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.

Top 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.

• 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.

• 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.

• 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.

Verified 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.

• 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.