History

Jupiter’s magic square

Posted on by John

Albrecht Dürer’s engraving Melencolia I contains an interesting magic square toward the top right corner.

Here’s a close-up of the magic square:

The square has the following properties:

  • Every row, column, and diagonal sums to 34.
  • The four squares in the center sum to 34.
  • The four squares in the corners sum to 34.
  • Each quadrant sums to 34.
  • The year the engraving was made, 1514, appears in the bottom row.

I’d seen all this years ago, but this week I learned something else about this square.

Magic squares of different sizes were traditionally associated with planets in the solar system. … the 4 × 4 square in Melancolia is Jupiter’s … One suggestion for Dürer’s use of the square is that it reflected the mystical belief that Jupiter’s joyfulness could counteract the sense of melancholy that pervades the engraving.

From The Number Mysteries.

Regarding “Jupiter’s joyfulness,” here’s the etymology of jovial from Online Etymology Dictionary.

1580s, from Fr., from It. joviale, lit. “pertaining to Jupiter,” from L. Jovialis “of Jupiter,” from Jovius (used as gen. of Juppiter) “Jupiter,” Roman god of the sky. The meaning “good-humored, merry,” is from astrological belief that those born under the sign of the planet Jupiter are of such dispositions. In classical L., the compound Juppiter replaced Old L. Jovis as the god’s name. Related: Jovially.

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History is strange

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From historian Patrick Allitt of Emory University:

History is strange, it’s alien, and it won’t give us what we would like to have. If you hear a historical story and at the end you feel thoroughly satisfied by it and find that it perfectly coincides with your political inclinations, it probably means that you’re actually listening to ideology or mythology. History won’t oblige us, and much of its challenge and interest comes from its immovable differentness from us and our own world.

The first FORTRAN program

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The first FORTRAN compiler shipped this week in 1957. Herbert Bright gives his account of running his first FORTRAN program with the new compiler here.

(Bright gives the date as Friday, April 20, 1957, but April 20 fell on a Saturday that year. It seems more plausible that he correctly remembered the day of the week—he says it was late on a Friday afternoon—than that he remembered the day of the month, so it was probably Friday, April 19, 1957.)

For more history, see Twenty Five Years of FORTRAN by J. A. N. Lee written in 1982.

Thanks to On This Day in Math for the story.

Augustine, Leibowitz, and evolution

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The following paragraph is from the science fiction novel A Canticle for Leibowitz:

A fourth century bishop and philosopher. He [Saint Augustine] suggested that in the beginning God created all things in their germinal causes, including the physiology of man, and that the germinal causes inseminate, as it were, the formless matter—which then gradually evolved into the more complex shapes, and eventually Man. Has this hypothesis been considered?

A Canticle for Leibowitz is set centuries after a nuclear holocaust. The war was immediately followed by the “Simplification.” Survivors rejected all advanced technology and hunted down everyone who was even literate. At this point in the book, a sort of Renaissance is taking place. The question above is addressed to a scientist who is explaining some of the (re)discoveries taking place. The scientist’s response was

“I’m afraid it has not, but I shall look it up,” he said, in a tone that indicated he would not.

Was the reference to Augustine simply made up for the novel, or is there something in Augustine’s writings that the author is alluding to? If so, does anyone know what in particular he may be referring to? Is such a proto-Darwinian reading of Augustine fair?

Thomas Jefferson and preparing for meetings

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Here’s an interesting historical anecdote from Karl Fogel’s Producing Open Source Software on the value of preparing for meetings.

In his multi-volume biography of Thomas Jefferson, Jefferson and His Time, Dumas Malone tells the story of how Jefferson handled the first meeting held to decide the organization of the future University of Virginia. The University had been Jefferson’s idea in the first place, but (as is the case everywhere, not just in open source projects) many other parties had climbed on board quickly, each with their own interests and agendas.

When they gathered at that first meeting to hash things out, Jefferson made sure to show up with meticulously prepared architectural drawings, detailed budgets for construction and operation, a proposed curriculum, and the names of specific faculty he wanted to import from Europe. No one else in the room was even remotely as prepared; the group essentially had to capitulate to Jefferson’s vision, and the University was eventually founded more or less in accordance with his plans.

