In this diagram AB and AC are equal, each being an inch and a half, and the central square has vanished. The square of the

hypothenuse, BCDE, is seen to contain four right-angled triangles, each equal to ABC. The square of the base AB, which is ABFE, contains two such triangles, and AC, or its equal, AE, being squared, gives the square AEGD, which also contains two such triangles. The squares of AB and AC together equal four times the area of the triangle ABC; and the square of BC also equals four times the same triangle. Therefore the square of the hypothenuse of a right angle equals the sum of the squares of the other two sides, agreeably to the proposition.

PROPOSITION LXX.

If from any circle there be cut a segment of one diameter, the chord of half the arc of that segment is the square root of the diameter of the circle.

The diagram presents eight circles-three perfect and five broken-there not being room on the diagram to complete them. The diameter of the first circle, AB, is one, [one inch.] The diameter of the second, AC, is 2. The diameter of the third, AD, is 3, and so on, the diameters increasing by unity till the last circle, whose diameter is 8. These circles all touch a common point at A, and their centers are all in the same straight line, AD, produced. From all these circles, except the smallest, the chord GH cuts a segment, each segment having the same diameter, AB, which is 1.

From the second circle, whose diameter is 2, the segment cut off by the chord GH is seen to be half the circle; and the chord of half the arc of that segment is seen to be A1, and A1 is the square root of 2-viz., it is 1.4142+.

A2 is the chord of half the arc of the segment cut from the third circle, whose diameter is 3; and A2 is the square root of 3viz., it is 1.732+.

A3 is the chord of half the arc of the segment of the fourth circle, whose diameter is 4, and A3 is the square root of 4—that is, the chord A3, with perfect drawing and perfect measurement, would be just two inches.

A4 is the chord of the fifth circle, whose diameter is 5, and A4 is the square root of 5.

A5 is the chord of the sixth circle, whose diameter is 6, and A5 is the square root of 6.

A6 is the chord of the seventh circle, whose diameter is 7, and A6 is the square root of 7.

A7 is the chord of half the arc of the segment cut off by GH from the largest circle, whose diameter is 8, and A7 is the square root of 8-viz., it is 2.8284+.

PROPOSITION LXXI.

If from any circle there be cut any segment whatever, the chord of half the arc of that segment is the square root of the diameter of the circle multiplied by the diameter of the segment; or the chord is a mean proportional between the diameter of the circle and the diameter of the segment.

AC, the diameter of the second circle, is 2. Therefore the chord IK cuts segments from all the larger circles, each segment having a diameter of 2.

A8 is seen to be the chord of half the arc of the segment cut from the fourth circle. This circle has a diameter of four, which multiplied by 2, the diameter of the segment, makes 8; and the chord A8 is the square root of 8-viz., it is 2.8284+.

A9 is seen to be the chord of half the arc of the segment cut from the sixth circle. This circle has a diameter of 6, which multiplied by 2, the diameter of the segment, makes 12; and the chord A9 is the square root of 12-viz., it is 3.464+.

AD, which equals 3, is the diameter of the segments cut off by the chord passing through D; and A10 is seen to be the chord of half the arc of the segment cut from the sixth circle. This circle has a diameter of 6, which multiplied by 3, the diameter of the segment, makes 18; and A10 is the square root of 18.

Again: let the segments cut from all the circles each have a diameter of half of one, that is, equal to AE. If lines were drawn from A, terminating on the line EF at the points 1, 2, 3, 4, &c., they would be the chords of half the arcs of the segments thus cut off. To make the lines more distinct they are drawn from the point B, and are manifestly of the same length as they would be if drawn from A. Now, these lines are respectively the square roots of half the diameters of the circles on which they terminate :-that is, B1 is the square root of half of 1, B2 is the square root of half of 2, B3 is the square root of half of 3, B4, which terminates on the circle whose diameter is 4, is the square root of half the diameter, and B8, terminating on the circle whose diameter is 8, is the square root of 4. So that, if the diagram were perfectly drawn and perfectly measured, B8 would be just 2 inches.

In like manner, if the segments cut off had a diameter equal to one-fourth of the unit, the chords would be the square roots respectively of one-fourth of the diameters of the circles.