Page:Newton's Principia (1846).djvu/289



One may demonstrate also that case in the ascent of the body, where the force of gravity is less than can be expressed by DA² or AB² + BD², and greater than can be expressed by AB² - DB², and must be expressed by AB². But I hasten to other things.


 * The same things being supposed, I say, that the space described in the ascent or descent is as the difference of the area by which the time is expressed, and of some other area which is augmented or diminished in an arithmetical progression; if the forces compounded of the resistance and the gravity be taken, in a geometrical progression.

Take AC (in these three figures) proportional to the gravity, and AK to the resistance; but take them on the same side of the point A, if the



body is descending, otherwise on the contrary. Erect Ab, which make to DB as DB² to 4BAC: and to the rectangular asymptotes CK, CH, describe the hyperbola bN; and, erecting KN perpendicular to CK, the area AbNK will be augmented or diminished in an arithmetical progression, while the forces CK are taken in a geometrical progression. I say, therefore, that the distance of the body from its greatest altitude is as the excess of the area AbNK above the area DET.

For since AK is as the resistance, that is, as AP² $$\scriptstyle \times$$ 2BAP; assume any given quantity Z, and put AK equal to $$\scriptstyle \frac{AP^{2}+2BAP}{Z}$$; then (by Lem.