28 PHILOSOPHICAL TRANSACTIONS. [^ANNO ]6Q5. 



height thereof above the gun ; and in the other case, when the object is below 

 the horizon, the difference of the same, per 47, 1. Eucl. And I then showed 

 how to find the elevation proper for the gun so charged, viz. As the horizontal 

 distance of the object, to the the sum or difference of the hypothenusal distance 

 and perpendicular height :: so radius to the tangent of the elevation sought. 

 But I was not at that time aware that the aforesaid elevation did constantly 

 bisect the angle between the perpendicular and the object, as is demonstrated 

 from the difference and sum of the tangent and secant of any arcli being always 

 equal to the tangent and cotangent of the half complement thereof to a qua- 

 drant. Having discovered this, I think nothing can be more compendious, or 

 bids fairer to complete the art of gunnery, it being as easy to shoot with a 

 mortar at any object on demand, as if it were on the level ; neitlier is there 

 need of any computation, but only simply laying the gun to pass, in the middle 

 line between the zenith and the object, and giving it its due charge. Nor is 

 there any great need of instruments for tliis purpose : for if the muzzle of the 

 mortar be turned truly square to the bore of the piece, as it usually is, 

 or ought to be, a piece of looking-glass plate applied parallel to the muzzle, 

 will by its reflection give the true position of the piece;- the bombardeer having 

 no more to do, but to look perpendicularly down on the looking-glass, along 

 a small thread with a plummet, and to raise or depress the elevation of the piece, 

 till the object appear reflected on the same point of the speculum on which the 

 plummet f;ills; for the angle of incidence and reflection being equal, in this 

 case a line at right angles to the speculum, as is the axis of the chase of the 

 piece, will bisect the angle between the perpendicular and the object, according 

 as our proposition requires. So that it only remains by good and valid experi- 

 ments to be assured of the force of gunpowder, how to make and conserve it 

 equal, and to know the effect thereof in each piece ; that is, how far different 

 charges will throw the same shot out of it : which may most conveniently be 

 engraven on the outside thereof, as a standing direction to all gunners, who 

 shall from thenceforward have occasion to use that piece : and were this matter 

 well ascertained, it might be worth the while to make all mortars of the like 

 diameter, as near as may be alike in length of chase, weight, chamber, and all 

 other circumstances. 



This discovery, that the utmost range on an inclined plane, is when the axis 

 of the piece makes equal angles with the perpendicular and the object, com- 

 pared with what I have demonstrated of the same problem in the aforesaid 

 N° 17Q, leads to and discovers two very ready theorems; the one to find the 

 greatest horizontal range at 45° elevation, by any shot made on any inclined 

 plane, with any elevation of the piece whatever : and the other lo find the 



