Isaac Newton / NOO-t’n /
English Mathematician and Scientist
Category Ranking 2nd of 46
Overall Ranking 3rd out of 500
Issued by Great Britain in 1987 to celebrate the 300th anniversary of Newton’s monumental work, Principia mathematica, in which he gave a mathematical description of the laws of mechanics and gravitation, and applied these to planetary and lunar motion.
Isaac Newton is one of the top three mathematicians, with Archimedes and Gauss. Newton discovered differential calculus (at 23) and its relationship with integration (at 24). A bitter quarrel with Gottfried Leibniz, a German mathematician, ensued as to which of them had discovered calculus first. At 23, Newton also discovered what we today call the “generalized binomial theorem.” According to the mathematician William Dunham, “These discoveries were probably the most productive two years that any thinker—certainly any 23-year-old thinker—has ever experienced.(1) “... the calculus, ... next to Euclidean geometry, is the greatest creation in all of mathematics.”(2) The focus of mathematical scholarship has shifted from place to place throughout history, from the Pythagoreans at Crotona, to Plato’s Athenian Academy, to Alexandria, to Baghdad, and then to the Renaissance Italy of Galileo. Incredibly, in the mid-1660s, it came to rest in the modest rooms of a Trinity College student, and wherever Newton was, there too was the mathematical center of the world.(3)
In 1687, at 45, he presented three basic principles dealing with motion that are now called Newton’s three laws of motion. These are the fundamental laws for all physical motion. These “laws” are described in his book Philosophiae naturalis principia mathematica, (Mathematical principles of Natural Philosophy, 1687, 367 pages) and according to world renowned astrophysicist Stephen Hawking, Newton’s book is “surely the most influential book ever written in physics.”(4)
Mathematics–Calculus (Types of Problems)
Though it was to some extent the answer to problems already tackled by the Greeks, the calculus was created primarily to treat the major scientific problems of the seventeenth century.(5) At least a dozen of the greatest mathematicians of the seventeenth century and several dozen minor ones tackled the problems of the calculus. All of their contributions were crowned by the achievement of Newton and Leibniz.(6)
There were four major types of problems. The first was: Given the formula for the distance a body covers as a function of the time, to find the velocity and acceleration at any instant. And, conversely, given the formula describing the acceleration of a body as a function of the time, to find the velocity and the distance traveled. This problem arose directly in the study of motion, and the difficulty it posed was that the velocities and the acceleration of concern to the seventeenth century varied from instant to instant. In calculating an instantaneous velocity, for example, one cannot, as one can in the case of average velocity, divide the distance traveled by the time of travel because at a given instant, both the distance traveled and time are zero, and 0/0 is meaningless. Nevertheless, it was clear on physical grounds that moving objects do have a velocity at each instant of their travel. The inverse problem of finding the distance covered, knowing the formula for velocity, involves the corresponding difficulty; one cannot multiply the velocity at any one instant by the time of travel to obtain the distance traveled because the velocity varies from instant to instant.
The second type of problem was to find the tangent to a curve. Interest in this problem came from more than one source. It was a problem of pure geometry, and it was of great importance for scientific applications. Optics was one of the major scientific pursuits of the seventeenth century. The design of lenses was of direct interest to Fermat, Descartes, Christiaan Huygens (1629–1695), and Newton. To study the passage of light through a lens, one must know the angle at which the ray strikes the lens in order to apply the law of refraction. The significant angle is that between the ray and the normal to the curve (figure above), the normal being the perpendicular to the tangent. Hence the problem was to find either the normal or the tangent. Another scientific problem involving the tangent to a curve arose in the study of motion. The direction of motion of a moving body at any point of its path is the direction of the tangent to the path.
The third problem was that of finding the maximum or minimum value of a function. When a cannonball is shot from a cannon, the distance it will travel horizontally—the range—depends on the angle at which the cannon is inclined to the ground. One practical problem was to find the angle that would maximize the range. Early in the seventeenth century, Galileo determined that (in a vacuum) the maximum range is obtained for an angle of fire of 45º. He also obtained the maximum heights reached by projectiles fired at various angles to the ground. The study of the motion of the planets also involved maxima and minima problems, such as finding the greatest and least distances of a planet from the sun.
