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On December 25, 1642 a child was born to Hannah Ayscough in Wolsthorpe, England. Little did Hannah know, this infant would grow up and become one of the world’s most respected scientists and mathematicians. This person changed the way humans viewed the physical world around them and his findings are still studied in schools today. This child’s name was Sir Isaac Newton. Newton was raised in Wolsthorpe and attended school in Grantham at the Free Grammar School of King Edward VI (Westfall 55). Newton was not known for his brilliance at the grammar school, instead he was remembered for being absentminded. One story that depicts this is when Newton was riding home and dismounted his horse to walk it up a hill but forgot to remount the horse for the rest of the ride home (Westfall 64). His intellect was first recognized when he attended Cambridge University in 1661 where he met Isaac Barrow, the Lucasian math professor. Barrow saw the brilliance in Newton and when Newton returned to the university as a teacher he resigned from the position of Lucasian professor so that Newton may have it in 1669 (Knox 69). It was during this time period that Newton published his famous book the Principia, which explained how the planets moved about the sun, the laws of motion, and calculus. Because of the Principia, human’s views about science and math changed drastically. Today Sir Isaac Newton is remembered as one of the greatest physicist, with his most profound work on physics being the Principia.

Newton would not have been able to prove his physical theories without the use of mathematics. However, without his extensive background in mathematics Newton would not have been able to prove his theories. Before the Principia was published, Newton was a professor of math at Cambridge University. At Cambridge, he would lecture on geometrical optics but eventually moved to algebra and related topics (Knox 69). During this time Newton wrote a treatise that brought together most of his extraordinary discoveries including the determination of maxima and minima, the expression of the binomial theorem, and many other mathematical achievements, all of which could be resolved by motion (Knox 79). In fact, most of Newton’s early essays in pure mathematics often tend to be in the language and principles of the physics of motion (Cohen 52). During his time at Cambridge Newton, was able to develop his math skills and devote the proper time necessary to prove mathematically how our solar system works.

The Principia is a collection of three books, the first completed book was Book II in 1686 followed by Book I also in 1686. The third book was originally supposed to be suppressed but Halley convinced Newton to publish it and he did so in 1687 (Gjersten 455). The entire series was published as one on July 5, 1687 in London, edited by Halley and was 511 pages long (Gjersten 470). The Principia began what is now known as the Newtonian Revolution. Newton addressed science in a different fashion with the first step in his methodology being the rejection of hypotheses, the stress upon induction, the working sequence (induction precedes deduction), and the inclusion of meta-physical arguments in physics (Cohen 11). Thanks to this methodology, Newton was able to write a book that changed the world of science and created a new system of analysis called Newtonian method. The Newtonian method was founded in this book and consists of simple systems that become more and more complex. He then takes the findings and creates laws and principles and applies them to bigger systems (Cohen 66). Through this Newton was able to create a new method for examining complex systems that cannot be physically tested. Because of this new methodology Newton was able to see how the solar system worked and was able to prove it by experiment, observation and codified reasoning (Cohen 51). However, Newton did not just merely state his beliefs. He was able to create a mathematical analysis to substantiate his claims that the Earth traveled around the sun in an ellipses (Cohen 15). By combining physics and math to better understand the physical world, he was able to prove that Kepler had been right and that the earth revolved around the sun in an elliptical pattern. He proved this by combing physics and math in order to better understand the physical world. Also, he was only able to do this because of this ability to reduce complex physical situations to a mathematical simplicity and expand upon both of these to eventually understand physics (Cohen 56). Newton then went on to record his findings and published them in what was called the Principia. The Principia is in three books, with Book I focusing on motion in free space devoid of fluid resistance, Book II considers various conditions of fluid resistance and a variety of related topics, and Book III applies the results of Book I to the physics of outer space (Cohen and Smith 58). The three books of the Principia all serve a purpose, however Book I is the most widely recognized of the three.

