■^^OL. XVI.] PHILOSOPHICAL TRANSACTIONS. 271 



the right angle LDA, pi. Q, fig. 4, make DA, DF =p, or greatest range, DG 

 = b the horizontal distance, and DB, DC = h, the perpendicular height of the 

 object ; draw GB, and make DE = to it. Then with the radius AC and centre 

 E sweep an arch, which if the thing be possible, will intersect the line AD in H; 

 and the lineDH, being laid both ways from F, will give the points K and L, to 

 which draw the lines GL, GK; I say the angles LGD, KGD are the elevations 

 requisite to strike the object B. But note, that if B be below the horizon, its 

 descent DC ^ DB must be laid from A, so as to have AC = to AD -f DC. 

 Note also, that if in descents DH be greater than FD, and so K fall below D, 

 the angle K GD shall be the depression below the horizon. And this construc- 

 tion so naturally follows from the equation, that I shall need say no more 

 about it. 



Prop. XL To determine the force or velocity of a project, in every point of 

 the curve it describes. — To do this, we need no other praecognita than only the 

 third proposition, viz. that the velocity of falling bodies is double to that which 

 in the same time would have described the space fallen by an equable motion : 

 for the velocity of a project is compounded of the constant equal velocity of the 

 impressed motion, and the velocity of the fall, under a given angle, viz. the 

 complement of the elevation ; for instance, in fig. 2, in the time in which a 

 project would move from G to L, it descends from L to X, and by the third pro- 

 position has acquired a velocity, which in that time would have carried it by an 

 equable motion from L to Z, or twice the descent LX; and drawing the line 

 GZ, I say, the velocity in the point X, compounded of the velocities GL and 

 LZ, under the angle GLZ, is to the velocity impressed in the point G, as GZ 

 is to GL; this follows from the second axiom, and by the 20th and 2 1st prop, 

 lib. 1, conic, Midorgii, XO parallel and equal to GZ will touch the parabola in 

 the point X. So that the velocities, in the several points, are as the lengths of 

 the tangents to the parabola in those points, intercepted between any two dia- 

 meters ; and these again are as the secants of the angles, which those tangents 

 continued make with the horizontal line GB. From what is here laid down 

 may the comparative force of a shot, in any two points of the curve, be either 

 geometrically or arithmetically discovered. 



Carol. From hence it follows, that the force of a shot is always least at V 

 the vertex of the parabola; and that at equal distances from thence, as at T and 

 X, G and B, its force is always equal ; and that the least force in V is to that in 

 G and B, as radius to the secant of the angle of elevation FGB. 



The 1 0th proposition contains a problem, untouched by Torricelli, which is 

 of the greatest use in gunnery, and for the sake of which this discourse was 

 principally intended; it was first solved by Mr. Anderson, in his book of the 



