times the Earth is most brightly enlightened by the Moon, at those times, on the contrary, the Moon is least enlightened by the Earth; and contrariwise. Let these few words on this subject suffice in this place; for I will consider it more fully in my System of the Universe, where, by very many arguments and experimental proofs, there is shown to be a very strong reflexion of the Sun’s light from the Earth, for the benefit of those who urge that the Earth must be separated from the starry host, chiefly for the reason that it has neither motion nor light, for I will prove that the Earth has motion, and surpasses the Moon in brightness, and is not the place where the dull refuse of the universe has settled down; and I will support my demonstration by a thousand arguments taken from natural phenomena.

[Sidenote: Stars. Their appearance in the telescope.]

Hitherto I have spoken of the observations which I have made concerning the Moon’s body; now I will briefly announce the phenomena which have been, as yet, seen by me with reference to the Fixed Stars. And first of all the following fact is worthy of consideration:—The stars, fixed as well as erratic, when seen with a telescope, by no means appear to be increased in magnitude in the same proportion as other objects, and the Moon herself, gain increase of size; but in the case of the stars such increase appears much less, so that you may consider that a telescope, which (for the sake of illustration) is powerful enough to magnify other objects a hundred times, will scarcely render the stars magnified four or five times. But the reason of this is as follows:—When stars are viewed with our natural eyesight they do not present themselves to us of their bare, real size, but beaming with a certain vividness, and fringed with sparkling rays, especially when the night is far advanced; and from this circumstance they appear much larger than they would if they were stripped of those adventitious fringes, for the angle which they subtend at the eye is determined not by the primary disc of the star, but by the brightness which so widely surrounds it. Perhaps you will understand this most clearly from the well-known circumstance that when stars rise just at sunset, in the beginning of twilight, they appear very small, although they may be stars of the first magnitude; and even the planet Venus itself, on any occasion when it may present itself to view in broad daylight, is so small to see that it scarcely seems to equal a star of the last magnitude. It is different in the case of other objects, and even of the Moon, which, whether viewed in the light of midday or in the depth of night, always appears of the same size. We conclude therefore that the stars are seen at midnight in uncurtailed glory, but their fringes are of such a nature that the daylight can cut them off, and not only daylight, but any slight cloud which may be interposed between a star and the eye of the observer. A dark veil or coloured glass has the same effect, for, upon placing them before the eye between it and the stars, all the blaze that surrounds them leaves them at once. A telescope also accomplishes the same result, for it removes from the stars their adventitious and accidental splendours before it enlarges their true discs (if indeed they are of that shape), and so they seem less magnified than other objects, for a star of the fifth or sixth magnitude seen through a telescope is shown as of the first magnitude only.

The difference between the appearance of the planets and the fixed stars seems also deserving of notice. The planets present their discs perfectly round, just as if described with a pair of compasses, and appear as so many little moons, completely illuminated and of a globular shape; but the fixed stars do not look to the naked eye bounded by a circular circumference, but rather like blazes of light, shooting out beams on all sides and very sparkling, and with a telescope they appear of the same shape as when they are viewed by simply looking at them, but so much larger that a star of the fifth or sixth magnitude seems to equal Sirius, the largest of all the fixed stars.[13]

[13] The immense distance of stars makes it impossible for them to be magnified by any telescope, however powerful; the apparent or spurious disc is an optical effect, which depends on the telescope used, and is smallest with the largest aperture.

[Illustration: Orion’s Belt and Sword; 83 Stars]

[Illustration: Pleiades; 36 Stars

Galileo: “Sidereus Nuncius.”]

[Sidenote: Telescopic Stars: their infinite multitude. As examples, Orion’s Belt and Sword and the Pleiades are described as seen by Galileo.]

But beyond the stars of the sixth magnitude you will behold through the telescope a host of other stars, which escape the unassisted sight, so numerous as to be almost beyond belief, for you may see more than six other differences of magnitude, and the largest of these, which I may call stars of the seventh magnitude, or of the first magnitude of invisible stars, appear with the aid of the telescope larger and brighter than stars of the second magnitude seen with the unassisted sight. But in order that you may see one or two proofs of the inconceivable manner in which they are crowded together, I have determined to make out a case against two star-clusters, that from them as a specimen you may decide about the rest.

As my first example I had determined to depict the entire constellation of Orion, but I was overwhelmed by the vast quantity of stars and by want of time, and so I have deferred attempting this to another occasion, for there are adjacent to, or scattered among, the old stars more than five hundred new stars within the limits of one or two degrees. For this reason I have selected the three stars in Orion’s Belt and the six in his Sword, which have been long well-known groups, and I have added eighty other stars recently discovered in their vicinity, and I have preserved as exactly as possible the intervals between them. The well-known or old stars, for the sake of distinction, I have depicted of larger size, and I have outlined them with a double line; the others, invisible to the naked eye, I have marked smaller and with one line only. I have also preserved the differences of magnitude as much as I could.

As a second example I have depicted the six stars of the constellation Taurus, called the Pleiades (I say six intentionally, since the seventh is scarcely ever visible), a group of stars which is enclosed in the heavens within very narrow precincts. Near these there lie more than forty others invisible to the naked eye, no one of which is much more than half a degree off any of the aforesaid six; of these I have noticed only thirty-six in my diagram. I have preserved their intervals, magnitudes, and the distinction between the old and the new stars, just as in the case of the constellation Orion.

[Sidenote: The Milky Way consists entirely of stars in countless numbers and of various magnitudes.]

The next object which I have observed is the essence or substance of the Milky Way. By the aid of a telescope any one may behold this in a manner which so distinctly appeals to the senses that all the disputes which have tormented philosophers through so many ages are exploded at once by the irrefragable evidence of our eyes, and we are freed from wordy disputes upon this subject, for the Galaxy is nothing else but a mass of innumerable stars planted together in clusters. Upon whatever part of it you direct the telescope straightway a vast crowd of stars presents itself to view; many of them are tolerably large and extremely bright, but the number of small ones is quite beyond determination.

[Illustration: Star-cluster in Orion’s Head]

[Illustration: Star-cluster of Praesepe in Cancer

Galileo: “Sidereus Nuncius,” Venice, 1610.]

[Sidenote: Nebulæ resolved into clusters of stars: as examples, the nebulæ in Orion’s Head and Præsepe.]

And whereas that milky brightness, like the brightness of a white cloud, is not only to be seen in the Milky Way, but several spots of a similar colour shine faintly here and there in the heavens, if you turn the telescope upon any of them you will find a cluster of stars packed close together. Further—and you will be more surprised at this,—the stars which have been called by every one of the astronomers up to this day nebulous, are groups of small stars set thick together in a wonderful way, and although each one of them on account of its smallness, or its immense distance from us, escapes our sight, from the commingling of their rays there arises that brightness which has hitherto been believed to be the denser part of the heavens, able to reflect the rays of the stars or the Sun.

I have observed some of these, and I wish to subjoin the star-clusters of two of these nebulæ. First, you have a diagram of the nebula called that of Orion’s Head, in which I have counted twenty-one stars.

The second cluster contains the nebula called Præsepe, which is not one star only, but a mass of more than forty small stars. I have noticed thirty-six stars, besides the Aselli, arranged in the order of the accompanying diagram.

[Sidenote: Discovery of Jupiter’s satellites, Jan. 7, 1610: record of Galileo’s observations during two months.]

I have now finished my brief account of the observations which I have thus far made with regard to the Moon, the Fixed Stars, and the Galaxy. There remains the matter, which seems to me to deserve to be considered the most important in this work, namely, that I should disclose and publish to the world the occasion of discovering and observing four PLANETS, never seen from the very beginning of the world up to our own times, their positions, and the observations made during the last two months about their movements and their changes of magnitude; and I summon all astronomers to apply themselves to examine and determine their periodic times, which it has not been permitted me to achieve up to this day, owing to the restriction of my time. I give them warning however again, so that they may not approach such an inquiry to no purpose, that they will want a very accurate telescope, and such as I have described in the beginning of this account.

On the 7th day of January in the present year, 1610, in the first[14] hour of the following night, when I was viewing the constellations of the heavens through a telescope, the planet Jupiter presented itself to my view, and as I had prepared for myself a very excellent instrument, I noticed a circumstance which I had never been able to notice before, owing to want of power in my other telescope, namely, that three little stars, small but very bright, were near the planet; and although I believed them to belong to the number of the fixed stars, yet they made me somewhat wonder, because they seemed to be arranged exactly in a straight line, parallel to the ecliptic,[15] and to be brighter than the rest of the stars, equal to them in magnitude. The position of them with reference to one another and to Jupiter was as follows (Fig. 1).

[14] The times of Galileo’s observations are to be understood as reckoned from sunset.

[15] The satellites of Jupiter revolve in planes very nearly, although not exactly, coincident with that of the equator of the planet, which is inclined 3° 5´ 30´´ to the orbit of the planet, and the plane of the orbit is inclined 1° 18´ 51´´ to the ecliptic.

On the east side there were two stars, and a single one towards the west. The star which was furthest towards the east, and the western star, appeared rather larger than the third.

I scarcely troubled at all about the distance between them and Jupiter, for, as I have already said, at first I believed them to be fixed stars; but when on January 8th, led by some fatality, I turned again to look at the same part of the heavens, I found a very different state of things, for there were three little stars all west of Jupiter, and nearer together than on the previous night, and they were separated from one another by equal intervals, as the accompanying illustration (Fig. 2) shows.

At this point, although I had not turned my thoughts at all upon the approximation of the stars to one another, yet my surprise began to be excited, how Jupiter could one day be found to the east of all the aforesaid fixed stars when the day before it had been west of two of them; and forthwith I became afraid lest the planet might have moved differently from the calculation of astronomers, and so had passed those stars by its own proper motion. I therefore waited for the next night with the most intense longing, but I was disappointed of my hope, for the sky was covered with clouds in every direction.

But on January 10th the stars appeared in the following position with regard to Jupiter; there were two only, and both on the east side of Jupiter, the third, as I thought, being hidden by the planet (Fig. 3). They were situated just as before, exactly in the same straight line with Jupiter, and along the Zodiac.

When I had seen these phenomena, as I knew that corresponding changes of position could not by any means belong to Jupiter, and as, moreover, I perceived that the stars which I saw had been always the same, for there were no others either in front or behind, within a great distance, along the Zodiac,—at length, changing from doubt into surprise, I discovered that the interchange of position which I saw belonged not to Jupiter, but to the stars to which my attention had been drawn, and I thought therefore that they ought to be observed henceforward with more attention and precision.

Accordingly, on January 11th I saw an arrangement of the following kind (Fig. 4), namely, only two stars to the east of Jupiter, the nearer of which was distant from Jupiter three times as far as from the star further to the east; and the star furthest to the east was nearly twice as large as the other one; whereas on the previous night they had appeared nearly of equal magnitude. I therefore concluded, and decided unhesitatingly, that there are three stars in the heavens moving about Jupiter, as Venus and Mercury round the Sun; which at length was established as clear as daylight by numerous other subsequent observations. These observations also established that there are not only three, but four, erratic sidereal bodies performing their revolutions round Jupiter, observations of whose changes of position made with more exactness on succeeding nights the following account will supply. I have measured also the intervals between them with the telescope in the manner already explained. Besides this, I have given the times of observation, especially when several were made in the same night, for the revolutions of these planets are so swift that an observer may generally get differences of position every hour.

Jan. 12.—At the first hour of the next night I saw these heavenly bodies arranged in this manner (Fig. 5). The satellite[16] furthest to the east was greater than the satellite furthest to the west; but both were very conspicuous and bright; the distance of each one from Jupiter was two minutes. A third satellite, certainly not in view before, began to appear at the third hour; it nearly touched Jupiter on the east side, and was exceedingly small. They were all arranged in the same straight line, along the ecliptic.

[16] Galileo continues to call these bodies stars, perhaps meaning “Medicean stars,” throughout the description of their configurations, but as he had now detected their nature, it is more convenient to call them satellites, the term introduced by Kepler.

Jan. 13.—For the first time four satellites were in view in the following position with regard to Jupiter (Fig. 6).

There were three to the west, and one to the east; they made a straight line nearly, but the middle satellite of those to the west deviated a little from the straight line towards the north. The satellite furthest to the east was at a distance of 2´ from Jupiter; there were intervals of 1´ only between Jupiter and the nearest satellite, and between the satellites themselves, west of Jupiter. All the satellites appeared of the same size, and though small they were very brilliant, and far outshone the fixed stars of the same magnitude.

Jan. 14.—The weather was cloudy.

Jan. 15.—At the third hour of the night the four satellites were in the state depicted in the next diagram (Fig. 7) with reference to Jupiter.

All were to the west, and arranged nearly in the same straight line; but the satellite which counted third from Jupiter was raised a little to the north. The nearest to Jupiter was the smallest of all; the rest appeared larger and in order of magnitude; the intervals between Jupiter and the three nearest satellites were all equal, and were of the magnitude of 2´ each; but the satellite furthest to the west was distant 4´ from the satellite nearest to it. They were very brilliant, and not at all twinkling, as they have always appeared both before and since. But at the seventh hour there were only three satellites, presenting with Jupiter an appearance of the following kind (Fig. 8). They were, that is to say, in the same straight line to a hair; the nearest to Jupiter was very small, and distant from the planet 3´; the distance of the second from this one was 1´; and of the third from the second 4´ 30´´. But after another hour the two middle satellites were still nearer, for they were only 30´´, or less, apart.

Jan. 16.—At the first hour of the night I saw three satellites arranged in this order (Fig. 9). Jupiter was between two of them, which were at a distance of 0´ 40´´ from the planet on either side, and the third was west of Jupiter at a distance of 8´. The satellites near to Jupiter appeared brighter than the satellite further off, but not larger.

Jan. 17, after sunset 0 hours 30 minutes, the configuration was of this kind (Fig. 10). There was one satellite only to the east, at a distance of 3´ from Jupiter; to the west likewise there was only one satellite, distant 11´ from Jupiter. The satellite on the east appeared twice as large as the satellite to the west; and there were no more than these two. But four hours after, that is, nearly at the fifth hour, a third satellite began to emerge on the east side, which, before its appearance, as I think, had been joined with the former of the two other satellites, and the position was of this kind (Fig. 11). The middle satellite was very near indeed to the satellite on the east, and was only 20´´ from it; and was a little towards the south of the straight line drawn through the two extreme satellites and Jupiter.

Jan. 18, at 0 h. 20 m. after sunset, the appearance was such as this (Fig. 12). The satellite to the east was larger than the western one, and was at a distance from Jupiter of 8´, the western one being at a distance of 10´.

Jan. 19.—At the second hour of the night the relative position of the satellites was such as this (Fig. 13); that is, there were three satellites exactly in a straight line with Jupiter, one to the east, at a distance of 6´ from Jupiter; between Jupiter and the first satellite to the west in order, there was an interval of 5´; this satellite was 4´ off the other one more to the west. At that time I was doubtful whether or no there was a satellite between the satellite to the east and Jupiter, but so very close to Jupiter as almost to touch the planet; but at the fifth hour I saw this satellite distinctly, by that time occupying exactly the middle position between Jupiter and the eastern satellite, so that the configuration was thus (Fig. 14). Moreover, the satellite which had just come into view was very small; yet at the sixth hour it was nearly as large as the rest.

Jan. 20: 1 h. 15 m.—A similar arrangement was seen (Fig. 15). There were three satellites, so small as scarcely to be distinguishable; their distances from Jupiter, and from one another, were not more than 1´. I was doubtful whether on the western side there were two satellites or three. About the sixth hour they were grouped in this way (Fig. 16). The eastern satellite was twice as far away from Jupiter as before, that is 2´; on the western side, the satellite in the middle was distant from Jupiter 0´ 40´´, and from the satellite still further to the west 0´ 20´´; at length, at the seventh hour, three satellites were seen on the western side (Fig. 17). The satellite nearest to Jupiter was distant from the planet 0´ 20´´; between this one and the satellite furthest to the west there was an interval of 40´´, but between these another satellite was in view slightly southward of them, and not more than 10´´ off the most westerly satellite.

Jan. 21: 0 h. 30 m.—There were three satellites on the east side; the satellites and Jupiter were at equal distances apart (Fig. 18). The intervals were by estimation 50´´ each. There was also one satellite on the west, distant 4´ from Jupiter. The satellite on the east side nearest to Jupiter was the least of all.

Jan. 22: 2 h.—The grouping of the satellites was similar (Fig. 19). There was an interval of 5´ from the satellite on the east to Jupiter; from Jupiter to the satellite furthest to the west 7´. The two interior satellites on the western side were 0´ 40´´ apart, and the satellite nearer to Jupiter was 1´ from the planet. The inner satellites were smaller than the outer ones, but they were situated all in the same straight line, along the ecliptic, except that the middle of the three western satellites was slightly to the south of it, but at the sixth hour of the night they appeared in this position (Fig. 20). The satellite on the east was very small, at a distance from Jupiter of 5´ as before; but the three satellites on the west were separated by equal distances from Jupiter and from each other; and the intervals were nearly 1´ 20´´ each. The satellite nearest Jupiter appeared smaller than the other two on the same side, but they all appeared arranged exactly in the same straight line.

Jan. 23, at 0 h. 40 m. after sunset, the grouping of the satellites was nearly after this fashion (Fig. 21). There were three satellites with Jupiter in a straight line along the ecliptic, as they always have been; two were on the east of the planet, one on the west; the satellite furthest to the east was 7´ from the next one, and this satellite 2´ 40´´ from Jupiter; Jupiter was 3´ 20´´ from the satellite on the west; and they were all of nearly the same size. But at the fifth hour the two satellites which had been previously near Jupiter were no longer visible, being, as I suppose, hidden behind Jupiter, and the appearance presented was such as this (Fig. 22).

Jan. 24.—Three satellites, all on the east side, were visible, and nearly, but not quite, in the same straight line with Jupiter, for the satellite in the middle was slightly to the south of it (Fig. 23). The satellite nearest to Jupiter was 2´ distant from the planet; the next in order 0´ 30´´ from this satellite, and the third was 9´ further off still; they were all very bright. But at the sixth hour two satellites only presented themselves to view in this position, namely in the same straight line with Jupiter exactly, and the distance of the nearest to the planet was lengthened to 3´; the next was 2´ further off, and unless I am mistaken, the two satellites previously observed in the middle had come together, and appeared as one.

Jan. 25, at 1 h. 40 m., the satellites were grouped thus (Fig. 24). There were only two satellites on the east side, and these were rather large. The satellite furthest to the east was 5´ from the satellite in the middle, and it was 6´ from Jupiter.

Jan. 26, at 0 h. 40 m., the relative positions of the satellites were thus (Fig. 25). Three satellites were in view, of which two were east and the third west of Jupiter; this one was distant 3´ from the planet. On the east side the satellite in the middle was at a distance of 5´ 20´´; the further satellite was 6´ beyond; they were arranged in a straight line, and were of the same size. At the fifth hour the arrangement was nearly the same, with this difference only, that the fourth satellite was emerging on the east side near Jupiter. It was smaller than the rest, and was then at a distance of 0´ 30´´ from Jupiter; but was raised a little above the straight line towards the north, as the accompanying figure shows (Fig. 26).

Jan. 27, 1 h. after sunset, a single satellite only was in view, and that on the east side of Jupiter in this position (Fig. 27). It was very small, and at a distance of 7´ from Jupiter.

Jan. 28 and 29.—Owing to the intervention of clouds, I could make no observation.

Jan. 30.—At the first hour of the night the satellites were in view, arranged in the following way (Fig. 28). There was one satellite on the east side, at a distance of 2´ 30´´ from Jupiter; and there were two satellites on the west, of which the one nearer to Jupiter was 3´ off the planet, and the other satellite 1´ further. The places of the outer satellites and Jupiter were in the same straight line; but the satellite in the middle was a little above it to the north. The satellite furthest to the west was smaller than the rest.

On the last day of the month, at the second hour, two satellites on the east side were visible, and one on the west (Fig. 29). Of the satellites east of the planet, the one in the middle was 2´ 20´´ distant from Jupiter; and the satellite further to the east was 0´ 30´´ from the middle satellite; the satellite on the west was at a distance of 10´ from Jupiter. They were in the same straight line nearly, and would have been exactly so, only the satellite on the east nearest to Jupiter was raised a little towards the north. At the fourth hour, the two satellites on the east were still nearer together, for they were only 20´´ apart (Fig. 30). The western satellite appeared rather small in these two observations.

Feb. 1.—At the second hour of the night the arrangement was similar (Fig. 31). The satellite furthest to the east was at a distance of 6´ from Jupiter, and the western satellite 8´. On the east side there was a very small satellite, at a distance of 20´´ from Jupiter. They made a perfectly straight line.

Feb. 2.—The satellites were seen arranged thus (Fig. 32). There was one only on the east, at a distance of 6´ from Jupiter. Jupiter was 4´ from the nearest satellite on the west; between this satellite and the satellite further to the west there was an interval of 8´; they were in the same straight line exactly, and were nearly of the same magnitude. But at the seventh hour four satellites were there—two on each side of Jupiter (Fig. 33). Of these satellites, the most easterly was at a distance of 4´ from the next; this satellite was 1´ 40´´ from Jupiter; Jupiter was 6´ from the nearest satellite on the west, and this one from the satellite further to the west, 8´; and they were all alike in the same straight line, drawn in the direction of the Zodiac.

Feb. 3: 7 h.—The satellites were arranged in the following way (Fig. 34):—The satellite on the east was at a distance of 1´ 30´´ from Jupiter; the nearest satellite on the west, 2´, and there was a long distance, 10´, from this satellite to the satellite further to the west. They were exactly in the same straight line, and of equal magnitude.

Feb. 4: 2 h.—Four satellites attended Jupiter, two on the east and two on the west, arranged in one perfectly straight line, as in the adjoining figure (Fig. 35). The satellite furthest to the east was at a distance of 3´ from the next satellite. This one was 0´ 40´´ from Jupiter; Jupiter 4´ from the nearest satellite on the west; and this one from the satellite further to the west 6´. In magnitude they were nearly equal; the satellite nearest to Jupiter was rather smaller in appearance than the rest. But at the seventh hour (Fig. 36) the eastern satellites were at a distance of only 0´ 30´´ apart. Jupiter was 2´ from the nearest satellite on the east; and from the satellite on the west, next in order, 4´; this one was distant 3´ from the satellite further to the west. They were all equal in magnitude, and in a straight line, drawn in the direction of the ecliptic.

Feb. 5.—The sky was cloudy.

Feb. 6.—Two satellites only appeared, with Jupiter between them, as is seen in the accompanying figure (Fig. 37). The satellite on the east was 2´ from Jupiter, and that on the west 3´. They were in the same straight line with Jupiter, and were equal in magnitude.

Feb. 7.—There were two satellites by the side of Jupiter, and both of them on the east of the planet, arranged in this manner (Fig. 38). The intervals between the satellites and Jupiter were equal, and of 1´ each; and a straight line would go through them and the centre of Jupiter.

Feb. 8: 1 h.—Three satellites were there, all on the east side of Jupiter, as in the diagram (Fig. 39). The nearest to Jupiter, a rather small one, was distant from the planet 1´ 20´´; the middle one was 4´ from this satellite, and was rather large; the satellite furthest to the east, a very small one, was at a distance of 0´ 20´´ from this satellite. It was doubtful whether there was one satellite near to Jupiter or two, for sometimes it seemed that there was another satellite by its side towards the east, wonderfully small, and only 10´´ from it. They were all situated at points in a straight line drawn in the direction of the Zodiac. At the third hour the satellite nearest to Jupiter was almost touching the planet, for it was only distant 10´´ from it; but the others had become further off, for the middle one was 6´ from Jupiter. At length, at the fourth hour, the satellite which was previously the nearest to Jupiter joined with the planet and disappeared.

Feb. 9: 0 h. 30 m.—There were two satellites on the east side of Jupiter, and one on the west, in an arrangement such as this (Fig. 40). The satellite furthest to the east, which was a rather small one, was distant 4´ from the next satellite; the satellite in the middle was larger, and at a distance of 7´ from Jupiter. Jupiter was distant 4´ from the western satellite, which was a small one.

Feb. 10: 1 h. 30 m.—A pair of satellites, very small, and both on the east of the planet, were visible, in the following position (Fig. 41). The further satellite was distant from Jupiter 10´, the nearer 0´ 20´´, and they were in the same straight line; but at the fourth hour the satellite nearest to Jupiter no longer appeared, and the other seemed so diminished that it could scarcely be kept in sight, although the atmosphere was quite clear, and the satellite was further from Jupiter than before, since its distance was now 12´.

Feb. 11: 1 h.—There were two satellites on the east, and one on the west (Fig. 42). The western satellite was at a distance of 4´ from Jupiter. The satellite on the east, nearest to the planet, was likewise 4´ from Jupiter; but the satellite further to the east was at a distance from this one of 8´; they were fairly clear to view, and in the same straight line; but at the third hour the fourth satellite was visible near to Jupiter on the east, less in magnitude than the rest, separated from Jupiter by a distance of 0´ 30´´, and slightly to the north out of the straight line drawn through the rest (Fig. 43). They were all very bright and extremely distinct, but at 5 h. 30 m. the satellite on the east nearest to Jupiter had moved further away from the planet, and was occupying a position midway between the planet and the neighbouring satellite further to the east. They were all in the same straight line exactly, and of the same magnitude, as may be seen in the accompanying diagram (Fig. 44).

Feb. 12: 0 h. 40 m.—A pair of satellites on the east, a pair likewise on the west, were near the planet (Fig. 45). The satellite on the east furthest removed from Jupiter was at a distance of 10´, and the further of the satellites on the west was 8´ off. They were both fairly distinct. The other two were very near to Jupiter, and very small, especially the satellite to the east, which was at a distance of 0´ 40´´ from Jupiter. The distance of the western satellite was 1´. But at the fourth hour the satellite which was nearest to Jupiter on the east was visible no longer.

Feb. 13: 0 h. 30 m.—Two satellites were visible in the east, two also in the west (Fig. 46). The satellite on the east near Jupiter was fairly distinct; its distance from the planet was 2´. The satellite further to the east was less noticeable; it was distant 4´ from the other. Of the satellites on the west, the one furthest from Jupiter, which was very distinct, was parted from the planet 4´. Between this satellite and Jupiter intervened a small satellite close to the most westerly satellite, being not more than 0´ 3´´ off. They were all in the same straight line, corresponding exactly to the direction of the ecliptic.

Feb. 15 (for on the 14th the sky was covered with clouds), at the first hour, the position of the satellites was thus (Fig. 47); that is, there were three satellites on the east, but none were visible on the west. The satellite on the east nearest to Jupiter was at a distance of 0´ 50´´ from the planet; the next in order was 0´ 20´´ from this satellite, and the furthest to the east was 2´ from the second satellite, and it was larger than the others, for those nearer to Jupiter were very small. But about the fifth hour only one of the satellites which had been near to Jupiter was to be seen; its distance from Jupiter was 0´ 30´´. The distance of the satellite furthest to the east from Jupiter had increased, for it was then 4´ (Fig. 48). But at the sixth hour, besides the two situated as just described on the east, one satellite was visible towards the west, very small, at a distance of 2´ from Jupiter (Fig. 49).

Feb. 16: 6 h.—Their places were arranged as follows (Fig. 50); that is, the satellite on the east was 7´ from Jupiter, Jupiter 5´ from the next satellite on the west, and this 3´ from the remaining satellite still further to the west. They were all of the same magnitude nearly, rather bright, and in the same straight line, corresponding accurately to the direction of the Zodiac.

Feb. 17: 1 h.—Two satellites were in view, one on the east, distant 3´ from Jupiter; the other on the west, distant 10´ (Fig. 51). The latter was somewhat less than the satellite on the east; but at the sixth hour the eastern satellite was nearer to Jupiter, being at a distance of 0´ 50´´, and the western satellite was further off, namely 12´. At both observations they were in the same straight line with Jupiter, and were both rather small, especially the eastern satellite in the second observation.

Feb. 18: 1 h.—Three satellites were in view, of which two were on the west and one on the east; the distance of the eastern satellite from Jupiter was 3´, and of the nearest satellite on the west 2´; the remaining satellite, still further to the west, was 8´ from the middle satellite (Fig. 52). They were all in the same straight line exactly, and of about the same magnitude. But at the second hour the satellites nearest to the planet were at equal distances from Jupiter, for the western satellite was now also 3´ from the planet. But at the sixth hour the fourth satellite was visible between the satellite on the east and Jupiter, in the following configuration (Fig. 53). The satellite furthest to the east was at a distance of 3´ from the next in order; this one was at a distance of 1´ 50´´ from Jupiter; Jupiter was at a distance of 3´ from the next satellite on the west, and this 7´ from the satellite still further to the west. These were nearly equal in magnitude, only the satellite on the east nearest to Jupiter was a little smaller than the rest, and they were all in the same straight line parallel to the ecliptic.

Feb. 19: 0 h. 40 m.—Two satellites only were in view, west of Jupiter, rather large, and arranged exactly in the same straight line with Jupiter, in the direction of the ecliptic (Fig. 54). The nearer satellite was at a distance of 7´ from Jupiter and of 6´ from the satellite further to the west.

Feb. 20.—The sky was cloudy.

Feb. 21: 1 h. 30 m.—Three satellites, rather small, were in view, placed thus (Fig. 55). The satellite to the east was 2´ from Jupiter; Jupiter was 3´ from the next, which was on the west; and this one was 7´ from the satellite further to the west. They were exactly in the same straight line parallel to the ecliptic.

Feb. 25: 1 h. 30 m. (for on the three previous nights the sky was overcast).—Three satellites appeared, two on the east, which were at a distance of 4´ apart, the same as the distance of the nearer satellite from Jupiter; on the west there was one satellite at a distance of 2´ from Jupiter. They were exactly in the same straight line in the direction of the ecliptic (Fig. 56).

Feb. 26: 0 h. 30 m.—A pair of satellites only were present, one on the east, distant 10´ from Jupiter; the other was on the west, at a distance of 6´ (Fig. 57). The eastern satellite was slightly smaller than the western. At the fifth hour three satellites were visible; for, besides the two already noticed, a third satellite was in view, on the west, near Jupiter, very small, which previously had been hidden behind Jupiter, and it was at a distance of 1´ from the planet (Fig. 58).

But the satellite on the east was seen to be further off than before, being at a distance of 11´ from Jupiter. On this night, for the first time, I determined to observe the motion of Jupiter and the adjacent planets (his satellites) along the zodiac, by reference to some fixed star; for there was a fixed star in view, eastwards of Jupiter, at a distance of 11´ from the satellite on the east, and a little to the south, in the following manner (Fig. 59).

Feb. 27: 1 h. 4 m.—The satellites appeared in the following configuration. The satellite furthest to the east was at a distance of 10´ from Jupiter; the next in order was near Jupiter, being at a distance of 0´ 30´´ from the planet. The next satellite was on the western side, at a distance of 2´ 30´´ from Jupiter; and the satellite further to the west was at a distance of 1´ from this. The two satellites near to Jupiter appeared small, especially the satellite on the east; but the satellites furthest off were very bright, particularly that on the west, and they made a straight line in the direction of the ecliptic exactly. The motion of the planets towards the east was plainly seen by reference to the aforesaid fixed star, for Jupiter and his attendant satellites were nearer to it, as may be seen in the accompanying figure (Fig. 60). At the fifth hour the satellite on the east, near to Jupiter, was 1´ from the planet.

Feb. 28: 1 h.—Only two satellites were visible, one on the east, at a distance of 9´ from Jupiter, and another on the west, at a distance of 2´; they were both rather bright, and in the same straight line with Jupiter, and a straight line drawn from the fixed star perpendicular to this straight line fell upon the satellite on the east, as in the figure (Fig. 61). At the fifth hour a third satellite was seen at a distance of 2´ from Jupiter, on the east, in the position shown in the figure (Fig. 62).

March 1: 0 h. 40 m.—Four satellites, all on the east of the planet, were seen; the satellite nearest to Jupiter was 2´ from the planet; the next 1´ from this; the third was 0´ 20´´ from the second, and was brighter than the others; and the satellite still further to the east was at a distance of 4´ from it, and was smaller than the others (Fig. 63). They made a straight line very nearly, only the satellite third from Jupiter was slightly above the line. The fixed star formed with Jupiter and the most easterly satellite an equilateral triangle, as in the figure.

March 2: 0 h. 40 m.—Three satellites were in attendance, two on the east and one on the west, in the configuration shown in the diagram (Fig. 64). The satellite furthest to the east was 7´ from Jupiter; from this satellite the next was distant 0´ 30´´, and the satellite on the west was separated from Jupiter by an interval of 2´. The satellites furthest off were brighter and larger than the remaining satellite, which appeared very small. The satellite furthest to the east seemed to be raised a little towards the north, out of the straight line drawn through the other satellites and Jupiter.

The fixed star already noticed was at a distance of 8´ from the western satellite, that is, from the perpendicular drawn from that satellite to the straight line drawn through all the system, as shown in the figure given.

These determinations of the motion of Jupiter and the adjacent planets (his satellites) by reference to a fixed star, I have thought well to present to the notice of astronomers, in order that any one may be able to understand from them that the movements of these planets (Jupiter’s satellites) both in longitude and in latitude agree exactly with the motions [of Jupiter] which are extracted from tables.

These are my observations upon the four Medicean planets, recently discovered for the first time by me; and although it is not yet permitted me to deduce by calculation from these observations the orbits of these bodies, yet I may be allowed to make some statements, based upon them, well worthy of attention.

[Sidenote: Deductions from the previous observations concerning the orbits and periods of Jupiter’s satellites.]

And, in the first place, since they are sometimes behind, sometimes before Jupiter, at like distances, and withdraw from this planet towards the east and towards the west only within very narrow limits of divergence, and since they accompany this planet alike when its motion is retrograde and direct, it can be a matter of doubt to no one that they perform their revolutions about this planet, while at the same time they all accomplish together orbits of twelve years’ length about the centre of the world. Moreover, they revolve in unequal circles, which is evidently the conclusion to be drawn from the fact that I have never been permitted to see two satellites in conjunction when their distance from Jupiter was great, whereas near Jupiter two, three, and sometimes all (four), have been found closely packed together. Moreover, it may be detected that the revolutions of the satellites which describe the smallest circles round Jupiter are the most rapid, for the satellites nearest to Jupiter are often to be seen in the east, when the day before they have appeared in the west, and contrariwise. Also the satellite moving in the greatest orbit seems to me, after carefully weighing the occasions of its returning to positions previously noticed, to have a periodic time of half a month.[17] Besides, we have a notable and splendid argument to remove the scruples of those who can tolerate the revolution of the planets round the Sun in the Copernican system, yet are so disturbed by the motion of one Moon about the Earth, while both accomplish an orbit of a year’s length about the Sun, that they consider that this theory of the constitution of the universe must be upset as impossible; for now we have not one planet only revolving about another, while both traverse a vast orbit about the Sun, but our sense of sight presents to us four satellites circling about Jupiter, like the Moon about the Earth, while the whole system travels over a mighty orbit about the Sun in the space of twelve years.

[17] In the edition of Galileo’s works published at Florence, 1854, there are given the tables of the hourly movements of the satellites of Jupiter, from which Galileo determined their periods of revolution. In the beginning of his treatise on floating bodies, Discorso intorno i Galleggianti, 1611-12, Galileo gives the times of rotation as approximately, (i.) 1 d. 18-1/2 h.; (ii.) 3 d. 13-1/3 h.; (iii.) 7 d. 4 h.; (iv.) 16 d. 18 h.; he also published configurations of the satellites calculated for March, April, and a part of May 1613. The periodic times of the satellites, as corrected by later observers, are, (i.) 1 d. 18 h. 28 m.; (ii.) 3 d. 13 h. 15 m.; (iii.) 7 d. 3 h. 43 m.; (iv.) 16 d. 16 h. 32 m.

[Sidenote: Explanation of the variations in brightness of Jupiter’s satellites.]

Lastly, I must not pass over the consideration of the reason why it happens that the Medicean stars, in performing very small revolutions about Jupiter, seem sometimes more than twice as large as at other times. We can by no means look for the explanation in the mists of the Earth’s atmosphere, for they appear increased or diminished, while the discs of Jupiter and neighbouring fixed stars are seen quite unaltered. That they approach and recede from the Earth at the points of their revolutions nearest to and furthest from the Earth to such an extent as to account for so great changes seems altogether untenable, for a strict circular motion can by no means show those phenomena; and an elliptical motion (which in this case would be nearly rectilinear) seems to be both untenable and by no means in harmony with the phenomena observed. But I gladly publish the explanation which has occurred to me upon this subject, and submit it to the judgment and criticism of all true philosophers. It is certain that when atmospheric mists intervene the Sun and Moon appear larger, but the fixed stars and planets less than they really are; hence the former luminaries, when near the horizon, are larger than at other times, but stars appear smaller, and are frequently scarcely visible; also they are still more diminished if those mists are bathed in light; so stars appear very small by day and in the twilight, but the Moon does not appear so, as I have previously remarked. Moreover, it is certain that not only the Earth, but also the Moon, has its own vaporous sphere enveloping it, for the reasons which I have previously mentioned, and especially for those which shall be stated more fully in my System; and we may consistently decide that the same is true with regard to the rest of the planets; so that it seems to be by no means an untenable opinion to place round Jupiter also an atmosphere denser than the rest of the ether,[18] about which, like the Moon about the sphere of the elements, the Medicean planets (Jupiter’s satellites) revolve; and that by the intervention of this atmosphere they appear smaller when they are in apogee; but when in perigee, through the absence or attenuation of that atmosphere, they appear larger. Want of time prevents my going further into these matters; my readers may expect further remarks upon these subjects in a short time.

[18] Modern astronomers agree in assigning an atmosphere to Jupiter, but consider it not extensive enough to affect the brightness of the satellites.—(WEBB, Celestial Objects for Common Telescopes.) Their absolute magnitudes are different, and their surfaces have been observed to be obscured by spots, which may account for the variations of their brightness. These spots, like the lunar spots, are probably due to variations of reflective power at different parts of their surfaces, for as they always turn the same face to Jupiter, they present different portions of their surfaces to us periodically, and it has been ascertained by observation that “these fluctuations in their brightness are periodical, depending on their position with respect to the Sun.”—(HERSCHEL, Outlines of Astronomy; ARAGO, Astronomie Populaire, 1854.)

Original Configurations of Jupiter’s Satellites observed by Galileo in the months of January, February, and March 1610, and published with the 1st edition of his book Sidereus Nuncius, Venice, 1610.

