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the present century have been made. He knows not even the terms in which the ideas of such writers as PONCELET, CHASLES, and SALMON, are expressed, and he is quite as much a stranger to the thought. In this part are presented the fundamental ideas concerning Loci, Symmetry, Maxima and Minima, Isoperimetry, the theory of Transversals, Anharmonic Ratio, Polars, Radical Axes, and other modern views concerning the circle.
PART IV. is Plane and Spherical Trigonometry, with the requisite Tables. While this Part, as a whole, is much more complete than the treatises in common use in our schools, it is so arranged that a shorter course can be taken by such as desire it. Thus, for a shorter course in Plane Trigonometry, see NOTE on page 55. In Spherical Trigonometry, the first three sections, either with or without the Introduction on Projection, will afford a very satisfactory elementary course.
A few words as to the manner in which this plan has been executed, may be important. In general, the Definitions are those usually given, with such slight alterations as have been suggested by reflection and experience. There are, however, a few exceptions. Among these is the definition of an Angle. I can but regard the attempt to define an angle as The difference in direction between two lines, or The amount of divergence, as needlessly vague, abstract, and perplexing to a student, as well as questionable on philosophical grounds. The definition given in the text will be seen to be, at bottom, the old one, the conception being slightly altered to bring it into more close connection with common thought, and also with the idea of an angle as generated by the revolution of a line. As to Parallels, and the definition of similarity, my experience as a teacher is decidedly in favor of retaining the old notions. So also in adopting a definition of a Trigonometrical Function, I am compelled to adhere to the geometrical conception. A ratio is a complex concept, and consequently not so easy of application as a simple one. For this reason, among others, I prefer the differential to the differential coefficient, in the calculus, and a line to a ratio, in Trigonometry. Moreover, I have found that students invariably rely upon the geometrical conception, even when first taught the other; hence I am not surprised that all our writers who define a trigonometrical function as a ratio, hasten to tell the pupil what it means, by giving him the geometrical illustrations. Nor are the superior facility which the geometrical conception affords for a full elucidation of the doctrine of the signs of the functions, and its admirable adaptation to fix these laws in the mind, considerations to be lost sight of in selecting the definition.
Surely no apology is needed, at the present day, for introducing the idea of motion into Elementary Geometry, notwithstanding the rigorous and disdainful manner with which its entrance was long resisted by the old Geometers. And, having admitted this idea, the conception of loci as generated by motion would seem to follow as a logical necessity. In like manner, I take it, the Infinitesimal method must come in. Its directness, simplicity, and necessity in applied mathematics, demand its recognition in the elements. In two or three instances, I have presented the reductio ad absurdum, where the methods are equivalents, and have always in presenting the infinitesimal method woven in the idea of limits, which I conceive to be fundamentally the same as the infinitesimal. Thus we bring the lower and higher mathematics into closer connection.
The order of arrangement in Plane Geometry (Chap. I.), is thought to be simple, philosophical, and practical. A glance at the table of contents will show what it is. This arrangement secures the very important result, that each section presents some particular method of proof, and holds the student to it, until it is familiar. True, it requires that a larger number of propositions be demonstrated from fundamental truths; but who will consider this an objection?
To such as consider it the sole province of geometrical demonstra tion, to convince the mind of the truth of a proposition, not a few theorems in these and ordinary pages must seem quite superfluous. To them, Prop. I., page 121, may afford some merriment. But those who, with myself, consider Geometry as a branch of practical logic, the aim of which is to detect and state the steps which actually lie between premise and conclusion, will see the propriety of such demonstrations; and for each individual of the other class, a separate treatise will be needed, since no two minds will intuitively grant exactly the same propositions.
To Ex-President HILL, of Harvard, I am indebted for the confirmation of an opinion which had been previously forming in my mind, that the study of Geometry as a branch of logic, should be preceded by a presentation of its leading facts. The works of COMPAGNON, TAPPAN, and our lamented countryman, CHAUVENET, have been within reach during the entire work of preparation, and this volume would have been different, in some respects, if any one of these able treatises had not appeared before it.
In the preparation of PART III. the works of ROUCHÉ et COMBEROUSSE and MULCAHY have been freely used. For the very concise and elegant form in which the principle of Delambre, for the pre
cise calculations of Trigonometrical Functions near their limits, is embodied in TABLE III., I am indebted to the recent work of President ELI T. TAPPAN, of Kenyon College, Ohio.
My long and intimate intercourse with Professor G. B. MERRIMAN, now of the department of Physics in the University, has been a source of great profit to me in the preparation of the entire work. His sound, practical judgment as a teacher of Geometry, and cultivated taste and skill as a Mathematician, have been ever at my service, and have done more than I can tell, in giving form to the work, both as respects its matter and its spirit.
UNIVERSITY OF MICHIGAN,
ANN ARBOR, January, 1872.