Dialogues Concerning Two New Sciences (Second to Fourth Days)

Summary

Dialogues Concerning Two New Sciences (Discorsi e dimostrazioni matematiche intorno a due nuove scienze) was printed at Leiden by the Elzevirs in 1638, since Galileo, condemned in 1633 and forbidden to publish, could not issue it in Italy. It gathers work on mechanics and motion that he had pursued since his years at Padua. The book is set in Venice over four days of conversation among the same three speakers as his earlier Dialogue on the world systems: Salviati, who speaks for 'our Academician' (Galileo); Sagredo, a quick and curious Venetian gentleman; and Simplicio, a well-read follower of Aristotle. This record covers the Second, Third and Fourth Days and the Appendix; the First Day appears in the library with Galileo's selected writings. The two new sciences are the strength of materials and the science of local motion. The Second Day treats the resistance of beams and rods to breaking. Starting from the lever, Salviati shows how a cantilever's strength depends on its length and cross-section, and draws the startling conclusion that similar structures are not equally strong: a large machine, building or animal is relatively weaker than a small one of the same shape. Hence giants would need monstrously thick bones, a small dog can carry several dogs its size, and nature makes the bones of birds and the stalks of grain hollow to gain strength without weight. The Third Day, in which the speakers read and discuss a Latin treatise by the Academician, defines uniform motion and then naturally accelerated motion, in which speed increases in proportion to time. Salviati declines to investigate the cause of acceleration and concentrates on describing it mathematically. He proves that distances fallen from rest increase as the square of the time, so that successive equal intervals cover distances in the ratio of the odd numbers, and he describes an inclined-plane experiment, timed with a water clock and repeated 'a full hundred times', that agreed with the law. Many propositions on motion along inclined planes and chords follow. The Fourth Day combines uniform horizontal motion with accelerated fall to show that a projectile's path is a parabola, works out tables of ranges and heights for different angles of elevation, and confirms the gunners' rule that the greatest range comes at 45 degrees. Galileo admits that air resistance disturbs these motions in ways no exact science can capture. The book is demanding, mixing conversation with Euclidean proofs in the style of the ancient geometers, but its dialogue makes the reasoning visible: errors are voiced, tested and corrected. Students can watch a new kind of natural philosophy take shape, one that idealizes, measures and demonstrates, and can ask what it gains and what it deliberately leaves aside when it sets questions of cause to one side.

Plot or Argument

• Second Day: the lever and the breaking of beams

  Sagredo reminds Salviati of the previous day's discussion of what holds solids together. Salviati turns to the resistance of solids to bending, granting Aristotle priority on the lever but giving Archimedes the first place for rigor of demonstration. He proves a version of the law of the lever and applies it to a beam fixed in a wall and loaded at its end, deriving how resistance to fracture varies with length, breadth and thickness.

• Second Day: why size matters

  Salviati shows that among similar solids the ratio of strength to weight is never the same, so that larger structures are relatively weaker and may collapse under their own weight. He applies this to ships, buildings, trees and animals: a giant would need disproportionately thick bones, while smaller bodies are relatively stronger. Simplicio raises the whale, and Salviati answers that water supports the bodies of fish.

• Second Day: beams of equal resistance and hollow solids

  Further propositions find the shape of a beam equally strong throughout its length, which turns out to be parabolic, and show how to draw a parabola. The day ends with hollow solids: bones of birds, reeds, straw and hollow lances are stronger than solid ones of the same weight and length, with the resistances in proportion to their diameters.

• Third Day: uniform motion

  Salviati reads the Academician's Latin treatise, which divides the subject into uniform motion, naturally accelerated motion and projectiles. Uniform motion is defined as equal distances in any equal times, and a series of theorems relates distances, times and speeds.

• Third Day: naturally accelerated motion and the law of fall

  The treatise defines natural acceleration as equal increments of speed in equal times. Salviati sets aside the dispute about the cause of acceleration. Sagredo proposes that speed grows with distance fallen; Salviati shows this to be impossible and remarks that the Author once held the same opinion. Theorem II proves that distances from rest are as the squares of the times, with successive intervals in the ratio 1, 3, 5, 7. Salviati describes the grooved-board experiment with a bronze ball and a water clock that confirmed the law on many inclinations.

• Third Day: inclined planes and chords

  A long sequence of theorems and problems treats descent along inclined planes and chords of a circle, including the equal time of descent along all chords ending at the lowest point of a vertical circle. Along the way Salviati states that motion on a horizontal plane, once imparted, would be maintained if causes of acceleration and retardation were removed. Sagredo praises the Academician for founding a new science on a very old subject.

• Fourth Day: the parabola of projectiles

  The treatise combines uniform horizontal motion with naturally accelerated fall and proves that the path is a semi-parabola. An objection follows: the proof assumes parallel lines of fall and ignores the medium. Salviati concedes that the results hold for projectiles near the earth's surface and that air resistance disturbs them in an infinite variety of ways, though by small amounts for heavy shot fired at moderate speeds.

