There are many milestone books in the history of science. True to the definition of “milestone,” these works mark pivotal achievements in mankind’s effort to understand natural law and the world in which we live. Among the very top tier of milestone science books is the *Discorsi*, published in 1638 by Galileo Galilei. It is his last and most significant contribution to science and is often referred to as the first physics “textbook.”

The *Discorsi* derives its fame as the published repository of Galileo’s life-long efforts to decipher the concept of motion and the “law of fall,” the mathematical characterization of free-falling bodies under the influence of gravity. Attempts to determine the velocity and distance profiles of falling bodies as a function of elapsed time of fall had yielded only conjecture over the centuries since Aristotle himself pondered the question. As Aristotle observed over two thousand years ago, “In order to know the natural world, one must first understand motion,” and, until Galileo and Isaac Newton came along, what we “knew” about motion was indeed largely conjecture – much of it erroneous!

It was clear that the force of gravity was the prime-mover causing bodies of mass to fall toward the earth’s center, but until Isaac Newton in 1687 established how gravity worked, no one really understood the mathematical details of gravitational attraction between any two bodies of mass. When we weigh ourselves on a scale, we are, in fact, measuring the force of gravitational attraction between the earth as one mass and our bodies as the other. One’s weight on the moon is less than that on earth because of the moon’s smaller mass. In empty space, we are weightless. Here, then, is the great question posed by the elusive “law of fall” whose answer had eluded man for centuries – until Galileo solved the riddle:

**What is the velocity attained by a free-falling body as a function of the elapsed time of fall? We know the velocity at the onset of fall: It is clearly zero at the instant of release. But what is its precise instantaneous velocity at each second of elapsed time thereafter? Furthermore, what is the distance of fall covered from the release point at each elapsed second of time?**

A falling body precisely one second into free-fall is already traveling with an instantaneous velocity of 32.2 feet per second. Making such measurements without modern instrumentation and high-speed strobe photography would be virtually impossible. In the early seventeenth century when photography was unimaginable and there certainly were no stopwatches or even accurate clocks, Galileo had to find a way to slow down the motion of a free-falling body. He “diluted” the effect of gravity by repeatedly rolling a small ball from a standing start down a long grooved, inclined plane. He then measured the distances covered by the slowly accelerating ball over successive constant-time intervals of an arbitrary “musical beat.” It is said that he used for a “timing clock” the steady cadence of a hummed Italian march for his equal timing intervals. Using multiple trials and noting the precise position of the rolling ball along the track on successive numbers of downbeats of the steady cadence, he was able to deduce the “law of fall.” Here is a figure illustrating the essence of Galileo’s brilliant inclined plane experiment. The figure is from chapter four of my book on motion, *The Elusive Notion of Motion: The Genius of Kepler, Galileo, Newton, and Einstein*.

**Here is the critical essence of what Galileo determined:**

**A body free-falling under gravity exhibits a constant acceleration value (near the earth’s surface). It remained for Galileo’s successors to determine the exact numerical value of that acceleration. Near the earth’s surface, a body of mass in free-fall attains an additional velocity of 32.2 feet per second for each additional second of elapsed time. Galileo’s determination that acceleration is constant in free-fall dictates two major conclusions:**

**-The velocity attained is proportional to the elapsed time of fall. Twice the elapsed time, twice the velocity, for example.**

**-The distance traveled is proportional to the square of the elapsed time of fall. Twice the elapsed time, four times the distance traveled.**

This is the celebrated “law of fall.” There might be a tendency for the casual reader to shrug-off Galileo’s achievement as no big deal in light of modern scientific achievements, but it was a VERY big deal for the progress of physics. Galileo, along with Johannes Kepler and Kepler’s experimentally formulated three laws of planetary motion, paved the way for the truly great reformation in mathematical physics initiated by Isaac Newton in his masterwork book of 1687, the *Principia*, universally acclaimed as the greatest scientific book ever published.

Image: Pierre Barge & Associates Auctions

**First-state presentation copy of Galileo’s Discorsi to the French Ambassador, Count Francois de Noailles who smuggled the manuscript from Florence, Italy, to Leiden, Holland, for publication in 1638. Auctioned in Paris for over $790,000 in April, 2017 by Pierre Barge & Associates, the book itself is dedicated to de Noailles.**

Here is perhaps the finest copy extant of Galileo’s *Discorsi E Dimostrazioni Matematiche intorno a due nuoue scienze*, otherwise known as *Discourses on Two New Sciences*. This one-of-a-kind presentation copy from 1638 was given to a friend of Galileo’s who smuggled the final portion of Galileo’s manuscript out of Florence, Italy, into Leiden, Holland for publication by the famed Elzevir Press. Galileo was being held in virtual house arrest within his villa outside of Florence by mandate from the Catholic Church and its Inquisition. Galileo had been accused of suspicion of heresy by the Church for his previous 1632 book, the *Dialogo*, which the Church felt promoted the Copernican “world system” which featured a sun-centered solar system. This went against established scripture which suggested that the earth was at the center of everything, according to the Church. Galileo famously maintained that religion’s role on earth should be to show the way to heaven; it should be the role of science, not the Church, to explain the clockwork of the heavens. The Church did not agree.

As for the “two new sciences” introduced by Galileo’s *Discorsi*: The first treatise in the book deals with what would today be called “Strength of Materials” and “Reliability Engineering.” This constituted a pioneering effort by Galileo in a new field of endeavor. The second treatise in the book involves those categories of physics known as “Mechanics,” and “Kinematics.” The centerpiece of the book is, of course, Galileo’s findings on motion and the long-delayed, finally published documentation of the “law of fall.” Most of the work presented in that section was done by Galileo as early as 1604. Galileo was in poor health and almost blind from glaucoma in 1638; accordingly, publication of this, his most important scientific work was, for him, a very high priority made complicated by the censorship of the Church. Galileo died in 1642, the year Isaac Newton came into this world. The torch had been passed.