mentary school. The thought of demands the thought of both numbers, 7 and 8, and the thought of their modification each through the other.

The methods in vogue in elementary schools are chiefly based on the idea that it is necessary to eliminate the ratio idea by changing one of the terms of the fraction to a qualitative unit and by this to change the thought to that of a simple number. Thus halves and quarters and cents and dimes are thought as individual things, and the fractional idea suppressed.

In the differential calculus ratio is most adequately expressed as the fundamental and true form of all quantity, number included. The differential of a and the differential of y are ratios.

The authors of this book have presented in an admirable manner this psychological view of number, and shown its application to the correct methods of teaching the several arithmetical processes. The shortcomings of the "fixed-unit" theory are traced out in all their consequences. The defects of a view which makes unity a qualitative instead of a quantitative idea are sure to appear in the methods of solution adopted.

Pupils studying music by the highest method learn thoroughly those combinations which involve double counterpoint. As soon as the hands are trained to readily execute such exercises the pupil can take up a sonata of Beethoven or a fugue of Bach, and soon become familiar with it. On the plan of the old lessons in counterpoint, the pupil found himself helpless before such a composition. His phrases furnished no key to the compositions of Bach or Beethoven, because the latter are constructed on a different counterpoint.

So the methods of teaching arithmetic by a "fixedunit" system do not lead towards the higher mathematics, but away from it. They furnish little, if any, training in thinking the ratio involved in the very idea of number.

The psychology of number requires that the meth. ods be chosen with reference to their power to train the mind of the pupil into this consciousness of the ratio idea. The steps should be short and the ascent gradual; but it should be continuous, so that the pupil constantly gains in his ability to hold in consciousness the unity of the two aspects of quantity, the unity of the discrete and the continuous, the unity of the multiplex and the simple unit.

Measurement is a process that makes these elements clear. The constituent unit becomes the including unit, and vice versa, through being measured and being made the measure of others. This, too, is involved in using the decimal system of numeration, and in understanding the different orders of units, each of which both includes constituent units and is included as a constituent of a higher unit.

The hint is obtained from this that the first lessons in arithmetic should be based on the practice of measuring in its varied applications.

Again, since ratio is the fundamental idea, one sees how fallacious are those theories which seek to lay a basis for mathematics by at first producing a clear and vivid idea of unity-as though the idea of quantity were to be built up on this idea. It is shown that such abstract unit is not yet quantity nor an element of quantity, but simply the idea of individuality, which is still

a qualitative idea, and does not become quantitative until it is conceived as composite and made up of constituent units homogeneous with itself.

The true psychological theory of number is the panacea for that exaggeration of the importance of arithmetic which prevails in our elementary schools. As if it were not enough that the science of number is indispensable for the conquest of Nature in time and space, these qualitative-unit teachers make the mistake of supposing that arithmetic deals with spiritual being as much as with matter; they confound quality with quantity, and consequently mathematics with metaphysics. Mental arithmetic becomes in their psychology "the discipline for the pure reason," although as a matter of fact the three figures of the regular syllogism are neither of them employed in mathematical reasoning. W. T. HARRIS.

WASHINGTON, D. C., June 25, 1895.

PREFACE.

Ir is perhaps natural that a growing impatience with the meagre results of the time given to arithmetic in the traditional course of the schools should result in attacks upon that study. While not all educational experts would agree that it is the "most useless of all subjects" taught, there is an increasing tendency to think of it and speak of it as a necessary evil, and therefore to be kept within the smallest possible bounds. However natural this reaction, it is none the less unwise when turned against arithmetic itself, and not against stupid and stupefying ways of teaching it. So conceived, the movement stands only for an aimless swing of the scholastic pendulum, sure to be followed by an equally unreasonable swing to the other extreme. If methods which cut across the natural grain of the mental structure and resist the straightforward workings of the mental machinery, waste time, create apathy and disgust, dull the power of quick perception, and cultivate habits of inaccurate and disconnected attention, what occasion for surprise? Because wrong methods breed bad results, it hardly follows that education can be made symmetrical by omitting a subject which stands par excellence for clear and clean cut methods