—————————+———————————+———————————————+——————————————— FIG. | DATE. | EAST. | WEST. —————————+———————————+———————————————+——————————————— 1 | Jan. 7 | | | • • ◯ • —————————+———————————+——————————————————————————————— 2 | 8 | | | ◯ • • • —————————+———————————+——————————————————————————————— 3 | 10 | | | • • ◯ —————————+———————————+——————————————————————————————— 4 | 11 | | | • • ◯ —————————+———————————+——————————————————————————————— 5 | 12 | | | • •◯ • —————————+———————————+—————————-————————————————————— 6 | 13 | | | • ◯ • ⠁ • —————————+———————————+——————————————————————————————— 7 | 15 | • | | ◯ • • • —————————+———————————+——————————————————————————————— 8 | 15 | | | ◯ • • • —————————+———————————+——————————————————————————————— 9 | 16 | | | • ◯ • • —————————+———————————+——————————————————————————————— 10 | 17 | | | • ◯ • —————————+———————————+——————————————————————————————— 11 | Jan. 17 | • • | | • ◯ —————————+———————————+——————————————————————————————— 12 | 18 | | | • ◯ • —————————+———————————+——————————————————————————————— 13 | 19 | | | • ◯ • • —————————+———————————+——————————————————————————————— 14 | 19 | | | • • ◯ • • —————————+———————————+——————————————————————————————— 15 | 20 | | | • ◯ • • —————————+———————————+——————————————————————————————— 16 | 20 | | | • ◯ • • —————————+———————————+——————————————————————————————— 17 | 20 | • ◯ • • | | • —————————+———————————+——————————————————————————————— 18 | 21 | | | • • ◯ • —————————+———————————+——————————————————————————————— 19 | 22 | • ◯• • | | • —————————+———————————+——————————————————————————————— 20 | 22 | | | • ◯ • • • —————————+———————————+——————————————————————————————— 21 | 23 | | | • • ◯ • —————————+———————————+——————————————————————————————— 22 | 23 | | | • ◯ —————————+———————————+——————————————————————————————— 23 | 24 | • • ◯ | | • —————————+———————————+——————————————————————————————— 24 | 25 | | | • • ◯ —————————+———————————+——————————————————————————————— 25 | 26 | | | • • ◯ • —————————+———————————+——————————————————————————————— 26 | 26 | • | | • • ◯ • —————————+———————————+——————————————————————————————— 27 | 27 | | | • ◯ —————————+———————————+——————————————————————————————— 28 | 30 | • | | • ◯ • —————————+———————————+——————————————————————————————— 29 | 31 | • | | • ◯ • —————————+———————————+——————————————————————————————— 30 | Jan. 31 | • | | • ◯ • —————————+———————————+——————————————————————————————— 31 | Feb. 1 | | | • • ◯ • —————————+———————————+——————————————————————————————— 32 | 2 | | | • ◯ • • —————————+———————————+——————————————————————————————— 33 | 2 | | | • • ◯ • • —————————+———————————+——————————————————————————————— 34 | 3 | | | • ◯ • • —————————+———————————+——————————————————————————————— 35 | 4 | | | • • ◯ • • —————————+———————————+——————————————————————————————— 36 | 4 | | | • • ◯ • • —————————+———————————+——————————————————————————————— 37 | 6 | | | • ◯ • —————————+———————————+——————————————————————————————— 38 | 7 | | | • • ◯ —————————+———————————+——————————————————————————————— 39 | 8 | | | • • • ◯ —————————+———————————+——————————————————————————————— 40 | 9 | | | • • ◯ • —————————+———————————+——————————————————————————————— 41 | 10 | | | • • ◯ —————————+———————————+——————————————————————————————— 42 | 11 | | | • • ◯ • —————————+———————————+——————————————————————————————— 43 | 11 | • | | • • ◯ • —————————+———————————+——————————————————————————————— 44 | 11 | | | • • • ◯ • —————————+———————————+——————————————————————————————— 45 | 12 | | | • • ◯ • • —————————+———————————+——————————————————————————————— 46 | 13 | | | • • ◯ • • —————————+———————————+——————————————————————————————— 47 | 15 | | | • • • ◯ —————————+———————————+——————————————————————————————— 48 | 15 | | | • • ◯ —————————+———————————+——————————————————————————————— 49 | Feb. 15 | | | • • ◯ • —————————+———————————+——————————————————————————————— 50 | 16 | | | • ◯ • • —————————+———————————+——————————————————————————————— 51 | 17 | | | • ◯ • —————————+———————————+——————————————————————————————— 52 | 18 | | | • ◯ • • —————————+———————————+——————————————————————————————— 53 | 18 | | | • • ◯ • • —————————+———————————+——————————————————————————————— 54 | 19 | | | ◯ • • —————————+———————————+——————————————————————————————— 55 | 21 | | | • ◯ • • —————————+———————————+——————————————————————————————— 56 | 25 | | | • • ◯ • —————————+———————————+——————————————————————————————— 57 | 26 | | | • ◯ • —————————+———————————+——————————————————————————————— 58 | 26 | | | • ◯ • • —————————+———————————+——————————————————————————————— 59 | 26 | • ◯ • • | | ⨀ Star. —————————+———————————+——————————————————————————————— 60 | 27 | • • ◯ • • | | Star ⨀ —————————+———————————+——————————————————————————————— 61 | 28 | • ◯ • | | Star ⨀ —————————+———————————+——————————————————————————————— 62 | 28 | | | • • ◯ • —————————+———————————+——————————————————————————————— 63 | Mar. 1 | • | | • • • ◯ | | Star ⨀ —————————+———————————+——————————————————————————————— 64 | 2 | • | | • ◯ • | | Star ⨀ —————————+———————————+———————————————————————————————

A PART OF THE PREFACE TO KEPLER’S DIOPTRICS

FORMING

A CONTINUATION OF GALILEO’S SIDEREAL MESSENGER.

In the preface to Kepler’s Dioptrics there are introduced letters of Galileo about the new and astonishing discoveries which he had made in the heavens by the aid of the telescope since the publication of his work, The Sidereal Messenger. The portion of the preface which refers to Galileo, containing these letters and Kepler’s remarks upon them, is added here, as continuing the original account of Galileo’s astronomical discoveries.

Extract from the Preface of Kepler’s Dioptrics. Augsburg, 1611.

[Sidenote: Kepler remarks on the importance of the application of the telescope to astronomical investigations as indicated by Galileo’s discoveries, published in his Sidereal Messenger.]

“The Sidereal Messenger” of Galileo has been for a long time in everybody’s hands, also my “Discussion, such as it is, with this Messenger,” and my Brief Narrative in confirmation of Galileo’s Sidereal Messenger, so any reader may briefly weigh the chief points of that Messenger and see the nature and the value of the discoveries made by the aid of the telescope, the theory of which I am intending to demonstrate in this treatise. Actual sight testified that there is a certain bright heavenly body which we call the Moon. It was demonstrated from the laws of optics that this body is round; also Astronomy, by some arguments founded upon optics, had built up the conclusion that its distance from the earth is about sixty semi-diameters of the earth. Various spots showed themselves in that body; and the result was a dubious opinion among a few philosophers, derived from Hecatæus’ account of the stories about the island of the Hyperboreans,[19] that the reflected images of mountains and valleys, sea and land, were seen there; but now the telescope places all these matters before our eyes in such a way that he must be an intellectual coward who, while enjoying such a view, still thinks that the matter is open to doubt. Nothing is more certain than that the southern parts of the moon teem with mountains, very many in number, and vast in size; and that the northern parts, inasmuch as they are lower, receive in most extensive lakes the water flowing down from the south. The conclusions which previously Pena published as disclosed by the aid of optics, started indeed from certain slight supports, rather than foundations, afforded by actual sight, but were proved by long arguments depending one upon another, so that they might be assigned to human reason rather than to sight; but now our very eyes, as if a new door of heaven had been opened, are led to the view of matters once hidden from them. But if it should please any one to exhaust the force of reasoning upon these new observations, who does not see how far the contemplation of nature will extend her boundaries, when we ask, “What is the use of the tracts of mountains and valleys, and the very wide expanse of seas in the moon?” and “May not some creature less noble than man be imagined such as might inhabit those tracts?”

[19] Diodorus Siculus, ii. 47.

With no less certainty also do we decide by the use of this instrument even that famous question, which, coeval with philosophy itself, is disputed to this day by the noblest intellects—I mean, whether the earth can move (as the theory of the Planets greatly requires) without the overthrow of all bodies that have weight, or the confusion of the motion of the elements? For if the earth were banished from the centre of the universe, some fear lest the water should leave the orb of the earth and flow to the centre of the universe; and yet we see that in the moon, as well as in the earth, there is a quantity of moisture occupying the sunken hollows of that globe; and although this orb revolves actually in the ether, and outside the centres not merely of the universe, but even of our earth, yet the mass of water in the moon is not at all hindered from cleaving invariably to the orb of the moon, and tending to the centre of the body to which it belongs. So, by this instance of the phenomena of the moon, the science of optics amends the received theory of mechanics, and confirms on this point my introduction to my Commentaries upon the Motions of the Planet Mars.[20]

[20] Kepler says in his introduction to his Commentaries upon the Motions of the Planet Mars, that the theory of gravitation depends on certain axioms, one of which is that “heavy bodies do not tend to the centre of the universe, supposing the earth to be placed there, because that point is the centre of the universe, but because it is the centre of the earth. So, wherever the earth be set, or whithersoever it be transported, heavy bodies have a continual tendency to it.” Kepler’s object in this work was to correct the methods for determining the apparent places of the planets according to the three theories then current—the Ptolemaic, the Copernican, and that of Tycho Brahe.

In 1593 the observed place of the planet Mars differed by nearly 5° from the place calculated for it. Kepler accordingly studied the motions of this planet, and “by most laborious demonstrations and discussions of many observations,” arrived at the conclusions known as Kepler’s first and second laws; according to which the Copernican system of eccentrics and epicycles was replaced by an ellipse whose centre and eccentricity were the same as the centre and eccentricity of the eccentric in the older method, and the Sun therefore was in one of the foci. Also the motion of the planet in its orbit was such that equal areas were described about the Sun by the radius vector of the planet in equal times.—KEPLER, Astronomia Nova αἰτιολογητός (Prague), 1609.

The followers of the Samian philosophy (for I may use this epithet to designate the philosophy originated by the Samians, Pythagoras and Aristarchus) have a strong argument against the apparent immobility of the earth provided in the phenomena of the moon. For we are taught by optics that if any one of us was in the moon, to him the moon, his abode, would seem quite immovable, but our earth and sun and all the rest of the heavenly bodies movable; for the conclusions of sight are thus related.

Pena has noticed how astronomers, using the principles of optics, have by most laborious reasoning removed the Milky Way from the elementary universe, where Aristotle had placed it, into the highest region of the ether; but now, by the aid of the telescope lately invented, the very eyes of astronomers are conducted straight to a thorough survey of the substance of the Milky Way; and whoever enjoys this sight is compelled to confess that the Milky Way is nothing else but a mass of extremely small stars.

Again, up to this time the nature of nebulous stars had been entirely unknown; but if the telescope be directed to one of such nebulous balls, as Ptolemy calls them, it again shows, as in the case of the Milky Way, three or four very bright stars clustered very close together.

Again, who without this instrument would have believed that the number of the fixed stars was ten times, or perhaps twenty times, more than that which is given in Ptolemy’s description of the fixed stars? And whence, pray, should we seek for conclusive evidence about the end or boundary of this visible universe, proving that it is actually the sphere of the fixed stars, and that there is nothing beyond, except from this very discovery by the telescope of this multitude of fixed stars, which is, as it were, the vaulting of the mobile universe? Again, how greatly an astronomer would go wrong in determining the magnitude of the fixed stars, except he should survey the stars all over again with a telescope, also may be seen in Galileo’s treatise, and we will also hereafter produce in proof a letter from a German astronomer.

But no words can express my admiration of that chapter of the Sidereal Messenger where the story is told of the discovery, by the aid of a very highly finished telescope, of another world, as it were, in the planet Jupiter. The mind of the philosopher almost reels as he considers that there is a vast orb, which is equal in mass to fourteen orbs like the earth (unless on this point the telescope of Galileo shall shortly reveal something more exact than the measurements of Tycho Brahe) round which circle four moons, not unlike this moon of ours; the slowest revolving in the space of fourteen of our days, as Galileo has published; the next to this, by far the brightest of the four, in the space of eight days, as I detected in last April and May; the other two in still shorter periods. And here the reasoning of my Commentaries about the Planet Mars, applied to a similar case, induces me to conclude also that the actual orb of Jupiter rotates with very great rapidity, most certainly faster than once in the space of one of our days; so that this rotation of the mighty orb upon its own axis is accompanied wherever it goes by the perpetual circuits of those four moons. Moreover, this sun of ours, the common source of heat and light for this terrestrial world as well as for that world of Jupiter, which we consider to be of the angular magnitude of 30´ at most, there scarcely subtends more than 6´ or 7´, and is found again in the same position among the fixed stars, having completed the zodiac in the interval, after a period of twelve of our years.[21] Accordingly, the creatures which live on that orb of Jupiter, while they contemplate the very swift courses of those four moons among the fixed stars, while they behold them and the sun rising and setting day by day, would swear by Jupiter-in-stone, like the Romans (for I have lately returned from those parts), that their orb of Jupiter remains immovable in one spot, and that the fixed stars and the sun, which are the bodies really at rest, no less than those four moons of theirs, revolve round that abode of theirs with manifold variety of motions. And from this instance now, much more than before from the instance of the moon, any follower of the Samian philosophy will learn what reply may be made to any one objecting to the theory of the motion of the earth as absurd, and alleging the evidence of our sight. O telescope, instrument of much knowledge, more precious than any sceptre! Is not he who holds thee in his hand made king and lord of the works of God? Truly

“All that is overhead, the mighty orbs With all their motions, thou dost subjugate To man’s intelligence.”

If there is any one in some degree friendly to Copernicus and the lights of the Samian philosophy, who finds this difficulty only, that he doubts how it can happen, supposing the earth to perform again and again her course among the planets through the ethereal plains, that the moon should keep so constantly by her side, like an inseparable companion, and at the same time fly round and round the actual orb of the earth, just like a faithful dog which goes round and round his master on some journey, now running in front, now deviating to this side or that, in ever-varying mazes, let him look at the planet Jupiter, which, as this telescope shows, certainly carries in its train not one such companion only, like the earth, as Copernicus showed, but actually four, that never leave it, though all the time hastening each in its own orbit.

[21] The degree of accuracy attained by Kepler and Galileo with their imperfect instruments will be appreciated by comparing these statements with the determinations of later astronomers. Jupiter is about 1300 times the size of the Earth. Its diameter is about 87,000 miles; time of rotation, 9 h. 55 m. 21 sec.; time of revolution, 4333 days nearly. The angular diameter of the sun, seen from Jupiter, is between 6´ and 7´. The times of revolution of the four satellites are, as already given: (i.) 1 d. 18 h. 28 m., (ii.) 3 d. 13 h. 15 m., (iii.) 7 d. 3 h. 43 m., (iv.) 16 d. 16 h. 32 m.

But enough has been said about these matters in my Discussion with the Sidereal Messenger. It is time that I should turn to those discoveries which have been made since the publication of Galileo’s Sidereal Messenger, and since my Discussion with it, by means of this telescope.

[Sidenote: Galileo’s discovery of Saturn’s ring (imperfectly).]

It is now just a year since Galileo wrote to Prague, and gave full notice that he had detected something new in the heavens beyond his former discoveries; and that there might not be any one who, with the intention of detracting from his credit, should try to pass himself off as an earlier observer of the phenomenon, Galileo gave a certain space of time for the publication of the new phenomena which any one had seen; he himself meanwhile described his discovery in letters transposed in this manner: s m a i s m r m i l m e p o e t a l e u m i b u n e n u g t t a u i r a s. Out of these letters I made an uncouth verse which I inserted in my Short Account in the month of September of last year:—

Salve umbistineum[22] geminatum Martia proles. Hail, twin companionship, children of Mars.

[22] Umbistineum. Apparently this is some German word with a Latin ending, such as um-bei-stehn; Kepler fancied that Galileo had discovered two satellites of Mars.

But I was a very long way from the meaning of the letters; it contained nothing to do with Mars; and, not to detain you, reader, here is the solution of the riddle in the words of Galileo himself, the author of it:[23]—

“Di Firenze li 13 di Novembre 1610.—Ma passando ad altro, giacchè il Sig. Keplero ha in questa sua ultima narrazione stampate le lettere che io mandai trasposte a Vostra Signoria Illustrissima e Reverendissima venendomi anco significato come Sua Maestà ne desidera il senso, ecco che io lo mando a Vostra Signoria Illustrissima per participarlo con Sua Maestà col Sig. Keplero e con chi piacerà a Vostra Signoria Illustrissima bramando io che lo sappia ognuno. Le lettere dunque combinate nel lor vero senso dicono così,

Altissimum planetam tergeminum observavi.

Questo è, che Saturno con mia grandissima ammirazione ho osservato essere non una stella sola, ma tre insieme, le quali quasi si toccano; e sono trà di loro totalmente immobili, e constituite in questa guisa o◯o. Quella di mezzo è assai più grande delle laterali; sono situate una da oriente, l’altra da occidente, nella medesima linea retta a capello; non sono giustamente secondo la dirittura del Zodiaco, ma l’occidentale si eleva alquanto verso Borea; forse sono parallele all’Equinoziale. Se si guarderanno con un occhiale che non sia di grandissima moltiplicazione, non appariranno tre stelle ben distinte, ma parrà, che Saturno sia una stella lunghetta in forma di un’oliva, così, ☾☽. Ma servendosi di un occhiale che moltiplichi più di mille volte in superficie, si vedranno tre globi distintissimi, che quasi si toccano, non apparendo trà essi maggior divisione di un sottil filo oscuro. Or ecco trovata la corte a Giove, e due Servi a questo Vecchio che l’aiutano a camminare nè mai se gli staccano dal fianco. Intorno agli altri Pianeti non ci è novità alcuna, ec.”

[23] The text of the four letters of Galileo followed here is that given in the edition of Galileo’s works published at Florence, 1842-56; that in the edition of Kepler’s Dioptrics, published at Augsburg, 1611, is very inaccurate. These letters were written to Giuliano de’ Medici, ambassador of the Grand-Duke of Tuscany to the Emperor Rudolf II. at Prague.

Although these words differ little from Latin, yet I will translate them that no difficulty may hinder my reader from following me. Thus then Galileo writes:—“But to come now to my second topic. Since Kepler has published in that recent ‘Narrative’ of his the letters which I sent as an anagram to your illustrious Lordship, and since an intimation has been given me that his Majesty desires to be taught the meaning of those letters, I send it to your illustrious Lordship, that your Lordship may communicate it to his Majesty, to Kepler, and to any one your Lordship may wish.

“The letters when joined together as they ought to be, say this,

‘Altissimum planetam tergeminum observavi,’

‘I have observed the most distant of the planets to have a triple form.’

“For in truth I have found out with the most intense surprise that the planet Saturn is not merely one single star, but three stars very close together, so much so that they are all but in contact one with another. They are quite immovable with regard to each other, and are arranged in this manner, o◯o. The middle star of the three is by far greater than the two on either side. They are situated one towards the east, the other towards the west, in one straight line to a hair’s-breadth; not, however, exactly in the direction of the Zodiac, for the star furthest to the west rises somewhat towards the north; perhaps they are parallel to the equator. If you look at them through a glass that does not multiply much, the stars will not appear clearly separate from one another, but Saturn’s orb will appear somewhat elongated, of the shape of an olive, thus, ☾☽. But if you should use a glass which multiplies a surface more than a thousand times, there will appear very distinctly three orbs, almost touching one another; and they will be thought to be not farther apart than the breadth of a very fine and scarcely visible thread. So you see a guard of satellites has been found for Jupiter, and for the decrepit little old man two servants to help his steps and never leave his side. Concerning the rest of the planets I have found nothing new.”

So says Galileo; but if I may do so, I will not make an old dotard out of Saturn, and two servants for him out of his companion orbs, but rather out of those three united bodies I will make a triple Geryon, out of Galileo Hercules, and out of the telescope his club, armed with which, Galileo has conquered that most distant of the planets, drawn it out of the furthest recesses of nature, dragged it down to earth, and exposed it to the gaze of us all. It pleases me too, now that the nest has been found, to consider with curiosity what kind of brood must be in it, what kind of life, if there be any life there, between orbs which all but touch each other two and two, where not even

“a space Of sky extends not more than three ells wide,”[24]

but where there is scarcely a chink of a nail’s-breadth all round.

[24] Virgil, Eclog. iii. 105.

Do indeed the astrologers rightly ascribe to Saturn the guardianship of miners, who, accustomed to spend their lives, like moles, underground, seldom breathe the free air under the open sky? Although the darkness here is rather more supportable than in Saturn, because the sun, which appears there only as large as Venus appears to us on the earth, continually casts its rays through the spaces between the different orbs in such a way that those inhabitants who are situated on one orb are covered by the other as by a ceiling; while those on the latter orb, on the top of this roof of theirs, exposed as it is to the full light of the sun, receive a glare as if from very firebrands. But I must draw in the reins and check my mind in its enjoyment of the free fields of ether; for fear, perchance, later observations should report something different from the first account, something changed in course of time.[25]

[25] The completion of Galileo’s observations on Saturn depended on the improvement of astronomical instruments, as will be evident from the following sketch. Galileo made out the first indications of Saturn’s ring in 1610, as narrated in his letter, with a power of thirty; but in December 1612 he wrote to one of his friends, Marco Velseri, that he could no longer see these indications, and began to imagine that his telescope had deceived him, and apparently abandoned further researches. Hevelius in 1642 saw the ring more clearly, but figured it as two crescents attached to Saturn by their cusps. At length, in 1653, Huyghens provided himself with a power of one hundred, having made the lenses with his own hands, and immediately discovered the explanation of the phenomena which had baffled previous observers. He published his explanation of Saturn’s ring, and his discovery of the first satellite, in his Systema Saturnium, 1659. Cassini, with still more powerful instruments, discovered four more satellites in 1671, 1672, 1684. Sir William Herschel in 1789 detected two more, “which can only be seen with telescopes of extraordinary power and perfection, and under the most favourable atmospheric circumstances.”—(HERSCHEL, Outlines of Astronomy, § 548.) And the last of the eight satellites was discovered in 1848 by Lassell of Liverpool, and Bond of Cambridge, U.S., simultaneously.

At the end of his letter Galileo seemed to think that he had come to the end of his reports about the planets, and observations of new phenomena respecting them, but ever on the watch, that eye of his, that one not of Nature’s making—I mean his telescope—in a short time made more discoveries, concerning which read the following letter of Galileo:—

[Sidenote: Account of Galileo’s discovery of the phases of Venus.]

“Di Firenze li 11 di Decembre 1610.—Sto con desiderio, attendendo la risposta a due mie scritte ultimamente per sentire quello, che averà detto il Sig. Keplero della stravaganza di Saturno. Intanto mando [a Vostra Signoria Illustrissima e Reverendissima] la cifra di un altro particolare osservato da me nuovamente, il quale si tira dietro la decisione di grandissime controversie in Astronomia, ed in particolare contiene in se un gagliardo argomento per la constitutione [Pitagorica e Copernicana] dell’Universo; e a suo tempo pubblicherò la deciferazione ed altri particolari. Spero che averò trovato il metodo per definire i periodi dei quattro Pianeti Medicei, stimati con gran ragione quasi inesplicabili dal Sig. Keplero, al quale piacerà, ec.

“Le lettere trasposte sono queste:

“Haec immatura a me jam frustra leguntur, o.y.”

Which may be translated thus:—

“I am anxiously looking for an answer to my last two letters, that I may learn what Kepler says about the marvels of Saturn’s orb. In the meantime I send him a riddle concerning a certain new and splendid observation which tends to decide great controversies in astronomy; and especially contains a fine argument in favour of the Pythagorean and Copernican system of the universe. At the proper time I will publish the solution of the riddle, and some other particulars. I hope that I have found a method of determining the periodic times of the four Medicean planets, which Kepler, not without very good reason, thought inexplicable, etc.

“The letters turned into an anagram, are these:

“Haec immatura a me jam frustra leguntur, o.y.”

So far Galileo. But if, reader, this letter has filled you with a desire to know the meaning contained in those letters, then you must read another letter of Galileo which follows.

[Illustration: Fig. 1.

E the Earth (centre of universe). S the Sun. C centre of eccentric. D centre of planet’s epicycle. VV' stationary points. s v v' projections of SVV' on the ecliptic of which E is the centre.]

[Illustration: Fig. 2.

S the Sun, centre of solar system. v e positions of planet and Earth at conjunction. VV' stationary points of planet. EE' corresponding positions of the Earth.]

But before you do so, I should like you to notice, by the way, what Galileo says about the Pythagorean and Copernican system of the universe. For he points to my Mystery of the Universe,[26] published fourteen years ago, in which I took the dimensions of the Planetary orbits according to the astronomy of Copernicus, who makes the sun immovable in the centre, and the earth movable both round the sun and upon its own axis; and I showed that the differences of their orbits corresponded to the five regular Pythagorean figures, which had been already distributed by their author among the elements of the world, though the attempt was admirable rather than happy or legitimate, and for which figures’ sake Euclid wrote the whole of his Geometry. Now, in that Mystery you may find a sort of combination of Astronomy and Euclid’s Geometry, and through this combination a most thorough completion and finishing of them both; and this was the reason why I waited with intense longing to see what sort of an argument Galileo would produce in favour of the Pythagorean system of the universe. After this explanation, Galileo’s letter about this argument was as follows:—

[26] Kepler, in his Mystery of the Universe, endeavoured to connect the orbits of the planets with the five regular solids, thus: If in a sphere (i.) a cube be inscribed, and in the cube a sphere (ii.); and in that sphere a tetrahedron, and in the tetrahedron a sphere (iii.); and in that sphere a dodecahedron, and in the dodecahedron a sphere (iv.); and in that sphere an icosahedron, and in the icosahedron a sphere (v.); and in that sphere an octahedron, and in the octahedron a sphere (vi.), the diameters of these six spheres will be proportional to the diameters of the orbits of Saturn, Jupiter, Mars, the Earth, Venus, and Mercury respectively; or, as Kepler expresses it, the common centre of these spheres represents the position of the Sun, and the six spheres represent the spheres of the planets.

By these considerations, however, Kepler was led to enunciate his third law, that the squares of the periodic times of planets are proportional to the cubes of their mean distances from the sun.—KEPLER, Prodromus Dissertationum Mathematicarum continens Mysterium Cosmographicum, etc. (Tübingen, 1596.)

“Illustrissimo e Reverendissimo Signore mio colendissimo.

“È tempo che io deciferi a Vostra Signoria Illustrissima e Reverendissima e per lei al Sig. Keplero le lettere trasposte le quali alcune settimane sono le inviai; è tempo dico, giacchè sono interamente chiaro della verità del fatto, sicchè non ci resta un minimo scrupolo, o dubbio. Sapranno dunque come circa a tre mesi fa vedendosi Venere vespertina la cominciai ad osservar diligentemente coll’occhiale, per veder col senso stesso quello di che non dubitava punto l’intelletto. La vidi dunque sul principio di figura rotonda, pulita e terminata, ma molto picciola; di tal figura si mantenne sino che cominciò ad avvicinarsi alla sua massima digressione, ma tra tanto andò crescendo in mole. Cominciò poi a mancare dalla rotondità nella sua parte orientale ed avversa al Sole, e in pochi giorni si ridusse ad esser un mezzo cerchio perfettissimo, e tale si mantenne, senza punto alterarsi, finchè incominciò à ritirarsi verso il Sole, allontanandosi dalla tangente. Ora va calando dal mezzo cerchio, e si mostra cornicolata, e anderà assottigliandosi sino all’occultazione, riducendosi allora con corna sottilissime. Quindi passando all’ apparizione mattutina, la vedremo pur falcata, e sottilissima e colle corna avverse al Sole; anderà poi crescendo fino alla massima digressione, dove apparirà semicircolare, e tale senza alterarsi si manterrà molti giorni, e poi dal mezzo cerchio passerà presto al tutto tondo, e così rotonda si conserverà poi per molti mesi. Il suo diametro adesso è circa cinque volte maggiore di quello, che si mostrava nella sua prima apparizione vespertina; dalla quale mirabile esperienza abbiamo sensata, e certa dimostrazione di due gran questioni state fin qui dubbie trà i maggiori ingegni del Mondo. L’una è, che i Pianeti tutti son di lor natura tenebrosi (accadendo anco a Mercurio l’istesso, che a Venere). L’altra, che Venere necessarissimamente si volge intorno al Sole, come anco Mercurio, cosa, che degli altri Pianeti, fu creduta da’ Pitagorici, dal Copernico, dal Keplero e da’ loro seguaci, ma non sensatamente provata, come ora in Venere, ed in Mercurio.

“Averanno dunque il Sig. Keplero, e gli altri Copernicani da gloriarsi di aver creduto e filosofato bene, sebbene ci è toccato, e ci è per toccare ancora ad esser reputati dall’ università dei Filosofi in libris, per poco intendenti, e poco meno che stolti.

“Le parole dunque, che mandai trasposte, e che dicevano,

Haec immatura a me jam frustra leguntur, o.y.

dicono ordinate

Cynthiae figuras aemulatur mater amorum.

“Cioè, che Venere imita le figure della Luna. Osservai tre notti sono l’ecclisse, nella quale non vi è cosa notabile, solo si vede il taglio dell’ombra indistinto, confuso e come annebbiato, e questo per derivare essa ombra dalla Terra lontanissima da essa Luna. Voleva scrivere altri particolari, ma essendo stato trattenuto molto da alcuni gentiluomini, ed essendo l’ora tardissima, son forzato a finire. Favoriscami salutare in mio nome i SS. Keplero, Asdale e Segheti, ed a Vostra Signoria Illustrissima con ogni reverenza bacio le mani, e dal Signore Dio gli prego felicità. Di Firenze il primo di Gennaio 1610. Ab Incarnatione.

“Di Vostra Signoria Illustrissima e Reverendissima Servidore obbligatissimo.

Galileo Galilei.”

Such is Galileo’s letter; but let me give you the substance of it:—

“It is time for me to disclose the method of reading the letters which some weeks since I sent you as an anagram. It is time now, I mean, after I have become quite certain about the matter, so much so that I have no longer even a shadow of doubt. You must know then that about three months ago, when the star of Venus could be seen, I began to look at it through a telescope with great attention, so that I might grasp with my physical senses an idea which I was entertaining as certain. At first then you must know the planet Venus appeared of a perfectly circular form, accurately so, and bounded by a distinct edge, but very small; this figure Venus kept until it began to approach its greatest distance from the sun, and meanwhile the apparent size of its orb kept on increasing. From that time it began to lose its roundness on the eastern side, which was turned away from the sun, and in a few days it contracted its visible portion into an exact semicircle; that figure lasted without the smallest alteration until it began to return towards the sun where it leaves the tangent drawn to its epicycle.[27] At this time it loses the semicircular form more and more, and keeps on diminishing that figure until its conjunction, when it will wane to a very thin crescent. After completing its passage past the sun, it will appear to us, at its appearance as a morning star, as only sickle-shaped, turning a very thin crescent away from the sun; afterwards the crescent will fill up more and more until the planet reaches its greatest distance from the sun, in which position it will appear semicircular, and that figure will last for many days without appreciable variation. Then by degrees, from being semicircular it will change to a full orb, and will keep that perfectly circular figure for several months; but at this instant the diameter of the orb of Venus is about five times as large as that which it showed at its first appearance as an evening star.

“From the observation of these wonderful phenomena we are supplied with a determination most conclusive, and appealing to the evidence of our senses, of two very important problems, which up to this day were discussed by the greatest intellects with different conclusions. One is that the planets are bodies not self-luminous (if we may entertain the same views about Mercury as we do about Venus). The second is that we are absolutely compelled to say that Venus (and Mercury also) revolves round the sun, as do also all the rest of the planets. A truth believed indeed by the Pythagorean school, by Copernicus, and by Kepler, but never proved by the evidence of our senses, as it is now proved in the case of Venus and Mercury. Kepler therefore and the rest of the school of Copernicus have good reason for boasting that they have shown themselves good philosophers, and that their belief was not devoid of foundation; however much it has been their lot, and may even hereafter be their lot, to be regarded by the philosophers of our times, who philosophise on paper, with an universal agreement, as men of no intellect, and little better than absolute fools.

“The words which I sent with their letters transposed, and which said,

Haec immatura a me jam frustra leguntur, o.y.

when reduced to their proper order, read thus,

Cynthiae figuras aemulatur mater amorum: The mother of the Loves rivals the phases of Cynthia:

that is,

Venus imitates the phases of the Moon.

“Three days ago I observed an eclipse of the moon, but not anything worthy of special notice occurred in it. Only the edge of the shadow appeared indistinct, blurred, and hazy; the cause of the phenomenon no doubt is that the shadow has its origin at the earth, at a great distance from the body of the moon.

“I have some other particulars, but I am prevented by time from writing about them, etc.”

So writes Galileo.

[27] In the Ptolemaic system the earth’s centre was regarded as the centre of the universe, and the movements of the heavenly bodies were explained by eccentrics and epicycles. The sun was conceived to describe a circle about a point not exactly coinciding with the centre of the earth, called the sun’s eccentric. The planets described epicycles (circles) whose centres described eccentrics (circles), and the centres of these eccentrics coincided with the centre of the sun’s eccentric. In the case of Mercury and Venus the centre of the epicycle was always on the line drawn from the centre of the eccentric to the sun’s centre. In the case of the other planets the construction was more complicated. The stationary points were determined by drawing tangents from the earth’s centre (or the observer) to the epicycle, as in the figure (1).—(GASSENDI, Institutio Astronomica, 1647.) This will explain Kepler’s description of the stationary points as the points where the planet leaves the tangent to its epicycle, supposing that he uses the terms of the current (i.e. Ptolemaic) astronomy. Copernicus placed the sun instead of the earth at the centre of the universe, but to determine the positions of the planets at any given time with as much accuracy as was attainable with the Ptolemaic system, he was obliged to use a similar method of eccentrics and epicycles, so that Kepler’s expression may be understood to describe the stationary points according to the Copernican theory, though it is still strange that he should not recognise the elliptical form of the planetary orbits, which he had lately demonstrated after most laborious reasoning in his Commentaries on the Motion of the Planet Mars, 1609. Galileo’s own expression seems to describe the stationary points according to the Copernican system, as would be expected, as the points where the planet leaves the tangent drawn to its orbit from the earth (Fig. 2).

What now, dear reader, shall we make out of our telescope? Shall we make a Mercury’s magic-wand to cross the liquid ether with, and, like Lucian,[28] lead a colony to the uninhabited evening star, allured by the sweetness of the place? or shall we make it a Cupid’s arrow, which, entering by our eyes, has pierced our inmost mind, and fired us with a love of Venus? For what language is too strong for the marvellous beauty of this orb, if, having no light of its own, it can attain simply by the borrowed light of the sun to such splendour, as Jupiter has not, nor the moon, though enjoying a proximity to the sun as close as Venus; for if the moon’s light be compared with the light of Venus, it will be seen to be certainly greater on account of the apparent magnitude of the moon, but, in comparison with the light of Venus, dull, dead, and leaden. O truly golden Venus! Will any one doubt any more that the whole orb of Venus is wrought most smoothly out of pure unalloyed gold, since its surface, when only placed in the sunlight, reflects a splendour so intense! And here let me add my experiments about the alteration of the light of Venus on blinking the eye, which I have examined in the part of my Astronomy which treats of Optics. Reasoning will be able to conclude nothing else but this, that the orb of Venus turns on its own axis with an exceedingly swift rotation, displaying one after another different parts of its surface which are more or less capable of retaining the sun’s light.[29]

[28] Lucian, Ver. Hist. i. 12.

[29] The first scientific determination of the period of the rotation of Venus was made by Dominique Cassini in 1666, from observations of spots on the planet, and concluded to be about 24 hours; but in 1726 Bianchini deduced a period of 24 d. 8 h. from similar observations. The true period is considered to be 23 h. 21 m., determined by Schroeter by a series of observations lasting from 1788 to 1793 on the periodicity of the deformation of the horns of Venus.—(ARAGO, Astronomie Populaire, 1854.)

Kepler’s statements can only be regarded as anticipations of phenomena not yet actually observed.

But enough of my own conclusions. Let us now hear as an epilogue Galileo’s conclusions built up out of all the observations which he has made with his telescope, and announced from time to time. Thus he writes once more:—

[Sidenote: Galileo’s conclusions with regard to the inherent nature of the brightness of the stars.]

“Illustrissimo e Reverendissimo Signore mio colendissimo.

“Ho ricevuto gusto, e contento particolarissimo nella lettura dell’ ultima di Vostra Signoria Illustrissima e Reverendissima delli 7 stante, ed in particolare in quella parte dove ella m’accenna la favorevole inclinazione dell’ Illustriss. Sig. Cons. Wackher, verso di me, la quale io infinitamente stimo, ed apprezzo; e poichè quella ha principalmente origine dall’ aver io incontrate osservazioni necessariamente dimostranti conclusioni per avanti tenute vere da sua Signoria Illustrissima per confermarmi maggiormente il possesso di grazia tanto pregiata da me, prego Vostra Signoria Illustrissima e Reverendissima a fargli intendere per mia parte come conforme alla credenza di Sua Signoria Illustrissima ho dimostrazione certa, che siccome tutti i Pianeti ricevono il lume dal Sole, essendo per se stessi tenebrosi, e opachi; così le Stelle fisse risplendono per lor natura, non bisognose dell’ illustrazione de’ raggi solari, li quali, Dio sa, se arrivino a tanta altezza, più di quello, che arrivi a noi il lume di una di esse fisse. Il principal fondamento del mio discorso è nell’ osservare io molto evidentemente con gli occhiali che quei Pianeti di mano in mano, che si trovano più vicini a noi, o al Sole, ricevono maggiore splendore, e più illustremente ce lo riverberano; e perciò Marte perigeo, e a noi vicinissimo si vede assai più splendido, che Giove; benchè a quello di mole assai inferiore; e difficilmente se gli può coll’ occhiale levare quella irradiazione, che impedisce il vedere il suo disco terminato, e rotondo; il che in Giove non accade, vedendosi esquisitamente circolare. Saturno poi per la sua gran lontananza si vede esattamente terminato, sì la Stella maggiore di mezzo, come le due piccole laterali; ed appare il suo lume languido, ed abbacinato e senza niuna irradiazione, che impedisca il distinguere i suoi tre piccoli globi terminatissimi. Ora poichè apertamente veggiamo, che il Sole molto splendidamente illustra Marte vicino, e che molto più languido è il lume di Giove (sebbene senza lo strumento appare assai chiaro, il che accade per la grandezza, e candore della Stella) languidissimo, e fosco quello di Saturno, come molto più lontano, quali doveriano apparirci le Stelle fisse lontane indicibilmente più di Saturno, quando il lume loro derivasse dal Sole? Certamente debolissime, torbide e smorte. Ma tutto l’opposito si vede, perocchè se rimireremo per esempio il Cane, incontreremo un fulgore vivissimo, che quasi ci toglie la vista, con una vibrazione di raggi tanto fiera, e possente, che in comparazione di quello rimangono i Pianeti, e dico Giove e Venere stessa, come un impurissimo vetro appresso un limpidissimo e finissimo diamante. E benchè il disco di esso Cane apparisca non maggiore della cinquantesima parte di quello di Giove, tuttavia la sua irradiazione è grande, e fiera in modo, che l’istesso globo tra i proprii crini s’implica, e quasi si perde, e con qualche difficoltà si distingue; dove che Giove (e molto più Saturno) si vedono e terminati, e di una luce languida, e per così dire quieta. E per tanto io stimo, che bene filosoferemo, referendo la causa della scintillazione delle Stelle fisse, al vibrare, che elle fanno dello splendore proprio e nativo dall’intima loro sustanza; dove che nella superficie de’ Pianeti termina più presto, e si finisce l’illuminazione, che dal Sole deriva, e si parte. Se io sentirò qualche particolare questione ricevuta dal medesimo Sig. Wackher, non resterò d’affaticarmivi intorno, per dimostrarmi, quale io sono desiderosissimo di servire un tanto Signore, e non già con isperanza di aggiungere al termine conseguito dal suo discorso, perchè benissimo comprendo, che a quanto sia passato per lo finissimo cribro del giudizio di esso, e del Sig. Keplero, non si può aggiungere di squisitezza; nè io pretenderei altro, che col dubitare, e mal filosofare, eccitar loro al ritrovamento di nuove sottigliezze. Gl’ingegni singolari, che in gran numero fioriscono nell’Alemagna, mi hanno lungo tempo tenuto in desiderio di vederla, il qual desiderio ora si raddoppia per la nuova grazia dell’Illustrissimo Sig. Wackher, la quale mi farebbe divenir grande ogni picciola occasione, che mi si presentasse. Ma ho di soverchio occupata Vostra Signoria Illustrissima e Reverendissima. Degnisi per fine di offerirmi e dedicarmi devotissimo servidore all’Illustrissimo Sig. Wackher, salutando anco caramente il Sig. Keplero, ed a lei con ogni riverenza bacio le mani, e dal Signore Dio le prego somma felicità.

“[Di Firenze li 26] di Marzo 1611. Di Vostra Signoria Illustrissima e Reverendissima obbligatissimo Servidore,

GALILEO GALILEI.”

When translated, the meaning is as follows:—

“Your last letter has exceedingly pleased me, especially that part which assures me of the friendly feeling entertained towards me by the most illustrious Imperial Counsellor, Wagher, which I for my part highly appreciate. And since the cause of this friendliness is, that I have incontestably demonstrated by some observations of mine certain conclusions which he had long held as true, I wish to confirm my possession of favour, which I value so much, and accordingly I ask you to give him this piece of news from me; that I have most conclusive arguments ready, showing clearly that, just as he holds, all the planets receive their light from the sun, being by constitution bodies dark and devoid of light;[30] but that the fixed stars shine by their own proper light, not needing to be illuminated by the sun’s rays, since God knows whether they reach the very remote region of the fixed stars with intensity even equal to the small intensity with which the rays of the fixed stars come down to us.

“My demonstration depends chiefly on this fact, that with the telescope I have distinctly observed that the planets receive greater brightness, and reflect it more intensely, in proportion as each one is nearer to us and to the sun. So Mars in perigee, that is, when nearest to the earth, greatly surpasses Jupiter in brightness, although in actual size it is far inferior to Jupiter; and in consequence it is difficult to receive the effulgence of this planet in the telescope, for it is so great as to prevent the eye from being able to distinguish the circular termination of the planet’s disc. This does not happen in the case of Jupiter, for it appears quite circular. The next planet, Saturn, on account of its great distance likewise—for indeed it is the most remote of the planets,—appears bounded by a well-defined edge, both the greater orb in the middle and the two small orbs at its sides. Indeed, it appears to shine with a faint and delicate light, without any effulgence to prevent an observer recognising the well-defined termination of its three orbs. Since, then, we see that Mars, the nearest of the three, is illumined by the sun with very great splendour, and that the light of Jupiter, at a greater distance, is much more faint (although without the use of an instrument it appears tolerably bright, which is due to the size and brilliancy of its body), and that the light of Saturn, at the greatest distance, is most faint, and almost watery, of what kind, do you think, would appear the light of the fixed stars, which are at an immeasurable distance further from the sun than Saturn, if they only received light from the sun? Most certainly, extremely feeble, indistinct, and pallid. And yet we find quite the contrary; for, let us look with our eyes at the Dog-Star, for example. We shall encounter a most vivid brilliancy, which almost pricks the eye with the rapid sparkling of its rays, of such intensity that, in comparison with it, the planets, even Jupiter, and Venus too, are as thoroughly outshone as common and bad glass compared with a highly polished and most sparkling diamond. And although the orb of the Dog-Star appears no larger than the fiftieth part of Jupiter’s disc, nevertheless its brilliancy is great and very strong; so that the form of its disc, which you expect to see, hides itself among the rays of its own refulgence, envelops itself in them, and almost disappears; and it is not distinguished without some difficulty from the rays which surround it. Whereas Jupiter, and still more Saturn, are seen well defined; and their light is without intensity, and, if I may say so, quiescent. Wherefore I think that we shall rightly apply our philosophy if we refer the cause of the twinkling of the fixed stars to vibrations of a brilliancy, which is their own, belonging to their constitution, and inherent in their substance, and say, on the other hand, that the illumination of the planets, which is derived from the sun, and distributed to the world, is limited to their surface.”

