SEC. V. OF NATURAL PHILOSOPHY. 299 the areas YwYZ. X/zwY, proportional to them, will be also equal ; and the lines SX, SY, SZ, that is, AH, EM, Q,T continually proportional, as they ought to be. And if the lines SA, SE, SQ,5 obtain any other order in the series of continued proportionals, the lines AH, EM, Q,T, because of the proportional hyperbolic areas, will obtain the same order in another series of quantities continually proportional. PROPOSITION XXII THEOREM XVII. Let the density of any fluid be proportional to the compression, and its parts be attracted downwards by a gravitation reciprocally propor tional to the squares of the distances from the centre : I say, that if the distances be taken in harmonic progression, the densities of the fluid at those distances will be in a geometrical progression. Let S denote the centre, and SA, SB, SC, SD, SE, the distances in geometrical progression. Erect the perpendiculars AH, BI, CK, (fee., which shall be as the densities of c the fluid in the places A, B, C, D, » E, (fee., and the specific gravities thereof in those places will be as AH BI ,^-, (fee. Suppose these V LN M I. K i V V V V7 ^ n w r / £ L / / /t SA2'SB2'SC2' gravities to be uniformly continued, the first from A to B, the second from B to C, the third from C to I), &c. And these drawn into the altitudes AB, BC, CO, DE, (fec.; or, which is the same thing/into the distances SA, ATT r>T OT7" SB, SC, (fee., proportional to those altitudes, will give -~-r-, ^=5, -~~, (fee.. the exponents of the pressures. Therefore since the densities are as th^ sums of those pressures, the differences AH — BI, BI — CK, (fee., of tb,« densities will be as the differences of those sums ~-r~, ^, ~~, (fee. With the centre S, and the asymptotes SA, S#, describe any hyperbola, cutting the perpendiculars AH, BI, CK, (fee., in a, 6, c, (fee., and the perpendicu lars H/, I//,, K?#, let fall upon the asymptote Sv, in h, i, k ; and the dif ferences of the densities tu, uw, (fee., will be as „ A , ^^, (fee. And the SA; SB; rectangles tu X th, uw X uij (fee., or tp, uq, (fee., as that is, as Aa, Bb, (fee. AH X th BI X ui — , (fee. SA SB For, by the nature of the hyperbola, SA is to AH or St as th to Ar, and therefore — pri — is equal to Aa . And, by a like SA