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Prop. XI. Prob. 3.] A Shot being made mi an Inclined Plain, having the Ho~riyOntal Dißance of the Object it ßrikes, veith the Elcvation of the Picee, andthe Angle at the Gun betneen the Object and the Perpendicular, to find the greateßHorizontal Rytnge of ■ that Piece, laden xoith the fame Charge j that is, half theLatus Rectum of all the Parabola: made mth the fame Impetus.

Take Half the Diftance of the Object front the Nadir, and take the Dif-tcrencc of the given Elevation front that Half ; the Verfed Sine of twicethat Difference Subftract from the. Verfed Sine of the Diftance of the Objectfront the Zenith ■ Then (hall the Difference of thofe Verfed Sines be to theSine of the Diftance of the Object front the Tjenieh, as the HorizontalDiftance of the Object struck to the greatest Horizontal Range at 45°.

Prop. XII. Prob. 4 ] Having the greateß Horizontal Range of a Gun, theHorizontal Dißancc and Angle of Inclination of an Olject ■ to the Perpendicular,to find the two Elevations nccejfary to jbrike that Olject.

Halvc the Diftance of the Object from the Nadir, this' Half is alwayscqual to the Half Sum of the two Elevations we feek. Then siy, As theGreatest Horizontal Range,is to the Horizontal Diftance of the Object, lö isthe Sine of the Angle of Inclination, or Diftance of the Object front thePerpendicular, to a 41h Proportional ; which 4dl being Substractcd fromthe Verfed Sine of the Diftance of the Object from the Tjcnith, leaves theVerfed Sine of the Difterence of \hc. Elevations föught ; which Elevations aretherefore had, by Addingand.Substracting that Half Difference to, and front,the aforesiid Half Sum..

n«;i75>. f. 18. Prop. XIII.] To determine the Force or Velociry of a Prcjecl, in every Point ofthe Curve it describes.

To do this, we need no ot'ner Prctcognita, but only the Third Propoßtion,viz. That the Velocity of Falling Bodies, is double to that which in the samt Timswould have deferibed the Space fallen by an Eijuable Motion : For the Velocityof a Project is compounded of the constant equal Velocity of the intprefledMotion, and the Vclocity of the Fall, under a given Angle, G-. TheCom-rtz. 175. p'cment of the Elevation For mstance, In the Time wherein a Project wouldMove from G to L, it Defcends from L to X, and by the Third Propoßtionhas Acquired a Velocity, which in that Time would have carried it by an Equ-able Motion from L to Z, or twice the Delcent LX; and drawing theLine GZ, I siy the Velocity in the Point X, compounded of the Veloci ticsGL, and LZ, under the Angle GLZ, is to th 0. Velocity impreft in thePoint G, as GZ, is to. G L ; this follows from our Second Axiom ; and bythe zoth and zith Prop. Lib. 1. Conic. Midorgii, X O, Parallel and Equal toG Z, (hall touch the Parabola in the Point X. So that the Velocities in thefeveral Points, are as the Lengths of the • 'Pangent s to the Parabola in thofePoints, interceptcd between any two Diameters: And thefe again are as theSecants of the Angles, which thofe Tangents continued make with the Hori-zontal Liue GB. From what.is heie laid down, may the comparative Force