SEC. III.] OF NATURAL PHILOSOPHY. 283 SCHOLIUM. 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 PROPOSITION XIV. THEOREM XL The same things being supposed, 1 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 re sistance 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 \* "1 \ B A K QP body is descending, otherwise on the contrary. Erect A b, which make to DB as DB2 to 4BAC : and to the rectangular asymptotes CK, CH, de scribe the hyperbola 6N ; and, erecting KN perpendicular to CK, the area A/AK will be augmented or diminished in an arithmetical progression, while the forces CK are taken in a geometrical progression. I say, there fore, that the distance of the body from its greatest altitude is as the excess of the area A6NK above the area DET. For since AK is as the resistance, that is, as AP2 X 2BAP ; assume any given quantity Z, and put AK equal to then (by Lem,