Iga PHILOSOPHICAL TRANSACTIONS. [aNNO \6g7. 



a point A be taken, whose distance AC from the asymptote hp is equal to the 

 subtangent hs ; and from the points h and b, any how taken in the asymptote, 

 equally distant from the point c, if ordinates hl, pg are erected to the loga- 

 rithmic curve, to half the sum of which hd or pf are made equal ; the points 

 D and p will be in the catenaria corresponding to the right line ac. 



Make ab=x, and therefore cb or dh, tlie half sum of the ordinates hl, 

 PG, will be a^x ; let the half difference of the same be called i/. Then ml 

 = a -\- X -\- 7/ , and pg =a -\- x — y. And since, from the nature of the loga- 

 rithmic curve, CA is a mean proportional between these, it will be aa -f- 2ax 



+ XX — 1/1J = aa, and therefore y = \/2ax -[• xx. So that bl = a -{- x -{- 



'^ lax -j- XX, and pg =: a + .r — V lax + xx. Therefore the fluxion of hl, 



-or Im, is °'^ '^'^T^- — —, And because of similar triangles /otl and lhs, 



it is lh : HS :: /m : mL. Whence mx. or dS, the fluxion of bd, is equal to 



"^ That is, the curve ad, derived from the logarithmic curve in the 



foregoing manner, is of such a nature, that if its axis be called x, and its 



fluxion 3.\ the fluxion of the ordinate bd will be . But this is the 



V -lax + XX 



very property of the catenaria to which a belongs, as demonstrated in Prop. 1 

 of this. Therefore the curve fad, above described, is no other than the cate- 

 naria. a. E. D. 



Corol. 1 . — As by the help of the logarithms the catenaria may be described ; 

 so, on the contrary, by means of the catenaria, which is constructed by nature 

 herself, the logarithm of a given number, or rather of a given ratio, may be 

 found. As supposing ca to be unity, whose logarithm is equal to o, let us find 

 the logarilhiTi of the number cq, or of the ratio between ca and cq. To the 

 right lines co and ca let the third proportional be cv, and let half the sum of 

 CQ and cv be cb. The ordinate to the catenaria from b, that is bd, is the loga- 

 rithm required. The reason is plain from the proposition. 



Corol. 1 ■ — On the contrarv, if from the logarithm given, ch or cp, the cor- 

 respondent number hl or pg were required, or the ratio hl to ca, or pg to ca ; 

 from H or p let a perpendicular be raised, meeting the catenaria in d or f ; and 

 let CK be made equal to hd or pf, that is, to cb, and let it be terminated at the 

 horizontal line apv. Then will ab be the semidifFerence of tiie lines required, 

 LH, GP, as HD or CR is their semisum, by what is demonstrated above about the 

 nature of the catenaria. For in three quantities that are geometrically propor- 

 tional, such as HL, ca, pg, the sijuare of the half sum o( the extremes, lessened 

 by the square of the mean, is equal to the square of the half difference of the 



