ALGEBRAS, SPACES, LOGICS. 517 



+ , , \f . 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 " Coordinate Geometry," involving an 

 algebra of form. Sprouting from a numerical stem, this soon tran- 

 scends merely metrical limits with a beautiful power of giving demon- 

 strations 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 opin- 

 ion, you should know that you do so because the child is almost iden- 

 tical 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 dptfytoc, 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 dic- 

 tionaries give it in probably its real imported sense, as meaning the 

 great rules of arithmetic. So Johnson in his old dictionary gives algo- 

 rithm, or algorism, as the six operations of arithmetic ; and the " Edin- 

 burgh Encyclopaedia " has it as the rules of arithmetic, or the art of 

 computing in some special way, and, finally, as the principles and nota- 

 tion 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 illustra- 

 tions of simplicity and seeming insignificance, let me call your atten- 

 tion 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 a b. But 

 suppose the number of rows should be equal to the number of columns, 



