“Herbart’s A, B, C, of Sense Perception,” together with a number of minor educational works, has been translated into English.[30] It abounds in shrewd observations and ingenious devices, yet as a whole it represents one of those side excursions, which, though delightful to genius, is not especially useful to the world. To drill children into the habit of resolving a landscape into a series of triangles, may indeed be possible, but like any other schematization of the universe, is too artificial to be desirable. Nevertheless, a limited use of the devices mentioned in this section might tend to quicken an otherwise torpid mind. [27] McLellan & Dewey, “The Psychology of Number,” International Education Series, D. Appleton & Co., New York, 1895. [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.