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It opens with a mathematical exposition of "the method of first and last ratios",  a geometrical form of infinitesimal calculus. The second section establishes relationships between centripetal forces and the law of areas now known as Kepler's second law Propositions 1—3 ,  and relates circular velocity and radius of path-curvature to radial force  Proposition 4 , and relationships between centripetal forces varying as the inverse-square of the distance to the center and orbits of conic-section form Propositions 5— Propositions 11—31  establish properties of motion in paths of eccentric conic-section form including ellipses, and their relation with inverse-square central forces directed to a focus, and include Newton's theorem about ovals lemma Propositions 43—45  are demonstration that in an eccentric orbit under centripetal force where the apse may move, a steady non-moving orientation of the line of apses is an indicator of an inverse-square law of force.
Book 1 contains some proofs with little connection to real-world dynamics.
But there are also sections with far-reaching application to the solar system and universe:. Propositions 57—69  deal with the "motion of bodies drawn to one another by centripetal forces". This section is of primary interest for its application to the Solar System , and includes Proposition 66  along with its 22 corollaries:  here Newton took the first steps in the definition and study of the problem of the movements of three massive bodies subject to their mutually perturbing gravitational attractions, a problem which later gained name and fame among other reasons, for its great difficulty as the three-body problem.
Propositions 70—84  deal with the attractive forces of spherical bodies.
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The section contains Newton's proof that a massive spherically symmetrical body attracts other bodies outside itself as if all its mass were concentrated at its centre. This fundamental result, called the Shell theorem , enables the inverse square law of gravitation to be applied to the real solar system to a very close degree of approximation. Part of the contents originally planned for the first book was divided out into a second book, which largely concerns motion through resisting mediums.
Just as Newton examined consequences of different conceivable laws of attraction in Book 1, here he examines different conceivable laws of resistance; thus Section 1 discusses resistance in direct proportion to velocity, and Section 2 goes on to examine the implications of resistance in proportion to the square of velocity. Book 2 also discusses in Section 5 hydrostatics and the properties of compressible fluids; Newton also derives Boyle's law. Newton compares the resistance offered by a medium against motions of globes with different properties material, weight, size.
In Section 8, he derives rules to determine the speed of waves in fluids and relates them to the density and condensation Proposition 48;  this would become very important in acoustics. He assumes that these rules apply equally to light and sound and estimates that the speed of sound is around feet per second and can increase depending on the amount of water in air.
Less of Book 2 has stood the test of time than of Books 1 and 3, and it has been said that Book 2 was largely written on purpose to refute a theory of Descartes which had some wide acceptance before Newton's work and for some time after. According to this Cartesian theory of vortices, planetary motions were produced by the whirling of fluid vortices that filled interplanetary space and carried the planets along with them. Book 3, subtitled De mundi systemate On the system of the world , is an exposition of many consequences of universal gravitation, especially its consequences for astronomy.
It builds upon the propositions of the previous books, and applies them with further specificity than in Book 1 to the motions observed in the Solar System. Here introduced by Proposition 22,  and continuing in Propositions 25—35  are developed several of the features and irregularities of the orbital motion of the Moon, especially the variation.
Newton lists the astronomical observations on which he relies,  and establishes in a stepwise manner that the inverse square law of mutual gravitation applies to Solar System bodies, starting with the satellites of Jupiter  and going on by stages to show that the law is of universal application. In Book 3 Newton also made clear his heliocentric view of the Solar System, modified in a somewhat modern way, since already in the mids he recognised the "deviation of the Sun" from the centre of gravity of the Solar System. The sequence of definitions used in setting up dynamics in the Principia is recognisable in many textbooks today.
Newton first set out the definition of mass. The quantity of matter is that which arises conjointly from its density and magnitude. A body twice as dense in double the space is quadruple in quantity.
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This quantity I designate by the name of body or of mass. This was then used to define the "quantity of motion" today called momentum , and the principle of inertia in which mass replaces the previous Cartesian notion of intrinsic force. This then set the stage for the introduction of forces through the change in momentum of a body. Curiously, for today's readers, the exposition looks dimensionally incorrect, since Newton does not introduce the dimension of time in rates of changes of quantities.
He defined space and time "not as they are well known to all". Instead, he defined "true" time and space as "absolute"  and explained:.
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Only I must observe, that the vulgar conceive those quantities under no other notions but from the relation they bear to perceptible objects. And it will be convenient to distinguish them into absolute and relative, true and apparent, mathematical and common. To some modern readers it can appear that some dynamical quantities recognised today were used in the Principia but not named.
The mathematical aspects of the first two books were so clearly consistent that they were easily accepted; for example, Locke asked Huygens whether he could trust the mathematical proofs, and was assured about their correctness. However, the concept of an attractive force acting at a distance received a cooler response. In his notes, Newton wrote that the inverse square law arose naturally due to the structure of matter.
However, he retracted this sentence in the published version, where he stated that the motion of planets is consistent with an inverse square law, but refused to speculate on the origin of the law.
Huygens and Leibniz noted that the law was incompatible with the notion of the aether. From a Cartesian point of view, therefore, this was a faulty theory. Newton's defence has been adopted since by many famous physicists—he pointed out that the mathematical form of the theory had to be correct since it explained the data, and he refused to speculate further on the basic nature of gravity.
The sheer number of phenomena that could be organised by the theory was so impressive that younger "philosophers" soon adopted the methods and language of the Principia. Perhaps to reduce the risk of public misunderstanding, Newton included at the beginning of Book 3 in the second and third editions a section entitled "Rules of Reasoning in Philosophy. The four Rules of the edition run as follows omitting some explanatory comments that follow each :.
