Pax bG=PA× AV, a constant quantity. Wherefore Pax bG must also be constant; and, consequently, since the factor Pa increases as the point of the curve gets further from PAV, on either side, so the other factor bG must decrease, and the curve approach nearer and nearer to EF without the possibility of ever meeting it.

In like manner, using the similar triangles b'aG', and the same triangle PaA, it may be shown that EAF is an asymptote to the inferior conchoid.

Property 2.-The equation of the superior conchoid BVB' is x2+2bx3 + (b2 — a2 + y2) x2 -2a2bxa2b2.

2bx3 +

The equation of the inferior conchoid CV'C' is x (b2 — a2 + y2) x2 + 2a2bx a2b2, where x represents AH or bG, and y represents bH or AG in the superior conchoid, or their correspondents in the inferior.

Scholium.-When AV is greater than AP, V' will fall below P, and the curve passing through C, P, V', C', and the several points b', as thus formed, is called the nodated conchoid, and the part P, V'', b', b', etc., is called a node. The student would find it interesting to construct the figure under this condition.

THE CISSOID OF DIOCLES.*

To construct the cissoid.

H

A

M

e

d

C B с Ꭰ E G' H I

Y

h g f Construction.-Draw XY perpendicular to AB, the diameter of the given circle AVPBp, and from A to the line XY draw any number of lines AC, AD, AE, AG', etc., Ac, Ad, Ae, etc., on each side of AB, cutting the circumference of the circle in the points R, P, Q, etc., y, z, p, etc. Then lay the chord AR on CA from C to L; the chord AP on DA from D to 0; the chord AT on G'A from G' to K, and so proceed to lay the intercepted chord of every line on that line from its intersection with XY towards A, and the curve passing through A; and these several points L, O, Q, K, etc.

* See Bonnycastle's Algebre, vol. ii. p. 401, London edition, 1820.

thus given successively from A, both ways, will be the cissoid of Diocles.

The circle AVBp is called the generating circle; AB the axis of the two branches, which meet in a cusp at A, and pass through the middle points Q and p of the two semicircles, drawing continually nearer and nearer to the directrix XY as it extends farther from AB; and the directrix XY is hence their common asymptote.

a, AG=x, GO

Draw PH and OG parallel to XY; put AB y; then the equation of the curve is (ax)y2.

THE QUADRATRIX OF DINOSTRATUS.

Construction.-Let AVB be a semicircle of which the centre is C. At C, perpendicular to AB, draw a line, on which lay CMthe diameter AB, and draw NML parallel to AB. Divide the

number on the radius CV will be the same part of the radius CV. that from B to the same number on the quadrant BV is of the quadrant BV; that is, C5: B5:: rad. CV: quad. BV:: diameter

CM: semicircle BVA. Also, C7: B7 :: rad. CV: quad. BV, and C14: B14::CM:BVA. Join C with each of the points 1, 2, 3, etc. on the semicircle, and parallel to AB, through the point 1 on the radius, draw a line to meet the line C1 in E. Through the point 2 on the radius draw a line parallel to AB to meet the line C2 in F. So, through each point on CM, successively, parallel to AB, draw a line to meet the line drawn from C to the point of the same number on the semicircle, in the points G, H, I, J, V, R, S, O, etc., and the curve passing through these several points E, F, G, H, etc., taken in order, is called the quadratrix.

The circle AVB completed is called the generating circle; the line NML is called the directrix, which is also, as will be readily seen from the mode of construction, the asymptote to the quadratrix. For, suppose there be taken on MC a distance Mp a thousandth or a ten-thousandth part of the distance C1, and on the semicircle Ap a like part of B1, join Cp, and produce it to meet a line parallel to MLN, through p, on MC, which will give a point in the curve, and this point will become more and more remote from M as the parts Mp and Ap are made smaller, and the limit is the extension of the two parallels CA and MN, which will never meet. Therefore LMN, produced, is the asymptote of the curve.

Putting the radius CB-a, CD, the base of the quadratrix, = b, the arc B5z, and CT=y, the equation of the quadratrix is ay = bz.

