diameter of the circle, which is equal to the excess of the greater of the given circles above the less.

PROP. LXXXVII.

102. PROBLEM. To find a circle to which a given circle shall have the same ratio, as that which one given straight line has to another.

Find (E. xii. 6.) a fourth proportional (L) to the two given straight lines (A) and (B) and to the diameter (D) of the given circle; find, also, (E. xiii. 6.) a mean proportional (M) between the diameter (D) of the given circle, and the fourth proportional (L) first found;

therefore (E. xx. 6. Cor. 2.) D2: M2 :: D : L; and (constr.) D: L:A B; : therefore (E. xi. 5.) D2 M2 :: A: B; therefore (E. ii. 12.) the given circle has to a circle described on M, as a diameter, the same ratio as that which A has to B.

PROP. LXXXVIII.

103. THEOREM. If, in any given circle, two chords cut each other at right angles, the four circles described upon their segments, as diameters, shall, together, be equal to the given circle.

For (Supp. 1. 3.) the squares of the four segments are, together, equal to the square of the diameter: It is manifest, therefore, from E. xviii. 5. and E. ii. 12., that the circles described on the four segments of the chords are, together, equal to the given .circle.

PROP. LXXXIX.

104. THEOREM. A circle is equal to the half of the rectangle contained by its semi-diameter and by a straight line which is equal to its circumference.

Let ABCD be a circle, and let F be the half of

the rectangle contained by its semi-diameter and by a straight line equal to its circumference: The circle ABCD is equal to the rectangle F.

For if it be not equal, it is either greater, or less, than it. If it be possible, let F< the circle ABCD; therefore (E. ii. 12.) a polygon ABCDE may be inscribed in the circle, which shall be greater than F.

Find (E. i. 3.) the centre G, and from G draw (E. xii. 1.) GH perpendicular to any side CD, of ABCDE, and join G, D. Then it may be assumed that the circumference of the circle is greater than the perimeter of the inscribed figure ABCDE; and (E. xvii. and xix. 1.) GD > GH; therefore the rectangle contained by the circumference and the semi

diameter of the circle is greater than that contained by GH, and the perimeter of ABCDE, which latter rectangle (E. xli. 1. and E. i. 2.) is the double of the polygon ABCDE; therefore F> ABCDE; and it is also less; which is absurd.

But, if it be possible, let F be greater than the circle. Then (E. ii. 12.) a polygon KLMNX may be described about the circle, which shall be less than F; join the centre G, and any of the points of contact Q; and since it may be assumed that the perimeter of KLMNX is greater than the circumference of the circle, the rectangle contained by the perimeter of KLMNX and GO, which rectangle is the double of KLMNX, is greater than the rectangle contained by the circumference of the circle and GO; therefore the circumscribed polygon KLMNX > F; and it is also less; which is absurd. Therefore, the circle ABCD can neither be greater, nor less, than F; that is, it is equal to F.

105. COR. The circumferences of circles are to one another as their semi-diameters.

PROP. XC.

106. THEOREM. A circle is a mean proportional between any regular polygon, described about it, and a similar polygon, the perimeter of which is equal to the circumference of the circle.

For if there be taken a straight line (P) equal to the perimeter of the regular polygon described about the circle, and another straight line (p) equal to the perimeter of the similar polygon, or (hyp.) equal to the circumference of the circle, then (E. xx. 6. and

E. xxii. 6.) the polygon, described about the circle, is to the similar polygon, as P2 is to p3: But (Supp. ii. 4. Cor. 2.) the polygon, described about the circle, is the half of the rectangle contained by P and the circle's semi-diameter; and (Supp. Ixxxix. 6.) the circle is the half of the rectangle contained by p; and by the circle's semi-diameter; therefore (E. i. 6.) that polygon is to the circle, as P is to p; and it has been shewn to be to the similar polygon, as P2 is to p; therefore it has to the similar polygon a ratio, the duplicate of that which it has to the circle; therefore the circle is a mean proportional between the two similar polygons.

A

SUPPLEMENT

TO THE

ELEMENTS OF EUCLID.

PART II.

BOOK I.

PROP. I.

1. THEOREM. IF two chords of a circle cut each other at right angles, either pair of opposite arches, intercepted between them, is equal to the semi-circumference of the circle.

In the circle ABC, let the chords AB, CD, cut

each other at right angles: AC+DB shall be equal

to the circumference of the circle.