LOGARITHMS. 
886 
index; and a table of natural fines for the fame parts to 
fifteen places, with the tangents and fecants to ten places, 
and the methods of conftrufting them. He defigned alio 
to have publiflied a treatife concerning the ufes and ap¬ 
plication of them, but died before this could be accom- 
plifhed. On his death-bed he recommended this work to 
Henry Gellibrand, profefl’or of aftronomy in Grefham col¬ 
lege, in which office he had fucceeded Mr. Gunter. Mr. 
Briggs’s tables above mentioned were printed at Gouda, 
and published in 1633; and the fame year Mr. Gellibrand 
added a preface with the application of logarithms to 
-plane and f'pherical trigonometry, the whole being deno¬ 
minated Trigonometria Britannica-, and, befides the arcs in 
degrees and hundredth parts, has another table contain¬ 
ing the minutes and feconds anfweringto the feverai hun¬ 
dredth parts in the fir ft column. The Trigonometria Arti- 
fcialis of Vlacq contains the logarithmic fines and tangents 
to ten places of figures, to which is added Briggs’s firft ta¬ 
ble of logarithms from 1 to 20,000, befides the index: 
the whole preceded by a defcription of the tables, and 
the application of them to plane and fpherical trigonome¬ 
try, chiefly extracted from Briggs’s Trigonometria Bri¬ 
tannica already mentioned. In 1635, Mr. Gellibrand alfo 
publifhed a work, intitled An Inftitution Trigonometrical, 
containing the logarithms of the firft 10,000 numbers, 
■with the natural fines, tangents, and fecants; and the lo¬ 
garithmic fines and tangents for degrees and minutes, all 
to feven places of figures befides the index; likewife other 
tables proper for navigation, with the ufes of the whole. 
Mr. Gellibrand died in 1636. See his article, vol. viii. 
p. 292. 
Sherwin’s Mathematical Tables, publifhed in 3 vo. form 
the moft complete colleftion of any; containing, befides 
the logarithms of all numbers to 101,000,. the fines, tan¬ 
gents, fecants, and verfed fines both natural and logarithmic, 
to every minute of the quadrant. The firft edition was 
printed in 1706; but the third, publifhed in 174.2 and re¬ 
viled by Gardiner, is looked upon to be fuperior to any 
other. The fifth and laft edition, publiflied in 1771, is fo 
incorrect, that no dependence can be placed upon it. 
“Tables of Logarithms from 1 to 102,100, and for the 
Sines and Tangents to every ten feconds of each degree 
in the quadrant; as.alfo for the Sines of the firft 72 mi¬ 
nutes to every fingle fecond, with other ufeful and ne~ 
ceffary Tables. By Gardiner; Lond. 1742.” This work 
contains a table of logiftical logarithms, and three fmaller 
tables to be ufed for finding the logarithms of numbers 
to twenty places of figures. Only a ftnall number of thefe 
tables was printed, and that by fubfcription; and they 
are now in the highelt efteem for accuracy and uf'efuinefs. 
An edition of thefe tables was printed at Avignon in 
France in 1770, with the addition of fines and tangents 
for every fingle fecond in the firft four degrees, and a fmall 
table of hyperbolic logarithms, taken from a treatife upon 
fluxions by the late Mr. Thomas Simfon. The tables are 
to feven places of figures, but fomewhat lefs correft than 
thofe publilhed by Gardiner himfelf. 
Different Methods of Constructing Logarithms. 
Napier's Method as explained by Maclaurin. 
The logarithms firft thought of by lord Napier were 
r.ot adapted to the natural feries of arithmetical numbers, 
x, 2, 3, &c. becaufe he did not then intend to adapt them 
to every kind of arithmetical calculation, but only to that 
particular operation which had called for their immediate 
conltruftion, viz. the ftiortening of trigonometrical ope¬ 
rations : he explained the generation of logarithms, there¬ 
fore, in a geometrical way. Both logarithms, and the 
quantities to which they correfpond, in his way may be 
luppofed to proceed from the motion of a point; which, 
if it moves over equal fpaces in equal times, will produce 
a line increafing equally ; but if, inftead of moving over 
equal fpaces in equal times, the point defcribes fpaces 
proportional to its diftances from a certain term, the 
line produced by it will then increafe proportionally. 
