93 



tion, in many inftances, how far the demonflration of a law by the 

 introduftion of impoflible quantities, exceeds in evidence, a conclufion 

 obtained by indu(ftion. 



On Kepler's own Solution of his Problem. 



Kepler having fatisfied himfelf, that the orbit of Mars was an el- 

 llpfe, and that equal areas were defcribed in equal times about the Sun, 

 in one of the foci, reduces the problem of finding the cocequate or true 

 anomaly from the mean, to this,* " Aream femicirculi ex quocunque 

 " punfto diametri in data ratione fecare," and obferves " mihi fufficit 

 " credere folvi a priori non poffe propter arcus et finus in^oysnur. 

 " Erranti mihi quicunque viam monftrabit, is erit mihi magnus Apol- 

 « lonius." 



Accordingly, he himfelf has recourfe to a tentative method of folu- 

 lion ;t he affumes the excentric anomaly, and then computes the mean 

 anomaly ; the error of the mean anomaly fo computed, he applies to 

 the firft alTumed excentric anomaly, and with the excentric anomaly fo 

 correfted, he repeats the operation as often as neceffary. This mode 

 of computing the excentric anomaly is derived from the equation, 



?n ^: c + e s, c 



%vhich equation follows from the equable defcription of areas. 



From this equation it is evident that the error of the afluraed va- 

 lue of c diiFers from the error of the computed value of w, only by 

 a quantity to which c has always a greater ratio than i : e. There- 

 fore, regarding the value of e in all the planets, the excentric anomaly 

 by repeating the operations, rapidly converges to its true value. The 

 ^ excentric 



* Kepler de Motu Stellje Mariis, p. 300. 

 t Kep. Epitome, Aft. p. 695. 



