266 THE MATHEMATICAL PRINCIPLES [BOOK li. (because tD is given), as ^f. But joD8 is AD3 + Ap2, that is, AD2 -h AD X AA-, or AD X Gk ; and (/Dp is 1 AD X pq. Therefore tDv, the BO particle of the sector, is as ^ , ; that is, as the least decrement pq of the velocity directly, and the force Gk which diminishes the velocity, inversely ; and therefore as the particle of time answering to the decrement of the ve locity. And, by composition, the sum of all the particles tDv in the sector AD/ will be as the sum of the particles of time answering to each of the lost particles pq of the decreasing velocity Ap, till that velocity, being di minished into nothing, vanishes; that is, the whole sector AD/ is as the whole time of ascent to the highest place. Q.E.D. CASE 2. Draw DQV cutting off the least particles TDV and PDQ of the sector DAV, and of the triangle DAQ ; and these particles will be to each other as DT2 to DP2, that is (if TX and AP are parallel), as DX2 to DA2 or TX2 to AP2 ; and, by division, as DX2 — TX2 to DA2 AP2. But. from the nature of the hyperbola, DX2 — TX2 is AD2 ; and, by the supposition, AP2 is AD X AK. Therefore the particles are to each other as AD2 to AD2 — AD X AK ; that is, as AD to AD — AK or AC ; and to CK : and therefore the particle TDV of the sector is PQ therefore (because AC and AD are given) as CK that is, as the increment of the velocity directly, and as the force generating the increment inverse ly ; and therefore as the particle of the time answering to the increment. And, by composition, the sum of the particles of time, in which all the par ticles PQ of the velocity A I3 are generated, will be as the sum of the par ticles of the sector ATI) ; that is, the whole time will be as the whole sector. Q.E.D. COR. 1. Hence if AB be equal to a fourth part of AC, the space which a body will describe by falling in any time will be to the space which the body could de scribe, by moving uniform]}' on in the same time with its greatest velocity AC, as the area ABNK, which ex presses the space described in falling to the area ATD, which expresses the time. For since AC is to AP as AP _ to AK, then (by Cor. 1, Lem. II, of this Book) LK is to PQ as 2AK to AP, that is, as 2AP to AC, and thence LK is to ^PQ as AP to JAC or AB ; and KN is to AC or AD as AB tc