VOL. XIX.] PHILOSOPHICAL TRANSACTIONS. 2Q 



elevations proper to strike a given object, with any force greater than what 

 suffices to reach it with the aforesaid middle elevation. Both which being per- 

 formed by one single proportion, may be very serviceable to such as are con- 

 cerned in the practice of gunnery, but are unwilling to trouble themselves with 

 tedious and difficult rules. The two propositions are these : 



Prop. I. — A shot being made on an inclined plane; having the horizontal 

 distance of the object it strikes, with the elevation of the piece, and the angle 

 at the gun between the object and the perpendicular : to find the greatest hori- 

 zontal range of that piece, laden with the same charge ; that is, half the latus 

 rectum of all the uarabolae made with the same impetus. 



Rule. — Take half the distance of the object from the nadir, and take the 

 difference of the given elevation from that half ; the versed sine of that differ- 

 ence subtract from the versed sine of the distance of the object from the 

 zenith : then shall the difference of those versed sines, be to the sine of the 

 distance of the object from the zenith, as the horizontal distance of the object 

 struck, to the greatest horizontal range at 45^. 



Prop. 2. — Having the greatest horizontal range of a gun, the horizontal 

 distance and angle of inclination of an object to the perpendicular ; to find the 

 two elevations necessary to strike that object. 



Rule. — Halve the distance of the object from the nadir, this half is always 

 equal to the half sum of the two elevations we seek. Then say, As the greatest 

 horizontal range is to the horizontal distance of the object, so is the sine of the 

 angle of inclination, or distance of the object from the perpendicular, to a 

 4th proportional ; which 4th being subtracted from the versed sine of the 

 distance of the object from the zenith, leaves the versed sine of half the differ- 

 ence of the elevations sought ; which elevations are therefore had by adding 

 and subtracting that half difference to and from the aforesaid half sum. 



I shall not need to speak of the facility of these solutions, I shall only ob- 

 serve that they are both general, without exception or caution, and derived 

 from the knowledge that these two elevations are equidistant above and below 

 the line bisecting the angle between the object and the zenith. 



An Account of Books, viz. I. The Mathematical TForks of Dr. John JVallis, 

 Savilian Professor of Geometry at Oxford, and F. R, S. 2 Fols. fol. Oxon, 

 N°2l6, p. 73. 



In the former of them are contained, 1. His Inaugural Oration, when he 

 entered on that employment, Oct. .31, 1649. 2. His Mathesis Universalis, or 

 Opus Arithmeticum ; ccjniainingnot only numeral arithmetic, but the specious 

 and algebraic, or the calculus geometricus, with many discourses or smaller tracts 



