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use of, and distinguishes time, space, place, and motion, into absolute and relative, real and apparent, mathematical and vulgar: showing the necessity of such distinction. To these definitions are subjoined the laws of motion, with several corollaries from them; as concerning the composition and resolution of any direct force out of, or into any oblique forces, by which the powers of all sorts of mechanical engines are demonstrated; the laws of the reflection of bodies in motion after their collision; and the like.
These necessary præcognita being delivered, our author proceeds to consider curves generated by the composition of a direct impressed motion with a gravitation or tendency towards a centre : and having demonstrated that in all cases the areas at the centre, described by a revolving body, are proportional to the times, he shows how, from the curve described, to find the law or rule of the decrease or increase of the tendency or centripetal forces as he calls it, in different distances from the centre.
of this there are several examples : as, if the curve described be a circle passing through the centre of tendency ; then the force or tendency towards that centre is in all points as the fifth power, or squared-tube, of the distance from it reciprocally: if in the proportional spiral, reciprocally as the cube of the distance : if in an ellipse about the centre of it, directly as the distànce. If in any of the conic sections about the focus, then he demonstrates that the vis centripeta, or tendency towards that focus, is in all places reciprocally as the square of the distance from it; and that according to the velocity of the impressed motion, the curve described is an hyperbola; if the body moved be swift to a certain degree, then a parabola; if slower, an ellipse, or a circle in one case. From this sort of tendency or gravitation it follows, likewise, that the squares of the times of the periodical revolutions are as the cubes of the radii or transverse axes of the ellipses.
All which being found to agree with the phænomena of the celestial motions, as discovered by the great sagacity and diligence of Kepler, our author extends himself upon the consequences of this sort of vis centripeta ; showing how to find the conic section which a body shall describe when projected with any velocity in a given line, supposing the quantity of the said force known: and laying down several neat constructions to determine the orbs, either from the focus given, and two points or tangents; or, without it, by five points or tangents, or any number of points and tangents, making together five. He then shows how, from the time given, to find the point in a given orbit answering to it; which he performs accurately in the parabola, and, by concise approximations, comes as near as he pleases in the ellipse and hyperbola : all which are problems of the highest concern in astronomy.
Next he lays down the rules of the perpendicular descent of bodies towards the centre, particularly in the case where the tendency to it is reciprocally as the square of the distance; and generally in all other cases, supposing a general quadrature of curve lines : upon which supposition, likewise, he delivers a general method of discovering the orbits described by a body moving in such a tendency towards a centre, increasing or decreasing in any given relation to the distance from the centre ; and then with great subtilty he determines in all cases the motion of the apses, or of the points of greatest distance from the centre, in all these curves, in such orbits as are nearly circular. Showing the apses fixed, if the tendency be reciprocally as the square of the distance ; direct in motion, in any ratio between the square and the cube ; and retrograde, if under the square : which motion he determines exactly from the rule of the increase or decrease of the vis centripeta.
Next the motion of bodies in given surfaces is considered, as likewise the oscillatory motion of pendules ; where it is shown how to make a pendulum vibrate always in equal times, though the centre or point of tendency be never so near ; to which, the demonstration of Mr. Huygens de Cycloide is but a corollary. And in another proposition is shown the velocity in each point, and the time spent in each part of the arch described by the vibrating body. After this, the effects of two or more bodies, towards each of which there is a tendency, is considered ; and it is made out that two bodies, so drawing or attracting each other, describe about the common centre of gravity curve lines, like to those they seem to describe about each other. And of three bodies, attracting each other, reciprocally as the square of the distance between their centres, the various consequences are considered and laid down, in several corollaries of great use in explaining the phænomena of the moon's motions, the flux and reflux of the sea, the precession of the equinoctial points, and the like.
This done, our author, with his usual acuteness, proceeds to examine into the causes of this tendency or centripetal force, which, from undoubted arguments, is shown to be in all the great bodies of the universe. Here he finds that if a sphere be composed of an infinity of atoms, each of which have a power which decreases in duplicate proportion of the distance between them ; then the whole congeries shall have the like tendency towards its centre, decreasing, in spaces without it, in duplicate proportion of the distances from the centre; and decreasing within its surface, as the distance from the centre directly; so as to be greatest on the surface, and nothing at the centre : and though this might suffice, yet to complete the argument, there is laid down a method to determine the forces of globes composed of particles whose tendencies to each other decrease in any other ratio of the distances; which speculation is carried on likewise to other bodies not spherical, whether finite or indeterminate. Lastly, is proposed a method of explaining the refractions and reflections of transparent bodies, from the same principles; and several problems solved of the greatest concern in the art of dioptrics.
