Philosophia Naturalis Principia Mathematica, Autore Is. Newton Trin. Coll. Contab. Soc. Matheseos Professore Lucasiano, & Societatis Regalis Sodali. 4to. Londini. Prostat apud plures Bibliopolas.
This incomparable Author having at length been prevailed upon to appear in publick, has in this Treatise given a most notable instance of the extent of the powers of the Mind; and has at once shewn what are the Principles of Natural Philosophy, and so far derived from them their consequences, that he seems to have exhausted his Argument, and left little to be done by those that shall succeed him. His great skill in the old and new Geometry, helped by his own improvements of the latter, (I mean his method of infinite Series) has enabled him to master those Problems, which for their difficulty would have still lain unresolved, had one less qualified than himself attempted them.
This Treatise is divided into three Books, whereof the two first are entitled de Motu Corporum, the third de Systemate Mundi.
The first begins with definitions of the Terms made use of, and distinguishes Time, Space, Place and Moffon into absolute and relative, real and apparent, Mathematical and vulgar: shewing the necessity of such distinction. To these definitions are subjoyned, the Laws of Motion with several Corollaries therefrom; as concerning the composition and resolution of any direct force out of, or into any oblique forces, (whereby 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 Praecognita being delivered, our Author proceeds to consider the Curves generated by the composition of a direct impressed motion with a gravitation or tendency towards a Center: and having demonstrated that in all cases the Areas at the Center, described by a revolving Body, are proportional to the Times; he shews how from the Curve described, to find the Law or Rule of the decrease or increase of the Tendency or Centripetal forces (as he ca]ls it) in differing distances from the Center. Of this there are several examples: as if the Curve described be a Circle passing through the Center of tendency; then the force or tendency towards that Center is in all points as the fifth power or squared-cube of the distance therefrom reciprocally. If in the proportional Spiral, reciprocally as the cube of the distance. If in an Ellipse about the Center thereof directly as the distance. If in any of the Conick Sections about the Focus thereof; then he demonstrates that the Vis Centripeta, or tendency towards that Focus, is in all places reciprocally as the square of the distance therefrom; 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 ~llipse or 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 ~llipses. All which being found to agree with the Phenomena 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; shewing how to find the Conick Section which a Bodie shall describe when cast 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. Then he shews how from the Time given to find the Point in a given Orb answering thereto; 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 Center, particularly in the case where the tendency thereto is reciprocally as the square of the distance; and generally in all other cases, supposing a general quadrature of Curve lines: upon which supposion likewise he delivers a general method of discovering the Orbs described by a Body moving in such a tendency towards a Center, increasing or decreasing in any given relation to the distance from the Center; and then with great subtilty he determines in all cases the Motion of the Apsides (or of the Points of greatest distance from the Center in all these Curves, in such Orbs as are nearly Circular. Shewing the Apsides fixt, if the tendency be reciproca]ly 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 is shewn how to make a Pendulum Vibrate always in equal times, tho’ the center or point of tendency be never so near; to which, the Demonstration of Mr. Hugens de Cycloide is but a Coro]]~ry. And in another Proposition is shewn 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 ’tis made out that two Bodies, so drawing or attracting each other, describe about the common center of Gravity, Curve Lines, like to those they seem to describe about one another. And of three Bodies, attracting each other, reciproca]ly as the Square of the distance between their Centers, the various Consequences are considered and laid down, in several Corollarys of great use in explicating the Phenomena of the Moons 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 Conatus accendendi ad invicem, which decreases in duplicate Proportion of the Distance between them; then the whole Congeries shall have the like tendency towards its Center, decreasing, in Spaces without it, in duplicate Proportion of the Distances from the Center; and decreasing, within its Surface, as the distance from the Center directly; so as to be greatest on the Surface, and nothing at the Center: and tho’ this might suffice, yet to compleat the Argument, there is laid down a Method to determine the forces of Globes composed of Particles whose Tendencies to each other do decrease in any other Ratio of the Distances: Which Speculation is carryed 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 Dioptricks.
Hitherto our Author has considered the Effects of compound Motions in Mediis non resistentibus, or wherein a Body once in Motion would move equably in direct Line, if not diverted by a supervening Attraction or tendency toward some other Body. Here is demonstrated what would be the consequence of a resistence from a Medium, either in the simple or duplicate Ratio of the Velocity, or else between both: and to compleat 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 plain of the Horizon, shall make a Project move in any curve Line assigned; which is the 10th. Prop. Lib. II. Then the circular Motion of Bodies in resisting Media is determined, and ’tis shown under what Laws of decrease of Density, the Circle will become a proportional Spiral. Next the density and compression of Fluids is considered, and the Doctrine of Hydrostaticks demonstrated; and here ’tis proposed to the Contemplation of Natural Philosophers, whether the surprizing Phenomena of the Elasticity of the Air and some other Fluids may not arise from their being composed of Particles which flie each other; which being rather a Physical than Mathematical Inquiry, our Author forbears to Discuss.
Next the Opposition of the Medium and its Effects on the Vibrations of the Pendulum is considered, which is followed by an Inquiry into the Rules of the Opposition to Bodies, as their Bulk, Shape, or Density may be varyed: Here with great exactness is an Account given of several Experiments tried with Pendula, in order to verify the aforegoing Speculation, and to determine the quantity of the Airs Opposition to Bodies moving in it.
From hence is 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 alltogether impossible.
The III. and last book is entituled de SystemateMundi, wherein the Demonstrations of the two former Books are applyed to the Explication of the principal Phenomena 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 Centers proportionate 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 Earths Axis is made out,) together with the retrocession of the Moons Nodes, the Quantity and inequalities of whose Motion are here exactly stated a 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 1680-1 the Motion thereof 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 Trajector~ of Comets, by an easy Geometrical Construction; upon supposition that those Curves are Parabolick, or so near it that the Parabola may serve without sensible Error; tho’ it be more probabl, saith our Author, that these Orbs 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 Lemma’s of General use in Geometry, and several new Methods applyed, 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.
Philosophical Transactions of the Royal Society of London, Vol. XV, 1685-86, pp. 291-97 [Edmund Halley].