The Calculus A Wine Keg of Infinitesimals

The CalculusA Wine Keg of Infinitesimals

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

Archimedes

By viewing any curve as a succession of infinitely short segments, or any area as an accumulation of infinitely fine slices, the Greeks—particularly Archimedes—had

solved a number of specific problems concerning rates of change.


Leibniz

Copernicus

Kepler

Mathematicians of the 16th and 17th Centuries also used infinitesimal methods, though seldom with rigorous Greek proofs.


http://en.wikipedia.org/wiki/Image:Nikolaus_Kopernikus.jpg

http://en.wikipedia.org/wiki/Image:Gottfried_Wilhelm_von_Leibniz.jpg

Kepler

Kepler, for instance, had employed infinitesimals to give vintners a formula for gauging the volume of wine kegs.


John Wallis

In Descartes’ time and in the 15 years after his death, his compatriot, Pierre de Fermat, and the Englishman, John Wallis, had begun to cast infinitesimals

in the helpful analytic molds of equations.


y

xT

P

Q R

N MO

Then, in about 1663, Newton’s professor at Cambridge, Isaac Barrow, became the first man to realize that the tangent problem and the area problem are two

sides of the same coin—in effect, that integration is the reverse of differentiation.


http://en.wikipedia.org/wiki/Image:Isaac_Barrow.jpg

When Newton first began to unite all these preliminary insights in the single well-knit structure of calculus, he showed Barrow some of his early results.

Barrow was so enthusiastic that he generously let it be known about Cambridge that Newton had done what he himself had failed to do.


http://en.wikipedia.org/wiki/Image:Isaac_Barrow.jpg

A few years later, in 1669, when he was retiring, he was instrumental in getting Newton, then 26, appointed as his successor to the Lucasian professorship of

mathematics at the university—one of the most desirable chairs of mathematical scholarship in the academic world.


Thereafter, honors and inspirations came to Newton in a steady stream. Over the next four decades he formulated the law of gravitation and used it to explain the

movements of the planets, moon and tides;


…analyzed the color spectrum of light;


…constructed the first modern reflecting telescope;


http://en.wikipedia.org/wiki/Image:NewtonsTelescopeReplica.jpg

…performed innumerable alchemistic experiments;


…tried to reconcile with Scripture the date of 4004 B.C., which was currently accepted as the time of Adam’s creation;


…served as a member of Parliament;


…was appointed warden and then master of the British mint;


Royal Society Mace

…knighted by Queen Anne in 1705 and was repeatedly elected president of Britain’s select scientific club, the Royal Society, from 1703 until his death in

1727.


Strangely enough, Newton revealed his monumental discoveries to only a few scientific cronies. Many explanations have been given for his inordinate

secretiveness.


It has been said that he was always too busy with new ideas to find time to write up old ones, and that he had a passionate distaste for the wranglers and criticism which inevitably raged around scientific pronouncements in

those days.


Then, too, he was just not much of a talker. While he was in Parliament, his only recorded utterance was a request to open the window.


Edmund Halley

On one occasion the astronomer Edmund Halley came to him, after a discussion with England’s most eminent scientists, to ask if he knew what path a planet

would take around the sun if the only force influencing it was a force that diminishes according to the square of its distance from the sun.


Newton immediately gave the answer: the path would be elliptical. When asked how he knew, he explained casually that he had worked out the problem years

before as a graduate student.


In other words, he had worked out the fundamental law of the universe and told nobody about it. Encouraged by Halley to re-create his original calculations, he

went on to produce his masterwork, the Principia.


Newton’s Principia is generally recognized as the most influential, conclusive andrevolutionary scientific work ever to appear in print. In it, he not only explained

why the solar system works the way it does but also laid down the laws of dynamics which are still the chief ingredients of practical engineering physics—of

missile shots or throughway construction.


Most of these laws Newton had worked out through calculus, but like Archimedes before him, he chose to present his finished work in universally understood

mathematics—as a lengthy Greek proof, couched almost entirely in the terms of classical geometry.


Not even the skillful coaxing of Halley could convince Newton to publish his calculus—not, that is, until another mathematician, the German Gottfried Wilhelm

von Leibniz, had independently re-created the entire mental machinery.


Leibniz invented calculus 10 years after Newton, in 1675, and in 1684 published his account of it, 20 years before Newton was to give the first published

explanation of his own version.


Like Newton, Leibniz was as successful and practical as the mathematics he originated. The son of a well-to-do university professor, he learned Greek and Latin by the age of 12, attended university, took a law degree, and went on to become the counsel to kings and princes in a illustrious career that sometimes

verged on shady.


He traveled all over Europe tracing dubious lineages to establish the dubious rights of prince-lings to vacant thrones.


He formulated many of our modern principles of international power politics—including the phrase “balance of power.”


cos xdx∫

During trips to Paris, he studied algebra and analytic geometry under the great physicist Christian Huygens. And while jogging along in coaches on diplomatic missions, he created new mathematics simply for pleasure, including his own

version of calculus.


y

dydxdy

x

dx

Although Newton accomplished far more with calculus than Leibniz did, Leibniz had a superior notation for it—one he polished so carefully that we still use it. It

was Leibniz who first wrote derivatives as dy/dx or dx/dy—forms that suggest the fractional rate-of-change measurements to which they apply.


y

dydxdy

x

dx

Newton wrote the derivative of y as and the derivative of x as . The dots in Newton’s symbolism led rebellious 19th Century Cambridge students to protest

against the “dotage” of English notation and to advocate the “pure d-ism” of continental notation.

y x


Unfortunately, Newton and Leibniz in their later years became embroiled in a chauvinistic dispute as to who had invented what first. The result was that scholars on the continent, helped by Leibniz’s notation, went on to develop

calculus further, while English mathematicians, hampered by the less felicitous notation devised by Newton, foundered in a morass of perplexities.


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The Calculus A Wine Keg of Infinitesimals