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A28 PHILOSOPHICAL TRANSACTIONS. [aNNO I67O. 



But alb is the same with ult, which is given; we have therefore the angle abd, 

 under which the arch a bis seen from d. 



After the same manner it will be demonstrated, that the difference of the 

 angles tly and tdi/, is equal to the sum of the angles ltd and lyd; as also that 

 the difference of the angles b Ic and bdc is equal to the sum of the angles Ibd 

 and led. And since Ibdis the half of ltd, and led half of lyd; the sum of Ibd 

 and led will be equal to half the sum of the angles ltd and lyd, that is, the dif- 

 ference of the angles blc and bdc will be equal to half the difference of the angles 

 ily and tdy, the former of which is the apparent interval of the two observations, 

 and the latter, that of the mean motion. Therefore their difference being given, 

 we have the difference of the angles blc and bdc. But blc is the same with the 

 given angle tly; therefore the angle bdc is also given, under which the arch 

 b c is seen from d. 



Whence it appears, that from the given mean and apparent intervals of two 

 observations, the angles are given -under which any arches of the circle abc are 

 seen from d, intercepted between the lines of the true motion. Therefore, by 

 Herigon's Theory of the Planets, 1. 1, c. 3, prop. 12, schol. 1, so many seg- 

 ments of a circle may be described, which may contain the angles under which 

 these arches are seen from d, all which segments will intersect each other mu- 

 tually in d; so that, after this manner, the apogees and eccentricities of the pla- 

 nets may be found by a geometrical delineation, by any number of observations: 

 and circles are as easily drawn as right lines. 



But to grant what is true, viz. that the geometrical delineation of M. Cassini 

 is somewhat more expeditious ; yet, should we aim at that accuracy which astro- 

 nomers desire, it might be feared it would require very large diagrams, and be- 

 come more operose than the calcillus itself. But if we make use of this, we 

 shall find both methods to be equivalent. 



It now remains that we examine the hypothesis. 



The invention of elliptical orbits is undoubtedly owing to Kepler; but the 

 determining the degrees of the acceleration and retardation with which the pla- 

 nets move, is no less necessary for completing the hypothesis, than the defin- 

 ing the orbit itself. Though nothing to this purpose is to be observed in Cas- 

 sini or his interpreter; and from the construction of the problem and its solution 

 it is manifest, that he supposes a planet seems to move from the superior focus 

 with an equable motion: and Kepler himself was of this opinion, as appears 

 from his writings. But, when he found that this did not agree with his obser- 

 vations, he changed his mind, and maintained that the line of the true motion 

 of a planet described equal elliptical areas in equal times ; and that there is no 

 point from which a planet is seen to move with an exactly equable motion. 



