LOGARITHMS. 
ms 
equal to zero ; then the infinitely-long area APZ is equal 
to ATX AP, or double the triangle APT. 
To find the Content of the Solid formed by the Revolution of 
the Curve o.bout its Axis A Z.~ The fluxion of the lolid S 
py^x-py-y^—payy, where/’=3 f 14159, Se c. and the cor- 
reft fluent is S = ' : >X (AP 2 — y 2 )-\py AT X (AP 2 
— NO 2 ), which is half the difference between two cylin¬ 
ders of the common altitude a, or AT, and the radii of 
their bafes AP, NO. And hence, fuppofing the axis in¬ 
finite towards Z, and confequently.the ordinate at its ex¬ 
tremity zero, the content of the infinitely-long folid will 
-be equal to } pa X AP 2 == \p X AT X AP 2 , or half the 
cylinder on the fame bafe and its altitude A T. 
This curve greatly facilitates the conception of loga¬ 
rithms, and affords a very obvious proof of the very im¬ 
portant property of their fluxions, or very fmall incre¬ 
ments; namely, that the fluxion of a number is to the 
fluxion of its logarithm as the number is to the fubtan- 
gent. As alfo this property, that if their numbers be 
taken very nearly equal, fo that their ratios may differ but 
a little from a ratio of equality, their difference will be 
very nearly proportional to the logarithm of the ratio of 
thefe numbers to each other; which follows from the lo¬ 
garithmic arcs being very little different from their chords 
when they are taken veiy fmaii. The conftant fubtan- 
gent of this curve is, what Cotes calls, “ the modulus of 
the fyftem of logarithms.” This curve has been treated 
of by a great number of very eminent mathematicians, 
as Huygens, Le Seur, Keil, Bernouilli, Emerfon, &c. 
See the latter author’s Treatife on Curve Lines, page 19. 
Logarithmic, or logistic, Spiral, is a curve con- 
ftruCted as follows : Divide the arch of a circle, into any 
A u equal parts AB, B D, DE, &c. and 
vT> upon the radii drawn to the points of 
)>TE division take C b, C d, Ce, See. in a 
geometrical progreflion ; 16 is the curve 
A bde, Sec. the logarithmic fpiral; fo 
called, becaufe it is evident that A B, 
AD, A E, Sec. being arithmeticals, 
are as the logarithms of C A, C b, C d, 
C e, Sec. which are geometrical ; and 
a fpiral, becaufe it winds continually about the centre C, 
coming continually nearer, but without ever really falling 
into it. In the Phil. Tranf. Dr. Halley has happily ap¬ 
plied this curve to the divifion of the meridian line in 
Mercator’s chart. See alfo Cotes’s Harmonia Menf. Guido 
Grando’s Demontr. Theor. Huygen. the A6ta Erudit. 1691. 
Emerfon’s Curves, Sec. 
Logarithmic Lines. —For many mechanical purpefes 
it is convenient to have the logarithms of numbers laid 
down on feales, as well as the logarithmic lines and tan¬ 
gents; by which means, computations may be carried on 
by mere menfuration with compaffes. Lines of this kind 
are always put on the common Gunter’s fCale ; but, as 
thefe inffruments muff be extended to a very great length 
in order to contain any conffderable quantity of numbers, 
it becomes an objefl of importance to thorten them. Such 
an improvement has been made by Mr. William Nichol- 
fon, and publiflied in the 77th volume of the Philofophi- 
cal Tranfaftions. The principles on which the conftruc- 
tion of his inffruments depends are as follow: 
If two geometrical feries of numbers, having the fame 
common ratio, be placed in order with the terms oppolite 
to each other, the ratio between any term in one feries 
and its oppofite in the other will be conftant: Thus, 
2 6 18 54 162, Sec. 
3 9 27 81 243, Sec. Then, 
2 3 6 9 18 27 54 81 162 243, Sec. 
where it is evident, that each of the terms in the upper 
feries is exactly two-thirds of the correfponding one in 
the lower. 
