Cauchy on Permutations and the Origin of Group Theory

 Augustin Louis Cauchy
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.

Cauchy's Memoire Sur le Nombre des Valeurs

 Cauchy's Memoire Sur le Nombre des Valeurs, page 1

Memoire Sur le Nombre des Valeurs qu'une Fonction peut acquerir, lorsqu'on y permute de toutes les manieres possibles les quantites qu'elle renferme
(Paper on the number of values that a function can return when the variables it contains are permuted in all possible ways)
by A. L. CAUCHY, Engineer of bridges and roads
Journal de l'École polytechnique, Tome X
January 1815 (p 1-28)

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.

Reading Euler's Introductio in Analysin Infinitorum

 Introductio in Analysin Infinitorum Title Page

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.

Descartes and La Géométrie

 Rene Descartes 1648
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.

Wisconsin State Journal Death Spiral

 State Journal Gay Marriage Aug 27 2014

The predecessor of the Wisconsin State Journal was founded in Madison in 1839 when Wisconsin was a rude territory, and as Whigs they endorsed William Henry Harrison in 1840 - Wikipedia has a great write-up on this paper over the years. The State Journal played a heroic role as anti-slavery crusaders in the Glover case in Milwaukee in 1854, helping to defy the Fugitive Slave Act and spring recaptured slave Joshua Glover from jail (I remember reading about it circa 1970 in Eric Foner's Free Soil, Free Labor, Free Men and feeling oddly proud of my hometown paper). Like many Whigs, including Abraham Lincoln, they became Republicans and duly endorsed the Great Emancipator. They became a progressive and even muckraking organ in the period 1890-1916. Since then they've been a reliable right ring mouthpiece, recent timid attempts to transcend that heritage notwithstanding. They were virulent jingoists in the First World War, scurrilously attacking Fighting Bob La Follette. They were shameful McCarthyites from the get-go, endorsing him in every one of his state-wide campaigns, including the first run for Senate in 1944, when they were one of four papers to do so and the only one outside the Appleton area. They broke their workers' strike in 1977 (I was on the board of the strikers' paper, the Press Connection).

Newton and Kepler's Laws

 Issac Newton 1689
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.

Intersecting Chord Theorem for Ellipses

 Brackenridge Figure 5.12
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 and Pi

 Archimedes by Fetti

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.

Stephanie Miller in Madison

 Stephanie Miller

Stephanie Miller is a radio talk show host with a devoted national following, nowhere more than in Madison, Wisconsin. She's on the local progressive radio station, 92.1 the Mic, from 8:00 AM to 11:00 AM CST and has been for close to ten years (the poor thing starts at 6:00 AM in LA, aka insane o'clock). She's done shows in Madison at the Barrymore Theater four or five times and always sells out far ahead of time. Some friends and I went the first time - she was broadcasting the show from Madison, so out of bed at 6:00 and on the street at 6:30 to join the stream of people passing my door, two blocks down from the theater.

Bill O'Reilly had just said Madison people communed with Satan and a good number of the patrons had little Satan hats on that lit up the dark theater - blink, blink, blink all around. Stephanie has this running gag that she's a sot, box wine her favorite, so she starts in on that and a good 25% of the audience raised their beer cups in salute. That set her back for a moment, which was funny right there - you're in Wisconsin now, baby!

Gauss and the Fast Fourier Transform

 Gauss Stamp

The Fast Fourier Transform (FFT) is a modern algorithm to compute the Fourier coefficients of a finite sequence. Fourier will forever be known by his assertion in 1807 that any function could be expressed as a linear combination of sines and cosines, its Fourier series. "Any" was a little ambitious, counter-examples coming to the fore in due time. A fair amount of mathematics from that time to this has been devoted to refining Fourier's insight and studying trigonometric series, a subject that led Georg Cantor to founding set theory. Piecewise smoothness is sufficient for pointwise convergence on \( [-\pi, \pi] \):

\[ f(x) = {a_0 \over 2} + \sum_{j=1}^\infty \left( a_j \cos jx + b_j \sin jx \right), \]

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