24 &c &c And by cor. 5 prop. 1 fect. 1, the area will be 1 x :a + ax + ¿ßx2 + zyx3+id** + 1x3 &c 1Bx- 1yx2-78d x3 ¿yx++; dx2+37&x3 + ** -- + TEX 76x? &c &c &c Where L represents the length of the base. Which area will be had accurately true when the differences are continued till one order of them become equal to nothing, for then the feries will break off and terminate; that is, an area is quadrable whenever one of the orders of the differences of its equidiftant ordinates confifts of a series of nothings, or any other equals, or of a series of arithmeticals; but if an order of the differences never become equal to nothing, the area will be expreffed by an infinite feries. When the terms or ordinates are taken near to one another, the differences will the fooner become equal to nothing, or nearly fo; and if any order of differences, and confequently its fucceeding ones, be rejected as inconfiderable, we shall have an approximate value of the area, and that the nearer to the truth as the more of the differences a, B, y, d, e, &c, are used. Thus, if there be only one ordinate, or if a, ß, y, &c, be rejected, the area will be aL. jected, x will be If there be two ordinates, or ß, y, 8, &c, be re1, and the area will be a+b (a + α) × L = X L. will 2 If there be three ordinates, or y, d, e, &c, be rejected, be 2, and the area will be (a + a + zB — 1B) × 1 = (a + a + 1B) × L = (a + 4b+c) × L. If there be four ordinates, or d, e, &c, be rejected, x will be 3, and the area will be 2 a + 23 a + 2 B + 2 x 1 × 1 = (a + 3 a +36 + y) × L -36-37 + 1/2 =(a+3b+3c+ d) × ÷ L. And thus by putting x equal to every number fucceffively, we fhall have the following table of areas, anfwering to the refpective number of ordinates fet oppofite to them; of which every expreffion is more accurate than the preceding ones, and in which A reprefents the fum of the first and last terms, в the fum of the fecond and last but one, c the fum of the third and laft but two, &c, and the laft of the letters denotes the double of the middle term when the number of terms is odd, alfo L denotes the length of the whole bafe, or the distance between the firft and laft terms. Taking the third example to the last propofition, in which are given the five perpendiculars 10, 11, 14, 16, 16, and the diftance between the firft and laft = 20; we shall have A 10 + 16 = 26, B = II +16=27, c = 14 X 228, and L = 20. Hence by the rule for five ordinates 74 +32B+6c XL= 182864+168 area as before, nearly. SCHOLIUM. 90 X 2 269 the When there are many ordinates given, the cafe may be reduced to fewer, by adding together the first and laft ordinates, the fecond and last but one, the third and laft but two, and fo on, and confidering the fums as a new fet of terms upon a bafe equal to K k 2 half half that of the former. Or there may be added together the two first and two laft terms into one fum, then the four terms next to thefe, and fo on; or we may add three at the beginning to three at the end, then the next fix, and fo on ; always diminishing the bafe in proportion to the number of terms that are added into each fum. And the content will be nearly the fame in each cafe. EXAMPLE. Taking the fourth example to the last propofition, in which are given the eleven ordinates 10 10 I O 10 10, 11, 12, 1, 12, 15, 18, 17, 18, 17, 10, and the diftance of the firft and laft = 1; we fhall have 10+ 0 = 15, 1+1=14354069, IO To ΙΟ उ 10 17 IO ΙΟ = 1.3888889, 19 + 9 = 1.357466, 12 + 18 13392857, and 1 x 2 = 1. B = 1°4354069 + 1°3392857 = 2•7746926, C= 1.3888889 +1357466 = 2.7463549, and L = 288 · 2 Then, by the rule for fix ordinates, we shall have 19A +75B + 50c XL = 6931476, which is much nearer to the truth than the number found by the former rule, the true number being 69314718, and is the hyperbolic logarithm of 2. SEC SECTION III. OF THE RELATION BETWEEN THE AREAS CENTERS OF GRAVITY OF THEIR IF any Line, Right or Curved, or Any Plane Figure, whether it be bounded by Right Lines or Curves, revolve about an Axe in the Plane of the Figure; the Surface or Solid generated will be refpectively equal to the Surface or Solid whofe bafe is the Given Line or Figure, and its Height equal to the Arc defcribed by the Center of Gravity of the Jaid Generating Line or Figure; and confequently the Content will be found by drawing the Generating Line or Figure into the Arc defcribed by its Center of Gravity. DEMONSTRATION. Let AFHD be the figure generated by the given line or plane ABD; through c the center of gravity of which draw DCAE perpendicular to the axe of revolution, and meeting HGFE in E; and let every point of the base be reduced to AD by means of perpendiculars to it. The figure AFHD gene rated, is equal to all the AF, CG, DH, &c. But, by fimi lar figures, all the AF, CG, DH, &c, are as all the Kk 3 EA, |
