For, the figures being similar, are to each other as the squares of their like sides, AB2 to FG2 (th. 88). But, by the last theorem, the sides AB, FG, are as the diameters AL, FM; and therefore the squares of the sides AB to FG2, as the squares of the diameters AL2 to FM (th. 74). Consequently the polygons ABCDE, FGHIK, are also to each other as the squares of the diameters AL2 to гM2 (ax. 1). Q. E. D.

[See fig. th. xc.]

THEOREM XCII.

THE circumferences of all circles are to each other as their diameters*.

The truth of theorems 92, 93, and 94, may be established more satisfactorily than in the text, upon principles analogous to those of the two last notes.

THEOREM. The area of any circle ABD is equal to the rectangle contained by the radius, and a straight line equal to half the circumference.

B

E

N

A

GFLD

H

If not, let the rectangle be less than the circle ABD, or equal to the circle FNH: and imagine ED drawn to touch the interior circle in F, and meet the circumference ABD in E and D. Join CD, cutting the arc of the interior circle in K. Let FH be a quadrantal arc of the inner circle, and from it take its half, from the remainder its half, and so on, until an arc Fi is obtained, less than FK. Join c1, produce it to cut ED in L, and make FG FL: so shall LG be the side of a regular polygon circumscribing the circle FNH. It is manifest that this polygon is less than the circle ABD, because it is contained within it. Because the triangle GCL is half the rectangle of base GL and altitude cr, the whole polygon of which GCL is a constituent triangle, is equal to half the rectangle whose base is the perimeter of that polygon and altitude cr. But that perimeter is less than the circumference ABD, because each portion of it, such as GL, is less than the corresponding arch of circle having radius CL, and therefore, a fortiori, less than the corresponding arch of circle with radius CA. Also CE is less than CA. Therefore the polygon of which one side is GL, is less than the rectangle whose base is half the circumference ABD and altitude CA; that is, (by hyp.) less than the circle FNH, which it contus: which is absurd. Therefore, the rectangle under the radius and half the circumference is not less than the circle ABD. And by a similar process it may be shown that it is not greater. Consequently, it is equal to that rectangle. Q. E. d.

THEOREM. The circumferences of two circles ABD, abd, are as their radfi.

Let D, d, denote the diameters of two circles, and c, c, their circumferences;

then will Dd::c: c, or D:c::d: c.

For (by theor. 90), similar polygons inscribed in circles have their perimeters in the same ratio as the diameters of those circles.

Now as this property belongs to all polygons, whatever the number of the sides may be; conceive the number of the sides to be indefinitely great, and the length of each indefinitely small, till they coincide with the circumference of the circle, and be equal to it, indefinitely near. Then the perimeter of the polygon of an infinite number of sides, is the same thing as the circumference of the circle. Hence it appears that the circumferences of the circles, being the same as the perimeters of such polygons, are to each other in the same ratio as the diameters of the circles. Q. E. D.

THEOREM XCIII.

THE areas or spaces of circles, are to each other as the squares of their diameters, or of their radii.

Let A, a, denote the areas or spaces of two circles, and D, d, their diameters; then A: a : : D2 : d2.

For (by theorem 91) similar polygons inscribed in circles are to each other as the squares of the diameters of the circles.

away its half, and then the half of the remainder, and so on, until there be obtained an arc ed less than eg; and from d draw ad parallel to fg, it will be the side of a regular polygon inscribed in the circle abd, yet evidently greater than the circle ihk, because each of its constituent triangles, as acd contains the corresponding circular sector cno. Let AD be the side of a similar polygon inscribed in the circle ADB, and join ac, CD, similarly to ac, cd. The similar triangles ACD, acd, give ac: ac :: AD: ad, and :: perim. of polygon in ABD perim. of polygon in abd. But, by the preceding theorem, Ac: ac :: circumf. ABD : circumf. abd. The perimeters of the polygons are, therefore, as the circumferences of the circles. But, this is impossible; because, (by hyp.) the perim. of polygon in ABD is less than the circumf.; while, on the contrary, the perim. of polygon in adb is greater than the circumf. ihk. Consequently, ac is not to ac, as circumf. ADB, to a circumference less than adb. And by a similar process it may be shown, that ac is not to ac, as the circumf. abd, to a circumference less than ABD. Therefore ac : ac :: circumf. ABD: circumf. abd. Q. E. D.

= TR,

Corol. Since by this theorem, we have c:c:: R: 7, or, if c= c = r; and, by the former, area (A): area (a) :: ARC Arc: we have A: α :: πR3: πr2 :: R2: r2 :: D2: d2 :: c2: c2.

