194 HISTORY OF SCIENCE. 



Also triangle A B s is equal to triangle B c s, and therefore triangle SEE 

 has the same area as triangle A B s. Now, again, suppose that when 

 the body had arrived at E, another impulse were to act upon it from 

 s, directed along the line E s, it would follow the diagonal of another 

 parallelogram, and at the end of the interval of time would be at F, 

 instead of at G, where, without the impulse towards s, it would .have 

 arrived, G E being equal to B E. The same reasoning as before shows 

 that triangle EPS equals triangle E B s. Thus, it is proved that a line 

 joining the point s to the moving body, and moving with it, would 

 sweep out equal areas in equal times. Now, we are to suppose that 

 the intervals of time at which the impulses act are made shorter, and 

 again shorter, and so on continually. The number of triangles would 

 be increased ; but, however greater their number, the same reasoning 

 would hold : equal areas would be swept out in equal times. In the 

 limit, the boundary A B E F would become a curve, and the series of 

 impulses would coalesce into a continuous force acting from the centre 

 s, while the area swept out by the line joining s and the body would 

 always be proportional to the times. There is only one point of this 

 reasoning which involves any difficulty, and that consists in the assump- 

 tion that the relations which are established for the elementary quan- 

 tities remain unaltered, as we pass from them to their limit, as we 

 pass, for instance, here from the polygonal to the curvilinear path. In 

 the case before us the ideas are sufficiently simple ; but Newton had 

 provided for any difficulty on this score by previously discussing various 

 instances of limits under the title of " Prime and Ultimate Ratios." 



. This ' theorem, considered by itself, would have been sufficiently 

 curious, but Newton investigated it with a view to the higher purpose 

 of establishing a whole system of doctrines concerning central forces. 

 He demonstrated the inverse proposition, namely, that if a body move 

 in one plane, so that the areas described by a line drawn to a fixed 

 point are proportional to the time of describing them, the body is acted 

 on by a force tending to that point. Newton, in fact, investigated all 

 the cases of curvilinear motion, and examined the mathematical con- 

 ditions of all problems arising from it. It was in this that the vast 

 resources of Newton's geometry were particularly displayed. The very 

 idea of motion about a centre of force was first accurately conceived 

 by him, and he gave in the equality of the areas the only criterion of 

 the existence of such forces. As a body projected in the direction of 

 a straight line, and at the same time exposed to the action of a central 

 force, will neither proceed in that straight line nor pass directly towards 

 the centre of force, the nature of the particular intermediate curve 

 it will describe will depend upon the ratio between the intensities of 

 the original impulsive force and the deflecting or central force. In 

 the investigation of the various possible cases, Newton displayed the 

 fertility of his genius and his knowledge of the stores of mathematical 

 truths which the labours of former times had accumulated. The pro- 



