[28] Speer, W. W., “The New Arithmetic,” Ginn & Co., Boston, 1896.
[29] Smith, David Eugene, “The Teaching of Elementary Mathematics,” Ch. V, The Macmillan Co., New York, 1900.
[30] Eckoff, William J., “Herbart’s A, B, C, of Sense Perception,” International Education Series, D. Appleton & Co., New York, 1896.
[254.] But now a subject comes up that, on account of the difficulties it causes, calls for special consideration, namely, that of logarithms. It is easy enough to explain their use, and to render the underlying concept intelligible as far as necessary in practice—arithmetical corresponding to geometrical series, the natural numbers being conceived of as a geometrical series. But scientifically considered, logarithms involve fractional and negative exponents, as also the application of the Binomial Theorem. The latter, to be sure, is merely an easy combinatory formula so far as integral positive exponents are concerned, but, limited to these, is here of comparatively little use.
Now, since trigonometry in its main theorems is independent of logarithms, but is little applied without their aid, the question arises whether beginners should necessarily be given a complete and vigorously scientific course in logarithms, the highly beneficial instruction in trigonometry being postponed until after the successful completion of such a course, or whether the practical use of logarithms is to be permitted before accurate insight into underlying principles has been gained.
Note.—The difficulty encountered in this subject—undoubtedly one of those difficulties most keenly felt in teaching mathematics—is after all only an illustration of the injurious consequences of former sins of omission. If the geometrical imagination were not neglected, there would be ample opportunity, not only for impressing far more deeply the concept of proportion, demanded even by elementary arithmetic, but also for developing early the idea of function. The object lessons mentioned above have already illustrated the dependence of tangents and secants on angles. When these relations of dependence have become as familiar as may be expected after a half year’s instruction, sines and cosines also are taken up. But it is not sufficient to leave the matter here. Somewhat later, about the time when mensuration is introduced, the squares and cubes of natural numbers must be emphasized, and very soon committed to memory. Next it should be pointed out how by finding the differences of squares and cubes respectively, and then adding these differences, the original numbers may be obtained again. A similar treatment should be accorded to figurate numbers.
Small wooden disks, like checker-pawns, commend themselves for the purpose. By means of them various figures are found. The pupils are asked to indicate how many disks they need to construct one or the other kind of figures. A further step will be to show the increase of squares and cubes corresponding to the increase of the root, and to make this information serve as the preparation for the elementary parts of differential calculus. Now the time has come for passing on to the consideration of consecutive values of the roots, which are found to differ by quantities of continuously decreasing smallness as one progresses continuously through the number system. And so, after the logarithms of 1, 10, 100, 1000, etc., also of 1/10, 1/100, etc., have been gone over many times, forward and backward, the conception is finally reached of the interpolation of logarithms.
[255.] In schools where practical aims predominate, logarithms should be explained by a comparison of the arithmetical with the geometrical series, and the practical application will immediately follow. But even where recourse is had to Taylor’s Theorem and the Binomial Theorem, the gain to the beginner will not usually be very much greater. Not as though these theorems, together with the elements of differential calculus, could not be made clear; the real trouble lies in the fact that much of what is comprehended is not likely to be retained in the memory. The beginner, when he comes to the application, still has the recollection of the proof and of his having understood it. Indeed, with some assistance he would be able, perhaps, to again retrace step by step the course of the demonstration. But he lacks perspective; and in his application of logarithms it is of no consequence to him by what method they have been calculated.
What has been said here of logarithms may be applied more generally. The value of rigid demonstrations is fully seen only when one has made himself at home in the field of concepts to which they belong.
It is customary in American schools to take up elementary algebra and elementary geometry upon the completion of arithmetic, both algebra and geometry being anticipated to some extent in the later stages of arithmetic. The following paragraphs from the pen of David Eugene Smith[31] indicate some of the advance in algebra since Herbart’s time:—