Astronomical and Nautical Collections. 175 



the angle of elevation, in the circle, of which the diameter is the 

 height due to the velocity ; and that its horizontal range veill be 

 four times the corresponding sine. Hence it is obvious, that when 

 the direction is nearly horizontal, the radius of curvature of the 

 path of the projectiles will be equal to twice the diameter of that 

 circle, or to twice the height duo to the velocity, since the 

 chord is twice as great, and the verse sine the same as in the 

 circle, so that the radius must be quadruple. 



B. It is also well known, that the tangent of a parabola in- 

 tersects its diameter at a distance above the vertex, equal to the 

 length of the absciss below it; so that the portion of the absciss 

 below the vertex is half of the part cut off by the tangent. 



C. The horizontal ordinate of the parabola, flowing uniformly 

 with the time, is always proportional to the vertical velocity, 

 and the difference of any two proximate ordinates, compared 

 with their length, and the evanescent interval between them, 

 will always give the distance of the intersection of the tangent, 

 according to the common method of finding the tangents of 

 curves ; that is, as the difference of the velocities is to the whole 

 velocity, so is the difference of the absciss to the part cut off by 

 the tangent, or to twice the absciss reckoned from the vertex, 



D. Now the velocity of light, considered as a projectile, must 

 be supposed to vary directly as the refractive density ; so that 

 we have only to determine what proportion the variation of the 

 refractive density of the atmosphere, in the height of a foot or 

 a yard, bears to the whole refractive density, and to increase 

 the foot or the yard in the same proportion, and we shall obtain 

 the measure of twice the height due to the velocity, or of the 

 radius of the circle of curvature of the ray of light moving ho- 

 rizontally through such an atmosphere. 



E. The velocity of light in a vacuum, and in the atmosphere 

 at 50°, with the barometer at 30, varies in the ratio of 3540 to 

 3541 ; the height of a homogeneous atmosphere, under these 

 circumstances, is 27,000 feet; and the temperature descends 

 about 1° for every 300 feet that we ascend. Consequently the 

 velocity varies ^j^ . -jj-rl^jf^ in every foot, as far as the diminution 

 of pressure is coDcerned, and -j^o • rho • vaT ^^ ^^ ^^ deducted 



