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Philosophiae Naturalis PRINCIPIA Mathematica - 1687
Newton’s greatest effort succeeded in putting together Galileo’s and Kepler’s intellectual achievements, fitting them into modern physics.  

Newton started his work emphasizing a sequence of physical magnitudes, stating the three universal laws of motion. Newton also assumed necessary an understanding of:
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Euclid’s: "Elements"   
Apollonius’s: "On Conic Sections"  
Galileo’s: "Two New Sciences: Discourses and mathematical demonstrations concerning two new sciences", "On Equable Motion", propositions 1-2-4; "On Naturally Accelerated Motion", propositions 1-2; "On the Motion of Projectiles", proposition 1
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Kepler’s thought was not directly recalled in his works: Newton achieved true results using "definitions" and his three laws of motion. He did not use his laws as starting hypotheses. 

Newton had a cosmological vision of mechanics; on the contrary, more simply, Galileo had a terrestrial vision. A projectile, falling, traces an elliptical orbit. Should it have a sufficient speed it would continue to orbit around the Earth. Newton in his "Principia" - Book I - Scholium - quoted: «… If the center depart to infinity and the ellipse be turned into a parabola, the body will move in this parabola …» 
 
1st law Law of Inertia ("vis insita") – Every body continues in its state of resting or of moving uniformly in a straight line, except insofar as it is driven by impressed forces to alter its state - [a reasonable hypothesis Newton admitted was taken from Galileo]. 
The contemporaries of the "Lucasian Professor" stated that the pure concept of inertia had been put forth by Descartes. Newton did not accept this assertion. To share his mechanics - metaphysical and mechanical - with a partisan of the scientific method was thoroughly impossible for him. 
It was true, indeed, that the Cartesian space, with its well known axes from which semi-axes extended to infinity, consented a rectilinear uniform motion (difficult, however, to make it compatible with the ever-present hypothesis of the vortices). 
Galileo stated that a body persisted in its state of unperturbed motion onto a spherical plane about the Earth: 
« ... indeed, after having removed all external impediments, a heavy body positioned onto a spherical plane concentric to the Earth will show indifferent to the state of rest and to those motions towards a part whatever of the horizon. It will stay in that state into which it was originally placed; thus meaning, if placed into a state of rest, it will maintain it and, on the contrary, if pushed, for instance towards West, it will continue to move towards this direction: same as a ship that, in a calm sea, would continue to circumnavigate our globe, once pushed, even once, by some kind of an impulse. On the contrary, should it be resting, it would never cease to remain in such state. This may happen if, in the first case, all extrinsic impediments cannot be removed and, in the second case, if no outside motion occurs ... » (*)
Besides, Galileo’s space is limited and, philosophically speaking, cannot take into account rectilinear trajectories extending to infinity:  « ... Of places where the new star maybe placed, [Tycho’s Nova - 1572], some are obviously impossible, others can be considered. It is absolutely impossible to place it above the fixed stars, because the space above is infinite. Such a site does not exist. If it existed, the star placed there would be invisible for us ... » (**)
Truly, both these personages gave a great contribution to the concept of inertia. Galileo, in an almost exhaustive way, except for some final contradictions. Newton in a total correct way.
2nd law The Change of Motion is proportional to the Motive Force, and takes place following the straight line in which that force is impressed – an absolute novelty - [reasonable hypothesis]
3rd law To an action there is always a contrary and equal reaction; or, the mutual actions of two bodies upon each other are always equal and directed to contrary parts - an absolute novelty - [reasonable hypothesis]
These three laws (reasonable working hypotheses carried out through the observation of the phenomena, induction and deduction) together with the Definitions, were all that Newton needed to demonstrate the validity of the universal gravitation. Some observable data (Phenomena) confirmed the soundness of his thinking.  

Newton was also the first to develop, independently of Leibnitz, infinitesimal calculus. He could have described universal gravitation briefly, in a few pages, instead of using a "geometrical method". Most probably he adopted this method to make himself understood by more readers, taking also into consideration the mathematical knowledge of his contemporaries.  

How did Newton put together Forces and Geometry to gain results? 

Newton had very clearly in mind that both velocity and force were vectors; they had an impulse, a direction, a tendency and could be represented by Sagittae [Newton used simple segments to subtend orientation]. Additionally he understood how vectors were arranged, therefore, from a hypothetical variation of the velocity [he considered two distinct points: P and Q] on a curve, he found the direction of the acceleration, and using the second law, also of the force.  
With the help of Euclid and of Apollonius Newton found a ratio between the force and the radius vector from focus to points. He made the two points coincide together: what was left was the inverse proportionality of the force with the square of the radius vector.  

Newton expressed himself as follows:
.«… centripetal force will be inversely as the solid
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.in our thought experiment of taking the limit, we imagine Q to be .moving towards P.…» [Problem 6, Theorem 5] 

.«... if a body revolve in an ellipse, it is required to find the law of .centripetal force tending to the focus ["umbilicum"] of the ellipse ....» [Proposition 11] 

.Let the focus of the ellipse be S and let P be a point moving in an .ellipse, SP is the radius vector. Newton explained his discovery in .Section 3, Proposition 11, with these words:
.«... let this identity:
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.Therefore [see Problem 6, Theorem 5] the centripetal force is .inversely as:
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.that is inversely in the duplicate ratio of the distance SP. Q.E.I. .(quod erat inveniendum – that which was to have been found) ....» [Proposition 11]  (*) ... L is constant ...
 
 
with modern words, [e.g. mass = quantity of matter] the final conclusion is:
Thesis  or 
4th law		 Universal gravitation: force is directly proportional to the product of the masses and, inversely to the squares of the distances. The law includes an invariable factor "universal constant", Cavendish measured it for the first time with the aid of a torsion balance.
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(*)
 
 
L is the "latus rectum" for a given ellipse: 
 
where BC is the semi minor-axis and AC is the semi major-axis, L is constant for a given ellipse.
  (*) Galilei, Galileo, «Seconda lettera del sig. Galileo Galilei al sig. Marco Velseri delle macchie solari». from Liber Liber. 
(**) Galilei, Galileo, «Dialogo sopra i due massimi sistemi del mondo». from Liber Liber. 
 

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