Mechanical Illustrations of the Planetary Motions. 317 



suppose onr imaginary circumference to be divided into por- 

 tions equal to the spaces through which any point moves, in 

 its rotation, in given times. Let us also imagine a vertical 

 plane to touch the circle in its lowest point, and the circle, 

 with the points marked upon it, to be orthographically pro- 

 jected upon that plane, 

 as in the annexed dia- 

 gram. Again, from the 

 points thus projected up- 

 on the circumference of 

 the ellipse, let perpendi- 

 culars be drawn to a line 

 touching the ellipse in 

 its lowest point. These 

 perpendiculars will be 

 equal to the abscissae of the ellipse for the projected points, 

 and set off upon the conjugate axis. Let the same distances 

 be set off upon the tangent line which were previously set off 

 upon the circumference of the circle : these will be the dis- 

 tances through which any point in the circumference would 

 move in the given times, if allowed to advance in a rectilineal 

 direction. Through these points let vertical lines be drawn 

 equal to the spaces through which the lowest point in the 

 circumference would descend in the same times, if the rota- 

 tion were stopped and the top allowed to fall, turning on its 

 pivot. The curve connecting the lower extremities of these 

 vertical lines will be an approximation to the parabola, and, 

 in fact, for a small portion at the vertex, may be regarded 

 as a parabola, the vertical lines being equal to its abscissae. 

 Now, it is a familiar law in dynamics, that, if two forces act 

 upon the same body in the same direction, the resulting force 

 is the sum of the two ; but if in opposite directions, the dif- 

 ference ; and forces are measured by the motions which they 

 produce in the same mass and in the same time. In the pre- 

 ceding diagram, the perpendiculars on the upper side of the 

 horizontal line show the spaces through which the lowest 

 point in the circumference would be raised, in the given 

 times, by the rotatory motion alone ; those under the hori- 

 zontal line show the spaces through which it would fall in 



NEW SERIES. VOL. I. NO. II. APRIL 1855. T 



