Notwithstanding the great simplicity of the construction of these problems, we cannot obtain numerical solutions for practice with equal simplicity, except when the line of position is horizontal, as in Fig. 2. This indeed is the most general case, and there are few situations so abrupt as to deviate very far from this case, the greatest height of a fortress commonly bearing but a small proportion to the distance of the mortar.
When AB is a horizontal plane, as in Fig. 2. the arch EDA is a semicircle.
In this case the maximum range A b is equal to AC, the radius of the circle, and equal to twice the height FA necessary for acquiring the velocity of the projection.
This greatest range is obtained by elevating the mor tar 45 degrees from the horizon.
The ranges with different directions, are proportional to the sines of twice the angles of elevation. For draw ing GC, DL, dl, perpendicular to EA, and drawing the radii CD and Cd, we have CG equal to the range A b, and I d equal to the range AB. Now CG is the sine of the angle ACG, which is double of GAB, and l d is the sine of AC d, which is double of AE d, which is equal to the elevation d AB ; and the same is true of all other elevations. We may always employ this analogy as radius to the sine of twice the angle of elevation, so is twice the height necessary for acquiring the velocity to the range of the projection on a horizontal plane.
The height to which the projectile rises above the hori zontal plane is as the square of the sine of elevation. For OV, the axis of the parabola, is 1th of DB or LA ;— and FA, the height to which the projectile would rise straight upwards, is ith of EA. Now EA : LA=EA2: AED,=rad. 2 elevation. Therefore FA : : sin.' elevation;' also VO : v DAB : d AB, &c.
The times of the flights are as the sines of the elevation. For the velocities in the directions AD, A d, being the same, the times of describing AD and A d uniformly will be as AD and A d. Now AD and A d are as the sines of the angles AED and AE d, which are equal to the angles DAB and d AB. Now the times of describing AD and A d
uniformly with the velocity of projection, are the same with the times of describing the parabolas AVB and A v B.
When the object to be struck is on an inclined plane, AB, ascending, as in Fig. 3. the arch EDA is less than a semi circle ; and when it is on a descending plane, as in Fig. 4. EDA is greater than a semicircle. This considerably em barrasses the process for obtaining the direction, when the ' impetus and the object are given, or conversely. • It has been much canvassed by the many authors who deliver theories of gunnery, and the parabola affords many very pretty methods of solving the problem. Dr Halley's, in the Philosophical Traksactions, No. 179, is peculiarly ele gant. Air Thomas Simpson's also, in No. 486, is extreme ly ingenious and comprehensive, and has been reduced to a very elegant simplicity by Frisius in his Cosmographia. But neither of these methods slim so distinctly the connec tion between the different circumstances of the motions, or keep the general principle so much in view, as the one here given ; and all the arithmetical operations which final ly result from them, are precisely similar to those deduced from our construction.
The following method, suggested by the simple construc tion now given, is probably as easy and as expeditious as any. • Draw the horizontal line HA a, Fig. 3. and 4. cutting the vertical drawn through Bin K; let C be the centre of the circular arch EDA. Join AC, and draw GC, cutting the verticals through A and B in the points,/ and g. Also draw CD and C d. Let fi represents the angle of position EA B, and d the angle of direction pAB, which the axis of the piece makes with the line of pusiOon AB. Also let z be the angle EAD which the axis makes with the vertical. Let r express the range AB, and f the fall FA necessary for communicating the velocity of projection. Then the parameter of the parabola at the point of projection is 4f, =AE, and using S to express the sine, We have AB: DB=S, EAD : S, : S, d.