Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/222

 and by the table of logarithms the area AIKP will be given, and equal thereto the area OPA, which ubducted from the triangle OPS will leave the area cut off APS. And by applying 2APS - 2A, or 2A - 2APS, the double difference of the area A that was to be cut off, and the area APS that is cut off to the line SN that is let fall from the focus S. perpendicular upon the tangent TI, we hall have the length of the chord PQ. Which chord PQ is to be incribed between A and P. if the area APS that is cut off be greater than the area A that was to be cut off, but towards the contrary ide of the point P, if otherwie: and the point Q will be the place of the body more accurately. And by repeating the computation the place may be found perpetually to greater and greater accuracy.

And by uch computations we have a general analytical revolution of the problem. But the particular calculus that follows, is better fitted for atronomical purpoes. Suppoing AO, OB, OD, (Pl. 14. Fig. 5.) to be the emi-axes of the ellipis, and L its latus rectum, and D the difference betwixt the leer emi-axis OD, and $$\scriptstyle \frac 12$$L, the half of the latus rectum: let an angle Y be found, whoe line may be to the radius, as the rectangle under that difference D and AO + OD the half um of the axes, to the figure of the greater axis AB. Find alo an angle Z, whoe ine may be to the radius, as the double rectangle under the ditance of the foci SH and that difference D to triple the quare of half the greater emi-axis AO. Thoe angles being once found, the place of the body may be thus determined. Take the angle T proportional to the time in which the arc BP