## A Brief History of Calculus

##### "Biographical history, as taught in our public schools, is still largely a history of boneheads: ridiculous kings and queens, paranoid political leaders, compulsive voyagers, ignorant generals -- the flotsam and jetsam of historical currents. The men who radically altered history, the great scientists and mathematicians, are seldom mentioned, if at all." - Martin Gardner

## A Brief History of Calculus

### From early to modern times

### THE ANCIENTS|THE FORERUNNERS|THE EARLY MODERNS|THE LATER MODERNS

### The Ancients

### Pythagoras(c. 580 - 500 B.C.)

Though not much is known of this mysterious man, it is almost certain that mathematics began with him. Pythagoras led a half-religious, half-mathematical group who kept most of their discoveries a secret. The Pythagoreans credited all their work to their leader and their mottos became "Everything is number" and "He [Pythagoras] himself has said it". Pythagoras came up with the idea of a mathematical proof, as well as his famous Pythagorean Theorem relating the sides of a right triangle to its hypotenuse. The Pythagoreans discovered irrational numbers, which to them was a disaster because the existance of irrational numbers went against their beliefs. However, this discovery led to opportunities for mathematicians to come."...three fifths of him genius and two fifths sheer fudge" - J.R.Lowell

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Euclid (c. 300 B.C.)

Few facts have been pieced together about Euclid and in fact not everyone is convinced that Euclid was one man. Most people believe that Euclid was the leader of a group of mathematicians in Egypt who wrote *The Elements*, a collection of books on geometry that organized all that was known on mathematics at his time. Among the topics covered in the 13 books are: plane geometry, number theory, irrationals, and the volume of a cone. In recent centuries many of the assumptions made by Euclid have been proven to be false, however it has been said that the books have had a greater influence on the human mind than almost any other work. Euclid was younger than Plato but older than Archimedes."There is no royal road to geometry"-Euclid

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Archimedes (c.287-212 B.C.)

Archimedes was one of the three greatest mathematicians of all time. Though he became famous for many of his inventions, inlcuding the "Screw of Archimedes" and many war machines he designed for the King of Greece, his true passion was for pure mathematics. Archimedes found the sum of a geometric series in such a way as to indicate that he understood the concept of limits. He also, among other things, calculated pi; used the sum of infinte rectangles to find the area under a curve; and found the volume and surface area of a sphere. Archimedes was killed by a roman soldier when the soldier walked through a drawing of some circles in the dust and Archimedes got mad at him. "He was a great civilization all by himself"- George F. Simmons

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### The Forerunners

Descartes

Mersenne

Pierre Fermat(1601-1665)

A lawyer by day and a mathematician by night, Fermat was a man of obvious genius who never attempted to publish his findings during the course of his life. Luckily Fermat corresponded with Mersenne and other mathematicians in Paris and revealed his discoveries. Though Newton and Leibniz are said to be the inventors of calculus, Fermat certainly had a hand in it. Fermat found a method of finding maxima and minima which students today would recognize as setting the derivative equal to zero. Fermat also invented analytic geometry and modern number theory. Fermat once left a note in the margin of a book stating a theorem (Fermat's Last Theorem) but offering no proof, and to this day no mathematician has been able to find a proof."And perhaps, posterity will thank me for having shown it that the ancients did not know everything."- Pierre Fermat

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Pascal

Huygens

### The Early Moderns

Sir Isaac Newton(1642-1727)

Newton actually discovered calculus between 1665 and 1667 after his university closed due to an outbreak of the Plague. Newton was only 22 at the time, and he preferred not to publish his discoveries. Meanwhile, in Germany, Leibniz discovered Calculus independently and he was very open with his findings. This led to a bitter dispute between the two mathematicians later known as the "Great Sulk". Today it is well known that both men discovered calculus independantly of the other, Leibniz about 8 years after Newton. Newton is best known for his work in physics, and especially his three laws of motion."If I have made any valuable discoveries, it has been owing more to patient attention than to any other talents"- Sir Isaac Newton

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Gottfried Wilhelm Leibniz (1646-1716)

Though Leibniz is credited with discovering calculus after Newton he is thought to be the true founder of modern European mathematics. Not only was he a great mathematician, he was also a philosopher, scientist, logician, diplomat and a lawyer. Leibniz is well known for introducing notations that are still used in Calculus today, such as 'dy/dx' and the integral symbol. It is even attributed to him that the '=' symbol is used throughout the world."It is rare to find learned men who are clean, do not stink, and have a sense of humour"- the Dutchess of Orleans, on Leibniz

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The Bernoulli Brothers

Euler(1707-1783)

