TRACT 37.

OF GUNNERY.

269

lower pitch, the shot has much less quantity of the resistingair to pass through, than in the higher ones. On these ac-counts the angle for the greatest horizontal range, may beat all degrees between 45° and 30°, the slowest motions andlargest shot being almost at 45°, but gradually more andmore below that degree, as the shot is smaller and the ve-locity greater, till at length, with the most rapid motionsand the smallest shot, the angle is little above 30°.

117. From all these circumstances it partly appears, howdifficult a thing it must he to institute a calculation that may-produce, with any tolerable degree of practical utility, all,or indeed any of the circumstances of a projectile’s path.But when it is further considered that the formula by whichthe resistance is expressed, in terms of the velocity, is ne-cessarily rather complex, consisting of two different powersof the velocity at least, the attempt will soon evince that theaccomplishment of the required object, is a thing on thevery verge of impossibility. Hence, all that has been everdone in this matter, by the ablest mathematicians that everexisted, with the aid of all the resources of the most refinedmethods of analysis, has amounted to nothing more than im-perfect and rather distant approximations ; nor is much morenow to be accomplished, or exfi^cted, on the circumstancesof the path, since the correct and general expression of theresistance has been obtained ; and so the case must continueto remain, till many and accurate experiments can he ob-tained, on the ranges and times of flight of projectiles, withall degrees of elevation and velocity; but which, it is to befeared, are not likely soon to be expected. The approxima-tions on this head, that have been made by Newton, andRobins, and Euler , and professor Robison, appear to ap-proach nearest to practical usefulness, and of some of thesewe shall endeavour to avail ourselves on this point; makinghowever several necessary alterations and corrections inthem, from the more correct nature and quantity of the re-sistance which we have been enabled to derive from ourmore correct and extensive experiments^