In the Land of Invented Languages - Arika Okrent [10]
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—voilà!—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