Note here, that in Descents, if the Tangent exceed the Quote, as it does when ph is more than bb, the direction of the lower Elevation will be below the Horizon, and if ph = bb, it must be directed Horizontal, and the Tangent of the upper Elevation will be pr⁄b: Note likewise, that if 4bb + 4ph in Ascents, or 4bb - 4ph in Descents, be equal to pp, there is but one Elevation that can hit the Object, and its Tangent is pr⁄2b. And if 4bb + 4ph in Ascents, or 4bb - 4ph in descents, do exceed pp, the Object is without the reach of a Project cast with that Velocity, and so the thing impossible.
From this Equation 4bb ∓ 4ph = pp are determined the utmost limits of the reach of any Project, and the Figure assigned, wherein are all the heights upon each Horizontal distance beyond which it cannot pass; for by reduction of that Equation, h will be found = ¼p - bb⁄p in heights, and bb⁄p - ¼p in descents; from whence it follows, that all the Points h are in the Curve of the Parabola, whose Focus is the Point from whence the Project is cast, and whose Latus rectum, or Parameter ad Axem is = p. Likewise from the same Equation may the least Parameter or Velocity be found capable to reach the Object proposed; for bb = ¼pp ∓ ph being reduced, ½p will be = √ bb + hh ± h { in ascents } | {in descents} which is the Horizontal Range at 45 degrees, of a Project cast with the least Velocity that would just reach the Object, and the Elevation requisite will be easily had; for dividing the so found Semi-parameter by the Horizontal distance given b, the Quote into Radius will be the Tangent of the Elevation sought. This Rule may be of good use to all Bombardiers and Gunners, not only that they may use no more Powder than is necessary, to cast their Bombs into the place assigned, but that they may shoot with much more certainty, for that a small Error committed in the Elevation of the Piece, will produce no sensible Difference in the fall of the Shot: For which Reasons the French Engineers in their late Sieges have used Mortar-pieces inclin'd constantly to the Elevation of 45, proportioning their Charge of Pouder according to the distance of the Object they intend to strike on the Horizon.
And this is all that need to be said concerning this Problem of shooting upon Heights and Descents. But if a Geometrical Construction thereof be required; I think I have one that is as easy as can be expected, which I deduce from the foregoing Analytical Solution, viz. t⁄r = p⁄2b ± √ ¼pp ± ph - bb / bb , and 'tis this, having made the right Angle GDF, (Tab. 5. Fig. 3.) make DF = ½p, or greatest Range, and GD = b the Horizontal Distance, and DB = h the perpendicular heighth of the Object; to be laid upwards from D, if the Object be above the Horizon, or downwards if below it. Parallel to GD draw FA, and make it equal to GB the Hypothenusal Distance of the Object; and with the Center A and Radius FB = ½p ± h, sweep an Arch, which shall if the thing be possible, intersect the indeterminate Perpendicular DF in two Points K and L, to which draw the Lines, GL, GK; I say, the Angles DGK, DGL, are the Elevations requisite to strike the Object B.
Demonstration. The Square of FK or FL, is equal to FBq - GBq: or (½p ± h)2 - bb - hh or ¼pp ± ph - bb, and therefore √ ¼pp ± ph - bb is = FK = FL, and by Consequence DK, DL = ½p ± √ ½pp ± ph - bb . And as DG: DK and DL :: Radius: Tangents sought, which coincides with our Algebraical Expression thereof.
Prop. XI. To determine the Force or Velocity of a Project, in every Point of the Curve it describes.
To do this we need no other Præcognita, but 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 wherein a Project would move from G to L, it descends from L to X, and by the third Proposition 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 impress'd in the Point G, as GZ is to GL; this follows from our second Axiom, and by the 20 and 21 Prop. lib. 1. conic. Midorgii, XO parallel and equal to GZ shall 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 Diameters: 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 discover'd.
Corollary.
From hence it follows, that the force of a Shot is always least at U, or the Vertex of the Parabola, and that at equal distances therefrom, as at T and X, G and B its force is always equal, and that the least force in U is to that in G and B, as Radius to the Secant of the Angle of Elevation FGB.
These Propositions considered, there is no question relating to Projects, which, by the help of them, may not easily be Solved; and tho' it be true that most of them are to be met withal, in Galileus, Torricellius and others, who have taken them from those Authors, yet their Books being Foreign, and not easy to come by, and their Demonstrations long and difficult, I thought it not amiss to give the whole Doctrine here in English, with such short Analytical Proof of my own, as might be sufficient to evince their Truth.
The Tenth Proposition contains a Problem, untouch'd by Torricellius, 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 Genuine Use and Effects of the Gun, Printed in the Year 1674; but his Solution required so much Calculation, that it put me upon search, whether it might not be done more easily, and thereupon in the Year 1678 I found out the Rule I now Publish, and from it the Geometrical Construction: Since which time there has a large Treatise of this Subject, Intituled, L'Art dejetter les Bombes, been Published by Monsieur Blondel, wherein he gives the Solutions of this Problem by Messieurs Buot, Romer and de la Hire: But none of them being the same with Mine, or, in my Opinion, more easy, and most of them more Operose, and besides mine finding the Tangent, which generally determines the Angle better than its Sine, I thought my self obliged to Print it for the use of all such, as desire to be informed in the Mathematical part of the Art of Gunnery.