, , √. In the same century Vieta introduced letters as symbols for known as well as for unknown quantities, and by this great advance not only laid the foundation for the general theory of equations, but rendered possible the birth of new algebras, children of the first.
The next step, a vast one, was definitely accomplished, when, in 1637, Descartes published his "Coördinate Geometry," involving an algebra of form. Sprouting from a numerical stem, this soon transcends merely metrical limits with a beautiful power of giving demonstrations projective, positional, descriptive. It matters not whether you prefer to think of this as a new algebra or as a new application of the first algebra of natural number. But, if you take the second opinion, you should know that you do so because the child is almost identical with the parent in formal algorithm. And there is a word coming into general use in pure science, yet whose present meaning is scarcely to be gained from dictionaries. It is an interesting word both in its birth and growth. When the Greek learning passed to the Arabs, so did the word ὰριθμὁς, as it has come to us in arithmetic. When the Arab and Moorish learning passed into Europe, the al was confounded with the following word, and from the Spaniards came the g between them. Thus, when the Indian numerals were introduced, this word came with them, and the new figures were denominated (by Chaucer, for example) augrime (or algorithm) figures; and rightly enough as being used according to an algorithm, for the old mathematical dictionaries give it in probably its real imported sense, as meaning the great rules of arithmetic. So Johnson in his old dictionary gives algorithm, or algorism, as the six operations of arithmetic; and the "Edinburgh Encyclopædia" has it as the rules of arithmetic, or the art of computing in some special way, and, finally, as the principles and notation of any calculus. Here we see it has sprouted and come very nearly into its present acceptation, in which I would define it as the fundamental operations of an algebra with their assumed laws and notation. In the algebra of natural number there are seven such, for we put in one more since the days of Samuel Johnson. As illustrations of simplicity and seeming insignificance, let me call your attention a moment to the three direct operations, which you have always known.
Suppose in counting we make a mark for each thing and connect them by Stifel's sign of addition, 1 1 1 . . . . . 1. Then, if we go over them one, by one we have a mark to register our result. But, even without taking the trouble to count them, we can say they will amount to some number and call it "a." But suppose we have to count a lot of the same sort of rows all equal, we know that an actual count will give for each the same number which we have called "a," and we will get a as many times as we have rows; that is, a number of times, say b times, and the grand total will be a taken b times, or ab. But suppose the number of rows should be equal to the number of columns,
