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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. 1736 Excerpt: ... yyy, and x = whence x x =z 9 fy, and by substituting this in the general Equa4 x x tion z = x x-f y y, in the room of x x, we shall have z z =z?l-JLZ y y; and because 1 x=y whence 4 #=4)'3. 4-x x If we substitute this last Quantity 4y in the room of 4 x x in the last Equation, we shall have z, k = 9 y y-y + y 4 4 1 whence x = 4y y + a a; whence the Fluxion of the Hyperbolic Space D E C A will be y x 4y--aa, the same with the Fluxion of the Curve of the Parabola; whence the Rectification of the Parabolic Curve depends upon the Quadrature of the Hyperbolic Space s whence the Rectification of a Curve may be compar'd with the Quadrature of a Curve, by supposing the Fluxion of the Curve to be rectified as an Ordinate, and the variable Quantity in that Fluxion as an Abscisle to that Ordinate. a a-4-2 y y X y y; whence = f-xy, for the Fluxion of the. a--jy Curve Line AC; and If we make the Parameter, which is supposed equal to a, to be equal to Unity, we shall have-- z = y x '-.;but the Root of 1 + zyy is 1 +j +yy S?f. and the Root of r +yy is 1-f-these there fore being divided by each other, will produce 1-j-yyy &c. and this being multiplied by j/, will produces + % y yy, whose Fluents-f y c-wil1 be the Length of the Curve Line A C of the Hyperbola. 1 EXAMPLE V. Ls T ft be required to find the Length of any Arch of the 3Equi-angular, or Logarithmical Spiral. L E T the Lines B E and F G touch the Spiral in the Points B and F, and draw the Lines A E and A G, so as to form right Angles with the Tangents B E and F G, and on A, as a Centre, describe the Arch Q.F. Put Afj, AC=, and B = AB--A F = _y, and FG = r. Now since, from the Nature of the Curve, any Radius, A F, forms the fame Angle A F G with the Tangent F G, there-/k Hh a for...

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