S 90 , LOGARITHMS, 
4. In any number of proportionals regularly increafing, 
the means divide the proportion of the extremes into one 
more than their own number. Thus, in the feries 2, 4, 
8, 16, the proportion of the extremes 2 and 16 is by the 
two means 4 and 8 divided into three proportions, viz. 
that betwixt 2 and 4, 4 and 8, 8 and 16. In like manner, 
in the feries 3- 6, 18, 54, 162, 486, the proportion betwixt 
3 and 486 is divided by the four means into the five pro¬ 
portions of 3 to 6,'6 to 18, 18 to 54, 54 to 162, and 162 
to 486. 
5. The proportion betwixt any two terms is divifible 
into any number of parts, until thefe become lefs than 
any afiignable quantity. Thus the proportion of 2 to 8 is 
divifible, by multiplying the two together and extracting 
the fquare root, into two parts by the number 4; by mul¬ 
tiplying 2 and 4 together, and extracting the fquare root, 
and doing the fame with 4 and ,8, the proportion would 
be divided into four parts, viz. a. 3/8. 4.3/32.8; or in 
numbers, 2 : 2'8i3, See. : 4 : $’655, Sec. : 8. 
6. By dividing the ratios in this manner, the elementary 
part will become at lad fo final!, that it may be denomi¬ 
nated by the mere difference of terms of that element. 
This is evident from the diminution of the ratios or pro¬ 
portions already inftanced; for the proportion between 2 and 
s - Si3 is only 1-406, See. and, if we were to find a mean pro¬ 
portional betwixt 2 and 2-813, the ratio betwixt that pro¬ 
portional and 2 would be much lefs. But it mu ft always 
be remembered, that fuch evanefeent quantities , as they are 
called, cannot give us any conclufion with abfolute ex- 
aftnefs, however they may anfwer every ufeful pnrpofe to 
us; for it is evident that neither mean proportional nor 
ratio can ever be found exaftly ; and therefore the error 
accumulated in all the operations niuft become very cor.fi- 
derable, if any circumftance fliall happen to make it appear. 
Mr. Briggs's Method. 
The methods principally made ufe of by this gentleman 
were publifhed in Napier’s poftbumous work. Having 
fuppofed o to be the logarithm of 1, and 1 with any num¬ 
ber of ciphers annexed, fuppofe 10, to be the logarithm 
of 10, this number is to be divided ten times by 5, which 
in a logarithmic number is equivalent to the extraction of 
the root of the fifth power; by which means he obtains 
the following numbers, viz. 2 with nine ciphers to it; 4 
with eight ciphers; 8 with feven ciphers; 16 with fix ci¬ 
phers; 32 with five ciphers; 64 with four; 128000, 25600, 
5120, and 1024. Dividing this laft logarithm ten times 
by 2, we have a geometrical feries of ten numbers ; the 
firft of which is 512, and the laft 1. Thus 20 logarithms 
are obtained ; but the labour of finding the numbers be¬ 
longing to them is fo excefiive, that it is furprifing how 
it could be undergone by any body. To obtain thofe 
eorrefponding to the firft ten logarithms, the fifth root 
niuft be extruded ten times, and the fquare root as often, to 
obtain the numbers eorrefponding to the others. The power 
from which thele extractions is made, mull originally be 
3 with a number of ciphers annexed. Other logarithms 
might be formed from thefe by adding them, and multi¬ 
plying their eorrefponding numbers; but, as this method, 
befides its excefiive labour, would produce only an antiloga- 
ritkmic canon like that of Mr. Dodfon already mentioned, 
other more eafy and proper methods were thought of. 