• Fourth Day: tables of range and the Appendix

  Salviati derives the speed needed for a given range, proves that range is greatest at an elevation of 45 degrees and equal for elevations equally above and below it, and presents tables of altitudes and ranges. Discussion turns to the force of percussion and to a hanging chain, which Salviati says fits a parabola almost perfectly, giving a quick way to draw one. The day closes with a request to keep the book for further study, and an appendix of earlier theorems on centers of gravity of solids ends the volume.

Characters

• Salviati, speaker; spokesman for the Academician

  Named for Galileo's Florentine friend Filippo Salviati (died 1614). He expounds and defends the Academician's demonstrations, reads the Latin treatise aloud, and corrects others' errors patiently.

• Sagredo, speaker; intelligent layman

  Named for the Venetian nobleman Giovanfrancesco Sagredo (died 1620). Quick, curious and sometimes mistaken, he proposes the plausible but false rule that speed grows with distance fallen and delights in mathematical proof.

• Simplicio, speaker; Aristotelian

  A learned follower of Aristotle, named after the ancient commentator Simplicius. In these days he is more willing than in Galileo's earlier Dialogue to learn geometry, admitting he has studied 'a little Euclid' to follow the proofs.

• The Academician, absent author

  The unnamed author of the Latin treatise on motion read in the Third and Fourth Days, understood to be Galileo himself; Salviati reports conversations with him and experiments made in his company.

Lessons

• Why giants cannot simply be scaled up

  The Second Day's argument that similar bodies are not equally strong overturns a common assumption: that a model built larger in the same proportions will behave the same. Weight grows faster than the cross-section that bears it, so large structures and animals need thicker supports or stronger materials. The argument joins geometry, engineering and biology, and it shows that the created world has an order of proportion that limits what can be made, not only what can be imagined.

  • Using Salviati's reasoning, explain why 'a small dog could probably carry on his back two or three dogs of his own size'.
  • Why does Simplicio raise the whale, and is Salviati's answer about water satisfactory?
  • What everyday objects or structures show Galileo's principle that larger is relatively weaker?

• Describing motion before explaining it

  In the Third Day Salviati sets aside the dispute over what causes falling bodies to accelerate and proposes instead to define accelerated motion and demonstrate its properties. This methodological choice, to describe mathematically before assigning causes, was a turning point from Aristotelian natural philosophy. Students should weigh both what it gained (precise, testable laws) and what it postponed (the question of why), which Newton and later physicists took up again.

  • What theories of the cause of acceleration does Salviati list, and why does he say examining them 'is not really worth while'?
  • Is a description of how something happens a complete account of it? What would Aristotle say is missing?
  • Can a scientist legitimately bracket questions of cause and purpose without denying that they exist?

• A plausible error, plainly refuted

  Sagredo suggests that a falling body's speed grows in proportion to the distance fallen, a view so natural that Simplicio also accepts it, and Salviati reveals that the Author once held it. Salviati then shows it to be impossible. The scene teaches intellectual humility and the value of a conversation in which mistakes are spoken aloud and tested, rather than hidden. It also shows that the plausible and the true can differ, and that careful reasoning is the means of telling them apart.

  • Why did the proportionality of speed to distance seem so obvious to Sagredo and Simplicio?
  • What does it reveal about Galileo that he has Salviati admit the Author once shared the fallacy?
  • How should a student respond when shown that a firmly held opinion is false?

• Experiment and demonstration together

  The grooved board, bronze ball and water clock of the Third Day are among the most famous experiments in the history of science, but they come after the mathematical proof, as confirmation. In the Fourth Day Sagredo exclaims that rigid demonstration fills him with wonder and delight, yet Galileo also admits that air resistance escapes exact law. The book thus models a partnership between reasoned proof and careful measurement, each correcting the other, and shows delight as a fitting response to discovering order in nature.

  • How did Salviati measure time in the inclined-plane experiment, and how precise did he claim it was?
  • Why does the experiment come after the theorem rather than before it? Would the reverse order be better?
  • What does Galileo concede about air resistance in the Fourth Day, and does this weaken his science or strengthen its honesty?

Context

The Two New Sciences was published in 1638 by Louis Elzevir at Leiden in the Dutch Republic, beyond the reach of the Roman Inquisition, which had condemned Galileo in 1633 and forbidden the printing of his works. Galileo, under house arrest at Arcetri and losing his sight, dedicated the book to the French ambassador François de Noailles. Its two sciences, the strength of materials and the mathematical description of motion, were foundational for later engineering and for the mechanics of Huygens and Newton.

