# LOGARITHMIC

, or Logistic Curve, a curve so called from its properties and uses, in explaining and constructing the Logarithms, because its ordinates are in geometrical progression, while the abscisses are in arithmetical progression; so that the abscisses are as the Logarithms of the corresponding ordinates. And hence| the curve will be constructed in this manner: Upon any right line, as an axis, take the equal parts AB, BC, CD, &c, or the arithmetical progression AB, AC, AD, &c; and at the points A, B, C, D, &c, erect the perpendicular ordinates AP, BQ, CR, DS, &c, in a geometrical progression; so is the curve line drawn through all the points P, Q, R, S, &c, the Logarithmic, or Logistic Curve; so called, because any absciss AB, is <*> the Logarithm of its ordinate BQ. So that the axis ABC &c is an asymptote to the curve.

Hence, if any absciss AN = *x,* its ordinate
NO = *y,* AP = 1, and *a* = a certain constant quantity,
or the modulus of the Logarithms; then the
equation of the curve is *x* = *a* × log. of *y* = log. *y*^{2}.

And if the fluxion of this equation be taken, it will
be ; which gives this proportion,
but in any curve the subtangent AT;
and therefore the subtangent of this curve is everywhere
equal to the constant quantity *a,* or the modulus of the
Logarithms.

*To find the Area contained between two ordinates.* Here
the fluxion of the area A^{.} or *yx*^{.} is ;
and the correct fluent is . That
is, the area APON between any two ordinates, is equal
to the rectangle of the constant subtangent and the
difference of the ordinates. And hence, when the
absciss is infinitely long, or the farther ordinate equal
to nothing, then the infinitely long area APZ is equal
AT × AP, or double the triangle APT.

*For the Solid formed by the curve revolved about its
axis* AZ. The fluxion of the solid is , where *p* is = 3.1416; and the correct
fluent is ,
which is half the difference between two cylinders of
the common altitude *a* or AT, and the radii of their
bases AP, NO. And hence supposing the solid insinitely
long towards Z, where *y* or the ordinate is nothing,
the infinitely long solid will be equal to
, or half the cylinder
on the same base and its altitude AT.

It has been said that Gunter gave the first idea of a curve whose abscisses are in arithmetical progression, while the corresponding ordinates are in geometrical progression, or whose absciss are the Logarithms of their ordinates; but I do not find it noticed in any part of his writings. This curve was afterwards considered by others, and named the Logarithmic or Logistic Curve by Huygens in his Dissertatio de Causa Gravitatis, where he enumerates all the principal propertics of it, shewing its analogy to Logarithms. Many other learned men have also treated of its properties; particularly Le Seur and Jacquier, in their Comment on Newton's Principia; Dr. John Keill, in the elegant little Tract on Logarithms subjoined to his edition of Euclid's Elements; and Francis Maseres Esq. Cursitor Baron of the Exchequer, in his ingenious Treatise on Trigonometry: see also Bernoulli's Discourse in the Acta Eruditorum for the year 1696, pa. 216; Guido Grando's Demonstratio Theorematum Huygeneanorum circa Logisticam seu Logarithmicam Lineam; and Emerson on Curve Lines, pa. 19.—It is indeed rather extraordinary that this curve was not sooner announced to the public, since it results immediately from Napier's manner of conceiving the generation of Logarithms, by only supposing the lines which represent the natural numbers as placed at right angles to that upon which the Logarithms are taken.

This curve greatly facilitates the conception of Logarithms to the imagination, and affords an almost intuitive proof of the very important property of their fluxions, or very small increments, namely, that the fluxion of the number is to the fluxion of the Logarithm, as the number is to the subtangent; as also of this property, that if three numbers be taken very nearly equal, so that their ratios may differ but a little from a ratio of equality, as the three numbers 10000000, 10000001, 10000002, their differences will be very nearly proportional to the Logarithms of the ratios os those numbers to each other: all which follows from the Logarithmic arcs being very little different from their chords, when they are taken very small. And the constant subtangent of this curve is what was afterwards by Cotes called the Modulus of the System of Logarithms.

Logarithmic, or *Logistic, Spiral,* a curve constructed
as follows. Divide the arch of a circle into any
equal parts AB, BD, DE, &c;
and upon the radii drawn to the
points of division take C*b,* C*d,* C*e,*
&c, in a geometrical progression;
so is the curve A*bde* &c the Logarithmic
Spiral; so called, because
it is evident that AB, AD, AE,
&c, being arithmeticals, are as the
the Logarithms of CA, C*b,* C*d,* C*e,*
&c, which are geometricals; and a Spiral, because it
winds continually about the centre C, coming continually
nearer, but without ever really falling into it.

In the Philos. Trans. Dr. Halley has happily applied this curve to the division of the meridian line in Merc<*>tor's chart. See also Cotes's Harmonia Mens., Guido Grando's Demonst. Theor. Huygen., the Acta Erudit. 1691, and Emerson's Curves, &c.