THE COMPUTATION OF LOGARITHMS, &c. 219 



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

 without any trouble with the modulus. 



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

 lowing example, in which the common logarithm of 61 is computed from those of 

 60 and 62. 



Z 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 

 sum 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, which 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=l, and transposing, 

 we get 



2 V2 x2 4 x* 6 ^6 ' / ' 



— a formula 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, occurred to me some time ago, and I then be- 

 lieved 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 volxmie (1793) of the 

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

 guished \^Titer. My investigation is altogether different from those given by him, 

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

 means of a method of integration 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 teniis of its tangent to the radius 1, 



d tan-i .r = t-^-^ , and therefore tan-^ x = I ^ . 



