and the triangles DEF, DEM, on the same base DE, are as their altitudes FH, DM, or DE that is, triangle ABC : triangle ABK :: CG : AB, and triangle DEF : triangle DEM :: FH : DE. But it has been shown that ca : AB :: FH : DE; theref. of equality A ABC : AABK :: ADEF : ADEM, or alternately, as A ABC : ADEF : : A ABK : ADEM. But the squares AL, DN, being the double of the triangles ABH, DEM, have the same ratio with them; therefore the AABC : A DEF :: squarc AL : Square DN. Q. E. D. THEOREM LXXXIX. All Similar Figures are to each other, as the Squares of their Like Sides. LET ABCDT, FGHIK, be D any two similar figures, the K like sides being AB, FG, and C H BC'gh, and so on in the same order: then will the figure А. B ABCDE be to the figureFGHIK, G as the square of AB to the square of FG, or as AB to FG%. For, draw BE, BD, GK, GI, dividing the figures into an equal number of triangles, by lines from two equal angles B and a The two figures being similar (by suppos.), they are equi. angular, and have their like sides proportional (def. 70). Then, since the angle A is = the angle F, and the sides AB, AE, proportional to the sides FG, FK, the triangles ABE; FGk, are equiangular (th. 86). In like manner, the two triangles BCD,'GHI. having the angle c= the angle , and the sides BC, CD, proportional to the sides gu, hi, are also equiangular. Also, if from the equal angles AED, FKI, there be taken the equal angles AEB, FKG, there will remain the equals BED, GKI; and if from the equal angles CDE, HIK, be taken away the equals cob, hig, there will remain the equals BDE, GIK; so that the two triangles BDE, GIK, having two angles equal, are also cquiangular. Hence each triangle of the one figure, is equiangular with each corresponding triangle of the other. But equiangular triangles are similar, and are proportional to the squares of their like sides (th. 88). Therefore Therefore the A ABE : À FGK :: AB? : FG2, and ABCD : A GHI :: BC? : GH2; and A BDE : A GIK : ; DE2 : 16. But as the two polygons are similar, their like sides are proportional, and consequently their squares also proportional ; so that all the ratios, AB? to FG2, and Bc? to GH2, and De’ to IK%, are equal among themselves, and consequently the corresponding triangles also, ABE to FGk, and BCD to Ghi, and BDE 10 GIK, have all the same ratio, viz. that of AB? 10 FG% : and hence all the antecedents, or the figure ABCDE, have to all the consequents, or the figure FGHIK, still the same ratio, yiz. that of ABS t0 FG(th. 72). Q. E. D. THEOREM XC. Similar Figures Inscribed in Circles, have their Like Sides, and also their Whole Perimeters, in the Same Ratio as the Diameters of the Circles in which they are Inscribed. LET ABCDE, FGAIK, D M м B one figure be to the like side Gp, GH, &c, of the other figure, or the whole perimeter AB + BC + &c, of the one figure, to the whole perimeter FG + G + &c, of the other figure, as the diameter Al to the diameter Fm. For, draw the two corresponding diagonals AC, FH, as also the lines BL, GM. Then, since the polygons are similar, they are equiangular, and their like sides have the same ratio (def. 70); therefore the two triangles ABC, FGH, have the angle B = the angle g, and the sides AB, BC, proportional to the two sides FG, GH, consequently these two triangles are equiangular (th. 86), and have the angle ACB = FHG. But the angle ACB = ALB, standing on the same arc AB ; and the angle FHG= FMG, standing on the same arc FG ; therefore the angle ALB = FMG (th. 1). And since the angle ABL = FGM, being both right angles, because in a semicircle ; therefore the two triangles ABL, FGM, having wo angles equal, are equiangular ; and consequently their like sides are proportional (th. 84); hence AB : F6 : : the diameter al : the diameter fm. In like manner, each side bc, cd, &c, has to each side GH, HI, &c, the same ratio of Al to Fm ; and consequently the sums of them are still in the same ratio ; viz, AB + BC + GD, &c : FG + Go + ni, &c : : the diam. Al : the diam. FM (th. 72). Q. E. D. THEOREM XCI. Similar Figures Inscribed in Circles, are to cach other as the Squares of the Diameters of those Circles. L LET ABCDE, FGHIK, I M be two similar figures inscribed in the circles C whose diameters are al and rm; then the surface of the polygon ABCDE A B will be to the surface of the polygon FGHIK, as all to FM”. For, the figures being similar, are to each other as the squares of their like sides, AB? to FG(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 ro', as the squares of the diameters Al? to FM3 (th. 74). Consequently the polygons ABCDE, FGHik, are also to each other as the squares of the diameters AL? LO FM? (ax. 1). Q. E. D. THEOREM XCII. The Circumferences of all Circles are to each other as their Diameters, LET D, d, denote the diameters of two circles, and C, C, their circumferences ; then will D :d :: cic, 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, diheir diameters; then A: a : : D?: d2. For (by theorem 91) similar polygons inscribed in circles are to each other as the squares of the diameters of the circles. 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 ap. proach, 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. Corul. 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 ABC, of which the altitude is the perpendicular cp from the centre to the base AB. AD B 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; conse 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. ADB 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 ce 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 ihe rectangle of the radius and half the circumference. Q. E. D. OF PLANES AND SOLIDS, DEFINITIONS. Def. 88. The common Section of two Planes, is the line in which they meet, to 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 Plane 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 cach other in the same point. 94. Similar |