...been faid of Circles in the fourth Book. Wherefore all fimilar Figures, &c. COROLLARY I. •• I. Circles are to each other as the Squares of their Radii or Diameters .- forfuppofe a Circle whbfe Radius is r, and then another Circle greater/ or"'lfefs;' than...

...or -\7lDt. For C=27tR. §342 Substituting in (1), S= | X %7tR XR, S=7tR*. Also 8 = 345. COR. 2. — The areas of circles are to each other as the squares of their radii or as the squares of their diameters. Let S and S' denote the areas of two circles, R and R1 their radii,...

...SA*xAB; ---- x -- = -- xAB, 3 5 )5 the sum of all the series of squares, ma", nc*, py2, Sfe. But as circles are to each other as the squares of their radii or diameters ; it evidently follows that -^ SA1 x.7854xAB will he the solidity of one-fourth of the spindle...

...as their radii. By the same construction, a similar train of reasoning would shew, that the surfaces of circles are to each other as the squares of their radii. We need not enter upon any farther details respecting this proposition, particularly as it forms a...

...circle whose radius is OB, is 2 equal to nx OB. But nx CA: nx OB: : CA : OB; therefore, the surfaces of circles are to each other as the squares of their radii, which agrees with the preceding theorem. 292. Scholium. We have already said, that the problem of the...

...whose radius is OB, is — 2 equal to nx OB. But n X CA: n X OB: : CA : OB; therefore, the surfaces of circles are to each other as the squares of their radii, which agrees with the preceding theorem. 292. Scholium. We have already said, that the problem of the...

...whose radius is OB, is — 3 equal to nx OB. But k nx CA: nx OB: : CA : OB; therefore, the surfaces of circles are to each other as the squares of their radii, which agrees with the preceding theorem. 292. Scholium. We have already said, that the problem of the...

...demonstrated. 1. Two similar polygons are to each other as the squares of their homologous sides. 2. Two circles are to each other as the squares of their radii or diameters. 3. The surfaces of two spheres are to each, other as the squares of their radii. 4. The...

...the circle, whose radius is OB, will be equal to * x OB2 ; but nx CA2 : nx OB2 : : CA2 : OB2 ; hence the areas of circles are to each other as the squares of their radii, which agrees with the preceding theorem. Scholium. We have already observed, that the problem of the...

...large engine has less rubbing surface, or less friction, according to its power, than a small one. 3. The areas of circles are to each other as the squares of their diameters, or as .7854 to 1 : a circle twice the diameter contains four times the area. EXAMPLE...