Numbers before computers
Seven steps from a reed in a pond to a program for the Bernoulli numbers. Each is a way of making numbers do more work for less effort, and by the end the methods are something a machine could carry out.
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Step 1: 九章算術·句股（節選） The Nine Chapters on the Mathematical Art: Base and Upright (excerpt)
今有池方一丈，葭生其中央，出水一尺。引葭赴岸，適與岸齊。問：水深、葭長各幾何？Suppose there is a square pond ten feet on a side, with a reed growing in the middle that stands one foot above the water. If the reed is pulled towards the bank, it just reaches the edge. How deep is the water, and how long is the reed?
Base three, upright four, hypotenuse five, then a log, a vine, a reed in a pond and a broken bamboo: the Nine Chapters states the rule once and lets ordinary things test it, and every answer can be checked with a little arithmetic; Euclid proves the same truth from first principles.
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Step 2: Elements, Book I: First Principles, Propositions 1 and 47
A point is that which has no part.
Twenty-three definitions, five postulates and five common notions, then every step citing the earlier step it rests on, until Proposition 47 reaches the right triangle again, proved rather than shown; Master Sun cares less about proof than about a procedure that never fails.
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Step 3: 孫子算經（選） The Mathematical Classic of Master Sun (selections)
今有物，不知其數。三三數之，賸二；五五數之，賸三；七七數之，賸二。問：物幾何？Suppose there are some things whose number is not known. Counted in threes, two are left over; counted in fives, three are left over; counted in sevens, two are left over. How many things are there?
Count in threes, fives and sevens, set down seventy, twenty-one and fifteen for each one left over, then take away one hundred and five: a recipe anyone can carry out without knowing why it works, which is what a program is; centuries earlier, Archimedes needed new names for numbers before he could count the sand.
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Step 4: The Sand-Reckoner (opening and conclusion)
There are some, king Gelon, who think that the number of the sand is infinite in multitude; and I mean by the sand not only that which exists about Syracuse and the rest of Sicily but also that which is found in every region whether inhabited or uninhabited.
To show King Gelon that the grains of sand are very many but not infinite, Archimedes builds names for numbers large enough to count a universe filled with sand, and pushes every assumption against himself so that the result holds anyway; Leibniz goes the other way, down to two digits.
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Step 5: Explication de l'arithmétique binaire Explanation of Binary Arithmetic
Les nombres étant réduits aux plus simples principes, comme 0 et 1, il paroit partout un ordre merveilleux.When numbers are reduced to the simplest principles, such as 0 and 1, a marvellous order appears everywhere.
With only 0 and 1 ‘a marvellous order appears everywhere’, and Leibniz, who saw Fuxi’s hexagrams through letters from Beijing, ends by dreaming of a script in which reasoning itself could be done by calculation; Boole begins to write that script.
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Step 6: An Investigation of the Laws of Thought — Chapter I: Nature and Design of this Work The Laws of Thought, Chapter I: Nature and Design of this Work
The design of the following treatise is to investigate the fundamental laws of those operations of the mind by which reasoning is performed
Boole sets out to find the laws by which the mind reasons and write them as a calculus, with one firm rule for symbols: a meaning once fixed must never quietly change in mid-argument; eleven years earlier, Lovelace had seen what a machine might do with symbols that are not numbers at all.
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Step 7: Notes by the Translator (Notes A and G) Notes on the Analytical Engine (Notes A and G)
The Analytical Engine has no pretensions whatever to originate any thing. It can do whatever we know how to order it to perform.
Lovelace sees that an engine which ‘weaves algebraical patterns just as the Jacquard-loom weaves flowers and leaves’ need not stop at number and might even compose music, then sets out the first published program and warns against exaggerating what the machine can do: this is where numbers meet the machine, and where the trail ends.