12 



AN ACCOUNT OF THE EARLY MATHEMATICAL AND 



Fig. 1. 



since the velocities of the two bodies when at D are equal 

 (Principia Lib, \,prop. 16, cor, 4 J, the times of describing, 

 and, consequently, the 

 heat acquired in Do?, 

 Dc, will be as Dd : De ; 

 that is, from similar 

 triangles, as DC : SD, 

 or as the area of the el- 

 lipse to the area of the 

 circle. But the whole 

 quantities received in 

 one revolution are as 

 the quantities received in the z. DSc?; therefore the truth of 

 the lemma is evident. 



Solution. " To determine the comet's orbit from the data 

 and the laws of centripetal forces, it will be as P : P :: (128:^)^ : 

 cube of the semitransverse axis of its orbit = 25*4314769 = a; 

 and the eccentricity = semitransverse — perihelion distance 

 = 24-9829669 = b; from which the conjugate axis = 4-755142 

 = c; — then from the first lemma it will heas \/ a : ^ I :: sg : 

 sq -^ ^ a = the heat which would be received by the comet 

 in one revolution round the sun in a circle whose diameter = 

 transverse axis of its orbit, where s denotes the number of 

 seconds in a year. But, by the second lemma, c : a :: sq -r- 

 y/~a : sqs/ a -^ c = the heat received by the comet in one 

 revolution in its proper orbit; half of which = 16733512 g' = 

 the quantity of heat received in its passage from aphelion to 

 perihelion, as required. Also, the heat of the comet in peri- 

 helion will be to the mean heat of the earth as 4-97113 : 1, 

 nearly." A second solution, by Mr. Dalton, " without the 

 lemmas," is also inserted in this Diary, but since it merely 

 confirms the accuracy of the preceding general expression for 

 the heat, it need not be added. 



In the Gentleman's Diary for the same year his contri- 



