To the extent possible under law, the Text Creation Partnership has waived all copyright and related or neighboring rights to this keyboarded and encoded edition of the work described above, according to the terms of the CC0 1.0 Public Domain Dedication (http://creativecommons.org/publicdomain/zero/1.0/). This waiver does not extend to any page images or other supplementary files associated with this work, which may be protected by copyright or other license restrictions. Please go to http://www.textcreationpartnership.org/ for more information.
IF any one had rather proceed herein by indivisibles, as in the precedent Prop. having divided the ax EF (Fig. 140.) again into three equal parts, and assuming the values of the lines determined in the hyperbola, viz. eb for the abscissa EF, ob for the transverse ax, 〈 math 〉〈 math 〉 for the Latus Rectum, oecd+eecd for the square of the semiordinate FG, &c. the lowest and greatest circle of the diameter HG will be as oecd+eecd, and, if you make
as the Latus Transv. to the Latus Rectum, so the □ DfE ob−〈 math 〉〈 math 〉 made of ob+⅔eb into ⅔ eb (i. e. ⅔oebb+〈 math 〉〈 math 〉eebb) to a fourth; there will come out ⅔oecd+〈 math 〉〈 math 〉eecd for the second circle of the diam. hg; and by the same inference (as ob to 〈 math 〉〈 math 〉 so ob+⅓eb into ⅓ eb to a fourth) for the third circle of the diameter HG ⅓oecd+〈 math 〉〈 math 〉eecd; so that these indivisibles [for which here and in the precedent also the partial circumscribed cylinders may be assumed] proceed in a double series of numbers, the first in a simple arithmetical progression 3, 2, 1, the latter in a du∣plicate Arithmetical progression of squares 9, 4, 1; and the same if you make further new bisections, will necessarily hap∣pen ad Infinitum, (the former numbers e. g. in the first bise∣ction will be 〈 math 〉〈 math 〉 ⅚ 〈 math 〉〈 math 〉 oecd the latter 〈 math 〉〈 math 〉 eecd, &c) it is manifest from the consectaries of Prop. 21. lib. 1. that the whole cylinder HK will in like manner be expressed by a double series of parts answering, in numbers to the indivisibles of the conoid made by any bisection, but in magnitude to the greatest of them all, and in the sum of its first series of parts will be to the sum of the first in the conoid, both being infinite, as 2 to 1 or 3 to 1 ½ oecd, by Consect. 9. of the said Prop. 21. and the sum of its latter to the sum of the former in the conoid will be as 3 to 1 eecd and so the whole cylinder to the whole conoid as 3 oecd+3 eecd to 1 ½ oecd + eecd i. e. (dividing by ecd) as 3o+3e to 1 ½o+e i. e. mul∣tiplying both sides by b) as 3ob+3eb to 1 ½ob+eb; and consequently the cone (which is ⅓ of the cylinder) to the co∣noid as ob+eb to 1 ½ob+eb. Q.E.D.