g 8 feet; and the area or surface of the figure ABDNG = 650 square feet; Then 102: 82:: 650 : 650 x 64 416 square feet, the area or sur face of abdng. 103. The Perimeters of similar right lined plane figures are in the same ratio as their homologous sides. (See the figures to the preceding Theorem.) For the angles being equal, each to each, and the sides about the equal angles respectively proportional; we have AG: ag :: GN: gn :: ND: nd :: DB : db :: BA : ba; therefore AG ag :: sum of all the antecedents AG +GN + ND+DB + BA (the perimeter): sum of all the consequents ag + gn + nd + db + ba (the perimeter). 104. The perimeters of similar Polygons (ABDNG, abdng inscribed in circles, have the same ratio as the diameters (AP, ap,) of those circles. But the angle APG is equal to the angle ABG; and the angle apg equal to the angle abg (70). And the angles AGP, agp, being right ones (72), the triangles APG, apg, are there. fore equi-angular. Hence AP ap: AG ag perim. of polyg. ABDNG : perim. of polyg, abdng (103). Corol. Hence it appears that the circumferences of circles have the same ratio as their diameters. For conceive regular polygons of the like number of sides to be inscribed in both circles; then it follows that those polygons will be similar, and that their perimeters are in the same ratio as the diameters of the circles, let the number of sides be what they will. If now we suppose the number of sides to be continually augmented and their lengths diminished, it is manifest that at last, the differences between the perimeters and the circumferences of the circles, will be less than any assignable quantities; consequently the ultimate ratio of the perimeters and that of the circumfer ences must be equal. 105. The areas or surfaces of similar polygons inscribed in circles are in the duplicate ratio, or as the squares of the diameters of the circles: (See the figures to the preceding Theorem). For the triangles APG, apg, being similar, we have (101), AP ap2 triang. APG triang. apg :: AG2 : ag2 :: polyg. ABDNG: polyg. abdng (102). Corol. Hence, if we suppose (as in the last Theorem) the circumference of a circle to be the perimeter of a regular polygon, consisting of an infinite or rather an indefinite number of indefinitely short sides, it follows that the surfaces or areas of circles will be as the squares of their diameters. And because the circumferences are directly proportional to the diameters (104, corol.) the areas will be as the squares of the circumferences also. 106. The area or surface of a polygon (ABDNG) is equal to a rectangle under half the perimeter and (CO) the distance of its centre from the sides. The centre of a regular polygon is a point equally distant from all its sides; and is the same as the centre of the inscribed, or circumscribing circle. per BA G Suppose lines are drawn from the centre. to the angular points; then the polygon will be divided into as many equal triangles as it has sides. And because those triangles are isosceles, CO will bisect AG and be pendicular to it (46): therefore the area of the triangle ACG is half the rectangle CO × AG (89, cor. 2), or CO AG; and the area of another of the triangles (GCN) is CO x GN, and so on: but the halves of all the sides. together make half the perimeter; therefore the rectangle CO half the perimeter, is the area of all the triangles or surface of the polygon. Corol. Hence it appears, that the area or surface of a circle is equal to a rectangle under the radius and a right line equal to half the circumference. For, if we conceive the circle to be a regular polygon of an indefinite number of indefinitely short sides, the distance (CO) of the centre (C) from the sides, will in that case, be the radius of the circle, and half the perimeter becomes half the circumference. 107. If semicircles (Q, R, S,) are described upon the sides of a right angled triangle (BCD), that which is upon the longest side (DB) will be equal to both the other twe taken together. For circles being similar, and in the same ratio as the squares' of their diameters (105, corol.) their halves must also be similar, and in like proportion, therefore (R. D B SR: CB2: CD2, and by composition Hence, if similar figures are described on the sides of a right-angled triangle, that on the longest side will be equal to the other two taken together. OF PLANES AND SOLIDS. DEFINITIONS. 108. A right line is perpendicular to a plane when it is at right angles to all the straight lines that can be drawn in that plane, from the point on which it insists. 109. The distance of a point from a Plane is a right line conceived to be drawn from that point perpendicular to the plane. Corol. From the two preceding Definitions, and Art. 48, it follows, that a perpendicular is the shortest line which can be drawn from any point to the Plane. R G 110. The inclination of one plane to another is measured by the inclination of two right lines in those planes, drawn from any point in their common intersection, and at right angles to the same: Thus if AB is the line of intersection of the two parallelograms AG, AD; and PR, PS are perpendicular A to AB, the inclination of the planes or parallelograms is the angle included by the lines PS, PR. P B 111. Parallel planes are those which are not inclined to each other, or are every where at an equal perpendicular distance. 112. A solid angle is that which is made by the meeting of more than two plane angles, which are not in the same plane, in one point. 113. Similar solid figures are such as have all their solid angles equal, each to each, and which are contained by the same number of similar planes. 114. A Prism is a solid whose ends are parallel, equal, and like plane figures, and its sides, connecting those ends, are parallelograms. Thus AB is a triangular prism, its ends being the parallel and equal triangles AOC, DGB. 115. D An upright prism is that which has the planes of the sides perpendicular to he ends or base. Thus AB is an upright prism; the sides, or parallelograms CG, GA, CD, being perpendicular to the ends or triangles AOC, DGB. 116. A Parallelopiped, or Parallelopipedon, is a prism bounded by six parallelograms, whereof the opposite ones are parallel, equal, and like to each other. 117. A rectangular parallelopipedon, or prism, is that whose bounding planes are all rectangles, and which stand at right angles one to another. 118. When all the bounding planes are squares, the prism or rectangular parallelopipedon, is called a Cube. 119. A Pyramid is a solid whose base is any right lined plane figure, and whose sides are triangles having all their vertices united in a point above the base, called the vertex of the pyramid. Thus AOCV is a triangular pyramid, its base being the triangle AOC, and its vertex V. 120. A Cylinder ABCD (sometimes called a round prism) is a solid conceived to be generated by the rotation of a rectangle SBCR about one of its sides SR, supposed at rest: which side SR is called the axis of the cylinder. D A |