Newton began the work on Book I in 1685 when he worked on the notion of ‘force,’ beginning with the idea of motion being compounded of an ‘inherent’ force (Knox 107). He also investigated the subject of impact and tried to quantify the change in a body’s ‘motion’ engendered by impact, and based this assumption on a body moving or at rest was at the passive recipient of some externally impressed ‘force’ (Knox 77). Newton wanted to better understand how the physical world functioned and this led him to his experiments with force and motion. This led to his studies of the solar system and by the end of 1685, Newton stumbled upon the realization that in our cosmos all bodies attracted every other body and that this applied to objects near the surface of the earth as much as it did to planets millions of miles apart (Knox 109). Thus Newton realized that if he could understand the physical motions of this planet he would have a better understanding of the cosmos. This is why Newton opens Book I by defining terms he uses. One of those terms, mass, is still in use today, along with his three laws of motion (Cohen and Smith 58). The idea of mass is important because it gave Newton a way to measure objects in an environment where there is no gravity. After the definition of weight Newton begins describing his three laws of motion. The first law states, a particle will stay at rest or continue at a constant velocity unless acted upon by an external unbalanced force, an example of this is how a ball that is laying on the ground will continue laying on the ground unless someone picks it up and throws it. The second law states, the net force on an object is equal to the mass of the object multiplied by its acceleration, and the third and final law says every action has an equal and opposite reaction (Krantz 126). An example of the third law is a plane taking off, in order for the plane to go in one direction it has to apply the force going the opposite direction to push it forward. These laws were so profound that they are still used today in high schools in the study of physics. Also these laws would be the stepping stones for Newton’s work in Book I of the Principia. After defining these laws, Newton makes the claim of equivalence between inertia, the property of matter by which it retains its state of rest or its velocity along a straight line so long as it is not acted upon by an external force, and gravitational mass of bodies (Gjersten 495).This led to Newton’s theory of centripetal force and the inverse square law. Newton stated that if bodies which move in any curved line, such as the planets, which observe Kepler’s area law are under the influence of a centripetal force and that given Kepler’s harmonic law, ‘centripetal forces will be inversely as the square of the radii’ (Gjersten 492). Thus, Newton has shown that bodies moving under the influence of a centripetal force will obey Kepler’s law and sweep out equal areas in equal times (Gjersten 492). Once he figured this out, Newton was able to show that the force by which the earth attracted the moon was a kind of ‘centripetal force’ that ‘very nearly’ obeyed the inverse square law and he made the conclusion that the sun attracted other bodes in the solar system in proportion to their mass (Knox 108). Because of Newton’s investigation into motion we have a better understanding of physics and a closer understanding of the earth and other planets revolution around the sun in an elliptical shape. However, he did not stop there with his proofs. After theorizing about how the planets orbited, Newton invented a new mathematical system to help bolster his statement.

Newton was able to create experiments about the earth’s revolution around the sun but it is his second book of the Principia where he invented calculus to help prove his claim (Krantz 125). Newton was forced to do this so that people realized that he was not just theorizing about physics but that he could actually prove it, “For this reason I now go on to explain the motion of bodies that mutually attract one another, considering centripetal forces as attractions, although perhaps-if we speak in the language of physics-they might more truly be called impulses. For we are here concerned with mathematics; and therefore, putting aside any debates concerning physics, we are using familiar language so as to be more easily understood by mathematical readers (Cohen 73).” He knew that unless he was able to prove his theories mathematically no one would be able to take his claims seriously. This is why books one and two of the Principia, as Newton states, are primarily mathematical and not physical (Cohen Pg. 70). Book one opens with section one on the theory of limits and section two is concerned with centripetal forces acting on point masses and section three deals with motions in conic sections (Cohen 70). However, Newton’s use of mathematical systems in relation to physics is best illustrated in book two of the Principia, in which the first three sections deal with mathematical constructs (Cohen 83). In corollary one, Newton shows how to compute the magnitude of this extraneous force from the motion of the apsides, either of two points in an eccentric orbit one being farthest from the center of attraction and the other being nearest the center, proving that centripetal force is exactly as the inverse square of the distance that the apsides can be at rest (Cohen 87). Through this he shows that the earth’s force on the moon is as the inverse square of the distance. To find the inverse square law Newton combined the Keplerian third law (r3/T2=constant) with the centrifugal rule (v2/r) to get the inverse square law (Bechler 28). Since, by Kepler’s third law (r3/T2) is a constant for the solar system, so that 4π2 (r3/T2) is a constant, if follows that v2/r is proportional to 1/r2 (Bechler 28).
Hence, at the earth’s surface this force would be 60x60 greater than at the moon’s orbit (Cohen 88). Newton then states that, “every attraction is mutual hence each planet will gravitate toward all its satellites…and the sun toward all the primary planets (Cohen 88),” meaning that two objects will gravitate towards one another and one will swing around the other in an elliptical pattern. Newton then concludes that, “the gravity that is directed toward every planet is inversely as the square of the distance of places from its centre (Cohen 88).” Newton’s ability to create a new system of math helped him prove his theories on motion and centripetal force.

