Thus geometry is founded upon mechanical procedure, and is nothing else but that part of universal mechanics that accurately sets forth and demonstrates the art of measuring. (查看原文)
For the whole difficulty of philosophy appears to turn upon this: that from the phenomena of motion we may investigate the forces of nature, and then from these forces we may demonstrate the rest of the phenomena. (查看原文)
Quantities, as well as the ratios of quantities, which in any finite time you please constantly tend towards equality, and before the end of that time approach nearer to each other than by any given difference you please, become ultimately equal.
If you deny it, let them become ultimately unequal, and let their ultimate difference be D. Therefore, they cannot approach nearer to each other than by the given difference D contrary to the hypothesis. (查看原文)
The hypothesis of indivisibles is harder, and that method is accordingly thought to be less geometrical....I don't wish these to be understood as indivisibles, but always as divisible evanescents, not as the sums and ratios of determinate parts, but always as the limits of sums and ratios..... (查看原文)
All the sides of similar figures that correspond to each other mutually, curvilinear as well as rectilinear, are proportional, and the areas are in duplicate ratio of the sides. (查看原文)
If any arc ACB given in position be subtended by a chord AB, and at some point A, in the middle of continuous curvature, it be touched by a straight line AD produced by a straight line AD produced both ways, and then if points A, B approach each other and come together, I say that the angle BAD, contained by the chord and the tangent, is diminished in infinitum and ultimately vanishes. (查看原文)
For if that angle does not vanish, the arc ACB will contain with the tangent AD an angle equal to a rectilinear angle, and for that reason the curvature at point A will not be continuous, contrary to the hypothesis. (查看原文)
If given straight lines AR, BR, together with the arc ACB, the chord AB, and the tangent AD, form the there triangles RAV, RACB, RAD, and then the points A and B approach each other, I say that the ultimate form of the evanescent triangles is similarity, and the ultimate ratio equality. (查看原文)
For while point B approaches point A, let AB, AD, AR, be understood to be extended to the distant points b,d, and r, and rbd to be drawn parallel to RD, and let the arc Acb be always similar to the arc ACB. And when the points A, B come together, the angle bAd will vanish, and therefore the three triangles rAb, rAcb, rAd, always finite, will coincide and for that reason are similar and equal. Hence, RAB, RACB, RAD, always similar and proportional to these, will ultimately become similar and equal to each other. (查看原文)
The space which a body describes when any finite force whatever urges [it], whether that force be determinate and immutable, or whether it be continually increased or continually decreased, are at the very beginning of the motion in the duplicate ratio of the times. (查看原文)
And hence it is easily deduced that when bodies describe similar parts of similar figures in proportional times, the deviations that are generated by any equal forces you please similarly applied to the bodies, and that are measured by the distanced of the bodies from those places of the similar figures to which the same bodies would have arrived in the same proportional times without those forces, are as the squares of the times in which they are generated, very nearly. (查看原文)
