PREFACE.

THE controlling purpose of the present work is not science, but

education. The following principles, therefore, will be found

fundamental to its design and execution:

1. That the peculiar advantages of mathematics in forming habits of close reasoning, the chief benefits to be derived from them in a harmonious development of the powers, are comprised in geometry.

2. That, while mathematical instruction per se is freely conceded to leave a chasm in mental culture, none can hope for the highest efficiency of his faculties without the training which this study is fitted to give, and which no other can supply.

3. That this branch possesses a twofold utility—absolute, so far as its cultivation is immediately conducive to mental improvement; relative, so far as its cultivation is necessary for the prosecution of other branches of knowledge.

4. That its most considerable absolute good is the correction of a certain vice and the formation of its corresponding virtue; the vice being the habit of mental distraction; the virtue, the habit of continuous attention.

5. That the utmost effectiveness of the study as a means of discipline requires a specific adaptation, based upon the initial idea that every beginner is intellectually an infant. Demonstrative reasoning is a mode of energizing to which he has thus far been a total stranger, and he finds himself in a new sphere of being and action. In thought, as in locomotion, he must creep before he walks, and 'be fed as he is able to bear it.' The true ideal is to render the subject intelligible, yet to avoid an enervat

ing manner of teaching it.

Doubtless it was because his book was not a class-book that Euclid gave his famous answer to the king who asked him to make geometry easier.

Accordingly, I have endeavored at the outset, by a process of induction, to fix clearly in the student's mind the distinction between the invisible entities of geometry and the visible representations of them, ignorance of which is a fruitful source of discouragement and failure. Clear perceptions must precede an intelligent exercise of the faculty of comparison.

Definitions are introduced only as they are demanded for immediate use. The learner is not held for several days to the mechanical task of memorizing a series of abstractions, and so made to acquire at the start a distaste for the whole subject, but, with every needed preparation, is brought quickly to the applications. Ill-digested and (for the time) useless information may tend to intellectual fat, but never to intellectual muscle.

Throughout, especially in the earlier chapters, questions are asked with a view to giving the student a clear idea of the essential steps in the demonstration, assisting his memory, and cultivating precision, as well as independence, of thought and expression. At suitable places, also, remarks of a suggestive character are inserted, to aid him in an economic and successful employment of his energies.

Numerical examples, selected for the application of principles to the actual business of life, as well as theorems of which original demonstration is required, are appended, with liberal suggestions, to many sections and at the end of every chapter; of sufficient difficulty to stimulate to success, yet not so difficult as to depress by failure. Exercises must be neither so multiplied in number, nor so difficult in kind, as to defeat their end. Let them be progressively adapted to the abilities of the average mind, and they will best accomplish their purpose with all,- keener interest, skill in the use of theory, and self-reliant activity.

Occasionally, in the first stages, the same theorem is required to be demonstrated from a figure differently lettered, unlettered, or differently placed. It is certainly beneficial to the student to see the truths of science put in different lights. They thus obtain in his mind their natural order an order independent of all arbitrary and accidental associations.

The Reductio ad Absurdum, which to the mere beginner is a 'stumbling block' and 'foolishness,' is deferred until sufficient strength has been acquired in the use of the direct method. A happy effect of this arrangement is a new incentive in the novelty and beauty of the indirect, as well as the clear demonstration, within several sentences, of theorems which have hitherto occupied from a half to an entire page.

The subject of Constructions is strongly emphasized in artistic execution and disciplinary results. Nor is its relation to the practical arts neglected. Such allusions cannot fail to render the study both more interesting and more useful.

There has been constant care to simplify both language and subject matter, yet never at the sacrifice of accuracy or rigor. In presence of the many attempts to abbreviate, it should be remembered that real brevity is to be obtained, not by sparing words to the author, but by sparing time and labor to the reader.

No theorem is stated in connection with its converse. Separate topics are made separate lessons. Notation has been greatly reduced. Where it has been customary to use many letters in designating the diagram, I have, where possible, used but one, it being easier for the mind to grasp a single point than to embrace several, or to trace a broken line.

The diagrams are large and distinct, and are so placed as to enable the student readily to refer to them at every step. In the few cases where it becomes necessary to turn the page in reading a demonstration, the diagram is repeated.

In general, great pains have been taken to secure convenience,