Hitherto our author has considered the effects of compound motions in non-resisting media, or wherein a body once in motion would move equally in a direct line, if not diverted by a supervening attraction or tendency towards some other body. Here is demonstrated what would be the consequence of a resistance from a medium, either in the simple or duplicate ratio of the velocity, or else between both: and to complete this argument, is laid down a general method of determining the density of the medium in all places, which, with a uniform gravity tending perpendicularly to the plane of the horizon, shall make a project move in any curve line assigned; which is the 10th prop. lib. 2. Then the circular motion of bodies in resisting media is determined, and it is shown under what laws of decrease of density the circle will become a proportional spiral. Next, the density and compression of Auids is considered, and the doctrine of hydrostatics demonstrated; and here it is proposed to the contemplation of natural philosophers, whether the surprising phænomena of the elasticity of the air, and some other fluids, may not arise from their being composed of particles which fly each other ; which being rather a physical than mathematical enquiry, our author forbears to discuss.
Next, the opposition of the medium, and its effects on the vibrations of the pendulum, are considered, which is followed by an enquiry into the rules of the opposition to bodies, as their bulk, shape, or density may be varied : here with great exactness is an account given of several experiments tried with pendula, in order to verify the foregoing speculation,
and to determine the quantity of the air's opposition to bodies moving in it.
From hence he proceeded to the undulation of fluids, the laws whereof are here laid down, and by them the motion and propagation of light and sound are explained. The last section of this book is concerning the circular motion of fluids, wherein the nature of their vortical motions is considered ; and from thence the Cartesian doctrine of the vortices of the celestial matter carrying with them the planets about the sun is proved to be altogether impossible.
The third and last book is entitled Of the System of the World, wherein the demonstrations of the two former books are applied to the explication of the principal phænomena of nature: here the verity of the hypothesis of Kepler is demonstrated; and a full resolution given to all the difficulties that occur in the astronomical science; they being nothing else but the necessary consequences of the sun, earth, moon, and planets, having all of them a gravitation or tendency towards their centres proportional to the quantity of matter in each of them, and whose force abates in duplicate proportion of the distance reciprocally.
Here, likewise, are indisputably solved the appearances of the tides, or flux and reflux of the sea ; and the spheroidical figure of the earth and Jupiter determined, from which the precession of the equinoxes, or rotation of the earth's axis, is made out, together with the retrocession of the moon's nodes, the quantity and inequalities of whose motion are here exactly stated à priore. Lastly, the theory of the motion of comets is attempted with such success, that in an example of the great comet which appeared in 1689, its motion is computed as exactly as we can pretend to give the places of the primary planets; and a general method is here laid down to state and determine the trajectorius of comets, by an easy geometrical construction; upon supposition that those curves are parabolic, or so near it that the parabola may serve without sensible error ; though it be more probable, says our author, that these orbits are elliptical, and that after long periods comets may return again. But such ellipses are, by reason of the immense distance of the foci, and smallness of the latus rectum, in the parts near the sun where comets appear, not easily distinguished from the curve of the parabola ; as is proved by the example produced.
The whole book is interspersed with lemmas of general use in geometry, and several new methods applied, which are well worth the considering ; and it may be justly said, that so
many and so valuable philosophical truths, as are herein discovered and put past dispute, were never yet owing to the capacity and industry of any one man whatever.
On the Effects of a Burning Speculum, lately made in
Germany. - [1687.] The outer circle of this concave burning speculum is near three Leipsic ells in diameter, and was made of a copper plate scarcely twice as thick as the back of an ordinary knife; and may therefore be easily removed from place to place, and ordered for use. The polish of it is very good, and represents, by distinct reflections, all those appearances which arise from its concave figure.
The force of this speculum in burning is incredible. For, 1. A piece of wood, put in the focus, flames in a moment, so as a fresh wind can hardly put it out. 2. Water, applied in an earthen vessel, presently boils ; and the vessel being held there some time, the water evaporates all away. 3. A piece of tin, or lead three inches thick, as soon as it is put in the focus, melts away in drops, and held there a little time is in a perfect fluor, so as in two or three minutes to be quite pierced through. 4. A plate of iron or steel placed in the focus is immediately seen to be red hot; and soon after a hole is burnt through. 5. Copper, silver, and the like, applied to the focus, melt in five or six minutes. 6. Things not apt to melt, as stones, brick, and the like, soon become red hot like iron. 7. Slate at first is red hot, but in a few minutes turns into a fine sort of black glass, of which if any part be taken in the tongs, and drawn out, it runs into glass threads. 8. Tiles which had suffered the most intense heat of fire, in a little time melt down into a yellow glass; as do, 9. Pot-shreads, not only well burnt at first, but much used in the fire, into a blackish-yellow glass. 10. Pumice-stone, said to be that of burning mountains, in this solar fire, melts into a white transparent glass. 11. A piece of a very strong crucible put in the focus in eight minutes was melted into a glass. 12. Bones turned into a kind of opake glass, and a clod of earth into a yellow or greenish glass.
It was tried what effect the ms of the full moon, concentred with this speculum, would have, at the time when she was at her greatest altitude; but there was not found any degree of heat, though the light was much increased.