The next was by finding continually geometrical means, 
firft between 10 and 1, and then between 10 and that mean, 
and fo on, taking the arithmetical means between their cor- 
relponding - logarithms. The operation is alfo facilitated 
by various properties of numbers and their logarithms, as 
that the products and quotients of numbers correfpond to 
the fums and differences of their logarithms; that the powers 
and roots of numbers anfwer to the products and quotients 
of the logarithms by the index of the power or root. Thus 
having tne logarithm of 2, we can have thole of 4, 16, 
256, &c. by multiplying the logarithms by 2, and fquaring 
the numbers, to as great an extent, in that feries, as we 
pleale. If we have aifo that of 3, we can have not only 
thofe of 9, 81, 8561, &c. but of 6, 18, 27, and all poflibl'e 
products of the powers of 2 and 3 into one another, or 
into the numbers themfelves. The following property 
may alfo be of ufe, viz. that, if the logarithms of any two 
numbers are given, and each number be railed to the 
power denoted by the index of the other, the products 
will be equal. Thus, 
Logarithms, 0123 4 5 6 
Nat. Numb, 1 2 4 8 16 .32 64 
Let the two numbers be 4 and 5 6 ; it is plain, that if we 
raife 4 to the fourth power and 16 to the fquare, the pro¬ 
ducts will be the fame; for 16X16 256, and 4X4 = 
26; 16X4 = 64; and 64X4 = 256. 
Another method mentioned by Mr. Briggs depends 
upon this property, that the logarithm of any number in 
this fcale is 1 lefs than the number of places or figures 
contained in that power of that number whofe exponent 
is the logarithm of 10, at leaft as to integral numbers, 
for Mr. Briggs has (hewn that they really differ by a frac¬ 
tion. To this Mr. Hutton adds the following; viz. That 
of any two numbers, as the greater is to the lefs, fo is the 
velocity of the increment or decrement of the logarithms 
at the greater; “that is (fays he), in our modern nota¬ 
tion, As X. Y ■. y : x -, where x and y are the fluxions of 
X and Y.” 
In the treatife written upon the conftruftion of loga¬ 
rithms by Mr. Briggs himfelf, he obferves, that they may 
be conftrufted chiefly by the two methods already men¬ 
tioned, concerning which he premifes feveral lemmata 
reflecting the powers of numbers and their indices, and 
how many places of figures are in the produCfs of num¬ 
bers. He obferves, that thefe products will coniift of as 
many figures as there are in both faCtors, unlefs the firft 
figures in each factor be exprefled in one figure only, 
which fometimes happens; and then there will commonly 
be one figure lefs in the product than in the two factors. 
He obferves, alfo, that if in any feries of geometricals we 
take two terms, and raife one to the power denoted by 
the index of the other, or any- number raifed to the power, 
denoted by the logarithm of the other, the product will 
be equal to this latter number raifed to the power deno¬ 
minated by the logarithm of the former. Hence, if one 
of the numbers be 20, whofe logarithm is 1 with any 
number of ciphers, then any number raifed to the power 
whofe index is the logarithm of that number, that is, the 
logarithm of any number in this fcale where a is the lo¬ 
garithm of 10, is the index of that power of 10 which is 
equal to the given number. But the index of any inte¬ 
gral power of 10 is one lefs than the number of places of 
figures it contains. Thus the fquare of 10, or 100, con¬ 
tains three places of figures, which is more by one than 2 
the index of the power; 1000, the cube of 20, contains 
four places, which is one more than the index, 3, of the 
power. Hence, as the number of places of the powers of 
20 are always exaftly one more than the indices of thofe 
powers, it follows, that the places of figures in the powers 
of any other number which is no integral power of 10 
will not always be exaftly one lefs in number than the 
indices of the powers. 
From thefe two properties is deduced the following rule 
for finding the logarithms of many prime numbers: Find 
the 20th, 200th, 1000th, or any other, power of a num¬ 
ber, fuppofe 2, with the number of places of figures in it; 
then that number of figures fhall always exceed the loga¬ 
rithm of 2, although the excefs will be conftantly lefs 
than 2 ; whence, by proceeding to very high powers, vve 
fliall at laft be able to obtain the logarithm of the number 
to great exaftnefs. Thus, the logarithm of 2, found by 
other methods, is known to be 30102999566583, See. The 
tenth power of 2 is 1024; which containing four places of 
figures, gives 4 for the logarithm of 2, which exceeds it ? 
though not quite by 1. The 20th power of 2, confiding 
of the 10th power multiplied into itfelf, by its number of 
places ought to give the logarithm of 4; and, according 
to the rule already laid, down, fliould contain eight places 
.1 of 
