46(j PHILOSOPHICAL TRANSACTIONS. [ANNO 1784. 



periodical time of the revolving body being given, the density of the cen- 

 tral body will be given also. See Newton's Prin. 3, pr. 8, cor. J. 3. But the 

 density of any central body being given, and the velocity any other body would 

 acquire by falling towards it from an infinite height, or, which is the same thing, 

 the velocity of a comet revolving in a parabolic orbit, at its surface, being given, 

 the quantity of matter, and consequently the real magnitude of the central body, 

 would be also given. 



4. Let us now suppose the particles of light to be attracted in the same 

 manner as all other bodies with which we are acquainted ; that is, by forces 

 bearing the same proportion to their vis inertias ; of which there can be no 

 reasonable doubt, gravitation being, as far as we know, or have any reason to 

 believe, a universal law of nature. On this supposition then, if any one of the 

 fixed stars, whose density was known by the abovementioned means, should be 

 large enough sensibly to affect the velocity of the light issuing from it, we should 

 have the means of knowing its real magnitude, &c. 



5. It has been demonstrated by Newton, in prop. 39, b. 1, that if a right 

 line be drawn, in the direction of which a body is urged by any forces whatever, 

 and there be erected at right angles to that line perpendiculars every where pro- 

 portional to the forces at the points at which they are erected respectively, the 

 velocity acquired by a body beginning to move from rest, in consequence of 

 being so urged, will always be proportional to the square root of the area de- 

 scribed by the aforesaid perpendiculars. And hence, 6. If such a body, instead 

 of beginning to move from rest, had already some velocity in the direction of 

 the same line, when it began to be urged by the aforesaid forces, its velocity 

 would then be always proportional to the square root of the sum or difference of 

 the aforeseid area, and another area, whose square root would be proportional to 

 the velocity which the body had before it began to be so urged ; that is, to the 

 square root of the sum of those areas, if the motion acquired was in the same 

 direction as the former motion, and the square root of the difference, if it was 

 in a contrary direction. See cor. 2, to the abovesaid proposition. 



7. In order to find, by the foregoing proposition, the velocity which a body 

 would acquire by falling towards any other central body, according to the com- 

 mon law of gravity, let c, in fig. 4, pi. 7, represent the centre of the central 

 body, towards which the falling body is urged, and let ca be a line drawn from 

 the point c, extending infinitely towards A. If then the line rd be supposed to 

 represent the force, by which the falling body would be urged at any point d, the 

 vciocity which it would have acquired by falling from an infinite height to the 

 place d would be the same as that which it would acquire by falling from d to c 

 with the force rd, the area of he infinitely extended hyperbolic space adrb, 

 where rd is always inversely proportional to tire square of dc, being equal to the 



