17TH CENTURY MATHEMATICS

Mary Jane R. De Ade

BSED II

History and Philosophy of Mathematics

17TH CENTURY MATHEMATICS

SIR ISAAC NEWTONGOTTFRIED LEIBNIZ

17TH CENTURY MATHEMATICS

SIR ISAAC NEWTON

(1643-1727)

SIR ISAAC NEWTON

In the heady atmosphere of 17th Century England, with the expansion of the British Empire in full swing, grand old universities like Oxford and Cambridge were producing many great scientists and mathematicians. But the greatest of them all was undoubtedly Sir Isaac Newton.

SIR ISAAC NEWTON

Physicist, mathematician, astronomer, natural philosopher, alchemist and theologian, Newton is considered by many to be one of the most influential men in human history.

SIR ISAAC NEWTON

His 1687 publication, the "Philosophiae Naturalis Principia Mathematica" (usually called simply the "Principia"), is considered to be among the most influential books in the history of science, and it dominated the scientific view of the physical universe for the next three centuries.

SIR ISAAC NEWTON

Although largely synonymous in the minds of the general public today with gravity and the story of the apple tree, Newton remains a giant in the minds of mathematicians everywhere (on a par with the all-time greats like Archimedes and Gauss), and he greatly influenced the subsequent path of mathematical development.

SIR ISAAC NEWTON

Over two miraculous years, during the time of the Great Plague of 1665-66, the young Newton developed a new theory of light, discovered and quantified gravitation, and pioneered a revolutionary new approach to mathematics: infinitesimal calculus.

SIR ISAAC NEWTON

His theory of calculus built on earlier work by his fellow Englishmen John Wallis and Isaac Barrow, as well as on work of such Continental mathematicians as Rene Descartes, Pierre de Fermat, Bonaventura Cavalieri, Johann van Waveren Hudde and Gilles Personne de Roberval.

SIR ISAAC NEWTON

Unlike the static geometry of the Greeks, calculus allowed mathematicians and engineers to make sense of the motion and dynamic change in the changing world around us, such as the orbits of planets, the motion of fluids, etc.

SIR ISAAC NEWTON

The initial problem Newton was confronting was that, although it was easy enough to represent and calculate the average slope of a curve (for example, the increasing speed of an object on a time-distance graph), the slope of a curve was constantly varying, and there was no method to give the exact slope at any one individual point on the curve i.e. effectively the slope of a tangent line to the curve at that point.

SIR ISAAC NEWTON

Intuitively, the slope at a particular point can be approximated by taking the average slope (“rise over run”) of ever smaller segments of the curve. As the segment of the curve being considered approaches zero in size (i.e. an infinitesimal change in x), then the calculation of the slope approaches closer and closer to the exact slope at a point.

Differentiation (derivative) approximates the slope of a curve as the interval approaches zero

SIR ISAAC NEWTON

SIR ISAAC NEWTON

Without going into too much complicated detail, Newton (and his contemporary Gottfried Leibniz independently) calculated a derivative function f ‘(x) which gives the slope at any point of a function f(x ).

SIR ISAAC NEWTON

This process of calculating the slope or derivative of a curve or function is called differential calculus or differentiation (or, in Newton’s terminology, the “method of fluxions” - he called the instantaneous rate of change at a particular point on a curve the "fluxion", and the changing values of x and y the "fluents").

SIR ISAAC NEWTON

For instance, the derivative of a straight line of the type f(x) = 4x is just 4; the derivative of a squared function f(x) = x2 is 2x; the derivative of cubic function f(x) = x3 is 3x2, etc. Generalizing, the derivative of any power function f(x) = xr is rxr-1 .

SIR ISAAC NEWTON

Other derivative functions can be stated, according to certain rules, for exponential and logarithmic functions, trigonometric functions such as sin(x), cos(x), etc. so that a derivative function can be stated for any curve without discontinuities. For example, the derivative of the curve f(x) = x4 - 5p3 + sin(x2) would be f ’(x) = 4x3 - 15x2 + 2xcos(x2).

