VI PHILOSOPHICAL TRANSACTIONS. [aNNO 1763. 



HI &c. and K^yh &c. fig. 2, pi. 1, and let y ^=pjcf, and v =■ -^ pa^~^ x — 



« X ( »- 1) X (« - 2) ha "- 3^ _|_ » X (» - 1) X (n - 2) X (n - 3) X ( » - ^) «-3 

 30x2x3 ^ "^ 42x2x3x4x5 P^ 



5 _ n X (n — 1) X (w — 2) X (n — 3) X (n — 4) X (n — 5) X (» — 6) n _ 7 ^ 

 "" 30x2x3x4x5x6x7 ^° '^ + 



5» X (»- 1) X («-J) X (n - 3) X (« - 4) X (n - 5) X (» - 6) X <•« - 7) X (n - 8) 



66 X 2 X 3x4x5x6x7x8xy ~ 



n - 9 9 6 91 X w X (w- 1) X (»-2) X (n- 3) X (»- 4) x (>»-5) x (n -6) x (»-7) 

 Z'" 2730 X2x3x4x5x6x7 x"8 X 9 X loim 



ij^-J)Mn - 9) X (« -JO)^ ^^ n - 1 i^n + &^ ^^e last term of which ought to 



be ar or ar , according as n is an even or an odd number. 



Let ar = ap = a, bisect ap in t, and draw the lines et^; and if ae, em, am, 

 be joined; then will the triangle aem be equal to the area tps^t. 



Again, bisect tp, at in Rand v, and draw rg, cvy ; and joining ac, ce, eg, 

 GM, then will the two triangles ace + egm = the area \TSy\. 



In like manner, if the parts av, vt, tr, rp, be again bisected in w, u, s, q, 

 and there be drawn the lines bw(3, ud, sf, qh; and joining ab, bc, cd, de, ep, 

 KG, GH, km; then will the 4 triangles abc + cde + efg + ghm be = the area 

 wvy(3w. And so on. 



Cor. 1 . If the curve be abc, and cm be a conic parabola, or y = pix^; then will 

 V = i-pax, and AByJ &c. will be a right line; and the proposition is the same 

 with the known proposition of the quadrature of the parabola. 



Cor. 2. Uy = po:^; it will be ?; = -^pax, and AByi &c. again a right line. 



Cor. 3. Given a curve, whose equation is y = pxi„, there can be found an- 

 other curve, whose dimensions are 2n — 1, in which the sum of the triangles, at 

 every bisection, will be respectively equal to the sum of the triangles of the 

 given curve. 



To these may be added, that if, instead of bisecting the abscissa ap, it be 

 divided into unequal parts in any other ratio; the sums of the triangles of the 

 curve abcd &c. will be equal to each of the divisions of , the segments of the 

 curve xQyi. 



XCVII. Second Paper concerning the Parallax oj the Sun determined from the 

 Observations of the late Transit oJ Venus ; in which this Subject is treated of 

 more at Length, and the Quantity of the Parallax more fully ascertained. By 

 James Short, M. A., F. R. S. p. 300, 



In the last volume of the Memoirs of the Royal Academy at Paris, for the 

 year 1761, there is a memoir by M. Pingrc, who went to the island of Rodrigues, 

 and observed the transit of Venus there; in this memoir M. Pingre endeavours 



