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 XI. 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 × 2BAP; assume any given quantity Z, and put AK equal to ; then (by Lem.[Pg 284] II of this Book) the moment KL of AK will be equal to or , and the moment KLON of the area AbNK will be equal to or . CASE 1. Now if the body ascends, and the gravity be as AB2 + BD2 BET being a circle, the line AC, which is proportional to the gravity, will be , and DP2 or AP2 + 2BAP + AB2 + BD2 will be AK × Z + AC × Z or CK × Z; and therefore the area DTV will be to the area DPQ as DT2 or DB2 to CK × Z. CASE 2. If the body ascends, and the gravity be as AB2 - BD2, the line AC will be , and DT2 will be to DP2 as DF2 or DB2 to BP2 - BD2 or AP2 + 2BAP + AB2 - BD2, that is, to AK × Z + AC × Z or CK × Z. And therefore the area DTV will be to the area DPQ as DB2 to CK × Z. CASE 3. And by the same reasoning, if the body descends, and therefore the gravity is as BD2 - AB2, and the line AC becomes equal to ; the area DTV will be to the area DPQ as DB2 to CK × Z: as above.