These are the scientific conclusions in Galileo’s letter; the rest I omit.

[30] Proctor (Other Worlds than Ours, 1875) has given some reasons for believing that Jupiter and Saturn shine in part with their own light, owing to their great internal heat.

You see then, studious reader, how the subtle mind of Galileo, in my opinion the first philosopher of the day, uses this telescope of ours like a sort of ladder, scales the furthest and loftiest walls of the visible world, surveys all things with his own eyes, and, from the position he has gained, darts the glances of his most acute intellect upon these petty abodes of ours—the planetary spheres I mean,—and compares with keenest reasoning the distant with the near, the lofty with the deep.

VALE ET DEUM IN OPERIBUS SUIS CELEBRARE NUNQUAM DESINE.

KEPLER, Narratio.

Edinburgh University Press:

THOMAS AND ARCHIBALD CONSTABLE, PRINTERS TO HER MAJESTY.

Letter to the Grand Duchess Christina (1615)

LETTER TO THE GRAND DUCHESS CHRISTINA

THE Ancient and Modern DOCTRINE OF Holy Fathers, AND Iudicious Divines, CONCERNING The rash citation of the Testimony of SACRED SCRIPTURE, in Conclusions meerly Natural, and that may be proved by Sensible Experiments, and Necessary Demonstrations. Written, some years since, to Gratifie The most SERENE CHRISTINA LOTHARINGA, Arch-Dutchess of TVSCANY; By GALILAEO GALILAEI, A Gentleman of Florence, and Chief Philosopher and Mathematician to His most Serene Highness the Grand DVKE.

And now rendred into English from the Italian, BY THOMAS SALUSBURY.

Naturam Rerum invenire, difficile; & ubi inveneris, indicare in vulgus, nefas.

Plato.

LONDON, Printed by WILLIAM LEYBOURN, 1661.

TO Her most Serene HIGHNES THE Gran Duchess Mother.

SOme years since, as Your most Serene Highness well knoweth, I did discover many particulars in Heaven that had been unseen and unheard of untill this our Age; which, as well for their novelty, as for certain consequences which depend upon them, clashing with some Physical Propositions commonly received by the Schools, did stir up against me no small number of such as professed the vulgar Philosophy in the Universities; as if I had with my own hand newly placed these things in Heaven to obscure and disturb Nature and the Sciences: who forgetting that the multitude of Truths contribute, and concur to the investigation, augmentation, and establishment of the Arts, and not to their diminution, and destruction; and at the same time shewing themselves more affectionate to their own Opinions, than to Truth, went about to deny, and to disprove those Novelties; of which their very sense, had they but pleased to have intentsly beheld them, would have rendered them thorowly assured. And to this purpose they alledged sundry things, and published certain Papers fraughted with vain discourses; and which was a more gross errour, interwoven with the attestations of the Sacred Scriptures, taken from places by them not rightly understood, and which did not any thing concern the point for which they were produced Into which errour perhaps they would not have run, if they had but been advertised of a most profitable Document which S. Augustine giveth us, concerning our proceeding warily, in making positive determinations in points that

are obscure and hard to be understood by the meer help of ratiocination; where treating (as we) of a certain natural conclusion concerning Celestial Bodies, he thus writes: But now having evermore a respect to the moderation of pious Gravity, we ought to believe nothing unadvisedly in a doubtful point; lest we conceive a prejudice against that, in favour to our Errour, which Truth hereafter may discover to be no wise contrary to the Sacred Books either of the Old or New Testament.

It hath since come to pass, that Time hath by degrees discovered to every one the truths before by me indicated: and together with the truth of the fact, a discovery hath been made of the difference of humours between those who simply and without passion did refuse to admit such like Phaenomena for true, and those who to their incredulity had added some discomposed affection: For as those who were better grounded in the Science of Astronomy, and Natural Philosophy, became satisfied upon my first intimation of the news; so all those who stood not in the Negative, or in doubt for any other reason, but because it was an unlookt-for-Novelty, and because they had not an occasion of seeing a sensible experiment thereof, did by degrees come to satisfie themselves: But those, who besides the love they bore to to their first Errour, have I know not what imaginary interess to render them disaffected; not so much towards the things, as towards the Author of them, not being able any longer to deny them, conceal themselves under an obstinate silence; and being exasperated more than ever by that whereby those others were satisfied and convinced, they divert their thoughts to other projects, and seek to prejudice me some other wayes: of whom I profess that I would make no more account than I have done of those who heretofore have contradicted me (at whom I alwaies laugh, as being assured of the issue that the business is to have) but that I see that those new Calumnies and Persecutions do not determine in our greater or lesser Learning (in which I will scarce pretend to any thing) but extend so far as to attempt to asperse me with Crimes which ought to be, and are more abhorred by me than Death it self: Nor ought I to content my self that they are known to be unjust by those onely who know me and them, but by all men whatsoever. They persisting therefore in their first Resolution, Of ruining me and whatsoever is mine, by all imaginable waies; and knowing how that I in my Studies of Astronomy and Philosophy hold, as to the Worlds Systeme, That the Sun, without changing place, is situate in the Centre of the Conversion of the Celestial Orbes; and that the Earth, convertible about its own Axis, moveth it self about the Sun: And moreover understanding, that I proceed to maintain this Position,

not onely by refuting the Reasons of Ptolomy and Aristotle, but by producing many on the contrary; and in particular, some Physical pertaining to Natural Effects, the causes of which perhaps can be by no other way assigned; and others Astronomical depending upon many circumstances and encounters of new Discoveries in Heaven, which manifestly confute the Ptolomaick Systeme, and admirably agree with and confirm this other Hypothesis: and possibly being ashamed to see the known truth of other Positions by me asserted, different from those that have been commonly received; and therefore distrusting their defence so long as they should continue in the Field of Philosophy: for th[…]se respects, I say, they have resolved to try whether they could make a Shield for the fallacies of their Arguments of the Mantle of a feigned Religion, and of the Authority of the Sacred Scriptures, applyed by them with little judgment to the confutation of such Reasons of mine as they had neither understood, nor so much as heard.

And first, they have indeavoured, as much as in them lay, to divulge an opiniou thorow the Universe, that those Propositions are contrary to the Holy Letters, and consequently Damnable and Heretical: And thereupon perceiving, that for the most part, the inclination of Mans Nature is more prone to imbrace those enterprizes, whereby his Neighbour may, although unjustly, be oppressed, than those from whence he may receive just incouragement; it was no hard matter to find those Complices, who for such (that is, for Damnable and Heretical) did from their Pulpits with unwonted confidence preach it, with but an unmerciful and less considerate injury, not only to this Doctrine, and to its followers, but to all Mathematicks and Mathematicians together. Hereupon assuming greater confidence, and vainly hoping that that Seed which first took root in their unsound mindes, might spread its branches, and ascend towards Heaven, they went scattering rumours up and down among the People, That it would, ere long be condemned by Supreme Authority: and knowing that such a Censure would supplant not onely these two Conclusions of the VVorlds Systeme, but would make all other Astronomical and Physical Observations that have correspondence and necessary connection therewith to become damnable, to facilitate the business they seek all they can to make this opinion (at least among the vulgar) to seem new, and peculiar to my self, not owning to know that Nicholas Copernicus was its Authour, or rather Restorer and Confirmer: a person who was not only a Catholick, but a Priest, Canonick, and so esteemed, that there being a Dispute in the Lateran Council, under Leo X. touching the correction of the Ecclesiastick Calender,

he was sent for to Rome from the remotest parts of Germany, for to assist in this Reformation, which for that time was left imperfect, onely because as then the true measure of the Year and Lunar Moneth was not exactly known: whereupon it was given him in charge by the Bishop of Sempronia, at that time Super-intendent in that Affair, to search with reiterated studies and pains for greater light and certainty, touching those Coelestial Motions. Upon which, with a labour truly Atlantick and with his admirable Wit, setting himself again to that Study, he made such a progress in these Sciences, and reduced the knowledge of the Coelestial Motions to such exactnesse, that he gained the title of an Excellent Astronomer. And, according unto his Doctrine, not only the Calendar hath been since regulated, but the Tables of all the Motions of the Planets have also been calculated: and having reduced the said Doctrine into six Books, he published them to the World at the instance of the Cardinal of Capua, and of the Bishop of Culma. And in regard that he had re-assumed this so laborious an enterprize by the order of The Pope; he dedicated his Book De Revolutionibus Coelestibus to His Successour, namely Paul III. which, being then also Printed, hath been received by The Holy Church, and read and studied by all the World, without any the least umbrage of scruple that hath ever been conceived at his Doctrine; The which, whilst it is now proved by manifest Experiments and necessary Demonstrations to have been well grounded, there want not persons that, though they never saw that same Book intercept the reward of those many Labours to its Authour, by causing him to be censured and pronounced an Heretick; and this, only to satisfie a particular displeasure conceived, without any cause, against another man, that hath no other interest in Copernicus, but only as he is an approver of his Doctrine.

Now in regard of these false aspersions, which they so unjustly seek to throw upon me, I have thought it necessary for my justification before the World (of whose judgment in matters of Religion and Reputation I ought to make great esteem) to discourse concerning those Particulars, which these men produce to scandalize and subvert this Opinion, and in a word, to condemn it, not only as false, but also as Heretical; continually making an Hipocritical Zeal for Religion their shield; going about moreover to interest the Sacred Scriptures in the Dispute, and to make them in a certain sense Ministers of their deceiptful purposes: and farthermore desiring, if I mistake not, contrary to the intention of them, and of the Holy Fathers to extend (that I may not say abuse) their Authority, so as that even in Conclusions meerly Natural, and not de Fide, they would have us altogether

leave Sense and Demonstrative Reasons, for some place of Scripture which sometimes under the apparent words may contain a different sense. Now I hope to shew with how much greater Piety and Religious Zeal I proceed, than they do, in that I propose not, that the Book of Copernicus is not to be condemned, but that it is not to be condemned, as they would have it; without understanding it, hearing it, or so much as seeing it; and especially he being an Author that never treateth of matters of Religion or Faith; nor by Reasons any way depending on the Authority of Sacred Scriptures whereupon he may have erroniously interpreted them; but alwaies insists upon Natural Conclusions belonging to the Celestial Motions, handled with Astronomical and Geometrical Demonstrations. Not that he had not a respect to the places of the Sacred Leaves, but because he knew very well that his said Doctrine being demonstrated, it could not contradict the Scriptures, rightly, and according to their true meaning understood. And therefore in the end of his Epistle Dedicatory, speaking to The Pope, he saith thus; If there should chance to be any Mataeologists, who though ignorant in all the Mathematicks, yet pretending a skill in those Learnings, should dare, upon the authority of some place of Scripture wrested to their purpose, to condemn and censure this my Hypothesis, I value them not, but shall slight their inconsiderate Judgement. For it is not unknown, that Lactantius (otherwise a Famous Author, though mean Mathematician) writeth very childishly touching the Form of the Earth, when he scoffs at those who affirm the Earth to be in Form of a Globe. So that it ought not to seem strange to the Ingenious, if any such should likewise now deride us. The Mathematicks are written for Mathematitians, to whom (if I deceive not my self) these Labours of mine shall seem to add something, as also to the Common-weale of the Church, whose Government is now in the hands of Your Holiness.

And of this kinde do these appear to be who indeavour to perswade that Copernicus may be condemned before his Book is read; and to make the World believe that it is not onely lawfull but commendable so to do, produce certain Authorities of the Scripture, of Divines, and of Councils; which as they are by me had in reverence, and held of Supream Authority, insomuch that I should esteem it high temerity for any one to contradict them whilst they are used according to the In stitutes of Holy Church, so I believe that it is no errour to speak, so long as one hath reason to suspect that a person hath a desire, for some concern of his own, to produce and alledge them, to purposes different from those that are in the most Sacred intention of The Holy Church. Therefore I not onely protest (and my sincerity shall manifest it

self) that I intend to submit my self freely to renounce those errors, into which, through ignorance I may run in this Discourse of matters pertaining to Religion; but I farther declare, that I desire not in these matters to engage dispute with any one, although it should be in points that are disputable: for my end tendeth onely to this, That if in these considerations, besides my own profession, amongst the errours that may be in them, there be any thing apt to give others an hint of some Notion beneficial to the Holy Church, touching the determining about the Copernican Systeme, it may be taken and improved as shall seem best to my Superiours: If not, let my Book be torn and burnt; for that I do neither intend, nor pretend to gain to my self any fruit from my writings, that is not Pious and Catholick. And moreover, although that many of the things that I observe have been spoken in my own hearing, yet I shall freely admit and grant to those that spake them, that they never said them, if so they please, but confess that I might have been mistaken: And therefore what I say, let it be supposed to be spoken not by them, but by those which were of this opinion.

The motive therefore that they produce to condemn the Opinion of the Mobility of the Earth, and Stability of the Sun, is, that reading in the Sacred Leaves, in many places, that the Sun moveth, that the Earth standeth still; and the Scripture not being capable of lying, or erring, it followeth upon necessary consequence, that the Position of those is Erronious and Heretical, who maintain that the Sun of it self is immoveable, and the Earth moveable.

Touching this Reason I think it fit in the first place, to consider, That it is both piously spoken, and prudently affirmed, That the Sacred Scripture can never lye, when ever its true meaning is understood: Which I believe none will deny to be many times very abstruce, and very different from that which the bare sound of the words signifieth. Whence it cometh to pass, that if ever any one should constantly confine himself to the naked Grammatical Sence, he might, erring himself, make not only Contradictions and Propositions remote from Truth to appear in the Scriptures, but also gross Heresies and Blasphemies: For that we should be forced to assign to God feet, and hands, and eyes, yea more corporal and humane affections, as of Anger, of Repentance, of Hatred, nay, and sometimes the Forgetting of things past, and Ignorance of those to come: Which Propositions, like as (so the Holy Ghost affirmeth) they were in that manner pronounced by the Sacred Scriptures, that they might be accommodated to the Capacity of the Vulgar, who are very rude and unlearned; so likewise, for the sakes of those that deserve to be distinguished

from the Vulgar, it is necessary that grave and skilful Expositors produce the true senses of them, and shew the particular Reasons why they are dictated under such and such words. And this is a Doctrine so true and common amongst Divines, that it would be superfluous to produce any attestation thereof.

Hence me thinks I may with much more reason conclude, that the same holy Writ, when ever it hath had occasion to pronounce any natural Conclusion, and especially, any of those which are more abstruce, and difficult to be understood, hath not failed to observe this Rule, that so it might not cause confusion in the mindes of those very people, and render them the more contumacious against the Doctrines that were more sublimely mysterious: For (like as we have said, and as it plainly appeareth) out of the sole respect of condescending to Popular Capacity, the Scripture hath not scrupled to shadow over most principal and fundamental Truths, attributing, even to God himself, qualities extreamly remote from, and contrary unto his Essence. Who would positively affirm that the Scripture, laying aside that respect, in speaking but occasionally of the Earth, of the Water, of the Sun, or of any other Creature, hath chosen to confine it self, with all rigour, within the bare and narrow literal sense of the words? And especially, in mentioning of those Creatures, things not at all concerning the primary Institution of the same Sacred Volume, to wit, the Service of God, and the salvation of Souls, and in things infinitely beyond the apprehension of the Vulgar?

This therefore being granted, methinks that in the Discussion of Natural Problemes, we ought not to begin at the authority of places of Scripture; but at Sensible Experiments and Necessary Demonstrations: For, from the Divine Word, the Sacred Scripture and Nature did both alike proceed; the first, as the Holy Ghosts Inspiration; the second, as the most observant Executrix of Gods Commands: And moreover it being convenient in the Scriptures (by way of condescension to the understanding of all men) to speak many things different, in appearance; and so far as concernes the naked signification of the words, from absolute truth: But on the contrary, Nature being inexorable and immutable, and never passing the bounds of the Laws assigned her, as one that nothing careth whether her abstruse reasons and methods of operating be, or be not exposed to the Capacity of Men; I conceive that that, concerning Natural Effects, which either Sensible Experience sets before our eyes, or Necessary Demonstrations do prove unto us, ought not, upon any account, to be called into question, much

less condemned upon the testimony of Texts of Scripture, which may, under their words, couch Senses seemingly contrary thereto; In regard that every Expression of Scripture is not tied to so strict conditions, as every Effect of Nature: Nor doth God less admirably discover himself unto us in Nature's Actions, than in the Scriptures Sacred Dictions. Which peradventure Tertullian intended to express in those words: We conclude, God is known; first, by Nature, and then again more particularly known by Doctrine: by Nature, in his Works; by Doctrine, in his Word preached.

But I will not hence affirm, but that we ought to have an extraordinary esteem for the Places of Sacred Scripture, nay, being come to a certainty in any Natural Conclusions, we ought to make use of them, as most apposite helps to the true Exposition of the same Scriptures, and to the investigation of those Senses which are necessarily conteined in them, as most true, and concordant with the Truths demonstrated.

This maketh me to suppose, that the Authority of the Sacred Volumes was intended principally to perswade men to the belief of those Articles and Propositions, which, by reason they surpass all humane discourse, could not by any other Science, or by any other means be made credible, than by the Mouth of the Holy Spirit it self. Besides that, even in those Propositions, which are not de Fide, the Authority of the same Sacred Leaves ought to be preferred to the Authority of all Humane Sciences that are not written in a Demonstrative Method, but either with bare Narrations, or else with probable Reasons; and this I hold to be so far convenient and necessary, by how far the said Divine Wisdome surpasseth all humane Judgment and Conjecture. But that that self same God who hath indued us with Senses, Discourse, and Understanding hath intended, laying aside the use of these, to give the knowledg of those things by other means, which we may attain by these, so as that even in those Natural Conclusions, which either by Sensible Experiments or Necessary Demonstrations are set before our eyes, or our Understanding, we ought to deny Sense and Reason, I do not conceive that I am bound to believe it; and especially in those Sciences, of which but a small part, and that divided into Conclusions is to be found in the Scripture: Such as, for instance, is that of Astronomy, of which there is so small a part in Holy Writ, that it doth not so much as name any of the Planets, except the Sun and the Moon, and once or twice onely Venus under the name of Lucifer. For if the Holy Writers had had any intention to perswade People to believe the Dispositions and Motions of the Coelestial Bodies; and that consequently we are still to derive that knowledge

from the Sacred Books they would not, in my opinion, have spoken so little thereof, that it is as much as nothing, in comparison of the infinite admirable Conclusions, which in that Science are comprized and demonstrated. Nay, that the Authours of the Holy Volumes did not only not pretend to teach us the Constitutions and Motions of the Heavens and Stars, their Figures, Magnitudes, and Distances, but that intentionally (albeit that all these things were very well known unto them) they forbore to speak of them, is the opinion of the Most Holy & Most Learned Fathers: and in S. Augustine we read the following words.

It is likewise commonly asked, of what Form and Figure we may believe Heaven to be, according to the Scriptures: For many contend much about those matters, which the greater prudence of our Authors hath forborn to speak of, as nothing furthering their Learners in relation to a blessed life; and, (which is the chiefest thing) taking up much of that time which should be spent in holy exercises. For what is it to me whether Heaven, as a Sphere, doth on all sides environ the Earth, a Mass ballanced in the middle of the World; or whether like a Dish it doth onely cover or overcast the same? But because belief of Scripture is urged for that cause, which we have oft mentioned, that is, That none through ignorance of Divine Phrases, when they shall find any thing of this nature in, or hear any thing cited out of our Bibles which may seem to oppose manifest Conclusions, should be induced to suspect their truth, when they admonish, relate, & deliver more profitable matters Briefly be it spoken, touching the Figure of Heaven, that our Authors knew the truth: But the H. Spirit would not, that men should learn what is profitable to none for salvation.

And the same intentional silence of these sacred Penmen in determining what is to be believed of these accidents of the Celestial Bodies, is again hinted to us by the same Father in the ensuing 10. Chapter upon the Question, Whether we are to believe that Heaven moveth, or standeth still, in these words: There are some of the Brethren that start a question concerning the motion of Heaven, Whether it be fixed, or moved: For if it be moved (say they) how is it a Firmament? If it stand still, how do these Stars which are held to be fixed go round from East to West, the more Northern performing shorter Circuits near the Pole; so that Heaven, if there be another Pole, to us unknown, may seem to revolve upon some other Axis; but if there be not another Pole, it may be thought to move as a Discus? To whom I reply, that

these points require many subtil and profound Reasons, for the making out whether they be really so, or no; the undertakeing and discussing of which is neither consistent with my leasure, nor their duty, vvhom I desire to instruct in the necessary matters more directly conducing to their salvation, and to the benefit of The Holy Church.

From which (that we may come nearer to our particular case) it necessarily followeth, that the Holy Ghost not having intended to teach us, whether Heaven moveth or standeth still; nor whether its Figure be in Form of a Sphere, or of a Discus, or distended in Planum: Nor whether the Earth be contained in the Centre of it, or on one side; he hath much less had an intention to assure us of other Conclusions of the same kinde, and in such a manner, connected to these already named, that without the dedermination of them, one can neither affirm one or the other part; which are, The determining of the Motion and Rest of the said Earth, and of the Sun. And if the same Holy Spirit hath purposely pretermitted to teach us those Propositions, as nothing concerning his intention, that is, our salvation; how can it be affirmed, that the holding of one part rather than the other, should be so necessary, as that it is de Fide, and the other erronious? Can an Opinion be Heretical, and yet nothing concerning the salvation of souls? Or can it be said that the Holy Ghost purposed not to teach us a thing that concerned our salvation? I might here insert the Opinion of an Ecclesiastical Person, raised to the degree of Eminentissimo, to wit, That the intention of the Holy Ghost, is to teach us how we shall go to Heaven, and not how Heaven goeth.

But let us return to consider how much necessary Demonstrations,

and sensible Experiments ought to be esteemed in Natural Conclusions; and of what Authority Holy and Learned Divines have accounted them, from whom amongst an hundred other attestations, we have these that follow: We must also carefully heed and altogether avoid in handling the Doctrine of Moses, to avouch or speak any thing affirmatively and confidently which contradicteth the manifest Experiments and Reasons of Philosophy, or other Sciences. For since all Truth is agreeable to Truth, the Truth of Holy Writ cannot be contrary to the solid Reasons and Experiments of Humane Learning.

And in St. Augustine we read: If any one shall object the Authority of Sacred Writ, against clear and manifest Reason, he that doth so, knows not what he undertakes: For he objects

against the Truth, not the sense of the Scripture (which is beyond his comprehension) but rather his own; not what is in it, but what, finding it in himself, he fancyed to be in it.

This granted, and it being true, (as hath been said) that two Truths cannot be contrary to each other, it is the office of a Judicious Expositor to study to finde the true Senses of Sacred Texts, which undoubtedly shall accord with those Natural Conclusions, of which manifest Sense and Necessary Demonstrations had before made us sure and certain. Yea, in regard that the Scriptures (as hath been said) for the Reasons alledged, admit in many places Expositions far from the Sense of the words; and moreover, we not being able to affirm, that all Interpreters speak by Divine Inspiration; For (if it were so) then there would be no difference between them about the Senses of the same places; I should think that it would be an act of great prudence to make it unlawful for any one to usurp Texts of Scripture, and as it were to force them to maintain this or that Naturall Conclusion for truth, of which Sence, & Demonstrative, and necessary Reasons may one time or other assure us the contrary. For who will prescribe bounds to the Wits of men? Who will assert that all that is sensible and knowable in the World is already discovered and known? Will not they that in other points disagree with us, confess this (and it is a great truth) that Eaquae scimus, sint minima pars corum quae ignoramus? That those Truths which we know, are very few, in comparison of those which we know not? Nay more, if we have it from the Mouth of the Holy Ghost, that Deus tradidit Mundum disputationi eorum, ut non inveniat homo opus, quod operatus est Deus ab initio ad finem: One ought not, as I conceive, to stop the way to free Philosophating, touching the things of the World, and of Nature, as if that they were already certainly found, and all manifest: nor ought it to be counted rashness, if one do not sit down satisfied with the opinions now become as it were commune; nor ought any persons to be displeased, if others do not hold, in natural Disputes to that opinion which best pleaseth them; and especially touching Problems that have, for thousands of years, been controverted amongst the greatest Philosophers, as is the Stability of the Sun, and Mobility of the Earth, an opinion held by Pythagoras, and by his whole Sect; by Heraclides Ponticus, who was of the same opininion; by Phylolaus, the Master of Plato; and by Plato himself, as Aristotle relateth, and of which Plutarch writeth in the life of Numa, that the said Plato, when he was grown old, said, It is a most absurd thing to think otherwise: The same was believed by Aristarchus Samius, as we have it in Archimedes; and probably by Archimedes himself;

by Nicetas the Philosopher, upon the testimony of Scicero, and by many others. And this opinion hath, finally, been amplified, and with many Observations and Demonstrations confirmed by Nicholaus Copernicus. And Seneca, a most eminent Philosopher, in his Book De Cometis, advertizeth us that we ought, with great diligence, seek for an assured knowledge, whether it be Heaven, or the Earth, in which the Diurnal Conversion resides.

And for this cause, it would probably be prudent and profitable counsel, if besides the Articles which concern our Salvation, and the establishment of our Faith (against the stability of which there is no fear that any valid and solid Doctrine can ever rise up) men would not aggregate and heap up more, without necessity: And if it be so, it would certainly be a preposterous thing to introduce such Articles at the request of persons who, besides that we know not that they speak by inspiration of Divine Grace, we plainly see that there might be wished in them the understanding which would be necessary first to enable them to comprehend, and then to discuss the Demonstrations wherewith the subtiler Sciences proceed in confirming such like Conclusions. Nay, more I should say, (were it lawful to speak my judgment freely on this Argument) that it would haply more suit with the Decorum and Majesty of those Sacred Volumes, if care were taken that every shallow and vulgar Writer might not authorize his Books (which are not seldome grounded upon foolish fancies) by inserting into them Places of Holy Scripture, interpreted, or rather distorted to Senses as remote from the right meaning of the said Scripture, as they are neer to derirision, who not without ostentation flourish out their Writings therewith. Examples of such like abuses there might many be produced, but for this time I will confine my self to two, not much besides these matters of Astronomy: One of which, is that of those Pamphlets which were published against the Medicean Planets, of which I had the fortune to make the discovery; against the existence of which there were brought many places of Sacred Scripture: Now, that all the World seeth them to be Planets, I would gladly hear with what new interpretations those very Antagonists do expound the Scripture, and excuse their own simplicity. The other example is of him who but very lately hath Printed against Astronomers and Philosophers, that the Moon doth not receive its light from the Sun, but is of its own nature resplendent: which imagination he in the close confirmeth, or, to say better, perswadeth himself that he confirmeth by sundry Texts of Scripture, which he thinks cannot be reconciled unlesse his opinion should be true and necessary. Neverthelesse,

the Moon of it self is Tenebrose, and yet it is no lesse lucid than the Splendor of the Sun.

Hence it is manifest, that these kinde of Authors, in regard they did not dive into the true Sence of the Scriptures, would (in case their authority were of any great moment) have imposed a necessity upon others to believe such Conclusions for true as were repugnant to manifest Reason, and to Sense. Which abuse Deus avertat, that it do not gain Countenance and Authority; for if it should, it would in a short time be necessary to proscribe and inhibit all the Contemplative Sciences. For being that by nature the number of such as are very unapt to understand perfectly both the Sacred Scriptures, and the other Sciences is much greater than that of the skilfull and intelligent; those of the first sort superficially running over the Scriptures, would arrogate to themselves an Authority of decreeing upon all the Questions in Nature, by vertue of some Word by them misunderstood, and produced by the Sacred Pen-men to another purpose: Nor would the small number of the Intelligent be able to repress the furious Torrent of those men, who would finde so many the more followers, in that the gaining the reputation of Wise men without pains or Study, is far more grateful to humane Nature, than the consuming our selves with restless contemplations about the most painfull Arts. Therefore we ought to return infinite thanks to Almighty God, who of his Goodness freeth us from this fear, in that he depriveth such kinde of persons of all Authority and, reposeth the Consulting, Resolving, and Decreeing upon so important Determinations in the extraordinary Wisdom and Candor of most Sacred Fathers; and in the Supream Authority of those, who being guided by his Holy Spirit, cannot but determin Holily: So ordering things, that of the levity of those other men, there is no account made. This kinde of men are those, as I believe, against whom, not without Reason, Grave, and Holy Writers do so much inveigh; and of whom in particular S. Hierom writeth: This (Scilicet the Sacred Scripture) the talking old woman, the doting old man, the talkative Sophister, all venture upon, lacerate, teach, and that before they have learnt it. Others induced by Pride, diving into hard words, Philosophate amongst Women, touching the Holy Scriptures. Others (Oh shameful!) Learn of Women what they teach to Men; and, as if this were nothing, in a certain facility of words, I may say of confidence, expound to others what they understand not themselves. I forbear to speak of those of my own Profession, who, if after Humane Learning they chance to attain to the Holy Scriptures, and tickle the ears of the people with affected and Studied expressions, they affirm that all they say, is to be entertained as the Law of God;

and not stooping to learn what the Prophets and Apostles held, they force incongruous testimonies to their own Sense: As if it were the genuine, and not corrupt way of teaching to deprave Sentences, and Wrest the Scripture according to their own singular and contradictory humour.

I will not rank among these same secular Writers any Theologists, whom I repute to be men of profound Learning, and sober Manners, and therefore hold them in great esteem and veneration: Yet I cannot deny but that I have a certain scruple in my mind, and consequently am desirous to have it removed, whilst I hear that they pretend to a power of constraining others by Authority of the Scriptures to follow that opinion in Natural Disputations, which they think most agreeth with the Texts of that: Holding withall, that they are not bound to answer the Reasons and Experiments on the contrary: In Explication and Confirmation of which their judgement they say, That Theologie being the Queen of all the Sciences, she ought not upon any account to stoop to accomodate her self to the Positions of the rest, less worthy, and inferior to her: But that they ought to refer themselves to her (as to their Supream Emperess) and change and alter their Conclusions, according to Theological Statutes and Decrees. And they further add, That if in the inferior Science there should be any Conclusion certain by vertue of Demonstrations or experiments, to which there is found in Scripture another Conclusion repugnant; the very Professors of that Science ought of themselves to resolve their Demonstrations, and discover the falacies of their own Experiments, without repairing to Theologers and Textuaries, it not suiting (as hath been said) with the dignity of Theologie to stoop to the investigation of the falacies of the inferior Sciences: But it sufficeth her, to determine the truth of the Conclusion with her absolute Authority, and by her infallibility. And then the Natural Conclusions in which they say that we ought to bide by the meer Authority of the Scripture, without glossing, or expounding it to Senses different from the Words, they affirm to be Those of which the Scripture speaketh alwaies in the same manner; and the Holy Fathers all receive, and expound to the same Sense.

Now as to these Determinations, I have had occasion to consider some particulars (which I will purpose) for that I was made cautious thereof, by those who understand more than I in these businesses, and to whose judgements I alwaies submit my self. And first I could say, that there might possibly a certain kinde of equivocation interpose, in that they do not distinguish the preheminences whereby Sacred Theologie meriteth the Title of Queen.

For it might be called so, either because that that which is taught by all the other Sciences, is found to be comprized and demonstrated in it, but with more excellent means, and with more sublime Learning; in like manner, as for example; The Rules of measuring of Land, & of Accountantship are much more excellently contained in the Arithmatick and Geometry of Euclid, than in the Practises of Surveyours and Accomptants: Or because the Subject about which Theologie is conversant, excelleth in Dignity all the other Subjects, that are the Matters of other Sciences: As also because its Documents are divulged by nobler waies. That the Title and Authority of Queen belongeth to Theologie in the first Sense, I think that no Theologers will affirm, that have but any in-sight into the other Sciences; of which there are none (as I believe) that will say that Geometry, Astronomy Musick, and Medicine are much more excellently and exactly contained in the Sacred Volumes, than in the Books of Archimedes, in Ptolomy, in Boetius, and in Galen. Therefore it is probable that the Regal Preheminence is given her upon the second account, namely, By reason of the Subject, and the admirable communicating of the Divine Revelations in those Conclusions which by other means could not be conceived by men, and which chiefly concern the acquist of eternal Beatitude. Now if Theologie being conversant about the loftiest Divine Contemplation, and residing for Dignity in the Regal Throne of the Sciences, (whereby she becometh of highest Authority) descendeth not to the more mean and humble Speculations of the inferior Sciences: Nay; (as hath been declared above) hath no regard to them, as not concerning Beatitude; the Professors thereof ought not to arrogate to themselves the Authority to determin of Controversies in those Professions which have been neither practised nor studied by them. For this would be as if an Absolute Prince, knowing that he might freely command, and cause himself to be obeyed, should (being neither Phisitian nor Architect) undertake to administer Medicines, and erect Buildings after his own fashion, to the great endangering af the lives of the poor Patients, and to the manifest destruction of the Edifices.

Again, to command the very Professors of Astronomy, that they of themselves see to the confuting of their own Observations and Demonstrations, as those that can be no other but Falacies and Sophismes, is to enjoyn a thing beyond all possibility of doing: For it is not onely to command them that they do not see that which they see, and that they do not understand that which they understand; but that in seeking, they finde the contrary of that which they happen to meet with. Therefore before that this is to be done, it would be necessary that they were

shewed the way how to make the Powers of the Soul to command one another, and the inferior the Superior; so that the imagination and will might, and should believe contrary to what the Intellect understands: I still mean in Propositions purely Natural, and which are not de Fide, and not in the Supernatural, which are de Fide.

I would entreat these Wise and Prudent Fathers, that they would withal diligence consider the difference that is between Opinable and Demonstrative Doctrines: To the end, that well weighing in their minds with what force Necessary Illations oblige, they might the better ascertain themselves, that it is not in the Power of the Professors of Demonstrative Sciences to change their Opinions at pleasure, and apply themselves one while to one side, and another while to another; and that there is a great difference between commanding a Methametitian or a Philosopher, and the disposing of a Lawyer or a Merchant; and that the demonstrated Conclusions touching the things of Nature and of the Heavens cannot be changed with the same facility, as the Opinions are touching what is lawful or not in a Contract, Bargain, or Bill of Exchange. This difference was well understood by the Learned and Holy Fathers, as their having been at great pains to confute many Arguments, or to say better, many Philosophical Fallacies, doth prove unto us; and as may expresly be read in some of them, and particularly we have in S. Augustine the following words: This is to be held for an undoubted Truth, That we may be confident, that whatever the Sages of this World have demonstrated touching Natural Points, is no waies contrary to our Bibles: And in case they teach any thing in their Books that is contrary to the Holy Scriptures, we may without any scruple conclude it to be most false; And according to our ability let us make the same appear: And let us so keep the Faith of our Lord, in whom are hidden all the Treasures of Wisdom; that we be neither seduced with the Loquacity of false Philosophy, nor scared by the superstition of a counterfeit Religion.

From which words, I conceive that I may collect this Doctrine, namely, That in the Books of the Wise of this World, there are contained some Natural truths that are solidly demonstrated, and others again that are barely taught; and that as to the first sort, it is the Office of wise Divines to shew that they are not contrary to the Sacred Scriptures; As to the rest, taught, but not necessarily demonstrated, if they shall contain any thing contrary to the Sacred Leaves, it ought to be held undoubtedly false, and such it ought by all possible waies to be demonstrated.

If therefore Natural Conclusions veritably demonstrated, are

not to be postposed to the Places of Scripture, but that it ought to be shewn how those Places do not interfer with the said Conclusions; then its necessary before a Physical Proposition be condemned, to shew that it is not necessarily demonstrated; and this is to be done not by them who hold it to be true, but by those who judge it to be false. And this seemeth very reasonable, and agreeable to Nature; that is to say, that they may much more easily find the fallacies in a Discourse, who believe it to be false, than those who account it true and concludent. Nay, in this particular it will come to passe, that the followers of this opinion, the more that they shall turn over Books, examine the Arguments, repeat the Observations, and compare the Experiments, the more shall they be confirmed in this belief. And your Highness knoweth what happened to the late Mathematick Professor in the University of Pisa, Who betook himself in his old age to look into the Doctrine of Copernicus, with hope that he might be able solidly to confute it (for that he held it so far to be false, as that he had never studied it) but it was his fortune, that as soon as he had understood the grounds, proceedings, and demonstrations of Copernicus, he found himself to be perswaded, and of an opposer became his most confident Defender. I might also nominate other Mathematicians, who being moved by my last Discoveries, have confessed it necessary to change the formerly received Constitution of the World, it not being able by any means to subsist any longer.

If for the banishing this Opinion and Hypothesis out of the World, it were enough to stop the mouth of one alone, as it may be they perswade themselves who measuring others judgements by their own, think it impossible that this Doctrine should be able to subsist and finde any followers, this would be very easie to be done, but the business standeth otherwise: For to execute such a determination, it would be necessary to prohibite not onely the Book of Copernicus, and the Writings of the other Authors that follow the same opinion, but to interdict the whole Science of Astronomy; and which is more, to forbid men looking towards Heaven, that so they might not see Mars and Venus at one time neer to the Earth, and at another farther off, with such a difference that the latter is found to be fourty times, and the former sixty times bigger in surface at one time than at another; and to the end, that the same Venus might not be discovered to be one while round, and another while forked, with most subtil hornes: and many other sensible Observations which can never by any means be reconciled to the Ptolomaick Systeme, but are unanswerable Arguments for the Copernican.

But the prohibiting of Copernicus his Book, now that by many

new Observations, and by the application of many of the Learned to the reading of him, his Hypothesis and Doctrine doth every day appear to be more true, having admitted and tolerated it for so many years, whilst he was lesse followed, studied, and confirmed, would seem, in my judgment, an affront to Truth, and a seeking the more to obscure and suppresse her, the more she sheweth her self clear and perspicuous.

The abolishing and censuring, not of the whole Book, but onely so much of it as concerns this particular opinion of the Earths Mobility, would, if I mistake not, be a greater detriment to souls, it being an occasion of great scandal, to see a Position proved, and to see it afterwards made an Heresie to believe it.

The prohibiting of the whole Science, what other would it be but an open contempt of an hundred Texts of the Holy Scriptures, which teach us, That the Glory, and the Greatnesse of Almighty God is admirably discerned in all his Works, and divinely read in the Open Book of Heaven? Nor let any one think that the Lecture of the lofty conceits that are written in those Leaves finish in only beholding the Splendour of the Sun, and of the Stars, and their rising and setting, (which is the term to which the eyes of bruits and of the vulgar reach) but there are couched in them mysteries so profound, and conceipts so sublime, that the vigils, labours, and studies of an hundred and an hundred acute Wits, have not yet been able thorowly to dive into them after the continual disquisition of some thousands of years. But let the Unlearned believe, that like as that which their eyes discern in beholding the aspect of a humane body, is very little in comparison of the stupendious Artifices, which an exquisite and curious Anatomist or Philosopher finds in the same when he is searching for the use of so many Muscles, Tendons, Nerves, and Bones; and examining the Offices of the Heart, and of the other principal Members, seeking the seat of the vital Faculties, noting and observing the admirable structures of the Instruments of the Senses, and, without ever making an end of satisfying his curiosity and wonder, contemplating the Receptacles of the Imagination, of the Memory, and of the Understanding; So that which represents it self to the meer sight, is as nothing in comparison and proportion to the strange Wonders, that by help of long and accurate Observations the Wit of Learned Men discovereth in Heaven. And this is the substance of what I had to consider touching this particular.

