OF RECTIFICATIONS; OR, TO FIND THE LENGTHS OF CURVE LINES. 60. RECTIFICATION, is the finding the length of a curve line, or finding a right line equal to a proposed curve. By art. 10 it appears, that the elementary triangle Eae, formed by the increments of the absciss, ordinate, and curve, is a right-angled triangle, of which the increment of the curve is the hypothenuse; and therefore the square of the latter is equal to the sum of the squares of the two former; that is, Ee2 = Ea2 + ae2. Or, substituting, for the increments, their proportional fluxions, it is żż = ✰✰ +ÿÿ, or ¿ =+; where z denotes any curve line AE, x its absciss AD, and y its ordinate DE. Hence this rule. RULE. 61. From the given equation of the curve put into fluxions, find the value of * or j2, which value substitute instead of it in the equation =√x2+j2; then the fluents, being taken, will give the value of z, or the length of the curve, in terms of the absciss or ordinate. EXAMPLES. EXAM. 1. To find the length of the arc of a circle, in terms both of the sine, versed sine, tangent, and secant. The equation of the circle may be expressed in terms of the radius, and either the sine, or the versed sine, or tangent, or secant, &c, of an arc. Let therefore the radius of the circle be CA or CE = r, the versed sine AD (of the arc AF)=x, the right sine DE=y, the tangent TE=t, and the secant CTS; then, by the nature of the circle, there arise these equations, viz. Then, by means of the fluxions of these equations, with the general fluxional equation 2 = 2+j, are obtained the following fluxional forms, for the fluxion of the curve; the fluent of any one of which will be the curve itself; viz. Hence the value of the curve, from the fluent of each of these, gives the four following forms, in series, viz. putting d2r the diameter, the curve is Now, it is evident that the simplest of these series, is the third in order, or that which is expressed in terms of the tangent. That form will therefore be the fittest to calculate an example by in numbers. And for this purpose it will be convenient to assume some arc whose tangent, or at least the square of it, is known to be some small simple number. Now, the arc of 45 degrees, it is known, has its tangent equal to the radius; and therefore, taking the radius. r = 1, and consequently the tangent of 45°, or t, = 1 also, in this case the arc of 45° to the radius 1, or the arc of the quadrant to the diameter 1, will be equal to the infinite series 1 + ÷ − ÷ + } - &c. But as this series converges very slowly, it will be proper to take some smaller arc, that the series may converge faster; such as the arc of 30 degrees, the tangent of which is, or its square : which being substituted in t2 = the series, the length of the arc of 30° comes out 3.3 5.32 7.33 9.34 pute these terms in decimal numbers, after the first, the succeeding terms will be found by dividing, always by 9, and these quotients again by the absolute numbers 3, 5, 7, 9, &c; and lastly, adding every other term together, into two sums, the one the sum of the positive terms, and the other the sum of the negative ones; then lastly, the one sum taken from the other, leaves the length of the arc of 30 degrees; whicła being the 12th part of the whole circumference when the radius is 1, or the 6th part when the diameter is 1, consequently 6 times that arc will be the length of the whole circumference to the diameter 1. Therefore, multiplying the first term by 6, the product is 12 3.4641016; and hence the operation will be conveniently made as follows: +Terms. EXAM. 2. To find the length of a parabola. EXAM. 3. To find the length of the semicubical parabola, whose equation is ar2 = y3. EXAM. 4. To find the length of an elliptical curve: EXAM. 5. To find the length of an hyperbolic curve. OF QUADRATURES; OR, FINDING THE AREAS OF CURVES. 62. The Quadrature of Curves, is the measuring their areas, or finding a square, or other right-lined space, equal to a proposed curvilineal one. By art. 9 it appears, that any flowing quantity being drawn into the fuxion of the line along which it flows, or in the direction of its motion, there is produced the fluxion of the quantity generated by the flowing. That is, Dd X DE or yx is the fluxion of the area ADE. Hence this rule. Hence the value of the curve, from the fluent of each of these, gives the four following forms, in series, viz. putting d2r the diameter, the curve is Now, it is evident that the simplest of these series, is the third in order, or that which is expressed in terms of the tangent. That form will therefore be the fittest to calculate an example by in numbers. And for this And for this purpose it will be convenient to assume some arc whose tangent, or at least the square of it, is known to be some small simple number. Now, the arc of 45 degrees, it is known, has its tangent equal to the radius; and therefore, taking the radius. r = 1, and consequently the tangent of 45°, or t, = 1 also, in this case the arc of 45° to the radius 1, or the arc of the quadrant to the diameter 1, will be equal to the infinite series 1 ÷ ÷ ÷ − ÷ ÷ ÷ − &c. But as this series converges very slowly, it will be proper to take some smaller arc, that the series may converge faster; such as the arc of 30 degrees, the tangent of which is, or its square : which being substituted in the series, the length of the arc of 30° comes out pute these terms in decimal numbers, after the first, the suc ceeding terms will be found by dividing, always by 3, anc these quotients again by the absolute numbers 3, 5, 7, 9, &c ̧ and lastly, adding every other term together, into two sun. the one the sum of the positive terms, and the other the sur of the negative ones; then lastly, the one sum taken fro the other, leaves the length of the arc of 30 degrees; whic being the 12th part of the whole circumference when th radius is 1, or the 6th part when the diameter is 1, cons quently 6 times that arc will be the length of the whole ci cumference to the diameter 1. Therefore, multiplying t first term by 6, the product is 12 = 3.4641016; a hence the operation will be conveniently made as follows + Ter |