Hence, conceiving the number of the sides of the polygons to be increased more and more, or the length of the sides to become less and less, the polygon approaches nearer and nearer to the circle, till at length, by an infinite approach, they coincide, and become in effect equal; and then it follows, that the spaces of the circles, which are the same as of the polygons, will be to each other as the squares of the diameters of the circles. Q. E. D.

Corol. The spaces of circles are also to each other as the squares of the circumferences; since the circumferences are in the same ratio as the diameters (by theorem 92).

THEOREM XCIV.

THE area of any circle, is equal to the rectangle of half its circumference and half its diameter.

Conceive a regular polygon to be inscribed in the circle; and radii drawn to all the angular points, dividing it into as many equal triangles as the polygon has sides, one of which is ABC, of which the altitude is the perpendicular CD from the centre to the base AB.

ADB

Then the triangle ABC, being equal to a rectangle of half the base and equal altitude (th. 26, cor. 2), is equal to the rectangle of the half base AD and the altitude CD; consequently the whole polygon, or all the triangles added together which compose it, is equal to the rectangle of the common altitude CD, and the halves of all the sides, or the half perimeter of the polygon.

Now conceive the number of sides of the polygon to be indefinitely increased; then will its perimeter coincide with the circumference of the circle, and consequently the altitude CD will become equal to the radius, and the whole polygon equal to the circle. Consequently the space of the circle, or of the polygon in that state, is equal to the rectangle of the radius and half the circumference.

Q. E. D.

338

OF PLANES AND SOLIDS.

DEFINITIONS.

DEF. 88. The Common Section of two Planes, is the line in which they meet, or cut each other.

89. A Line is Perpendicular to a Plane, when it is perpendicular to every line in that plane which meets it.

90. One Plåne is Perpendicular to Another, when every line of the one, which is perpendicular to the line of their common section, is perpendicular to the other.

91. The Inclination of one Plane to another, or the angle they form between them, is the angle contained by two lines, drawn from any point in the common section, and at right angles to the same, one of these lines in each plane.

92. Parallel Planes, are such as being produced ever so far both ways, will never meet, or which are every where at an equal perpendicular distance.

93. A Solid Angle, is that which is made by three or more plane angles, meeting each other in the same point.

94. Similar Solids, contained by plane figures, are such as have all their solid angles equal, each to each, and are bounded by the same number of similar planes, alike placed.

95. A Prism, is a solid whose ends are parallel, equal, and like plane figures, and its sides, connecting those ends, are parallelograms.

96. A Prism takes particular names according to the figure of its base or ends, whether triangular, square, rectangular, pentagonal, hexagonal, &c.

97. A Right or Upright Prism, is that which has the planes of the sides perpendicular to the planes of the ends or base.

98. A Parallelopiped, or Parallelopipedon, is a prism bounded by six parallelograms, every opposite two of which are equal, alike, and parallel.

99. A Rectangular Parallelopidedon, is that whose bounding planes are all rectangles, which are perpendicular to each other.

100. A Cube, is a square prism, being bounded by six equal square sides or faces, and are perpendicular to each other.

101. A Cylinder is a round prism, having circles for its ends; and is conceived to be formed by the rotation of a right line about the circumferences of two equal and parallel circles, always parallel to the axis.

102. The Axis of a Cylinder, is the right

line joining the centres of the two parallel circles, about which the figure is described.

103. A Pyramid, is a solid, whose base is any right-lined plane figure, and its sides triangles, having all their vertices meeting together in a point above the base, called the vertex of the pyramid.

104. A pyramid, like the prism, takes particular names from the figure of the base.

105. A Cone, is a round pyramid, having a circular base, and is conceived to be generated by the rotation of a right line about the circumference of a circle, one end of which is fixed at a point above the plane of that circle.

106. The Axis of a cone, is the right line, joining the vertex, or fixed point, and the centre of the circle about which the figure is described.

107. Similar Cones and Cylinders, are such as have their altitudes and the diameters of their bases proportional.

108. A Sphere, is a solid bounded by one curve surface, which is every where equally distant from a certain point within, called the Centre. It is conceived to be generated by the rotation of a semicircle about its diameter, which remains fixed.

109. The Axis of a Sphere, is the right line about which the semicircle revolves; and the centre is the same as that of the revolving semicircle.

110. The Diameter of a Sphere, is any right line passing through the centre, and terminated both ways by the surface. 111. The Altitude of a solid, is the perpendicular drawn from the vertex to the opposite side or base.