1 a 'one half the circumference of the circle. Therefore be- cause the angle of a semicircle is [by A B, E c [by 47. 1.] will be equal to С the square of the diameter of the cir- squares of AD, DB will be equal to the square of A B, and the squares of ED, Dc equal to the square of E C. Therefore [by adding equals to equals) the squares of A D, DB, DE, DC will be equal to the squares of A B, E C. But the squares of A B, E c have been proved to be equal to the square of the diameter. Therefore the squares of A D, BD, E D, CD will be equal to the square of the diameter of the circle A BCE. Therefore, &c. Which was to be demonstrated. PRO P. XV. THEOR. If from the vertex of an equilateral triangle ereated upon the diameter of a circle, a right line be drawn to any point in that diameter, and from that point le drawn a right line perpendicular to the diameter. meeting the circumference of the circle ; the Square of that right line, together with the square of that perpendicular, will be equal to the square of the diameter of the circle. Let A MB be a circle whose diaineter is A B, and centre c; and let A D B be an equilateral triangle described upon the diameter A B. Let p be any point taken in A B. Let the right line p D be drawn, and let P M be perpendicular to the diameter A B, meeting the circumference of the circle in m: I say the two squares of P D, PM will be equal to the square of the diameter A B of the circle. For join C D. Then because a c is equal to cb; A D equal to DB, and c d is common; the angles A CD, B C D [by 4. 1.] will be equal to che another; and so each of them will be a right angle. Therefore the square of A D will [by 47. 1.] be M be equal to the two squares of ac, CD. And since A D is equal to A B, by reason of the equilateral triangle, and [by 4. 2.] four times the square of the femidiameter A c is equal to the square of the diameter AB : Therefore four times the square of a C, will be equal to the two squares of A C, CD; and taking away the common square of ac from both, there will remain thrice the square of A C equal to the square of cd. Again, because [by 35. 3.] the {quare of P M is equal to the rectangle under AP, PB (for if PM be D continued to meet the circumference in N, MN l by 3. 3.] will be bisected in P) and since A B is divided equally in c and unequally in P; and fo [by 5. 2.) the rectangle В A under A P, P B, together with the P с square of PC, is equal to the square of a c: Therefore the squares of N PC, PM will be equal to the square of Ac. Wherefore since it has been proved that the square of cp is equal to thrice the square of A C, if equals be added to equals, the three squares of C D, PC, PM will be equal to four times the square of a c, that is, to the square of AB. But because [by 47. 1.] the square of PD is equal to the two squares of p C, CD, by adding again equals to equals, the squares of PC, PM, CD, PD will be equal to the squares of A B, PC, cd, and taking away the squares of pc, cp from both, there will remain the squares of PD, P M equal to the square of the diameter A B. Therefore, &c. Which was to be demonstrated. р SCHOLIUM. There are many other elegant and useful theorems concerning the equalities of the squares and rectangles of right lines drawn in and about a circle, a few of which I shall add at the end of the sixth book, because their demonstrations are porter and much easier from the proportionality of the sides of equiangular triangles, and the equality of the rectangles under the means and extremes, than by the propositions of the second book. I shall here only just mention two theorems; one of Mr. Huygens's, whose demonstration is too long to be fribjoin'd. If a circle be described about the centre * of gravity of any right lined figure, and right lines be drawn from all the angles of the figure to any point of the circumference of the circle, the sum of the squares of all those lines will always be of the same magnitude. And the following theorem, which was found out by me fome years ago, and communicated by me to several people fome years ago, viz. The square of the number expressing the area of any trapezium inscribed in a circle, will be equal to the product of the four numerical differences between the number expresing half the sum of the four sides, and each of the numbers expressing the sides. The demonstration, which I bave by me, cannot here be conveniently annex'd. That point upon which if the figure any how refts, fuppofing it to be heavy, it would continue witbout altering its fituarion. EUCLI D's EUCLI D's E L E M E N T S, BOOK IV. DEFINITIONS. I, A Right lined figure is faid to be inscribed in a right , lined figure, when every angle of the figure inscribed touches every side of the figure in which it is inscribed. 2. A right lined figure is said to circumscribe a right lined figure, when every side of the circumscribed figure touches every angle of the figure about which it is circụmscribed. 3. A right lined figure is said to be inscribed in a circle, when every angle of the inscribed figure touches the circumference of the circle. 4. A right lined figure is said to be circumscribed about a circle, when every side of the circumscribed figure touches the circumference of the circle. 5. A circle is said to be inscribed in a right lined figure, when the circunference of the circle touches every fide of the figure in which it is inscribed. 6. A circle is said to be circumscribed about a right lined figure, when the circumference of the circle touches every angle of the figure about which it is circumfcribed. 7. A right line is said to be applied in a circle, when the extremes of that line are in the circumference of the circle, PROPO PROPOSITION I. PROBL E M. To apply a right line in a given circle equal to a given right line which is not greater than the diameter of the circle a Let the given circle be A B C, and D a given line not greater than the diameter of the circle : it is required to apply a right line in the circle A B C equal to the given right line d. Draw the diameter BC of the circle ABC; then if Bc be equal to D, the thing re quired will be done already ; с E B for the right line Bc is applied in the circle ABC equal to the given right line D. But if not D Bc is greater than D, and [by 3. 1.] make c E. equal to D, and from the centre c with the distance c E [by 3. poft.] describe the circle A E F, and [by 1. post.) join c A. Then because the point c is the centre of the circle AEF, CA will be equal to CE: But d is equal to the right line ce; and therefore D will be equal to ca. Wherefore the right line ca is applied in the circle ABC, equal to the given right line D, which is not greater than the diameter of the circle. Which was to be done. • If it be required to accommodate or apply a right line less than the diameter of a given circle within the circle equal to a given right line which shall be parallel to another given right line, the thing may be done thus : Let ABC be the given circle, D the centre, and ef the right line, to which the given right line less than the diameter of the circle is to be accommodated LBN E in the circle equal ; and let G be the E right line to which the applied right D line is to be parallel. Thro' the centre A HD draw [by 31. 1.] a diameter AC of the circle parallel to the right line : T Then if the right line E F be equal M to the diameter, the thing required is G done. But if ef be less than the diameter, |