32 NEWTON S PRINCIPIA. 



which are described by a line drawn from it to any 

 point are proportional to the portions of the straight line 

 through which the body moves, (that is to the time, since, 

 moving equably, it moves through equal spaces in equal 

 times,) because those triangles, having the same altitude, 

 are to one another in the proportions of their bases. S 

 being the point and AO the line of motion, SAB is to 

 SB c as AB to B c. If then at B a force acts in the line 

 S B, drawing the body towards S, it will move in the 

 diagonal B C of a parallelogram of which the sides are 

 B c and B V, the line through which the deflecting force 



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would make it move if the motion caused by the other 

 force ceased. Cc therefore is parallel to VB, and the 

 triangle SBC is equal to the triangle SBc; consequently 

 the motion through A B and B C, or the times, are as the 

 two triangles SAB and SBC : and so it may be proved if 

 the force acting towards S again deflects the body at C, 

 making it move in the diagonal C D. If, now, instead of 

 this deflecting force acting at intervals A, B, C, it acts at 

 every instant, the intervals of time become less than any 

 assignable time, and then the spaces A B, B C, CD will 

 become also indefinitely small and numerous, and they will 

 form a curve line ; and the straight lines drawn from any 

 part of that curve to S will describe curvilinear areas, as 

 the body moves in the curve ABCD, those areas being 

 proportional to the times. So conversely, if the triangles 



