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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 Transactions ; wherein he has demonstrated those Series after a new Way, and shews 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, easily and expeditiously, the Logarithms of large Numbers.
Let z be an odd Number, whose Logarithm is sought; then fall the Numbers 2-1 and z+be even, and accordingly their Logarithms, and the Difference of the Logarithms, will be had, which let be called y. Therefore, also, 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
tan 42 2423 36ozs
2522029 &c. thall be equal to the Logarithm of the Ratio, which the Geometrical Mean between the Numbers Z-I and zti has to the Arithmetical Mean, viz. to the Number z.
If the Number exceeds 1000, the first Term of the Series is fufficient for producing the Logarithm to
42 13 or 14 Places of Figures, and the second 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 so this Series is of great Ųle 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 same as the Logarithm of 2 with the Index 4 prefixed to it; and the Difference of the Logarithms of 20000 and 20002 is the same 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
A a 3
shall be .
0.00000 00005 42814 And if the Logarithm of the 4.30105 17093 02416 Geometrical Mean be added
4.30105 17098 45230 to the Quotient, the Sum will be the Logarithm of 20001. Wherefore it is manifest, that to have the Logarithm to 14 Places of Figures, there is no Neceflity 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 firit Fi. gures 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 mostly confift in setting down the Numbers.
Note, This Series is easily deduced from that found
out by Dr. Halley; and those who have a mind to be informed more in this Matter, let them consult his above-named Treatise.
APPEN D I X.
T is needless here to write a Prefatory Discourse,
а setting forth the Use and Invention of Logarithms,
since the Author has supplied that, in his Preface to the Treatise of the Nature and Arithmetic of Logarithms annexed to these Elements : It is enough to inform the Reader, that my chief Design in writing this Appendix was, to render their Construction easy, by investing various Theorems for that Purpose, and illustrating them by proper Examples; all which is performed in the actual Operation of making the Logarithms of the first 10 Numbers, and of the prime Number 101, which is more than sufficient to inform the meanest Capacity how to examine or construct the whole Table. I have also shewn how, from the Logarithm given, to find its correspond ing Number ; and the Investigation of the Series omitted by the Author in Page 357, for expeditiously finding the Logarithms of large Numbers. As to those Se. ries exhibited by him in his Trigonometrical Trearise, Page 287, for making the Sines and Cofines; I muft declare," that I have exceeded my first 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 shewed their Investigations too, from which those of our Author easily 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 IF =.
A a 4
'Then, the Triangles KGE, KHL, are similar, be, cause LH is parallel to EG; the Triangles ĶLH FLE,are fimilar, because the Angles KLH,FL E,are equal, and the Angles KHL, FE L, are both Right Angles; and the Triangles FLE, FEI, are similar, because the Angles FEL, FIE, are both Right Angles, and the Angle F is common : Therefore the Triangles FIE and KG E are similar, whence, KG (or v rybewys) : KE (or r)::FI (or s): F E (ora) i
so Now 'y-sar
16r5 &c. by extracting the Square Root : And if r's be divided by that Series, the Quotient,
5565 sis +ärr + 8,4+
c. will be the
166 Fluxion of the Arc; therefore the Fluent thereof, viz. 53
+ 2.3rr 2.4.5757 18.104.22.168,6*22.214.171.124.918 st +
126.96.36.1997 188.8.131.52.7.759 +
+ 2.332' 184.108.40.2064220.127.116.11.6.776. 18.104.22.168.22.214.171.1249
&c. will be equal to the Arc of a Circle whose Radius is r, and Sine's. But ifr be put equal to Unity, then
s?, &c. will ex
. 2.3 126.96.36.199 188.8.131.52.6.7 press the Length of the Arc a.
Let it be required to find the Length of the Arc of 30 Degrees to 6 Places of Decimals, the Radius being Unity.
Here's=, and ss=*; whence the Operation may be as follows:
23487 19531 3559
721 13312 305
25235984 Hence the Length of the Arc of 30 Degrees is 2523598+. Now if this Arc be multiplied by 6, we Ihall have the Length of the Arc of the Semicircle in such Parts as the Radius is 1, or of the whole Circumference in such Parts as the Diameter is 1, viz. 3,14159+.
But there is no Series so easy to be retained in the Memory, and so readily put in Practice, for obtaining the Ratio of the Diameter of the Circle to its Circumference, as that which is derived from the Tangent. For if t be put equal to the Tangent of any Arc, then a=i-13 + 45-47 +17, &c.