fore the area of that circle being drawn into the fluxion of the axis, will produce the fluxion of the solid. That is, AD X area of the circle BCF, whose radius is DE, or diameter BE, is the fluxion of the solid, by art. 9. 67. Hence, if AD = x, DE = y, c = 3·1416; because cy is equal to the area of the circle BCF; therefore cy's is the fluxion of the solid. Consequently if, from the given equation of the curve, the value of either y or r be found, and that value substituted for it in the expression cy's, the fluent of the resulting quantity, being taken, will be the solidity of the figure proposed. EXAMPLES. EXAM. 1. To find the solidity of a sphere, or any segment. The equation to the generating circle being y = ax - x2, where a denotes the diameter, by substitution, the general fluxion of the solid cy2, becomes caxx cx2x, the fluent of which gives cax2 - cx3, or cx2 (3a -2x), for the solid content of the spherical segment BAE, whose height Ad is x. When the segment becomes equal to the whole sphere, then x = a, and the above expression for the solidity, becomes ca3 for the solid content of the whole sphere. And these deductions agree with the rules before given and demonstrated in the Mensuration of Solids. EXAM. 2. To find the solidity of a spheroid. : TO FIND LOGARITHMS. 68. It has been proved, art. 23, that the fluxion of the hyperbolic logarithm of a quantity, is equal to the fluxion of the quantity divided by the same quantity. Therefore, when any quantity is proposed, to find its logarithm; take the fluxion of that quantity, and divide it by the same quantity; then take the fluent of the quotient, either in a series or otherwise, and it will be the logarithm sought; when corrected as usual, if need be; that is, the hyperbolic logarithm. 69. But, for any other logarithm, multiply the hyperbolic logarithm, above found, by the modulus of the system, for the logarithm sought. Note Note. The modulus of the hyperbolic logarithms, is 1; and the modulus of the common logarithms, is 43429448190 &c; and, in general, the modulus of any system, is equal to the logarithm of 10 in that system divided by the number 2-3025850929940&c, which is the hyp. log. of 10. Also, the hyp. log. of any number, is in proportion to the com. log. of the same number, as unity or 1 is to 43429&c, or as the number 2.302585&c, is to 1; and therefore, if the common log. of any number be multiplied by 2.302585&c, it will give the hyp. log. of the same number; or if the hyp. log. be divided by 2-302585&c, or multiplied by 43429&c, it will give the common logarithm. Denoting any proposed number z, whose logarithm is required to be found, by the compound expression a + x a -, the fluxion of the number ż, is-, and the fluxion a Then the fluent of these terms give the logarithm of z Now, for an example in numbers, suppose it were required to compute the common logarithm of the number 2. This will be best done by the series, a + x a-x x a =, and a Making = 2, gives a = 3x; conseq. =, which is the constant factor for every succeeding term; also, 2m = 2 × 43429448190 = 868588964; therefore the calculation will be conveniently made, by first dividing this number by 3, then the quotients successively by 9, and lastly these quotients in order by the respective numbers 1, 3, 5, 7, 9, &c, and after that, adding all the terms together, as follows: Sum of the terms gives log. 2 = 301029995 TO FIND THE POINTS OF INFLEXION, OR OF CONTRARY FLEXURE IN CURVES. changes from concave to convex, or from convex to concave, on the same side of the curve. Such as the point e in the annexed figures, where the former of the two is concave towards towards the axis AD, from A to E, but convex from E to F; and on the contrary, the latter figure is convex from A to E, and concave from E to F. 71. From the nature of curvature, as has been remarked before at art. 28, it is evident, that when a curve is concave towards an axis, then the fluxion of the ordinate decreases, or is in a decreasing ratio, with regard to the fluxion of the absciss; but, on the contrary, that it increases, or is in an increasing ratio to the fluxion of the absciss, when the curve is convex towards the axis; and consequently those two fluxions are in a constant ratio at the point of inflexion, where the curve is neither convex nor concave; that is, & is toj in a constant ratio, or * oris a constant quantity. But constant quantities have no fluxion, or their fluxion is equal to nothing; so that in this case, the fluxion of ز or of is equal to nothing. And hence we have this general rule: نو 72. Put the given equation of the curve into fluxions; from which find either or Then take the fluxion of this ratio, or fraction, and put it equal to O or nothing; and from this last equation find also the value of the same * or j Then put this latter value equal to the former, which will form an equation; from which, and the first given equation of the curve, x and y will be determined, being the absciss and ordinate answering to the point of inflexion in the curve, as required. EXAMPLES. EXAM. 1. To find the point of inflexion in the curve whose equation is ar2 = a2y + x2y. This equation in fluxions is 2axx = a2y + 2xyx + x2j, a2+x2 which gives -2ax-2xy Then the fluxion of this quantity made = 0, gives 2xx (ax-xy) = (a2 + x2) x (ax-xy-xj); a2 + x2 being put equal the former, gives x2a-y a-y ; and hence 2x2 = a2 or 3x2 = a2, and x = a, the absciss. Hence also, from the original equation, ax2 2 = a, the ordinate of the point of in flexion sought. EXAM. 2. To find the point of inflexion in a curve defined by the equation ay = a/ax2 + xx. EXAM. 3. To find the point of inflexion in a curve defined by the equation ay2 = a2x+x3. ABCE FGHI D EXAM. 4. To find the point of inflexion in the Conchoid of Nicomedes, which is generated or constructed in this manner: From a fixed point P, which is called the pole of the conchoid, draw any number of right lines PA, PB, PC, PE, &c, cutting the given line FD in the points F, G, H, I, &c: then make the distances FA, GB, HC, IE, &c, equal to each other, and equal to a given line; then the curve line ABCE &C, will be the conehoid; a curve so called by its inventor Nicomedes. TO FIND THE RADIUS OF CURVATURE OF CURVES. 73. THE Curvature of a Circle is constant, or the same in every point of it, and its radius is the radius of curvature. But the case is different in other curves, every one of which has its curvature continually varying, either increasing or decreasing, and every point having a degree of curvature peculiar to itself; and the radius of a circle which has the same curvature with the curve at any given point, is the radius of curvature at that point; which radius it is the business of this chapter to find. 74. Let AEe be any curve, concave towards its axis AD; draw an ordinate De to the point E, where the curvature is to be found; and suppose Ec perpendicular to the curve, and equal to the radius of curvature sought, or equal to the radius of a circle having the same curvature there, and with that radius describe the said equally B d D G |