Galileo writes that for angles of projection of a projectile at angles (450+θ) and (450-θ), the horizontal ranges described by the projectile are in the ratio of (if θ>450)

For a projectile launched with velocity u at an angle θ, the horizontal range is given byR = u2sin2 θg(i) for θ1 = 450-θ         R1 = u2sin(900-2θ)g = u2gcos2θ(ii) For θ2 = 450 + θR2=u2sin(900+2θ)g=u2gcos2θ