their ratios by means of their multiples. For instance, the magnitude A is considered to have to the magnitude B the ratio of 2 to 3 when 3A = 2B. This system has the undeniable advantage of admitting commensurable and incommensurable quantities to be treated on a uniform plan. But it has the disadvantage of not according with the natural and customary way of thinking of the subject. When we say that the magnitude A is to B as 2 to 3, we mean that if A is represented by the number 2, or is divided into 2 parts, B will be represented by 3 of those parts. The author has considered it more important to base the subject on natural and customary modes of thought than to adopt a system simple and rigorous, but not so based. The mode in which he has endeavored to avoid the difficulty, and to render the natural system as rigorous and nearly as simple as the other, will be seen by an examination of the chapter on Proportion. VIII. Another difficult subject is the fundamental relations of lines and planes in space. In presenting it the author has been led to follow more closely the line of thought in Euclid than that in modern works. At the same time he is not fully satisfied with his treatment, and conceives that improvements are yet to be made. A collection of notes on the fundamental principles of geometry upon which the work has been based will be found in the Appendix. The author believes, from some trials, that the study of geometry as here presented can be advantageously commenced at the age of twelve or thirteen years. No especial knowledge of algebra is required for the first three books, but a previous familiarity with symbolic notation will facilitate the study of the second and following books, and may be found necessary to their advantageous use. From the fourth book onward a knowledge of simple equations is sometimes presupposed. SYMBOLS AND ABBREVIATIONS USED IN DEMONSTRATIONS. WHEN a step of a demonstration leads to a relation of two lines or other magnitudes, the relation is expressed by symbols. = equals states that two magnitudes are equal. I parallel to: states that two lines are parallel. I perpendicular: states that two lines are perpendicular to each other. coincides with, or falls upon: states that two points, lines, surfaces, or figures coincide with each other. In recitation, the teacher may find it advantageous to have the student recite the reasoning orally, but write the conclusion of each step on the blackboard. In this case symbols or abbreviations of the more common words will shorten the work. The following are recommended, though others are frequently used: These abbreviations are not generally used in the printed book, the author believing that the full word, in its usual form, will make a stronger impression on the mind of the beginner than any symbolic representation of it. |