The facts that construction went far over budget, and that many of his ideas did not, for various reasons, work out in the end, were all things Jefferson probably knew perfectly well would happen. His purpose was strategic: to show up at the meeting with something so substantive that everyone else would have to fall into the role of simply proposing modifications to it, so that the overall shape, and therefore schedule, of the project would be roughly as he wanted.

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Military intelligence from serial numbers

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During World War II, America and her allies needed to estimate the number of Panzer V tanks Germany had produced. The solution was simple: Look at the serial numbers of the captured tanks. If you assume the tanks had been sequentially numbered — as in fact they were — you could view the serial numbers of the captured tanks as random samples from the entire range. You could then use statistics to estimate the range and hence the number of tanks produced. More details available here.

A few years later America tried to use the serial number trick to estimate the number of Soviet strategic bombers. This time the trick backfired.

In 1958, American military intelligence believed the USSR would soon have four hundred Bison and three hundred Bear bombers capable of striking the American heartland. Their evidence was the high serial number of a Bison that had flown at a May Day parade in Moscow. In fact, the Soviets knew the Americans were watching, and intentionally inflated that number. — Rocket Men, page 118.

The Panzer estimate was accurate because the Allies had hundreds of data points, enough to support the assumption that the tanks were sequentially numbered and to make a good estimate of the total number.

The Bison bomber was only one data point, but it was consistent with what intelligence services (wrongly) believed. At that time, the US had grossly over-estimated the military capabilities of the USSR. According to Rocket Men, Khrushchev turned down US offers to cooperate in space exploration because he feared that such cooperation would give the US a more accurate assessment of his country’s military.

Related post: Selection bias and bombers

The grand unified theory of 19th century math

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The heart of 19th century math was the study of special functions arising from mathematical physics.

It is well known that the central problem of the whole of modern mathematics is the study of the transcendental functions defined by differential equations.

The above quote was the judgment of  Felix Klein (of Klein bottle fame) in 1893. The differential equations he had in mind were the second order differential equations of mathematical physics.

Special functions were the core of 19th century math, and hypergeometric series were the grand unifying theory of special functions. (Not every special function is hypergeometric, but quite a few are.) And yet they’re hardly taught any more. I never heard of hypergeometric series in college, even though I studied differential equations and applied math. Later I encountered hypergeometric functions first in combinatorics and only later in differential equations.

It’s odd that what was “the central problem of the whole of modern mathematics” could become almost a lost art a century later. How could this be? I believe part of the explanation is that special functions, and hypergeometric function in particular, fall between two stools: too advanced for undergraduate programs but not a hot enough of a research area for graduate programs.

Related: Consulting in differential equations

Earliest personal account of slavery

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According to William R. Cook, there is only one ancient account of slavery written by a slave that still survives: a letter written by Saint Patrick. We have many ancient documents that were written by slaves, but not documents about their experience of being a slave.

Patrick was born in Britain. He was kidnapped at age 16 and became a slave in Ireland. He served as a slave for six years before escaping and returning to Britain. Later he returned to Ireland as a missionary. Although there are many legends surrounding Patrick, historians generally agree that his autobiographical letter, now known as the Confession of St. Patrick, is authentic.

I was surprised to hear that there are no other extant autobiographies of slaves since there were many literate slaves in antiquity. Obviously slaves were not given the liberty to write about whatever they pleased, and slave owners would be unlikely to request candid biographies of their chattel. Still, I imagine some slaves wrote autobiographies, perhaps secretly. But it makes sense that such documents would not likely be preserved.

The lack of first-hand accounts of slavery may contribute to our rosy mental image of classical history. When we think of ancient Greece, we think of Plato and Aristotle, not the anonymous slaves who made up perhaps 40% of the population of classical Athens.

[Update December 2014: The information above comes from a Teaching Company course by William Cook. The original link is dead, and I don’t remember now which of his courses it was from.]

Computing before Fortran

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In the beginning was Fortran. Or maybe not.

It’s easy to imagine that no one wrote any large programs before there were compilers, but that’s not true. The SAGE system, for example, involved 500,000 lines of assembly code and is regarded as one of the most successful large computer systems ever developed. Work on SAGE began before the first Fortran compiler was delivered in 1957.

The Whirlwind computer that ran SAGE had a monitor, keyboard, printer, and modem. It also had a light gun, a precursor to the mouse. It’s surprising that all this came before Fortran.

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