LOGARITHMS. 
to do but interpolate the several times 
through ten intervals. 
Now the void places may he filled up by 
the following theorem. Let n be a number, 
whose logarithm is wanted ; let x be the 
difference between that and the two nearest 
nunibers, equally distant on each side, whose 
logarithms are already found ; and let d be 
half the d ifference of their logarithms : then 
the required logarithm of the number n will 
be had by adding 
the logarithm of the lesser number; for if 
the numbers are represented by Cp, CG, 
C P (fig. 16.) and the ordinates p s, P Q, be 
raised ; if n be wrote for C G, and x for 
G P, or G p, the area p s Q P, or — + 
f_-|- — &c. will be to the area p s H G, 
an- ' 3n’ 
as the difference between the logarithms of 
the extreme numbers, or 2 d, is to the dif- 
ference between the logarithms of the les- 
ser, and of the middle one; which, therefore. 
will be 
d X 
u 
dx^ 
Sn' 
* u ' 
dx^ 
12 » 3 ’ 
&c. 
the middle number and the greater; and 
this although the numbers should not be in 
arithmetical progression. Also by parsuing 
the steps of this method, rules may be easily 
discovered for the construction of artificial 
sines .and tangents, without the help of the 
natural tables. Thus far the great Newton, 
who says, in one of his letters to M. Leib- 
nitz, that he was so much delighted with 
the construction of logarithms, at his first 
setting out in those studies, that he was 
ashamed to tell to how many places of 
figures he had carried them at that time ; 
and this was before the year 1666 ; because, 
he says, the plague made him lay aside 
those studies, and think of other tilings. 
Dr. Keil, in his Treatise of Logarithms, 
at the end of his Commandine’s Euplid, 
gives a series, by means of which may be 
found easily and expeditiously the loga- 
rithms of large numbers. Thus, let z be an 
odd number, whose logarithnj is sought; 
then shall the numbers z — 1 and z-j-l be 
even, and accordingly their logarithms, and 
the difference of the logaritlims will be had, 
' which let be called y. Therefore, also the 
logarithm of a number, which is a geome- 
trical mean betw'een z — 1 and a-f-l, will 
be given, riz. equal to half the sum of the 
logarithms. Now the series y X 
d X • 
The two first terras d -I of this series, 
being sufficient for the construction of a 
canon of logarithms, even to 14 places of 
figures, provided the number, whose loga- 
rithm is to be found, be less than 1000 ; 
which cannot be very troublesome, because 
X is either 1 or 2 : yet it is not necessary to 
interpolate all the places by help of this 
rule, since the logarithms of numbers, which 
are produced by the multiplication or divi- 
sion of tlie number last found, may be ob- 
tained by tlie numbers ^ whose logarithms 
were had before, by the addition or sub- 
traction of their logarithms. Moreover, by 
the difference of their logarithms, and by 
their second and third differences, if neces- 
sary, the void places may be supplied more 
expeditiously, the rule beforegoing being to 
be applied only where the continuation of 
some full places is wanted, in order to ob- 
tain these differences. 
By the same method rules may be found 
for the intercalation of logarithms, when of 
three numbers the logarithm of the lesser 
and of the middle number are given, or of 
1 I 
24 z^ ‘”15120 
13 
&c. shall be 
25200 z’ 
equal to the logarithm of the ratio, which 
the geometrical mean between the numbers 
z — 1 and z 1, has to the arithmetical 
mean, viz. to the number z. If the number 
exceeds 1000, the first term of the series, viz. 
~. is sufficient for producing the logarithm 
4z 
to 13 or 14 places of figures, and the second 
term will give the logarithm to 20 places of 
figures. But if z be greater than 10000, the 
first term wilt exhibit the logarithm to 18 
places of figures : and so this series is of 
great use in filling up the chiliads omitted 
by Mr. Briggs. For example, it is required 
to find the logarithm of 20001 : the loga- 
rithm of 20000 is the same as the logarithm' 
of 2, with the index 4 prefixed to it ; and 
the difference of the logarithms of 20000 and 
20001, is the same as the difference of the 
logarithms of tlie numbers 10000 and 10001, 
viz. 0.000043427$, &c. And if this differ- 
ence be divided by 4 z, or 80004, the quo- 
V 
tient — shall be 
4z 
