C^7} 
If future Indaftry /hall ever produce Logarithmick Tables 
to many more places^ chan now we have them 5 the aforefaid 
Theorems will be of more ufe to deduce the correfpondent 
Natural Numbers to all the places thereof. In order co make 
the firft Chiliad ferve all Ufes^ I was defirous to contra(5l this 
Series^ wherein all the Powers of / are prefent^ into one^ where- 
in each alternate Power might be wanting ; but found it 
neither fo fimple or uniform as the other. Yet the firft ftep 
thereof is I conceive moft commodious for Practice, and with- 
al exad enough for Numbers not exceeding fourteen places^ 
(iich as are Mr. Briggsh large Table of Logarithms ; and there- 
a I 
fore I recommend it to common Ule. It is thus: a + n 
hi 
or b — J'j^i.'i be, the Number anfwering to the Logarithm 
given^ differing from the truth by but one half of the third 
ftep of the former Series, But that which renders it yet more 
eligible is^ that with equal facilitv it ferves for Briggs's or any 
other fort of Logarithms, with the only variation of writing 
I a I hi 
~ inftead of that is, a -f - r? ^ — , . ^ > , 
. j-a4-\la ~h — \lh 
or 7-7— and . , which are eafily refolved into 
Analogies, ^ix.. 
As 43429 &c. — 4 / to 43429 : : So is totheNum= 
or As 45429 &;c. -1-1^^043429—1/:: Sois^J her fought 
If more ftepsof this Seriei bedefired^it will be found as follows, 
^ J_ — — 'A ; &c. as may eafily be demon- 
"^" ^ — I — /' 1—2/ ^ ^ 
ftrated by working out the Dlvifions in each ftep, and colle- 
ding the Quotes^ whofe Sum will be found to agree with our 
former Sivks. 
Thus I hope I have cleared up the Doft-rine of Logarithms, 
and fhewn txheir Conftrudion and Ufe independent from the 
Hjperkla, whofe Affe(^ions have hitherto been made ufe of 
for this purpofe, though this be a matter purely Arithmetical, 
nor properly demonftrable from the Principles of Geometry. 
Nor have I been obliged to have recoiirfe to the Method of 
IndivifibleSj or the Arithmetick of Intinices^ the whole being 
no other than an eafie Corollary to Mr. Ncwtonh General 
Theorem for forming Roots and Powers. V. A 
