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The following chapter is meant to provide brief biographies of the mathematicians that significantly contributed to the development of the Calculus.

**4.2
Gregory of St. Vincent (1584-1667)**

A Jesuit teacher in Rome and Prague, he later became a tutor in the
court of Philip IV of Spain. He tried to "square the circle"
(constructing a square equal in area to circle using only a straight edge
and compass) throughout his life, and discovered several interesting theorems
while doing so. He discovered the expansion for log(1 + *x*) for ascending
powers of *x*. Eventually, he thought he had squared the circle, but
his method turned out to be equivalent to the modern method of integration.
He successfully integrated *x*^{-1} in a geometric form which
is equivalent to the natural logarithm function.

** **

**4.3
Rene Descartes (1596-1650)**

Rene Descartes was a philosopher of great acclaim. The idea that humans may make mistakes in reasoning is the foundation of his philosophy. He cast aside all traditional beliefs and tried to build his philosophy from the ground up, based on his reasoning alone. In his search for a base on which he might begin to build his reconstructed view of the world, he doubted the reality of his own existence. The existence of his doubt persuaded him to formulate the famous maxim, "I think, therefore I am."

The precision and clarity of mathematics and mathematical reasoning impressed Descartes. He hoped to make use of it in the development of his philosophy and thereby reduce the susceptibility to flaws of his own reasoning. He spent a number of years studying mathematics and developing systematic methods for distinguishing between truth and falsehood. His contribution to the field of geometry can be thought of as an example of how his methods can be applied to reveal new truths, but to the mathematicians to follow him, Descartes’ analytical geometry was powerful tool in its own right. Descartes’ work in geometry laid the foundation for the calculus that was to come after him.

Due to his wariness of mistakes in reasoning, Descartes’ tended to de-emphasize his formal education and instead focus on learning by first-hand experience. His philosophy of experience led him to travel outside of his native France, serve in the military, and eventually live in Holland. During his time in Holland, Descartes tutored Princess Elisabeth, but devoted most of his time to contemplation of his philosophy and his writing. He was summoned to Sweden in 1646 to tutor Queen Christine, but the Swedish winters were too difficult for him and Descartes died in 1650.

**4.4
Bonaventura Cavalieri (1598-1647)**

Cavalieri became a Jesuate (not a Jesuit as is frequently stated) at an early age and it was because of that that he was made a professor of mathematics at Bologna in 1629. He held the position until his death in 1647. Cavalieri published tables of sines, tangents, secants, and versed sines along with their logarithms out to eight decimal places, but his most well known contribution is in the invention of the principle of indivisibles. His principle of indivisibles, developed by 1629, was first published in 1635 and was again published in 1653, after his death, with some corrections. The principle of indivisibles is based on the assumption that any line can be divided up into an infinite number of points, each having no length, a surface may be divided into an infinite number of lines, and a volume can be divided into an infinite number of surfaces.

**4.5
Pierre de Fermat (1601-1665)**

Pierre de Fermat was born in France, near Montauban, in 1601, and he died at Castres on January 12, 1665. Fermat was the son of a leather merchant, and he was educated at home. He became a councilor for the local parliament at Toulouse in 1631, a job where he spent the rest of his life. Fermat’s life, except for a dispute with Descartes, was peaceful and unremarkable. The field of mathematics was a hobby for Fermat. He did not publish much during his lifetime regarding his findings. Some of his most important contributions to mathematics were found after his death, written in the margins of works he had read or contained within his notes. He did not seem to intend for any of his work to be published, for he rarely gave any proof with his notes of his discoveries. Pierre de Fermat’s interests were focused in three areas of mathematics: the theory of numbers, the use of geometry of analysis and infinitesimals, and probability. Math was a hobby for Fermat – his real job was as a judge. Judges of the day were expected to be aloof (so as to resist bribery), so he had a lot of time for his hobby.

**4.6
Gilles Persone de Roberval (1602-1675)**

Held chair of Ramus at the Collège Royale for 40 years from 1634.
He developed a method of indivisibles similar to that of Cavalieri, but
did not disclose it. Roberval became involved in a number of disputes about
priority and credit; the worst of these concerned cycloids. He developed
a method to find the area under a cycloid. Some of his more useful discoveries
were computing the definite integral of sin *x*, drawing the tangent
to a curve, and computing the arc length of a spiral. Roberval called a
cycloid a trochoid, which is Greek for wheel.

A professor of geometry at Oxford, he had several very important publications, which advanced the field of indivisibles. He studied the works of Cavalieri, Descartes, Kepler, Roberval, and Torricelli. He introduced ideas in calculus that went beyond those he read of. He discovered methods to evaluate integrals that were later used by Newton in his work on the binomial theorem. Wallis was the first to use the modern symbol for infinity. It is interesting to note that Wallis rejected the idea that negative numbers were less than nothing but accepted the notion that they were greater than infinity.

Pascal was a French student of Desargues. Etienne Pascal, Blaise’s father, kept him away from mathematical texts early in his life until Blaise was twelve, when he studied geometry on his own. After this, Etienne, himself a mathematician, urged Blaise to study. Pascal invented an adding machine, to aid in his father’s job as a tax collector. Its development was hindered by the units of currency used in France and England at the time. Two hundred and forty deniers equaled one livre, which is a difficult ratio for conversion.

Pascal turned to religion at the age of twenty-seven, ceasing to work on any mathematical problems. When he had to administer his father’s estate for a time, he returned to studying the pressure of gasses and liquids, which got him into many arguments because he believed that there is a vacuum above the atmosphere; an unpopular belief at the time. During this time he also founded the theory of probability with Fermat. Late in his life, he turned to the study of the cycloid when he had a toothache. The tooth ache went away immediately upon pondering a cycloid, and he took this as a sign to study more on the subject of cycloids.

