Prop. VI. The Horizontal Distances of Projections made with the same Velocity, at several Elevations of the Line of Direction, are as the Sines of the doubled Angles of Elevation.

Let GB (Fig. 1) the Horizontal Distance be = z, the Sine of the Angle of Elevation, FGB, be = s, its Co-sine = c, Radius = r, and the Parameter = p. It will be as c to s; so z to sz⁄c = FB = GC, and by reason of the Parabola psz⁄c = to the Square of CB, or GF; Now as c to r, so is z to zr⁄c = GF, and its Square zzrr⁄cc will be therefore = to psz⁄c: Which Equation reduced will be psc⁄rr = z. But by the former Lemma 2sc⁄r is equal to the Sine of the double Angle, whereof s is the Sine: Wherefore 'twill be as Radius to Sine of double the Angle FGB, so is half the Parameter, to the Horizontal Range or Distance sought; and at the several Elevations, the Ranges are as the Sines of the double Angles of Elevation, Q. E. D.

Corollary.

Hence it follows, that half the Parameter is the greatest Randon, and that that happens at the Elevation of 45 Degrees, the Sine of whose double is Radius. Likewise that the Ranges equally distant above and below 45 are equal, as are the Sines of all double Arches, to the Sines of their doubled Complements.

Prop. VII. The Altitudes of Projections made with the same Velocity, at several Elevations, are as the versed Sines of the doubled Angles of Elevation: As c is to s; so is psc⁄rr = GB to pss⁄rr = BF: and UK = RU = BF/4, the Altitude of the Projection = psc⁄4rr. Now by the foregoing Lemma 2ss⁄r = to the versed Sine of the double Angle, and therefore it will be as Radius, to versed Sine of double the Angle FGB, so an 8th of the Parameter to the height of the Projection VK; and so these heights at several Elevations, are as the said versed Sines, Q. E. D.

Corollary.

From hence it is plain, that the greatest Altitude of the perpendicular Projection is a 4th of Parameter, or half the greatest Horizontal Range; the versed Sine of 180 Degrees being = 2r.

Prop. VIII. The Lines GF, or Times of the Flight of a Project cast with the same Degree of Velocity at different Elevations, are as the Sines of the Elevations.

As c is to r; so is psc⁄rr = GB by the 6 Prop. to ps⁄r GF; that is, as Radius to Sine of Elevation, so the Parameter to the Line GF; so the Lines GF are as the Sines of Elevation, and the Times are proportional to those Lines; wherefore the Times are as the Sines of Elevation: Ergo constat propositio.

Prop. IX. Problem. A Projection being made as you please, having the Distance and Altitude, or Descent, of an Object, through which the Project passes, together with the Angle of Elevation of the Line of Direction; to find the Parameter and Velocity, that is (in Fig. 1.) having the Angle FGB, GM, and MX.