It may not be uninstructive to subjoin a sketch of the manner in which Newton established the same celebrated results, starting from principles of motion diametrically opposed to Kepler's, and it need scarcely be added, reasoning upon them in a manner not less different. For this purpose, a very few prefatory remarks will be found sufficient.
The different motions seen in nature are best analysed and classified by supposing that every body in motion, if left to itself, will continue to move forward at the same rate in a straight line, and by considering all the observed deviations from this manner of moving, as exceptions and disturbances occasioned by some external cause. To this supposed cause is generally given the name of Force, and it is said to be the first law of motion, that, unless acted on by some force, every body at rest remains at rest, and every body in motion proceeds uniformly in a straight line. Many employ this language, without perceiving that it involves a definition of force, on the admission of which, it is reduced to a truism. We see common instances of force in a blow, or a pull from the end of a string fastened to the body: we shall also have occasion presently to mention some forces where no visible connexion exists between the moving body and that towards which the motion takes place, and from which the force is said to proceed.
_c_ C +-------------+ \ / \ \ / \ \ / \ \ / \ \ / \ \ / \ +-------------+ B C' ]
A second law of motion, founded upon experiment, is this: if a body have motion communicated to it in two directions, by one of which motions alone it would have passed through a given space in a given time, as for instance, through BC´ in one second, and by the other alone through any other space B_c_ in the same time, it will, when both are given to it at the same instant, pass in the same time (in the present instance in one second) through BC the diagonal of the parallelogram of which BC´ and B_c_ are sides.
/ S \ / /|\ \ / / | \ \ / / | \ \ / / | \ \ / / | \ \ / / | \ \ --------+-+------+------+-+ A B C D E ]
Let a body, acted upon by no force, be moving along the line AE; that means, according to what has been said, let it pass over the equal straight lines AB, BC, CD, DE, &c., in equal times. If we take any point S not in the line AE, and join AS, BS, &c., the triangles ASB, BSC, &c. are also equal, having a common altitude and standing on equal bases, so that if a string were conceived reaching from S to the moving body (being lengthened or shortened in each position to suit its distance from S), this string, as the body moved along AE, would sweep over equal triangular areas in equal times.