Page:Newton's Principia (1846).djvu/272

(because tD is given), as $\scriptstyle {\frac {qDp}{pD^{2}}}$. But pD² is AD² + Ap², that is, AD² + AD $\scriptstyle \times$ Ak, or AD $\scriptstyle \times$ Ck; and qDp is ½AD $\scriptstyle \times$ pq. Therefore tDv, the particle of the sector, is as $\scriptstyle {\frac {pq}{Ck}}$; that is, as the least decrement pq of the velocity directly, and the force Ck which diminishes the velocity, inversely; and therefore as the particle of time answering to the decrement of the velocity. And, by composition, the sum of all the particles tDv in the sector ADt 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 diminished into nothing, vanishes; that is, the whole sector ADt 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 $\scriptstyle \times$ AK. Therefore the particles are to each other as AD² to AD² - AD $\scriptstyle \times$ AK; that is, as AD to AD - AK or AC to CK: and therefore the particle TDV of the sector is $\scriptstyle {\frac {PDQ\times AC}{CK}}$; and therefore (because AC and AD are given) as $\scriptstyle {\frac {PQ}{CK}}$; that is, as the increment of the velocity directly, and as the force generating the increment inversely; 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 particles PQ of the velocity AP are generated, will be as the sum of the particles of the sector ATD; 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 describe, by moving uniformly on in the same time with its greatest velocity AC, as the area ABNK, which expresses 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 ½PQ as AP to ¼AC or AB; and KN is to AC or AD as AB to