PREFACE TO THE SEVENTH EDITION. xix

ly found, I have endeavoured to supply this imperfection, by propositions M, N, O, P, added to the Sixth Book. Propositions N and O, while they connect propositions M and P, the latter of which is the theorem in question, they serve also, when introduced into Trigonometry, to find the angles of a triangle when its sides are known. (See Plane Trigonometry, Prop. vii. and viii.)

Although hardly any demonstration can be more simple and elegant than that of Prop. 47. Book I., given by Euclid; yet I have been tempted, by the peculiar neatness of one which I had once regarded as new, but have since found in Elémens de Géométrie par M. CLAIRAUT, to insert it among the Notes (p. 451). I have, in addition, also added some remarks on the theorems in the First Book, that relate to the equality of certain figures, and which lead to curious and important conclusions respecting equal rectilineal figures, however dissimilar they may be in form,

In the Plane Trigonometry, new demonstrations, deduced from Prop. N and O of Book VI., have been given of Prop. VII. and VIII.; and two Propositions, viz. IX. and X., have been now added; from which I have deduced a second solution to Case III, of oblique-angled triangles (see page 344), in addition to that formerly in the work.

I have given a solution to Case IV. in a Note on the fourth solution (page 346), that seems to be preferable to the ordinary one, when the three sides are given, and all the angles are required.

Such are the principal changes and additions which have been made in this Edition. How far

they are improvements, candid and intelligent teachers and students will determine.

COLLEGE OF EDINBURGH,

11th Nov. 1826.

WILLIAM WALLACE.

ELEMENTS

OF

GEOMETRY.

BOOK I.

DEFINITIONS.

I.

"A POINT is that which has position, but not magni- See Notes.

"tude *."

II.

A line is length without breadth.

"COROLLARY.

The extremities of a line are points; "and the intersections of one line with another are "also points."

III.

"If two lines are such that they cannot coincide in any “ two points, without coinciding altogether, each of "them is called a straight line."

"COR. Hence two straight lines cannot inclose a space. "Neither can two straight lines have a common seg"ment; for they cannot coincide in part, without coinciding altogether."

66

IV.

A superficies is that which has only length and breadth. "COR. The extremities of a superficies are lines; and "the intersections of one superficies with another are "also lines."

The definitions marked with inverted commas are different from those of EUCLID.

B

V.

Book 1. A plane superficies is that in which any two points being taken, the straight line between them lies wholly in that superficies.

VI.

A plane rectilineal angle is the inclination of two straight lines to one another, which meet together, but are not in the same straight line.

A

D

E

6

N. B. • When several angles are at one point B, any one of them is expressed by three letters, of which the letter that is at the vertex of the angle, that is, at the point in which the straight lines that contain the angle ❝ meet one another, is put between the other two letters, ' and one of these two is somewhere upon one of those straight lines, and the other upon the other line: Thus 'the angle which is contained by the straight lines AB, 'CB, is named the angle ABC, or CBA; that which is 'contained by AB, BD is named the angle ABD, or 'DBA, and that which is contained by DB, CB is called 'the angle DBC, or CBD; but if there be only one angle at a point, it may be expressed by a letter placed at that point, as the angle at E.’

VII.

When a straight line standing on
another straight line makes the
adjacent angles equal to one an-
other, each of the angles is called
a right angle; and the straight
line which stands on the other
is called a perpendicular to it.

IT

VIII.

An obtuse angle is that which is greater than a right angle. Book I.

IX.

An acute angle is that which is less than a right angle.

X.

A figure is that which is inclosed by one or more boundaries." The space contained within a figure is called the Area of the Figure."

XI.

A circle is a plane figure contained by one line, which is called the circumference, and is such that all straight lines drawn from a certain point within the figure to the circumference, are equal to one another.

XII.

This point is called the centre of the circle.

XIII.

A diameter of a circle is a straight line drawn through the centre, and terminated both ways by the circumference.

XIV.

A semicircle is the figure contained by a diameter and the part of the circumference cut off by the diameter.