yOL. XVI.] PHILOSOPHICAL TRANSACTIONS. 267 



a body, put in motion, will move on continually in a right line with an equable 

 motion, unless some other force or impediment intervene, by which it is acce- 

 lerated, or retarded, or deflected. 2dly. That a body being agitated by two 

 motions at a time, does by their compounded forces pass through the same 

 points as it would do, if the two motions were divided and acted successively. 

 As, for instance, suppose a body moved in the line GF, (fig. 2, pi. Q.) from 

 G to R, and there stopping, by another impulse suppose it moved, in a space 

 of time equal to the former, from R towards K, to V. I say, the body shall 

 pass through the point V, though these two several forces, acted both at the 

 same time. 



Prop. V. The motion of all projects is in the curve of a parabola. For let 

 the line GRF, in fig. 2, be the line in which the projectile is directed, and in 

 which, by the first axiom, it would move equal spaces in equal times, were it 

 not deflected downwards by the force of gravity. Let GB be the horizontal 

 line, and GC perpendicular to it. Then the line GRF being divided into equal 

 parts, answering to equal spaces of time, let the descents of the projectile be 

 laid down in lines parallel to GC, proportional to the squares of the lines GS, 

 GR, GL, GF, or as the squares of the times, as from S to T, from R to V; 

 from L to X, and from F to B, and draw the lines TH, VD, XY, BC parallel 

 to GF ; I say the points T, V, X, B, are points in the curve described by the 

 projectile, and that the curve is a parabola. By the second axiom, they are 

 points in the curve, and the parts of the descent GH, GD, GY, GC, ^ to 

 ST, RV, LX, FB, being as the squares of the times, by the 2d prop, that is, 

 as the squares of the ordinates, HT, DV, YX, BC, equal to GS, GR, GL, 

 GF, the spaces measured in those times; and there being no other curve but 

 the parabola, whose parts of the diameter are as the squares of the ordinates, 

 it follows that the curve described by a project, can be no other than a parabola : 

 and saying, as RV the descent in any time is to GR or DV the direct motion 

 in the same time, so is DV to a third proportional ; that third will be the line 

 "ailed by all writers of conies the parameter of the parabola to the diameter GC, 

 which is always the same in projects cast with the same velocity ; and the velo- 

 city being defined by the number of feet moved in a second of time, the para- 

 meter will be found by dividing the square of the velocity by l6 feet 1 inch, the 

 fall of a body in the same time. 



Lemma. The sine of the double of any arch is equal to twice the sine of 

 that arch into its cosine, divided by radius ; and the versed sine of the double 

 of any arc, is equal to the square of its sine divided by radius. For, let the 

 arch BC, in fig. 3, be double the arch BF, and A the centre ; draw the radii 

 AB, AF, AC, and the chord BDC, and let fall BE perpendicular to AC : then 



M M 2 