In Pascal’s *Pensées*, one of his large religious papers,
Pascal made a famous statement known as Pascal’s Wager: "If God does
not exist, one will lose nothing by believing in him, while if he does
exist, one will lose everything by not believing." His conclusion
was that "...we are compelled to gamble..."

**4.9
Christiaan Huygens (1629-1695)**

Descartes took interest in Huygens at an early age and influenced his mathematical education. He developed new methods of grinding and polishing telescope lenses, and using a lens he made, he was able to see the first moon of Saturn. He was the one to discover the shape of the rings around Saturn using his improved telescopes. Huygens patented the first pendulum clock, which was able to keep more accurate time than current clocks because he needed a way to keep more accurate time for his astronomical observations. He was elected to the Royal Society of London and also to the Académie Royale des Sciences in France. Leibniz was a frequent visitor to the Académie and learned much of his mathematics from Huygens. Throughout his life he worked on pendulum clocks to determine longitude at sea.

In one of his books, he describes the descent of heavy bodies in a vacuum
in which he shows that the *cycloid* is the *tautochrone*, which
means it is the shortest path. He also shows that the force on a body moving
in a circle of radius *r* with a constant velocity of *v* varies
directly as *v ^{2}* and inversely as

An Englishman, he was ordained and later made a professor of geometry at Gresham College in London. Barrow developed a method for determining tangents that closely approached the methods of calculus. He was also the first to discover that differentiation and integration were inverse operations. He thought that algebra should be part of logic instead of mathematics, which hindered his search for analytic discoveries. Barrow published a method for finding tangents, which turned out to be an improvement on Fermat’s method of tangents. He worked with Cavalieri, Huygens, Gregory of St. Vincent, James Gregory, Wallis, and Newton.

**4.11
James Gregory (1638-1675)**

A Scotsman, he was familiar with the mathematics of several countries. Gregory worked with infinite series expansion, and infinite processes in general. He sought to prove, through infinite processes, that one could not square the circle, but Huygens, who was regarded as the leading mathematician of the day, believed that pi could be expressed algebraically, and many questioned the validity of Gregory’s methods. Two hundred years later, it was proved that Gregory was right.

Much of his work was expressed in geometric terms, which was more difficult to follow than if it has been expressed algebraically. Because of this, Newton was the first to invent Calculus, even though Gregory knew all the important elements of Calculus, they were not expressed in a form that was easily understandable. Gregory only has the infinite series for arctangent attributed to him, even though he also discovered the infinite series for tangent, arcsecant, cosine, arccosine, sine, and arcsine. Using his infinite series for arctangent, he was able to find an expansion for p /4 several years before Leibniz.

**4.12
Sir Isaac Newton (1642 - 1727)**

Newton’s father was a farmer, and it was intended that he follow in the family business. Instead of running the farm, an uncle decided that he should attend college, specifically Trinity College in Cambridge, where the uncle has attended college. Newton’s original objective was to obtain a law degree. He attended Barrow’s lectures and originally studied geometry only as a means to understand astronomy. In 1665, Trinity College closed down because of the plague in England. During the year it was closed he made several important discoveries. He developed the foundation for his integral and differential calculus, his universal theory of gravitation and also some theories about color. Upon his return to Trinity, Barrow resigned the Lucasian chair in 1669, and recommended Newton for the position. He continued to work on optics and mathematical problems until 1693, when he had a nervous breakdown. He took up a government position in London, ceasing all research. In 1708, Newton was knighted by Queen Anne; he was honored for all his scientific work. He was elected president to the Royal Society in 1703 and held the position until his death in 1727.

**4.13
Gottfried Wilhelm Leibniz (1646 - 1716)**

Leibniz was a law student at the University of Leipzig and the University
of Altdorf. In 1672, he traveled to Paris in order to try and dissuade
Louis XIV from attacking German areas. He stayed in Paris until 1676. During
this time he continued to study law but also studied physics and mathematics
under Huygens. During this time, he developed the basic version of his
calculus. In 1676, he moved to Hanover, where he spent the rest of his
life. In one of his manuscripts dated November 21, 1675, he used the current-day
notation for the integral, and also gave the product rule for differentiation.
By 1676, he had discovered the power rule for both integral and fractional
powers. In 1684 he published a paper containing the now-common *d*
notation, the rules for computing derivatives of powers, products and quotients.
One year later, Newton published his *Principia*. Because Newton’s
work was published after Leibniz’s, there was a great dispute over who
discovered the theories of calculus first which went on past their deaths.

**4.14
Leonhard Euler (1707 - 1783)**

Euler was the son of a Lutheran minister and was educated in his native town under the direction of John Bernoulli. He formed a life-long friendship with John Bernoulli’s sons, Daniel and Nicholas. Euler went to the St. Petersburg Academy of Science in Russia with Daniel Bernoulli at the invitation of the empress. The harsh climate in Russia affected his eyesight; he lost the use of one eye completely in 1735. In 1741, Euler moved to Berlin at the command of Frederick the Great. While in Berlin, he wrote over 200 articles and three books on mathematical analysis. Euler did not get along well with Frederick the Great, however, and he returned to Russia in 1766. Within three years, he had become totally blind. Even though he was blind, he continued his work and published even more works. Euler produced a total of 886 books and papers through his life. After he died, the St. Petersburg Academy continued to publish his unpublished papers for 50 years. Euler used the notations f(x), i for the square root of -1, p for pi, S for summation, and e for the base of a natural logarithm. Euler died in 1783 of apoplexy.

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