Rule 1: We are to admit no more causes of natural things than such as are both true and sufficient to explain their appearances. Rule 2: Therefore to the same natural effects we must, as far as possible, assign the same causes. Rule 3: The qualities of bodies, which admit neither intensification nor remission of degrees, and which are found to belong to all bodies within the reach of our experiments, are to be esteemed the universal qualities of all bodies whatsoever.
Rule 4: In experimental philosophy we are to look upon propositions inferred by general induction from phenomena as accurately or very nearly true, not withstanding any contrary hypothesis that may be imagined, till such time as other phenomena occur, by which they may either be made more accurate, or liable to exceptions. This section of Rules for philosophy is followed by a listing of 'Phenomena', in which are listed a number of mainly astronomical observations, that Newton used as the basis for inferences later on, as if adopting a consensus set of facts from the astronomers of his time.
Both the 'Rules' and the 'Phenomena' evolved from one edition of the Principia to the next. Rule 4 made its appearance in the third edition; Rules 1—3 were present as 'Rules' in the second edition, and predecessors of them were also present in the first edition of , but there they had a different heading: they were not given as 'Rules', but rather in the first edition the predecessors of the three later 'Rules', and of most of the later 'Phenomena', were all lumped together under a single heading 'Hypotheses' in which the third item was the predecessor of a heavy revision that gave the later Rule 3.
From this textual evolution, it appears that Newton wanted by the later headings 'Rules' and 'Phenomena' to clarify for his readers his view of the roles to be played by these various statements. The first rule is explained as a philosophers' principle of economy. The second rule states that if one cause is assigned to a natural effect, then the same cause so far as possible must be assigned to natural effects of the same kind: for example respiration in humans and in animals, fires in the home and in the Sun, or the reflection of light whether it occurs terrestrially or from the planets.
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An extensive explanation is given of the third rule, concerning the qualities of bodies, and Newton discusses here the generalisation of observational results, with a caution against making up fancies contrary to experiments, and use of the rules to illustrate the observation of gravity and space. Isaac Newton's statement of the four rules revolutionised the investigation of phenomena.
With these rules, Newton could in principle begin to address all of the world's present unsolved mysteries. He was able to use his new analytical method to replace that of Aristotle, and he was able to use his method to tweak and update Galileo 's experimental method.
The re-creation of Galileo's method has never been significantly changed and in its substance, scientists use it today. The General Scholium is a concluding essay added to the second edition, and amended in the third edition, Here Newton used what became his famous expression Hypotheses non fingo , "I formulate no hypotheses",  in response to criticisms of the first edition of the Principia. Newton's gravitational attraction, an invisible force able to act over vast distances , had led to criticism that he had introduced " occult agencies" into science.
Newton also underlined his criticism of the vortex theory of planetary motions, of Descartes, pointing to its incompatibility with the highly eccentric orbits of comets, which carry them "through all parts of the heavens indifferently".
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Newton also gave theological argument. From the system of the world, he inferred the existence of a Lord God, along lines similar to what is sometimes called the argument from intelligent or purposive design. It has been suggested that Newton gave "an oblique argument for a unitarian conception of God and an implicit attack on the doctrine of the Trinity ",   but the General Scholium appears to say nothing specifically about these matters.
In January , Edmond Halley , Christopher Wren and Robert Hooke had a conversation in which Hooke claimed to not only have derived the inverse-square law, but also all the laws of planetary motion. Wren was unconvinced, Hooke did not produce the claimed derivation although the others gave him time to do it, and Halley, who could derive the inverse-square law for the restricted circular case by substituting Kepler's relation into Huygens' formula for the centrifugal force but failed to derive the relation generally, resolved to ask Newton.
Halley's visits to Newton in thus resulted from Halley's debates about planetary motion with Wren and Hooke, and they seem to have provided Newton with the incentive and spur to develop and write what became Philosophiae Naturalis Principia Mathematica. Halley was at that time a Fellow and Council member of the Royal Society in London positions that in he resigned to become the Society's paid Clerk. Halley then had to wait for Newton to 'find' the results, but in November Newton sent Halley an amplified version of whatever previous work Newton had done on the subject.
This took the form of a 9-page manuscript, De motu corporum in gyrum Of the motion of bodies in an orbit : the title is shown on some surviving copies, although the lost original may have been without title. Newton's tract De motu corporum in gyrum , which he sent to Halley in late , derived what are now known as the three laws of Kepler, assuming an inverse square law of force, and generalised the result to conic sections.
It also extended the methodology by adding the solution of a problem on the motion of a body through a resisting medium. The contents of De motu so excited Halley by their mathematical and physical originality and far-reaching implications for astronomical theory, that he immediately went to visit Newton again, in November , to ask Newton to let the Royal Society have more of such work.
He mentioned the publicity surrounding the Mayan calendar that many said showed the world was coming to an end in Ofer then spoke about enlightenment. To illustrate his view, he mentioned the Greek philosopher Socrates. He was condemned by those who did not understand his views were based on enlightenment.
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Ofer said that Plato, a student of Socrates, used an allegory about a cave in his The Republic to illustrate this point. Ofer explained how this allegory about people shackled and facing only one direction believed that what they saw was the world they lived in. However, the allegory speaks of others who can move about and see the world for what it is. There is more to understanding than the knowledge one gains from universities or science. Even though science continues to search for smaller and smaller matter, there is more than that to existence.
In their pursuit, they do not see that the world was created by something. Although some may call it god, it is life; a stream of light that functions like DNA and goes in a cycle. He then mentioned the impact of the classical elements fire, air, water, earth, etc.