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ចយ 18t៨៨ : Berwick & Smith, Boston, U.S.A.

PREFACE.

The present work is a result of the Author's experience

in teaching Geometry to Junior Classes in the University for a series of years. It is not an edition of 5 Euclid's

“ Elements," and has in fact little relation to that cele

brated ancient work except in the subject matter.

The work differs also from the majority of modern

treatises on Geometry in several respects.

The point, the line, and the curve lying in a common

plane are taken as the geometric elements of Plane

Geometry, and any one of these or any combination of them is defined as a geometric plane figure. Thus a triangle is not the three-cornered portion of the plane

inclosed within its sides, but the combination of the

three points and three lines forming what are usually

termed its vertices and its sides and sides produced.

This mode of considering geometric figures leads naturally to the idea of a figure as a locus, and consequently prepares the way for the study of Cartesian Geometry. It requires, however, that a careful distinction be drawn between figures which are capable of superposition and those which are equal merely in area. The properties of congruence and equality are accordingly carefully distinguished.

The principle of motion in the transformation of geometric figures, as recommended by Dr. Sylvester,

and as a consequence the principle of continuity are

freely employed, and an attempt is made to generalize

all theorems which admit of generalization.

An endeavour is made to connect Geometry with

Algebraic forms and symbols, (1) by an elementary

study of the modes of representing geometric ideas in

the symbols of Algebra, and (2) by determining the

[blocks in formation]

In dealing with proportion the method of measures is employed in preference to that of multiples as being equally accurate, easier of comprehension, and more in line with elementary mathematical study. In dealing

with ratio I have ventured, when comparing two finite

lines, to introduce Hamilton's word tensor as seeming to

me to express most clearly what is meant.

After treating of proportion I have not hesitated to employ those special ratios known as trigonometric functions in deducing geometric relations.

In the earlier parts of the work Constructive Geometry

is separated from Descriptive Geometry, and short

descriptions are given of the more important geometric drawing-instruments, having special reference to the

geometric principles of their actions.

Parts IV. and V. contain a synthetic treatment of the

theories of the mean centre, of inverse figures, of pole

and polar, of harmonic division, etc., as applied to the

line and circle; and it is believed that a student who

becomes acquainted with these geometric extensions in this their simpler form will be greatly assisted in the wider discussion of them in analytical conics. Throughout the whole work modern terminology and modern

processes have been used with the greatest freedom, regard being had in all cases to perspicuity.

As is evident from what has been said, the whole

intention in preparing the work has been to furnish the student with that kind of geometric knowledge which

may enable him to take up most successfully the modern

works on Analytical Geometry.

N. F. D.

QUEEN'S COLLEGE,

KINGSTON, CANADA.

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