SIR ISAAC NEWTON

Having established the derivative function for a particular curve, it is then an easy matter to calculate the slope at any particular point on that curve, just by inserting a value for x. In the case of a time-distance graph, for example, this slope represents the speed of the object at a particular point.

SIR ISAAC NEWTON

The “opposite” of differentiation is integration or integral calculus (or, in Newton’s terminology, the “method of fluents”), and together differentiation and integration are the two main operations of calculus.

SIR ISAAC NEWTON

Newton’s Fundamental Theorem of Calculus states that differentiation and integration are inverse operations, so that, if a function is first integrated and then differentiated (or vice versa), the original function is retrieved.

SIR ISAAC NEWTON

The integral of a curve can be thought of as the formula for calculating the area bounded by the curve and the x axis between two defined boundaries.

SIR ISAAC NEWTON

For example, on a graph of velocity against time, the area “under the curve” would represent the distance travelled. Essentially, integration is based on a limiting procedure which approximates the area of a curvilinear region by breaking it into infinitesimally thin vertical slabs or columns.

SIR ISAAC NEWTON

In the same way as for differentiation, an integral function can be stated in general terms: the integral of any power f(x) = xr is xr+1⁄r+1, and there are other integral functions for exponential and logarithmic functions, trigonometric functions, etc, so that the area under any continuous curve can be obtained between any two limits.

Integration approximates the area under a curve as the size of the samples approaches zero

SIR ISAAC NEWTON

SIR ISAAC NEWTON

Newton chose not to publish his revolutionary mathematics straight away, worried about being ridiculed for his unconventional ideas, and contented himself with circulating his thoughts among friends.

SIR ISAAC NEWTON

After all, he had many other interests such as philosophy, alchemy and his work at the Royal Mint. However, in 1684, the German Leibniz published his own independent version of the theory, whereas Newton published nothing on the subject until 1693.

SIR ISAAC NEWTONAlthough the Royal Society, after due deliberation, gave credit for the first discovery to Newton (and credit for the first publication to Leibniz), something of a scandal arose when it was made public that the Royal Society’s subsequent accusation of plagiarism against Leibniz was actually authored by none other Newton himself, causing an ongoing controversy which marred the careers of both men.

SIR ISAAC NEWTON

Despite being by far his best known contribution to mathematics, calculus was by no means Newton’s only contribution.

He is credited with the generalized binomial theorem, which describes the algebraic expansion of powers of a binomial (an algebraic expression with two terms, such as a2 - b2);

SIR ISAAC NEWTON

he made substantial contributions to the theory of finite differences (mathematical expressions of the form f(x + b) - f(x + a));

he was one of the first to use fractional exponents and coordinate geometry to derive solutions to Diophantine equations (algebraic equations with integer-only variables);

SIR ISAAC NEWTON

he developed the so-called “Newton's method” for finding successively better approximations to the zeroes or roots of a function;

he was the first to use infinite power series with any confidence; etc.

SIR ISAAC NEWTON

In 1687, Newton published his “Principia” or “The Mathematical Principles of Natural Philosophy”, generally recognized as the greatest scientific book ever written. In it, he presented his theories of motion, gravity and mechanics, explained the eccentric orbits of comets, the tides and their variations, the precession of the Earth's axis and the motion of the Moon.

SIR ISAAC NEWTON

Later in life, he wrote a number of religious tracts dealing with the literal interpretation of the Bible, devoted a great deal of time to alchemy, acted as Member of Parliament for some years, and became perhaps the best-known Master of the Royal Mint in 1699, a position he held until his death in 1727.

SIR ISAAC NEWTON

In 1703, he was made President of the Royal Society and, in 1705, became the first scientist ever to be knighted. Mercury poisoning from his alchemical pursuits perhaps explained Newton's eccentricity in later life, and possibly also his eventual death.

