James has to say upon the subject of primary work: "It is ... in the association of concretes that the child's mind takes most delight. Working out results by rule of thumb, learning to name things when they see them, drawing maps, learning languages, seem to me the most appropriate activities for children under thirteen to be engaged in. . . . I feel pretty confident that no man will be the worse analyst or reasoner or mathematician at twenty for lying fallow in these respects during his entire childhood."1

Approximations-There is a feeling among many teachers that some virtue attaches to the carrying of a result to a large number of decimal places, and hence this is rather encouraged among pupils. As a matter of fact the contrary is usually the case in practice. If the diameter of a circle has been measured correctly to 0.001 inch there is no use in attempting to compute the circumference to more than three decimal places, and 3.1416 is a better multiplier than 3.14159. The result should be cut off at thousandths and the labor of extending it beyond that place should be saved.

Now since we rarely use decimals beyond 0.001 except in scientific work, and since no result can be more exact than the data, and since even our scientific measurements rarely give us data beyond three or four decimal places, the practical operations are the contracted

1 Letter to F. A. Walker, in the latter's monograph, p. 22,

ones, those which are correct to a given number of places. For this reason, in this age of science, approximate methods are of great value in the higher grades which precede the study of physics. The following are types of such work: 1

For the same reason the practical use of a small logarithmic table is of great value in the computations of elementary physics. Two or three lessons suffice to explain the use of the tables and to justify the laws of operation, a small working table can be bought for five cents, and the field of physics affords abundant practice.

Reviews However much reviews may fail from their stupidity, as is apt to be the case with "set reviews," a skilful teacher is always reviewing in connection with the advance work. But there is one season when a review is essential, a brisk running

1 The explanations are given in any higher arithmetic, e.g. Beman and Smith, p. 8, 11.

over of the preceding work that the pupil may take his bearings, and this is at the opening of the school year. Such a refreshening of the mind, such a lubricating of the mental machinery, gets one ready for the year's work. Complaints which teachers generally make of poor work in the preceding grade are not unfrequently due to the one complaining; the effects of the long vacation have been forgotten; the engine is rusty and it needs oiling before the serious start is made.

In these reviews the same correctness of statement is necessary as in the original presentation, though not always the same completeness. To let a child say that 2 + 3 × 2 is 10 (instead of 8) is to sow tares which will grow up and choke the good wheat. To let him see forms like

2 ft. x 3 ft. = 6 sq. ft., 45°÷ 15 = 3 hrs.,

or to let him hear expressions like "As many times as 2 is contained in $10," "2 times greater than $3," etc., is to take away a large part of the value that mathematics should possess.

CHAPTER VI

THE GROWTH OF ALGEBRA

Egyptian algebra - Reserving for the following chapter the question of the definition of algebra, we may say that the science is by no means a new one. Or rather, to be more precise, the idea of the equation is not new, for this is only a part of the rather undefined discipline which we call algebra. In the oldest of extant deciphered mathematical manuscripts, the Ahmes papyrus to which reference has already been made, the simple equation appears. It is true that neither symbols nor terms familiar in our day are used, but in the so-called hau computation the linear equation with one unknown quantity is solved. Symbols for addition, subtraction, equality, and the unknown quantity are used. The following is an example of the simpler problems which Ahmes gives, his twenty-fourth: "Hau (literally heap), its seventh, its whole, it makes 19," which put in modern symbols means + x = 19. Somewhat more difficult

χ

7

problems are also given, like the following (his thirty-first): "Hau, its, its , its, its whole, it makes 33,"

It must be said, however, that Ahmes had no notion of solving the equation by any of our present algebraic methods. His was rather a "rule of false position," as it was called in medieval times, - guessing at an answer, finding the error, and then modifying the guess accordingly.1 Ahmes also gives some work in arithmetical series and one example in geometric.

Greek algebra-Algebra made no further progress, so far as now known, among the Egyptians. But in the declining generations of Greece, long after the "golden age" had passed, it assumed some importance. As already stated, the Greek mind had a leaning toward form, and so it worked out a wonderful system of geometry and warped its other mathematics accordingly. The fact that the sum of the first n odd numbers is n2, for example, was discovered or proved by a geometric figure; square root was extracted with reference to a geometric diagram; figurate numbers tell by their name that geometry entered into their study.

So we find in Euclid's "Elements of Geometry" (B.C., C. 300) formulae for (a + b)2 and other simple algebraic relations worked out and proved by geometric figures. Hence Euclid and his followers knew

1 Besides Eisenlohr's translation already mentioned, see Cantor, I, p. 38. A short sketch is given in Gow's History of Greek Mathematics,