In the next place, as to those that adde, That those Natural Propositions of which the Scripture still speaks in one constant tenour, and which the Fathers all unanimously receive in the same sense, ought to be accepted according to the naked and

literal sense of the Words, without glosses and interpretations; and received and held for most certain and true; and that consequently the Mobility of the Sun, and Stability of the Earth, as being such, are de Fide to be held for true, and the contrary opinion to be deemed Heretical. I shall propose to consideration, in the first place, That of Natural Propositions, some there are, of which all humane Science and Discourse can furnish us only with some plausible opinion, and probable conjecture rather than with any certain and demonstrative knowledge; as for example, whether the Stars be animated: Others there are, of which we have, or may confidently believe that we may have, by Experiments, long Observations, and Necessary Demonstrations an undubitable assurance; as for instance, whether the Earth and Heavens move, or not; whether the Heavens are Spherical, or otherwise. As to the first sort, I doubt not in the least, that if humane Ratiocinations cannot reach them, and that consequently there is no Science to be had of them, but only an Opinion or Belief, we ought fully and absolutely to comply with the meer Verbal Sense of the Scripture: But as to the other Positions, I should think (as hath been said above) That we are first to ascertain our selves of the fact it self, which will assist us in finding out the true senses of the Scriptures; which shall most certainly be found to accord with the fact demonstrated, for two truths can never contradict each other. And this I take to be a Doctrine orthodox and undoubted, for that I finde it written in Saint Augustine, who speaking to our point of the Figure of Heaven, and what it is to be believed to be, in regard that which Astronomers affirm concerning it seemeth to be, contrary to the Scripture, (they holding it to be rotund, and the Scripture calling it as it were a Curtain, determineth that we are not at all to regard that the Scripture contradicts Astronomers; but to believe its Authority, if that which they say shall be false, and founded only on the conjectures of humane infirmity: but if that which which they affirm be proved by indubitable Reasons, this Holy Father doth not say, that the Astronomers are to be enjoyned, that they themselves resolving and renouncing their Demonstrations do declare their Conclusion to be false, but saith, that it ought to be demonstrated, That what is said in Scripture of a Curtain is not contrary to their true Demonstrations. These are his words:

But some object; How doth it appear, that the saying in our Bibles, Who stretcheth out the Heaven as a Curtain, maketh not against those who maintain the Heavens to be in figure of a Sphere? Let it be so, if that be false which they affirme: For that is truth which is spoke by Divine Authority, rather than

that which proceeds from Humane Infirmity. But if peradventure they should be able to prove their Position by such Experiments as puts it out of question, it is to be proved, that vvhat is said in Scripture concerning a Curtain, doth in no vvise contradict their manifest Reasons.

He proceedeth afterwards to admonish us that we ought to be no less careful and observant in reconciling a Text of Scripture with a demonstrated Natural Proposition, than w[…]th another Text of Scripture which should sound to a contrary Sense. Nay methinks that the circumspection of this Saint is worthy to be admired and imitated, who even in obscure Conclusions, and of which we may assure our selves that we can have no knowledge or Science by humane demonstration, is very reserved in determining what is to be believed, as we see by that which he writeth in the end of his second Book, de Genesi ad Litteram, speaking, whether the Stars are to be believed animate: Which particular, although (at present) it cannot easily be comprehended, yet I suppose in our farther Progress of handling the Scriptures, we may meet with some more pertinent places, upon which it will be permitted us (if not to determin any thing for certain, yet) to suggest somewhat concerning this matter, according to the dictates of Sacred Authority. But novv, the moderation of pious gravity being alwaies observed, vve ought to receive nothing rashly in a doubtful point, least perhaps vve reject that out of respect to our Errour, vvhich hereafter Truth may discover, to be in no vvise repugnant to the Sacred Volumes of the Old and Nevv Testament.

By this and other places (if I deceive not my self) the intent of the Holy Fathers appeareth to be, That in Natural questions, and which are not de Fide, it is first to be considered, whether they be indubitably demonstrated, or by sensible Experiments known; or whether such a knowledge and demonstration is to be had; which having obtained, and it being the gift of God, it ought to be applyed to find out the true Sences of the Sacred Pages in those places, which in appearance might seem to speak to a contrary meaning: Which will unquestionably be pierced into by Prudent Divines, together with the occasions that moved the Holy Ghost, (for our exercise, or for some other reason to me unknown) to veil it self sometimes under words of different significations.

As to the other point, Of our regarding the Primary Scope of those Sacred Volumes, I cannot think that their having spoken alwaies in the same tenour, doth any thing at all disturb this Rule. For if it hath been the Scope of the Scripture by way of condescention to the capacity of the Vulgar at any time, to express

a Proposition in words, that bear a sense different from the Essence of the said Proposition; why might it not have observed the same, and for the same respect, as often as it had occasion to speak of the same thing? Nay I conceive, that to have done otherwise, would but have encreased the confusion, and diminished the credit that these Sacred Records ought to have amongst the Common People.

Again, that touching the Rest and Motion of the Sun and Earth, it was necessary, for accommodation to Popular Capacity, to assert that which the Litteral sense of the Scripture importeth, experience plainly proveth: For that even to our dayes people far less rude, do continue in the same Opinion upon Reasons, that if they were well weighed and examined, would be found to be extream trivial, and upon Experiments, either wholly false, or altogether besides the purpose. Nor is it worth while to go about to remove them from it, they being incapable of the contrary Reasons that depend upon too exquisite Observations, and too subtil Demonstrations, grounded upon Abstractions, which, for the comprehending of them, require too strong an Imagination. Whereupon, although that the Stability of Heaveu, and Motion of the Earth should be more than certain and demonstrated to the Wise; yet nevertheless it would be necessary, for the conservation of credit amongst the Vulgar, to affirm the contrary: For that of a thousand ordinary men, that come to be questioned concerning these particulars, its probab[…]e that there will not be found so much as one that will not answer that he thinketh, and so certainly he doth, that the Sun moveth, and the Earth standeth still. But yet none ought to take this common Popular Assent to be any Argument of the truth of that which is affirmed: For if we should examine these very men touching the grounds and motives by which they are induced to believe in that manner; and on the other side should hear what Experiments and Demonstrations perswade those few others to believe the contrary, we should finde these latter to be moved by most solid Reasons, and the former by simple appearances, and vain and ridiculous occurrences. That therefore it was necessary to assign Motion to the Sun, and Rest to the earth, lest the shallow capacity of the Vulgar should be confounded, amused, and rendred obstinate and contumacious, in giving credit to the principal Articles, and which are absolutely de fide, it is sufficiently obvious. And if it was necessary so to do, it is not at all to be wondred at, that it was with extraordinary Wisdom so done, in the Divine Scriptures.

But I will alledge further, That not onely a respect to the Incapacity of the Vulgar, but the current Opinion of those times

made the Sacred Writers, in the points that were not necessary to salvation, to accommodate themselves more to the received use, than to the true Essence of things: Of which S. Hierom treating, writeth: As if many things were not spoken in the Holy Scriptures according to the judgement of those times in which they were acted, and not according to that which truth contained. And elsewhere, the same Saint: It is the custome for the Pen-men of Scripture, to deliver their Judgments in many things, according to the common received opinion that their times had of them. And S. Thomas Aquinas in Job upon those words, Qui extendit Aquilonem super vacuum, & appendit Terram super nihilum: Noteth that the Scripture calleth that space Vacuum and Nihilum, which imbraceth and invironeth the Earth, and which we know, not to be empty, bat filled with Air, Neverthelesse, saith he, The Scripture to comply with the apprehension of the Vulgar, who think that in that same space there is nothing, calleth it Vacuum and Nihilum. Here the words of S. Thomas, Quod de superiori Haemisphaerio Coeli nihil nobis apparet, nisi spatium aëre plenum, quod vulgares homines reputant Vacuum; loquitur enim secundum existimationem vulgarium hominum, prout est mos in Sacra Scriptura. Now from this Place I think one may very Logically argue, That the Sacred Scripture for the same respect had much more reason to phrase the Sun moveable, and the Earth immoveable. For if we should try the capacity of the Common People, we should find them much more unapt to be perswaded of the stability of the Sun, and Motion of the Earth, than that the space that environeth it is full of Air. Therefore if the sacred Authors, in this point, which had not so much difficulty to be beat into the capacity of the Vulgar, have notwithstanding forborn to attempt perswading them unto it, it must needs seem very reasonable that in other Propositions much more abstruse they have observed the same stile. Nay Copernicus himself, knowing what power an antiquated custome and way of conceiving things become familiar to us from our infancy hath in our Fancy, that he might not increase confusion and difficulty in our apprehensions, after he had first demonstrated, That the Motions which appear to us to belong to the Sun, or to the Firmament, are really in the Earth; in proceeding afterwards to reduce them into Tables, and to apply them to use, he calleth them the Motions of the Sun, and of the Heaven that is above the Planets; expresly terming them the Rising and Setting of the Sun and Stars; and mutations in the obliquity of the Zodiack, and variations in the points of the Equinoxes, the Middle Motion, Anomalia, Prosthaphaeresis of the Sun; and such other things; which do in reality belong to the Earth: But because

being joyned to it, and consequently having a share in every of its motions, we cannot immediately discern them in her, but are forced to refer them to the Celestial Bodies in which they appear; therefore we call them as if they were made there, where they seem to us to be made. Whence it is to be noted how nenessary it is to accommodate our discourse to our old and accustomed manner of understanding.

That, in the next place, the common consent of Fathers, in receiving a Natural Proposition of Scripture, all in the same sense ought to Authorize it so far, as to make it become a matter of Faith to believe it to be so, I should think that it ought at most to be understood of those Conclusions onely, which have been by the said Fathers discussed, and sifted with all possible diligence, and debated on the one side, and on the other, and all things in the end concurring to disprove the one, and prove the other. But the Mobility of the Earth, and Stability of the Sun, are not of this kinde; For, that the said Opinion was in those times totally buried, and never brought amongst the Questions of the Schools, and not considered, much less followed by any one: So that it is to be believed that it never so much as entered into the thought of the Fathers to dispute it, the Places of Scripture, their own Opinion, and the assent of men having all concurred in the same judgement, without the contradiction of any one, so far as we can finde.

Besides, it is not enough to say that the Fathers all admit the stability of the Earth, &c. Therefore to believe it is a matter of Faith: But its necessary to prove that they have condemned the contrary Opinion: For I may affirm and bide by this, That their not having occasion to make satisfaction upon the same, and to discuss it, hath made them to omit and admit it, onely as current, but not as resolved and proved. And I think I have very good Reason for what I say; For either the Fathers did make reflection upon this Conclusion as controverted, or not: If not, then they could determin nothing concerning it, no not in their private thoughts; and their incogitance doth not oblige us to receive those Precepts which they have not, so much as in their intentions enjoyned. But if they did reflect and consider thereon, they would long since have condemned it, if they had judged it erroneous; which we do not find that they have done. Nay, after that some Divines have began to consider it, we find that they have not deem'd it erroneous; as we read in the Commentaries of Didacus a Stunica upon Job, in Cap. 9, v. 6. on the words, Qui commovet Terram de loco suo, &c. Where he at large discourseth upon the Copernican Hypothesis, and concludeth, That the Mobility of the Earth, is not contrary to Scripture.

Withal, I may justly question the truth of that determination, namely, That the Church enjoyneth us to hold such like Natural

Conclusions as matters of Faith, onely because they bear the stamp of an unanimous Interpretation of all the Fathers: And I do suppose that it may possibly be, that those who hold in this manner, might possibly have gone about in favour of their own Opinion, to have amplified the Decretal of the Councils; which I cannot finde in this case to prohibit any other, save onely, Perverting to Senses contrary to that of Holy Church, or of the concurrent consent of Fathers, those places, and those onely that do pertain either to Faith or Manners, or concern our edification in the Doctrine of Christianity: And thus speaks the Council of Trent.

Sess. 4. But the Mobility or Stability of the Earth, or of the Sun, are not matters of Faith, nor contrary to Manners, nor is there any one, that for the stablishing of this Opinion, will pervert places of Scripture in opposition to the Holy Church, or to the Fathers: Nay, Those who have writ of this Doctrine, did never make use of Texts of Scripture; that they might leave it still in the breasts of Grave and Prudent Divines to interpret the said Places, according to their true meaning.

And how far the Decrees of Councills do comply with the Holy Fathers in these particulars, may be sufficiently manifest, in that they are so far from enjoyning to receive such like Natural Conclusions for matters of Faith, or from censuring the contrary Opinions as erronious; that rather respecting the Primitive and primary intention of the Holy Church, they do adjudge it unprofitable to be busied in examining the truth thereof. Let your Highness be pleased to hear once again what S. Augustine answers to to those Brethren who put the Question, Whether it be true that Heaven moveth, or standeth still? To these I answer, That Points of this nature require a curious and profound examination, that it may truly appear whether they be true or false; a work inconsistent with my leasure to undertake or go thorow with, nor is it any way necessary for those, whom we desire to inform of things that more nearly concern their own salvation and The Churches Benefit.

But yet although in Natural Propositions we were to take the resolution of condemning or admitting them from Texts of Scripture unanimously expounded in the same Sense by all the Fathers, yet do I not see how this Rule can hold in our Case; for that upon the same Places we read several Expositions in the Fathers;

Dionysius Areopagita saying, That the Primum Mobile, and not the Sun stand still. Saint Augustine is of the same Opinion;

All the Celestial Bodies were immoveable. And with them concurreth Abulensis. But which is more, amongst the Jewish Authors (whom Josephus applauds) some have held, That

The Sun did not really stand still, but seemed so to do, during the short time in which Israel gave the overthrow to their Enemies. So for the Miracle in the time of Hezekiah, Paulus Burgensis is of opinion that it was not wrought on the Sun, but on the Diall. But that, in short, it is necessary to Glosse and Interpret the words of the Text in Joshua, when ever the Worlds Systeme is in dispute, I shall shew anon. Now finally, granting to these Gentlemen more than they demand, to wit, That we are wholly to acquiesce in the judgment of Judicious Divines, and that in regard that such a particular Disquisition is not found to have been made by the Ancient Fathers, it may be undertaken by the Sages of our Age, who having first heard the Experiments, Observations, Reasons, and Demonstrations of Philosophers and Astronomers, on the one side, and on the other (seeing that the Controversie is about Natural Problems, and Necessary Dilemma's, and which cannot possibly be otherwise than in one of the two manners in controversie) they may with competent certainty determine what Divine Inspirations shall dictate to them. But that without minutely examining and discussing all the Reasons on both sides; and without ever comming to any certainty of the truth of the Case, such a Resolution should be taken, Is not to be hoped from those who do not stick to hazzard the Majesty and Dignity of the Sacred Scripture, in defending the reputation of their vain Fancies; Nor to be feared from those who make it their whole businesse, to examine with all intensness, what the Grounds of this Doctrine are; and that only in an Holy Zeal for Truth, the Sacred Scriptures, and for the Majesty, Dignity, and Authority, in which every Christian should indeavour to have them maintained. Which Dignity, who seeth not that it is with greater Zeal desired and procured by those who, absolutely submitting themselves to the Holy Church, desire, not that this, or that opinion may be prohibited, but onely that such things may be proposed to consideration, as may the more ascertain her in the safest choice, than by those who being blinded by their particular Interest, or stimulated by malitious suggestions, preach that she should, without more ado, thunder out Curses, for that she had power so to do: Not considering that all that may be done is not alwayes convenient to be done. The Holy Fathers of old were not of this opinion, but rather knowing of how great prejudice, and how much against the primary intent of the Catholick Church, it would be to go about from Texts of Scripture to decide Natural Conclusions, touching which, either Experiments or necessary Demonstrations, might in time to come evince the contrary, of that which the naked sense of the Words soundeth, they have

not only proceeded with great circumspection, but have left the following Precepts for the instruction of others. In points obscure and remote from our Sight, if we come to read any thing out of Sacred Writ, that, with a Salvo to the Faith that we have imbued, may correspond with several constructions, let us not so farre throw our selves upon any of them with a precipitous obstinacy, as that if, perhaps the Truth being more diligently search't into, it should justly fall to the ground, we might fall together with it: and so shew that we contend not for the sense of Divine Scriptures, but our own, in that we would have that which is our own to be the sense of Scriptures, vvhen as vve should rather desire the Scriptures meaning to be ours.

He goeth on, and a little after teacheth us, that no Proposition can be against the Faith, unlesse first it be demonstrated false; saying, Tis not all the while contrary to Faith, until it be disproved by most certain Truth, which if it should so be, the Holy Scripture affirm'd it not, but Humane Ignorance supposed it. Whereby we see that the senses which we impose on Texts of Scripture, would be false, when ever they should disagree with Truths demonstrated. And therefore we ought, by help of demonstrated Truth, to seek the undoubted sense of Scripture: and not according to the sound of the words, that may seem true to our weaknesse, to go about, as it were, to force Nature, and to deny Experiments and Necessary Demonstrations.

Let Your Highnesse be pleased to observe farther, with how great circumspection this Holy Man proceedeth, before he affirmeth any Interpretation of Scripture to be sure, and in such wise certain, as that it need not fear the encounter of any difficulty that may procure it disturbance, for not contenting himself that some sense of Scripture agreeth with some Demonstration, he subjoynes. But if right Reason shall demonstrate this to be true, yet is it questionable whether in these words of Sacred Scripture the Pen-man would have this to be understood, or somewhat else, no lesse true. And in case the Context of his Words shall prove that he intended not this, yet will not that which he would have to be understood be therefore false, but most true, aad that which is more profitable to be knovvn.

But that which increaseth our wonder concerning the circumspection,

wherewith this Pious Author proceedeth, is, that not trusting to his observing, that both Demonstrative Reasons, and the sense that the words of Scripture and the rest of the Context both precedent and subsequent, do conspire to prove the same thing, he addeth the following words.

But if the Context do not hold forth any thing that may disprove this to be the Authors Sense, it yet remains to enquire, Whether the other may not be intended also. And not yet resolving to accept of one Sense, or reject another, but thinking that he could never use sufficient caution, he proceedeth: But if so be we finde that the other may be also meant, it vvill be doubted which of them he would have to stand; or which in probability he may be thought to aim at, if the true circumstances on both sides be weighed. And lastly, intending to render a Reason of this his Rule, by shewing us to what perils those men expose the Scriptures, and the Church; who, more respecting the support of their own errours, than the Scriptures Dignity, would stretch its Authority beyond the Bounds which it prescribeth to it self, he subjoyns the ensuing words, which of themselves alone might suffice to repress and moderate the excessive liberty, which some think that they may assume to themselves: For it many times falls out, that a Christian may not so fully understand a Point concerning the Earth, Heaven, and the rest of this Worlds Elements; the Motion, Conversion, Magnitude, and Distances of the Stars, the certain defects of the Sun and Moon, the Revolutions of Years and Times, the Nature of Animals, Fruits, Stones, and other things of like nature, as to defend the same by right Reason, or make it out by Experiments. But its too great an absurdity, yea most pernicious, and chiefly to be avoided, to let an Infidel finde a Christian so stupid, that he should argue these matters; as if they were according to Christian Doctrine; and make him (as the Proverb saith) scarce able to contain his laughter, seeing him so far from the Mark. Nor is the matter so much that one in an errour should be laught at, but that our Authors should be thought by them that are without, to be of the same Opinion, and to the great prejudice of those, whose salvation we wait for, sensured and rejected as unlearned. For when they shal confute any one of the Christians in that matter, vvhich they themselvs thorovvly understand, and shall thereupon express their light esteem of our Books; hovv shall these Volumes be believed touching the Resurrection of the Dead, the Hope of eternal Life, and the Kingdom of Heaven; vvhen, as to these Points vvhich admit of present Demonstration, or undoubted Reasons, they conceive them to be falsly vvritten.

And how much the truly Wise and Prudent Fathers are displeased with these men, who in defence of Propositions which they do not understand, do apply, and in a certain sense pawn Texts of Scripture, and afterwards go on to encrease their first Errour, by producing other places less understood than the former. The same Saint declareth in the expressions following: What trouble and sorrow weak undertakers bring upon their knowing Brethren, is not to be expressed; since vvhen they begin to be told and convinced of their false and unsound Opinion, by those vvho have no respect for the Authority of our Scriptures, in defence of vvhat through a fond Temerity, and most manifest falsity, they have urged; they fall to citing the said Sacred Books for proof of it, or else repeat many vvords by heart out of them, vvhich they conceive to make for their purpose; not knovving either what they say, or vvhereof they affirm.

In the number of these we may, as I conceive, account those, who, being either unwilling or unable to understand the Demonstrations and Experiments, wherewith the Author and followers of this Opinion do confirm it, run upon all occasions to the Scriptures, not considering that the more they cite them, and the more they persist in affirming that they are very clear, and do admit no other senses, save those which they force upon them, the greater injury they do to the Dignity of them (if we allowed that their judgments were of any great Authority) in case that the Truth coming to be manifestly known to the contrary, should occasion any confusion, at least to those who are separated from the Holy Church; of whom yet she is very solicitous, and like a tender Mother, desirous to recover them again into her Lap Your Highness therefore may see how praeposterously those Persons proceed, who in Natural Disputations do range Texts of Scripture in the Front for their Arguments; and such Texts too many times, as are but superficially understood by them.

But if these men do verily think, & absolutely believe that they have the true sence of Such a particular place of Scripture, it must needs follow of consequence, that they do likewise hold for, certain, that they have found the absolute truth of that Natural Conclusion, which they intend to dispute: And that withal, they do know that they have a great advantage of their Adversary, whose Lot it is to defend the part that is false; in regard that he who maintaineth the Truth, may have many sensible experiments, and many necessary Demonstrations on his side; whereas his Antagonist can make use of no other than deceitful appearances, Paralogisms and Sophisms. Now if they keeping within natural bounds, & producing no other Weapons but those of Philosophy, pretend however, to have so much advantage of their Enemy; why do they afterwards

in coming to engage, presently betake themselves to a Weapon inevitable & dreadful to terrifie their Opponent with the sole beholding of it? But if I may speak the truth, I believe that they are the first that are affrighted, and that perceiving themselves unable to bear up against the assaults of theit Adversary, go about to find out ways how to keep them far enough off, forbidding unto them the use of the Reason which the Divine Bounty had vouchsafed them, & abusing the most equitable Authority of sacred Scripture, which rightly understood and applyed, can never, according to the common Maxime of Divines, oppose the Manifest Experiments, or Necessary Demonstrations. But these mens running to the Scriptures for a Cloak to their inability to comprehend, not to say resolve the Reasons alledged against them, ought (if I be not mistaken) to stand them in no stead: the Opinion which they oppose having never as yet been condemned by Holy Church. So that if they would proceed with Candor, they should either by silence confess themselves unable to handle such like points, or first consider that it is not in the power of them or others, but onely in that of the Pope, and of Sacred Councils to censure a Position to be Erroneous: But that it is left to their freedome to dispute concerning its falsity. And thereupon, knowing that it is impossible that a Proposition should at the same time be True and Heretical; they ought, I say, to imploy themselves in that work which is most proper to them, namely, in demonstrating the falsity thereof: whereby they may see how needlesse the prohibiting of it is, its falshood being once discovered, for that none would follow it: or the Prohibition would be safe, and without all danger of Scandal. Therefore first let these men apply themselves to examine the Arguments of Copernicus and others; and leave the condemning of them for Erroneous and Heretical to whom it belongeth: But yet let them not hope ever to finde such rash and precipitous Determinations in the Wary and Holy Fathers, or in the absolute Wisdome of him that cannot erre, as those into which they suffer themselves to be hurried by some particular Affection or Interest of their own. In these and such other Positions, which are not directly de Fide, certainly no man doubts but His Holiness hath alwayes an absolute power of Admitting or Condemning them, but it is not in the power of any Creature to make them to be true or false, otherwise than of their own nature, and de facto they are.

Therefore it is in my judgment more discretion to assure us first of the necessary and immutable Truth of the Fact, (over which none hath power) than without that certainty by condemning one part to deprive ones self of that authority of freedome

to elect, making those Determinations to become necessary, which at present are indifferent and arbitrary, and rest in the will of Supreme Authority. And in a word, if it be not possible that a Conclusion should be declared Heretical, whilst we are not certain, but that it may be true, their pains are in vain who pretend to condemn the Mobility of the Earth and Stability of the Sun, unlesse they have first demonstrated it to be impossible and false.

It remaineth now, that we consider whether it be true, that the Place in Joshuah may be taken without altering the pure signification of the words: and how it can be that the Sun, obeying the command of Joshuah, which was, That it should stand still, the day might thereupon be much lengthened. Which businesse, if the Celestial Motions be taken according to the Ptolomaick Systeme, can never any wayes happen, for that the Sun moving thorow the Ecliptick, according to the order of the Signes, which is from East to West (which is that which maketh Day and Night) it is a thing manifest, that the Sun ceasing its true and proper Motion, the day would become shorter and not longer; and that on the contrary, the way to lengthen it would be to hasten and velocitate the Suns motion; insomuch that to cause the Sun to stay above the Horizon for some time, in one and the same place, without declining towards the West, it would be necessary to accelerate its motion in such a manner as that it might seem equal to that of the Primum Mobile, which would be an accelerating it about three hundred and sixty times more than ordinary. If therefore Joshuah had had an intention that his words should be taken in their pure and proper signification, he would have bid the Sun to have accelerated its Motion so, that the Rapture of the Primum Mobile might not carry it to the West: but because his words were heard by people which haply knew no other Celestial Motion, save this grand and common one, from East to West, stooping to their Capacity, and having no intention to teach them the Constitution of the Spheres, but only that they should perceive the greatness of the Miracle wrought, in the lengthening of the Day, he spoke according to their apprehension. Possibly this Consideration moved Dionysius Areopagita to say that in this Miracle the Primum Mobile stood still, and this stopping, all the Celestial Spheres did of consequence stay: of which opinion is S. Augustine himself, and Abulensis at large confirmeth it. Yea, that Joshua's intention was, that the whole Systeme of the Celestial Spheres should stand still, is collected from the command he gave at the same time to the Moon, although that it had nothing to do in the lengthening of the day; and under the injunction laid upon the

Moon, we are to understand the Orbes of all the other Planets, passed over in silence here, as also in all other places of the Sacred Scriptures; the intention of which, was not to teach us the Astronomical Sciences. I suppose therefore, (if I be not deceived) that it is very plain, that if we allow the Ptolemaick Systeme, we must of necessity interpret the words to some sense different from their strict signification. Which Interpretation (being admonished by the most usefull precepts of S. Augustine) I will not affirm to be of necessity this above-mentioned, since that some other man may haply think of some other more proper, and more agreeable Sense.

But now, if this same passage may be understood in the Copernican Systeme, to agree better with what we read in Joshuah, with the help of another Observation by me newly shewen in the Body of the Sun; I will propound it to consideration, speaking alwaies with those safe Reserves; That I am not so affectionate to my own inventions, as to prefer them before those of other men, and to believe that better and more agreeable to the intention of the Sacred Volumes cannot be produced.

Supposing therefore in the first place, that in the Miracle of Joshuah, the whole Systeme of the Celestial Revolutions stood still, according to the judgement of the afore-named Authors: And this is the rather to be admitted, to the end, that by the staying of one alone, all the Constitutions might not be confounded, and a great disorder needlesly introduced in the whole course of Nature: I come in the second place to consider how the Solar Body, although stable in one constant place, doth nevertheless revolve in it self, making an entire Conversion in the space of a Month, or thereabouts; as I conceive I have solidly demonstrated in my Letters Delle Machie Solari: Which motion we sensibly see to be in the upper part of its Globe, inclined towards the South; and thence towards the lower part, to encline towards the North, just in the same manner as all the other Orbs of the Planets do. Thirdly, If we respect the Nobility of the Sun, and his being the Fountain of Light, by which, (as I necessarily demonstrate) not onely the Moon and Earth, but all the other Planets (all in the same manner dark of themselves) become illuminated; I conceive that it will be no unlogicall Illation to say, That it, as the Grand Minister of Nature, and in a certain sense the Soul and Heart of the World, infuseth into the other Bodies which environ it; not onely Light, but Motion also; by revolving in it self: So that in the same manner that the motion of the Heart of an Animal ceasing, all the other motions of its Members would cease; so, the Conversion of the Sun ceasing, […] Conversions of all the Planets would stand still. And though

I could produce the testimonies of many grave Writers to prove the admirable power and influence of the Sun, I will content my self with one sole place of Holy Dionisius Areopagita in his Book de Divinis Nominibus; who thus writes of the Sun: His Light gathereth and converts all things to himself, which are seen, moved, illustrated, wax hot, and (in a word) those things which are preserved by his splendor: Wherefore the Sun is called […], for that he collecteth and gathereth together all things dispersed. And a little after of the Sun again he adds; If this Sun which we see, as touching the Essences and Qualities of those things which fall vvithin our Sense, being very many and different; yet if he vvho is one, and equally bestovves his Light, doth renew, nourish, defend, perfect, divide, conjoyn, cherish, make fruitfull, encrease, change, fix, produce, move, and fashion all living creatures: And every thing in this Vniverse at this Pleasure, is partaker of one and the same Sun; and the causes of many things which participate of him, are equally auticipated in him: Certainly by greater reason; &c. The Sun therefore being the Fountain of Light and, Principle of Motion, God intending, that at the Command of Joshua, all the Worlds Systeme, should continue many hours in the same state, it sufficeth to make the Sun stand still, upon whose stay (all the other Conversions ceasing) the Earth, the Moon, the Sun did abide in the same Constitution as before, as likewise all the other Planets: Nor in all that time did the Day decline towards Night, but it was miraculously prolonged: And in this manner, upon the standing still of the Sun, without altering, or in the least disturbing the other Aspects and mutual Positions of the Stars, the Day might be lengthned on Earth; which exactly agreeth with the Litteral sense of the Sacred Text.

But that of which, if I be not mistaken, we are to make no small account, is, That by help of this Copernican Hypothesis, we have the Litteral, apert, and Natural Sense of another particular that we read of in the same Miracle; which is, That the Sun stood still in Medio Caeli: Upon which passage grave Divines raise many questions, in regard it seemeth very probable, That when Joshuah desired the lengthning of the Day, the Sun was near setting, and not in the Meridian; for if it had been in the Meridian, it being then about the Summer Solstice, and consequently the dayes being at the longest, it doth not seem likely that it was necessary to pray for the lengthning of the day, to prosecute Victory in a Battail, the space of seven hours and more, which remained to Night, being sufficient for that purpose. Upon which Grave Divines have been induced to think that the Sun was near setting: And so the words themselves seem to

sound, saying, Ne movearis Sol, ne movearis. For if it had been in the Meridian, either it had been needless to have asked a Mircale, or it would have been sufficient to have onely praid for some retardment. Of this opinion is Cajetan, to which subscribeth Magaglianes, confirming it by saying, that Joshua had that very day done so many other things before his commanding the Sun, as were not possibly to be dispatch't in half a day. Whereupon they are forced to read the Words in Medio Coeli (to confess the truth) with a little harshness, saying that they import no more than this: That the Sun stood still, being in our Hemisphere,

that is, above the Horizon. But (if I do not erre) we shall avoid that and all other harsh expositions, if according to the Copernican Systeme we place the Sun in the midst, that is, in the Centre of the Coelestial Orbes, and of the Planetary Conversions, as it is most requisite to do. For supposing any hour of the day (either Noon, or any other, as you shall please neerer to the Evening) the Day was lengthened, and all the Coelestial Revolutions stayed by the Suns standing still, In the midst, that is, in the Centre of Heaven, where it resides: A Sense so much the more accomodate to the Letter (besides what hath been said already) in that, if the Text had desired to have affirmed the Suns Rest to have been caused at Noon-day, the proper expression of it had been to say, It stood still at Noon-day, or in the Meridian Circle, and not in the midst of Heaven: In regard that the true and only Middle of a Spherical Body (as is Heaven) is the Centre.

Again, as to other places of Scripture, which seem contrary to this position, I do not doubt but that if it were acknowledged for True and Demonstrated those very Divines who so long as they repute it false, hold those places incapable of Expositions that agree with it would finde such Interpretations for them, as should very well suit therewith; and especially if to the knowledge of Divine Learning they would but adde some knowledge of the Astronomical Sciences: And as at present, whilst they deem it false they think they meet in Scripture only with such places as make against it, if they shall but once have entertained another conceipt thereof, they would meet peradventure as many others that accord with it, and haply would judge, that the Holy Church doth very appositly teach, That God placed the Sun in the Centre of Heaven, and that thereupon by revolving it in it self, after the manner of a Wheel, He contributed the ordinary Courses to the Moon and other Erratick Stars, whilst that she Sings,

Coeli Deus sanctissime,

Qui lucidum Centrum Poli,

Candore pingis igneo,

Augens decoro lumine,

Quarto die, qui flammeam

Solis rotam constituens

Lunae ministras ordinem,

Vagosque cursus Syderum.

They might say, that the Name of Firmament very well agreeth, ad literam, to the Starry Sphere, and to all that which is above the Planetary Conversions; which according to this Hypothesis is altogether firme and immoveable. Ad litteram (the Earth moving circularly) they might understand its Poles, where it's said, Nec dum Terram fecerat, & flumina, & Cardines Orbis Terrae, Which Cardines or Hinges seem to be ascribed to the Earth in vain, if it be not to turn upon them.

FINIS.

Two New Sciences: First Day (1638)

DIALOGUES CONCERNING TWO NEW SCIENCES: FIRST DAY

FIRST DAY

INTERLOCUTORS: SALVIATI, SA- GREDO AND SIMPLICIO

ALV. The constant activity which you Venetians display in your famous arsenal suggests to the studious mind a large field for investigation, especially that part of the work which involves mechanics; for in this department all types of instruments and machines are constantly being cons tru (fled by many artisans, among whom there must be some who, partly by inherited experience and partly by their own observations, have become highly expert and clever in explanation. SAGR. You are quite right. Indeed, I myself, being curious by nature, frequently visit this place for the mere pleasure of observing the work of those who, on account of their superiority over other artisans, we call "first rank men." Conference with them has often helped me in the investigation of certain effects including not only those which are striking, but also those which are recondite and almost incredible. At times also I have been put to confusion and driven to despair of ever explaining something for which I could not account, but which my senses told me to be true. And notwithstanding the fact that what the old man told us a little while ago is proverbial and commonly accepted, yet it seemed to me altogether false, like many another saying which is current among the ignorant; for I think they introduce these expressions in order to give the appearance of knowing something about matters which they do not understand.

Salv.

[So]

SALV. You refer, perhaps, to that last remark of his when we asked the reason why they employed stocks, scaffolding and bracing of larger dimensions for launching a big vessel than they do for a small one; and he answered that they did this in order to avoid the danger of the ship parting under its own heavy weight [vasta mole], a danger to which small boats are not subject?

SAGR. Yes, that is what I mean; and I refer especially to his last assertion which I have always regarded as a false, though current, opinion; namely, that in speaking of these and other similar machines one cannot argue from the small to the large, because many devices which succeed on a small scale do not work on a large scale. 'Now, since mechanics has its foundation in geometry, where mere size cuts no figure, I do not see that the properties of circles, triangles, cylinders, cones and other solid figures will change with their size. If, therefore, a large machine be constructed in such a way that its parts bear to one another the same ratio as in a smaller one, and if the smaller is sufficiently strong for the purpose for which it was designed, I do not see why the larger also should not be able to withstand any severe and destructive tests to which it may be subjected.

SALV. The common opinion is here absolutely wrong. Indeed, it is so far wrong that precisely the opposite is true, namely, that many machines can be constructed even more perfectly on a large scale than on a small; thus, for instance, a clock which indicates and strikes the hour can be made more accurate on a large scale than on a small. There are some intelligent people who maintain this same opinion, but on more reasonable grounds, when they cut loose from geometry and argue that the better performance of the large machine is owing to the imperfections and variations of the material. Here I trust you will not charge me with arrogance if I say that imperfections in the material, even those which are great enough to invalidate the clearest mathematical proof, are not sufficient to explain the deviations observed between machines in the concrete and in the abstract. Yet I shall say it and will affirm that, even if the imperfections did

FIRST DAY 3 did not exist and matter were absolutely perfect, unalterable and free from all accidental variations, still the mere fact that it is matter makes the larger machine, built of the same material and in the same proportion as the smaller, correspond with exactness to the smaller in every respect except that it will not be so strong or so resistant against violent treatment; the larger the machine, the greater its weakness. Since I assume * matter to be unchangeable and always the same, it is clear that we are no less able to treat this constant and invariable property in a rigid manner than if it belonged to simple and pure mathematics. Therefore, Sagredo, you would do well to change the opinion which you, and perhaps also many other students of mechanics, have entertained concerning the ability of machines and structures to resist external disturbances, thinking that when they are built of the same material and maintain the same ratio between parts, they are able equally, or rather proportionally, to resist or yield to such external disturbances and blows. For we can demonstrate by geometry that the large machine is not proportionately stronger than the small. Finally, we may say that, for every machine and structure, whether artificial or natural, there is set a necessary limit beyond which neither art nor nature can pass; it is here understood, of course, that the material is the same and the proportion preserved. .

SAGR. My brain already reels. My mind, like a cloud momentarily illuminated by a lightning-flash, is for an instant filled with an unusual light, which now beckons to me and which.now suddenly mingles and obscures strange, crude ideas. From what you have said it appears to me impossible to build two similar structures of the same material, but of different sizes and have them proportionately strong; and if this were so, it would

not be possible to find two single poles made of the same wood which shall be alike in strength and resistance but unlike in size.

SALV. So it is, Sagredo. And to make sure that we understand each other, I say that if we take a wooden rod of a certain length and size, fitted, say, into a wall at right angles, i. e., parallel parallel to the horizon, it may be reduced to such a length that it will just support itself; so that if a hair's breadth be added to its length it will break under its own weight and will be the only rod of the kind in the world.* Thus if, for instance, its length be a hundred times its breadth, you will not be able to find another rod whose length is also a hundred times its breadth and which, like the former, is just able to sustain its own weight and no more: all the larger ones will break while all the shorter ones will be strong enough to support something more than their own weight. And this which I have said about the ability to support itself must be understood to apply also to other tests; so that if a piece of scantling [corrente] will carry the weight of ten similar to itself, a beam [trove] having the same proportions will not be able to support ten similar beams.

Please observe, gentlemen, how facts which at first seem improbable will, even on scant explanation, drop the cloak which has hidden them and stand forth in naked and simple beauty. Who does not know that a horse falling from a height of three or four cubits will break his bones, while a dog falling from the same height or a cat from a height of eight or ten cubits will suffer no injury? Equally harmless would be the fall of a grasshopper from a tower or the fall of an ant from the distance of the moon. Do not children fall with impunity from heights which would cost their elders a broken leg or perhaps a fractured skull? And just as smaller animals are proportionately stronger and more robust than the larger, so also smaller plants are able to stand up better than larger. I am certain you both know that an oak two hundred cubits [braccia] high would not be able to sustain its own branches if they were distributed as in a tree of ordinary size; and that nature cannot produce a horse as large as twenty ordinary horses or a giant ten times taller than an

ordinary man unless by miracle or by greatly altering the proportions of his limbs and especially of his bones, which would have to be considerably enlarged over the ordinary. Likewise the current belief that, in the case of artificial machines the very

* The author here apparently means that the solution is unique.

[Trans.]

FIRST DAY 5 large and the small are equally feasible and lasting is a manifest error. Thus, for example, a small obelisk or column or other solid figure can certainly be laid down or set up without danger of breaking, while the very large ones will go to pieces under die slightest provocation, and that purely on account of their own weight. And here I must relate a circumstance which is worthy"^ of your attention as indeed are all events which happen contrary to expectation, especially when a precautionary measure turns out to be a cause of disaster. A large marble column was laid out so that its two ends rested each upon a piece of beam; a little later it occurred to a mechanic that, in order to be doubly sure of its not breaking in the middle by its own weight, it would be wise to lay a third support midway; this seemed to all an excellent idea; but the sequel showed that it was quite the opposite, for not many months passed before the column was found cracked and broken exactly above the new middle support.

SIMP. A very remarkable and thoroughly unexpected accident, especially if caused by placing that new support in the middle.

SALV. Surely this is the explanation, and the moment the cause is known our surprise vanishes; for when the two pieces of the column were placed on level ground it was observed that one of the end beams had, after a long while, become decayed and sunken, but that the middle one remained hard and strong, thus causing one half of the column to project in the air without any support. Under these circumstances the body therefore behaved differently from what it would have done if supported only upon the first beams; because no matter how much they might have sunken the column would have gone with them. This is an accident which could not possibly have happened to a ~ small column, even though made of the same stone and having a length corresponding to its thickness, i. e., preserving the ratio between thickness and length found in the large pillar.

SAGR. I am quite convinced of the facts of the case, but I do not understand why the strength and resistance are not multiplied in the same proportion as the material; and I am the more puzzled puzzled because, on the contrary, I have noticed in other cases that the strength and resistance against breaking increase in a larger ratio than the amount of material. Thus, for instance, if two nails be driven into a wall, the one which is twice as big as the other will support not only twice as much weight as the other, but three or four times as much.

SALV. Indeed you will not be far wrong if you say eight times as much; nor does this phenomenon contradict the other even though in appearance they seem so different.

SAGR. Will you not then, Salviati, remove these difficulties and clear away these obscurities if possible: for I imagine that this problem of resistance opens up a field of beautiful and useful ideas; and if you are pleased to make this the subject of to-day's discourse you will place Simplicio and me under many obligations.

SALV. I am at your service if only I can call to mind what I learned from our Academician * who had thought much upon this subject and according to his custom had demonstrated everything by geometrical methods so that one might fairly call this a new science. For, although some of his conclusions had been reached by others, first of all by Aristotle, these are not the most beautiful and, what is more important, they had not been proven in a rigid manner from fundamental principles. Now, since I wish to convince you by demonstrative reasoning rather than to persuade you by mere probabilities, I shall suppose that you are familiar with present-day mechanics so far as it is needed in our discussion. First of all it is necessary to consider what happens when a piece of wood or any other solid which coheres firmly is broken; for this is the fundamental fact, involving the first and simple principle which we must take for granted as well known.

• To grasp this more clearly, imagine a cylinder or prism, AB, made of wood or other solid coherent material. Fasten the upper end, A, so that the cylinder hangs vertically. To the lower end, B, attach the weight C. It is clear that however great they may be, the tenacity and coherence [tenacita e

* I. e. Galileo: The author frequently refers to himself under this name. [Trans.]

