nxn and — 2x-1=-mxm - 1 :: + n x22+mxm-1 Confequently (from the laft, and laft but two Steps) n x n − 1 NF. PROP. IV. A PEP B Of all the Cones that can be infcribed in a given Sphere; 'tis required to find that which has the greatest convex Surface. This Propofition amounts to no more than this; To determine the Point E in the Diameter AB, of the Circle A BF, fo that the Rectangle, comprehended under AF and F E, be the greateft of all the like Rectangles comprehended under A N and N p. For if we imagine the Semi-circle A FB to revolve about its Axis A B, it is evident that the Semi-circle defcribes a Sphere, and the Ld As AFE and A Np defcribe Cones infcribed in the fame Sphere, whofe Surfaces are proportional to the refpective Rectangles A Fx FE, and A NxNp. Suppofe the Diameter of the Sphere, viz. AB=27, the unknown Quantity AE=x? And the indefinitely short Line pE Ep=a: = Then (by the Property of a Circle) : 2 r x - xx: =EF, and 2rx-x x + x x := V2rx=FA; and 2 r xx xx+xx:= √2rx= √:2rx - X0 X0: √: 4rrxx 21x3: Maximum. Quere x, it being an affirmative Quantity. increas'd diminish'd Suppofex (AE) to be by the indefinitely little Quantity a (E); then y: 4 rrx x2rx3: (=AFX FE) will become : 4 rrxx-2 rx3 +8 rr xa+6rxxa+ 4rraa-6rxa a zra3: (=ANxN); But the former being a Maximum, is therefore greater than the latter, 2 3 and the and confequently (each of them being more than Nothing) the Square of the former, viz. 4 r r x x 2 r x3 is Square of the latter, viz. 4rrx x − 2 r x3 + 8 rrxa + 6 rxxa + 4 rra2 · 6 rx a2 + 2 ra : Wherefore +6 r x x + 6 r x x - 4 rras = ± 8 rrx. 4rra € Hence 6rx a Let CF be a Chord, whofe Length must not exceed CB, and let one End of the Chord C be made faft at C, and to the other End faften the Pulley F, about which fufpend the Weight D, by the Chord D F B, which muft be of a fufficient Length, fastening one End thereof at B, and let the Points B and C be in the fame horizontal Line BC: And fuppofe both the Chords and the Pulley to be without Weight; 'tis required to find the loweft Defcent of the Weight D. The Chords and the Pulley being without Weight, or the Weight D fufficiently heavy, it is evident that the Weight D will defcend below the horizontal Line CB, as low as the Chords CF and BFD will permit; and therefore DFE will reprefent the greatest Defcent of the Weight D. Suppofe the known Quantities DFBb, CBc, and CFd: And fuppofe the unknown and variable Quantity CE; then is EFy: ddxx, and FBy: d'd +cc2cx:, and DFE b-v: dd + cc-2cx: + √ : d d x x : Maximum. Quere x. augmented xx:= Suppofe x to be diminished by the indefinitely fmall Quantity a, then bv: dd+cc-2cx: +√:dd-xx: will become by: dd+cc-2 cx +2ca:+v: dd - x x +2xa-aa:; But, fince the former is a Maximum, it is, *K k of of Confequence, greater than the latter; that is (by Evolution) b = √ : d d + c c − 2 cx : + √ : d d − x x : — b √: dd+cc−2cx: - ; x d d + cc - 2 cx'x + 2 ca 2 -2 2 xdd+cc-2cx − x x : + × d d — x x 2 I when reduc'd, gives + 2 Which laft Step, хха 2 xdd — xx and V:dd. - 00 - 30 ; Confequently x x : √:dd-xx: C √:dd+cc 2c x: 5 Wherefore ccd dccxx d d x x + ccxx — 2 c x3·· 2 c x3 — 2 c2 x x - d d x x + c c d do; which Equation being divided by x-c (0) gives 2 cx xd dx - c d d CHA P. III. Of Quadratures. PROP. I. Glv Iven the Bafe A C, and Perpendicular A B of the Ld A AC-b BA=pS LdAABC, Required. Preparation. f e F Z A Suppofe the AB to be divided into an indefinite Number of equal Parts BD, DE, EF, &c. and Z A ; and fuppofe each of thofe Parts = a; then thro' the Points B, D, E, F, &c. and Z, draw the Lines Bb, DDD, Eee, Fff, &c. and Z 23 ||s AC; and thro' the Points, wherein the faid Lines interfect the Line BC, viz. D, E, F, &c. z and C, draw the Lines b De, o Ef, eff, &c. 2a, and 3 C || sBA: Then the Sum of the s BbD, DoCE, E e FF, $50. and Z3 CA is called the Sum of the circumfcribings, because their Sum circumfcribes the A; and the Sum of the SD De E, EEƒF, &c. and Z za Á, is called the Sum of the infcribed s. Now the first (or leaft), fecond, third, 5c. of the circumfcribings, being equal to the firft, fecond, third, &c. of the infcribed as refpectively, and the Number of the circumfcribings being (as manifeftly it is) Number of equal А В Parts, into which AB is, by Suppofition, divided = a (= 용). and by one more than the Number of the in fcribed s; the Sum therefore of the Areas of the circumfcribings is, by the Area of the greatest of them, more than the Sum of the Areas of the infcribeds. |