438 THE MATHEMATICAL PRINCIPLES [BOOK III N and TNr let there be described an ellipsis NHwL ; and in the time in which the sun recedes from the node through the arc N0, if there be drawn the right line Tba, the area of the sector NTa will be the exponent of the sum of the motions of the sun and node in the same time. Let, there fore, the extremely small arc aA. be that which the right line T/w, revolv ing according to the aforesaid law, will uniformly describe in a given particle of time, and the extremely small sector TAa will be as the sum of the velocities with which the sun and node are carried two different ways in that time. Now the sun's velocity is almost uniform, its ine quality being so small as scarcely to produce the least inequality in the mean motion of the nodes. The other part of this sum, namely, the mean quantity of the velocity of the node, is increased in the recess from the gyzygies in a duplicate ratio of the sine of its distance from the sun (by Cor. Prop. XXXI, of this Book), and, being greatest in its quadratures with the sun in K, is in the same ratio to the sun's velocity as SK to TS. that is, as (the difference of the squares of TK and TH, or) the rectangle KHM to TH2. But the ellipsis NBH divides the sector AT«, the expo nent of the sum of these two velocities, into two parts ABba and BTb, proportional to the velocities. For produce BT to the circle in 0, and from the point B let fall upon the greater axis the perpendicular BG, which being produced both ways may meet the circle in the points F and f; and because the space ABba is to the sector TBb as the rectangle AB£ to BT2 (that rectangle being equal to the difference of the squares of TA nnd TB, because the right line A3 is equally cut in T, and unequally in B), therefore when the space ABba is the greatest of all in K, this ratio will be the same as the ratio of the rectangle KHM to HT2. But the greatest mean velocity of the node was shewn above to be in that very