Chapter 11
The people history skipped
This chapter is not an appendix to the story. Several of these people did work that is in the standard curriculum under somebody else's name, and the reasons they are missing are documented rather than vague.
Sophie Germain
Ranked by what a sixteen-year-old will remember:
- Her parents confiscated her candles, her fire, and her clothes to stop her studying at night. They gave up when they found her asleep in the library at dawn with the ink frozen solid in the inkwell (Source 410, OConnor and Robertson).
- She got into mathematics because she read that Archimedes was killed by a Roman soldier for refusing to stop doing geometry, and decided anything worth dying over was worth doing (Source 410, OConnor and Robertson).
- She corresponded with Gauss for three years as "Monsieur LeBlanc". She revealed herself only because Napoleon's army occupied Gauss's town and she was afraid he would die the way Archimedes did, so she sent a French general to check on a man she had never met (Source 410, OConnor and Robertson). Gauss's letter on learning the truth is verified.
- The astronomer Lalande told her to put down Laplace and read his Astronomy for Ladies, a book with not one equation in it. He apologized in writing on 4 November 1797. She never forgave him (Source 410, OConnor and Robertson).
- She won the Paris Academy prize for elastic plate theory (1815 or 1816 in different sources). Seventy-two names are carved around the first stage of the Eiffel Tower, a structure that stands up because of elasticity theory. Hers is not one of them (Source 410, OConnor and Robertson).
Mary Somerville
Mechanism of the Heavens (1831) was her translation-and-expansion of Laplace, and it went into use at Cambridge: George Peacock wrote to her on 14 February 1832 about its adoption there (Source 412, Somerville). Whether the adoption was formal and required, or a strong recommendation, is not settled by the source; use Peacock's own wording rather than the phrase "required text".
- She was asked to translate a book that, by the commissioner's own estimate, fewer than twenty people in England understood. She agreed on condition that if she failed, the manuscript would go in the fire (Source 412, Somerville).
- Poisson told her husband, at a dinner in Paris where the whole table was congratulating her, that there were "not twenty men in France" who could read her book (Source 412, Somerville).
- Her mathematics books were left to Girton, the first Cambridge college for women, a college that did not exist when she was refused an education (Source 412, Somerville).
Her best line for a nervous student: do not confuse the mathematics required to discover something with the mathematics required to understand it (Source 413, Somerville). A student intimidated by calculus is, in Somerville's diagnosis, making exactly that mistake.
The usual story about the word "scientist" is wrong in two places. The word first appears in print in an anonymous 1834 Quarterly Review article now attributed to William Whewell. The article reviews On the Connexion of the Physical Sciences, not Mechanism of the Heavens, and Whewell does not claim the coinage, he credits "some ingenious gentleman" (Source 411, Whewell; Source 442, OConnor and Robertson). So "Whewell coined the word" is a sentence this book will not write. The word appears nowhere in Somerville's memoir. What is true and better: the word covering every chemist, geologist, and astrophysicist on Earth was born in a book review, as a joke the reviewer thought had not caught on, in the same paragraph as the rejected alternative "nature-poker", and the book under review was written by a woman who had never been allowed to attend a university (Source 411, Whewell).
Whewell was the nineteenth century's naming service. When Faraday needed words for the parts of an electrolysis cell, Whewell supplied anode, cathode, and ion (Source 442, OConnor and Robertson).
Ada Lovelace
Note G of her 1843 translation-and-notes contains an algorithm for computing Bernoulli numbers.
- She states in print that she chose the harder Bernoulli formula on purpose, to show the machine off (Source 415, Menabrea (trans. and notes by Lovelace)).
- The Analytical Engine was never built. The algorithm in Note G was, for over a century, a program with no computer (Source 415, Menabrea (trans. and notes by Lovelace)).
- The Engine's control system came from the silk-weaving industry: punched cards from the Jacquard loom. Her sentence about the engine weaving algebraical patterns as the loom weaves flowers and leaves is the origin of the whole "software as fabric" family of metaphors (Source 415, Menabrea (trans. and notes by Lovelace)).
What she did and did not do is a live scholarly dispute. Present both sides; do not pick one. And note the connection: Somerville was Lovelace's mentor and the person who brought her to Babbage's house (Source 416, OConnor and Robertson).
Sofia Kovalevskaya
The obstacle was never the mathematics. It was a registrar (Source 374, O'Connor). Universities would not admit her, so Weierstrass taught her privately. She solved a partial differential equations problem well enough to have her name permanently attached to Cauchy's, and she was not allowed in the building.
