or twice BD, to BE; that is, as Radius to Co-sine , so twice Sine to Sine of the double Arch . And as AB to BD, so twice BD or BC to EC, that is, as Radius to Sine , so twice that Sine , to the Versed Sine of the double Arch ; which two Analogies resolved into Equations , are the Propositions contained in the Lemma to be proved.
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