The fourth type of problem included: finding the lengths of curves–for example, the distance covered by a planet in a given period of time; the areas bounded by curves; volumes bounded by surfaces; centers of gravity of bodies; and the gravitational attraction that an extended body, a planet for example, exerts on another body. The Greeks (see discussion under Archimedes) had used the method of exhaustion to find some areas and volumes. Despite the fact that they used it for relatively simple areas and volumes, they had to apply much ingenuity because the method lacked generality. Nor did they often come up with numerical answers. Descartes’s analytic geometry of course rendered the information in numeric terms by assigning numeric coordinates to every point. This new rendering of geometry was very important in enabling Newton and Leibniz to discover the calculus. Interest in finding lengths, areas, volumes, and centers of gravity was revived when the work of Archimedes became known in Europe. The method of exhaustion was first modified gradually, and then radically by the invention of the calculus.(7) The method of exhaustion is the geometric forerunner of the modern notion of “limit,” which in turn lies at the heart of the calculus.(8)
Mathematics–Calculus (Basic Concepts)
Calculus begins by trying to solve the following problem: What is the slope of the tangent line for a particular curve at a given point? It turns out that this problem is exactly the same problem as finding the speed of an object if we are given its position function. The slope of the tangent line to the curve y = f (x) (read y is a function of x or y varies as function of x is described) is given by the following expression, known as the derivative:
slope of tangent line = derivative = instantaneous velocity = f’ (x) = dy/dx
= lim f(x+∆x) – f(x)
The operation “lim” means to take the limit of the expression as x moves very close to zero, but we do not ever let x actually equal zero.
We can derive a set of rules that make it possible to find the derivatives (think also instantaneous velocity) of different functions. We call this process differentiating the function.
If y = c then the derivative of y is zero expressed as: y’ = 0
Another rule: If y = cx then the derivative of y is c, expressed as: y’ = c
It is often useful to reverse the process of differentiation. This process is called integration or anti-differentiation. An integral represents not only the antiderivative but also the area under a curve. Integrals or integral calculus can be used for more general problems, such as volumes, surface areas, or the center of mass. The beauty and importance of calculus is that it provides a systematic way for the exact calculation of many areas, volumes, and other quantities that were beyond the methods of the early Greeks.(9)
The greatest achievement of the seventeenth century was the calculus.(10) From this source there stemmed major new branches of mathematics: differential equations, infinite series, differential geometry, the calculus of variations, and functions of complex variables. Indeed, the beginnings of some of these subjects were already present in the works of Newton and Leibniz. The eighteenth century was devoted largely to the development of some of these branches of analysis.(11)
Mathematics–Calculus, Algebra (Binomial Theorem)
Newton used his new technique of integration from the calculus and his discovery of the binomial theorem to come up with an even more accurate estimate of pi, to 16 decimal places.(12) The binomial theorem dealt with expanding expressions of the form (a + b)n. One can simply multiply (a + b) by itself as many times as indicated by n, but this becomes tedious–say, when n is 12. Blaise Pascal had discovered a way to generate the coefficients in a binomial expansion from the array now known as “Pascal’s triangle”:
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
For example, (a + b)4 = (1)a4 + 4a³b + 6a²b² + 4ab³ + (1)b4, which uses the fourth row of Pascal’s triangle – 1, 4, 6, 4, 1 as the coefficients in the binomial expansion of (a + b)4.
At 23, Newton devised a formula for generating the binomial coefficients directly without the cumbersome process of constructing the triangle down to the necessary row. Using integration and the binomial theorem, Newton presented the value of pi to 16 decimal places, based on the 20-term binomial expansion of 1- x. In the above expansion of a + b, 4a³b is a term as is 4ab³.
In his major treatise, Philosophiae Naturalis Principia Mathematica (Mathematical Principles of Natural Philosophy, 1687, 367 pages), Newton gave a mathematical description of the laws of mechanics and gravitation, and applied these to planetary and lunar motion. He used his newly discovered calculus to maximum effect. For most purposes, Newtonian mechanics has survived even the introduction of Albert Einstein’s relativity theory and quantum mechanics, to both of which it stands as a good approximation. Quantum mechanics is a mathematical form of quantum theory dealing with the motion and interaction of (especially subatomic) particles and incorporating the concept that these particles can also be regarded as waves. As a physicist, Newton had the single greatest influence on theoretical physics until Einstein. His Mathematical Principles was one of the most important single works in the history of modern science.(13) Again, as pointed out before, throughout the history of mathematics, each new development builds from the work of previous individuals. In this case, “it was the pure mathematics of Apollonius that made possible, some 1,800 years later, the Principia of Newton.”(14)
It is important to recognize that Newton's masterpiece may not have come about without the extraordinary efforts of Edmund Halley who visited Newton at Cambridge in August 1684. Halley suggested the project in the first place; he averted suppression of the third book; bore all the expenses of printing and binding, corrected the proofs, and laid his own work entirely aside to see Newton's Principia through the press.