 Newton's Method for approximating the roots of a

curve by successive interactions after an initial guess

SIR ISAAC NEWTON

17TH CENTURY MATHEMATICS

GOTTFRIED LEIBNIZ (1646-1716)

GOTTFRIED LEIBNIZ

The German polymath Gottfried Wilhelm Leibniz occupies a grand place in the history of philosophy. He was, along with René Descartes and Baruch Spinoza, one of the three great 17th Century rationalists, and his work anticipated modern logic and analytic philosophy. Like many great thinkers before and after him, Leibniz was a child prodigy and a contributor in many different fields of endeavor.

GOTTFRIED LEIBNIZ

But, between his work on philosophy and logic and his day job as a politician and representative of the royal house of Hanover, Leibniz still found time to work on mathematics.

GOTTFRIED LEIBNIZ

He was perhaps the first to explicitly employ the mathematical notion of a function to denote geometric concepts derived from a curve, and he developed a system of infinitesimal calculus, independently of his contemporary Sir Isaac Newton. He also revived the ancient method of solving equations using matrices, invented a practical calculating machine and pioneered the use of the binary system.

GOTTFRIED LEIBNIZ

Like Newton, Leibniz was a member of the Royal Society in London, and was almost certainly aware of Newton’s work on calculus. During the 1670s (slightly later than Newton’s early work), Leibniz developed a very similar theory of calculus, apparently completely independently.

GOTTFRIED LEIBNIZ

Within the short period of about two months he had developed a complete theory of differential calculus and integral calculus (

see the section on Newton for a brief description and explanation of the development of calculus).

GOTTFRIED LEIBNIZ

Unlike Newton, however, he was more than happy to publish his work, and so Europe first heard about calculus from Leibniz in 1684, and not from Newton (who published nothing on the subject until 1693). When the Royal Society was asked to adjudicate between the rival claims of the two men over the development of the theory of calculus, they gave credit for the first discovery to Newton, and credit for the first publication to Leibniz.

GOTTFRIED LEIBNIZ

However, the Royal Society, by then under the rather bias presidency of Newton himself, later also accused Leibniz of plagiarism, a slur from which Leibniz never really recovered. Ironically, it was Leibniz’s mathematics that eventually triumphed, and his notation and his way of writing calculus, not Newton’s clumsier notation, is the one still used in mathematics today.

Leibniz’s and Newton’s

notation for Calculus

GOTTFRIED LEIBNIZ

GOTTFRIED LEIBNIZ

In addition to calculus, Leibniz re-discovered a method of arranging linear equations into an array, now called a matrix, which could then be manipulated to find a solution.

GOTTFRIED LEIBNIZ

A similar method had been pioneered by Chinese mathematicians’ almost two millennia earlier, but had long fallen into disuse. Leibniz paved the way for later work on matrices and linear algebra by Carl Friedrich Gauss.

GOTTFRIED LEIBNIZ

He also introduced notions of self-similarity and the principle of continuity which foreshadowed an area of mathematics which would come to be called topology.

GOTTFRIED LEIBNIZ

During the 1670s, Leibniz worked on the invention of a practical calculating machine, which used the binary system and was capable of multiplying, dividing and even extracting roots, a great improvement on Pascal’s rudimentary adding machine and a true forerunner of the computer.

GOTTFRIED LEIBNIZ

He is usually credited with the early development of the binary number system (base 2 counting, using only the digits 0 and 1), although he himself was aware of similar ideas dating back to the I Ching of Ancient China.

GOTTFRIED LEIBNIZ

Because of the ability of binary to be represented by the two phases "on" and "off", it would later become the foundation of virtually all modern computer systems, and Leibniz's documentation was essential in the development process.

Binary Number System

GOTTFRIED LEIBNIZ

GOTTFRIED LEIBNIZ

Leibniz is also often considered the most important logician between Aristotle in Ancient Greece and George Boole and Augustus De Morgan in the 19th Century. Even though he actually published nothing on formal logic in his lifetime, he enunciated in his working drafts the principal properties of what we now call conjunction, disjunction, negation, identity, set inclusion and the empty set.

17TH CENTURY MATHEMATICS

“I can calculate the motion of heavenly bodies but not

the madness of people”-Isaac Newton

“Music is the pleasure the human mind experiences from counting without being aware that it is counting.”-Gottfried Wilhelm Leibniz

THANK YOU!!