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IN adding to the number of text-books on Geometry the author aims not so much to provide new material for college preparatory work in the subject, as to furnish if possible to the student a more suggestive method of study and a more graphic form of written demonstration.

Our American text-books have practically agreed upon the essential theorems in geometric demonstration, and novelties in subject-matter are unnecessary. It is believed, however, that the subject may yet acquire a new attractiveness and be more promptly mastered if a few important principles are magnified.

Experience shows that a student's attention should be directed to the geometric relation itself, rather than to the description of the relation to the properties of the figures used in illustrating the geometric fact, rather than to verbal discussions of hypothesis and conclusion. Valuable as the argumentative drill may be, the study of Geometry has a mathematical value which exceeds even the charm of its logic.

Geometric figures should speak for themselves carry their own hypotheses and suggest their own inquiries. By a simple method of marking the lines many of the equalities between the parts of figures are shown; others will readily suggest

themselves. A student should not only appreciate these symbols but should use them constantly, so that in meeting new problems he may instinctively transfer to the figure as much as is possible of the hypothesis, and from the data thus graphically before him proceed to the discussion and the solution.

The propositions of Interpretation given on pages 52-56, 84-86, 116-118, 180-182 illustrate the emphasis laid upon this feature of the text.

In the written demonstration attention is paid to the logical arrangement of major and minor steps; the important or principal steps are placed in columns by themselves, and the secondary steps, or supporting reasons, are placed in brackets beside the primary steps. These minor steps are often omitted in the later books and should be supplied when necessary by the student. The chain of reasoning is kept distinct, and is condensed in form and phraseology, even at the sacrifice of literary form.

The use of such methods has satisfied the author that the concentration of the student's attention upon the few main steps of proof in each demonstration, simplifies the mastery of the principles involved, and in written work facilitates such examination and criticism by a class, and by the instructor, as is rarely possible in a full-text, disarranged form of proof.

Lack of confidence is a common complaint among students in attacking and solving original exercises. It has been the author's privilege to find several hundred pupils acquiring confidence and facility in original demonstration, largely through the use of graphic figures and a condensed, wellarranged form of written proof.

The exercises have been selected and devised with much care; the list furnishes no mathematical puzzles and no problems beyond the grasp of beginners in the study of Geometry.

A brief discussion of Logarithms as required for the solution of simple Geometric problems is appended to the text, - also tables of Logarithms of Numbers, and Metric Tables.

The attention of teachers is respectfully directed to the features above mentioned, to the topical summary on page 172, and to the college examination papers on page 217; criticism is cordially invited.

Opportunity is here taken to express to my colleague, Mr. George T. Eaton, my appreciation of his assistance in the preparation of this work.

Andover, Mass., April, 1896.

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