Newton's Inverse Square Law

Eight key definitions underlie the three laws of the Newton’s mathematical mode. In the first two definitions, Newton defines quantity of matter (mass) as the product of volume and density and quantity of motion, or momentum, as the product of mass and velocity. The remainder of Newton’s definitions lay out the Newtonian concept of a force. Definition three

The remained of Newton’s first book uses these three axioms to create a powerful mathematical model of force. Only three propositions from this book, Propositions 2,4,45, are necessary for the demonstration of the existence of a universal gravitation force. In Proposition 2, Newton shows that any body describing a curve about a point with the area described being proportional to time – that is, an object following Kepler’s Second Law of planetary motion – is attracted to the center of the curve by a centripetal force. He does this by approximating the curve by a series of equal area triangles with a common vertex at the center of the curve, which Euclid proved true. By the first Law of Motion, a body moving along the edges of these triangles would continue in a straight line unless it was acted upon by a force. Because it instead shifts direction at the corner of each triangle, the Second Law holds that a force proportional to the change upon it at these points. That change of motion is parallel to the line from the point where the body would have been had no force acted upon it to the next corner of the triangle. This line is parallel to the line from the first corner to the center, and therefore the force must be directed to the center. Since this proof continues to hold as the number of triangles increases to infinity and the path becomes a true curve, a centripetal force must constantly influence a body following such a path. Corollary IV of Proposition 1 extends this statement by showing that the magnitude of this force at different points along the curve is equal to the versed sines of arcs described in equal times.

Newton then proceeds to extern a gravitational attraction to all bodies and to quantify its strength in Propositions 6 and 7. Newton believed that all bodies descend from equal heights in equal times when the effects of air resistance are removed. By Proposition 5, the same is true around any other planet. But because the accelerative quantity of the gravitational force on any body at a given height is constant, the motive quantity of the force must be proportional to the body’s mass. This further implies that the net gravitational force on any body is equal to the sum of the gravitational force on its parts. If this were not the case, then either objects at equal heights would not fall in equal times, or the differing accelerations of different parts of a body would have to be precisely balanced so as to cancel one another out. The complexity of this second scenario contradicts the first rule of reasoning, so the force of gravity must be proportional to the both the mass of the attracted object and the inverse square of the distance to the object attracting it. By the Third Law of Motion which Newton applies in Proposition 7, gravitational force must be proportional to the product of the both the masses of the bodies involved divided by the square of the distance between them.

describes the vis insita, the innate force of matter which maintains an object in its state of rest. It is distinguished from an impressed force, which is transient, external to a body and causes a change in a body’s momentum. A centripetal force, such as gravity will prove to be, is an impressed force that draws bodies toward a point. A centripetal force can be quantified by three measures: its absolute quantity, which m

Some topics in this essay:
Isaac Newton, Third Law, Corollary Corollary, Law Motion, IV Proposition, Moon Newton, Earth Newton’s, Rules III, Corollary Proposition, Motion Newton, force proportional, inverse square, centripetal force, gravitational force, third law, square force, inverse square force, book 3, law motion, square distance, proposition 4, third law motion, inverse square distance, corollary iv proposition, kepler’s third law,