26 PHILOSOPHICAL TRANSACTIONS. [aNNO I695. 



— j^Ul + t't^* rW ^* &c. wherein as / is less, the series will converge the 



swifter. And if the first 20000 logarithms be given to J 4 places, there is rarely 

 occasion for the three first steps of this series, to find the number to as many 

 places. But for Vlacq's great canon of 100000 logarithms, which is made but 

 to ten places, there is scarcely ever need for more than the first step a + al or 

 a + malm one case, or else b — bl or b — mbl in the other, to have the 

 number true to as many figures as those logarithms consist of. 



If future industry shall ever produce logarithmic tables to many more places, 

 than now we have them ; the aforesaid theorems will be of more use to deduce 

 the correspondent natural numbers to all the places thereof. In order to make 

 the first chiliad serve all uses, I was desirous to contract this series, wherein all 

 the powers of / are present, into one, wherein each alternate power might be 

 wanting; but found it neither so simple nor uniform as the other. Yet the first 

 step thereof is I conceive most commodious for practice, and withal exact 

 enough for numbers not exceeding 14 places, such as are Mr. Briggs's large 

 table of logarithms ; and therefore I recommend it to common use. It is thus: 



a + -■ _ ^ or b — , . will be the number answering to the logarithm 



given, differing from the truth by only one half of the third step of the former 

 series. But that which renders it yet more eligible is, that with equal facility it 

 serves for Briggs's or any other sort of logarithms, with the only variation of 



writing — instead of 1, that is, 



./ , , - a + Ma -b — \ lb 



a + 7 and b j s or — ; and —. — - , 



- - il i + W --4^ ~ + ^l 



m m m ' m ' ' 



which are easily resolved into analogies, viz. 



As 4342Q&C. — 4- / to 43420 + i / :: So is a > ^ ,, , ,^ 



A .o.r,!^fi I i it^Aci.r,^^ , I c • ; r to the number sought, 

 or As 43429 &c. + -^- / to 43429 — 4 / :: 00 is ) ° 



If more steps of this series be desired, it will be found as follows, 

 a -\ _ , ■ — 'f-^i + 7^0/ ^^- ^s ™^y easily be demonstrated by work- 

 ing out the divisions in each step, and collecting the quotes, whose sum will be 

 found to agree with our former series. 



Thus I hope I have cleared up the doctrine of logarithms, and shown their 

 construction and use independent of the hyperbola, whose affections have 

 hitherto been made use of for this purpose, though this be a matter purely 

 arithmetical, nor properly demonstrable from the principles of geometry. Nor 

 have I been obliged to have recourse to the method of indivisibles, or the arith- 

 metic of infinites, the whole being no other than an easy corollary to Mr. 

 Newton's general theorem for forming roots and powers. 



