AN EXPLANATION

OF THE SYMBOLS EMPLOYED IN THIS TREATISE,

AB, or AB ...... a straight line, of which the points

AB

.........

AB2

ABX CD

2AB, &c.

Δ

.........

......

denoted by A and B are the extremities.

a circular arch, of which the points denoted by A and B are the extremities.

a square, having AB for one of its sides.

a rectangle, of which AB and CD are adjacent sides. ...... the double, &c. of AB. denotes a triangle.

A:B......... the ratio of A to B.

A:B::C:D ... the ratio of A to B is equivalent

to the ratio of C to D.

........ therefore.

A

SUPPLEMENT

TO THE

ELEMENTS OF EUCLID.

BOOK I.

PROP. I.

1. PROBLEM. A GIVEN plane rectilineal angle being divided into any number of equal angles, to divide the half of it into the same number of angles, all equal to one another.

Bisect (E.* 9, 1.) the given angle: And, first, if it be divided into an odd number of equal parts, it is evident that the middle part is thereby bisected. Bisect, therefore, each of the remaining

* In this and the following references, the letter E is used to indicate Euclid's Elements; the letter S, in like manner, refers to this Supplement; the former of the subsequent numbers points out the Proposition, and the latter the Book, intended to be quoted.

B

equal parts, on either side of that middle part, and the half of the given angle will, manifestly, be divided into as many equal parts as the given angle itself.

Again, if the given angle be divided into an even number of equal parts, it is plain that the straight line which bisects it, will have the half of that number of equal parts, on each side of it. Bisect, therefore, each of the equal parts, on either side of that line; and the half of the given angle will thereby be divided, as before, into as many equal parts as the given angle itself.

PROP. II.

2. PROBLEM. From the vertex of a given scalene triangle, to draw, to the base, a straight line which shall exceed the less of the two sides, as much as it is itself exceeded by the greater.

Let ABC be the given scalene triangle, and let AB be greater than AC: It is required to draw,