y = 16t 2

The Calculus A Simple Gift from Galileo

The Calculus Benjamin David (and the Staff Editors of Life Science Library, Mathematics)

The way these rules work, and the reason for their enormous usefulness, can best be illustrated by applying them to the classically simple equation y=16t2, devised by the renowned Italian astronomer and physicist, Galileo Galilei.

The Calculus Benjamin David (and the Staff Editors of Life Science Library, Mathematics)

This brief, unpretentious expression is one of the most versatile in all physics because it shows how gravity acts on a freely falling object—an elevator run amuck, a hailstone or a jumper descending to ground.

The Calculus Benjamin David (and the Staff Editors of Life Science Library, Mathematics)

Since almost all movements and changes on earth are heavily influenced by gravity, the equation of free fall indirectly plays a part in innumerable human actions—from taking a step or lobbing a tennis ball to lifting a steel girder or launching an astronaut into orbit.

The Calculus Benjamin David (and the Staff Editors of Life Science Library, Mathematics)

Timing an object as it falls from a given height is the most straightforward method of gauging the effects of gravity.

The Calculus Benjamin David (and the Staff Editors of Life Science Library, Mathematics)

It was this technique which Galileo used, about 1585, to arrive at his free-fall equation. According to legend, Galileo dropped small cannon balls from the colonnades of the leaning tower of Pisa. According to his own account, he used the less fanciful means of timing cannon balls as they rolled down a ramp.

The Calculus Benjamin David (and the Staff Editors of Life Science Library, Mathematics)

In any event, Galileo ascertained that the equation for free fall was y = 16t2, with y representing the distance fallen in feet and t the elapsed time in seconds after the start of the fall.

The Calculus Benjamin David (and the Staff Editors of Life Science Library, Mathematics)

y = 16t 2

By differentiating this equation twice—so as to shave away successive layers of change and inconsistency—Newton uncovered the essential nature of gravitation.

The Calculus Benjamin David (and the Staff Editors of Life Science Library, Mathematics)

y = 16t 2 d 2 16 t ( ) dt y ' = 32t

Differentiating the equation once, he found that the speed with which a jumper is falling at any moment equals 32 times the number of seconds which he is falling.

The Calculus Benjamin David (and the Staff Editors of Life Science Library, Mathematics)

y = 16t 2 d 2 16 t ( ) dt y ' = 32t d ( 32t ) dt y " = 32 Differentiating the equation a second time, he found that the jumper’s acceleration—the rate of increase in his speed—is always 32 feet second, every second.

The Calculus Benjamin David (and the Staff Editors of Life Science Library, Mathematics)

g = 32

ft s2

The fact that in the free-fall equation acceleration equals a constant number, 32, indicates that end of the trail. This 32 need not be differentiated further; it does not change, and its rate of change is zero. It represents a law of nature: that every free-falling object falls to earth with a constant acceleration of 32 feet per second, every second.

The Calculus Benjamin David (and the Staff Editors of Life Science Library, Mathematics)

Having ascertained this fact by calculus, Newton was able to set his mathematical sights far beyond the earth and to deduce the law of universal gravitation—one of the most important results ever to be achieved by mathematics.

The Calculus Benjamin David (and the Staff Editors of Life Science Library, Mathematics)

It is the law which governs the movements of all celestial bodies—from human beings in orbit to entire systems of stars.

The Calculus Benjamin David (and the Staff Editors of Life Science Library, Mathematics)

Looking back with awe on what a little deduction could accomplish in the mind of Isaac Newton, later thinkers have ranked him as the greatest physicists and one of the greatest mathematicians the world has ever known.

The Calculus Benjamin David (and the Staff Editors of Life Science Library, Mathematics)

Albert Einstein wrote: “Nature to him was an open book, whose letters he could read without effort.”

The Calculus Benjamin David (and the Staff Editors of Life Science Library, Mathematics)

Newton himself said: “I do not know what I may appear to the work; but to myself I seem to have been only like a boy playing on the seashore, and diverting myself in now and then finding a smoother pebble or a prettier shell than ordinary, whilst the great ocean of truth lay all undiscovered before me.”

The Calculus Benjamin David (and the Staff Editors of Life Science Library, Mathematics)

Newton began to use his astounding inventiveness while still a child, to build toys for himself, including a wooden water clock that actually kept time and a flour mill worked by a mouse.

The Calculus Benjamin David (and the Staff Editors of Life Science Library, Mathematics)

His brilliance did not really catch fire, however, until he read Euclid at the age of 19. The story goes that he rushed impatiently on to Descartes’ relatively abstruse La Géométrie. Thereafter his progress was meteoric.

The Calculus Benjamin David (and the Staff Editors of Life Science Library, Mathematics)

Five years later, while still a graduate student at Cambridge, he had already worked out the basic operations of calculus—the rules of integration and differentiation, which he called the laws of “fluxions and fluents.”

The Calculus Benjamin David (and the Staff Editors of Life Science Library, Mathematics)

Newton put together his great invention and applied it in a preliminary way to the problems of motion and gravitation in a two-year burst of creativity, while rusticating during the epidemic of plague which swept England in 1665 and 1666.

The Calculus Benjamin David (and the Staff Editors of Life Science Library, Mathematics)

In retrospect it seems as if the whole framework of modern science arose from his mind as miraculously as a jinni in a bottle.

The Calculus Benjamin David (and the Staff Editors of Life Science Library, Mathematics)

But as Newton himself said, he “stood on the shoulders of giants.” Many men had wrestled with the same problems; it was his genius to fuse their separate inspirations.

The Calculus Benjamin David (and the Staff Editors of Life Science Library, Mathematics)

The twin processes of differentiation and integration in calculus, for instance, were rooted in two classic questions of Greek antiquity: how to construct a tangent line (a line that just touches a curve at a given point), and how to calculate an area which is bounded on one side by a curve.

The Calculus Benjamin David (and the Staff Editors of Life Science Library, Mathematics)

The problem of the tangent, or “touching” line, was equivalent to the problem of finding the slope of a curve at any point and therefore of finding the derivative of an equation.

The Calculus Benjamin David (and the Staff Editors of Life Science Library, Mathematics)

The area problem was equivalent to the problem of integrating the equation that gives the rate of growth of an area.

The Calculus Benjamin David (and the Staff Editors of Life Science Library, Mathematics)