Algebra

a branch of mathematics. Sir Isaac Newton aimed to indicate at least the origin of the subject by calling it "Universal Arithmetic." Of the several differences between arithmetic and algebra, two may be mentioned. Arithmetic stops at zero; algebra goes farther. If we count backward in arithmetic, we say four, three, two, one, zero; here arithmetic stops. In algebra we may continue: four, three, two, one, zero, minus one, minus two, minus three, and so on indefinitely, using minus to indicate quantities on the other side of zero. In algebra the signs + and - have been adopted to indicate the positive and negative quantities, as they are called. The quantities that correspond to arithmetical quantities are known as positive. Those of the opposite nature are called negative.

The nature of algebraic quantities may be illustrated further by reference to multiplication. If we use the terms of a descending series for multiplicands and employ a constant multiplier, we shall find that our products also form a descending series, and that they run into negative numbers, as for example:

Lest a false impression be given, it should be remembered that -2, for instance, is not to be regarded simply as two less than nothing. We may illustrate from the idea of property. A man who has nothing has zero. A hundred dollars more than zero is property. A hundred dollars less than nothing is a debt, and a debt is not only something, but it is a serious consideration. In the same way -2 is a real quantity.

We may take a negative quantity for multiplicands and use the terms of a descending series for multipliers:

As the products form an ascending series, we see that the product of two negative factors is a positive quantity.

A second respect in which algebra differs from arithmetic is the consideration of unknown quantities. In algebra we may add, subtract, multiply, and divide quantities without knowing or needing to know what they are. For example:

A farmer sold his sheep for m dollars and gained y dollars. What did they cost him?

Ans. (m-y) dollars.

A boy who earns b dollars a day spends x dollars a week. What can he save in three weeks' time?

Ans. (18b-3x) dollars.

The earliest traces of algebra are found among the Hindus. The following problem illustrates the flowery style of the Hindus:

"The square root of one-half the number of bees in a swarm has flown out upon a jessamine bush, eight-ninths of the whole swarm remained behind; one female bee flies around a male bee that is buzzing within a lotus flower into which he was allured in the night by its sweet odors, but is now imprisoned in it. Tell the number of bees.

Ans. 72."

The Egyptians and Babylonians had a knowledge of the elements of algebra. The following problem is from an Egyptian papyrus roll in the British Museum. It dates 2000 B. C. and is itself a copy of some older manuscript at that:

"Heap, its two-thirds, its one-half, its one-seventh, its whole, it makes 97."

The Egyptians seem to have used heap for our x to denote an unknown quantity.

The Greeks made progress in geometry, but did not advance beyond other ancient people in algebra. The Arabs gathered up what was known, probably from India as well as the Mediterranean world. Through them the subject was introduced to the western world. About 1228 algebra attracted the attention of Italian scholars. It was in an elementary stage. The most learned had not thought of algebra beyond the simplest quadratic equations.

The earliest printed algebra--in Latin, of course--appeared in 1494. It was prepared by an Italian friar, Lucas de Burgo. The earliest English algebra appeared at Cambridge. It was written by Thomas Recorde. He called his volume "The Whetstone of Wit." The old textbooks seem elementary and crude. The signs +, -, X, /, =, are all modern. Exponents and the symbols for square root are devices that have been adopted later.

It was then the practice among the cultivators of algebra, when they advanced a step, to conceal it carefully from their contemporaries, and to challenge them to resolve arithmetical questions, so framed as to require for their solution a knowledge of their own new-found rules. In this spirit did Ferreus make a secret of his discovery: he communicated it, however, to a favorite scholar, a Venetian named Florido. About the year 1535, this person, having taken up his residence at Venice, challenged Tartalea of Brescia, a man of great ingenuity, to a trial of skill in the resolution of problems by algebra. Florido framed his questions so as to require for their solution a knowledge of the rule which he had learned from his preceptor Ferreus; but Tartalea had, five years before this time, advanced further than Ferreus, and was more than a match for Florido. He therefore accepted the challenge, and a day was appointed when each was to propose to the other thirty questions. Before this time came, Tartalea had resumed the study of cubic equations, and had discovered the solution of two cases in addition to two which he knew before. Florido's questions were such as could be resolved by the single rule of Ferreus; while, on the contrary, those of Tartalea could only be resolved by one or other of three rules, which he himself had found, but which could not be resolved by the remaining rule, which was also that known to Florido. The issue of the contest is easily anticipated; Tartalea resolved all his adversary's questions in two hours, without receiving one answer from him in return.--Britannica.

I was just going to say, when I was interrupted, that one of the many ways of classifying minds is under the heads of arithmetical and algebraical intellects. All economical and practical wisdom is an extension or variation of the following arithmetical formula: 2 + 2 = 4. Every philosophical proposition has the more general character of the expression a + b = c. We are mere operatives, empirics, and egotists, until we learn to think in letters instead of figures.--Holmes, Autocrat of the Breakfast Table.