Researches in the Theory and Calculus of Operations. 237 



areas. The areas and times represent each other; and as any 

 area has its equivalent square, the times of description are 

 measured by such squares constituting the equivalent of the 

 area of the circle, and therefore expressible in squares of 

 the radius. In uniform motion, the greater the distance, the 

 longer the time of transit. Therefore in two circles of radii* 

 r, r f , and times /, t f , the ratios of the squares of the radii 

 to the squares of the times are to each other inversely as the 



radii, giving the proportion —: — :: r':r; whence ^ 



the squares of the times proportional to the cubes of the 

 distances. Substitution of an inscribed ellipse for the circle 

 leaves unchanged the relation between the radius and the 

 time of description (Delambre). This leads to the following 



Example. 



Miles. 



35393000 = Distance of Mercury from the Sun =r. 

 7.5489174= log r; 

 22.6467522=3 . log r = log r 3 . 

 5.1251954 =log t\ 



27.7719476 = log r*. h. 



91430000 = Distance of Earth from the Sun=r / . 

 7.9610887 = log /; 

 23.8832661 = 3 . log /=log r'\ 

 3.8886614 = log t\ 



27/7719275 = log r'\ f. (Lockyer). 



Days. 



365.2563 = Revolution of Earth around the Sun = ^ 

 2.5625977 =log t; 

 5.1251954 



2 . log t'= log t'\ 



87.9692 = Revolution of Mercury around the Sun = t. 

 1.9443307 = log t\ 

 3.8886614 = 2. log * = log f. 



22.6467522 

 23.8832661 



2.7634861= log 



.'3 > 



0.0580078 



: ratio of cubes 

 of distances. 



3.8886614 

 5.1251954 



2.7634660= log ^ 



0.0580051 = ratio of squares 

 of times. 



