LOGARITHMS. 



than 100, you have nothing to do but in- 

 terpoiate the several times through ten 

 intervals. 



Now the void places may be filled up 

 by the following theorem. Let ?i be a 

 number, whose logarithm is wanted ; let 

 x i>e the difference between that and the 

 two nearest numbers, equally distant on 

 each side, whose logarithms are already 

 found: and let d be halt' the difference of 

 their logarithms : then the required loga- 

 rithm of the number n will be had by add- 



d x dx3 

 ing d -f -f- , &c. to the logarithm 



of the lesser number: for if the numbers 

 are represented by C p, C G, C P (fig. 

 16.) and the ordinates/s, P Q, be raised ; 

 if n be wrote for C G, and x for G P, or 



G p, the area p s Q P, or -f- j-J--~. 



&c. will be to the area p s H G, as the 

 difference between the logarithms of the 

 extreme numbers, or 2 d, is to the differ- 

 ence between the logarithms of the lesser, 

 and of the middle one ; which, therefore, 

 dx_ dx* dxj 



n "*" 2*1 3n 



will be 



The two first terms d -f - of this se- 

 zn 



sufficient for the construction 



greater; and this although the numbers 

 should not be in arithmetical progression. 

 Also by pursuing the steps of this me- 

 thod, rules may be easily discovered for 

 the construction of artificial sines and 

 tangents, without the help of the natural 

 tables. Thus far the great Newton, who 

 says, in one of his letters to M. Leibnitz, 

 that he was so much delighted with the 

 construction of logarithms, at his first 

 setting out in those studies, that he was 

 ashamed to tell to how many places of 

 figures he had carried them at that time : 

 and this was before the year 1666 ; 

 because, he says, the plague made him 

 lay aside those studies, and think of other 

 things. 



Dr. Keil, in his Treatise of Logarithms, 

 at the end of his Commandine's Euclid, 

 gives a series, by means of which may be 

 found easily and expeditiously the loga- 

 rithms of large numbers. Thus, let z be 

 an odd number, whose logarithm is 

 sought : then shall the numbers z 1 and 

 S -f- 1 be even, and accordingly their 

 logarithms, and the difference of the lo- 

 garithms will be had, which let be called 

 y. Therefore, also the logarithm of a 

 number, which is a geometi-ical mean be- 

 tween z 1 and z -}- 1, will be given, 

 viz. equal to half the sum of the loga- 

 rithms. Now the series y X 7 K"> 



181 . li o -U 11 V 



;c. shall be 



;, because x is either 1 or 2: yet it is 

 not necessary to interpolate all the places 

 by lielp of this rule, since the logarithms 

 of numbers, which are produced by the 

 multiplication or division of the number 

 lasv found, may be obtained by the num- 

 bers whose logarithms were had before, 

 by the addition or subtraction of their 

 logarithms. Moreover, by the difference 

 of their logarithms, and by their second 

 and third differences, if necessary, the 

 void places may be supplied more expe- 

 ditiously, the rule beforegoing being to 

 be applied only where the continuation of 

 some full places is wanted, in order to ob- 

 tain these differences. 



By the same method rules may be found 

 for the intercalation of logarithms, when 

 of three numbers the logarithm of the 

 lesser and of the middle number are giv- 

 en, or of the middle number and the 



1000, the first term of the 

 se ries, viz. , is sufficient for producing 



the logarithm to 13 or 14 places of figures, 

 and the second term will give the loga- 

 "thm to 20 places of figures. But if z be 

 greater than 10000, the first term will ex- 

 hibiUhe logarithm to 18 places of figures : 

 and so this series is of great use in filling 

 "P the chiliads omitted by Mr. Bri 

 For example, it is required to find 

 logarithm of 20001 : the logarithm of 

 20000 is the same as the logarithm of 2, 

 w th the index 4 prefixed to it; and the 

 difference of the logarithms of 20000 and 

 20001, is the same as the difference of 

 the logarithms of the numbers 10000 and 

 10001, viz. 0.0000434272, &c. And if this 

 difference be divided by 4 z, or 80004, 



,, auot : ent JL shall be 

 tlie ( l uotient 4z shali bc 



