The Archimedeans - A Very Short History of Mathematics

A Very Short History of Mathematics

BY W. W. O. SCHLESINGER AND A. R. CURTIS

This paper was read to the Adams Society (St. John's College Mathematical Society) at their 25th anniversary dinner, Michaelmas Term, 1948.

MATHEMATICS is very much older than History, which begins* in +1066, as is well known; for the first mathematician of any note was a Greek named Zeno, who was born in -494, just 1,559 years earlier. Zeno is memorable for proving three theorems: (i) that motion is impossible; (ii) that Achilles can never catch the tortoise (he failed to notice that this follows from his first theorem); and (iii) that half the time may be equal to double the time. This was not considered a very good start by the other Greeks, so they turned their attention to Geometry.

* See Sellars and Yeatman, "1066 and all that," or any similar standard work.

Euclid, about -300, invented Geometry, including Pythagoras' theorem which is how it got the name. He also invented parallel lines, which have really been of more use to railways than to mathematicians. Most people already know more about Euclid than we do.

Archimedes (-286 to -211) is very memorable for taking a bath. Unfortunately he forgot to get dressed afterwards, in spite of his principles.

From this time onward there was an open interval, the other end point of which was Descartes (1596 to 1650), who was divinely inspired to invent analytical geometry, and was once found sitting inside a stove to keep himself warm. He also discovered that he existed, and, moreover, he was able to prove it.

Newton (1642 to 1727) was very memorable indeed, chiefly for having just missed living in St. John's. To console himself he invented the Calculus.

Newton is also memorable for having been admired by Taylor, who invented Maclaurin's series and admired Newton. However, Taylor lived in St. John's and so was luckier than Newton.

The next important mathematician is the Bernoullis. In spite of his having invented numbers, nobody knows how many of him there were, and he lived all over the century. He was called Nicholas, Jacob and John, and one of him was called Daniel.

Euler (1707 to 1783), Langrange (1736 to 1813), and Laplace (1749 to 1827) are all famous for inventing equations. Only one of Laplace's equations is well known, but this is enough for anyone. It makes electricity and hydrodynamics much easier for people who don't have to solve it. Euler and Lagrange both went about varying things, which caused the calculus of variations. This was both memorable and regrettable.

Gauss invented so many things that it just isn't true. These included the magnetism of the earth, the theory of equations, Cauchy's theorem and the Cauchy-Riemann equations. In fact, whenever anyone invented anything in the first half of the 19th century, Gauss had invented it twenty years earlier, and was still alive to tell him so. He was born in 1777, died in 1855, and lived all the years in between. He was very memorable, and a good thing.

Cauchy's theorem is very important, but is much harder to prove now than it was when Gauss invented it.

Lobatchewski (1793 to 1856) must have failed an examination in geometry when he was at school, for he made things harder for everyone by inventing non-Euclidean geometry - just to get his revenge, of course. This was especially bad for the railways, since it made parallel lines so much more difficult.

Hamilton (1805 to 1865) was an Irishman . When he had learnt 13 languages before he had left school, he decided that there was no future in this, and look up mathematics. He invented Hamilton's priniciple, the Hamiltonian, the Hamilton-Jacobi theorem, and the Hamilton-Cayley theorem, but not the Hamilton Academicals. Towards the end of his life he also invented quaternions, but nobody except himself ever fell in love with them.

Weierstrass (1815 to 1897) is memorable because of Sonja Kowalewski (1850 to 1891), who, of course, is memorable because of Weierstrass. He said that if you put infinitely many things into a small space, some of them would be pretty close together.

The most memorable of all mathematicians was John Couch Adams (1819 to 1890). He had the good fortune to live in St. John's, and was named after this society. He discovered Neptune just after Leverrier, and would have discovered it before if the Astronomer Royal had kept his eyes open.

Charles Lutwidge Dodgson was a minor Oxford mathematician who must not be confused with Lewis Carroll, whom he impersonated when sending copies of his works to Queen Victoria. They lived contemporaneously.

The chief problem treated by Carroll was that of the Cheshire cat. His treatment is essentially unsound, however, since he says: "... this time the cat vanished quite slowly, beginning with the end of the tail, and ending with the grin, which remained some time after the rest of it had gone."* It is obvious that, by the time the tail had disappeared, the cat would be a Manx cat. This is a contradiction, since it was a Cheshire cat, by hypothesis. Carroll also discussed the increased angular velocity of the world if everybody minded his own business.

* Lewis Carroll, Alice's Adventures in Wonderland.

Riemann (1826 to 1866) invented the tensor calculus, and thus caused the theory of relativity.

In 1895 Bertrand Russell stated the following theorem: the class of all classes which are not members of themselves is either a member of itself or not. Whichever it is, it is the other. This a contradiction, and the end of mathematics.

Eureka, 12.

Reproduced from Eureka 27 pages 36-38.