# Isaac Newton and the Mathematization of the World System

June 27, 2021Newton, at the age of forty-five, published “Mathematical Principles of Natural Philosophy” in London in 1687. “Principles of Mathematics” is presented as three chapters or three books.

The first book deals with problems related to the science of motion independently of the resistances of the mediums, from a purely mathematical point of view. The second book is devoted mainly to the motions of bodies in resistive environments, particularly those in which the opposition in oblique shots varies, such as velocity, as the square of velocity, or a combination of both. Newton also raised problems about the form of the rigid body that would have the least resistance and discussed the theoretical verification of Torricelli’s law of flow. This second book contains a sharp critique of the Cartesian vortex hypothesis. The style of this critique illustrates perfectly the contrast between Cartesian geometric cosmology and Newton’s physico-mathematical deductive structure. The third book revisits the conclusions of the first two books and applies them to physical problems (movements of the planets and the Moon, shape of the Earth, theory of tides..)

“Principles of Mathematics” opens with two preambles: “Definitions and “Axioms or Laws of Motion.” The “Definitions” section specifically defines the following concepts: quantity of matter (“quantity of matter is the measure obtained from both the density and volume of the body”), motion quantity (movement quantity is the value obtained from both the velocity of the object and the quantity of matter.) melting force (via impressa) this force is the effect that will change the state of an object at rest or in uniform linear motion.

This set of definitions ends with an explanation that gives the very famous definitions of absolute space and absolute time:

1) Absolute, real and mathematical time flows unchanged, without being dependent on any external factors other than itself and its own nature; It can also be called time. Relative, apparent, and everyday time gives an external and sensible measure of duration by virtue of movement, and is commonly used in place of real time; such as hour, day, month and year.

2) Absolute space always exists and is immobile by itself, regardless of any external thing. Relative space is the moving dimension or measure of absolute space; It is determined by our senses by its position vis-a-vis objects and is commonly perceived as still space.

The chapter “Axioms or Laws of Motion” brings together for the first time the three great laws of mechanics very closely to what we know them today. The first law expresses the principle of inertia, that is, the conservation of uniform linear motion: “Every body either remains at rest or maintains uniform linear motion unless an external force acts on it.”

The second law is stated: “The change of motion is proportional to the force acting, and it goes along a straight line in the direction in which the motion acts.” This law should not be confused with the law expressed in differential terms that we know today as “Newton’s law.” Newton speaks here of “change of motion” without any indication of the time when this change occurs. If this law were to be written in modern terms, the closest expression would be: F = D (m v); where F is the acting force, m is the mass and v’ is the velocity; D (m v) represents the “change of motion”. From this point of view, it can be said that the acting force is not a force in the modern sense of the term, but a push.

The third law is about the equality of action and reaction; “To every action there is always an equal reaction to it; that is, the mutual influences exerted by two bodies on one another are always equal and opposite.” This third law, which was not included in the first drafts at the time of writing the Principles of Mathematics in 1685, allowed Newton to formulate the law of universal gravitation in its full scope in Book 111.

So, on the basis of these “Definitions” and these “Axioms or Laws of Motion”, the motions of bodies under the influence of centripetal forces acquired a mathematical existence. While achieving this goal, Newton incorporated the mathematical methods of classical geometry originating from Euclid into his work; but with these methods, while enriching his book with many results based on the study of conics (Chapters IV and V of Book 1), on the other hand, a chapter containing reasonings about infinitesimal geometry has been added to the first chapter of Book 1, which he called “The Method of First and Last Causes”. . It is interesting, however, to see (with the exception of Chapter II of Book II) that Newton did not use the “fluid method”, of which he had first principles, from around 1670 on in his proofs.