In the third book of the Principia, Newton ties physics and math together to explain why the moon revolves around the earth and the earth around the sun. Once Newton had proved that centripetal force is used in the planets elliptical pattern, he then needed to prove why the planets are not sucked in by the sun’s gravitational pull. He explains this using his third law of motion. Newton stated that since the earth and the sun had their own gravitational pull then their pull on one another was mutual and equal thus neither body was at rest and both bodies revolved around a common center (Bechler 44). In other words, Newton has seen that a consequence of the third law of motion is that if the sun pulls on the earth, then the earth must also pull on the sun and with a force of equal magnitude (Bechler 44) keeping them in their place. Through the use of math Newton was able to determine the laws of physics and how they relate to the solar system.

Newton’s writings in the Principia changed the way scientists analyzed the world. Scientists became more math oriented and tried to prove all their theories mathematically. In the subject of physics, Newton radically restructured the principles and concepts of motion along the lines of mass, acceleration, and force (Cohen 127), plus the elaboration of a system of the world operating in terms of the new dynamics, in which universal gravity is the governing force and inertia is a primary or essential property of matter (Cohen 127). The Newtonian style was a key to the revolution in physics (Cohen 127). Newton used the idea of starting small and building upon it, the Principia was set up in this manner. Newton opens with the laws of physics and demonstrates how these laws affect particles on earth, such as ocean waves and how the pull of the moon and earth’s gravity makes the waves. He then expands his thought process to show how these same concepts of centripetal force affect the rotation of the earth around the moon and finally the earth and the sun (Bechler 30). He had recognized both the role of mass in inertial physics and the distinction between mass as a resistance to acceleration (what we call inertial mass) and as a determinant of force in a gravitational field (gravitational mass) and this showed how the one and the same universal force serves to account for the motion of planets around the sun, of both real and artificial satellites around planets, and of comets. However Newton would not have been able to prove any of this without the use of math. Newton produced such an astonishing revolution in science by applying mathematics to natural phenomena. Newton’s achievement was not merely to guess or even to know the inverse-square law, but rather to use it to demonstrate elliptical orbits and to develop a system of the world based upon it (Cohen 130). This could not have been done by experiment and observation, by induction, or by philosophical speculation, but only by mathematics. And the key to applying mathematics to the world of nature was the Newtonian style, in which by stages there could be added the conditions that would bring the original imagined system and mathematical construct into congruence with the realities of experience. The mathematics needed for this task was a new mathematics, the fluxional calculus, embodied in the continual use of limits and of infinite series (Cohen 130). With the knowledge of the first law of objects in motion will stay in motion, and the fact that we are rooted on earth and not floating, shows that gravity is an active force upon us. We then know that in order to travel into space we need to exert more force than is being applied to us by gravity and Newton’s calculus helps us to find the necessary force. Once in space, we use the moon’s gravitational field to land on it. As one can see, Newton’s laws of physics and calculus serve an important role in space travel. It is difficult to think of any other scientific book that had ever been written which embodied so complete a change in the state of knowledge concerning the physics of the heavens and the earth (Cohen 130).

Isaac Newton was one of the world’s greatest mathematicians and physicists. The greatest work of his is the Principia where he demonstrates the laws of motion, calculus, and proved Kepler’s theory that the planets revolve around the sun in an ellipses. This revolutionary work changed the way humans viewed the earth and changed the way scientists examined it. After the printing of his book scientists began attacking science from a new direction and with greater success. His methodology and approach of starting small and building on his findings and applying math to science is still used today. Students are still taught in high school and college Newton’s three laws of motion, calculus, and how math can prove the science of physics. Newton created a revolution in two subject fields. With one book, Newton was able to change the world and our outlook on it. Newton changed the way humans look at the world, and how we study it.



  Isaac Newton and the Principles of the Heavens


Page Author: Brian K. Davis

Saturday, 11-Feb-2012 14:19


Bechler, Z. Contemporary Newtonian Research. Dordrecht, Holland: D. Reidel Pub., 1982. Print.

Cohen, I. Bernard, and George E. Smith, eds. The Cambridge Companion to Newton. Cambridge [u.a.: Cambridge Univ., 2002. Print.

Cohen, I. Bernard. The Newtonian Revolution: With Illustrations of the Transformation of Scientific Ideas. Cambridge [Eng.: Cambridge UP, 1980. Print.

Gjertsen, Derek. The Newton Handbook. London: Routledge & Kegan Paul, 1986. Print.

Knox, Kevin C., and Richard Noakes, eds. From Newton to Hawking: A History of Cambridge University's Lucasian Professors of Mathematics. Cambridge, U.K.: Cambridge UP, 2003. Print.

Krantz, Steven G. An Episodic History of Mathematics: Mathematical Culture through Problem Solving. [Washington, DC]: Mathematical Association of America, 2010. Print.

Westfall, Richard S. Never at Rest: A Biography of Isaac Newton. Cambridge [Eng].: Cambridge UP, 1980. Print.


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