THE COMPUTATION OF LOGARITHMS, &e. 219 



ing it, without employing any of the terms in the vinculmn, and consequently 

 without any trouble with the modulus. 



The facility of the process by means of formula (7) wiU appear from the fol- 

 lowing example, in which the common logai-ithm of 61 is computed from those of 

 60 and 62. 



/ 62= 1.792391689 



/60 = 1.778151250 



2)3.570542939 



Half sum = 1.785271469 



Difference = 0.014240439 



Now 61 X 4 = 244, and dividing the difference by this, we get 0.000058362 ; the 

 simi of which and of the half sum, found above, is 1.785329831, the logarithm of 

 61. This is true in all its figures except the last, wliich ought to be 5. 



It may be proper to remark, that when x is large, its logarithm will be ob- 

 tained very readily by means of formula (3) ; as, by taking n=\, and transposing, 

 we get 



— a fonnula which will give the logarithms of whole numbers above 2000, true 

 for seven or more decimals, by means of the logarithms of the two numbers im- 

 mediately preceding and following, without any term of the series. 



II. 



A series which gives the rectification of the circle with greater ease than any 

 other with which I am acquainted, occmxed to me some time ago, and I then be- 

 Meved it to be new. I have lately found, however, that the same series was dis- 

 covered by EuLER, and that it appeared in the eleventh voliune (1793) of the 

 NoxiU. Acta of the Petersburgh Academy, with two investigations by that distin- 

 guished writer. My investigation is altogether different from those given by him, 

 and is very sunple — perhaps more so than either of his. It is obtained, also, by 

 means of a method of integi-ation which may be employed with advantage in 

 many other instances : and though, as might be expected, several things in my 

 paper are anticipated in Euler's, yet mine contains others which are not to be 

 found in his. For these reasons, I shall present the paper in almost exactly the 

 same state in which it was before I saw the article by Euler. 



If we put tan"' x to denote the circular arc, whose tangent is x, we have, by 

 the formula for the differential of the arc in tenns of its tangent to the radius 1, 



d tan-' .r = r-T—-5 > aud therefore tan-' .r = / ,'^„ . 



