Volume 1

The mathematical principles of natural philosophy / by Sir Isaac Newton; translated into English by Andrew Motte: to which are added, Newtonʼs System of the world; a short comment on and defense of, the Principia, by William Emerson; with the laws of the moonʼs motion according to gravity, by John Machin.

Newton, Isaac, 1642-1727. Principia. English

Date:: 1819

Credit: The mathematical principles of natural philosophy / by Sir Isaac Newton; translated into English by Andrew Motte: to which are added, Newtonʼs System of the world; a short comment on and defense of, the Principia, by William Emerson; with the laws of the moonʼs motion according to gravity, by John Machin. Source: Wellcome Collection.

9/384 page 3

$Conür. Hence if to the re&angular áfymptotes AC, CH, the hyperbola BG is defcribed, and AB, DG, be drawn per- pendicular to the afymptote AC, and both the velocity of the body; and the refiftance of the medium, at the very beginning of the motion, be expreffed by any &iven line AC, and, after fome timie 1s elápfed, by the indefinite line DC; the time may be éxpreffed by the area ABGD, atid the fpace defcrib- ed in that time by the line AD. For if that area, by the motion of the point D, be uniformly increafed in the fame manner as thé timé, the right line DC will decreafe in a geo- metrical ratio in thé fanie manner asthe velocity; and the parts of the right line AC, deferibed in equal times, will de- creafe in thé fame ratio. PROPOSITION [I]. PROBLEM T. | To define the motion of a body which, in a fimilar medium, üfcends or defcends in a right line, and is refijted in the ratio of its velocity, and acted upon by an dee iil force of gravity. (Pl 1,Fig.2. ^ The body afeénding, let ie gravity be expounded by any eiven rectangle BACH ; and the refiftance of the medium, at the beginning of the aféent, by the re&angle BADE, taken on. thé contrary fide of the right line AB. Through the point B, with the reétanguldr afymptotes AC, CH, defcribe an Hy Hee bola, cutting the perpéndiculars DE, de, in G, g; and the body afcending will if the time DGgd defcribe the fpace EGse ; in the tirié DGBA, the {pace of the whole afcent EGB ; in the time ABKI, the fpace of defeent BFK; and in the time IKki the fpace of. defcent KFfk ; and the velocities of the bodies (proportional to the refiftance of the medium) in thefe periods of time will be ABED, ABed, 0, ABFI, ABfi, re- fpectively ; and the greateft velocity which the body can ac- quire by defcending will be BACH. .— For let the reétangle BACH (Pl. 1, Fig. 3) be refolved into jnnumierable ré&angles Ak, Kl, ia, Mi, &c. which fhall - be as the increments of the velocities produced in fo many — equal times; then will 0; Ak; Al, Am, An, &c. bé as the . whole velocities, and therefore (by fuppofition) as the refift- ances of the medium in the beginning of each of the equal$

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