FIRST DAY 7 coerenza] between the parts of this solid, so long as they are not infinite, can be overcome by the pull of the weight C, a weight which can be increased indefinitely until finally the solid breaks like a rope. And as in the case of the rope whose strength we know to be derived from a multitude of hemp threads which compose it, so in the case of the wood, we observe its fibres and filaments run lengthwise and render it much stronger than a hemp rope of the same thickness. But in the case of a stone or metallic cylinder where thej coherence seems to be still greater the cement which holds the parts together must be something other than filaments and fibres; and, yet even this can be broken by a strong pull.

SIMP. If this matter be as you say I can well understand that the fibres of the wood, being as long as the piece of wood itself, render it strong and resistant against large forces tending to break it. But how can one make a rope one hundred cubits long out of hempen fibres which are not more than two or three cubits long, and still give it so much strength ? Besides, I should be glad to hear your opinion as to the manner in which the parts of metal, stone, and other materials not showing a filamentous structure are Fig. i put together; for, if I mistake not, they exhibit even greater tenacity.

SALV. To solve the problems which you raise it will be necessary to make a digression into subjects which have little bearing upon our present purpose.

SAGR. But if, by digressions, we can reach new truth, what harm is there in making one now, so that we may not lose this knowledge, remembering that such an opportunity, once omitted, may not return; remembering also that we are not tied down to a fixed and brief method but that we meet solely for our own entertainment? Indeed, who knows but that we may thus

frequently frequently discover something more interesting and beautiful than the solution originally sought? I beg of you, therefore, to grant the request of Simplicio, which is also mine; for I am no less curious and desirous than he to learn what is the binding material which holds together the parts of solids so that they can scarcely be separated. This information is also needed to understand the coherence of the parts of fibres themselves of which some solids are built up.

SALV. I am at your service, since you desire it. The first question is, How are fibres, each not more than two or three cubits in length, so tightly bound together in the case of a rope one hundred cubits long that great force [violenza] is required to break it?

Now tell me, Simplicio, can you not hold a hempen fibre so tightly between your fingers that I, pulling by the other end, would break it before drawing it away from you? Certainly you can. And now when the fibres of hemp are held not only at the ends, but are grasped by the surrounding medium throughout their entire length is it not manifestly more difficult to tear them loose from what holds them than to break them? But in the case of the rope the very act of twisting causes the threads to bind one another in such a way that when the rope is stretched with a great force the fibres break rather than separate from each other.

At the point where a rope parts the fibres are, as everyone knows, very short, nothing like a cubit long, as they would be if the parting of the rope occurred, not by the breaking of the filaments, but by their slipping one over the other.

SAGR. In confirmation of this it may be remarked that ropes sometimes break not by a lengthwise pull but by excessive twisting. This, it seems to me, is a conclusive argument because the threads bind one another so tightly that the compressing fibres do not permit those which are compressed to lengthen the spirals even that little bit by which it is necessary for them to lengthen in order to surround the rope which, on twisting, grows shorter and thicker.

SALV. You are quite right. Now see how one fact suggests another

FIRST DAY 9 another. The thread held between the fingers does not yield

[S7l to one who wishes to draw it away even when pulled with considerable force, but resists because it is held back by a double compression, seeing that the upper finger presses against the lower as hard as the lower against the upper. Now, if we could retain only one of these pressures there is no doubt that only half the original resistance would remain; but since we are not able, by lifting, say, the upper finger, to remove one of these pressures without also removing the other, it becomes necessary to preserve one of them by means of a new device which causes the thread to press itself against the finger or against some other solid body upon which it rests; and thus it is brought about that the very force which pulls it in order to snatch it away compresses it more and more as the pull increases. This is accomplished by wrapping the thread around the solid in the manner of a spiral; and will be better understood by means of a figure. Let AB and CD be two cylinders between which is stretched the thread EF: and for the sake of greater clearness we will imagine it to be a small cord. If these two cylinders be pressed strongly together, the cord EF, when drawn by the end F, will undoubtedly stand a considerable pull before it slips between the two compressing solids. But if we remove one of these cylinders the cord, though remaining in contact with the other, will not thereby be prevented from slipping freely. On the other hand, if one holds the cord loosely against the top of the cylinder A, winds it in the spiral form AFLOTR, and then pulls it by the end R, it is evident that the cord will begin to bind the cylinder; the greater the number of spirals the more tightly will the cord be pressed against the cylinder by any given pull. Thus as the number of turns increases, the line of contact

Fig. 2 io THE TWO NEW SCIENCES OF GALILEO contact becomes longer and in consequence more resistant; so that the cord slips and yields to the tractive force with increasing difficulty.

Is it not clear that this is precisely the kind of resistance which one meets in the case of a thick hemp rope where the fibres form thousands and thousands of similar spirals? And, indeed, the binding effect of these turns is so great that a few short rushes woven together into a few interlacing spirals form one of the strongest of ropes which I believe they call pack rope [jtuftz]. SAGR. What you say has cleared up two points which I did not previously understand. One fact is how two, or at most three, turns of a rope around the axle of a windlass cannot only hold it fast, but can also prevent it from slipping when pulled by the immense force of the weight [forza del peso] which it sustains; and moreover how, by turning the windlass, this same axle, by mere friction of the rope around it, can wind up and lift huge stones while a mere boy is able to handle the slack of the rope. The other fact has to do with a simple but clever device, invented by a young kinsman of mine, for the purpose of descending from a window by means of a rope without lacerating the palms of his hands, as had happened to him shortly before and greatly to his discomfort. A small sketch will make this clear. He took a wooden cylinder, AB, about as thick as a walking stick and about one span long: on this he cut a spiral channel of about one turn and a half, and large enough to just receive the rope which he wished to use. Having introduced the rope at the end A and led it out again at the end B, he enclosed both the cylinder and the rope in a case of wood or tin, hinged along the side so that it Fig- 3 could be easily opened and closed. After he had fastened the rope to a firm support above, he could, on grasping and squeezing the case with both hands, hang by his arms. The pressure on the rope, lying between the case and the cylinder, was such that he could, at will, either grasp the case more

FIRST DAY ii more tightly and hold himself from slipping, or slacken his hold and descend as slowly as he wished.

SALV. A truly ingenious device! I feel, however, that for a complete explanation other considerations might well enter; yet I must not now digress upon this particular topic since you are waiting to hear what I think about the breaking strength of other materials which, unlike ropes and most woods, do not show a filamentous structure. The coherence of these bodies is, in my estimation, produced by other causes which may be grouped under two heads. One is that much-talked-of repugnance which nature exhibits towards a vacuum; but this horror of a vacuum not being sufficient, it is necessary to introduce another cause in the form of a gluey or viscous substance which j binds firmly together the component parts of the body.

First I shall speak of the vacuum, demonstrating by definite experiment the quality and quantity of its force [wrfft]. If you take two highly polished and smooth plates of marble, metal, or glass and place them face to face, one will slide over the other with the greatest ease, showing conclusively that there is nothing of a viscous nature between them. But when you attempt to separate them and keep them at a constant distance apart, you find the plates exhibit such a repugnance to separation that the upper one will carry the lower one with it and keep it lifted indefinitely, even when the latter is big and heavy.

This experiment shows the aversion of nature for empty space, even during the brief moment required for the outside air to rush in and fill up the region between the two plates. It is also observed that if two plates are not thoroughly polished, their contact is imperfect so that when you attempt to separate them slowly the only resistance offered is that of weight; if, however, the pull be sudden, then the lower plate rises, but quickly falls back, having followed the upper plate only for that very short interval of time required for the expansion of the small amount of air remaining between the plates, in consequence of their not fitting, and for the entrance of the surrounding air. This resistance which is exhibited between the two plates

12 THE TWO NEW SCIENCES OF GALILEO plates is doubtless likewise present between the parts of a solid, and enters, at least in part, as a concomitant cause of their coherence.

SAGR. Allow me to interrupt you for a moment, please; for I want to speak of something which just occurs to me, namely, when I see how the lower plate follows the upper one and how rapidly it is lifted, I feel sure that, contrary to the opinion of many philosophers, including perhaps even Aristotle himself, motion in a vacuum is not instantaneous. If this were so the two plates mentioned above would separate without any resistance whatever, seeing that the same instant of time would suffice for their separation and for the surrounding medium to rush in and fill the vacuum between them. The fact that the lower plate follows the upper one allows us to infer, not only that motion in a vacuum is not instantaneous, but also that, between the two plates, a vacuum really exists, at least for a very short time, sufficient to allow the surrounding medium to rush in and fill the vacuum; for if there were no vacuum there would be no need of any motion in the medium. One must admit then that a vacuum is sometimes produced by violent motion [violenza] or contrary to the laws of nature, (although in my opinion nothing occurs contrary to nature except the impossible, . and that never occurs).

But here another difficulty arises. While experiment convinces me of the correctness of this conclusion, my mind is not entirely satisfied as to the cause to which this effect is to be attributed. For the separation of the plates precedes the formation of the vacuum which is produced as a consequence of this separation; and since it appears to me that, in the order of nature, the cause must precede the effect, even though it appears to follow in point of time, and since every positive effect must have a positive cause, I do not see how the adhesion of two plates and their resistance to separation — adlual facts — can be referred to a vacuum as cause when this vacuum is yet to follow. According to the infallible maxim of the Philosopher, the non-existent can produce no effect.

Simp.

FIRST DAY 13

SIMP. Seeing that you accept this axiom of Aristotle, I hardly think you will reject another excellent and reliable maxim of his, namely, Nature undertakes only that which happens without resistance; and in this saying, it appears to me, you will find the solution of your difficulty. Since nature abhors a vacuum, she prevents that from which a vacuum would follow as a necessary consequence. Thus it happens that nature prevents the separation of the two plates.

SAGR. Now admitting that what Simplicio says is an adequate solution of my difficulty, it seems to me, if I may be allowed to resume my former argument, that this very resistance to a vacuum ought to be sufficient to hold together the parts either of stone or of metal or the parts of any other solid which is knit together more strongly and which is more resistant to separation. If for one effect there be only one cause, or if, more being assigned, they can be reduced to one, then why is not this vacuum which really exists a sufficient cause for all kinds of resistance?

SALV. I do not wish just now to enter this discussion as to whether the vacuum alone is sufficient to hold together the separate parts of a solid body; but I assure you that the vacuum which acts as a sufficient cause in the case of the two plates is not alone sufficient to bind together the parts of a solid cylinder of marble or metal which, when pulled violently, separates and divides. And now if I find a method of distinguishing this well known resistance, depending upon the vacuum, from every other kind which might increase the coherence, and if I show you that the aforesaid resistance alone is not nearly sufficient for such an effect, will you not grant that we are bound to introduce another cause? Help him, Simplicio, since he does not know what reply to make.

SIMP. Surely, Sagredo's hesitation must be owing to another reason, for there can be no doubt concerning a conclusion which is at once so clear and logical.

SAGR. You have guessed rightly, Simplicio. I was wondering whether, if a million of gold each year from Spain were not sufficient to pay the army, it might not be necessary to make i4 THE TWO NEW SCIENCES OF GALILEO make provision other than small coin for the pay of the soldiers.*

But go ahead, Salviati; assume that I admit your conclusion and show us your method of separating the action of the vacuum from other causes; and by measuring it show us how it is not sufficient to produce the effect in question.

SALV. Your good angel assist you. I will tell you how to separate the force of the vacuum from the others, and afterwards how to measure it. For this purpose let us consider a continuous substance whose parts lack all resistance to separation except that derived from a vacuum, such as is the case with water, a fact fully demonstrated by our Academician in one of his treatises. Whenever a cylinder of water is subjected to a pull and

offers a resistance to the separation of its parts this can be attrib-

A R uted to no other cause than the resistance of the

* vacuum. In order to try such an experiment I have invented a device which I can better explain by means of a sketch than by mere words. Let CABD represent the cross section of a cylinder either of metal or, preferably, of glass, hollow inside and accurately turned. Into this is introduced a perfectly fitting D cylinder of wood, represented in cross section by EGHF, and capable of up-and-down motion. Through the middle of this cylinder is bored a hole to receive an iron wire, carrying a hook at the end K, while the upper end of the wire, I, is provided with a conical head. The wooden cylinder is countersunk Fig. 4 at the top so as to receive, with a perfect fit, the conical head I of the wire, IK, when pulled down by the end K.

Now insert the wooden cylinder EH in the hollow cylinder AD, so as not to touch the upper end of the latter but to leave free a space of two or three finger-breadths; this space is to be filled

* The bearing of this remark becomes clear on reading what Salviati says on p. 1 8 below. [Trans.]

FIRST DAY 15 with water by holding the vessel with the mouth CD upwards, pushing down on the stopper EH, and at the same time keeping the conical head of the wire, I, away from the hollow portion of the wooden cylinder. The air is thus allowed to escape alongside the iron wire (which does not make a close fit) as soon as one presses down on the wooden stopper. The air having been allowed to escape and the iron wire having been drawn back so that it fits snugly against the conical depression in the wood, invert the vessel, bringing it mouth downwards, and hang on the hook K a vessel which can be filled with sand or any heavy material in quantity sufficient to finally separate the upper surface of the stopper, EF, from the lower surface of the water to which it was attached only by the resistance of the vacuum. Next weigh the stopper and wire together with the attached vessel and its contents; we shall then have the force of the vacuum [forza del vacuo]. If one attaches to a cylinder of marble

or glass a weight which, together with the weight of the marble or glass itself, is just equal to the sum of the weights before mentioned, and if breaking occurs we shall then be justified in saying that the vacuum alone holds the parts of the marble and glass together; but if this weight does not suffice and if breaking occurs only after adding, say, four times this weight, we shall then be compelled to say that the vacuum furnishes only one fifth of the total resistance [resistenza].

SIMP. No one can doubt the cleverness of the device; yet it presents many difficulties which make me doubt its reliability. For who will assure us that the air does not creep in between the glass and stopper even if it is well packed with tow or other yielding material ? I question also whether oiling with wax or turpentine will suffice to make the cone, I, fit snugly on its seat. Besides, may not the parts of the water expand and dilate? Why may not the air or exhalations or some other more subtile substances penetrate the pores of the wood, or even of the glass itself?

SALV. With great skill indeed has Simplicio laid before us the difficulties; and he has even partly suggested how to prevent the air

16 THE TWO NEW SCIENCES OF GALILEO air from penetrating the wood or passing between the wood and the glass. But now let me point out that, as our experience increases, we shall learn whether or not these alleged difficulties really exist. For if, as is the case with air, water is by nature expansible, although only under severe treatment, we shall see the stopper descend; and if we put a small excavation in the upper part of the glass vessel, such as indicated by V, then the air or any other tenuous and gaseous substance, which might penetrate the pores of glass or wood, would pass through the water and collect in this receptacle V. But if these things do not happen we may rest assured that our experiment has been performed with proper caution; and we shall discover that water does not dilate and that glass does not allow any material, however tenuous, to penetrate it.

SAGR. Thanks to this discussion, I have learned the cause of a certain effect which I have long wondered at and despaired of understanding. I once saw a cistern which had been provided with a pump under the mistaken impression that the water might thus be drawn with less effort or in greater quantity than by means of the ordinary bucket. The stock of the pump car-

ried its sucker and valve in the upper part so that the water was lifted by attraction and not by a push as is the case with pumps in which the sucker is placed lower down. This pump worked perfectly so long as the water in the cistern stood above a certain level; but below this level the pump failed to work. When I first noticed this phenomenon I thought the machine was out of order; but the workman whom I called in to repair it told me the defect was not in the pump but in the water which had fallen too low to be raised through such a height; and he added that it was not possible, either by a pump or by any other machine working on the principle of attraction, to lift water a hair's breadth above eighteen cubits; whether the pump be large or small this is the extreme limit of the lift. Up to this time I had been so thoughtless that, although I knew a rope, or rod of wood, or of iron, if sufficiently long, would break by its own weight when held by the upper end, it never occurred to me that

FIRST DAY 17 that the same thing would happen, only much more easily, to a column of water. And really is not that thing which is attracted in the pump a column of water attached at the upper end and stretched more and more until finally a point is reached where it breaks, like a rope, on account of its excessive weight ? SALV. That is precisely the way it works; this fixed elevation of eighteen cubits is true for any quantity of water whatever, be the pump large or small or even as fine as a straw. We may therefore say that, on weighing the water contained in a tube eighteen cubits long, no matter what the diameter, we shall obtain the value of the resistance of the vacuum in a cylinder of any solid material having a bore of this same diameter. And having gone so far, let us see how easy it is to find to what length cylinders of metal, stone, wood, glass, etc., of any diameter can be elongated without breaking by their own weight.

Take for instance a copper wire of any length and thickness; fix the upper end and to the other end attach a greater and greater load until finally the wire breaks; let the maximum load be, say, fifty pounds. Then it is clear that if fifty pounds of copper, in addition to the weight of the wire itself which may be, say, Vs ounce, is drawn out into wire of this same size we shall have the greatest length of this kind of wire which can sustain its own weight. Suppose the wire which breaks to be one cubit in length and Vs ounce in weight; then since it supports 50 Ibs. in addition to its own weight, i. e., 4800 eighths-of-anounce, it follows that all copper wires, independent of size, can sustain themselves up to a length of 4801 cubits and no more. Since then a copper rod can sustain its own weight up to a length of 4801 cubits it follows that that part of the breaking strength [resistenza] which depends upon the vacuum, comparing it with the remaining facflors of resistance, is equal to the weight of a rod of water, eighteen cubits long and as thick as the copper rod. If, for example, copper is nine times as heavy as water, the breaking strength [resistenza allo strapparsi] of any copper rod, in so far as it depends upon the vacuum, is equal to the weight of two cubits of this same rod. By a similar method one can find i8 THE TWO NEW SCIENCES OF GALILEO find the maximum length of wire or rod of any material which will just sustain its own weight, and can at the same time discover the part which the vacuum plays in its breaking strength.

SAGR. It still remains for you to tell us upon what depends the resistance to breaking, other than that of the vacuum; what is the gluey or viscous substance which cements together the parts of the solid? For I cannot imagine a glue that will not burn up in a highly heated furnace in two or three months, or certainly within ten or a hundred. For if gold, silver and glass are kept for a long while in the molten state and are removed from the furnace, their parts, on cooling, immediately reunite and bind themselves together as before. Not only so, but whatever difficulty arises with respect to the cementation of the parts of the glass arises also with regard to the parts of the glue; in other words, what is that which holds these parts together so firmly?

SALV. A little while ago, I expressed the hope that your good angel might assist you. I now find myself in the same straits. Experiment leaves no doubt that the reason why two plates cannot be separated, except with violent effort, is that they are held together by the resistance of the vacuum; and the same can be said of two large pieces of a marble or bronze column. This being so, I do not see why this same cause may not explain the coherence of smaller parts and indeed of the very smallest particles of these materials. Now, since each effect must have one true and sufficient cause and since I find no other cement, am I not justified in trying to discover whether the vacuum is not a sufficient cause?

SIMP. But seeing that you have already proved that the resistance which the large vacuum offers to the separation of two large parts of a solid is really very small in comparison with that cohesive force which binds together the most minute parts, why do you hesitate to regard this latter as something very different from the former?

SALV. Sagredo has already [p. 13 above] answered this question when he remarked that each individual soldier was being paid

FIRST DAY 19 paid from coin collected by a general tax of pennies and farthings, while even a million of gold would not suffice to pay the entire army. And who knows but that there may be other extremely minute vacua which affect the smallest particles so that that which binds together the contiguous parts is throughout of the same mintage? Let me tell you something which has just occurred to me and which I do not offer as an absolute fact, but rather as a passing thought, still immature and calling for more careful consideration. You may take of it what you like; and judge the rest as you see fit. Sometimes when I have observed how fire winds its way in between the most minute particles of this or that metal and, even though these are solidly cemented together, tears them apart and separates them, and when I have observed that, on removing the fire, these particles reunite with the same tenacity as at first, without any loss of quantity in the case of gold and with little loss in the case of other metals, even though these parts have been separated for a long while, I have thought that the explanation might lie in the fact that the extremely fine particles of fire, penetrating the slender pores of the metal (too small to admit even the finest particles of air or of many other fluids), would fill the small intervening vacua and would set free these small particles from the attraction which these same vacua exert upon them and which prevents their separation. Thus the particles are able to

move freely so that the mass [massa] becomes fluid and remains so as long as the particles of fire remain inside; but if they depart and leave the former vacua then the original attraction [attrazzione] returns and the parts are again cemented together.

In reply to the question raised by Simplicio, one may say that although each particular vacuum is exceedingly minute and therefore easily overcome, yet their number is so extraordinarily great that their combined resistance is, so to speak, multipled almost without limit. The nature and the amount of force \Jorza] which results [risulta] from adding together an immense number of small forces [debolissimi momenti] is clearly illustrated by the fact that a weight of millions of pounds, suspended by

20 THE TWO NEW SCIENCES OF GALILEO by great cables, is overcome and lifted, when the south wind carries innumerable atoms of water, suspended in thin mist, which moving through the air penetrate between the fibres of the tense ropes in spite of the tremendous force of the hanging weight. When these particles enter the narrow pores they swell the ropes, thereby shorten them, and perforce lift the heavy mass [mole].

SAGR. There can be no doubt that any resistance, so long as it is not infinite, may be overcome by a multitude of minute forces. Thus a vast number of ants might carry ashore a ship laden with grain. And since experience shows us daily that one ant can easily carry one grain, it is clear that the number of grains in the ship is not infinite, but falls below a certain limit. If you take another number four or six times as great, and if you set to work a corresponding number of ants they will carry the grain ashore and the boat also. It is true that this will call for a prodigious number of ants, but in my opinion this is precisely the case with the vacua which bind together the least particles of a metal.

SALV. But even if this demanded an infinite number would you still think it impossible?

SAGR. Not if the mass [mole] of metal were infinite; otherwise. . . .

SALV. Otherwise what? Now since we have arrived at paradoxes let us see if we cannot prove that within a finite extent it is possible to discover an infinite number of vacua. At the same time we shall at least reach a solution of the most remarkable of all that list of problems which Aristotle himself calls wonderful; I refer to his Questions in Mechanics. This solution may be no less clear and conclusive than that which he himself gives and quite different also from that so cleverly expounded by the most learned Monsignor di Guevara.*

First it is necessary to consider a proposition, not treated by others, but upon which depends the solution of the problem and from which, if I mistake not, we shall derive other new and remarkable facts. For the sake of clearness let us draw an

* Bishop of Teano; b. 1561, d. 1641. [Trans.]

FIRST DAY 21 accurate figure. About G as a center describe an equiangular and equilateral polygon of any number of sides, say the hexagon ABCDEF. Similar to this and concentric with it, describe another smaller one which we shall call HIKLMN. Prolong the

Fig. 5 side AB, of the larger hexagon, indefinitely toward S; in like manner prolong the corresponding side HI of the smaller hexagon, in the same direction, so that the line HT is parallel to AS; and through the center draw the line GV parallel to the other two. This done, imagine the larger polygon to roll upon

the line AS, carrying with it the smaller polygon. It is evident that, if the point B, the end of the side AB, remains fixed at the beginning of the rotation, the point A will rise and the point C will fall describing the arc CQ until the side BC coincides with the line BQ, equal to BC. But during this rotation the point I, on the smaller polygon, will rise above the line IT because IB is oblique to AS ; and it will not again return to the line IT until the point C shall have reached the position Q. The point I, having described the arc IO above the line HT, will reach the position

Oat

22 THE TWO NEW SCIENCES OF GALILEO

O at the same time the side IK assumes the position OP; but in the meantime the center G has traversed a path above GV and does not return to it until it has completed the arc GC. This step having been taken, the larger polygon has been brought to rest with its side BC coinciding with the line BQ while the side IK of the smaller polygon has been made to coincide with the line OP, having passed over the portion IO without touching it; also the center G will have reached the position C after having traversed all its course above the parallel line GV. And finally the entire figure will assume a position similar to the first, so that if we continue the rotation and come to the next step, the side DC of the larger polygon will coincide with the portion QX and the side KL of the smaller polygon, having first skipped the arc PY, will fall on YZ, while the center still keeping above the line GV will return to it at R after having jumped the interval CR. At the end of one complete rotation the larger polygon will have traced upon the line AS, without break, six lines together equal to its perimeter; the lesser polygon will likewise have imprinted six lines equal to its perimeter, but separated by the interposition of five arcs, whose chords represent the parts of HT not touched by the polygon : the center G never reaches the line GV except at six points. From this it is clear that the space traversed by the smaller polygon is almost equal to that traversed by the larger, that is, the line HT approximates the line AS, differing from it only by the length of one chord of one of these arcs, provided we understand the line HT to include the five skipped arcs.

Now this exposition which I have given in the case of these hexagons must be understood to be applicable to all other polygons, whatever the number of sides, provided only they are

similar, concentric, and rigidly connected, so that when the greater one rotates the lesser will also turn however small it may be. You must also understand that the lines described by these two are nearly equal provided we include in the space traversed by the smaller one the intervals which are not touched by any part of the perimeter of this smaller polygon.

Let

FIRST DAY 23

Let a large polygon of, say, one thousand sides make one complete rotation and thus lay off a line equal to its perimeter; at the same time the small one will pass over an approximately equal distance, made up of a thousand small portions each equal to one of its sides, but interrupted by a thousand spaces which, in contrast with the portions that coincide with the sides of the polygon, we may call empty. So far the matter is free from difficulty or doubt.

But now suppose that about any center, say A, we describe two concentric and rigidly connected circles; and suppose that from the points C and B, on their radii, there are drawn the tangents CE and BF and that through the center A the line AD is drawn parallel to them, then if the large circle makes one complete rotation along the line BF, equal not only to its circumference but also to the other two lines CE and AD, tell me what the smaller circle will do and also what the center will do. As to the center it will certainly traverse and touch the entire line AD while the circumference of the smaller circle will have measured off by its points of contact the entire line CE, just as was done by the above mentioned polygons. The only difference is that the line HT was not at every point in contact with the perimeter of the smaller polygon, but there were left untouched as many vacant spaces as there were spaces coinciding with the sides. But here in the case of the circles the circumference of the smaller one never leaves the line CE, so that no part of the latter is left untouched, nor is there ever a time when some point on the circle is not in contact with the straight line. How now can the smaller circle traverse a length greater than its circumference unless it go by jumps ?

SAGR. It seems to me that one may say that just as the center of the circle, by itself, carried along the line AD is constantly in contact with it, although it is only a single point, so the points on the circumference of the smaller circle, carried along by the motion of the larger circle, would slide over some small parts of the line CE.

SALV. There are two reasons why this cannot happen. First because

24 THE TWO NEW SCIENCES OF GALILEO because there is no ground for thinking that one point of contact, such as that at C, rather than another, should slip over certain portions of the line CE. But if such slidings along CE did occur they would be infinite in number since the points of contact (being mere points) are infinite in number: an infinite number of finite slips will however make an infinitely long line, while as a matter of fact the line CE is finite. The other reason is that as the greater circle, in its rotation, changes its point of contact continuously the lesser circle must do the same because B is the only point from which a straight line can be drawn to A and pass through C. Accordingly the small circle must change its point of contact whenever the large one changes : no point of the small circle touches the straight line CE in more than one point. Not only so, but even in the rotation of the polygons there was no point on the perimeter of the smaller which coincided with more than one point on the line traversed by that perimeter; this is at once clear when you remember that the line IK is parallel to BC and that therefore IK will remain above IP until BC coincides with BQ, and that IK will not lie upon IP except at the very instant when BC occupies the position BQ; at this instant the entire line IK coincides with OP and immediately afterwards rises above it.

SAGR. This is a very intricate matter. I see no solution. Pray explain it to us.

SALV. Let us return to the consideration of the above mentioned polygons whose behavior we already understand. Now in the case of polygons with 100000 sides, the line traversed by the perimeter of the greater, i. e., the line laid down by its 100000 sides one after another, is equal to the line traced out by the 100000 sides of the smaller, provided we include the 100000 vacant spaces interspersed. So in the case of the circles, polygons having an infinitude of sides, the line traversed by the continuously distributed [continuamente disposti] infinitude of sides is in the greater circle equal to the line laid down by the infinitude of sides in the smaller circle but with the exception that these latter alternate with empty spaces; and since the sides are not finite in number, but infinite, so also are the intervening

FIRST DAY 25 vening empty spaces not finite but infinite. The line traversed by the larger circle consists then of an infinite number of points which completely fill it; while that which is traced by the smaller circle consists of an infinite number of points which leave empty spaces and only partly fill the line. And here I wish you to observe that after dividing and resolving a line into a finite number of parts, that is, into a number which can be counted, it

is not possible to arrange them again into a greater length than that which they occupied when they formed a continuum [continuate] and were connected without the interposition of as many empty spaces. But if we consider the line resolved into an infinite number of infinitely small and indivisible parts, we shall be able to conceive the line extended indefinitely by the interposition, not of a finite, but of an infinite number of infinitely small indivisible empty spaces.

Now this which has been said concerning simple lines must be understood to hold also in the case of surfaces and solid bodies, it being assumed that they are made up of an infinite, not a finite, number of atoms. Such a body once divided into a finite number of parts it is impossible to reassemble them so as to occupy more space than before unless we interpose a finite number of empty spaces, that is to say, spaces free from the substance of which the solid is made. But if we imagine the body, by some extreme and final analysis, resolved into its primary elements, infinite in number, then we shall be able to think of them as indefinitely extended in space, not by the interposition of a finite, but of an infinite number of empty spaces. Thus one can easily imagine a small ball of gold expanded into a very large space without the introduction of a finite number of empty spaces, always provided the gold is made up of an infinite number of indivisible parts.

SIMP. It seems to me that you are travelling along toward those vacua advocated by a certain ancient philosopher.

SALV. But you have failed to add, "who denied Divine Providence," an inapt remark made on a similar occasion by a certain antagonist of our Academician.

Simp.

26 THE TWO NEW SCIENCES OF GALILEO

SIMP. I noticed, and not without indignation, the rancor of this ill-natured opponent; further references to these affairs I omit, not only as a matter of good form, but also because I know how unpleasant they are to the good tempered and well ordered mind of one so religious and pious, so orthodox and God-fearing as you.

But to return to our subject, your previous discourse leaves with me many difficulties which I am unable to solve. First among these is that, if the circumferences of the two circles are equal to the two straight lines, CE and BF, the latter considered as a continuum, the former as interrupted with an infinity of empty points, I do not see how it is possible to say that the line AD described by the center, and made up of an infinity of points, is equal to this center which is a single point. Besides, this building up of lines out of points, divisibles out of indivisibles, and finites out of infinites, offers me an obstacle difficult to avoid; and the necessity of introducing a vacuum, so conclusively refuted by Aristotle, presents the same difficulty.

SALV. These difficulties are real; and they are not the only ones. But let us remember that we are dealing with infinities and indivisibles, both of which transcend our finite understanding, the former on account of their magnitude, the latter because of their smallness. In spite of this, men cannot refrain from discussing them, even though it must be done in a roundabout way.

Therefore I also should like to take the liberty to present some of my ideas which, though not necessarily convincing, would, on account of their novelty, at least, prove somewhat startling. But such a diversion might perhaps carry us too far away from the subject under discussion and might therefore appear to you inopportune and not very pleasing.

SAGR. Pray let us enjoy the advantages and privileges which come from conversation between friends, especially upon subjects freely chosen and not forced upon us, a matter vastly different from dealing with dead books which give rise to many doubts but remove none. Share with us, therefore, the thoughts

FIRST DAY 27 which our discussion has suggested to you; for since we are free from urgent business there will be abundant time to pursue the topics already mentioned; and in particular the objections raised by Simplicio ought not in any wise to be neglected.

SALV. Granted, since you so desire. The first question was, How can a single point be equal to a line? Since I cannot do more at present I shall attempt to remove, or at least diminish, one improbability by introducing a similar or a greater one, just as sometimes a wonder is diminished by a miracle.*

And this I shall do by showing you two equal surfaces, together with two equal solids located upon these same surfaces as bases, all four of which diminish continuously and uniformly in such a way that their remainders always preserve equality among themselves, and finally both the surfaces and the solids terminate their previous constant equality by degenerating, the one solid and the one surface into a very long line, the other solid and the other surface into a single point; that is, the latter to one point, the former to an infinite number of points.

SAGR. This proposition appears to me wonderful, indeed; but let us hear the explanation and demonstration.

SALV. Since the proof is purely geometrical we shall need a figure. Let AFB be a semicircle with center at C; about it describe the rectangle ADEB and from the center draw the straight lines CD and CE to the points D and E. Imagine the radius CF to be drawn perpendicular to either of the lines AB or DE, and the entire figure to rotate about this radius as an axis. It is clear that the rectangle ADEB will thus describe a cylinder, the semicircle AFB a hemisphere, and the triangle CDE, a cone. Next let us remove the hemisphere but leave the cone and the rest of the cylinder, which, on account of its shape, we will call a "bowl." First we shall prove that the bowl and the cone are equal; then we shall show that a plane drawn parallel to the circle which forms the base of the bowl and which has the line DE for diameter and F for a center — a plane whose trace is GN — cuts the bowl in the points G, I, O, N, and the cone in the points H, L, so that the part of the cone indicated by CHL is always equal to * C/. p. 30 below. [Trans.]

28 THE TWO NEW SCIENCES OF GALILEO the part of the bowl whose profile is represented by the triangles

GAI and BON. Besides this we shall prove that the base of the cone, i. e., the circle whose diameter is HL, is equal to the circular

^ £ B surface which forms the base of this portion of the bowl, or as one might say, equal to a ribbon N whose width is GI. (Note by the way the nature of mathematical definitions which consist merely in the imposition of names or, if you prefer, abbrevi- Fig. 6 ations of speech established and introduced in order to avoid the tedious drudgery which you and I now experience simply because we have not agreed to call this surface a "circular band" and that sharp solid portion of the bowl a "round razor.") Now call them by

what name you please, it suffices to understand that the plane, drawn at any height whatever, so long as it is parallel to the base, i. e., to the circle whose diameter is DE, always cuts the two solids so that the portion CHL of the cone is equal to the upper portion of the bowl; likewise the two areas which are the bases of these solids, namely the band and the circle HL, are also equal. Here we have the miracle mentioned above; as the cutting plane approaches the line AB the portions of the solids cut off are always equal, so also the areas of their bases. And as the cutting plane comes near the top, the two solids (always equal) as well as their bases (areas which are also equal) finally vanish, one pair of them degenerating into the circumference of a circle, the other into a single point, namely, the upper edge of the bowl and the apex of the cone. Now, since as these solids diminish equality is maintained between them up to the very last, we are justified in saying that, at the extreme and final end of this diminution, they are still equal and that one is not infinitely greater than the other. It appears therefore that we may equate the circumference of a large circle to a single point. And this which is true of the solids is true also of the surfaces which form

FIRST DAY 29 form their bases; for these also preserve equality between themselves throughout their diminution and in the end vanish, the one into the circumference of a circle, the other into a single point. Shall we not then call them equal seeing that they are the last traces and remnants of equal magnitudes? Note also that, even if these vessels were large enough to contain immense celestial hemispheres, both their upper edges and the apexes of the cones therein contained would always remain equal and would vanish, the former into circles having the dimensions of the largest celestial orbits, the latter into single points. Hence in conformity with the preceding we may say that all circumferences of circles, however different, are equal to each other, and are each equal to a single point.

SAGR. This presentation strikes me as so clever and novel that, even if I were able, I would not be willing to oppose it; for to deface so beautiful a structure by a blunt pedantic attack would be nothing short of sinful. But for our complete satisfac-

tion pray give us this geometrical proof that there is always equality between these solids and between their bases; for it cannot, I think, fail to be very ingenious, seeing how subtle is the philosophical argument based upon this result.

SALV. The demonstration is both short and easy. Referring to the preceding figure, since IPC is a right angle the square of the radius 1C is equal to the sum of the squares on the two sides IP, PC; but the radius 1C is equal to AC and also to GP, while CP is equal to PH. Hence the square of the line GP is equal to the sum of the squares of IP and PH, or multiplying through by 4, we have the square of the diameter GN equal to the sum of the squares on IO and HL. And, since the areas of circles are to each other as the squares of their diameters, it follows that the area of the circle whose diameter is GN is equal to the sum of the areas of circles having diameters IO and HL, so that if we remove the common area of the circle having IO for diameter the remaining area of the circle GN will be equal to the area of the circle whose diameter is HL. So much for the first part. As for the other part, we leave its demonstration for the present, partly because

3o THE TWO NEW SCIENCES OF GALILEO because those who wish to follow it will find it in the twelfth proposition of the second book of De centra gravitatis solidorum by the Archimedes of our age, Luca Valerio,* who made use of it for a different object, and partly because, for our purpose, it suffices to have seen that the above-mentioned surfaces are always equal and that, as they keep on diminishing uniformly, they degenerate, the one into a single point, the other into the circumference of a circle larger than any assignable; in this fa eft lies our miracle. t

SAGR. The demonstration is ingenious and the inferences drawn from it are remarkable. And now let us hear something concerning the other difficulty raised by Simplicio, if you have anything special to say, which, however, seems to me hardly possible, since the matter has already been so thoroughly discussed.

SALV. But I do have something special to say, and will first of all repeat what I said a little while ago, namely, that infinity and indivisibility are in their very nature incomprehensible to us; imagine then what they are when combined. Yet if

we wish to build up a line out of indivisible points, we must take an infinite number of them, and are, therefore, bound to understand both the infinite and the indivisible at the same time. Many ideas have passed through my mind concerning this subject, some of which, possibly the more important, I may not be able to recall on the spur of the moment; but in the course of our discussion it may happen that I shall awaken in you, and especially in Simplicio, objections and difficulties which in turn will bring to memory that which, without such stimulus, would have lain dormant in my mind. Allow me therefore the customary liberty of introducing some of our human fancies, for indeed we may so call them in comparison with supernatural truth which furnishes the one true and safe recourse for decision in our discussions and which is an infallible guide in the dark and dubious paths of thought.

* Distinguished Italian mathematician; born at Ferrara about 1552; admitted to the Accademia dei Lincei 1612; died 1618. [Trans.} f C/. p. 27 above. [Trans.}

FIRST DAY 31

One of the main objections urged against this building up of continuous quantities out of indivisible quantities [continue cT indwisibili\ is that the addition of one indivisible to another cannot produce a divisible, for if this were so it would render the indivisible divisible. Thus if two indivisibles, say two points, can be united to form a quantity, say a divisible line, then an even more divisible line might be formed by the union of three, five, seven, or any other odd number of points. Since however these lines can be cut into two equal parts, it becomes possible to cut the indivisible which lies exactly in the middle of the line. In answer to this and other objections of the same type we reply that a divisible magnitude cannot be constructed out of two or ten or a hundred or a thousand indivisibles, but requires an infinite number of them.

SIMP. Here a difficulty presents itself which appears to me insoluble. Since it is clear that we may have one line greater than another, each containing an infinite number of points, we are forced to admit that, within one and the same class, we may have something greater than infinity, because the infinity of points in the long line is greater than the infinity of points in the short line. This assigning to an infinite quantity a value greater than infinity is quite beyond my comprehension.

SALV. This is one of the difficulties which arise when we attempt, with our finite minds, to discuss the infinite, assigning to it those properties which we give to the finite and limited; but

this I think is wrong, for we cannot speak of infinite quantities as being the one greater or less than or equal to another. To prove this I have in mind an argument which, for the sake of clearness, I shall put in the form of questions to Simplicio who raised this difficulty.

I take it for granted that you know which of the numbers are squares and which are not.

SIMP. I am quite aware that a squared number is one which results from the multiplication of another number by itself; thus 4, 9, etc., are squared numbers which come from multiplying 2, 3, etc., by themselves.

Salv.

32 THE TWO NEW SCIENCES OF GALILEO

SALV. Very well; and you also know that just as the products are called squares so the fadlors are called sides or roots; while on the other hand those numbers which do not consist of two equal fadlors are not squares. Therefore if I assert that all numbers, including both squares and non-squares, are more than the squares alone, I shall speak the truth, shall I not?

SIMP. Most certainly.

SALV. If I should ask further how many squares there are one might reply truly that there are as many as the corresponding number of roots, since every square has its own root and every root its own square, while no square has more than one root and no root more than one square.

SIMP. Precisely so.

SALV. But if I inquire how many roots there are, it cannot be denied that there are as many as there are numbers because every number is a root of some square. This being granted we must say that there are as many squares as there are numbers because they are just as numerous as their roots, and all the numbers are roots. Yet at the outset we said there are many more numbers than squares, since the larger portion of them are not squares. Not only so, but the proportionate number of squares diminishes as we pass to larger numbers. Thus up to 100 we have 10 squares, that is, the squares constitute i/io part of all the numbers; up to 10000, we find only i/ioo

part to be squares; and up to a million only i/iooo part; on the other hand in an infinite number, if one could conceive of such a thing, he would be forced to admit that there are as many squares as there are numbers all taken together.

SAGR. What then must one conclude under these circumstances ?