The Cauchy-Kovalevskaya theorem is the only major theorem in the classical canon named jointly for a French royalist man and a Russian radical woman, born sixty-one years apart, who never met (Source 374, O'Connor).
She took her doctorate from Göttingen in 1874 in absentia, won the Prix Bordin, and became a professor at Stockholm in 1889. Sources disagree on her Berlin arrival date, the Prix Bordin year, and the exact "first in northern Europe" formulation. She also wrote a novel and a memoir good enough to be translated and sold on their literary merits; her Swedish circle knew her as both (Source 417, Kovalevsky).
Grace Chisholm Young
This is the strongest single artifact in the whole credit theme. A letter from her husband, William Young, in plain words:
The fact is that our papers ought to be published under our joint names, but if this were done neither of us get the benefit of it. No. Mine the laurels now and the knowledge. Yours the knowledge only. Everything under my name now, and later when the loaves and fishes are no more procurable in that way, everything or much under your name. At present you cannot undertake a public career. You have your children. I can and do. (Source 419, OConnor and Robertson)
Guess before you read on
Read the middle of that letter again. The arrangement is explicitly temporary: his name now, hers later, once the loaves and fishes stop coming that way. Guess how many papers it took before they switched.
I have a guess
They never switched. Two hundred and twenty papers later it stood exactly as he had written it (Source 419, OConnor and Robertson).
If you took him at his word, so did she, and she was the one with everything riding on it. When somebody asks how work ends up credited to the wrong person, this is the answer in the participants' own handwriting.
Two hundred and twenty papers later, they had not switched (Source 419, OConnor and Robertson). When a student asks how work gets credited to the wrong person, this letter is the answer in the participants' own handwriting.
Emmy Noether
For four years she lectured at Göttingen under Hilbert's name because she was not allowed to hold a post. Students went to "Hilbert's" course and found Noether at the board (Source 418, OConnor and Robertson).
Noether's theorem is why a physics student can say why energy is conserved: because the laws do not change from one moment to the next (Source 418, OConnor and Robertson).
Two stories about her turn up everywhere and neither one could be verified here: Hilbert's retort to the faculty senate, "this is a university, not a bath-house", and Einstein's New York Times letter about her. Both are in Appendix K.1, which is where this book keeps the things it could not stand behind (Source 418, OConnor and Robertson).
Mary Cartwright
Chaos theory was found because radio engineers complained. During the Second World War the Radio Research Board circulated a plea for help with valve amplifiers whose output went strange. Cartwright and Littlewood took the van der Pol equation seriously and found solutions that never settle down (Source 420, OConnor and Robertson). That is chaotic dynamics, twenty years before Lorenz's weather model and thirty before anyone called it chaos.
She left school teaching because she was not allowed to experiment with how she taught, and then spent her research life on equations nobody could predict (Source 420, OConnor and Robertson).
Julia Robinson
Hilbert's Tenth Problem asked for a procedure to decide whether any polynomial equation in whole numbers has a whole-number solution. Robinson spent decades reducing the question to one missing ingredient. In 1970 a 22-year-old in Leningrad, Yuri Matiyasevich, supplied it, using Fibonacci numbers. The answer: no such procedure exists. Robinson wrote to him at once; they became friends and collaborators across the Cold War (Source 421, OConnor and Robertson).
Katherine Johnson and the NASA computers
The strongest single fact: in 1962 the most advanced computer in the United States produced a trajectory, and John Glenn refused to fly until a human being had done the same arithmetic by hand and agreed with it (Source 422, Shetterly). NASA's own words record his request to "get the girl". That is what checking your work means at 17,500 miles per hour.
NASA TN D-233 (Skopinski and Johnson, 1960), read in full for this book, runs the problem backwards. Everyone imagines rocket science as "we launch, where does it land?" Johnson and Skopinski wrote the reverse: pick the spot on Earth where you want the capsule to come down, count the orbits, and the equations hand you the compass bearing the rocket must point at when the engines cut out (Source 423, Skopinski and Johnson). Her own summary:
You tell me when you want it and where you want it to land, and I'll do it backwards and tell you when to take off. (Source 423, Skopinski and Johnson, quoted by MacTutor)
Her own pick for her best work was not the famous one. She said it was the Apollo rendezvous: getting the lunar module to find the command module again in orbit around the Moon (Source 422, Shetterly). And she got author credit on that report only because her co-author wanted to move to Houston and told their supervisor, who did not like women, that she should finish it since she had done most of it anyway (Source 422, Shetterly; Source 424, OConnor and Robertson).
