Let the _Arch_ BC (in _Fig. 2. Tab. 5._) be double the _Arch_ BF, and A the _Center_; draw the _Radii_ AB, AF, AC, and the _Chord_ BDC, and let fall BE perpendicular to AC, and the _Angle_ EBC, will be equal to the _Angle_ ABD, and the _Triangle_ BCE, will be like to the _Triangle_ BDA; wherefore it will be as AB to AD, so BC 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_ 2_sc_/_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_.