THE SECOND BOOK, IN GENERAL TERMS. PROPOSITION 1. THEOR. If there be two right lines, one of which is divided into any number of parts, the rectangle under the two lines is equal to the sum of the rectangles under the undivided line and the several parts of the divided line. Draw from either extremity of the divided line, a line at right angles to it and to the undivided line, and complete the rectangle: then through each point of bisection of the divided line draw a line parallel to the perpendicular, to meet the opposite side. Then it is evident that those parallel lines are to the perpendicular and,. to the undivided line, also that the whole rectangle is to a rect. under the given lines, and is divided into as many rectangles as there are segments of the divided line, and .. into rectangles under the undivided line and the several parts of divided line, N 2 This rectangle is to the square of one part with the rectangle under the parts. For it is the rectangle under the whole line and that part on which the is described; if then the be taken away, the remainder will be the rectangle under the parts, ... &c. &c. Otherwise thus: Assume a line to either part. Then, the rectangle under this assumed line and the whole line is to the sum of the rectangles under the assumed line and each of the parts, (prop. 1, b. 2.) but the rect. under the assumed line and part to it, is the ' of that part; and the rect. under assumed line and other part is to the rect. under the parts, .. &c. &c. From this prop. we may infer that the rect, under any two lines is to the of the less + the rect. under the less and difference. 2 Thus, in numbers 6 x 4 is to 4 + 4 x 2. From this and the former prop. may be infered that the of the lesser of any two lines the rect. under it and difference is to the of the greater rect. under it and difference. 2 the Thus, in numbers, suppose 8 and 6 are the given quantities: then 6 x 66 x 2 is to 8 x 8 8 X 2. PROP. 4, THEOR. If a right line be divided into any two parts, the square of the whole line is equal to the sum of the squares of the parts and twice the rectangle under the parts. On the given line describe a D', draw a diagonal, draw through the point of section a right line parallel to one side of the to meet the opposite side, and through the point in which this parallel line cuts the diagonal draw a line par. to the given line. The entire is divided into the squares about the diagonal, which are the rs of the parts, and the complements, each of which is a rectangle under the parts, &c. &c. 2 Otherwise thus: The of the whole line is to the sum of the rectangles under the whole line and each of the parts, (prop. 2, b. 2.) and each of those rectangles is to a2 of its respective part and a rect. under the parts, .. the the whole line, &c. &c. of Cor. Hence it is evident that the square of half the line is the fourth part of the square of the whole line, for when the line is bisected the rectangle under the parts is equal to the square of half the line. NOTE-This prop. may be thus reduced to numbers: suppose the given line 10 and the parts 6 and 4 feet, then 10 X 10 100 is to 6 x 6 + 4 X 4 + 6 × 4 + 4 X 6. = From this prop. it is evident if two right lines be given, that the of their sum exceeds the sum of their 'rs by two rectangles under them; and if the lines be , the of their sum is double the sum of their PROP. 5, THEOR. 2rs. If a right line be cut into equal and also into unequal parts, the rectangle under the unequal parts with the square of the intermediate part, is equal to the square of half the line. Describe on the half line which contains the intermediate part, a square; draw a diagonal; through the point of unequal section draw a line parallel to a side of the square, and through the point in which it cuts the diagonal draw a line par to the given line, to meet a line drawn through the other extremity of the giveu line, par. to a side of the 2. Then since the rectangles under the halves and lesser segment are and also the complemental rectangles; the rectangle under the unequal segment is to the gnomon formed by taking the of the intermediate part from the ☐ of half the given lue, ... if the 2 of the intermediate part be added to each, they will be = to one another. 2 Otherwise thus: The rectangle under the unequal segments is to a rect. under half and lesser seg. with a rect. under intermediate pari and lesser segmen; but the rect. under half and lesser seg. is to a rect. under intermediate ザー THE SECOND BOOK, IN GENERAL TERMS. PROPOSITION 1. THEOR. If there be two right lines, one of which is divided into any number of parts, the rectangle under the two lines is equal to the sum of the rectangles under the undivided line and the several parts of the divided line. Draw from either extremity of the divided line, a line at right angles to it and to the undivided line, and complete the rectangle: then through each point of bisection of the divided line draw a line parallel to the perpendicular, to meet the opposite side. Then it is evident that those parallel lines are to the perpendicular and,. to the undivided line, also that the whole rectangle is to a rect. under the given lines, and is divided into as many rectangles as there are segments of the divided line, and .. into rectangles under the undivided line and the several parts of divided line. N |