27S THE MATHEMATICAL PRINCIPLES [BOOK 11 common intersection of this hyperbolic curve and the circumference of the de scribed circle. Q.E.D. It is to be ob served that this operation is the same, whether the right line AKN be parallel to the horizon, or inclined thereto in any an gle : and that from two intersections H, //., there arise two angles NAH, NAA ; and that in mechanical practice it is suf ficient once to describe a circle, then to apply a ruler CH, of an indeterminate length, HO to the point C, that its part PH, intercepted between the circle and the right line FK, may bo equal to its part CE placed between the point C and the right line AK What has been said of hyperbolas may be easily applied to pir i'>;>l.i3. For if a parabola be re presented by XAGK, touched by a right line XV in the vertex X, and the ordinates IA, YG be as any powers XI", XV"; of the abscissas XI, XV ; draw XT, GT, AH, whereof let XT be parallel to VG, and let GT, AH touch the parabola in G and A : and a body projected from any place A, in the direction of the right line AH, with a due velocity, will describe this parabola, if the density of the medium in each of the places G be reciprocally as the tangent GT. In that case the velocity in G will be the same as would cause a body, moving in a nonresisting space, to describe a conic parabola, having G for its vertex, VG 2GT2 produced downwards for its diameter, and -. — for its latus nn — n X VG rectum. And the resisting force in G will be to the force of gravity as GT to 2nti — 2tt ~2~ VG. Therefore if NAK represent an horizontal line, and botli the density of the medium at A, and the velocity with which the body is projected, remaining the same, the angle NAH be any how altered, the lengths AH, AI, HX will remain; and thence will be given the vertex X of the parabola, and the position of the right line XI ; and by taking VG to IA as XVn to XI", there will be given all the points G of the parabola, through which the projectile will pass.