_Secondly_, That a _Body_ being agitated by two _Motions_ at a time, does by their _compounded Forces_ pass through the same _Points_, as it would do, were the two _Motions divided_ and acted _successively_. As for Instance, Suppose a _Body_ moved in the Line GF, (_Fig. 1. Tab. 5._) from G to R, and there stopping, by another _Impulse_, suppose it moved in a _Space_ of _Time_ equal to the former, from R towards K, to V. I say, the _Body_ shall pass through the Point to V, though these two _several Forces_ acted both in the _same time_. _Prop._ V. The _Motion_ of all _Projects_ is in the _Curve_ of a _Parabola_: Let the _Line_ GRF (in _Fig._ 1.) be the _Line_ in which the _Project_ is directed, and in which by the first _Axiom_ it would move equal _Spaces_ in equal _Times_, were it not deflected downwards by the Force of _Gravity_. Let GB be the _Horizontal Line_, and GC a _Perpendicular_ thereto. Then the _Line_ GRF being divided into equal Parts, answering to equal _Spaces_ of _Time_, let the _Descents_ of the _Project_ be laid down in _Lines parallel_ to GC, proportioned as the _Squares_ of the _Lines_ GS, GR, GL, GF, or as the _Squares_ of the _Times_, from S to T, from R to V, from L to X, and from F to B, and draw the _Lines_ TH, VD, XY, BC parallel to GF; I say, the Points T, V, X, B, are Points in the _Curve_ described by the _Project_, and that that _Curve_ is a _Parabola_. By the second _Axiom_, they are Points in the _Curve_; and the Parts of the _Descent_ GH, GD, GY, GC, = to ST, RV, LX, FB, being as the _Squares_ of the _Times_ (by the _Second Proposition_) that is, as the _Squares_ of the _Ordinates_, HT, DU, YX, BC, equal to GS, GR, GL, GF, the _Spaces_ measured in those Times; and there being no other _Curve_ but the _Parabola_, whose Parts of the _Diameter_ are as the _Squares_ of the _Ordinates_, it follows that the _Curve_ describ'd by a _Project_, can be no other than a _Parabola_: And saying, as RU the _Descent_ in any _time_, to GR or UD the _direct Motion_ in the same _time_, so is UD to a _third_ proportional; that _third_ will be the _Line_ call'd by all Writers of _Conicks_, the _Parameter_ of the _Parabola_ to the _Diameter_ GC, which is always the same in _Projects_ cast with the same _Velocity_: And the _Velocity_ being defined by the Number of _Feet_ moved in a _Second_ of Time, the _Parameter_ will be found by dividing the _Square_ of the _Velocity_, by 16 _Feet_, 1 _Inch_, the _Fall_ of a _Body_ in the same _Time_. _Lemma._ The _Sine_ of the double of any _Arch_, is equal to twice the _Sine_ of that _Arch_ into its _Co-sine_, divided by _Radius_; and the _versed Sine_ of the _double_ of any _Arch_ is equal to twice the _Square_ of the _Sine_ thereof divided by _Radius_.