The mechanics of the Principles was an exact quantitative description of the motion of visible bodies. It rested on Newton’s three laws of motion:
1. (Law of Inertia) In the absence of outside forces, the momentum of a system remains constant. An object that is sitting still will stay in place until it is pushed, and something that is moving will keep moving in a straight line until it is pushed to change its speed or direction. The reason why moving objects on Earth eventually slow down and stop is because they are acted on by the force called friction.
2. (Law of Constant Acceleration) If force acts on a body, the body accelerates in the direction of the force such that the greater the force, the greater the acceleration and the greater the mass of the body, the less the acceleration. In other words, the change of velocity, i.e., acceleration times the mass of the body, is proportional to the force impressed. In mathematical language, F = ma. The equation F = ma is probably the best-known differential equation and essentially the first example of a differential equation. From this equation all of the fundamental equations of dynamics can be derived by methods of calculus.(15)
3. (Law of Conservation of Momentum) If a force is exerted on a body, that body reacts with an equal and opposite force on the body that exerted the force. This is also stated this way: For every action, there is an equal and opposite reaction.
From these laws, Newton formulated the law of universal gravitation: Every particle of matter in the universe attracts every other particle with a force that is proportional to the product of their masses and inversely proportional to the square of the distance between their centers.
In symbols, the magnitude of the attractive force F is equal to G (the gravitational constant, a number the size of which depends on the system of units used and which is a universal constant) multiplied by the product of the masses (m1 and m2) and divided by the square of the distance R.
Newton’s law of universal gravitation on a Nicaraguan stamp from 1971.
Newton put forward the law in 1687 and used it to explain the observed motions of the planets and their moons, which had been reduced to mathematical form by Johannes Kepler early in the seventeenth century. Later he also confirmed his universal law through the movement of the tides and the orbits of comets.
Newton arrived by induction at his three simple laws of motion and his great fundamental generalization—the law of universal gravitation. The educated world at once was so impressed with his discovery that Newton was idolized, almost deified, in his own lifetime. This majestic new Universe, built upon a few simple assumptions, now made the Greek philosophers look like simple school boys. Nevertheless, as James A. Connor points out in his biography of Johannes Kepler:
Newton, in his darker moments, knew quite well what he owed to Kepler, who had brought him right up to the doorstep of his theory of gravitation, who had laid the foundation for his work in optics, and who, as Leibniz recognized and Newton dismissed, had set the stage for the invention of calculus. He knew what he owed to Kepler but would not acknowledge it.(16)
Ultimately, though, the revolution begun by Galileo at the beginning of the 1600s was triumphantly completed by Newton at the century’s end.(17)
His work Optics (1704), an account of his optical experiments and theories, includes the realization that white light is a mixture of all the colors of the rainbow. Light itself is a heterogeneous mixture of differently refractible rays. There was an exact correlation between color and “degree of refractibility”—the least refractible being red and the most refractible being a deep violet. It would be possible, then, to designate any color by a number indicating its degree of refractibility. Newton was the first person to analyze light in this way. This was the foundation for the science of spectroscopy.(18)
Newton was appointed Master of the Mint in 1699, entered Parliament as MP for Cambridge University in 1701, and in 1703 was elected president of the Royal Society.
(1) William Dunham, Journey Through Genius – The Great Theorems of Mathematics (John Wiley and Sons, Inc., New York, 1990), p. 164.
(2) Morris Kline, Mathematical Thought From Ancient to Modern Times (New York, 1972), p. 342.
(3) Dunham, p. 165.
(4) Stephen Hawking, A Brief History of Time: From the Big Bang To Black Holes (New York, 1988), p. 181.
(5) Kline, p. 342.
(6) Ibid., p. 344.
(7) Ibid., pp. 342-343.
(8) Dunham, p. 29.
(9) Encyclopaedia Britannica, Micropaedia, Volume 2, 1993, 15th Edition, p. 734.
(10) Kline, p. 400
(11) Ibid., p. 400.
(12) Dunham, p. 177.
(13) Encyclopaedia Britannica, Macropaedia, Volume 24, 1993, 15th Edition, p. 931.
(14) Carl B. Boyer, A History of Mathematics, 2nd Edition (New York, 1991), p. 152.
(15) Jan Gullberg, Mathematics: From the Birth of Numbers (New York, 1997), p. 895.
(16) James A. Connor, Kepler’s Witch: An Astronomer’s Discovery of Cosmic Order Amid Religious War, Political Intrigue, and the Heresy Trial of His Mother (New York, 2004), p. 5.
(17) Isaac Asimov, The Intelligent Man’s Guide to Science, Vol. 1 – The Physical Sciences (New York, 1960), p. 19.
(18) Daniel J. Boorstin, The Discoverers – A History of Man’s Search To Know His World And Himself (New York, 1985), pp. 402-403