1815.] Solution of a Prolletn of Col. Silas Titus. 55 



Now the circumferences of circles are constantly proportional to 

 tlieir radii : if, then, we designate by 2 tt the ratio of the circum- 

 ference to its radius, the series of circumferences will be expressed 



by the progression (O^)' x 2?: [V 1, V 2, V iJ, \/ 4, &c.] . . (8) 







But the areas of circles are as the squares of their radii ; so that 

 the series of circles will be expressed by the progression (O 1)". 2 tt 



o 



[I, 2, 3, 4,&c.] (9) 



In this series (by taking away the common factor (O 1)*. 2 tt) 1 







expresses the area of the central circle ; 2, 3, 4, &c. express those 

 of'the successive circles. If we take the difference of each of the 

 areas of these consecutivp circles, we shall have the areas of the 

 rings. Now these difterences are constantly equal to 1 ; conscr 

 quenily the areas of the rings are = 1. 



It follows from the above conclusions that if we reckon the 

 central circle for tlie first ring, the series of rings will be expressed 



by the common ordinals, 1st, 2d, 3d, 4th, &c (10) 



while the series of circles are expressed by the alsohite numbers 1, 

 2, 3, 4, &c ...' (11) 



These ordinal numbers follow the direct or inverse order: when 

 they follow the direct order, the areas of the rings are positive ; 

 when they follow the inveise order, they are negative. These 

 areas are constantly = 1, and are represented by the equation 

 + I =e±2/c,rV':ri ^j2) 



{k being any positive whole number.) 



And the negative areas by - 1 = e* ^^ ^' + *) '^ ^ - * (IS) 



These equations, which are fundamental, I thus demonstrate : 

 in every system of logarithms the logs, are exponents, and these 

 exponents are ordinal numbers, because they are the indexes of the 

 terms of a geometrical progression whose first term is 1. Now from 

 the principles demonstrated by Euler (Introductio in Analysin Infi- 

 nitorum. Cap. VIII. No. IVJ,) we may prove the truth of the two 

 following equations : — 



Log. (+ 1) = ± 2 ^ ^ V ^^ (14) 



Log. (- I) = ± 2 (/e + 1) TT ^/ - 1 (15) 



(Sec Lacaille's Lemons dc Mathematiques, Nos. 833, 834, 835.) 

 Let t be the base of the hvperbolic logarithm : we have I = 



log. f : ^^^ (16) 



Consequently log. (-h l)=±2^7r\/— 1 log. e (17) 



and log. (- 1) = ± 2 {k + 1) tt ^/'- Tlog. e (18) 



From the above we easily derive the equations (12), (13). The 

 following is an explanation of these equations : k can only be ua 



4 