SALV. So far^as I see we can only infer that the totality of all numbers is infinite, that the number of squares is infinite, and that the number of their roots is infinite; neither is the number of squares less than the totality of all numbers, nor the latter greater than the former; and finally the attributes "equal," "greater," and "less," are not applicable to infinite, but

FIRST DAY 33 but only to finite, quantities. When therefore Simplicio introduces several lines of different lengths and asks me how it is possible that the longer ones do not contain more points than the shorter, I answer him that one line does not contain more or less or just as many points as another, but that each line contains an infinite number. Or if I had replied to him that the points in one line were equal in number to the squares; in another, greater than the totality of numbers; and in the little one, as many as the number of cubes, might I not, indeed, have satisfied him by thus placing more points in one line than in another and yet maintaining an infinite number in each? So much for the first difficulty.

SAGR. Pray stop a moment and let me add to what has already been said an idea which just occurs to me. If the preceding be true, it seems to me impossible to say either that one infinite number is greater than another or even that it is greater than a finite number, because if the infinite number were greater than, say, a million it would follow that on passing from the million to higher and higher numbers we would be approaching the infinite; but this is not so; on the contrary, the larger the number to which we pass, the more we recede from [this property of] infinity, because the greater the numbers the fewer [relatively] are the squares contained in them; but the squares in infinity cannot be less than the totality of all the numbers, as we have just agreed; hence the approach to greater and greater numbers means a departure from infinity.*

SALV. And thus from your ingenious argument we are led to

conclude that the attributes "larger," "smaller," and "equal" have no place either in comparing infinite quantities with each other or in comparing infinite with finite quantities.

I pass now to another consideration. Since lines and all continuous quantities are divisible into parts which are themselves divisible without end, I do not see how it is possible

* A certain confusion of thought appears to be introduced here through a failure to distinguish between the number n and the class of the first n numbers; and likewise from a failure to distinguish infinity as a number from infinity as the class of all numbers. [Trans.]

34 THE TWO NEW SCIENCES OF GALILEO to avoid the conclusion that these lines are built up of an infinite number of indivisible quantities because a division and a subdivision which can be carried on indefinitely presupposes that the parts are infinite in number, otherwise the subdivision would reach an end; and if the parts are infinite in number, we must conclude that they are not finite in size, because an infinite number of finite quantities would give an infinite magnitude. And thus we have a continuous quantity built up of an infinite number of indivisibles.

SIMP. But if we can carry on indefinitely the division into finite parts what necessity is there then for the introduction of non-finite parts?

SALV. The very fact that one is able to continue, without end, the division into finite parts [in parti quante] makes it necessary to regard the quantity as composed of an infinite number of immeasurably small elements [di infiniti non quanti}. Now in order to settle this matter I shall ask you to tell me whether, in your opinion, a continuum is made up of a finite or of an infinite number of finite parts [parti quante].

SIMP. My answer is that their number is both infinite and finite; potentially infinite but actually finite [infinite, in potenza; e finite, in atto] ; that is to say, potentially infinite before division and actually finite after division; because parts cannot be said to exist in a body which is not yet divided or at least marked out; if this is not done we say that they exist potentially.

SALV. So that a line which is, for instance, twenty spans long is not said to contain actually twenty lines each one span in length except after division into twenty equal parts; before division it is said to contain them only potentially. Suppose the facts are as you say; tell me then whether, when the division is once made, the size of the original quantity is thereby increased, diminished, or unaffected.

SIMP. It neither increases nor diminishes.

SALV. That is my. opinion also. Therefore the finite parts [parti quante] in a continuum, whether adlually or potentially present, do not make the quantity either larger or smaller; but it is perfectly clear that, if the number of finite parts actually contained

FIRST DAY 35 contained in the whole is infinite in number, they will make the | magnitude infinite. Hence the number of finite parts, although existing only potentially, cannot be infinite unless the magnitude containing them be infinite; and conversely if the magnitude is ;

PI]

finite it cannot contain an infinite number of finite parts either actually or potentially.

SAGR. How then is it possible to divide a continuum without limit into parts which are themselves always capable of subdivision ?

SALV. This distinction of yours between actual and potential appears to render easy by one method what would be impossible by another. But I shall endeavor to reconcile these mattersx in another way; and as to the query whether the finite parts * of a limited continuum [continuo terminate] are finite or infinite in number I will, contrary to the opinion of Simplicio, , answer that they are neither finite nor infinite.

SIMP. This answer would never have occurred to me since I did not think that there existed any intermediate step between the finite and the infinite, so that the classification or distinction which assumes that a thing must be either finite or infinite is faulty and defective.

SALV. So it seems to me. And if we consider discrete quantities I think there is, between finite and infinite quantities, a third intermediate term which corresponds to every assigned number; so that if asked, as in the present case, whether the finite parts of a continuum are finite or infinite in number the best reply is that they are neither finite nor infinite but correspond to every assigned number. In order that this may be possible, it is necessary that those parts should not be included within a limited number, for in that case they would not correspond to a number which is greater; nor can they be infinite in number since no assigned number is infinite; and thus at the pleasure of the questioner we may, to any given line, assign a hundred finite parts, a thousand, a hundred thousand, or indeed any number we may please so long as it be not infinite. I grant, therefore, to the philosophers, that the continuum contains as ' many

36 THE TWO NEW SCIENCES OF GALILEO many finite parts as they please and I concede also that it contains them, either actually or potentially, as they may like; but I must add that just as a line ten fathoms [canne] in length contains ten lines each of one fathom and forty lines each of one cubit [braccia] and eighty lines each of half a cubit, etc., so it contains an infinite number of points; call them actual or potential, as you like, for as to this detail, Simplicio, I defer to your opinion and to your judgment.

SIMP. I cannot help admiring your discussion; but I fear that this parallelism between the points and the finite parts contained in a line will not prove satisfactory, and that you will not find it so easy to divide a given line into an infinite number of points as the philosophers do to cut it into ten fathoms or forty cubits; not only so, but such a division is quite impossible to realize in practice, so that this will be one of those potentialities which cannot be reduced to actuality.

SALV. The fact that something can be done only with effort or diligence or with great expenditure of time does not render it impossible; for I think that you yourself could not easily divide a line into a thousand parts, and much less if the number of parts were 937 or any other large prime number. But if I were to accomplish this division which you deem impossible as readily as another person would divide the line into forty parts would you then be more willing, in our discussion, to concede the possibility of such a division ?

SIMP. In general I enjoy greatly your method; and replying to your query, I answer that it would be more than sufficient if it prove not more difficult to resolve a line into points than to divide it into a thousand parts.

SALV. I will now say something which may perhaps astonish you; it refers to the possibility of dividing a line into its infinitely small elements by following the same order which one employs in dividing the same line into forty, sixty, or a hundred parts, that is, by dividing it into two, four, etc. He who thinks that, by following this method, he can reach an infinite number of points is greatly mistaken; for if this process were followed to eternity

FIRST DAY 37 eternity there would still remain finite parts which were un-| divided.

Indeed by such a method one is very far from reaching the goal of indivisibility; on the contrary he recedes from it and while he thinks that, by continuing this division and by multiplying the multitude of parts, he will approach infinity, he is, in my opinion, getting farther and farther away from it. My reason is this. In the preceding discussion we concluded that, in an infinite number, it is necessary that the squares and cubes should be as numerous as the totality of the natural numbers [tutti i numeri], because both of these are as numerous as their roots which constitute the totality of the natural numbers. Next we saw that the larger the numbers taken the more sparsely distributed were the squares, and still more sparsely the cubes; therefore it is clear that the larger the numbers to which we pass the farther we recede from the infinite number; hence it follows

that, since this process carries us farther and farther from the a end sought, if on turning back we shall find that any number , can be said to be infinite, it must be unity. Here indeed are I satisfied all those conditions which are requisite for an infinite number; I mean that unity contains in itself as many squares as I there are cubes and natural numbers [tutti i numeri].

SIMP. I do not quite grasp the meaning of this.

SALV. There is no difficulty in the matter because unity is at once a square, a cube, a square of a square and all the other powers [dignita] ; nor is there any essential peculiarity in squares or cubes which does not belong to unity; as, for example, the property of two square numbers that they have between them a mean proportional; take any square number you please as the first term and unity for the other, then you will always find a number which is a mean proportional. Consider the two square numbers, 9 and 4; then 3 is the mean proportional between 9 and I ; while 2 is a mean proportional between 4 and I ; between 9 and 4 we have 6 as a mean proportional. A property of cubes is that they must have between them two mean proportional numbers; take 8 and 27; between them lie 12 and 18; while between

38 THE TWO NEW SCIENCES OF GALILEO between I and 8 we have 2 and 4 intervening; and between I and 27 there lie 3 and 9. Therefore we conclude that unity is the only infinite number. These are some of the marvels which our imagination cannot grasp and which should warn us against the serious error of those who attempt to discuss the infinite by assigning to it the same properties which we employ for the finite, the natures of the two having nothing in common.

With regard to this subject I must tell you of a remarkable property which just now occurs to me and which will explain the vast alteration and change of character which a finite quantity would undergo in passing to infinity. Let us draw the straight line AB of arbitrary length and let the point C divide it into two unequal parts; then I say that, if pairs of lines be drawn, one from each of the terminal points A and B, and if the ratio between the lengths of these lines is the same as that between AC and CB, their points of intersection will all lie upon the circumference of one and the same circle. Thus, for ex-

ample, AL and BL drawn from A and B, meeting at the point L, bearing to one another the same ratio as AC to BC, and the pair AK and BK meeting at K also bearing to one another the same ratio, and likewise the pairs |EAI,BI,AH,BH,AG, BG, AF, BF, AE, BE, have their points of intersection L, K, I,H,G, F,E, allly- Fig. 7 ing upon the circumference of one and the same circle. Accordingly if we imagine the point C to move continuously in such a manner that the lines drawn from it to the fixed terminal points, A and B, always maintain the same ratio between their lengths as exists between the original parts, AC and CB, then the point C will, as I shall presently prove, describe a circle. And the circle thus described will increase

FIRST DAY 39 increase in size without limit as the point C approaches the middle point which we may call O; but it will diminish in size as C approaches the end B. So that the infinite number of points located in the line OB will, if the motion be as explained above, describe circles of every size, some smaller than the pupil of the eye of a flea, others larger than the celestial equator. Now if we move any of the points lying between the two ends O and B they will all describe circles, those nearest O, immense circles; but if we move the point O itself, and continue to move it according to the aforesaid law, namely, that the lines drawn from O to the terminal points, A and B, maintain the same ratio as the original lines AO and OB, what kind of a line will be produced ? A circle will be drawn larger than the largest of the others, a circle which is therefore infinite. But from the point O a straight line will also be drawn perpendicular to BA and extending to infinity without ever turning, as did the others, to join its last end with its first; for the point C, with its limited motion, having described the upper semi-circle, CHE, proceeds to describe the lower semicircle EMC, thus returning to the starting point. But the point O having started to describe its circle, as did all the other points in the line AB, (for the points in the other portion OA describe their circles also, the largest being those nearest the point O) is unable to return to its starting point because the circle it describes, being the largest of all, is infinite; in fact, it describes an infinite straight line as circumference of its infinite circle. Think now what a difference there is between a finite and an infinite circle since the latter changes character in such a manner that it loses not only its existence but also its possibility of existence; indeed, we already clearly understand that there can be no such thing as an infinite circle; similarly there can be no infinite sphere, no infinite body, and no infinite surface of any shape. Now what shall we say concerning this metamorphosis in the transition from finite to infinite? And why should we feel greater repugnance, seeing that, in our search after the infinite among numbers we found it in unity? Having broken up a solid into many parts, having reduced it to the finest of powder

4o THE TWO NEW SCIENCES OF GALILEO powder and having resolved it into its infinitely small indivisible atoms why may we not say that this solid has been reduced to a single continuum [un solo continuo] perhaps a fluid like water or mercury or even a liquified metal? And do we not see stones melt into glass and the glass itself under strong heat become more fluid than water?

SAGR. Are we then to believe that substances become fluid in virtue of being resolved into their infinitely small indivisible components ?

SALV. I am not able to find any better means of accounting for certain phenomena of which the following is one. When I take a hard substance such as stone or metal and when I reduce it by means of a hammer or fine file to the most minute and impalpable powder, it is clear that its finest particles, although when taken one by one are, on account of their smallness, imperceptible to our sight and touch, are nevertheless finite in size, possess shape, and capability of being counted. It is also true that when once heaped up they remain in a heap ; and if an excavation be made within limits the cavity will remain and the surrounding particles will not rush in to fill it; if shaken the particles come to rest immediately after the external disturbing agent is removed; the same effects are observed in all piles of

larger and larger particles, of any shape, even if spherical, as is the case with piles of millet, wheat, lead shot, and every other material. But if we attempt to discover such properties in water we do not find them; for when once heaped up it immediately flattens out unless held up by some vessel or other external retaining body; when hollowed out it quickly rushes in to fill the cavity; and when disturbed it fluctuates for a long time and sends out its waves through great distances.

Seeing that water has less firmness [consistenza] than the finest of powder, in fact has no consistence whatever, we may, it seems to me, very reasonably conclude that the smallest particles into which it can be resolved are quite different from finite and divisible particles; indeed the only difference I am able to discover is that the former are indivisible. The exquisite transparency

FIRST DAY 41 transparency of water also favors this view; for the most transparent crystal when broken and ground and reduced to powder loses its transparency; the finer the grinding the greater the loss; but in the case of water where the attrition is of the highest degree we have extreme transparency. Gold and silver when pulverized with acids [acque forti\ more finely than is possible with any file still remain powders,* and do not become fluids until the finest particles [gf indivisibili] of fire or of the rays of the sun dissolve them, as I think, into their ultimate, indivisible, and infinitely small components.

SAGR. This phenomenon of light which you mention is one which I have many times remarked with astonishment. I have, for instance, seen lead melted instantly by means of a concave mirror only three hands \palmi\ in diameter. Hence I think that if the mirror were very large, well-polished and of a parabolic figure, it would just as readily and quickly melt any other metal, seeing that the small mirror, which was not well polished and had only a spherical shape, was able so energetically to melt lead and burn every combustible substance. Such effects as these render credible to me the marvels accomplished by the mirrors of Archimedes.

SALV. Speaking of the effects produced by the mirrors of Archimedes, it was his own books (which I had already read and studied with infinite astonishment) that rendered credible to me all the miracles described by various writers. And if any doubt had remained the book which Father Buenaventura Cavalierif

has recently published on the subject of the burning glass [specchio ustorio] and which I have read with admiration would have removed the last difficulty.

SAGR. I also have seen this treatise and have read it with

* It is not clear what Galileo here means by saying that gold and silver when treated with acids still remain powders. [Trans.]

f One of the most active investigators among Galileo's contemporaries; born at Milan 1598; died at Bologna 1647; a Jesuit father, first to introduce the use of logarithms into Italy and first to derive the expression for the focal length of a lens having unequal radii of curvature. His " method of indivisibles" is to be reckoned as a precursor of the infinitesimal calculus. [Trans.]

42 THE TWO NEW SCIENCES OF GALILEO pleasure and astonishment; and knowing the author I was confirmed in the opinion which I had already formed of him that he was destined to become one of the leading mathematicians of our age. But now, with regard to the surprising effect of solar rays in melting metals, must we believe that such a furious adlion is devoid of motion or that it is accompanied by the most rapid of motions ?

SALV. We observe that other combustions and resolutions are accompanied by motion, and that, the most rapid; note the action of lightning and of powder as used in mines and petards; note also how the charcoal flame, mixed as it is with heavy and impure vapors, increases its power to liquify metals whenever quickened by a pair of bellows. Hence I do not understand how the adlion of light, although very pure, can be devoid of motion and that of the swiftest type.

SAGR. But of what kind and how great must we consider this speed of light to be ? Is it instantaneous or momentary or does it like other motions require time? Can we not decide this by experiment?

SIMP. Everyday experience shows that the propagation of light is instantaneous; for when we see a piece of artillery fired, at great distance, the flash reaches our eyes without lapse of time; but the sound reaches the ear only after a noticeable interval.

SAGR. Well, Simplicio, the only thing I am able to infer from this familiar bit of experience is that sound, in reaching our ear, travels more slowly than light; it does not inform me whether the coming of the light is instantaneous or whether, although extremely rapid, it still occupies time. An observation of this kind tells us nothing more than one in which it is claimed that "As soon as the sun reaches the horizon its light reaches our eyes"; but who will assure me that these rays had not reached this limit earlier than they reached our vision ?

SALV. The small conclusiveness of these and other similar observations once led me to devise a method by which one might accurately ascertain whether illumination, i. e., the propagation of light, is really instantaneous. The fact that the speed of sound

FIRST DAY 43

sound is as high as it is, assures us that the motion of light cannot fail to be extraordinarily swift. The experiment which I devised was as follows :

Let each of two persons take a light contained in a lantern, or other receptacle, such that by the interposition of the hand, the one can shut oft or admit the light to the vision of the other. Next let them stand opposite each other at a distance of a few cubits and practice until they acquire such skill in uncovering and occulting their lights that the instant one sees the light of his companion he will uncover his own. After a few trials the response will be so prompt that without sensible error [svario] the uncovering of one light is immediately followed by the uncovering of the other, so that as soon as one exposes his light he will instantly see that of the other. Having acquired skill at this short distance let the two experimenters, equipped as before, take up positions separated by a distance of two or three miles and let them perform the same experiment at night, noting carefully whether the exposures and occultations occur in the same manner as at short distances; if they do, we may safely conclude that the propagation of light is instantaneous; but if time is required at a distance of three miles which, considering the going of one light and the coming of the other, really amounts to six, then the delay ought to be easily observable. If the experiment is to be made at still greater distances, say eight or ten miles, telescopes may be employed, each observer adjusting one for himself at the place where he is to make the experiment at night; then although the lights are not large and are therefore invisible to the naked eye at so great a distance, they can readily be covered and uncovered since by aid of the telescopes, once adjusted and fixed, they will become easily visible.

Two New Sciences: First Day (1638) (part 2)

SAGR. This experiment strikes me as a clever and reliable invention. But tell us what you conclude from the results.

SALV. In facl: I have tried the experiment only at a short distance, less than a mile, from which I have not been able to ascertain with certainty whether the appearance of the opposite

44 THE TWO NEW SCIENCES OF GALILEO posite light was instantaneous or not; but if not instantaneous it is extraordinarily rapid — I should call it momentary; and for the present I should compare it to motion which we see in the lightning flash between clouds eight or ten miles distant from us. We see the beginning of this light — I might say its head and

source — located at a particular place among the clouds; but it immediately spreads to the surrounding ones, which seems to be an argument that at least some time is required for propagation; for if the illumination were instantaneous and not gradual, we should not be able to distinguish its origin — its center, so to speak — from its outlying portions. What a sea we are gradually slipping into without knowing it! With vacua and infinities and indivisibles and instantaneous motions, shall we ever be able, even by means of a thousand discussions, to reach dry land?

SAGR. Really these matters lie far beyond our grasp. Just think; when we seek the infinite among numbers we find it in unity; that which is ever divisible is derived from indivisibles; the vacuum is found inseparably connected with the plenum; indeed the views commonly held concerning the nature of these matters are so reversed that even the circumference of a circle turns out to be an infinite straight line, a fact which, if my memory serves me correctly, you, Salviati, were intending to demonstrate geometrically. Please therefore proceed without further digression.

SALV. I am at your service; but for the sake of greater clearness let me first demonstrate the following problem :

Given a straight line divided into unequal parts which bear to each other any ratio whatever, to describe a circle such that two straight lines drawn from the ends of the given line to any point on the circumference will bear to each other the same ratio as the two parts of the given line, thus making those lines which are drawn from the same terminal points homologous.

Let AB represent the given straight line divided into any two unequal parts by the point C; the problem is to describe a circle such

FIRST DAY such that two straight lines drawn from the terminal points, A and B, to any point on the circumference will bear to each other the same ratio as the part AC bears to BC, so that lines drawn from the same terminal points are homologous. About C as center describe a circle having the shorter part CB of the given line, as radius. Through A draw a straight line AD which

shall be tangent to the circle at D and indefinitely prolonged toward E. Draw the radius CD which will be perpendicular to AE. At B erect a perpendicular to AB; this perpendicular will intersect AE at some point since the angle at A is acute; call this point of intersection E, and from it draw a perpendicular t o AE which will intersect AB prolonged in F. Now I say the two straight lines FE and FC are equal. For if we join E and C, we shall have two FiS- 8 triangles, DEC and BEC, in which the two sides of the one, DE and EC, are equal to the two sides of the other, BE and EC, both DE and EB being tangents to the circle DB while the bases DC and CB are likewise equal; hence the two angles, DEC and BEC, will be equal. Now since the angle BCE differs from a right angle by the angle CEB, and the angle CEF also differs from a right angle by the angle CED, and since these differences are equal, it follows that the angle FCE is equal to CEF ;' consequently the sides FE and FC are equal. If we describe a circle with F as center and FE as radius it will pass through the point C; let CEG be such a circle. This is the circle sought, for if we draw lines from the terminal points A and B to any point on its circumference they will bear to each other the

46 THE TWO NEW SCIENCES OF GALILEO the same ratio as the two portions AC and BC which meet at the point C. This is manifest in the case of the two lines AE and BE, meeting at the point E, because the angle E of the triangle AEB is bisected by the line CE, and therefore AC : CB = AE : BE. The same may be proved of the two lines AG and BG terminating in the point G. For since the triangles AFE and EFB are similar, we have AF:FE=EF:FB, or AF:FC=CF:FB, and dividendo AC : CF = CB : BF, or AC : FG = CB : BF ; also componendo we have both AB : BG = CB : BF and AG : GB = CF : FB =AE:EB=AC:BC. Q. E. D.

Take now any other point in the circumference, say H, where the two lines AH and BH intersect; in like manner we shall have AC: CB=AH: HB. Prolong HB until it meets the circumference at I and join IF; and since we have already found that AB:BG=CB:BF it follows that the rectangle AB.BF is equal to the rectangle CB.BG or IB.BH. Hence AB : BH=IB:BF. But the angles at B are equal and therefore AH:HB=IF:FB=EF:FB=AE:EB.

Besides, I may add, that it is impossible for lines which maintain this same ratio and which are drawn from the terminal points, A and B, to meet at any point either inside or outside the circle, CEG. For suppose this were possible; let AL and BL be two such lines intersecting at the point L outside the circle: prolong LB till it meets the circumference at M and join MF. If AL:BL=AC:BC=MF:FB, then we shall have two triangles ALB and MFB which have the sides about the two angles proportional, the angles at the vertex, B, equal, and the two remaining angles, FMB and LAB, less than right angles (because the right angle at M has for its base the entire diameter CG and not merely a part BF: and the other angle at the point A is acute because the line AL, the homologue of AC, is greater than BL, the homologue of BC). From this it follows that the triangles ABL and MBF are similar and therefore AB:BL= MB:BF, making the rectangle AB.BF =MB.BL; but it has been demonstrated that the rectangle AB.BF is equal to CB.BG; whence it would follow that the rectangle MB.BL is equal to the rectangle

FIRST DAY 47 rectangle CB.BG which is impossible; therefore the intersection cannot fall outside the circle. And in like manner we can show that it cannot fall inside; hence all these intersections fall on the circumference.

But now it is time for us to go back and grant the request of Simplicio by showing him that it is not only not impossible to resolve a line into an infinite number of points but that this is quite as easy as to divide it into its finite parts. This I will do under the following condition which I am sure, Simplicio, you will not deny me, namely, that you will not require me to separate the points, one from the other, and show them to you,

one by one, on this paper; for I should be content that you, without separating the four or six parts of a line from one another, should show me the marked divisions or at most that you should fold them at angles forming a square or a hexagon: for, then, I am certain you would consider the division distinctly and actually accomplished.

SIMP. I certainly should.

SALV. If now the change which takes place when you bend a line at angles so as to form now a square, now an octagon, now a polygon of forty, a hundred or a thousand angles, is sufficient to bring into actuality the four, eight, forty, hundred, and thousand parts which, according to you, existed at first only potentially in the straight line, may I not say, with equal right, that, when I have bent the straight line into a polygon having an infinite number of sides, i. e., into a circle, I have reduced to actuality that infinite number of parts which you claimed, while it was straight, were contained in it only potentially? Nor can one deny that the division into an infinite number of points is just as truly accomplished as the one into four parts when the square is formed or into a thousand parts when the millagon is formed; for in such a division the same conditions are satisfied as in the case of a polygon of a thousand or a hundred thousand sides. Such a polygon laid upon a straight line touches it with one of its sides, i. e., with one of its hundred thousand parts; while the circle which is a polygon of an infinite number of sides touches

48 THE TWO NEW SCIENCES OF GALILEO touches the same straight line with one of its sides which is a single point different from all its neighbors and therefore separate and distinct in no less degree than is one side of a polygon from the other sides. And just as a polygon, when rolled along a plane, marks out upon this plane, by the successive contacts of its sides, a straight line equal to its perimeter, so the circle rolled upon such a plane also traces by its infinite succession of contacts a straight line equal in length to its own circumference. I am willing, Simplicio, at the outset, to grant to the Peripatetics the truth of their opinion that a continuous quantity [it continue] is divisible only into parts which are still further divisible so that however far the division and subdivision be continued no end will be reached; but I am not so certain that they will concede to me that none of these divisions of theirs can be a final one, as is surely the fact, because there always remains "another"; the final and ultimate division is rather one which resolves a continuous quantity into an infinite number of indivisible quantities, a result which I grant can never be reached by successive division into an ever-increasing number of parts. But if they employ the method which I propose for separating

and resolving the whole of infinity \tutta la infinita], at a single stroke (an artifice which surely ought not to be denied me), I think that they would be contented to admit that a continuous quantity is built up out of absolutely indivisible atoms, especially since this method, perhaps better than any other, enables us to avoid many intricate labyrinths, such as cohesion in solids, already mentioned, and the question of expansion and contraction, without forcing upon us the objectionable admission of empty spaces [in solids] which carries with it the penetrability of bodies. Both of these objections, it appears to me, are avoided if we accept the above-mentioned view of indivisible constituents.

SIMP. I hardly know what the Peripatetics would say since the views advanced by you would strike them as mostly new, and as such we must consider them. It is however not unlikely that they would find answers and solutions for these problems which

FIRST DAY 49

I, for want of time and critical ability, am at present unable to solve. Leaving this to one side for the moment, I should like to hear how the introduction of these indivisible quantities helps us to understand contraction and expansion avoiding at the same time the vacuum and the penetrability of bodies.

SAGR. I also shall listen with keen interest to this same matter which is far from clear in my mind; provided I am allowed to hear what, a moment ago, Simplicio suggested we omit, namely, the reasons which Aristotle offers against the existence of the vacuum and the arguments which you must advance in rebuttal.

SALV. I will do both. And first, just as, for the production of expansion, we employ the line described by the small circle during one rotation of the large one — a line greater than the circumference of the small circle — so, in order to explain contraction, we point out that, during each rotation of the smaller circle, the larger one describes a straight line which is shorter than its circumference.

For the better understanding of this we proceed to the consideration of what happens in the case of polygons. Employing

a figure similar to the earlier one, construct the two hexagons, ABC and HIK, about the common center L, and let them roll along the parallel lines HOM and ABc. Now holding the vertex I fixed, allow the smaller polygon to rotate until the side IK lies upon the parallel, during which motion the point K will describe the arc KM, and the side KI will coincide with IM. Let us see what, in the meantime, the side CB of the larger polygon has been doing. Since the rotation is about the point I, the terminal point B, of the line IB, moving backwards, will describe the arc Eb underneath the parallel cA so that when the side KI coincides with the line MI, the side BC will coincide with be, having advanced only through the distance Br, but having retreated through a portion of the line BA which subtends the arc B&. If we allow the rotation of the smaller polygon to go on it will traverse and describe along its parallel a line equal to its perimeter; while the larger one will traverse and describe a line less than its perimeter by as many times the length bB as there are

5o THE TWO NEW SCIENCES OF GALILEO are sides less one; this line is approximately equal to that described by the smaller polygon exceeding it only by the distance bE. Here now we see, without any difficulty, why the larger polygon, when carried by the smaller, does not measure off with its sides a line longer than that traversed by the smaller one; this is because a portion of each side is superposed upon its immediately preceding neighbor. Let us next consider two circles, having a common center at A, and lying upon their respective parallels, the smaller being tangent to its parallel at the point B ; the larger, at the point C. Here when the small circle commences to roll the point B

does not remain at rest for a while so as to allow Fig. 9 BC to move backward and carry with it the point C, as happened in the case of the polygons, where the point I remained fixed until the side KI coincided with MI and the line IB carried the terminal point B backward as far as b, so that the side BC fell upon be, thus superposing upon the line BA, the portion B&, and advancing by an amount Be, equal to MI, that is, to one side of the smaller polygon. On account of these superpositions, which are the excesses of the sides of the larger over the smaller polygon, each net advance is equal to one side of the smaller polygon and, during one complete rotation, these amount to a straight line equal in length to the perimeter of the smaller polygon.

But

FIRST DAY 51

But now reasoning in the same way concerning the circles, we must observe that whereas the number of sides in any polygon is comprised within a certain limit, the number of sides in a circle is infinite; the former are finite and divisible; the latter infinite and indivisible. In the case of the polygon, the vertices remain at rest during an interval of time which bears to the period of one complete rotation the same ratio which one side bears to the perimeter; likewise, in the case of the circles, the delay of each of the infinite number of vertices is merely instantaneous, because an instant is such a fraction of a finite interval as a point is of a line which contains an infinite number of points. The retrogression of the sides of the larger polygon is not equal to the length of one of its sides but merely to the excess of such a side over one side of the smaller polygon, the net advance being equal to this smaller side; but in the circle, the point or side C, during the instantaneous rest of B, recedes by an amount equal to its excess over the side B, making a net progress equal to B itself. In short the infinite number of indivisible sides of the greater circle with their infinite number of indivisible retrogressions, made during the infinite number of instantaneous delays of the infinite number of vertices of the smaller circle, together with the infinite number of progressions, equal to the infinite number of sides in the smaller circle — all these, I say, add up to a line equal to that described by the smaller circle, a line which contains an infinite number of infinitely small superpositions, thus bringing about a thickening or contraction without any overlapping or interpenetration of finite parts. This result could not be obtained in the case of a line divided into finite parts such as is the perimeter of any polygon, which when laid out in a straight line cannot be shortened except by the overlapping and interpenetration of its sides. This contraction of an infinite number of infinitely small parts without the interpenetration or overlapping of finite parts and the previously mentioned [p. 70, Nat. Ed.] expansion of an infinite number of indivisible parts by the interposition of indivisible vacua is, in my opinion, the most that can be said concerning the contraction and

52 THE TWO NEW SCIENCES OF GALILEO and rarefaction of bodies, unless we give up the impenetrability of matter and introduce empty spaces of finite size. If you find anything here that you consider worth while, pray use it; if not regard it, together with my remarks, as idle talk; but this remember, we are dealing with the infinite and the indivisible.

SAGR. I frankly confess that your idea is subtle and that it impresses me as new and strange; but whether, as a matter of fact, nature actually behaves according to such a law I am unable to determine; however, until I find a more satisfactory explanation I shall hold fast to this one. Perhaps Simplicio can tell us something which I have not yet heard, namely, how to explain the explanation which the philosophers have given of this abstruse matter; for, indeed, all that I have hitherto read concerning contraction is so dense and that concerning expansion so thin that my poor brain can neither penetrate the former nor grasp the latter.

SIMP. I am all at sea and find difficulties in following either path, especially this new one; because according to this theory an ounce of gold might be rarefied and expanded until its size would exceed that of the earth, while the earth, in turn, might be condensed and reduced until it would become smaller than a walnut, something which I do not believe; nor do I believe that you believe it. The arguments and demonstrations which you have advanced are mathematical, abstract, and far removed from concrete matter; and I do not believe that when applied to the physical and natural world these laws will hold.

SALV. I am not able to render the invisible visible, nor do I think that you will ask this. But now that you mention gold, do not our senses tell us that that metal can be immensely expanded ? I do not know whether you have observed the method

employed by those who are skilled in drawing gold wire, of which really only the surface is gold, the inside material being silver. The way they draw it is as follows: they take a cylinder or, if you please, a rod of silver, about half a cubit long and three or four times as wide as one's thumb; this rod they cover with gold-leaf which is so thin that it almost floats in air, putting on not

FIRST DAY 53 not more than eight or ten thicknesses. Once gilded they begin to pull it, with great force, through the holes of a draw-plate; again and again it is made to pass through smaller and smaller holes, until, after very many passages, it is reduced to the fineness of a lady's hair, or perhaps even finer; yet the surface remains gilded. Imagine now how the substance of this gold has been expanded and to what fineness it has been reduced.

SIMP. I do not see that this process would produce, as a consequence, that marvellous thinning of the substance of the gold which you suggest: first, because the original gilding consisting of ten layers of gold-leaf has a sensible thickness; secondly, because in drawing out the silver it grows in length but at the same time diminishes proportionally in thickness; and, since one dimension thus compensates the other, the area will not be so increased as to make it necessary during the process of gilding to reduce the thinness of the gold beyond that of the original leaves.

SALV. You are greatly mistaken, Simplicio, because the surface increases directly as the square root of the length, a fact which I can demonstrate geometrically.

SAGR. Please give us the demonstration not only for my own sake but also for Simplicio provided you think we can understand it. ,

SALV. I'll see if I can recall it on the spur of the moment. At the outset, it is clear that the original thick rod of silver and the wire drawn out to an enormous length are two cylinders of the same volume, since they are the same body of silver. So

that, if I determine the ratio between the surfaces of cylinders of the same volume, the problem will be solved. I say then,

The areas of cylinders of equal volumes, neglecting the bases, bear to each other a ratio which is the square root of the ratio of their lengths.

Take two cylinders of equal volume having the altitudes AB and CD, between which the line E is a mean proportional. Then I claim that, omitting the bases of each cylinder, the surface of the cylinder AB is to that of the cylinder CD as the length AB is

54 THE TWO NEW SCIENCES OF GALILEO is to the line E, that is, as the square root of AB is to the square root of CD. Now cut off the cylinder AB at F so that the altitude AF is equal to CD. Then since the bases of cylinders of equal volume bear to one another the inverse ratio of their heights, it follows that the area of the circular base of the cylinder CD will be to the area of the circular base of AB as the altitude BA is to DC: moreover, since circles are to one another as the squares of their diameters, the said squares will be to each other as BA is to CD. But BA is to CD as the square of » BA is to the square of E : and, therefore, these

'"""""^ four squares will form a proportion; and likewise their sides; so the line AB is to E as the diameter of circle C is to the diameter of the circle A. But the diameters are proportional to the circumferences and the circumferences are proportional to the areas of cylinders of equal height; hence the line AB is to E as the surface of the cylinder CD is to the surface of the cylinder AF. Now since the height AF is to AB as the surface of AF is to the surface of AB; and since the height AB is to the line E as the m surface CD is to AF, it follows, ex cequali in

Fig. 10 proportione perturbata* that the height AF is to E as the surface CD is to the surface AB, and convertendo, the surface of the cylinder AB is to the surface of the cylinder CD as the line E is to AF, i. e., to CD, or as AB is to E which is the square root of the ratio of AB to CD. Q. E. D.

If now we apply these results to the case in hand, and assume that the silver cylinder at the time of gilding had a length of only half a cubit and a thickness three or four times that of

one's thumb, we shall find that, when the wire has been reduced to the fineness of a hair and has been drawn out to a length of twenty thousand cubits (and perhaps more), the area of its surface will have been increased not less than two hundred times. Consequently the ten leaves of gold which were laid on

* See Euclid, Book V, Def. 20., Todhunter's Ed., p. 137 (London, 1877.)

[Trans]

FIRST DAY 55 have been extended over a surface two hundred times greater, assuring us that the thickness of the gold which now covers the surface of so many cubits of wire cannot be greater than one twentieth that of an ordinary leaf of beaten gold. Consider now what degree of fineness it must have and whether one could conceive it to happen in any other way than by enormous expansion of parts; consider also whether this experiment does not suggest that physical bodies [materie fisiche] are composed of infinitely small indivisible particles, a view which is supported by other more striking and conclusive examples.

SAGR. This demonstration is so beautiful that, even if it does not have the cogency originally intended, — although to my mind, it is very forceful — the short time devoted to it has nevertheless been most happily spent.

SALV. Since you are so fond of these geometrical demonstrations, which carry with them distinct gain, I will give you a companion theorem which answers an extremely interesting query. We have seen above what relations hold between equal cylinders of different height or length; let us now see what holds when the cylinders are equal in area but unequal in height, understanding area to include the curved surface, but not the upper and lower bases. The theorem is :

The volumes of right cylinders having equal curved surfaces are inversely proportional to their altitudes. Let the surfaces of the two cylinders, AE and CF, be equal but let the height of the latter, CD, be greater than that of the former, AB : then I say that the volume of the cylinder AE is to that of the cylinder CF as the height CD is to AB. Now since the surface of CF is equal to the surface of AE, it follows that the volume of CF is less than that of AE; for, if they were equal, the surface of CF would, by the preceding proposition, exceed that of AE, and the excess would be so much the greater if the volume of the cylinder CF were greater than that

of AE. Let us now take a cylinder ID having a volume equal to that of AE; then, according to the preceding theorem, the surface of the cylinder ID is to the surface of AE as the altitude

56 THE TWO NEW SCIENCES OF GALILEO

IF is to the mean proportional between IF and AB. But since one datum of the problem is that the surface of AE is equal to that of CF, and since the surface ID is to the surface CF as the altitude IF is to the altitude CD, it follows that CD is a ^ — ^ mean proportional between IF and AB. Not only so, but since the volume of the cylinder ID is equal to that of AE, each will bear the same ratio to the volume of the cylinder CF; but the volume ID is to the volume CF as the altitude IF is to the altitude CD; hence the volume of AE is to the volume of CF as the length IF is to the length CD, that is, as the length CD is to the length AB. Q. E. D.

This explains a phenomenon upon which the common people always look with wonder, namely, if we have a piece of stuff which has one side longer than the other, we can make from it a cornsack, using the customary wooden base, which will hold more when the short side of the cloth is used for the height of the sack and the long side is wrapped around the wooden base, than with the alternative arrangement. So that, for instance, from a piece of cloth which is six cubits on one side and twelve on the other, a sack can be made which will hold more when the side of twelve cubits is wrapped around the wooden base, leaving the sack six cubits high than when the six cubit side is put around the base making the sack twelve cubits high. From what has been proven above we learn not only the general fact that one sack holds more than the other, but we also get specific and particular information as to how much more, namely, just in proportion as the altitude of the sack diminishes the contents increase and vice versa. Thus if we use the figures given which make the cloth twice as long as wide and if we use the long side for the seam, the volume of the sack will be just one-half as great as with the opposite arrangement. Likewise if

FIRST DAY 57

if we have a piece of matting which measures 7 x 25 cubits and make from it a basket, the contents of the basket will, when the seam is lengthwise, be seven as compared with twenty-five when the seam runs endwise.

SAGR. It is with great pleasure that we continue thus to acquire new and useful information. But as regards the subject just discussed, I really believe that, among those who are not already familiar with geometry, you would scarcely find four persons in a hundred who would not, at first sight, make the mistake of believing that bodies having equal surfaces would be equal in other respects. Speaking of areas, the same error is made when one attempts, as often happens, to determine the sizes of various cities by measuring their boundary lines, forgetting that the circuit of one may be equal to the circuit of another while the area of the one is much greater than that of the other. And this is true not only in the case of irregular, but also of regular surfaces, where the polygon having the greater number of sides always contains a larger area than the one with the less number of sides, so that finally the circle which is a polygon of an infinite number of sides contains the largest area of all polygons of equal perimeter. I remember with particular pleasure having seen this demonstration when I was studying the sphere of Sacrobosco * with the aid of a learned commentary.

SALV. Very true! I too came across the same passage which suggested to me a method of showing how, by a single short demonstration, one can prove that the circle has the largest content of all regular isoperimetric figures; and that, of other

figures, the one which has the larger number of sides contains a greater area than that which has the smaller number.

SAGR. Being exceedingly fond of choice and uncommon propositions, I beseech you to let us have your demonstration.

SALV. I can do this in a few words by proving the following theorem:

The area of a circle is a mean proportional between any

* See interesting biographical note on Sacrobosco [John Holywood] in Ency. Brit., I ith Ed. [Trans.]

58 THE TWO NEW SCIENCES OF GALILEO two regular and similar polygons of which one circumscribes it and the other is isoperimetric with it. In addition, the area of the circle is less than that of any circumscribed polygon and greater than that of any isoperimetric polygon. And further, of these circumscribed polygons, the one which has the greater number of sides is smaller than the one which has a less number; but, on the other hand, that isoperimetric polygon which has the greater number of sides is the larger.

