5 i6 THE POPULAR SCIENCE MONTHLY. 



ALGEBEAS, SPACES, LOGICS. 



AX UNTECHNICAL ILLUSTRATION OF DEVELOPMENT IN PUKE SCIENCE. 

 By GEORGE BRUCE HALSTED, A. M., Ph. D. 



WHEN at the making of a new university a lot of specialists were 

 thrown together, I was impressed by their lack of information 

 in regard to the progress of the eldest of the family of sciences, mathe- 

 matics. One fellow, a graduate of the University of Virginia, said that, 

 from what had been taught him, he had come to believe mathematics 

 finished by Newton, and now he was puzzled by a talk of progress. 

 Another, an engineer thoroughly grounded in what the previous one 

 had considered all possible mathematic, asked what it could mean 

 this turning out of new algebras, this new geometrizing ? He had 

 heard that metaphysics was interminable, and knew that a pseudo- 

 philosopher could spin out metaphysic by the yard ; was this new 

 mathematic something of the same sort, or was it worth his looking 

 into ? and so on. Let me, then, try to give an untechnical illustration 

 of the fact that mathematic, though with a safe start of perhaps a 

 thousand years over the other sciences, may now lay claim to be more 

 than ever fundamentally and rapidly advancing, developing. From 

 the vast field of choice, let us, to fix the attention, confine ourselves 

 simply to what is involved in the addition of a single letter, s, to three 

 common words, algebra, space, logic ; that is, implied in getting a 

 plural to the ideas embodied in these words. 



Algebra has been and still is defined as universal arithmetic, and is 

 most commonly thought of as simply a generalized statement of the 

 truths about natural numbers. And historically such it was ; as such 

 it started, and was indeed a very gradual growth. In the first known 

 treatise on the subject by Diophantus, in the third or fourth century, 

 the few symbols employed are mere abbreviations for ordinary words. 

 The Arabians, who obtained their algebra from the Hindoos, did little 

 or nothing toward its extension, though it retains in its name an Arabic 

 touch, and the word algorithm, always, and now more than ever, asso- 

 ciated with it, has the Arabic al. It was after their treatises had been 

 carried into Italy by a merchant of Pisa, about 1200, that important 

 improvements began. About 1500 the first problem of the third 

 degree is said to have been solved. After that, Cardan first gave the 

 general solution of a cubic equation, and employed letters to denote 

 the unknown quantities, the given ones being still mere numbers. 

 Toward the middle of the sixteenth century algebra was introduced 

 into Germany, France, and England, by Stifel, Peletarius, and Robert 

 Recorde, respectively. Recorde endowed it with the symbol of rela- 

 tion = , and Stifel with the far more important symbols of operation, 