Again, if the point moves over fuch fpaces in equal 
times as are always in the farhe conftant ratio to tfrff 
lines from which they are fubd utted, or to the diltance 
of that point at the beginning of the lines from a given 
term in that line, the line fo produced will decreafe pro¬ 
portionally. Thus, let ac be to ao, cd to co, ef to Jo, and 
P 
fg to fo, always in a certain ratio, viz. that of Q R to 
Q S, and let us fuppofe the point p to fet out from a, de¬ 
fending the diftances ac, cd, de, &c. in equal fpaces of time, 
then will the line ao decreafe proportionally. 
In like manner, the line oa, increafes proportionally, 
0 a c d e f s 
1 -1- 1 -1-1- i -j 
P . 
if the point p , in equal times, defcribes the fpaces ac, cd, 
de, fg, Sec. fo that ac is to ao, cd to co, de to do, Se c. in a 
conftant ratio. If we now fuppofe a point P delcribing 
AC D E F G 
| - 1 - 1 -j -1 - i 
P 
the line AG with an uniform motion, while the pointy 
defcribes a line increafing or decreafing proportionally, 
the line AP, deferibed by P, with this uniform motion, 
in the fame time that oa, by increafing or decreafing pro¬ 
portionally, becomes equal to op, is the logarithm of op. 
Thus AC, AD, AE, See, are the logarithms of oc, od, ce , 
See. refpettively; and oa is the quantity whofe logarithm 
is fuppofed equal to nothing. 
We have here abftratted from numbers, that the doc¬ 
trine may be the more general; but it is plain, that if 
AC, AD, AE, Sec. be luppofed 1, 2, 3, Sec. in arithme¬ 
tic progreifion; oc, od, oe, Sec. will be in geometric pro- 
greflion ; and that the logarithm of oa t which may be taken 
for unity, is nothing. 
Lord Napier, in his firft fcheme of logarithms, fuppofes, 
that, while op increafes or decreafes proportionally, the 
uniform motion of the point P, by which the logarithm 
of op is generated, is equal to the velocity o ip at a-, that 
is, at the term of time when the logarithms begin to be 
generated. Hence logarithms, formed after this model, 
are called Napier's logarithms, and fometimes natural loga¬ 
rithms. 
When a ratio is given, the point p defcribes the differ¬ 
ence of the terms of the ratio at the fame time. When 
a ratio is duplicate of another ratio, the point p defcribes 
the difference of the terms in a double time. When a 
ratio is triplicate of another, it defcribes the difference of 
the terms in a triple time; and fo on. Alfo, when a ra¬ 
tio is compounded of two or more ratios, the point/! de¬ 
fcribes the difference of the terms of that ratio in a time 
equal to the fum of the times in which it defcribes the 
differences of the terms of the limple ratios of which it is 
compounded. And what is here faid of the times of the 
motion of p when op increafes proportionally, is to be ap¬ 
plied to the fpaces deferibed by P, in thole times, with 
its uniform motion. 
Hence the chief properties of logarithms are deduced. 
They are the meafurcs of ratios. The excels of the loga¬ 
rithm of the antecedent above the logarithm of the con- 
fequent, meafures the ratio of thofe terms. The meafure 
of the ratio of a greater quantity to a lefs is pofitive; 
as this ratio, compounded with any other ratio, increafes 
it. The ratio of equality, compounded with any other 
ratio, neither increafes nor diminifhes it; and its meafure 
is nothing. The meafure of the ratio of a lefs quantity 
to a greater is negative; as this ratio, compounded with 
any other ratio, diminilhes it. The ratio of any quantity 
A to unity, compounded with the ratio of unity to A, 
produces the ratio of A to A, or the ratio of equality; 
x and 