It is hard to believe that a man with 13 children who went blind for the last 17 years of his life was able to publish, on average, about 800 pages a year throughout his life. Euler contributed to every branch of pure and applied mathematics and even discovered some new ones. It has been said that any elementary or advanced calculus text printed after 1748 is basically a copy or a copy of a copy of Euler. Thanks to Euler we have symbols for pi and *e*. He also did extensive work on infinite series and amazed his teacher, Johann Bernoulli, by finding the sum of a particular series. Euler managed, with writers to help him, to publish even more work after going blind."[Upon losing the use of his right eye] Now I will have less distraction"-Euler

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Fourier(1768-1830)

A French mathematician, Fourier was torn between his father's desire for him to enter the priesthood and his real interest in mathematics. By the age of 14 mathematics won out. He became involved in the aftermath of the French Revolution and even became an acquaintance of Napolean's. At one point he was imprisoned and destined for the guillotine, but he was eventually freed instead. Fourier is best known for the series that bears his name. He also expanded the definition of a function. Riemann used the Fourier series to define a definite integral, and the series has also been used for many other applications to physics."Fourier is a mathematical poem"- Lord Kelvin

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### The Later Moderns

Johann Carl Friedrich Gauss(1777-1885)

Perhaps the greatest mathematician that ever lived, Gauss' talent was recognized from an early age by his elementary school teachers in Germany. At the age of 15 Gauss entered Brunswick Collegium Carolinum and there he independantly discovered many mathematical laws and theorems. Gauss made contributions to many areas including number theory, differential equations, conics, and differential geometry. His work never suffered, despite personal tragedy. Whithin one year his father, wife and son all passed away. Gauss is said to have discovered non-Euclidean geometry although he never published anything on the matter because he did not want to ruin his reputation. "If others would but reflect on mathematical truths as deeply and continuously as I have, they would make my discoveries."-Gauss

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Augustin Louis Cauchy(1789-1857)

For all his genius, Cauchy was strongly disliked by most of his contemporaries, and has even been described as a narrow-minded bigot. However, he did make some great contributions to calculus including the first proof of the convergence of a Taylor series as well as a rigorous treatment of limits, derivatives, and integrals. Cauchy came second to Euler in terms of productivity, filling 27 volumes with his discoveries."Cauchy is mad and there is nothing that can be done about him, although, right now, he is the only one who knows how mathematics should be done." - Abel

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Abel

Johann Peter Gustav Lejeune Dirichlet(1805-1859)

Dirichlet's idol was Gauss and supposedly he carried around Gauss'*Disquisitiones arithmeticae* wherever he went. Dirichlet is best known for his work on number theory and analysis. In 1829 he gave a definition of a function that is still used today. He said that *y* is a function of *x* when each value of *x* in a given interval has a unique value of *y*. Later in his life Dirichlet became a friend of Gauss, and succeeded him as a professor at the University of Gottingen.

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Liouville

Hermite

Lord Kelvin (William Thomson)(1824-1907)

Lord Kelvin gained fame and fortune when he invented the mirror galvanometer, a mechanism that could be used to translate morse code sent over the Atlantic Ocean. A man by the name of Whitehouse was working on the same project and in an effort to beat Thomson he secretly started using the instrument. In the end Thomson gained all the deserving credit. Thomson's father taught him mathematics and at the age of ten he entered university though he did not start university level courses until the age of 14. He applied math to heat flow using Fourier analysis (which involves trigonometric integrals) and this eventually led him to his most famous (and lucrative) discovery. Lord Kelvin also devised the absolute temperature scale that bears his name."When you are face to face with a difficulty, you are up against a discovery."- Lord Kelvin

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Bernhard Riemann(1826-1866)

Riemann showed an interest in mathematics form an early age but when he first entered university it was in the faculty of theology, to please his father. However, soon after, with permission from his father, he switched to mathematics at the University of Gottingen where Gauss was head of mathematics. A year later Riemann left and went to school in Berlin, for Gauss was unapproachable, especially to lowly first year students, and in Berlin he was accepted with open arms by Dirichlet. A few years later Riemann returned to Gottingen and impressed Gauss with a very famous lecture on geometry. He would eventually become a professor at the university. Riemann's definition of an integral is still used in virtually all text books today. Riemann died at the young age of 39 from tuberculosis and for this reason published a relatively small (yet important) amount of work. He left mathematicians to follow the Riemann Hypothesis, which has yet to be solved.

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##### "Mathematics is not a careful march down a well-cleared highway, but a journey into a strange wilderness, where the explorers often get lost. Rigour should be a signal to the historian that the maps have been made, and the real explorers have gone elsewhere." - W.S.Anglin

References: Simmons, G., *Calculus with analytical geometry* (McGraw-Hill: New York, 1985)

The Mactutor History of Mathematics archive http://www-gap.dcs.st-and.ac.uk/~history/index.html