*THE POPULAR SCIENCE MONTHLY.*

ALGEBRAS, SPACES, LOGICS. |

AN UNTECHNICAL ILLUSTRATION OF DEVELOPMENT IN PURE SCIENCE.

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, mathematics. 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, associated 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 relation , and Stifel with the far more important symbols of operation,