PROBLEM XLVII, To make a square equal to two given squares P and Q. Set two sides AB, BC, of the given squares perpendicular to each other. Join their extremities AC; so shall the square R, constructed on AC, be equal to the two P and Q taken together.* B R NOTE. Circles, or any other similar figures, are added in the same manner. For if AB and BC be the diameters of two circles, AC will be the diameter of a circle equal to the other two. And if AB and BC be the like sides of any two similar figures, then AC will be the like side of another sim ilar figure equal to the two former, and upon which the third figure may be constructed, by Problem Xlii. PROBLEM XLVIII. To make a square equal to the difference of two given On the side AC of the greater square, as a diameter, describe a semicircle; in which apply AB, the side of the less square. Join BC, and it will be the side of a square, equal to the difference between the two P and R, as required. B P R *For in a right angled triangle, the square of the hypote nuse is equal to the sum of the squares of the other two sides. Z z PROBLEM XLIX. To make a square equal to the sum of any number of squarts taken together. Draw two indefinite lines A m An, perpendicular to each other at the point A. On one of these set off AB the side of one of the given squares, and on £he other AC the side of another of them. Join BC, and it will be the side of a square equal to the two together. Then take AD equal to BC, and AE equal to the side of the third given square. So shall DE be the D B F side of a square equal to the sum of the three given squares. And so on continually, always setting more sides of the given squares on the line An, and the sides of the successive sums on the other line Am. Note. And thus any number of any kind of figures may be added together. PROBEM i. To construct the tines of the plane scale. The divisions on the plane scale are of two kinds ; one kind having relation merely to right lines, and the other to the circle and its properties. The former are called lines, or scales, of equal parts, and are either simple or diagonal. By the lines of the plane scale, we here mean the following lines, most of which commonly, and all of them sometimes, are drawn on a Plane Scale. 1. A LINE OF SCALE of EQUAL PARTS, marked E. P. 1. To construct plane diagonal scales. Draw any line, as A B, of any convenient length. Divide it into 11 equal parts.* Complete these into rectangles of a convenient height> by drawing parallel and perpendicular lines. Divide the altitude into 10 equal parts, if it be for a decimal scale for common' numbers, or into 12 equal parts, if it be for feet and inches; and through these points of division draw as many parrallel lines, the whole length of the scale. Then divide the length of the first division A C into 10 equal parts, both above and below; and connect these points of division by diagonal lines, and the scale is finished, after being numbered as you please. *Only 4 parts are here drawn for want of room. B Of the preceding three forms of scales for two figures the first is a decimal scale, for taking off common numbers consisting of two figures. The other two are duodecimal scales, and serve for feet and inches. In order to construct the other lines, describe a circumference with any convenient radius, and draw the diame ters A B, DE> at right angles to each other; continue BA at pleasure toward F; through D draw D G parrallel to BF; and draw the chords B D, BE, A D, A E. Circumscribe the circle with the square HMN, whose sides HM, MN, shall be parallel to AB, ED. 2. To construct the line of chords. Divide the arc A D into 90 equal parts; mark the 10th divisions with the figures 10, 20, 30, 40, 50, 60, 70, 80, 90; on D, as a centre, with the compasses, transfer the several divisions of the quadrantal arc to the chord A D, which, marked with the figures corresponding, will be a line of chords. Note. In the construction of this and the following seales, only the primary divisions are drawn; the intermediate ones are omitted> that the figure may not appear too much crowded, 4 |