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Commentaries on the Principia of Sir Isaac Newton Respecting His Theory ...
Sin vista previa disponible - 2020
A B D according acting angle applied approach areas bases become body cause centre centrifugal force centripetal forces chord circle circular coincide Commentaries concluding consequences consider continually Corol Corollary deduced demon demonstrated described determined diminished direction directly divided drawn duplicate ratio earth effect enunciated equal evanescent expressed follows further given gravitation greater hypothesis impulse inconsistent inference inversely Kepler's Analogy lemma length less light mean distances meet monstrated motion move nearer planet nearly Newton Newton's theory obvious orbit parallel periodic planets position Principia produced Prop properties proportional Proposition quantities radii radius reasoning represent respect revolving right line says seems sides similar spaces square-roots squares supposed taking tances tangent tion triangles true ultimate ratio vanish velocities versed sines wherefore whole
Página 36 - LEMMA X. THE SPACES WHICH A BODY DESCRIBES BY ANY FINITE FORCE URGING IT, WHETHER THAT FORCE IS DETERMINED AND IMMUTABLE, OR IS CONTINUALLY AUGMENTED OR CONTINUALLY DIMINISHED, ARE IN THE VERY BEGINNING OF THE MOTION ONE TO THE OTHER IN THE DUPLICATE RATIO OF THE TIMES. Let the times be represented by the lines AD, AE, and the velocities generated in those times by the ordinates DB, EC.
Página 26 - BF parallel to the tangent, always cutting any right line AF passing through A in F, this line BF will be ultimately in the ratio of equality with the evanescent arc ACB; because, completing the parallelogram AFBD, it is always in a ratio of equality with AD. COR. 2. And if through B and A more right lines are drawn, as BE, BD, AF, AG, cutting the tangent AD and its parallel BF; the ultimate ratio of all the abscissas AD, AE, BF, BG, and of the chord and arc AB, any one to any other, will be the...
Página 47 - SAFS, of those areas, are one to the other as the times in which they are described. Now let the number of those triangles be augmented, and their breadth diminished in infinitum; and (by Cor.
Página 45 - THE AREAS, WHICH REVOLVING BODIES DESCRIBE BY RADII DRAWN TO AN IMMOVABLE CENTRE OF FORCE DO LIE IN THE SAME IMMOVABLE PLANES, AND ARE PROPORTIONAL TO THE TIMES IN WHICH THEY ARE DESCRIBED.
Página 46 - E, &c., and makes the body, in each single particle of time, to describe the right lines CD, DE, EF, &c. , they will all lie in the same plane; and the triangle SCD will be equal to the triangle SBC, and SDE to SCD, and SEF to SDE. And therefore, in equal times, equal areas are described in one immovable plane: and, by composition, any sums SADS, SAFS, of those areas, are one to the other as the times in which they are described.
Página 27 - RAD, and the points A and B approach and meet, I say that the ultimate form of these evanescent triangles is that of similitude, and their ultimate ratio that of equality.
Página 91 - That the squares of the times of the revolutions of the planets are as the cubes of their mean distances from the sun.
Página 46 - THE TIMES IN WHICH THEY ARE DESCRIBED. For suppose the time to be divided into equal parts, and in the first part of that time let the body by its innate force describe the right line AB. In the second part of that time, the same would (by Law...
Página 82 - ... 72333, 38710. Their distances, according to Kepler and Bullialdus, scarcely differ by any sensible quantity, and where they differ most the distances drawn from the periodic times fall in between them. That the circum-terrestrial force likewise decreases in the duplicate proportion of the distances, I infer thus. The mean distance of the moon from the centre of the earth, is, in semi-diameters of the earth, according to...