Here's Looking at Euclid: Chapters Four–Five

He seems to have had some of the symptoms of high-functioning autism, for when he was taken to see the play Richard III, he was left nonplussed by the experience, although he notified his hosts that the actors had taken 5,202 steps and spoken 14,445 words.

Thirty years after the first cheap electronic calculators precipitated a widespread demise in mental calculation skills, a backlash is under way. Newspapers offer up daily math brainteasers, popular computer games with arithmetic puzzles sharpen our minds and—at the high end—lightning calculators compete in regular international tournaments.

The Mental Calculation World Cup was founded in 2004 by German computer scientist Ralf Laue and takes place every two years. It was the logical culmination of Laue’s two hobbies: mental arithmetic and collecting unusual records.

Conspicuously absent from proceedings in Leipzig, however, was perhaps the world’s most famous mathlete, the French student Alexis Lemaire, who prefers another yardstick to measure computational power.

This is a matter of deep controversy in the calculation milieu, mirroring the battle almost 200 years ago between Zerah Colburn and George Bidder, both exceptional at their own type of sum.

Lemaire has worked out algorithms, which he has not divulged, to calculate the other 14 digits in the final answer.

He was probably the last human calculator to have been employed as one. As computers developed, his skills became redundant, and in retirement he returned to showbiz, appearing frequently on TV.

The Greeks and the Chinese were hampered by cumbersome notation.

Leibniz had devised the formula using “the calculus,” a powerful type of mathematics in which a new understanding of infinitesimal amounts was used to calculate areas, curves and gradients.

Leibniz had devised the formula using “the calculus,” a powerful type of mathematics in which a new understanding of infinitesimal amounts was used to calculate areas, curves and gradients.

He encouraged his protégé William Shanks, an amateur mathematician who ran a boarding school in County Durham, to go even further.

His metaphysics was based on the belief that the world was made up of numbers and the harmonic proportions between them. The existence of a number that could not be described as a ratio diminished his position, at the very least, if it did not contradict him outright.

According to legend, their existence was first proved by the Pythagorean disciple Hippasus, which did not endear him to the Brotherhood; he was declared a heretic and was drowned at sea.

The early pi men may have hoped that after the initial chaos of 3.14159..., the noise would calm and a pattern ensue.

In the eighteenth century, they started to speculate about a special type of irrational called transcendental numbers.

These were numbers so mysterious and evasive that finite mathematics could not capture them.

Pi’s enigmatic attributes were, in the eighteenth and nineteenth centuries, revealed to be not only at the heart of ancient geometrical problems but also deeply rooted in new fields of science that were not always obviously related to circles.

The ubiquity of pi, however, has made it more than just a celebrity among numbers.

The record is held by Mats Bergsten, an actuary in Sweden in his late 50s, who has recited 9,778 digits while juggling with three balls.

When Akira Haraguchi recited 100,000 digits of pi by heart, he used a mnemonic technique, assigning syllables to each number from 0 to 9 and then translating pi’s decimals into words, which in turn formed sentences.

Entire poems—or “piems”—have been written under the constraint that the number of letters per word is determined by pi, usually with the convention that a 0 in the expansion requires a ten-letter word. The most ambitious piem is the Cadaeic Cadenza by Mike Keith, which follows pi for 3,835 digits.

It begins as a pastiche of Edgar Allan Poe:
One; A poem
A Raven
Midnights so dreary, tired and weary...

Silently pondering volumes extolling all by-now obsolete lore.

Pi has gone by this name only since 1706, when the Welshman William Jones introduced the symbol π in his book, the snappily titled A New Introduction to the Mathematics, for the Use of some Friends who have neither Leisure, Convenience, nor, perhaps, Patience, to search into so many different Authors, and turn over so many tedious Volumes, as is unavoidably required to make but tolerable progress in the Mathematics.

Euler was the most prolific mathematician of all time (he published 886 books), and he is possibly the one who most contributed to an understanding of pi.

Talking to the Chudnovskys was like wearing stereo headphones with an erratically alternating connection to both ears.

In ninth-century Baghdad, Muhammad ibn Musa al-Khwarizmi wrote a math primer entitled Hisab al-jabr w’al-muqabala, or Calculation by Restoration and Reduction.

Between the fifteenth and seventeenth centuries mathematical sentences moved from rhetorical to symbolic expression. Slowly, words were replaced with letters.

Within a few decades of Viète’s death, René Descartes published his Discourse on Method.

Were it not for issues of limited printing type, the Y-Factor could have become a phrase to describe intangible star quality and the African-American political leader might have gone by the name Malcolm Z.

When problems were expressed rhetorically, as in Egypt, mathematicians used ingenious, but rather haphazard, methods to solve them. These early problem solvers were like explorers stuck in a fog with few tricks to help them move about.

The surprise of the 1089 trick is that from a randomly chosen number we can always produce a fixed number. Algebra lets us see beyond the legerdemain, providing a way to go from the concrete to the abstract—from tracking the behavior of a specific number to tracking the behavior of any number.

Pierre de Fermat, a civil servant and judge living in Toulouse, was an enthusiastic amateur mathematician who filled his own copy of Arithmetica with numerical musings.

In any case, his cheeky sentence was fantastic bait to generations of mathematicians. The proposition became known as Fermat’s Last Theorem and was the most famous unsolved problem in math until the Briton Andrew Wiles cracked it in 1995.

Napier wrote a best-selling Protestant polemic in which he claimed the Pope was the Antichrist and predicted that the Day of Judgment would come between 1688 and 1700.

Hopp, a retired electrical engineer, is an extremely affable man with wispy eyebrows, blue eyes and luxuriant jowls.

Yet despite their inherent imprecision, Hopp said that—for his purposes as an engineer, at least—slide rules were accurate enough for most uses.

Herzstark, an Austrian whose father was Jewish, was given special dispensation to work on his calculating machine because he was known to the camp authorities as an engineering genius.

Having a slide rule at hand means you can work out estimates very quickly, and also have a more visual understanding of the numerical parameters of the flight. Flying is safer because of pilots’ dexterity with an early-seventeenth-century calculating machine.

One of the most obvious corollaries of Descartes’s insight that equations in x and y can be written as lines was the recognition that different types of equation produce different types of line.