NOTE 1.-This curve obtained much notoriety from the fact that if it were possible to form it accurately by a simple geometrical operation, it would enable mathematicians to determine the rectification and quadrature of the circle. It was from this property that the curve was called the quadratrix. It would also afford a means of dividing a given angle, or given arc, into any number of equal parts or in any given ratio.*

NOTE 2.-One principal difficulty in constructing the quadratrix is in finding the point D in CB so as to determine the base CD of the quadratrix. The most practical method I have been able to devise is to continue the semicircle below B, and also the line VC below AB. Then take B a half, a fourth, or as small a part as may be employed of B1, and lay it from B to below B. On VC, produced to X, lay C, a like part of C1, from C to . Join C and the points and on the arc, and through the points and on VX, parallel to CB, draw the lines from to d and from to d′, meeting the lines from C to and in d and d', which will be points in the curve, and in the curve continued below CB, and hence the curve must pass through the point D, and thus give CD, the base of the quadratrix, as accurately as can be obtained by mechanical means.

*See Bonnycastle's Algebra, vol. ii. p. 403, London edition, 1820.

abscissas AB, AC, AD, etc., Aa, Ab, Ac, etc., increasing in an arithmetical progression, having the intervening spaces AB, BC, CD, Aa, ab, bc, etc. all equal; and then, parallel to AH, through the points B, C, D, etc., a, b, c, d, etc., draw lines BI, CJ, DK, etc., ai, bj, ck, etc., whose lengths shall be in geometrical progression, increasing from AH towards Q by a common multiplier or ratio, and decreasing from A towards R by a common divisor of the same value. In the present figure AH was taken 12, the common difference = 5, and the ratio 14. Then the curve passing through the upper extremities of all these lines N, M, L, K, etc. will be the logarithmic curve.

By laying these respective distances on the lines indicated, the points I, J, K, L, etc. in the curve are obtained, and the curve can readily be drawn through the upper extremities of all these lines, beginning at N, and taking M, L, K, J, I, H, i, j, k, etc. in succession.

Scholium 1.—It is evident that the curve approaches nearer and nearer to RP without the possibility of ever arriving at it; for, however many times we may divide the value of AH by-14, or, which is the same thing, multiply it by 4, it must still give some value to lay on a perpendicular above RQ; hence the curve, though approaching

nearer and nearer, can never arrive at RQ, and RQ is the asymptote to the curve.

Scholium 2.-Since the ordinates AH, BI, CJ, etc. are in geometrical progression, the square of any one is equal to the product of the two adjacent ones, or of any two lines equally distant from it. Thus, AH2 Thus, AH2 = BI × ai : CJ × bj = DK × ck dl, etc.

EL X

Scholium 3.-Putting r = ratio (in this case 14), and AH we have BI: a.r; CJ=a. r2; DK= a.r3; EL

α

ai

r

α

α

α

a,

a. r*, etc.; and

; bj == 2; ck ==/; dl=, etc. If now we commence the 203

curve at H', where A'H' will be just equal to r,* and then lay off the arithmetical ratios, or common differences, A'P, Ph, hg, gf, fe, etc., and erect perpendiculars at these successive points, we have A'H'r; Pp' — A'H' × r = r2; hq = Pp' x r=r3; go = r2; fn Χ r5; em= =76, etc. Putting x to represent any number of these arithmetical differences from A' towards Q, calling A' one, P two, h three, etc., and putting y to represent the corresponding ordinate, we have y, which is the equation of the logarithmic curve; and from the form of its equation, the logarithmic curve is sometimes called the exponential curve.

Now, if we lay A'S A'P, then the ordinate Ss

1; and beginning at S, the ordinates will be 1, r, r2, μ3, pt, 2.5 etc. 1, 14, (14)2, (14)”, (14)*, (14)5, etc.; that is, Ss1, A'H' = 11, Pp' — (14)2, hq′ = (14)3, go= (14)*, ƒn = (14)3, fn (11)5, etc.

To find the distance AA' at which the ordinate A'H' of the curve will be equal to the ratio r, having AH and the arithmetical and geometrical ratios given, let x = the number of arithmetical differences in AA', then, by the hypothesis, we have A'H' (x+1). Whence, taking the logarithm of each side of the

a

Hence ar

11.136

1

10.136

the number of arithmetical spaces between A and A'; and

as each space in the figure is 5, we have AA' 10.136 X 5 50.68.