Her line about the era: "There were no textbooks, so we had to write them." A sixteen-year-old learning calculus from a textbook should know that the people who worked out how to put a human in orbit did it without one (Source 424, OConnor and Robertson).
A correction, and it matters because the claim is everywhere: the widely repeated statement that Johnson used "Euler's method" for the Friendship 7 calculations is absent from NASA's biography, from MacTutor, and from TN D-233 itself, which uses iterative correction of Kepler equations plus partial-derivative sensitivity analysis (Source 422, Shetterly; Source 423, Skopinski and Johnson). Three independent sources are silent on it. No source read for this book supports it.
Gladys West computed the geoid. Every phone map on Earth silently uses a model of the planet's shape that is not a sphere and not even an ellipsoid, but a lumpy surface, because gravity is stronger over mountain ranges and weaker over trenches, so "sea level" is not level. West's team computed that surface on a 1960s mainframe, one correction at a time (Source 426, OConnor and Robertson).
She did it for 42 years and never mentioned it to her friends. She said she "just thought it was my work". She found out it mattered when she was 87 (Source 426, OConnor and Robertson).
Mary Jackson took a demotion on purpose. After nearly twenty years as an engineer she looked at the promotion statistics for women at Langley, concluded the ceiling was structural, and took a lower-graded job whose entire purpose was to change hiring and promotion for the next generation (Source 425, Shetterly). That is a harder kind of courage than films usually show.
Seki Takakazu, and Japanese calculus
Seki Takakazu (関 孝和) was born, on the usual dating of 1642, which some sources dispute in favor of 1637, in the same year as Newton, on the other side of a sealed border, and ended his career as an accounts examiner for a military government, exactly as Newton ended his running the Royal Mint. Neither ever heard of the other (Source 429, OConnor and Robertson).
- He got into mathematics because a servant in the house noticed a nine-year-old was good at it.
- He had determinants ten years before Leibniz and the Bernoulli numbers before Bernoulli, and because Japan was closed under sakoku, none of it reached Europe. Every one of those results had to be discovered a second time (Source 429, OConnor and Robertson).
- His tombstone calls him "The Arithmetical Sage".
Takebe Katahiro, his pupil, computed π to more than forty decimal places in 1722 using a convergence-acceleration trick that Western numerical analysis rediscovered in 1927 (Richardson) and 1955 (Romberg) (Source 430, OConnor and Robertson). He also wrote a book whose stated purpose was to explain how to do research: twelve worked examples of investigation rather than twelve theorems, one of the earliest self-conscious accounts of mathematical method anywhere (Source 430, OConnor and Robertson).
And the misattribution runs in the other direction too. The enri, the "circle principle" that made Japanese calculus possible, was credited to Seki for two hundred years and has recently been handed back to Takebe (Source 430, OConnor and Robertson). Misattribution is not a Western monopoly.
Srinivasa Ramanujan
- Before Hardy, he wrote to three English mathematicians. One replied without understanding him and two did not reply at all. The most famous letter in the history of mathematics was the fourth attempt (Source 431, OConnor and Robertson).
- He learned what mathematics looks like on the page from a crammer's revision handbook of about 5,000 unproved formulae, which is why the world spent a century proving his statements for him (Source 431, OConnor and Robertson).
- He could not get into university because he was brilliant at mathematics and neglected everything else. The examination system that failed him is the direct ancestor of the one his readers are sitting (Source 431, OConnor and Robertson).
- Divergent series, the thing Cauchy, Abel, and Weierstrass spent a century declaring illegitimate, were the thing he was best at. Two decades after his death Hardy wrote Divergent Series and made the subject respectable again (Source 431, OConnor and Robertson).
One person left out, and why
Wang Zhenyi (王贞仪, 1768 to 1797) appears in many popular lists of women in mathematics. She is not in this book, for two reasons stated openly. First, the one peer-reviewed open-access study located (Tamboukou, EJLW 15, 2026) was robots-blocked and could not be read in full, so nothing about her could be sourced to the standard this book uses. Second, and more decisive, her work is Qing calendrical astronomy, not calculus (Source 449, Tamboukou). Including her would be padding.
One question before you go
Noether's theorem lets you say why energy is conserved, instead of only that it is. What property of the laws makes it true?
Show the answer
They do not change from one moment to the next. Symmetry in time gives you conservation of energy (Source 418, OConnor and Robertson). She worked at Göttingen for four years lecturing under Hilbert's name, because she was not allowed a post of her own, and students who signed up for his course found her at the board.
Integrals 3.6 has you computing work as an integral of force, and the number that comes out is only worth having because energy is conserved. Her theorem is what sits under the exercise.