[ 2 75 ] 
Since the logarithm of unity is univerfally deter- 
mined to be nothing; that of 2, 3, 4, 10, or any 
other number, confider’d as a root, is one ; that of 
4, S>, 16, 100, &c. confider’d as the fquare of that 
root, is two; andfo on ; it follows, that (in all cafes) 
the logarithm of a greater number will exceed that of 
its lelfer ; and each logarithm will have fome rela- 
tion to the excefs of its number, above unity, the 
number, whole logarithm is nothing : the terms of 
* the feries, therefore, which will reprefent the loo-a- 
rithm of any number, will confifi: of the powers^ of 
the excefs of that number, above unity, with fome, 
yet unknown, but conftant coefficients. 
That the logarithm of the fquare of any number 
is twice the logarithm of its root, is a well-known 
property of thole artificial numbers ; and therefore, 
the doubles of the particular terms of the afifumed 
feries will confidtute a feries, expreffing the logarithm 
of the fquare of the given number. 
But, by the fourth propofition of the fecond book 
of Euclid, the fquare of any quantity is equal the 
fum of the fquares of its two parts, more a double 
rectangle of thofe parts ; which, in this cafe (where 
the given number has been afiumed, to con fill: of 
unity, and an excefs) will be unity, more twice that 
excefs, more the fquare thereof. 
Iff therefore, the feveral powers of the compound 
quantity (twice the excefs of the given number above 
unity, more the fquare thereof) be multiplied by the 
above afiumed coefficients, and afterwards ranged 
under each other, according to the powers of the 
faid excefs; their fums will again exprels the loga- 
rithm of the fquare of the given number. 
Mm2 Now, 
