RECTIFICATION

, in Geometry, is the finding of a right line equal to a curve. The Rectification of curves is a branch of the higher geometry, a branch in which the use of the inverse method of Fluxions is especially useful. This is a problem to which all mathematicians, both ancient and modern, have paid the greatest attention, and particularly as to the Rectification of the circle, or finding the length of the circumference, or a right line equal to it; but hitherto without the perfect effect: upon this also depends the quadrature of the circle, since it is demonstrated that the area of a circle is equal to a right angled triangle, of which one of the sides about the right angle is the radius, and the other equal to the circumference: but it is much to be feared that neither the one nor the other will ever be accomplished. Innumerable approximations however have been made, from Archimedes, down to the mathematicians of the present day. See Circle and Circumference.

The first person who gave the Rectification of any curve, was Mr. Neal, son of Sir Paul Neal, as we find at the end of Dr. Wallis's treatise on the Cissoid; where he says, that Mr. Neal's Rectification of the curve of the semicubical parabola, was published in July or August, 1657. Two years after, viz in 1659, Van Haureat, in Holland, also gave the Rectification of the same curve; as may be seen in Schooten's Commentary on Des Cartes's Geometry.

The most comprehensive method of Rectification of curves, is by the inverse method of fluxions, which is thus: Let ACc be any curve line, AB an absciss, and BC a perpendicular ordinate; also bc another ordinate indefinitely near to BC; and Cd drawn parallel to the absciss AB. Put the absciss AB = x, the ordinate BC = y, and the curve AC = z: then is Cd = Bb = x. the fluxion of the absciss AB, and cd = y. the fluxion of the ordinate BC, also Cc = z. the fluxion of the curve AB. Hence because Ccd may be considered as a plane right-angled triangle, , or ; and therefore ; which is the fluxion of the length of any curve; and consequently, out of this equation expelling either x. or y., by means of the particular equation expressing the nature of the curve in question, the fluents of the resulting equation, being then taken, will give the length of the curve, in finite terms when it is | rectifiable, otherwise in an infinite series, or in a logarithmic or exponential &c expression, or by means of some other curve, &c.

Ex. 1. To rectify the common parabola.—In this case, the equation of the curve is , where a is half the parameter. The fluxion of this equation is , and hence ; this being substituted in the general equation , it becomes ; the correct fluents of which give , which is the length of the curve AC, when it is a parabola.

And the same might be expressed by an infinite series, by expanding the quantity √(aa + yy). See my Mensuration, pa. 361, 2d edit.

Ex. 2. To rectify the Circle.—The equation of the circle may be expressed either in terms of the sine, or versed sine, or tangent, or secant, &c, and the radius. Let therefore the radius of the circle be DA or DC = r, the versed sine AB = x, the right sine BC = y, the tangent CE = t, and the secant DE = s; then, by the nature of the circle, we have these equations, ; and by means of the fluxions of these equations, with the general equation , 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, the curve, putting d = 2r the diameter, is .

See my Mensur. 2d edit. pa. 118 &c, also most treatises on Fluxions.

It is evident that the simplest of these series is the third, or that which is expressed in terms of the tangent. It will therefore be the properest form to calculate an example by in numbers. And for this purpose it will be convenient to assume some arc whose tangent, or at least its square, is known to be some small finite number. Now the arc of 45° 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 quadrant to the diameter 1, will be = 1 - 1/3 + 1/5 - 1/7 + 1/9 &c. But as this series converges very slowly, some smaller arch must be taken, that the series may converge faster; such as the arc of 30°, whose tangent is = √(1/3) = .5773502, or its square ; and hence, after the first term, the succeeding terms will be found by dividing always by 3, and these quotients divided 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°, which is the 12th part of the whole circumference when the radius is 1, or the 6th part when the diameter is 1, and consequently 6 times that arc will be the length of the whole circumference to the diameter 1; therefore multiply the 1st term √(1/3) by 6, and the product is √(36/3) or √12 = 3.4641016; hence the operation will be conveniently made as follows:

+ Terms.- Terms.
1)3.4641016(3.4641016
3)1.1547005(0.3849002
5)3849002(769800
7)1283001(183286
9)427667(47519
11)142556(12960
13)47519(3655
15)15840(1056
17)5280(311
19)1760(93
21)587(28
23)196(8
25)65(3
27)22(<*>
+ 3.5462332-0.4046406
- 0.4046406
3.1415926the circumference.
Various other series for the Rectification of the circle may be seen in different parts of my Mensuration, as at pa. 121, 122, 137, 138, 422, &c. See also my paper on this subject in the Philos. Trans. vol. 66, pa. 476.

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Entry taken from A Mathematical and Philosophical Dictionary, by Charles Hutton, 1796.

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RECLINER
RECOIL
RECORDE (Robert)
RECTANGLE
RECTANGLED
* RECTIFICATION
RECTIFIER
RECTILINEAL
RED
REDANS
REDINTEGRATION