Mathematics

A. L. Cauchy 1789 - 1857

The history of any subject is like a great river, the central channel meandering but ultimately driving down to the sea, with side branches veering off here and there along the shore to make their own way. Some of those rivulets find their way back to the main course, others peter out in a swamp, the two exchanging places as blockages and torrential rains come into play. Powerful underground springs interact with topography and gravity to govern the web of flowing water in mysterious ways. And all changing year by year as Mark Twain said of the old Mississippi.

Cauchy's early work on permutations was pivotal in the literal sense of that word. He harkened back to the preoccupations and problems of predecessors immediate and remote, recasting them into a new framework whose foundations last to this day and constitute the essential basis of group theory. Granted, everything in sight is a permutation group and it took some time for an abstract group theory to form and emancipate itself from early origins, even to outgrow the connection with algebraic functions and the term substitution (Cauchy's term for permutation). Camille Jordan's path-breaking book in 1870, for example, was titled Traité des substitutions et des équations algébriques, Eugen Netto's in 1882 Substitutionentheorie und ihre Anwendung auf die Algebra.

MM. Lagrange and Vandermonde are, I believe, the first who considered functions of several variables relative to the number of values they can return, when these variables are substituted for each other.

Noted historian of mathematics Carl Boyer called Euler's Introductio in Analysin Infinitorum "the foremost textbook of modern times"^[1] (guess what is the foremost textbook of all times). Published in two volumes in 1748, the Introductio takes up polynomials and infinite series (Euler regarded the two as virtually synonymous), exponential and logarithmic functions, trigonometry, the zeta function and formulas involving primes, partitions, and continued fractions. That's Book I, and the list could continue; Book II concerns analytic geometry in two and three dimensions. The Introductio was written in Latin^[2], like most of Euler's work. This article considers part of Book I and a small part. The Introductio has been massively influential from the day it was published and established the term "analysis" in its modern usage in mathematics. It is eminently readable today, in part because so many of the subjects touched on were fixed in stone from that day till this, Euler's notation, terminology, choice of subject, and way of thinking being adopted almost universally.

René Descartes in 1648.

Everyone knows that Descartes founded analytic geometry with his little essay La Géométrie, published in 1637 as an appendix and proof-of-concept for a work on philosophy, Discourse on Method. As always, the story is more complex and interesting than that. The canonical modern work in English is History of Analytic Geometry^[1] by Carl Boyer, which includes this from his much-admired antecedent Gino Loria (1862-1954):

In truth, whoever studies thoroughly the treatise of Apollonius on Conics must confess the profound analogy it bears to an exposition of the properties of the curves of second degree by means of Cartesian coordinates; not only do the fundamental properties employed by the Greek geometer to distinguish the three curves one from the other translate into the canonical equations of the same in Descartes' method, but many of the reasonings given, when translated into the ordinary language of algebra, answer to elimination, solution of equations, transformation of coordinates, and the like. What we would however seek in vain in the Greek geometer is the concept of a system of axes, given a priori, of the figure to be studied.

Issac Newton in 1689, at the height of his powers (age 46).

Nature and Nature's Laws lay hid in Night:
God said, 'Let Newton be!' and all was Light.

- Alexander Pope

Kepler's laws of planetary motion are:

1. The orbit of a planet is an ellipse with the Sun at one focus.
2. A line from a planet to the Sun sweeps out equal areas in equal times.
3. The square of the time of revolution of a planet is proportional to the cube of the transverse axis of its elliptical orbit, with the same constant of proportionality for all planets.

PG and DK are conjugate diameters and PG bisects chords QQ' and DK. Book 1, Proposition 21 of Apollonius's Conics implies that (PV×VG)/QV² = PC²/DC².

This image and caption are from J. Bruce Brackenridge's The Key to Newton's Dynamics: The Kepler Problem and the Principia^[1] (p 114). Newton used the result to prove that elliptical orbits imply an inverse square law. Note that PG and DK are "conjugate diameters" of the ellipse, meaning that PG is parallel to the tangent at D (or K - the tangents at D and K are themselves parallel). The situation is symmetric, so it is equally correct to say that DK is parallel to the tangent at P (or G). As the diagram suggests, all the chords parallel to one conjugate diameter are bisected by the paired congugate diameter. Conjugate diameters are an old concept going back at least to Apollonius; modern too since they map to perpendicular diameters of a circle through an affine transformation.

Archimedes is one of greatest mathematicians of all time (his name \( A \rho \chi \iota \mu \acute{\eta} \delta \eta \varsigma \) means "master of thought" in Greek). He lived in the third century BC in Syracuse in Sicily (287 BC - 212 BC), then an outpost of Greek civilization. He has been highly regarded since his own time, which is perhaps why much of his work survives. Not all of it though, The Method being turned up in 1906 in the Archimedes Palimpsest (see Wikipedia's write-up as well). What survives is sufficient to measure his stature; he plainly anticipated calculus and knew as well as anyone today what a proof is, heir to the great classical school of Greek mathematics and Euclid. There are a few stories. One is that he was relaxing in his bath pondering the question of whether the king's gold crown had been adulterated and in an instant conceived the notion of buoyancy that bears his name. He was so excited, he jumped up and ran naked through the town shouting "εὕρηκα!" (Eureka - I have found it). It does seem fanciful, but is based on his surviving work On Floating Bodies.