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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
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.