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:-- “The great revival of learning known as the Renaissance, in the sixteenth century, saw algebra take a fresh start after several centuries of complete stagnation. Tartaglia solved the cubic equation, and a little later Ferrari solved the biquadratic. By the close of the sixteenth century Vieta had put the keystone in the arch of elementary algebra, the only material improvements for some time to come being in the way of symbolism. For the next two hundred years the struggle of algebraists was for a solution of the quintic equation, or, more generally, for a general solution of an equation of any degree. “The opening of the nineteenth century saw a few great additions to the theory of algebra. The first was the positive proof that the general equation of the fifth degree is insoluble by elementary algebra, a proof due to Abel. The second was the mastery of the number systems of algebra,--the complete understanding of the negative, the imaginary, the incommensurable, the transcendent. Other additions were in the line of the convergency of series, the approximation of the real roots of numerical equations, the study of determinants--all finding their way into the elements, together with the theories of forms and groups, which must soon begin to influence the earlier chapters of the subject.