But, since the law of centripetal force employed by nature is to be discovered from its symptoms, the indisputably elliptical orbit and the sesquialteral ratio of the periodic times and the distances from the centre of forces, the same great Newton solved not only the universal problem of determining the trajectory and the motion in it for any given centripetal force, but also its converse. After this universal problem had been solved the sequel was to find other [quantities] in the geometric figure that are measures of physical qualities; for example, that the periodic times in ellipses are in the sesquiplicate ratio of the transverse axes [the squares of the times are as the cubes of the axes], and as many other things similar to these as possible. Also, for instance, to compare this force, which we experience in the planets, with another given force near to us, namely gravity. But also the new philosophy was to concern itself with movable elliptical orbits, in which the line of apsides either advances or retires. Also, for instance, a more exact [theory] of rectilinear descent and of the motion of pendulous bodies than the Huygenian one, since that supposes the centre to be infinitely removed. Therefore also, other s different from the common one and variously devised according as the pendulum oscillates inside or outside the surface of the Earth. And let that suffice for this problem. But also on account of the mutual actions of bodies moving around a centre the orbits usually turn out to be deformed, and also an investigation of these actions and of the deformity arising from them, whence arise many minor inequalities of the planets, such as the motion of the nodes, the variation of maximum latitude, and other things in the moon.
Scottish astronomer and mathematician (1659-1708)
David Gregory (originally spelt Gregorie) FRS (3 June 1659 – 10 October 1708) was a Scottish mathematician and astronomer. He was professor of mathematics at the , and later at the University of Oxford, and a proponent of Isaac Newton's .
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Alternative Names:
David Gregorie
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Mr Issac Newton in addition to the geometric figure in any orbit of a projectile sought also to find the measure of the (tending to a given centre) of the body borne in that orbit, from whatever cause that force may arise, be it from a deeper mechanical one or from a law imposed by the supreme creator of all things. He inquires geometrically into the law of centripetal force of a body moved in the circumference of a circle with the force tending to a given point either on the circumference or anywhere outside it or inside it, or even infinitely removed. By the same method he seeks the law of centripetal force tending to the centre of a plane nautical spiral (that is one that the radii cut in a given angle) which will drive a body in that spiral. Also the law of centripetal force that would make a body rotate in an ellipse when the centre of the ellipse coincides with the centre of forces. If the ellipse is changed into a hyperbola and the centripetal force into a centrifugal one the same things apply to the hyperbola. Also the resolution of the same problem when the centre of forces coincides with either focus of the ellipse shows that the law of centripetal force is reciprocally in the duplicate ratio of the distance [as the inverse square of the distance]; others had long before shown that this was the one and only law that would satisfy the other phenomenon observed by Kepler in the motion of the planets. These results also apply to the hyperbola and the parabola when the centre of forces is situated in a focus of the conic section.
Since two or more mutually gravitating bodies describe orbits around a common immobile centre of gravity, and since by common consent there is an immense difference between the quantity of matter in the sun and that in the Earth, it is clear that neither the sun nor, much less, the sun in the company of five planets can revolve around an immobile Earth. Thus is shown not only the falsity but the impossibility of the .
[W]e have come into the age where questions that were once cosmographical are being transformed into geometrical problems. For it has now been shown not only that the areas which bodies driven in a circuit describe by the radii drawn to the centre of forces are in immobile planes and proportional to the times, but also conversely that every body moved in this way is impelled by a centripetal force that tends towards the aforesaid point. By this proposition alone the Ptolemaic system is destroyed, for the primary planets by radii drawn to the Earth describe areas in no way proportional to the times, while with the radii drawn to the sun it is established that they run over areas sufficiently proportional to the times.
