4 is the length and 5 the diagonal. What is the breadth? Its size is not known. 4 times 4 is 16. 5 times 5 is 25. You take 16 from 25 and there remains 9. What times what shall I take in order to get 9? 3 times 3 is 9. 3 is the breadth.
And expressed a little more abstractly by Euclid a couple millennia later:
In right-angled triangles the square on the side subtending the right angle is equal to the squares on the sides containing the right angle.
And Copernicus, over fifteen hundred years after that, taking advantage of the theorem to solve the position of Venus:
It has already been shown that in units whereof DG is 303, hypotenuse AD is 6947 and DF is 4997, and also that if you take DG, made square, out of both AD and FD, made square, there will remain the squares of both AG and GF.
This is how math was done. The clarity of your explanations depended on the vocabulary you chose, the order of your clauses, and your personal style, all of which could cause problems. Here, for example, is Urquhart, in his “voices of angels” trigonometry book, doing something somehow related to the Pythagorean theorem—it’s hard to tell:
The multiplying of the middle termes (which is nothing else but the squaring of the comprehending sides of the prime rectangular) affords two products, equall to the oblongs made of the great subtendent, and his respective segments, the aggregate whereof, by equation, is the same with the square of the chief subtendent, or hypotenusa.
It is possible to do mathematics like this, but the text really gets in the way. Wait, which sides are squared? What is taken out of what? What was that thing three clauses ago that I’m now supposed to add to this thing? Late-sixteenth-century scientists who were engaged in calculating the facts of the universe had a sense that the important ideas, the truths behind the calculations, were struggling against the language in which they were trapped. The astronomer Johannes Kepler had turned to musical notation (already well developed at that time) in an effort to better express his discoveries about the motions of the planets, yielding “the harmony of the spheres.” But musical notation could only go so far. The development of mathematical notation in this context was nothing short of revolutionary.
The notational innovations of the seventeenth century—symbols and variables instead of words, equations instead of sentences—not only made it easier to keep track of which thing was which in a particular calculation; they also made it easier to see fundamental similarities and differences, and to draw generalizations that hadn’t been noticed before. In addition, the notation was universal; it could be understood no matter what your national language was. The pace of innovation in science accelerated rapidly. Modern physics and calculus were born. It seemed that the truth was finally being revealed through this new type of language. A tantalizing idea took hold: just imagine what might be revealed if we could express all of our thoughts this way.
But how do you turn the world of discourse into math? Three primary strategies emerged from the competitive flurry of schemes whipped up by this challenge, two so superficial they allowed the illusion of success (leaving the egos of the authors undisturbed), and one so ambitious that those who attempted to implement it could only be humbled by the enormity of the task it revealed.
The first strategy was to simply use letters in a number-like way. When you combine the letters or do some sort of computation with them (the nature of that computation being very vaguely described), you get a word and—voila!—a language. This was Urquhart’s approach. He had tried a version of this strategy in his trigonometry book when he assigned letters to concepts, such as E for “side” and L for “secant,” and then formed words out of the letters to express statements like Eradetul, meaning “when any of the sides is Radius, the other of them is a Tangent, and the Subtendent a Secant.” He thought a similar approach could be used to make precise, definition-containing words for everything in the universe. All you needed was the right alphabet, and he claims to have devised one so perfect that not only can it generate distinct words for all possible meanings, but the words for stars will show you their exact position in the sky in degrees and minutes, the words for colors will show their exact mixture of light, shadow, and darkness, the names of individual soldiers will show their exact duty and rank. What’s more, in comparison with all other languages, it produces the best prayers, the most elegant compliments, the pithiest proverbs, and the most “emphatical” interjections. And besides all that, it is the easiest to learn. He stops short of claiming that it whitens your teeth and cures impotence, but he might as well have. His claims can’t be disproved, because he doesn’t provide any examples.
The second strategy was to turn words into numbers. This was the approach of Cave Beck, an Ipswich schoolmaster who published his invention (The Universal Character: By Which All the Nations in the World May Understand One Anothers Conceptions) in 1657. He assigned numbers to concepts: 1 was “to abandon,” 2 “to abash,” 3 “to abate,” 742 “to embroider,” q2126 “gogle-eyed,” r2654 “a loosenesse in the belly,” p2846 “hired mourners at funerals.” (Letters appearing before the numbers were used to indicate part of speech and grammatical concerns such as tense and gender.) He provided a pronunciation key for the numbers so that the language could be spoken out as words (for example, 7 is pronounced “sen”). Though the book opens with a series of poems (by his friends) praising Beck and his invention, his confidence is far less blustery than Urquhart’s; he presents his system as merely a practical tool for translating between languages. However, with an ambitious gleam in his eye, he adds that if it should happen to become a universal language that could unlock “Glorious Truths,” he will “judge this pains of mine happily bestowed.” He provides only one example of the language in action, the fifth commandment. Honor thy father and thy mother, “leb 2314 p2477 & pf2477,” to be pronounced, “Leb toreonfo, pee to-fosensen et pif tofosensen.”
There is an assumption in these approaches that all you have to do to build a perfect language is find the right set of symbols—whether letters, numbers, or line drawings. The focus on symbols was influenced by other, related popular pursuits of the time such as cryptography, shorthand, and kabbalism (seeking divine messages in patterns of letters in ancient texts). Another influence was the widespread interest in hieroglyphics and Chinese writing, which were believed to represent concepts more directly than alphabetic writing systems. But if your goal is to craft a language capable of mathematically exposing the truths of the universe, the form of the symbols you use is relatively unimportant. What is more important is that systematic relations obtain between the symbols. The number 1 stands for the concept of oneness, and 100 stands for the concept of onehundredness, but, more important, there is a relationship between oneness and onehundredness that is captured by the relationship between the symbols 1 and 100. And it is the same relationship that obtains between 2 and 200. In Beck’s system there is no such relationship between 1 (abandon) and 100 (agarick—a type of mushroom), and if you do find a way to read a relationship into them, it won’t be the same as the one between 2 (abate) and 200 (an anthem). The numbers are just labels for words. They might as well be words. Both Beck and Urquhart had a vague sense that symbols were capable of systematically capturing relationships between concepts, but they never did the hard work of applying this idea to language.
They could have learned a thing or two from the humble Francis Lodwick, a Dutchman living far from home in London whose 1647 book, A Common Writing, was signed simply “a Well-wilier to Learning.” In his preface he apologizes for the “harshnesse of [his] stile” and entreats “a more abler wit and Pen, to a compleate attyring and perfecting of the Subject.” His modesty was partly due to a feeling of inferiority, life- station-wise. He was a merchant with no formal education, which, in the opinion of the author of a later scheme, made him “unequal to the undertaking.” But his modesty was also of the hard-earned type—the modesty that all
