“This hasty glance at the development of the subject is sufficient to show how it has been revolutionized in modern times. To-day it is progressing as never before. The higher culture is beginning to affect the lower; determinants have found place in the beginner’s course; graphic methods, objected to as innovations by some who are ignorant of their prominence in the childhood of science, are reasserting their rights; the ‘imaginary’ has become very real; the inheritances of the algebra-teachers’ guild are being examined with critical eyes, and many an old problem and rule must soon go by the board. It is valuable to a teacher to see what changes have been wrought so that he may join in the movement to weed out the bad, to cling to the good, and to reach up into the realm of modern mathematics to see if, perchance, he cannot find that which is good and usable and light-shedding for the elementary work.” The true order of elementary mathematics, according to Dr. Smith, is substantially as follows:-- 1. Elementary operations of arithmetic. 2. Simple mensuration, correlation with drawing, the models in hand:-- Inductive geometry--the primitive form of the science. 3. Arithmetic of business and of science, using the simple equation with one unknown quantity wherever it throws light upon the subject. 4. Simple theory of numbers, the roots, series, logarithms. 5. Elementary algebra, including quadratic and radical equations. 6. Demonstrative plane geometry begun before the algebra is completed and correlated with it. 7. Plane trigonometry and its elementary applications. 8. Solid geometry. Trigonometry. Advanced algebra, with the elements of differentiation and integration. “The student should then take a rapid review of his elementary mathematics, including a course in elementary analytic geometry and the calculus. He would then be prepared to enter upon the study of higher mathematics.” [31] Compare Smith, David Eugene, “History of Modern Mathematics,” in Merriman & Woodworth’s “Higher Mathematics,” Wiley, New York, 1896. 256. Demonstrations taking a roundabout way through remote auxiliary concepts are a grave evil in instruction, be they ever so elegant. Such modes of presentation are rather to be selected as start from simple elementary notions. For with these conviction does not depend on the unfortunate condition requiring a comprehensive view of a long series of preliminary propositions. Thus Taylor’s Theorem can be deduced from an interpolation formula, and this, in turn, from the consideration of differences, for which nothing is needed beyond addition, subtraction, and knowledge of the permutation of numbers. The following account of imaginary and complex numbers by Dr. David Eugene Smith is so lucid that it is given at length:--