appears, that the times of vibrations of pendulums, in cycloidal arcs, are as the square roots of their lengths, whatever may be their weights; or

t: tvl : √l'.

And if the pendulum vibrate in small arcs of a circle, which it is evident will nearly coincide with the small cycloidal arcs at the vertex v, the times, in this case, will also be nearly isochronus; and, consequently, the above formulæ are equally applicable to the circle and the cycloid, as far as regards any practical purposes.

10. Lastly, the cycloid has the remarkable property, mentioned in the last Note, of being the curve of swiftest descent: that is, a body will descend through the arc of the curve APV, from A to v, in less time than it would pass over the right line AV, or by any other rout.

OF THE LOGARITHMIC CURVE.

If from any fixed point a, in the indefinite right line pa, there be taken a number of parts AB, ac, AD, &c. in arithmetical progression, and the perpendicular ordinates AE, BF, CG, &c, drawn from the points A, B, C, &c. be taken in geometrical progression, the line EFGH, &c. which passes through their extremities, is called the logarithmic curve (h).

(h) This curve, of which Gunter is commonly said to have given the first idea, greatly facilitates the conception of logarithms to the imagination, and affords an obvious proof of the very important property, that the fluxion of any number,

1. Hence, agreeably to the description of the figure, here laid down, the line pa of the abscissæ, which is called the axis, may be shown to be an asymptote to the curve, as follows:

Let the abscissæ Ab, AC, &c. taken on the left of AE, make a part of the same arithmetical progression with AB, AC, AD, &c. on the right; and the ordinates bf, cg, &c. a part of the same geometrical progression with AE, BF, CG, &c, but so as to decrease while the others increase.

Then, because the square of any term of a continued geometrical progression is equal to the rectangle of any two terms that are equally distant from it (Art M, Vol. 11), we shall have

bf× BF, or cg × CG, &c. =ae2,

And, consequently, since the square of AE is a constant quantity, any one of the rectangles bf x BF, cg× CG, &c. will also be constant, whatever may

or quantity, is to the fluxion of its logarithm as the number itself is to the subtangent.

It may be farther observed, that Huygens, in his Dissertatio de Causa Gravitatis, as well as several later writers, have treated very fully of all the principal properties of this curve, and shown their analogy to logarithms; but, as has been observed in the Introduction to my Treatise on Plane and Spherical Trigonometry, the doctrine of logarithms, though it may admit of a clear illustration in this way, has no necessary connexion with this or any other geometrical figure.

be the values of the ordinates, of which they are composed.

But, by the description of the figure, the ordinates bf, cg, &c. decrease as BF, CG, &c. increase; wherefore, the right line pa and the curve gEH must continually approach towards each other, without ever meeting; or otherwise the rectangles above mentioned could not be always equal to AE.

Whence, the axis pa is an asymptote to the curve, as was to be shown.

COR. As the abscissæ AB, AC, AD, &c. in the above figure, constitute a series of quantities in arithmetical progression, and their ordinates AE, BF, CG, &c. a corresponding series in geometrical progression, the former may be considered as a set of natural numbers that are analogous to the logarithms of the latter; from which property the curve has received its name,

2. The equation of the curve may, also, be readily derived from the above description of it in the following manner:

Put AE= 1, and Bra; then, from the nature of the figure, we shall have

1 : a :: a: CG, a : a2 :: a: DH, &c.; or CG = a, DH = a', &c.

Whence, if x be made to denote any number of equal parts AB, BC, CD, &c. of the axis ra, a will evidently be equal to the ordinate drawn from the extremity of the segment which is represented by x. And, consequently, if this ordinate be put equal to y, we shall have

for the equation required; from which, on account

of its form, the figure is sometimes called the exponential curve.

3. The subtangent of the logarithmic curve is always of the same length, from whatever point of the curve the tangent may be drawn.

For let ET, FV, be any two tangents to the curve KEFG, at the points E, F, which are indefinitely near each other; and having drawn the equidistant ordinates EB, FC, GD, make En, Fr, parallel to AD.

Then, since вC=CD, we shall have, from the description of the curve, BE CF :: CF: DG; or, by division, BE CF CF-BE DG-CF (Euc. v, 19).

But CF-BE=nF, and DG-CF=rG; wherefore BE: CF :: NF : rG; or, alternately, BE: nF ::

CF TG.

Also, because the triangle FnE is similar to EBt, and the triangle GrF to FCV, BT: NE

and cv rF :: CF : rg.

Hence, since BE NF: CF : rG, shown, we shall also have BT NE (Euc. v, 11).

BE: nf,

But CB being = CD, NE will be =rF; and, consequently, the other two terms of the proportion, or the subtangents BT, CV, will be equal.

And the same will be true for any other points in the curve.

Note. The constant subtangent of this curve is what was first called, by Cotes, the modulus of the system of logarithms.

4. The logarithmic space BELG, comprehended between any two ordinates BG, EL, is equal to the rectangle of the subtangent TE, and the difference LK, of those ordinates.

For let the ordinate DI be indefinitely near EL, and draw ir parallel to AE.

Then, if LT be a tangent to the curve at L, the triangles LTE, LIr, will be similar.

Whence, since EL ET FL : ri, we shall

have EL X ri = ET × rl.

But DI being indefinitely near EL, the rectangle EL × ED, or EL × ri, = the area of the space DELI; wherefore, also, ET x rL DELI.

Hence, since, as before shown, ET is a constant quantity, the sum of all the spaces DELI, or the whole area BELG is = ET × sum of all the Lr's, or

ET X LK.

COR. From this it appears, that the area of the whole logarithmic space, indefinite towards a, is double the triangle LET.

5. To this we may add, that the solid formed by the revolution of the infinitely long space FLEA about the axis AE, is equal to half a cylinder of the base LE, and of the altitude of the constant subtangent ET.