INTRODUCTION.

GEOMETRY is highly important, and growing in importance as a branch of mathematical study. This is not only true for mathematicians; but, what is extremely interesting to teachers, to practical men also. The vast development of machinery, of steam-power, of the applications of electricity, bring chapters of mathematical science into every-day use which have long been employed in physical investigation, but never before in the arts of life; and in the construction of all kinds of machinery and industrial devices the geometrical representation to the eye is of far more immediate and practical value than abstract calculations. The men who can draw are rapidly gaining on those who can merely calculate. In such matters as statistics the old processes are even reversed. We have not only an application of arithmetic to geometry, but geometrical representations of arithmetical results; there is not merely an algebraic geometry, but a graphic algebra.

In manual training schools geometry is fully as important a branch of mathematics as arithmetic, even for the future mechanic. The great defect in American mathematical training has been that arithmetic and algebra have been too much favored as against geometry. Teachers have delayed, and still delay, presenting even the elements of geometry till a great deal of algebra has been mastered; the meagre facts which must be stated before even mensuration can be intelligently treated have been reduced to the smallest compass and the most mechanical shape; and it is only because we confuse the difficulty of the subject with our stupid ways of teaching it that we tolerate the geometrical ignorance of our pupils.

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There are skilful mathematicians who are unaware how much better geometers early training would have made them, if the geometrical side of things had had fair play in their education. But a reform is impending; and Mr. Hopkins's text-book here presented is intended to promote it. I desire to call the attention of all earnest and progressive teachers to the heuristic method as here expounded.

The word heuristic is derived from the Greek; it means the method of discovery. Inventional is another word which has been somewhat similarly used; but, I think, in a narrower meaning.

In mathematics the "heuristic" is the same as the "development" method; that is, the method by which the pupil is led to see the theorems and their demonstrations for himself.

In attempting to use this book, the ordinary laws of good teaching must be followed. Consequently the pupils, however mature, must possess all the prior qualifications; they must be intelligent as to the subject-matter. The greatest difficulty in now teaching geometry by any method lies in that neglect of the elements to which I have before alluded. If a pupil reaches the age of fifteen (as the pupils of the Massachusetts grammar schools are said to do) without thorough and systematic training in the elements of the subject, nothing remains but to prefix a course of instruction in these elements to the study of demonstrative geometry. A very skilful and celebrated teacher was about 1875 mentioned to me by his pupils as using the heuristic method; and I at once wrote him to inquire how he presented the elements. His answer was that about six weeks were spent in working up, orally, the doctrine of form; that important part of geometry whose results are contained in the definitions and other preliminary matter. This was in America, with pupils who had passed the grammar school without learning these elements; and represents what seems to the writer to be the minimum of attention to be given to this part of the subject.