Let A and B be two similar polygons of which A circumscribes the given circle and B is isoperimetric with it. The area of the circle will then be a mean proportional between the areas of the polygons. For if we indicate the radius of the circle by AC and if we remember that the area of the circle is equal to that of a right-angled triangle in which one of the sides about the right angle is equal to the radius, AC, and the other to the circumference; and if likewise we remember that the area of the polygon A is equal to the area of a right-angled triangle one of

whose sides about the right angle has the same length as AC and the other is equal to the perimeter of the polygon itself; it is then

Fig. 12 manifest^that the circumscribed polygon bears to the circle the same ratio which its perimeter bears to the circumference of the circle, or to the perimeter of the polygon B which is, by hypothesis, equal to the circumference of the circle. But since the polygons A and B are similar their areas are to each other as the squares of their perimeters; hence the area of the circle A is a mean

FIRST DAY 59 mean proportional between the areas of the two polygons A and B. And since the area of the polygon A is greater than that of the circle A, it is clear that the area of the circle A is greater than that of the isoperimetric polygon B, and is therefore the greatest of all regular polygons having the same perimeter as the circle.

We now demonstrate the remaining portion of the theorem, which is to prove that, in the case of polygons circumscribing a given circle, the one having the smaller number of sides has a larger area than one having a greater number of sides; but that on the other hand, in the case of isoperimetric polygons, the one having the more sides has a larger area than the one with less sides. To the circle which has O for center and OA for radius draw the tangent AD; and on this tangent lay off, say, AD which shall represent one-half of the side of a circumscribed pentagon and AC which shall represent one-half of the side of a heptagon; draw the straight lines OGC and OFD; then with O as a center and OC as radius draw the arc ECI. Now since the triangle DOC is greater than the secftor EOC and since the secftor COI is greater than the triangle COA, it follows that the triangle DOC bears to the triangle COA a greater ratio than the secftor EOC bears to the secftor COI, that is, than the secftor FOG bears to the secftor GOA. Hence, componendo et permutando, the triangle DOA bears to the secftor FOA a greater ratio than that which the triangle COA bears to the secftor GOA, and also 10 such triangles DOA bear to 10 such secftors FOA a greater ratio than 14 such triangles COA bear to 14 such secftors GOA, that is to say, the circumscribed pentagon bears to the circle a greater ratio than does the heptagon. Hence the pentagon exceeds the heptagon in area.

But now let us assume that both the heptagon and the pentagon have the same perimeter as that of a given circle. Then I say the heptagon will contain a larger area than the pentagon. For since the area of the circle is a mean proportional between areas of the circumscribed and of the isoperimetric pentagons,

and since likewise it is a mean proportional between the circumscribed

6o THE TWO NEW SCIENCES OF GALILEO cumscribed and isoperimetric heptagons, and since also we have proved that the circumscribed pentagon is larger than the circumscribed heptagon, it follows that this circumscribed pentagon bears to the circle a larger ratio than does the heptagon, that is, the circle will bear to its isoperimetric pentagon a greater ratio than to its isoperimetric heptagon. Hence the pentagon is smaller than its isoperimetric heptagon. Q. E. D.

SAGR. A very clever and elegant demonstration! But how did we come to plunge into geometry while discussing the objections urged by Simplicio, objections of great moment, especially that one referring to density which strikes me as particularly difficult?

SALV. If contraction and expansion [condensazione e rarefazzione] consist in contrary motions, one ought to find for each great expansion a correspondingly large contraction. But our surprise is increased when, every day, we see enormous expansions taking pkce almost instantaneously. Think what a tremendous expansion occurs when a small quantity of gunpowder flares up into a vast volume of fire! Think too of the almost limitless expansion of the light which it produces! Imagine the contraction which would take place if this fire and this light were to reunite, which, indeed, is not impossible since only a little while ago they were located together in this small space. You will find, upon observation, a thousand such expansions for they are more obvious than contractions since dense matter is more palpable and accessible to our senses. We can take wood and see it go up in fire and light, but we do not see

them recombine to form wood; we see fruits and flowers and a thousand other solid bodies dissolve largely into odors, but we do not observe these fragrant atoms coming together to form fragrant solids. But where the senses fail us reason must step in; for it will enable us to understand the motion involved in the condensation of extremely rarefied and tenuous substances just as clearly as that involved in the expansion and dissolution of solids. Moreover we are trying to find out how it is possible to produce expansion and contraction in bodies which are capable of such changes without introducing vacua and without giving up

FIRST DAY 61 up the impenetrability of matter; but this does not exclude the possibility of there being materials which possess no such properties and do not, therefore, carry with them consequences which you call inconvenient and impossible. And finally, Simplicio, I have, for the sake of you philosophers, taken pains to find an explanation of how expansion and contraction can take place without our admitting the penetrability of matter and introducing vacua, properties which you deny and dislike; if you were to admit them, I should not oppose you so vigorously. Now either admit these difficulties or accept my views or suggest something better.

SAGR. I quite agree with the peripatetic philosophers in denying the penetrability of matter. As to the vacua I should like to hear a thorough discussion of Aristotle's demonstration in which he opposes them, and what you, Salviati, have to say in reply. I beg of you, Simplicio, that you give us the precise proof of the Philosopher and that you, Salviati, give us the reply.

SIMP. So far as I remember, Aristotle inveighs against the ancient view that a vacuum is a necessary prerequisite for motion and that the latter could not occur without the former. In opposition to this view Aristotle shows that it is precisely the phenomenon of motion, as we shall see, which renders untenable the idea of a vacuum. His method is to divide the argument into two parts. He first supposes bodies of different weights to move in the same medium; then supposes, one and the same body to move in different media. In the first case, he

supposes bodies of different weight to move in one and the same medium with different speeds which stand to one another in the same ratio as the weights ; so that, for example, a body which is ten times as heavy as another will move ten times as rapidly as the other. In the second case he assumes that the speeds of one and the same body moving in different media are in inverse ratio to the densities of these media; thus, for instance, if the density of water were ten times that of air, the speed in air would be ten times greater than in water. From this second supposition,

62 THE TWO NEW SCIENCES OF GALILEO tion, he shows that, since the tenuity of a vacuum differs infinitely from that of any medium filled with matter however rare, any body which moves in a plenum through a certain space in a certain time ought to move through a vacuum instantaneously; but instantaneous motion is an impossibility; it is therefore impossible that a vacuum should be produced by motion.

SALV. The argument is, as you see, ad hominem, that is, it is directed against those who thought the vacuum a prerequisite for motion. Now if I admit the argument to be conclusive and concede also that motion cannot take place in a vacuum, the assumption of a vacuum considered absolutely and not with reference to motion, is not thereby invalidated. But to tell you what the ancients might possibly have replied and in order to better understand just how conclusive Aristotle's demonstration is, we may, in my opinion, deny both of his assumptions. And as to the first, I greatly doubt that Aristotle ever tested by experiment whether it be true that two stones, one weighing ten times as much as the other, if allowed to fall, at the same instant, from a height of, say, 100 cubits, would so differ in speed that when the heavier had reached the ground, the other would not have fallen more than 10 cubits.

SIMP. His language would seem to indicate that he had tried the experiment, because he says: We see the heavier; now the word see shows that he had made the experiment.

SAGR. But I, Simplicio, who have made the test can assure you that a cannon ball weighing one or two hundred pounds, or even more, will not reach the ground by as much as a span ahead of a musket ball weighing only half a pound, provided both are dropped from a height of 200 cubits.

SALV. But, even without further experiment, it is possible to prove clearly, by means of a short and conclusive argument, that a heavier body does not move more rapidly than a lighter one provided both bodies are of the same material and in short such as those mentioned by Aristotle. But tell me, Simplicio, whether you admit that each falling body acquires a definite speed

FIRST DAY 63 speed fixed by nature, a velocity which cannot be increased or diminished except by the use of force [violenza] or resistance.

SIMP. There can be no doubt but that one and the same body moving in a single medium has a fixed velocity which is determined by nature and which cannot be increased except by the addition of momentum [impeto] or diminished except by some resistance which retards it.

SALV. If then we take two bodies whose natural speeds are different, it is clear that on uniting the two, the more rapid one will be partly retarded by the slower, and the slower will be somewhat hastened by the swifter. Do you not agree with me in this opinion ?

SIMP. You are unquestionably right.

SALV. But if this is true, and if a large stone moves with a speed of, say, eight while a smaller moves with a speed of four, then when they are united, the system will move with a speed less than eight; but the two stones when tied together make a stone larger than that which before moved with a speed of eight. Hence the heavier body moves with less speed than the lighter; an effect which is contrary to your supposition. Thus you see

how, from your assumption that the heavier body moves more rapidly than the lighter one, I infer that the heavier body moves more slowly.

SIMP. I am all at sea because it appears to me that the smaller stone when added to the larger increases its weight and by adding weight I do not see how it can fail to increase its speed or, at least, not to diminish it.

SALV. Here again you are in error, Simplicio, because it is not true that the smaller stone adds weight to the larger.

SIMP. This is, indeed, quite beyond my comprehension.

SALV. It will not be beyond you when I have once shown you the mistake under which you are laboring. Note that it is necessary to distinguish between heavy bodies in motion and the same bodies at rest. A large stone placed in a balance not only acquires additional weight by having another stone placed upon it, but even by the addition of a handful of hemp its weight is augmented

64 THE TWO NEW SCIENCES OF GALILEO augmented six to ten ounces according to the quantity of hemp. But if you tie the hemp to the stone and allow them to fall freely from some height, do you believe that the hemp will press down upon the stone and thus accelerate its motion or do you think the motion will be retarded by a partial upward pressure? One always feels the pressure upon his shoulders when he prevents the motion of a load resting upon him ; but if one descends just as rapidly as the load would fall how can it gravitate or press upon him ? Do you not see that this would be the same as trying to strike a man with a lance when he is running away from you with a speed which is equal to, or even greater, than that with which you are following him ? You must therefore conclude that, during free and natural fall, the small stone does not press upon the larger and consequently does not increase its weight as it does when at rest.

SIMP. But what if we should place the larger stone upon the smaller?

SALV. Its weight would be increased if the larger stone moved more rapidly; but we have already concluded that when the small stone moves more slowly it retards to some extent the speed of the larger, so that the combination of the two, which is a heavier body than the larger of the two stones, would move less rapidly, a conclusion which is contrary to your hypothesis. We infer therefore that large and small bodies move with the same speed provided they are of the same specific gravity.

SIMP. Your discussion is really admirable; yet I do not find it easy to believe that a bird-shot falls as swiftly as a cannon ball.

SALV. Why not say a grain of sand as rapidly as a grindstone? But, Simplicio, I trust you will not follow the example of many others who divert the discussion from its main intent and fasten upon some statement of mine which lacks a hair's-breadth of the truth and, under this hair, hide the fault of another which is as big as a ship's cable. Aristotle says that "an iron ball of one hundred pounds falling from a height of one hundred cubits reaches the ground before a one-pound ball has fallen a single cubit." I say that they arrive at the same time. You find, on making

FIRST DAY 65 making the experiment, that the larger outstrips the smaller by two finger-breadths, that is, when the larger has reached the ground, the other is short of it by two finger-breadths; now you would not hide behind these two fingers the ninety-nine cubits of Aristotle, nor would you mention my small error and at the same time pass over in silence his very large one. Aristotle declares that bodies of different weights, in the same medium, travel (in so far as their motion depends upon gravity) with speeds which are proportional to their weights ; this he illustrates by use of bodies in which it is possible to perceive the pure and unadulterated effect of gravity, eliminating other considerations, for example, figure as being of small importance [minimi momenti\, influences which are greatly dependent upon the medium which modifies the single effect of gravity alone. Thus we observe that gold, the densest of all substances, when beaten out into a very thin leaf, goes floating through the air; the same thing happens with stone when ground into a very fine powder. But if you wish to maintain the general proposition you will have to show that the same ratio of speeds is preserved in the

[no]

case of all heavy bodies, and that a stone of twenty pounds moves ten times as rapidly as one of two; but I claim that this is false and that, if they fall from a height of fifty or a hundred cubits, they will reach the earth at the same moment.

SIMP. Perhaps the result would be different if the fall took place not from a few cubits but from some thousands of cubits.

SALV. If this were what Aristotle meant you would burden him with another error which would amount to a falsehood; because, since there is no such sheer height available on earth, it is clear that Aristotle could not have made the experiment; yet he wishes to give us the impression of his having performed it when he speaks of such an effect as one which we see.

SIMP. In fact, Aristotle does not employ this principle, but uses the other one which is not, I believe, subject to these same difficulties.

SALV. But the one is as false as the other; and I am surprised that you yourself do not see the fallacy and that you do not perceive

66 THE TWO NEW SCIENCES OF GALILEO perceive that if it were true that, in media of different densities and different resistances, such as water and air, one and the same body moved in air more rapidly than in water, in proportion as the density of water is greater than that of air, then it would follow that any body which falls through air ought also to fall through water. But this conclusion is false inasmuch as many bodies which descend in air not only do not descend in water, but acftually rise.

SIMP. I do not understand the necessity of your inference; and in addition I will say that Aristotle discusses only those bodies which fall in both media, not those which fall in air but rise in water.

SALV. The arguments which you advance for the Philosopher are such as he himself would have certainly avoided so as not to aggravate his first mistake. But tell me now whether the density [corpulenza] of the water, or whatever it may be that retards the motion, bears a definite ratio to the density of air which is less retardative; and if so fix a value for it at your pleasure.

SIMP. Such a ratio does exist; let us assume it to be ten; then, for a body which falls in both these media, the speed in water will be ten times slower than in air.

SALV. I shall now take one of those bodies which fall in air but not in water, say a wooden ball, and I shall ask you to assign to it any speed you please for its descent through air.

SIMP. Let us suppose it moves with a speed of twenty.

SALV. Very well. Then it is clear that this speed bears to some smaller speed the same ratio as the density of water bears to that of air; and the value of this smaller speed is two. So that really if we follow exadlly the assumption of Aristotle we ought to infer that the wooden ball which falls in air, a substance ten times less-resisting than water, with a speed of twenty would fall in water with a speed of two, instead of coming to the surface from the bottom as it does; unless perhaps you wish to reply, which I do not believe you will, that the rising of the wood through the water is the same as its falling with a speed of two.

But

FIRST DAY 67

But since the wooden ball does not go to the bottom, I think you will agree with me that we can find a ball of another material, not wood, which does fall in water with a speed of two.

SIMP. Undoubtedly we can; but it must be of a substance considerably heavier than wood.

SALV. That is it exadtly. But if this second ball falls in water with a speed of two, what will be its speed of descent in air? If you hold to the rule of Aristotle you must reply that it will move at the rate of twenty; but twenty is the speed which you yourself have already assigned to the wooden ball; hence this and the other heavier ball will each move through air with the same speed. But now how does the Philosopher harmonize this result with his other, namely, that bodies of different weight move through the same medium with different speeds — speeds which are proportional to their weights? But without going into the matter more deeply, how have these common and

obvious properties escaped your notice ? Have you not observed that two bodies which fall in water, one with a speed a hundred times as great as that of the other, will fall in air with speeds so nearly equal that one will not surpass the other by as much as one hundredth part? Thus, for example, an egg made of marble will descend in water one hundred times more rapidly than a hen's egg, while in air falling from a height of twenty cubits the one will fall short of the other by less than four finger-breadths. In short, a heavy body which sinks through ten cubits of water in three hours will traverse ten cubits of air in one or two pulsebeats; and if the heavy body be a ball of lead it will easily traverse the ten cubits of water in less than double the time required for ten cubits of air. And here, I am sure, Simplicio, you find no ground for difference or objection. We conclude, therefore, that the argument does not bear against the existence of a vacuum; but if it did, it would only do away with vacua of considerable size which neither I nor, in my opinion, the ancients ever believed to exist in nature, although they might possibly be produced by force [violenza] as may be gathered from various experiments whose description would here occupy too much time.

Sagr.

68 THE TWO NEW SCIENCES OF GALILEO

SAGR. Seeing that Simplicio is silent, I will take the opportunity of saying something. Since you have clearly demonstrated that bodies of different weights do not move in one and the same medium with velocities proportional to their weights, but that they all move with the same speed, understanding of course that they are of the same substance or at least of the same specific gravity; certainly not of different specific gravities, for I hardly think you would have us believe a ball of cork moves with the same speed as one of lead; and again since you have clearly demonstrated that one and the same body moving through differently resisting media does not acquire speeds which are inversely proportional to the resistances, I am curious to learn what are the ratios actually observed in these cases.

SALV. These are interesting questions and I have thought much concerning them. I will give you the method of approach and the result which I finally reached. Having once established the falsity of the proposition that one and the same body moving through differently resisting media acquires speeds which are inversely proportional to the resistances of these media, and having also disproved the statement that in the same medium bodies of different weight acquire velocities proportional to their weights (understanding that this applies also to bodies which differ merely in specific gravity), I then began to combine these two facts and to consider what would happen if bodies of different weight were placed in media of different resistances; and I found that the differences in speed were greater in those media which were more resistant, that is, less yielding. This difference was such that two bodies which differed scarcely at all in their speed through air would, in water, fall the one with a speed ten times as great as that of the other. Further, there are bodies which will fall rapidly in air, whereas if placed in water not only will not sink but will remain at rest or will even rise to the top: for it is possible to find some kinds of wood, such as knots and roots, which remain at rest in water but fall rapidly in air.

SAGR. I have often tried with the utmost patience to add grains of sand to a ball of wax until it should acquire the same specific

FIRST DAY 69 specific gravity as water and would therefore remain at rest in this medium. But with all my care I was never able to accomplish this. Indeed, I do not know whether there is any solid substance whose specific gravity is, by nature, so nearly equal to that of water that if placed anywhere in water it will remain at rest.

SALV. In this, as in a thousand other operations, men are surpassed by animals. In this problem of yours one may learn much from the fish which are very skillful in maintaining their equilibrium not only in one kind of water, but also in waters which are notably different either by their own nature or by some accidental muddiness or through salinity, each of which produces a marked change. So perfectly indeed can fish keep their equilibrium that they are able to remain motionless in any position. This they accomplish, I believe, by means of an apparatus especially provided by nature, namely, a bladder located in the body and communicating with the mouth by means of a narrow tube through which they are able, at will, to expel a portion of the air contained in the bladder: by rising to the surface they can take in more air; thus they make themselves heavier or lighter than water at will and maintain equilibrium.

SAGR. By means of another device I was able to deceive some friends to whom I had boasted that I could make up a ball of wax that would be in equilibrium in water. In the bottom of a vessel I placed some salt water and upon this some fresh water; then I showed them that the ball stopped in the middle of the water, and that, when pushed to the bottom or lifted to the top, would not remain in either of these places but would return to the middle.

SALV. This experiment is not without usefulness. For when physicians are testing the various qualities of waters, especially their specific gravities, they employ a ball of this kind so adjusted that, in certain water, it will neither rise nor fall. Then in testing another water, differing ever so slightly in specific gravity \pcso\ the ball will sink if this water be lighter and rise if it be heavier. And so exact is this experiment that the addition yo THE NEW TWO SCIENCES OF GALILEO tion of two grains of salt to six pounds of water is sufficient to make the ball rise to the surface from the bottom to which it had fallen. To illustrate the precision of this experiment and also to clearly demonstrate the non-resistance of water to division, I wish to add that this notable difference in specific gravity can be produced not only by solution of some heavier substance, but also by merely heating or cooling; and so sensitive is water to this process that by simply adding four drops of another water which is slightly warmer or cooler than the six pounds one can cause the ball to sink or rise; it will sink when the warm water is poured in and will rise upon the addition of cold water. Now you

["Si can see how mistaken are those philosophers who ascribe to water viscosity or some other coherence of parts which offers resistance to separation of parts and to penetration.

SAGR. With regard to this question I have found many convincing arguments in a treatise by our Academician; but there is one great difficulty of which I have not been able to rid myself, namely, if there be no tenacity or coherence between the particles of water how is it possible for those large drops of water to stand out in relief upon cabbage leaves without scattering or spreading out?

SALV. Although those who are in possession of the truth are able to solve all objections raised, I would not arrogate to myself such power; nevertheless my inability should not be allowed to becloud the truth. To begin with let me confess that I do not understand how these large globules of water stand out and hold themselves up, although I know for a certainty, that it is not owing to any internal tenacity acting between the particles of water; whence it must follow that the cause of this effect is external. Beside the experiments already shown to prove that the cause is not internal, I can offer another which is very convincing. If the particles of water which sustain themselves in a heap, while surrounded by air, did so in virtue of an internal cause then they would sustain themselves much more easily when surrounded by a medium in which they exhibit less tendency to fall than they do in air; such a medium would be any fluid heavier

FIRST DAY 71 heavier than air, as, for instance, wine: and therefore if some wine be poured about such a drop of water, the wine might rise until the drop was entirely covered, without the particles of water, held together by this internal coherence, ever parting company. But this is not the fadl; for as soon as the wine touches the water, the latter without waiting to be covered scatters and spreads out underneath the wine if it be red. The cause of this effecl: is therefore external and is possibly to be found in the surrounding air. Indeed there appears to be a considerable antagonism between air and water as I have observed in the following experiment. Having taken a glass globe which had a mouth of about the same diameter as a straw, I filled it with water and turned it mouth downwards; neverthe-

less, the water, although quite heavy and prone to descend, and the air, which is very light and disposed to rise through the water, refused, the one to descend and the other to ascend through the opening, but both remained stubborn and defiant. On the other hand, as soon as I apply to this opening a glass of red wine, which is almost inappreciably lighter than water, red streaks are immediately observed to ascend slowly through the water while the water with equal slowness descends through the wine without mixing, until finally the globe is completely filled with wine and the water has all gone down into the vessel below. What then can we say except that there exists, between water and air, a certain incompatibility which I do not understand, but perhaps. . . .

SIMP. I feel almost like laughing at the great antipathy which Salviati exhibits against the use of the word antipathy; and yet it is excellently adapted to explain the difficulty.

SALV. Alright, if it please Simplicio, let this word antipathy be the solution of our difficulty. Returning from this digression, let us again take up our problem. We have already seen that the difference of speed between bodies of different specific gravities is most marked in those media which are the most resistant: thus, in a medium of quicksilver, gold not merely sinks to the bottom more rapidly than lead but it is the only substance

72 THE TWO NEW SCIENCES OF GALILEO substance that will descend at all; all other metals and stones rise to the surface and float. On the other hand the variation of speed in air between balls of gold, lead, copper, porphyry, and other heavy materials is so slight that in a fall of 100 cubits a ball of gold would surely not outstrip one of copper by as much as four fingers. Having observed this I came to the conclusion that in a medium totally devoid of resistance all bodies would fall with the same speed.

SIMP. This is a remarkable statement, Salviati. But I shall never believe that even in a vacuum, if motion in such a place were possible, a lock of wool and a bit of lead can fall with the same velocity.

SALV. A little more slowly, Simplicio. Your difficulty is not so recondite nor am I so imprudent as to warrant you in believing that I have not already considered this matter and found the proper solution. Hence for my justification and for your enlightenment hear what I have to say. Our problem is to find out what happens to bodies of different weight moving in a medium devoid of resistance, so that the only difference in speed is that which arises from inequality of weight. Since no medium except one entirely free from air and other bodies, be it ever so tenuous and yielding, can furnish our senses with the evidence we are looking for, and since such a medium is not available, we shall observe what happens in the rarest and least resistant media as compared with what happens in denser and more resistant media. Because if we find as a fact that the variation of speed among bodies of different specific gravities is less and less according as the medium becomes more and more yielding, and if finally in a medium of extreme tenuity, though not a perfect vacuum, we find that, in spite of great diversity of specific gravity [p£H>], the difference in speed is very small and almost inappreciable, then we are justified in believing it highly probable that in a vacuum all bodies would fall with the same speed. Let us, in view of this, consider what takes place in air, where for the sake of a definite figure and light material imagine an inflated bladder. The air in this bladder when surrounded by air

FIRST DAY 73 air will weigh little or nothing, since it can be only slightly compressed; its weight then is small being merely that of the skin which does not amount to the thousandth part of a mass of lead having the same size as the inflated bladder. Now, Simplicio, if we allow these two bodies to fall from a height of four or six cubits, by what distance do you imagine the lead will anticipate the bladder? You may be sure that the lead will not travel three times, or even twice, as swiftly as the bladder, although vou would have made it move a thousand times as rapidly.

SIMP. It may be as you say during the first four or six cubits of the fall; but after the motion has continued a long while, I believe that the lead will have left the bladder behind not only six out of twelve parts of the distance but even eight or ten.

SALV. I quite agree with you and doubt not that, in very long distances, the lead might cover one hundred miles while the

bladder was traversing one; but, my dear Simplicio, this phenomenon which you adduce against my proposition is precisely the one which confirms it. Let me once more explain that the variation of speed observed in bodies of different specific gravities is not caused by the difference of specific gravity but depends upon external circumstances and, in particular, upon the resistance of the medium, so that if this is removed all bodies would fall with the same velocity; and this result I deduce mainly from the fact which you have just admitted and which is very true, namely, that, in the case of bodies which differ widely in weight, their velocities differ more and more as the spaces traversed increase, something which would not occur if the effect depended upon differences of specific gravity. For since these specific gravities remain constant, the ratio between the distances traversed ought to remain constant whereas the fact is that this ratio keeps on increasing as the motion continues. Thus a very heavy body in a fall of one cubit will not anticipate a very light one by so much as the tenth part of this space; but in a fall of twelve cubits the heavy body would outstrip

74 THE TWO NEW SCIENCES OF GALILEO strip the other by one-third, and in a fall of one hundred cubits by 90/100, etc.

SIMP. Very well: but, following your own line of argument, if differences of weight in bodies of different specific gravities cannot produce a change in the ratio of their speeds, on the ground that their specific gravities do not change, how is it possible for the medium, which also we suppose to remain constant, to bring about any change in the ratio of these velocities ?

SALV. This objection with which you oppose my statement is clever; and I must meet it. I begin by saying that a heavy body has an inherent tendency to move with a constantly and uniformly accelerated motion toward the common center of gravity, that is, toward the center of our earth, so that during equal intervals of time it receives equal increments of momentum and velocity. This, you must understand, holds whenever all external and accidental hindrances have been removed; but of these there is one which we can never remove, namely, the medium which must be penetrated and thrust aside by the falling body. This quiet, yielding, fluid medium opposes motion

through it with a resistance which is proportional to the rapidity with which the medium must give way to the passage of the body; which body, as I have said,. is by nature continuously accelerated so that it meets with more and more resistance in the medium and hence a diminution in its rate of gain of speed until finally the speed reaches such a point and the resistance of the medium becomes so great that, balancing each other, they prevent any further acceleration and reduce the motion of the body to one which is uniform and which will thereafter maintain a constant value. There is, therefore, an increase in the resistance of the medium, not on account of any change in its essential properties, but on account of the change in rapidity with which it must yield and give way laterally to the passage of the falling body which is being constantly accelerated.

Now seeing how great is the resistance which the air offers to the slight momentum [momenta] of the bladder and how small that which it offers to the large weight [peso] of the lead, I am

FIRST DAY 75 am convinced that, if the medium were entirely removed, the advantage received by the bladder would be so great and that coming to the lead so small that their speeds would be equalized. Assuming this principle, that all falling bodies acquire equal speeds in a medium which, on account of a vacuum or something else, offers no resistance to the speed of the motion, we shall be able accordingly to determine the ratios of the speeds of both similar and dissimilar bodies moving either through one and the same medium or through different space-filling, and therefore resistant, media. This result we may obtain by observing how much the weight of the medium detracts from the weight of the moving body, which weight is the means employed by the falling body to open a path for itself and to push aside the parts of the medium, something which does not happen in a vacuum where, therefore, no difference [of speed] is to be expected from a difference of specific gravity. And since it is known that the effect of the medium is to diminish the weight of the body by the weight of the medium displaced, we may accomplish our purpose by diminishing in just this proportion the speeds of the falling bodies, which in a non-resisting medium we have assumed to be equal.

Thus, for example, imagine lead to be ten thousand times as heavy as air while ebony is only one thousand times as heavy.

Here we have two substances whose speeds of fall in a medium devoid of resistance are equal: but, when air is the medium, it will subtract from the speed of the lead one part in ten thousand, and from the speed of the ebony one part in one thousand, i. e. ten parts in ten thousand. While therefore lead and ebony would fall from any given height in the same interval of time, provided the retarding effect of the air were removed, the lead will, in air, lose in speed one part in ten thousand; and the ebony, ten parts in ten thousand. In other words, if the elevation from which the bodies start be divided into ten thousand parts, the lead will reach the ground leaving the ebony behind by as much as ten, or at least nine, of these parts. Is it not clear then that a leaden ball allowed to fall from a tower two hundred cubits high

76 THE TWO NEW SCIENCES OF GALILEO high will outstrip an ebony ball by less than four inches ? Now ebony weighs a thousand times as much as air but this inflated bladder only four times as much; therefore air diminishes the inherent and natural speed of ebony by one part in a thousand ; while that of the bladder which, if free from hindrance, would be the same, experiences a diminution in air amounting to one part in four. So that when the ebony ball, falling from the tower, has reached the earth, the bladder will have traversed only three-quarters of this distance. Lead is twelve times as heavy as water; but ivory is only twice as heavy. The speeds of these two substances which, when entirely unhindered, are equal will be diminished in water, that of lead by one part in twelve, that of ivory by half. Accordingly when the lead has fallen through eleven cubits of water the ivory will have fallen through only six. Employing this principle we shall, I believe, find a much closer agreement of experiment with our computation than with that of Aristotle.

In a similar manner we may find the ratio of the speeds of one and the same body in different fluid media, not by comparing the different resistances of the media, but by considering the excess of the specific gravity of the body above those of the media. Thus, for example, tin is one thousand times heavier than air and ten times heavier than water; hence, if we divide its unhindered speed into 1000 parts, air will rob it of one of these parts so that it will fall with a speed of 999, while in water its speed will be 900, seeing that water diminishes its weight by one part in ten while air by only one part in a thousand.

Again take a solid a little heavier than water, such as oak, a ball of which will weigh let us say 1000 drachms; suppose an

equal volume of water to weigh 950, and an equal volume of air, 2; then it is clear that if the unhindered speed of the ball is 1000, its speed in air will be 998, but in water only 50, seeing that the water removes 950 of the 1000 parts which the body weighs, leaving only 50.

Such a solid would therefore move almost twenty times as fast in air as in water, since its specific gravity exceeds that of water

FIRST DAY 77 water by one part in twenty. And here we must consider the fact that only those substances which have a specific gravity greater than water can fall through it — substances which must, therefore, be hundreds of times heavier than air; hence when we try to obtain the ratio of the speed in air to that in water, we may, without appreciable error, assume that air does not, to any considerable extent, diminish the free weight [assoluta gravitd], and consequently the unhindered speed [assoluta velocitd] of such substances. Having thus easily found the excess of the weight of these substances over that of water, we can say that their speed in air is to their speed in water as their free weight [totale gravitd] is to the excess of this weight over that of water. For example, a ball of ivory weighs 20 ounces; an equal volume of water weighs 17 ounces; hence the speed of ivory in air bears to its speed in water the approximate ratio of 20:3.

SAGR. I have made a great step forward in this truly interesting subject upon which I have long labored in vain. In order to put these theories into practice we need only discover a method of determining the specific gravity of air with reference to water and hence with reference to other heavy substances.

SIMP. But if we find that air has levity instead of gravity what then shall we say of the foregoing discussion which, in other respects, is very clever?

SALV. I should say that it was empty, vain, and trifling. But can you doubt that air has weight when you have the clear testimony of Aristotle affirming that all the elements have weight including air, and excepting only fire? As evidence of this he cites the fact that a leather bottle weighs more when inflated than when collapsed.

SIMP. I am inclined to believe that the increase of weight observed in the inflated leather bottle or bladder arises, not from the gravity of the air, but from the many thick vapors mingled with it in these lower regions. To this I would attribute the increase of weight in the leather bottle.

SALV. I would not have you say this, and much less attribute it to Aristotle; because, if speaking of the elements, he wished to persuade

78 THE TWO NEW SCIENCES OF GALILEO persuade me by experiment that air has weight and were to say to me: "Take a leather bottle, fill it with heavy vapors and observe how its weight increases," I would reply that the bottle would weigh still more if filled with bran; and would then add that this merely proves that bran and thick vapors are heavy, but in regard to air I should still remain in the same doubt as before. However, the experiment of Aristotle is good and the proposition is true. But I cannot say as much of a certain other consideration, taken at face value; this consideration was offered by a philosopher whose name slips me; but I know I have read his argument which is that air exhibits greater gravity than levity, because it carries heavy bodies downward more easily than it does light ones upward.

SAGR. Fine indeed! So according to this theory air is much heavier than water, since all heavy bodies are carried downward more easily through air than through water, and all light bodies buoyed up more easily through water than through air; further there is an infinite number of heavy bodies which fall through air but ascend in water and there is an infinite number of substances which rise in water and fall in air. But, Simplicio, the question as to whether the weight of the leather bottle is owing to thick vapors or to pure air does not affect our problem which is to discover how bodies move through this vapor-laden atmosphere of ours. Returning now to the question which interests me more, I should like, for the sake of more complete and thorough knowledge of this matter, not only to be strengthened in my belief that air has weight but also to learn, if possible, how great its specific gravity is. Therefore, Salviati, if you can satisfy my curiosity on this point pray do so.

SALV. The experiment with the inflated leather bottle of Aristotle proves conclusively that air possesses positive gravity and not, as some have believed, levity, a property possessed possibly by no substance whatever; for if air did possess this quality of absolute and positive levity, it should on compression

exhibit greater levity and, hence, a greater tendency to rise; but experiment shows precisely the opposite.

As

FIRST DAY 79

As to the other question, namely, how to determine the specific gravity of air, I have employed the following method. I took a rather large glass bottle with a narrow neck and attached to it a leather cover, binding it tightly about the neck of the bottle: in the top of this cover I inserted and firmly fastened the valve of a leather bottle, through which I forced into the glass bottle, by means of a syringe, a large quantity of air. And since air is easily condensed one can pump into the bottle two or three times its own volume of air. After this I took an accurate balance and weighed this bottle of compressed air with the utmost precision, adjusting the weight with fine sand. I next opened the valve and allowed the compressed air to escape ; then replaced the flask upon the balance and found it perceptibly lighter: from the sand which had been used as a counterweight I now removed and laid aside as much as was necessary to again secure balance. Under these conditions there can be no doubt but that the weight of the sand thus laid aside represents the weight of the air which had been forced into the flask and had afterwards escaped. But after all this experiment tells me merely that the weight of the compressed air is the same as that of the sand removed from the balance; when however it comes to knowing certainly and definitely the weight of air as compared with that of water or any other heavy substance this I cannot hope to do without first measuring the volume [qicantiia] of compressed air; for this measurement I have devised the two following methods.

According to the first method one takes a bottle with a narrow neck similar to the previous one; over the mouth of this bottle is slipped a leather tube which is bound tightly about the neck of the flask; the other end of this tube embraces the valve attached to the first flask and is tightly bound about it. This second flask is provided with a hole in the bottom through which an iron rod can be placed so as to open, at will, the valve above mentioned and thus permit the surplus air of the first to escape after it has once been weighed: but his second bottle must be filled with water. Having prepared everything in the manner

above

8o THE TWO NEW SCIENCES OF GALILEO above described, open the valve with the rod; the air will rush into the flask containing the water and will drive it through the hole at the bottom, it being clear that the volume [quantita] of water thus displaced is equal to the volume [mole e quantita] of air escaped from the other vessel. Having set aside this displaced water, weigh the vessel from which the air has escaped (which is supposed to have been weighed previously while containing the compressed air), and remove the surplus of sand as described above; it is then manifest that the weight of this sand is precisely the weight of a volume [mole] of air equal to the volume of water displaced and set aside; this water we can weigh and find how many times its weight contains the weight of the removed sand, thus determining definitely how many times heavier water is than air; and we shall find, contrary to the opinion of Aristotle, that this is not 10 times, but, as our experiment shows, more nearly 400 times.

The second method is more expeditious and can be carried out with a single vessel fitted up as the first was. Here no air is added to that which the vessel naturally contains but water is forced into it without allowing any air to escape; the water thus introduced necessarily compresses the air. Having forced into the vessel as much water as possible, filling it, say, three-fourths full, which does not require any extraordinary effort, place it upon the balance and weigh it accurately; next hold the vessel mouth up, open the valve, and allow the air to escape; the volume of the air thus escaping is precisely equal to the volume of water contained in the flask. Again weigh the vessel which will have diminished in weight on account of the escaped air; this loss in weight represents the weight of a volume of air equal to the volume of water contained in the vessel.

SIMP. No one can deny the cleverness and ingenuity of your devices; but while they appear to give complete intellectual satisfaction they confuse me in another direction. For since it is undoubtedly true that the elements when in their proper places have neither weight nor levity, I cannot understand how it is possible for that portion of air, which appeared to weigh, say, 4 drachms of sand, should really have such a weight in air as the sand

FIRST DAY 81 sand which counterbalances it. It seems to me, therefore, that the experiment should be carried out, not in air, but in a medium

in which the air could exhibit its property of weight if such it really has.

SALV. The objection of Simplicio is certainly to the point and must therefore either be unanswerable or demand an equally clear solution. It is perfectly evident that that air which, under compression, weighed as much as the sand, loses this weight when once allowed to escape into its own element, while, indeed, the sand retains its weight. Hence for this experiment it becomes necessary to select a place where air as well as sand can gravitate; because, as has been often remarked, the medium diminishes the weight of any substance immersed in it by an amount equal to the weight of the displaced medium; so that air in air loses all its weight. If therefore this experiment is to be made with accuracy it should be performed in a vacuum where every heavy body exhibits its momentum without the slightest diminution. If then, Simplicio, we were to weigh a portion of air in a vacuum would you then be satisfied and assured of the fact?

SIMP. Yes truly: but this is to wish or ask the impossible.

SALV. Your obligation will then be very great if, for your sake, I accomplish the impossible. But I do not want to sell you something which I have already given you; for in the previous experiment we weighed the air in vacuum and not in air or other medium. The fact that any fluid medium diminishes the weight of a mass immersed in it, is due, Simplicio, to the resistance which this medium offers to its being opened up, driven aside, and finally lifted up. The evidence for this is seen in the readiness with which the fluid rushes to fill up any space formerly occupied by the mass; if the medium were not affected by such an immersion then it would not react against the immersed body. Tell me now, when you have a flask, in air, filled with its natural amount of air and then proceed to pump into the vessel more air, does this extra charge in any way separate or divide or change the circumambient air? Does the vessel perhaps expand

82 THE TWO NEW SCIENCES OF GALILEO so that the surrounding medium is displaced in order to give more room? Certainly not. Therefore one is able to say that

this extra charge of air is not immersed in the surrounding medium for it occupies no space in it, but is, as it were, in a vacuum. Indeed, it is really in a vacuum; for it diffuses into the vacuities which are not completely filled by the original and uncondensed air. In facft I do not see any difference between the enclosed and the surrounding media: for the surrounding medium does not press upon the enclosed medium and, vice versa, the enclosed medium exerts no pressure against the surrounding one; this same relationship exists in the case of any matter in a vacuum, as well as in the case of the extra charge of air compressed into the flask. The weight of this condensed air is therefore the same as that which it would have if set free in a vacuum. It is true of course that the weight of the sand used as a counterpoise would be a little greater in vacuo than in free air. We must, then, say that the air is slightly lighter than the sand required to counterbalance it, that is to say, by an amount equal to the weight in vacuo of a volume of air equal to the volume of the sand.

At this point in an annotated copy of the original edition the following note by Galileo is found.

[SAGR. A very clever discussion, solving a wonderful problem, because it demonstrates briefly and concisely the manner in which one may find the weight of a body in vacuo by simply weighing it in air. The explanation is as follows: when a heavy body is immersed in air it loses in weight an amount equal to the weight of a volume [mole} of air equivalent to the volume [mole] of the body itself. Hence if one adds to a body, without expanding it, a quantity of air equal to that which it displaces and weighs it, he will obtain its absolute weight in vacuo, since, without increasing it in size, he has increased its weight by just the amount which it lost through immersion in air.

When therefore we force a quantity of water into a vessel which already contains its normal amount of air, without allowing any of this air to escape it is clear that this normal quantity of air will be compressed and condensed into a smaller space in order to make room for the water which is forced in: it is also clear that the volume of air thus compressed is equal to the volume of water added. If now the vessel be weighed

FIRST DAY 83 weighed in air in this condition, it is manifest that the weight of the water will be increased by that of an equal volume of air; the total weight of water and air thus obtained is equal to the weight of the water alone in vacuo.

Now record the weight of the entire vessel and then allow the compressed air to escape; weigh the remainder; the difference of these two weights will be the weight of the compressed air which, in volume, is equal to that of the water. Next find the weight of the water alone and add to it that of the compressed air; we shall then have the water alone in vacuo. To find the weight of the water we shall have to remove it from the vessel and weigh the vessel alone; subtrad this weight from that of the vessel and water together. It is clear that the remainder will be the weight of the water alone in air.]

SIMP. The previous experiments, in my opinion, left something to be desired : but now I am fully satisfied.

SALV. The facts set forth by me up to this point and, in particular, the one which shows that difference of weight, even when very great, is without effect in changing the speed of falling bodies, so that as far as weight is concerned they all fall with equal speed: this idea is, I say, so new, and at first glance so remote from fact, that if we do not have the means of making it just as clear as sunlight, it had better not be mentioned; but having once allowed it to pass my lips I must neglect no experiment or argument to establish it.

