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This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1748 Excerpt: ...Concave Solid V. Theorem LIX. The Complement of which to its circumfcribing Cylinder--v i s muft give the Content of the convex circumambient Solid P which is formed by the Rotation of the greater Area VMP about the fame Axis V Z. And (putting M Z =s d fixed, and M B =-y variable, whence E D = d---u) the Ultimator of their common curve Superfices will be-3--xd---v, reftoring-y for d in the Subject. Theorem LX.. L a Thirdly, Tfc/W/y, If the greater Area MVP (fee the laft Figure) be turned about its Semiordinate M P, as a fixed Axis, I call the Solid, fo generated, Convex M. In this Cafe put again MP=s =a, and VP=i; = ar (Thco. 51. 52): whence (Theo. 55.) the Area of the Bafe is-ar. and the Content of the circum b bribing Cylinder is ji a '. Now in the Concave Solid V it was juft now (hewn that all its Semiordinates D E, Z M (to the fame Axis.VZ = MP = a = s) were ir, zr, 3', %, 4', 5', &c. to a'. Therefore every E F, being here taken as a Semiordinate, as the Complements of all thofe to the greateft V P= ZM, muft give all fuch Semiordinates EF equal to a'---ir, a'---ar, ar--3', a'--4', ar--5', &c. decreafing ad tnfinitum, till all terminate in the Point M = ar--a' = o. Hence the Squares of all thefe Semiordinates (or Radii of their refpedlive Circles) area1'--2 ar x ir-f ir, atr--2 ar x z-f 2', a --2ar x 3' + 3 Y ar--2 ar x 4' + 4r, &c. And each multiplied by-are the Areas of their reipe&ive Circles. But it is evident, D i. that the Sum of all the firft Terms of thefe circular Areas is (a xMPxVP =3-a2r+I, the fame with the circumfcribing Cylinder. Confequently the Sums of all the reft, under their contrary Signs, muft be the Complement of the Convex Solid M to its circumfcribing Cylinder j which is the Content of the circumambient ...

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