Composition

Much of the content goes back decades. Galileo had studied motion and mechanics at Pisa and Padua (1589-1610), and his correspondence shows him working on the law of fall in the early 1600s, at one stage assuming, as Sagredo does in the Third Day, that speed grows with distance. After 1633 he assembled and completed the material at Arcetri, writing the treatise on motion in Latin and setting it within Italian dialogue. Henry Crew and Alfonso de Salvio's English translation (Macmillan, 1914) is the library's text; diagrams are omitted in this transcription.

Reception

The book's law of fall and parabolic trajectory were absorbed into the mechanics of the later seventeenth century, and its analysis of beams opened a field that later engineers revised, correcting Galileo's assumption about how stress is distributed across a breaking beam. Its scaling argument about bones became a classic of biology, taken up by D'Arcy Thompson in On Growth and Form (1917). The dialogue was included in volume 26 of the second edition of the Great Books of the Western World (1990).

Comparisons

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

  The First Day, printed in the library's selected writings, ranges over cohesion, the vacuum, infinity, the pendulum and the fall of bodies in different media; the Second to Fourth Days narrow to the two promised sciences and shift from wide-ranging conversation to sustained geometrical proof.

• Principia

  Newton's laws of motion generalize what Galileo demonstrated for falling bodies and projectiles, and supply the cause, universal gravitation, that Salviati deliberately set aside as 'not really worth while' to examine.

• Posterior Analytics and Physics

  Simplicio's views on motion derive from Aristotle's Physics, which explains motion by natures and causes and holds that heavier bodies fall faster; Galileo answers not by refuting every Aristotelian cause but by replacing causal inquiry with mathematical definition and measurement.

• The Method (Geometrical Solutions Derived from Mechanics)

  Salviati gives Archimedes 'the first place' for rigor in mechanics and borrows a lemma from On Spirals. The Method shows Archimedes himself using mechanical reasoning to discover geometrical truths, the reverse of Galileo's use of geometry to demonstrate mechanical ones.

• Elements, Books I-VI

  The propositions of the Third and Fourth Days rely on Euclid's theory of proportion; Simplicio has studied 'a little Euclid' to follow them, and Sagredo compares the new treatise to Book III of the Elements, whose few properties of the circle lead to many others.

• On the Motion of the Heart and Blood in Animals (De Motu Cordis)

  Harvey's De Motu Cordis (1628), published a decade earlier, also used quantitative argument and experiment to overturn long-held assumptions, in physiology rather than mechanics; both men studied at Padua, Galileo as a professor and Harvey as a medical student.

Quotes

Medical Lens

Not a medical book, but the Second Day contains a classic passage for the history of comparative anatomy and biomechanics: Galileo's argument that bones cannot scale in simple proportion, illustrated with a sketch of an enlarged bone, and his remark that the bones of birds are hollow for strength without weight. The Third Day's inclined-plane experiment measures time in fractions of a pulse-beat, a reminder of how physicians' and natural philosophers' timekeeping overlapped. Galileo enrolled in medicine at Pisa before turning to mathematics. No medical theory or practice is discussed.

• Second Day, after Proposition VIII

  Salviati argues that a giant or a very large animal would need disproportionately thick or harder bones, sketches a bone enlarged three times in length, and notes that smaller animals are relatively stronger.

  Anatomy Dissection
• Second Day, closing discussion of hollow solids

  Nature uses hollow structures, as in the bones of birds and in reeds, to increase strength without adding weight.

  Anatomy Dissection

Creative Prompts

• Classroom

  Rebuild the Third Day inclined-plane experiment with a grooved board, a ball and a homemade water clock, and record whether distances from rest grow as 1, 3, 5, 7 in equal times.

• Dialogue

  Write a Fifth Day in which Salviati, Sagredo and Simplicio discuss whether a film giant or an enormous insect could exist, using only arguments from the Second Day.

• Essay

  Salviati sets aside the cause of acceleration to describe its properties. Write an essay on whether this was a wise limitation, a loss, or both, drawing on the Third Day and Aristotle's Physics.

• Fiction

  Tell the story of the manuscript's journey from Arcetri to the Elzevir press at Leiden from the point of view of a messenger who does not understand the mathematics but knows the risk.

Further Reading

• Dialogue Concerning the Two Chief World Systems, Galileo Galilei

  The 1632 dialogue with the same three speakers, whose publication led to Galileo's trial.

• Galileo at Work: His Scientific Biography, Stillman Drake

  Traces the long development of the work on motion that culminated in the Two New Sciences.

• Galileo's Daughter, Dava Sobel

  Galileo's last years at Arcetri through the letters of his daughter, Sister Maria Celeste.

• The Origins of Modern Science, Herbert Butterfield

  A classic account of the change from Aristotelian to modern mechanics.

• The Galileo Affair: A Documentary History, Maurice A. Finocchiaro

  The trial documents that explain why the book was printed in Leiden.

• On Growth and Form, D'Arcy Wentworth Thompson

  The 1917 study of biological form that develops Galileo's argument about the scaling of bones.