After Kepler’s bold and fruitful efforts to advance natural philosophy by the help of geometry, there should have appeared any philosopher and particularly a geometer, namely Descartes, who should leave this one narrow path and try to investigate the causes of things logically, or rather, sophistically. What is to be said of him who while certainly learned in geometry would build his cosmic system (which he valued so highly and of which he boasted so grandiloquently) from vortices, without previously examining whether bodies carried around by a vortex at different distances from the centre would have periodic times whose squares were as the cubes of the distances from the centre? But he was intoxicated by easier and less composite laws, and, not applying his geometric ability in the slightest, fell into errors from which we were at length liberated by the aid of geometers.
After great and fruitful efforts both in the purer geometry and the more intricate and complex physics, the most skilful geometer Sir Christopher Wren, who among other luminaries of the University of Oxford graced this professorship, solved the following problem: To find the law of gravity or centripetal force by which several bodies moved around a common centre of forces are driven, given that the squares of the periodic times are as the cubes of the radii, as was observed by Kepler for the planets moved around the sun. The most renowned Wren found that the required law of gravity was such that the centripetal forces were reciprocally as the squares of the distances from the centre of forces, and that no other law would agree with what was observed.
[G]lory has been reserved to our era and to the English people, who since the instauration of the sciences have made such advances... And passing over the immense labours undergone by the most fruitful astronomers of our people... [H]ow easy and how exact... how geometrical, astronomy has been left to us by that most acute geometer... or astronomer, the Right Reverend Dr Seth sometime Bishop of Salisbury, who while he was among men adorned this chair. How geometrically and acutely he determined the positions and species of the orbit and other related matters, following Kepler and substituting as mean motion the angle at the other focus (which he accordingly called that of the mean motion) in place of the areas to the sun that the radius vector describes and as it were sweeps out. Content with this artifice he did not detain himself over the solution of Kepler’s problem, in which the division of the area of an ellipse in a given ratio by a straight line through a focus is required. But, being a most perspicacious man, he was conscious of what delays arose hence in the construction of tables, and, in order to show the world that astronomy was to be advanced by the help of geometry whatever hypotheses it depended upon, he accomplished the same astronomical problems geometrically from the circular hypothesis.
[W]hat sharpness of mind was employed by John Kepler... when, from there being just five regular solids... he inferred that the number of the planets was six, and by inscription of spheres within these solids and circumscription of spheres around them related the distances and ratios of the orbits. It can scarcely be said with what power of prophecy and by what labours he succeeded in arriving at that great theorem of the elliptical planetary orbits with a common focus at the sun... in such a way that the areas that the radius vector of the planet from the sun traverses are proportional to the times. Nevertheless... so great a man... owned himself unequal to... solving directly the problem of determining for a given time the place of the planet in the elliptical orbit. Here geometry, his goddess-mother, was of no avail... But... he brought forward a conjecture of great use, namely, that the squares of the periodic times are in the same ratio as the cubes of the distances between the planets and the sun. Finally, he discovered a marvellous property of bodies by which in the minimally resisting ether they seek each other and as it were attract. From this he also deduced the tides in a clear but brief discourse in his immortal Commentaries on the star Mars, and was as it were a prophet and a precursor of a great geometer born among the English.
In the past many very base Remus’s leapt over the walls of the astronomical city, but now the geometers have so fortified it with a ditch and a rampart that the portals of the sun receive those whom impartial Appollonius has loved and whom Kepler, Wren, Wallis and Newton have borne to the aetherial regions, and accordingly the profane, that is ungeometrical men, are exiled and depart from the grove and wander away over the whole heaven.
[W]ithin the memory of ourselves and... our fathers, philosophers began to extend the limits of geometry in order to found the kingdom of astronomy. This they have carried out... with such success that now no one can be received into astronomical citizenship who is not a visiting citizen in the most abstruse geometry and has not arisen from the patrician, that is the geometrical, family of philosophers.
Although in every age there have been those who cultivated astronomy, either by... observations... or by theories and systems made up according to the state of understanding of any period, or by a talent for exposition, yet the lucubrations of all these astronomers do not reveal the ways of the heaven any more than they reveal the skill and experience of their progenitors in geometrical matters.