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Numbers to any Number of Places may be had more expeditiously, and truer: Concerning which Series. Dr. Halley has written a learned Tract, in the Philofophical Tranfactions; wherein he has demonftrated those Series after a new Way, and fhews how to compute the Logarithms by them. But I think it may be more proper here to add a new Series, by Means of which may be found, eafily and expeditiously, the Logarithms of large Numbers.
Let z be an odd Number, whofe Logarithm is fought; then fhall the Numbers z -1 and z+1 be even, and accordingly their Logarithms, and the Difference of the Logarithms, will be had, which let be called y. Therefore, alfo, the Logarithm of a Number, which is a Geometrical Mean between -I and I, will be given, viz. equal to the Half Sum of the Logarithms. Now the Series
+ + 4% 2423 36025 1512027 2522029 c. thall be equal to the Logarithm of the Ratio, which the Geometrical Mean between the Numbers 2-1 and z+1 has to the Arithmetical Mean, viz. to the Number Z.
If the Number exceeds 1000, the firft Term of the Series is fufficient for producing the Logarithm to
13 or 14 Places of Figures, and the fecond Term will give the Logarithm to 20 Places of Figures. But, if z be greater than 10000, the first Term will exhibit the Logarithm to 18 Places of Figures; and fo this Series is of great Ufe in filling up the Logarithms of the Chiliads omitted by Briggs. For Example, it is required to find the Logarithm of 20001. The Logarithm of 20000 is the fame as the Logarithm of 2 with the Index 4 prefixed to it; and the Difference of the Logarithms of 20000 and 20002 is the fame as the Difference of the Logarithms of the Numbers 10000 and 10001, viz. 0.00004 34272 7687. And if this Difference be divided by 4%, or 80004, the Quotient
0.00000 00005 42814 4.30105 17093 02416 4.30105 17098 45230
And if the Logarithm of the Geometrical Mean be added to the Quotient, the Sum will be the Logarithm of 20001. Wherefore it is manifeft, that to have the Logarithm to 14 Places of Figures, there is no Neceffity of continuing out the Quotient beyond fix Places of Figures. But if you have a mind to have the Logarithm to 10 Places of Figures only, as they are in Vlacq's Table, the two firft Figures of the Quotient are enough. And if the Logagarithms of the Numbers above 20000 are to be found by this Way, the Labour of doing them will moftly confift in fetting down the Numbers.
Note, This Series is eafily deduced from that found out by Dr. Halley; and thofe who have a mind to be informed more in this Matter, let them confult his above-named Treatife.
T is needlefs here to write a Prefatory Difcourfe, fetting forth the Ufe and Invention of Logarithms,
fince the Author has fupplied that, in his Preface to the Treatife of the Nature and Arithmetic of Logarithms annexed to thefe Elements: It is enough to inform the Reader, that my chief Defign in writing this Appendix was, to render their Conftruction eafy, by invefting various Theorems for that Purpose, and illuftrating them by proper Examples; all which is performed in the actual Operation of making the Logarithms of the firft 10 Numbers, and of the prime Number 101, which is more than fufficient to inform the meaneft Capacity how to examine or construct the whole Table. I have also fhewn how, from the Logarithm given, to find its correfponding Number; and the Investigation of the Series omitted by the Author in Page 357, for expeditiously finding the Logarithms of large Numbers. As to thofe Series exhibited by him in his Trigonometrical Treatise, Page 287, for making the Sines and Cofines; I muft declare, that I have exceeded my firft Intentions, which were to give their Investigation only; but confidering, that as they depended upon the Newtonian Series without the Investigation of which our Author's Series could never be thoroughly understood; I thought it would therefore prove acceptable, if I fhewed their Investigations too, from which thofe of our Author eafily flow. In order to which, and to keep the Reader no longer in Suspense, let r be put for the Radius KE of the Circle ABCD; a for the Arc BE, whose Length is to be investigated; and s equal to the Sine EG: Then is FE a, and IFs.