SAGR. Not only this but also many other of your views are so far removed from the commonly accepted opinions and doctrines that if you were to publish them you would stir up a large number of antagonists; for human nature is such that men do not look with favor upon discoveries — either of truth or fallacy — in their own field, when made by others than themselves. They call him an innovator of doctrine, an unpleasant title, by which they hope to cut those knots which they cannot untie, and by subterranean mines they seek to destroy structures which patient artisans have built with customary tools.

But as for ourselves who have no such thoughts, the experiments and arguments which you have thus far adduced are fully satisfactory; however if you have any experiments' which are

84 THE TWO NEW SCIENCES OF GALILEO are more direct or any arguments which are more convincing we will hear them with pleasure.

SALV. The experiment made to ascertain whether two bodies, differing greatly in weight will fall from a given height with the same speed offers some difficulty; because, if the height is considerable, the retarding effect of the medium, which must be penetrated and thrust aside by the falling body, will be greater in the case of the small momentum of the very light body than in the case of the great force [violenza] of the heavy body; so that, in a long distance, the light body will be left behind; if the height be small, one may well doubt whether there is any difference; and if there be a difference it will be inappreciable.

It occurred to me therefore to repeat many times the fall through a small height in such a way that I might accumulate all those small intervals of time that elapse between the arrival of the heavy and light bodies respectively at their common terminus, so that this sum makes an interval of time which is not only observable, but easily observable. In order to employ the slowest speeds possible and thus reduce the change which the resisting medium produces upon the simple effect of gravity it occurred to me to allow the bodies to fall along a plane slightly inclined to the horizontal. For in such a plane, just as well as in a vertical plane, one may discover how bodies of different weight behave: and besides this, I also wished to rid myself of the resistance which might arise from contact of the moving body with the aforesaid inclined plane. Accordingly I took two balls, one of lead and one of cork, the former more than a hundred times heavier than the latter, and suspended them by means of two equal fine threads, each four or five cubits long. Pulling each ball aside from the perpendicular, I let them go at the same instant, and they, falling along the circumferences of circles having these equal strings for semi-diameters, passed beyond the perpendicular and returned along the same path. This free vibration [per lor medesime le andate e le tornate] repeated a hundred times showed clearly that the heavy body maintains so

nearly the period of the light body that neither in a hundred swings

FIRST DAY 85 swings nor even in a thousand will the former anticipate the latter by as much as a single moment [minima momenta], so perfectly do they keep step. We can also observe the effect of the medium which, by the resistance which it offers to motion, diminishes the vibration of the cork more than that of the lead, but without altering the frequency of either; even when the arc traversed by the cork did not exceed five or six degrees while that of the lead was fifty or sixty, the swings were performed in equal times.

SIMP. If this be so, why is not the speed of the lead greater than that of the cork, seeing that the former traverses sixty degrees in the same interval in which the latter covers scarcely six?

SALV. But what would you say, Simplicio, if both covered their paths in the same time when the cork, drawn aside through thirty degrees, traverses an arc of sixty, while the lead pulled aside only two degrees traverses an arc of four? Would not then the cork be proportionately swifter? And yet such is the experimental fact. But observe this: having pulled aside the pendulum of lead, say through an arc of fifty degrees, and set it free, it swings beyond the perpendicular almost fifty degrees, thus describing an arc of nearly one hundred degrees; on the return swing it describes a little smaller arc; and after a large number of such vibrations it finally comes to rest. Each vibration, whether of ninety, fifty, twenty, ten, or four degrees occupies the same time: accordingly the speed of the moving body keeps on diminishing since in equal intervals of time, it traverses arcs which grow smaller and smaller.

Two New Sciences: First Day (1638) (part 3)

Precisely the same things happen with the pendulum of cork, suspended by a string of equal length, except that a smaller number of vibrations is required to bring it to rest, since on account of its lightness it is less able to overcome the resistance of the air; nevertheless the vibrations, whether large or small, are all performed in time-intervals which are not only equal among themselves, but also equal to the period of the lead pendulum. Hence it is true that, if while the lead is traversing an arc of fifty degrees the cork covers one of only ten, the cork moves more slowly than the lead; but on the other hand it is also true that

86 THE TWO NEW SCIENCES OF GALILEO

that the cork may cover an arc of fifty while the lead passes over one of only ten or six; thus, at different times, we have now the cork, now the lead, moving more rapidly. But if these same bodies traverse equal arcs in equal times we may rest assured that their speeds are equal.

SIMP. I hesitate to admit the conclusiveness of this argument because of the confusion which arises from your making both bodies move now rapidly, now slowly and now very slowly, which leaves me in doubt as to whether their velocities are always equal.

SAGR. Allow me, if you please, Salviati, to say just a few words. Now tell me, Simplicio, whether you admit that one can say with certainty that the speeds of the cork and the lead are equal whenever both, starting from rest at the same moment and descending the same slopes, always traverse equal spaces in equal times?

SIMP. This can neither be doubted nor gainsaid.

SAGR. Now it happens, in the case of the pendulums, that each of them traverses now an arc of sixty degrees, now one of fifty, or thirty or ten or eight or four or two, etc. ; and when they both swing through an arc of sixty degrees they do so in equal intervals of time; the same thing happens when the arc is fifty degrees or thirty or ten or any other number; and therefore we conclude that the speed of the lead in an arc of sixty degrees is equal to the speed of the cork when the latter also swings through an arc of sixty degrees; in the case of a fifty-degree arc these speeds are also equal to each other; so also in the case of other arcs. But this is not saying that the speed which occurs in an arc of sixty is the same as that which occurs in an arc of fifty; nor is the speed in an arc of fifty equal to that in one of thirty, etc. ; but the smaller the arcs, the smaller the speeds; the facft observed is that one and the same moving body requires the same time for traversing a large arc of sixty degrees as for a small arc of fifty or even a very small arc of ten; all these arcs, indeed, are covered in the same interval of time. It is true therefore that the lead

and

FIRST DAY 87 and the cork each diminish their speed [moto] in proportion as their arcs diminish; but this does not contradict the fact that they maintain equal speeds in equal arcs.

My reason for saying these things has been rather because I wanted to learn whether I had correctly understood Salviati, than because I thought Simplicio had any need of a clearer explanation than that given by Salviati which like everything else of his is extremely lucid, so lucid, indeed, that when he solves questions which are difficult not merely in appearance, but in reality and in fact, he does so with reasons, observations and experiments which are common and familiar to everyone.

In this manner he has, as I have learned from various sources, given occasion to a highly esteemed professor for undervaluing his discoveries on the ground that they are commonplace, and established upon a mean and vulgar basis; as if it were not a most admirable and praiseworthy feature of demonstrative science that it springs from and grows out of principles wellknown, understood and conceded by all.

But let us continue with this light diet; and if Simplicio is satisfied to understand and admit that the gravity inherent [interna graviia] in various falling bodies has nothing to do with the difference of speed observed among them, and that all bodies, in so far as their speeds depend upon it, would move with the same velocity, pray tell us, Salviati, how you explain the appreciable and evident inequality of motion; please reply also to the objection urged by Simplicio — an objection in which I .concur — namely, that a cannon ball falls more rapidly than a bird-shot. From my point of view, one might expect the difference of speed to be small in the case of bodies of the same substance moving through any single medium, whereas the larger ones will descend, during a single pulse-beat, a distance which the smaller ones will not traverse in an hour, or in four, or even in twenty hours; as for instance in the case of stones and fine sand and especially that very fine sand which produces muddy water and which in many hours will not fall through as much as two cubits, a distance which stones not much larger will traverse in a single pulse-beat.

Salv.

88 THE TWO NEW SCIENCES OF GALILEO

SALV. The acflion of the medium in producing a greater retardation upon those bodies which have a less specific gravity has already been explained by showing that they experience a diminution of weight. But to explain how one and the same medium produces such different retardations in bodies which are made of the same material and have the same shape, but differ only in size, requires a discussion more clever than that by which one explains how a more expanded shape or an opposing motion of the medium retards the speed of the moving body. The solution of the present problem lies, I think, in the roughness and porosity which are generally and almost necessarily found in the surfaces of solid bodies. When the body is in motion these rough places strike the air or other ambient medium. The evidence for this is found in the humming which accompanies the rapid motion of a body through air, even when that body is as round as possible. One hears not only humming, but also hissing and whistling, whenever there is any appreciable cavity or elevation upon the body. We observe also that a round solid body rotating in a lathe produces a current of air. But what more do we need ? When a top spins on the ground at its greatest speed do we not hear a distinct buzzing of high pitch? This sibilant note diminishes in pitch as the speed of rotation slackens, which is evidence that these small rugosities on the surface meet resistance in the air. There can be no doubt, therefore, that in the motion of falling bodies these rugosities strike the surrounding fluid and retard the speed; and this they do so much the more in proportion as the surface is larger, which is the case of small bodies as compared with greater.

SIMP. Stop a moment please, I am getting confused. For although I understand and admit that f riction of the medium upon the surface of the body retards its motion and that, if other things are the same, the larger surface suffers greater retardation, I do not see on what ground you say that the surface of the smaller body is larger. Besides if, as you say, the larger surface suffers greater retardation the larger solid should move more slowly, which is not the fact. But this objection can be easily met

FIRST DAY 89 met by saying that, although the larger body has a larger surface, it has also a greater weight, in comparison with which the resistance of the larger surface is no more than the resistance of the small surface in comparison with its smaller weight; so that the speed of the larger solid does not become less. I therefore see no reason for expecting any difference of speed so long as the driving weight \gravita movente] diminishes in the same propor-

[133] tion as the retarding power [facolta ritardante] of the surface.

SALV. I shall answer all your objections at once. You will admit, of course, Simplicio, that if one takes two equal bodies, of the same material and same figure, bodies which would therefore fall with equal speeds, and if he diminishes the weight of one of them in the same proportion as its surface (maintaining the similarity of shape) he would not thereby diminish the speed of this body.

SIMP. This inference seems to be in harmony with your theory which states that the weight of a body has no effect in either accelerating or retarding its motion.

SALV. I quite agree with you in this opinion from which it appears to follow that, if the weight of a body is diminished in greater proportion than its surface, the motion is retarded to a certain extent; and this retardation is greater and greater in proportion as the diminution of weight exceeds that of the surface.

SIMP. This I admit without hesitation.

SALV. Now you must know, Simplicio, that it is not possible to diminish the surface of a solid body in the same ratio as the weight, and at the same time maintain similarity of figure. For since it is clear that in the case of a diminishing solid the weight grows less in proportion to the volume, and since the volume always diminishes more rapidly than the surface, when the same shape is maintained, the weight must therefore diminish more rapidly than the surface. But geometry teaches us that, in the case of similar solids, the ratio of two volumes is greater than the ratio of their surfaces; which, for the sake of better understanding, I shall illustrate by a particular case.

Take,

90 THE TWO NEW SCIENCES OF GALILEO

Take, for example, a cube two inches on a side so that each face has an area of four square inches and the total area, i. e., the sum of the six faces, amounts to twenty-four square inches; now imagine this cube to be sawed through three times so as to divide it into eight smaller cubes, each one inch on the side, each face one inch square, and the total surface of each cube six square inches instead of twenty-four as in the case of the larger

cube. It is evident therefore that the surface of the little cube is only one-fourth that of the larger, namely, the ratio of six to twenty-four; but the volume of the solid cube itself is only oneeighth; the volume, and hence also the weight, diminishes therefore much more rapidly than the surface. If we again divide the little cube into eight others we shall have, for the total surface of one of these, one and one-half square inches, which is onesixteenth of the surface of the original cube; but its volume is only one-sixty-fourth part. Thus, by two divisions, you see that the volume is diminished four times as much as the surface. And, if the subdivision be continued until the original solid be reduced to a fine powder, we shall find that the weight of one of these smallest particles has diminished hundreds and hundreds of times as much as its surface. And this which I have illustrated in the case of cubes holds also in the case of all similar solids, where the volumes stand in sesquialteral ratio to their surfaces. Observe then how much greater the resistance, arising from contact of the surface of the moving body with the medium, in the case of small bodies than in the case of large; and when one considers that the rugosities on the very small surfaces of fine dust particles are perhaps no smaller than those on the surfaces of larger solids which have been carefully polished, he will see how important it is that the medium should be very fluid and offer no resistance to being thrust aside, easily yielding to a small force. You see, therefore, Simplicio, that I was not mistaken when, not long ago, I said that the surface of a small solid is comparatively greater than that of a large one.

SIMP. I am quite convinced; and, believe me, if I were again beginning my studies, I should follow the advice of Plato and start

FIRST DAY 91 start with mathematics, a science which proceeds very cautiously md admits nothing as established until it has been rigidly demonstrated.

SAGR. This discussion has afforded me great pleasure; but Before proceeding further I should like to hear the explanation of i phrase of yours which is new to me, namely, that similar solids ire to each other in the sesquialteral ratio of their surfaces; for although I have seen and understood the proposition in which it s demonstrated that the surfaces of similar solids are in the

luplicate ratio of their sides and also the proposition which Droves that the volumes are in the triplicate ratio of their sides, ^et I have not so much as heard mentioned the ratio of the /olume of a solid to its surface.

SALV. You yourself have suggested the answer to your question and have removed every doubt. For if one quantity is the cube of something of which another quantity is the square does it not follow that the cube is the sesquialteral of the square? Surely. Now if the surface varies as the square of its linear dimensions while the volume varies as the cube of these dimensions may we not say that the volume stands in sesquialteral ratio to the surface?

SAGR. Quite so. And now although there are still some details, in connection with the subject under discussion, con- :erning which I might ask questions yet, if we keep making one digression after another, it will be long before we reach the main topic which has to do with the variety of properties found in the resistance which solid bodies offer to fracture; and, therefore, if you please, let us return to the subject which we originally proposed to discuss.

SALV. Very well; but the questions which we have already considered are so numerous and so varied, and have taken up so much time that there is not much of this day left to spend upon our main topic which abounds in geometrical demonstrations calling for careful consideration. May I, therefore, suggest that we postpone the meeting until to-morrow, not only for the reason just mentioned but also in order that I may bring with me

92 THE TWO NEW SCIENCES OF GALILEO me some papers in which I have set down in an orderly way the theorems and propositions dealing with the various phases of this subject, matters which, from memory alone, I could not present in the proper order.

SAGR. I fully concur in your opinion and all the more willingly because this will leave time to-day to take up some of my difficulties with the subject which we have just been discussing. One question is whether we are to consider the resistance of the medium as sufficient to destroy the acceleration of a body of very heavy material, very large volume, and spherical figure. I say spherical in order to select a volume which is contained within a minimum surface and therefore less subject to retardation.

Another question deals with the vibrations of pendulums which may be regarded from several viewpoints; the first is whether all vibrations, large, medium, and small, are performed in exactly and precisely equal times : another is to find the ratio of the times of vibration of pendulums supported by threads of unequal length.

SALV. These are interesting questions : but I fear that here, as in the case of all other facts, if we take up for discussion any one of them, it will carry in its wake so many other facts and curious consequences that time will not remain to-day for the discussion of all.

SAGR. If these are as full of interest as the foregoing, I would gladly spend as many days as there remain hours between now and nightfall; and I dare say that Simplicio would not be wearied by these discussions.

SIMP. Certainly not; especially when the questions pertain to natural science and have not been treated by other philosophers.

SALV. Now taking up the first question, I can assert without hesitation that there is no sphere so large, or composed of material so dense but that the resistance of the medium, although very slight, would check its acceleration and would, in time reduce its motion to uniformity; a statement which is strongly

FIRST DAY 93 strongly supported by experiment. For if a falling body, as time goes on, were to acquire a speed as great as you please, no such speed, impressed by external forces [motore esterno], can be so great but that the body will first acquire it and then, owing to the resisting medium, lose it. Thus, for instance, if a cannon ball, having fallen a distance of four cubits through the air and having acquired a speed of, say, ten units [gradi\ were to strike the surface of the water, and if the resistance of the water were not able to check the momentum [impeto] of the shot, it would either increase in speed or maintain a uniform motion until the bottom were reached: but such is not the observed fact; on the contrary, the water when only a few cubits deep hinders and diminishes the motion in such a way that the shot delivers to the bed of the river or lake a very slight impulse. Clearly

then if a short fall through the water is sufficient to deprive a cannon ball of its speed, this speed cannot be regained by a fall of even a thousand cubits. How could a body acquire, in a fall of a thousand cubits, that which it loses in a fall of four? But what more is needed? Do we not observe that the enormous momentum, delivered to a shot by a cannon, is so deadened by passing through a few cubits of water that the ball, so far from injuring the ship, barely strikes it? Even the air, although a very yielding medium, can also diminish the speed of a falling body, as may be easily understood from similar experiments. For if a gun be fired downwards from the top of a very high tower the shot will make a smaller impression upon the ground than if the gun had been fired from an elevation of only four or six cubits; this is clear evidence that the momentum of the ball, fired from the top of the ^ower, diminishes continually from the instant it leaves the barrel until it reaches the ground. Therefore a fall from ever so great an altitude will not suffice to give to a body that momentum which it has once lost through the resistance of the air, no matter how it was originally acquired. In like manner, the destructive effect produced upon a wall by a shot fired from a gun at a distance of twenty cubits cannot be duplicated by the fall of the same shot from any altitude however

94 THE TWO NEW SCIENCES OF GALILEO ever great. My opinion is, therefore, that under the circumstances which occur in nature, the acceleration of any body falling from rest reaches an end and that the resistance of the medium finally reduces its speed to a constant value which is thereafter maintained.

SAGR. These experiments are in my opinion much to the purpose; the only question is whether an opponent might not make bold to deny the fact in the case of bodies \moli] which are very large and heavy or to assert that a cannon ball, falling from the distance of the moon or from the upper regions of the atmosphere, would deliver a heavier blow than if just leaving the muzzle of the gun.

SALV. No doubt many objections may be raised not all of which can be refuted by experiment: however in this particular

case the following consideration must be taken into account, namely, that it is very likely that a heavy body falling from a height will, on reaching the ground, have acquired just as much momentum as was necessary to carry it to that height; as may be clearly seen in the case of a rather heavy pendulum which, when pulled aside fifty or sixty degrees from the vertical, will acquire precisely that speed and force which are sufficient to carry it to an equal elevation save only that small portion which it loses through friction on the air. In order to place a cannon ball at such a height as might suffice to give it just that momentum which the powder imparted to it on leaving the gun we need only fire it vertically upwards from the same gun; and we can then observe whether on falling back it delivers a blow equal to that of the gun fired at close range; in my opinion it would be much weaker. The resistance of the air would, therefore, I think, prevent the muzzle velocity from being equalled by a natural fall from rest at any height whatsoever.

We come now to the other questions, relating to pendulums, a subject which may appear to many exceedingly arid, especially to those philosophers who are continually occupied with the more profound questions of nature. Nevertheless, the problem is one which I do not scorn. I am encouraged by the example

FIRST DAY 95 example of Aristotle whom I admire especially because he did not fail to discuss every subject which he thought in any degree worthy of consideration.

Impelled by your queries I may give you some of my ideas concerning certain problems in music, a splendid subject, upon which so many eminent men have written: among these is Aristotle himself who has discussed numerous interesting acoustical questions. Accordingly, if on the basis of some easy and tangible experiments, I shall explain some striking phenomena in the domain of sound, I trust my explanations will meet your approval.

SAGR. I shall receive them not only gratefully but eagerly. For, although I take pleasure in every kind of musical instru-

ment and have paid considerable attention to harmony, I have never been able to fully understand why some combinations of tones are more pleasing than others, or why certain combinations not only fail to please but are even highly offensive. Then there is the old problem of two stretched strings in unison; when one of them is sounded, the other begins to vibrate and to emit its note; nor do I understand the different ratios of harmony [forme delle consonanze] and some other details.

SALV. Let us see whether we cannot derive from the pendulum a satisfactory solution of all these difficulties. And first, as to the question whether one and the same pendulum really performs its vibrations, large, medium, and small, all in exactly the same time, I shall rely upon what I have already heard from our Academician. He has clearly shown that the time of descent is the same along all chords, whatever the arcs which subtend them, as well along an arc of 1 80° (i. e., the whole diameter) as along one of 100°, 60°, 10°, 2°, ^°> or 4'. It is understood, of course, that these arcs all terminate at the lowest point of the circle, where it touches the horizontal plane.

If now we consider descent along arcs instead of their chords then, provided these do not exceed 90°, experiment shows that they are all traversed in equal times; but these times are greater for the chord than for the arc, an effect which is all the more remarkable

96 THE TWO NEW SCIENCES OF GALILEO remarkable because at first glance one would think just the opposite to be true. For since the terminal points of the two motions are the same and since the straight line included between these two points is the shortest distance between them, it would seem reasonable that motion along this line should be executed in the shortest time; but this is not the case, for the shortest time — and therefore the most rapid motion — is that employed along the arc of which this straight line is the chord.

As to the times of vibration of bodies suspended by threads of different lengths, they bear to each other the same proportion as the square roots of the lengths of the thread; or one might say the lengths are to each other as the squares of the times; so that if one wishes to make the vibration-time of one pendulum twice that of another, he must make its suspension four times as long. In like manner, if one pendulum has a suspension nine times as

long as another, this second pendulum will execute three vibrations during each one of the first; from which it follows that the lengths of the suspending cords bear to each other the [inverse] ratio of the squares of the number of vibrations performed in the same time.

SAGR. Then, if I understand you corredlly, I can easily measure the length of a string whose upper end is attached at any height whatever even if this end were invisible and I could see only the lower extremity. For if I attach to the lower end of this string a rather heavy weight and give it a to-and-fro motion, and if I ask a friend to count a number of its vibrations, while I, during the same time-interval, count the number of vibrations of a pendulum which is exactly one cubit in length, then knowing the number of vibrations which each pendulum makes in the given interval of time one can determine the length of the string. Suppose, for example, that my friend counts 20 vibrations of the long cord during the same time in which I count 240 of my string which is one cubit in length; taking the squares of the two numbers, 20 and 240, namely 400 and 57600, then, I say, the long string contains 57600 units of such length that my pendulum will contain 400 of them; and since the length of my

FIRST DAY 97 my string is one cubit, I shall divide 57600 by 400 and thus obtain 144. Accordingly I shall call the length of the string 144 cubits.

SALV. Nor will you miss it by as much as a hand's breadth, especially if you observe a large number of vibrations.

SAGR. You give me frequent occasion to admire the wealth and profusion of nature when, from such common and even trivial phenomena, you derive facts which are not only striking and new but which are often far removed from what we would have imagined. Thousands of times I have observed vibrations especially in churches where lamps, suspended by long cords, had been inadvertently set into motion; but the most which I could infer from these observations was that the view of those who think that such vibrations are maintained by the medium is highly improbable: for, in that case, the air must needs have considerable judgment and little else to do but kill time by pushing to and fro a pendent weight with perfect regularity. But I never dreamed of learning that one and the same body, when

[Hi]

suspended from a string a hundred cubits long and pulled aside through an arc of 90° or even i° or ^2°, would employ the same time in passing through the least as through the largest of these arcs; and, indeed, it still strikes me as somewhat unlikely. Mow I am waiting to hear how these same simple phenomena can \irnish solutions for those acoustical problems — solutions which will be at least partly satisfactory.

SALV. First of all one must observe that each pendulum has ts own time of vibration so definite and determinate that t is not possible to make it move with any other period [altro periodo] than that which nature has given it. For let any one :ake in his hand the cord to which the weight is attached and :ry, as much as he pleases, to increase or diminish the frequency frequenza] of its vibrations; it will be time wasted. On the other hand, one can confer motion upon even a heavy pendulum which is at rest by simply blowing against it; by repeating these Dlasts with a frequency which is the same as that of the pendulum one can impart considerable motion. Suppose that by the first

98 THE TWO NEW SCIENCES OF GALILEO first puff we have displaced the pendulum from the vertical b; say, half an inch; then if, after the pendulum has returned and about to begin the second vibration, we add a second puff, w shall impart additional motion; and so on with other bias provided they are applied at the right instant, and not when tl: pendulum is coming toward us since in this case the blast woul impede rather than aid the motion. Continuing thus wit many impulses [impulsi\ we impart to the pendulum sue momentum [impeto] that a greater impulse [forza] than that of single blast will be needed to stop it.

SAGR. Even as a boy, I observed that one man alone h giving these impulses at the right instant was able to ring bell so large that when four, or even six, men seized the rof and tried to stop it they were lifted from the ground, all < them together being unable to counterbalance the momentu] which a single man, by properly-timed pulls, had given it.

SALV. Your illustration makes my meaning clear and is quil as well fitted, as what I have just said, to explain the wonderfi phenomenon of the strings of the cittern [cetera] or of the spin<

[cimbalo], namely, the fact that a vibrating string will s< another string in motion and cause it to sound not only when tl: latter is in unison but even when it differs from the former by a octave or a fifth. A string which has been struck begins 1 vibrate and continues the motion as long as one hears tit sound [risonanza]; these vibrations cause the immediately su: rounding air to vibrate and quiver; then these ripples in the a expand far into space and strike not only all the strings of tli same instrument but even those of neighboring instrument; Since that string which is tuned to unison with the one plucke is capable of vibrating with the same frequency, it acquire: at the first impulse, a slight oscillation; after receiving tw< three, twenty, or more impulses, delivered at proper interval; it finally accumulates a vibratory motion equal to that of th plucked string, as is clearly shown by equality of amplitude i their vibrations. This undulation expands through the air an sets into vibration not only strings, but also any other bod whic

FIRST DAY 99 which happens to have the same period as that of the plucked string. Accordingly if we attach to the side of an instrument small pieces of bristle or other flexible bodies, we shall observe that, when a spinet is sounded, only those pieces respond that have the same period as the string which has been struck; the remaining pieces do not vibrate in response to this string, nor do the former pieces respond to any other tone.

If one bows the base string on a viola rather smartly and brings near it a goblet of fine, thin glass having the same tone [tuono] as that of the string, this goblet will vibrate and audibly resound. That the undulations of the medium are widely dispersed about the sounding body is evinced by the fact that a glass of water may be made to emit a tone merely by the friction of the finger-tip upon the rim of the glass; for in this water is produced a series of regular waves. The same phenomenon is observed to better advantage by fixing the base of the goblet upon the bottom of a rather large vessel of water filled nearly to the edge of the goblet; for if, as before, we sound the glass by friction of the finger, we shall see ripples spreading with the utmost regularity and with high speed to large distances about the glass. I have often remarked, in thus sounding a rather

large glass nearly full of water, that at first the waves are spaced with great uniformity, and when, as sometimes happens, the tone of the glass jumps an octave higher I have noted that at this moment each of the aforesaid waves divides into two; a phenomenon which shows clearly that the ratio involved in the octave [forma delV ottava] is two.

SAGR. More than once have I observed this same thing, much to my delight and also to my profit. For a long time I have Deen perplexed about these different harmonies since the explanations hitherto given by those learned in music impress me as not sufficiently conclusive. They tell us that the diapason, i. e. the octave, involves the ratio of two, that the diapente which we call the fifth involves a ratio of 3 :2, etc. ; because if the open string of a monochord be sounded and afterwards a bridge be placed in the middle and the half length be sounded one ioo THE TWO NEW SCIENCES OF GALILEO one hears theocftave; and if the bridge be placed at 1/3 the lengt of the string, then on plucking first the open string and afte wards 2/3 of its length the fifth is given ; for this reason they sa that the octave depends upon the ratio of two to one [coi tenuta tra'l due e Vuno\ and the fifth upon the ratio of three t two. This explanation does not impress me as sufficient t establish 2 and 3/2 as the natural ratios of theocftave and thefiftl and my reason for thinking so is as follows. There are thrc different ways in which the tone of a string may be sharpenec namely, by shortening it, by stretching it and by making thinner. If the tension and size of the string remain constar one obtains the odlave by shortening it to one-half, i. e., by sounc ing first the open string and then one-half of it; but if length an size remain constant and one attempts to produce the octave b stretching he will find that it does not suffice to double th stretching weight; it must be quadrupled; so that, if the fundc mental note is produced by a weight of one pound, four will b required to bring out the octave.

And finally if the length and tension remain constant, whii one changes the size * of the string he will find that in order t produce the odlave the size must be reduced to X tnat wnic gave the fundamental. And what I have said concerning th octave, namely, that its ratio as derived from the tension an size of the string is the square of that derived from the lengtl applies equally well to all other musical intervals [interval musici\. Thus if one wishes to produce a fifth by changing th length he finds that the ratio of the lengths must be sesquialtera in other words he sounds first the open string, then two-third of it; but if he wishes to produce this same result by stretching c thinning the string then it becomes necessary to square th ratio 3/2 that is by taking 9/4 [dupla sesquiquarta]; according!} if the fundamental requires a weight of 4 pounds, the highe note will be produced not by 6, but by 9 pounds; the same i true in regard to size, the string which gives the fundamental i larger than that which yields the fifth in the ratio of 9 to 4 In view of these facts, I see no reason why those wise philos * For the exact meaning of "size" see p. 103 below. [Trans.]

FIRST DAY 101 ophers should adopt 2 rather than 4 as the ratio of the octave, or why in the case of the fifth they should employ the sesquialteral ratio, 3/2, rather than that of 9/4. Since it is impossible to count the vibrations of a sounding string on account of its high frequency, I should still have been in doubt as to whether a string, emitting the upper octave, made twice as many vibrations in the same time as one giving the fundamental, had it not been for the following fact, namely, that at the instant when the tone jumps to the octave, the waves which constantly accompany the vibrating glass divide up into smaller ones which are precisely half as long as the former.

SALV. This is a beautiful experiment enabling us to distinguish individually the waves which are produced by the vibrations of a sonorous body, which spread through the air, bringing to the tympanum of the ear a stimulus which the mind translates into sound. But since these waves in the water last only so long as the friction of the finger continues and are, even then, not constant but are always forming and disappearing, would it not be a fine thing if one had the ability to produce waves which would persist for a long while, even months and years, so as to easily measure and count them?

SAGR. Such an invention would, I assure you, command my admiration.

SALV. The device is one which I hit upon by accident; my part consists merely in the observation of it and in the appreciation of its value as a confirmation of something to which I had given profound consideration; and yet the device is, in itself, rather common. As I was scraping a brass plate with a sharp iron

chisel in order to remove some spots from it and was running the chisel rather rapidly over it, I once or twice, during many strokes, heard the plate emit a rather strong and clear whistling sound; on looking at the plate more carefully, I noticed a long row of fine streaks parallel and equidistant from one another. Scraping with the chisel over and over again, I noticed that it was only when the plate emitted this hissing noise that any marks were left upon it; when the scraping was not accompanied by

102 THE TWO NEW SCIENCES OF GALILEO by this sibilant note there was not the least trace of such marks. Repeating the trick several times and making the stroke, now with greater now with less speed, the whistling followed with a pitch which was correspondingly higher and lower. I noted also that the marks made when the tones were higher were closer together; but when the tones were deeper, they were farther apart. I also observed that when, during a single stroke, the speed increased toward the end the sound became sharper and the streaks grew closer together, but always in such a way as to remain sharply defined and equidistant. Besides whenever the stroke was accompanied by hissing I felt the chisel tremble in my grasp and a sort of shiver run through my hand. In short we see and hear in the case of the chisel precisely that which is seen and heard in the case of a whisper followed by a loud voice; for, when the breath is emitted without the production of a tone, one does not feel either in the throat or mouth any motion to speak of in comparison with that which is felt in the larynx and upper part of the throat when the voice is used, especially when the tones employed are low and strong.

At times I have also observed among the strings of the spinet two which were in unison with two of the tones produced by the aforesaid scraping; and among those which differed most in pitch I found two which were separated by an interval of a perfect fifth. Upon measuring the distance between the markings produced by the two scrapings it was found that the space which contained 45 of one contained 30 of the other, which is precisely the ratio assigned to the fifth.

But now before proceeding any farther I want to call your attention to the fact that, of the three methods for sharpening a tone, the one which you refer to as the fineness of the string should be attributed to its weight. So long as the material of

the string is unchanged, the size and weight vary in the same ratio. Thus in the case of gut-strings, we obtain the octave by making one string 4 times as large as the other; so also in the case of brass one wire must have 4 times the size of the other; but if now we wish to obtain the octave of a gut-string, by use of brass

FIRST DAY 103 brass wire, we must make it, not four times as large, but four times as heavy as the gut-string: as regards size therefore the metal string is not four times as big but four times as heavy. The wire may therefore be even thinner than the gut notwithstanding the fact that the latter gives the higher note. Hence if two spinets are strung, one with gold wire the other with brass, and if the corresponding strings each have the same length, diameter, and tension it follows that the instrument strung with gold will have a pitch about one-fifth lower than the other because gold has a density almost twice that of brass. And here it is to be noted that it is the weight rather than the size of a moving body which offers resistance to change of motion [velocita del moto] contrary to what one might at first glance think. For it seems reasonable to believe that a body which is large and light should suffer greater retardation of motion in thrusting aside the medium than would one which is thin and heavy; yet here exactly the opposite is true.

Returning now to the original subject of discussion, I assert that the ratio of a musical interval is not immediately determined either by the length, size, or tension of the strings but rather by the ratio of their frequencies, that is, by the number of pulses of air waves which strike the tympanum of the ear, causing it also to vibrate with the same frequency. This fact established, we may possibly explain why certain pairs of notes, differing in pitch produce a pleasing sensation, others a less pleasant effect, and still others a disagreeable sensation. Such an explanation would be tantamount to an explanation of the more or less perfect consonances and of dissonances. The unpleasant sensation produced by the latter arises, I think, from the discordant vibrations of two different tones which strike the ear out of time [sproporzionatamente]. Especially harsh is the dissonance between notes whose frequencies are incommensurable; such a case occurs when one has two strings in unison and sounds one of them open, together with a part of the other

which bears the same ratio to its wnole length as the side of a square bears to the diagonal; this yields a dissonance similar to

104 THE TWO NEW SCIENCES OF GALILEO to the augmented fourth or diminished fifth [tritono o semidiapente].

Agreeable consonances are pairs of tones which strike the ear with a certain regularity; this regularity consists in the fact that the pulses delivered by the two tones, in the same interval of time, shall be commensurable in number, so as not to keep the ear drum in perpetual torment, bending in two different directions in order to yield to the ever-discordant impulses.

The first and most pleasing consonance is, therefore, the odtave since, for every pulse given to the tympanum by the lower string, the sharp string delivers two; accordingly at every other vibration of the upper string both pulses are delivered simultaneously so that one-half the entire number of pulses are delivered in unison. But when two strings are in unison their vibrations always coincide and the effect is that of a single string; hence we do not refer to it as consonance. The fifth is also a pleasing interval since for every two vibrations of the lower string the upper one gives three, so that considering the entire number of pulses from the upper string one-third of them will strike in unison, i. e., between each pair of concordant vibrations there intervene two single vibrations; and when the interval is a fourth, three single vibrations intervene. In case the interval is a second where the ratio is 9/8 it is only every ninth vibration of the upper string which reaches the ear simultaneously with one of the lower; all the others are discordant and produce a harsh effect upon the recipient ear which interprets them as dissonances.

SIMP. Won't you be good enough to explain this argument a little more clearly?

SALV. Let AB denote the length of a wave [lo spazio e la dilatazione d'una vibrazione] emitted by the lower string and CD that of a higher string which is emitting the odlave of AB; divide AB in the middle at E. If the two strings begin their motions at A and C, it is clear that when the sharp vibration has reached the end D, the other vibration will have travelled only as far as E, which, not being a terminal point, will emit no pulse; but there is a blow delivered at D. Accordingly when the one wave

FIRST DAY 105 wave comes back from D to C, the other passes on from E to B ; hence the two pulses from B and C strike the drum of the ear simultaneously. Seeing that these vibrations are repeated again and again in the same manner, we con- jv, E B elude that each alternate pulse from CD falls ' ' « in unison with one from AB. But each of the , - L

[I48] C D pulsations at the terminal points, A and B, is A £ 0 B constantly accompanied by one which leaves al- * ' ' ' ways from C or always from D. This is clear t i i because if we suppose the waves to reach A and c m D C at the same instant, then, while one wave Fig. 13 travels from A to B, the other will proceed from C to D and back to C, so that waves strike at C and B simultaneously; during the passage of the wave from B back to A the disturbance at C goes to D and again returns to C, so that once more the pulses at A and C are simultaneous.

Next let the vibrations AB and CD be separated by an interval of a fifth, that is, by a ratio of 3/2; choose the points E and O such that they will divide the wave length of the lower string into three equal parts and imagine the vibrations to start at the same instant from each of the terminals A and C. It is evident that when the pulse has been delivered at the terminal D, the wave in AB has travelled only as far as O; the drum of the ear receives, therefore, only the pulse from D. Then during the return of the one vibration from D to C, the other will pass from O to B and then back to O, producing an isolated pulse at B— a pulse which is out of time but one which must be taken into consideration.

Now since we have assumed that the first pulsations started from the terminals A and C at the same instant, it follows that the second pulsation, isolated at D, occurred after an interval of time equal to that required for passage from C to D or, what is the same thing, from A to O; but the next pulsation, the one at B, is separated from the preceding by only half this interval, namely, the time required for passage from O to B. Next while the one vibration travels from O to A, the other travels from C to

106 THE TWO NEW SCIENCES OF GALILEO

D, the result of which is that two pulsations occur simultaneously at A and D. Cycles of this kind follow one after another, i. e., one solitary pulse of the lower string interposed between two solitary pulses of the upper string. Let us no\\ imagine time to be divided into very small equal intervals then if we assume that, during the first two of these intervals, the disturbances which occurred simultaneously at A and C have travelled as far as O and D and have produced a pulse at D; anc if we assume that during the third and fourth intervals one disturbance returns from D to C, producing a pulse at C, while the other, passing on from O to B and back to O, produces i pulse at B ; and if finally, during the fifth and sixth intervals, the disturbances travel from O and C to A and D, producing c pulse at each of the latter two, then the sequence in which the pulses strike the ear will be such that, if we begin to count time from any instant where two pulses are simultaneous, the ea] drum will, after the lapse of two of the said intervals, receive c solitary pulse; at the end of the third interval, another solitary pulse; so also at the end of the fourth interval; and two intervals later, i. e., at the end of the sixth interval, will be hearc two pulses in unison. Here ends the cycle — the anomaly, so tc speak — which repeats itself over and over again.

SAGR. I can no longer remain silent; for I must express to yoi; the great pleasure I have in hearing such a complete explanatior of phenomena with regard to which I have so long been ir darkness. Now I understand why unison does not differ from z single tone; I understand why the ocftave is the principal harmony, but so like unison as often to be mistaken for it and alsc why it occurs with the other harmonies. It resembles unisor because the pulsations of strings in unison always occur simultaneously, and those of the lower string of the octave are always accompanied by those of the upper string; and among the lattei is interposed a solitary pulse at equal intervals and in such a manner as to produce no disturbance; the result is that such a harmony is rather too much softened and lacks fire. But the fifth is characterized by its displaced beats and by the interposition

FIRST DAY 107 tion of two solitary beats of the upper string and one solitary beat of the lower string between each pair of simultaneous pulses; these three solitary pulses are separated by intervals of time equal to half the interval which separates each pair of simultaneous beats from the solitary beats of the upper string. Thus the effect of the fifth is to produce a tickling of the ear drum such that its softness is modified with sprightliness, giving at the same moment the impression of a gentle kiss and of a bite.

SALV. Seeing that you have derived so much pleasure from these novelties, I must show you a method by which the eye may enjoy the same game as the ear. Suspend three balls of lead, or other heavy material, by means of strings of different length such that while the longest makes two vibrations the shortest will make four and the medium three; this will take place when the longest string measures 16, either in hand breadths or in any other unit, the medium 9 and the shortest 4, all measured in the same unit.

Now pull all these pendulums aside from the perpendicular and release them at the same instant; you will see a curious interplay of the threads passing each other in various manners but such that at the completion of every fourth vibration of the longest pendulum, all three will arrive simultaneously at the same terminus, whence they start over again to repeat the same cycle. This combination of vibrations, when produced on strings is precisely that which yields the interval of the octave and the intermediate fifth. If we employ the same disposition

of apparatus but change the lengths of the threads, always however in such a way that their vibrations correspond to those of agreeable musical intervals, we shall see a different crossing of these threads but always such that, after a definite interval of time and after a definite number of vibrations, all the threads, whether three or four, will reach the same terminus at the same instant, and then begin a repetition of the cycle.

If however the vibrations of two or more strings are incommensurable so that they never complete a definite number of vibrations at the same instant, or if commensurable they return only io8 THE TWO NEW SCIENCES OF GALILEO only after a long interval of time and after a large number oi vibrations, then the eye is confused by the disorderly succession of crossed threads. In like manner the ear is pained by an irregular sequence of air waves which strike the tympanum without any fixed order.

But, gentlemen, whither have we drifted during these many hours lured on by various problems and unexpected digressions ? The day is already ended and we have scarcely touched the subject proposed for discussion. Indeed we have deviated so far that I remember only with difficulty our early introduction and the little progress made in the way of hypotheses and principles for use in later demonstrations.

SAGR. Let us then adjourn for to-day in order that our minds may find refreshment in sleep and that we may return tomorrow, if so please you, and resume the discussion of the main question.

SALV. I shall not fail to be here to-morrow at the same hour, hoping not only to render you service but also to enjoy your company.

END OF THE FIRST DAY.