300 THE MATHEMATICAL PRINC. PLES [BOOK II. reasoning, — ^n~~ *s e(lua^ to ^, &c- But Aa> B^> ^c, &cv are continu ally proportional, and therefore proportional to their differences Aa — B&, B6 — Cc; &c., therefore the rectangles fy?, nq, &c., are proportional to those differences ; as also the sums of the rectangles tp + uq, or tp + uq -f w to the sums of the differences Aa — Cc or Aa — Da7. Suppose several of these terms, and the sum of all the differences, as Aa — F/, will be pro portional t? the sum of all the rectangles, as zthn. Increase the number of terms, and diminish the distances of the points A, B, C, (fee., in iiijinitum, and those rectangles will become equal to the hyperbolic area zthn. and therefore the difference Aa — F/ 19 proportional to this area. Take nowT any distances, as SA, SD, SF, in harmonic progression, and the dif ferences Aa — Da7, Da1 — F/ will be equal ; and therefore the areas thlx, xlnz, proportional to those differences will be equal among themselves, and the densities St, S:r, Sz, that is, AH, DL, FN, continually proportional. Q.E.D. COR. Hence if any two densities of the fluid, as AH and BI, be given, the area thiu, answering to their difference tu, will be given; and thence the density FN will be found at any height SF, by taking the area thnz to that given area thiu as the difference Aa — F/ to the difference Aa — Eh. SCHOLIUM. By a like reasoning it may be proved, that if the gravity of the particles of a fluid be diminished in a triplicate ratio of the distances from the centre ; and the reciprocals of the squares of the distances SA, SB, SC, &c., (namely, SA3 SA3 SA3 . opt' ^e ta^en m an arithmetical progression, the densities AH. BI, CK, &c., will be in a geometrical progression. And if the gravity be diminished in a quadruplicate ratio of the distances, and the reciprocals of the cubes of the distances (as ^-r^, SRS» sps' ^c'^ ^e ta^cn ^n ai>itnmeticai progression, the densities AH, BI, CK, &c., will be in geometrical pro gression. And so in irtfinitum. Again ; if the gravity of the particles of the fluid be the same at all distances, and the distances be in arithmetical progression, the densities will be in a geometrical progression as Dr. Halley has found. If the gravity be as the distance, and the squares of the distances be in arithmetical progression, the densities will be in geometri cal progression. And so in infinitum. These things will be so, when the density of the fluid condensed by compression is as the force of compres sion ; or, which is the same thing, when the space possessed by the fluid is reciprocally as this force. Other laws of condensation may be supposed, as that the cube of the compressing force may be as the biquadrate of the