2. The ratio of any two terms in one feries will be the 
Vol.XII. No. 882. 
fame with that between thofc which have an equal diltance 
in the other. 
3. In all fuch geometrical feries as have the fame ratio, 
the property above-mentioned takes place, though we com¬ 
pare the terms of any feries with thofe of another: Thus, 
{ 2 4 8 16 32 64, Sec. 
3 6 12 24 48 96, Sec. 
{ 4 8 16 32 64 128, Sec. 
5 10 20 40 80 160, Sec. where it is plain that 2, 4, 
3, 6 ; alfo 2, 4, 4, 8, and 2, 4, 5, 10, Sec. have the fame 
ratio with that of each feries. 
4. If the differences of the logarithms of the numbers 
be laid in order upon equiditlant parallel right lines, irs 
fuch a manner that a right line drawn acrofs the whole 
fliall interfeft it at divisions denoting numbers in geome¬ 
trical progreflion ? then, from the condition of the ar¬ 
rangement, and the property of this logarithmic line, it 
follows, iff, That every right line fo drawn will, by its 
interfeftions, indicate a geometrical feries of numbers. 
2dly, That fuch feries as are indicated by thefe right lines 
will have the fame common ratio. 3dly, That the feries 
thus indicated by two parallel right lines, fuppofed to. 
move laterally, without changing either their mutual difi- 
tance or parallelifm to themfelves, will have each the fame 
ratio, and in all feries indicated by fuch two lines, the 
ratio between an antecedent and confequent; the former 
taken upon one line, and the latter upon another, will be 
alfo the fame. 
The iff of thefe propofitions is proved in the following- 
manner. Let the lines AB, CD, EF, fig. 8, on the pre¬ 
ceding Plate, reprefent parts of the logarithmic line ar¬ 
ranged according to the proportion already mentioned ; 
and let GH be a right line palling through the points c, 
c, a, denoting numbers in geometrical progreflion ; then 
will any other line IK, drawn acrofs the arrangement, 
likewife pal’s through three points,y', d, b, in geometrical 
progreflion. From one of the points of interfeftion f in 
the laft-mentioned line IK, draw the line fg parallel to 
G H, and interfering the arrangement in the points 
i, /i; and the ratios of the numbers e,f c, i, will be equal, 
as well as of a, h ; becaufe the intervals on the logarith¬ 
mic line, or differences of the logarithms of thofe num¬ 
bers, are equal. Again, the point f, the line id, and the 
line hb, are in arithmetical progreflion, denoting the dif¬ 
ferences between the logarithms of the numbers them- 
felves; whence the quotients of the numbers are in geo¬ 
metrical progreflion. 
The 2d propolition is proved in a fimilar manner. For, 
as it was (hown that the line fg, parallel to GH, paffes 
through points of divifion denoting numbers in the fame 
continued ratio as thofe indicated by the line GH ; it may 
alfo be flrown, that the line LM, parallel to any other line 
IK, will pafs through a feries of points denoting numbers 
which have the fame continual ratio with thofe indicated 
by the line IK, to which it is parallel. 
The 3d propofition arifes from the parallelifm of the 
lines to their former fituation ; by which means they in¬ 
dicate numbers in a geometrical feries, having the fame 
common ratio as before; their diftance on the logarithmic 
line alfo remains unchanged; whence the differences be¬ 
tween the logarithms of the oppofite numbers, and of 
confequence their ratios, will always be conftant. 
5. Suppofing now an antecedent and confequent to be 
given in any geometrical feries, it will always be poflible 
to And them, provided the line be of unlimited length. 
Drawing two parallel lines, then, through each of the 
numbers, and fuppofing the lines to move without chang¬ 
ing their direction or parallel fituation, they will conti¬ 
nually deferibe new antecedents and confequents in the 
fame geometrical feries as before. 
6. Though the logarithmic line contain no greater range 
of numbers than from 1 to 10, it will not be found ne- 
ceflary for the purpofes of copmutation to repeat it. The 
only thing requifue is to have a Aider or beam with two 
io X fixe«t 
