( ) 
IV. A moft compendious and facile Method for 
ConfiruSling the Logarithms^ exemplified and 
demonftrated from the Nature of Numbers , 
without any regard^to the Hyperbolay with 
a Jpeedy Method for finding the ]>{umberfrdm 
the Logarithm gi<ven. By E. Hal ley. 
TH E Invention of the Logarithms is juftly efteetned 
one of the moft Ufeful Diftoveries in the Art of Num- 
bers, and accordingly has had an Univerfal Reception and 
Applaufe \ and the great Geometricians of this Age have not 
been wanting to cultivate this Subjed with all the Accuracy 
and Subtilty a matter of that conftquence doth require ; and 
they have demonftrated leveral very admirable Properties of 
theft Artificial NumberSj which have rendred their Conftru- 
<aion much more facile than by thofe opero(e Methods at firft 
ufed by their truly Noble Inventer the Lord Napeir^ and our 
worthy Country-man Mr. Briggs. 
But notwiihftanding all their Endeavours^ I find very few 
of thofe who make conftant ufe of Logarithms, to have at- 
tained an adequate Notion of them, to know how to make 
or examine them j or to underftand the extent of the ufe of 
them: Contenting themfelves with the Tables of them as 
they find them^without daring to queftion them, or caring to 
know how to redifie them, (hould they be found amifs,being 
I fuppofe under the apprehenfion of fome great difficulty 
therein. For the fake of fuch the following Trad is prin- 
cipally intended, but not without hopes however to produce 
fomething that may be acceptable to the moft knowing in 
thefe matters. 
But firft, it may be requifite to premife a definition of Lo- 
garithms, in order to render the enfiiing Difcourfe more clear, 
the rather becaule the ofd one Numerorum frofortionalium /equi 
diffmntes comites,{eQms too fcanty to define them fully. They 
may much more properly be faid to be Numeri Rationum Ex- 
^omntes : Wherein we cotifidQr ratip sls SL ^afititas fut gener^, 
beginning from the ratio of eqmli^jf^ or x to i = o j being 
Afe- 
