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1.5: Age of Enlightenment

The Age of Enlightenment is a term used to describe a phase in Western philosophy and cultural life in which reason was advocated as the primary source and legitimacy for authority. It developed simultaneously in Germany, France, Britain, the Netherlands, and Italy around the 1650’s and lasted until the French Revolution in 1789. The intellectual and philosophical developments led to moral, social, and political reforms. The principles of individual rights, reason, common sense, and deism were a revolutionary departure from the existing theocracy, autocracy, oligarchy, aristocracy, and the divine right of kings. It led to political revolutions in France and the United States. It marks a dramatic departure from the Early Modern period which was noted for religious authority, absolute state power, guild-based economic systems, and censorship of ideas. It opened a new era of rational discourse, liberalism, freedom of expression, and scientific method. This new environment led to tremendous advances in both science and mathematics in addition to music (Johann Sebastian Bach, Mozart), literature (Goethe), philosophy (Spinoza, Kant) and art (Rubens). Scientific development during the 17\(^{th}\) century included the pivotal advances made by Newton and Leibniz at the beginning of the revolutionary Age of Enlightenment, culminating in the development of variational calculus and analytical mechanics by Euler and Lagrange. The scientific advances of this age include publication of two monumental books "Philosophiae Naturalis Principia Mathematica" by Newton in 1687 and Mécanique analytique by Lagrange in 1788. These are the definitive two books upon which classical mechanics is built.


René Descartes (1596-1650) attempted to formulate the laws of motion in 1644. He talked about conservation of motion (momentum) in a straight line but did not recognize the vector character of momentum. Pierre de Fermat (1601-1665) and René Descartes were two leading mathematicians in the first half of the 17 century. Independently they discovered the principles of analytic geometry and developed some initial concepts of calculus. Fermat and Blaise Pascal (1623-1662) were the founders of the theory of probability. Fermat revived the principle of least time, which states that "light travels between two given points along the path of shortest time" and was used to derive Snell’s law in 1657. This enunciation of variational principles in physics played a key role in the historical development of the principle of least action that underlies the analytical formulations of classical mechanics.

\begin{equation}\sum_{i=1}^{N}(F_i-\dot p_i) \delta \dot r_i = 0\end{equation}

Isaac Newton (1642-1727) made pioneering contributions to physics and mathematics as well as being a theologian. At 18 he was admitted to Trinity College Cambridge where he read the writings of modern philosophers like Descartes, and astronomers like Copernicus, Galileo, and Kepler. By 1665 he had discovered the generalized binomial theorem, and began developing infinitessimal calculus. Due to a plague, the university closed for two years in 1665 during which Newton worked at home developing the theory of calculus that built upon the earlier work of Barrow and Descartes. He was elected Lucasian Professor of Mathematics in 1669 at the age of 26. From 1670 Newton focussed on optics leading to his "Hypothesis of Light" published in 1675 and his book "Opticks" in 1704. Newton described light as being made up of a flow of extremely subtle corpuscles that also had associated wavelike properties to explain diffraction and optical interference that he studied. Newton returned to mechanics in 1677 by studying planetary motion and gravitation that applied the calculus he had developed. In 1687 he published his monumental treatise entitled "Philosophiae Naturalis Principia Mathematica" which established his three universal laws of motion, the universal theory of gravitation, derivation of Kepler’s three laws of planetary motion, and was his first publication of the development of calculus which he called "the science of fluxions".


Newton’s laws of motion are based on the concepts of force and momentum, that is, force equals the rate of change of momentum. Newton’s postulate of an invisible force able to act over vast distances led him to be criticized for introducing "occult agencies" into science. In a remarkable achievement, Newton completely solved the laws of mechanics. His theory of classical mechanics and of gravitation reigned supreme until the development of the Theory of Relativity in 1905. The followers of Newton envisioned the Newtonian laws to be absolute and universal. This dogmatic reverence of Newtonian mechanics prevented physicists from an unprejudiced appreciation of the analytic variational approach to mechanics developed during the 17\(^{th}\) through 19\(^{th}\) centuries. Newton was the first scientist to be knighted and was appointed president of the Royal Society. Newton had an unpleasant character and was notorious for the heated disputes he provoked with other academics. Eventually he left academia and became active in politics. This led to his appointment as Warden of the Royal Mint where he conducted a major campaign against counterfeiting that sent several men to their death on the gallows.

Gottfried Leibniz (1646-1716) was a brilliant German philosopher, a contemporary of Newton, who worked on both calculus and mechanics. Leibniz started development of calculus in 1675, ten years after Newton, but Leibniz published his work in 1684, which was three years before Newton’s Principia. Leibniz made significant contributions to integral calculus and was responsible for the calculus notation currently used. He introduced the name calculus based on the Latin word for the small stone used for counting. Newton and Leibniz were involved in a protracted argument over who originated calculus. It appears that Leibniz saw drafts of Newton’s work on calculus during a visit to England. Throughout their argument Newton was the ghost writer of most of the articles in support of himself and he had them published under non-de-plume of his friends. Leibniz made the tactical error of appealing to the Royal Society to intercede on his behalf. Newton, as president of the Royal Society, appointed his friends to an "impartial " committee to investigate this issue, then he wrote the committee’s report that accused Leibniz of plagiarism of Newton’s work on calculus, after which he had it published by the Royal Society. Still unsatisfied he then wrote an anonymous review of the report in the Royal Society’s own periodical. This bitter dispute lasted until the death of Leibniz. When Leibniz died his work was largely discredited. The fact that he falsely claimed to be a nobleman and added the prefix von to his name, coupled with Newton’s vitriolic attacks, did not help his credibility. Newton is reported to have declared that he took great satisfaction in "breaking Leibniz’s heart." Studies during the 20\(^{th}\) century have largely revived the reputation of Leibniz and he is acknowledged to have made major contributions to the development of calculus.


Leibniz made significant contributions to classical mechanics. In contrast to Newton’s laws of motion, which are based on the concept of momentum, Leibniz devised a new theory of dynamics based on kinetic and potential energy that anticipates the analytical variational approach of Lagrange and Hamilton. Leibniz argued for a quantity called the "vis viva", which is Latin for "living force" that equals twice the kinetic energy. Leibniz argued that the change in kinetic energy is equal to the work done. In 1687 Leibniz proposed that the optimum path is based on minimizing the time integral of the vis viva which is equivalent to the action integral. Leibniz used both philosophical and causal arguments in his work which were equally acceptable during the Age of Enlightenment. Unfortunately for Leibniz, his analytical approach based on energies, which are scalars, appeared contradictory to Newton’s intuitive vectorial treatment of force and momentum. There was considerable prejudice and philosophical opposition to the variational approach which assumes that nature is thrifty in all of its actions. The variational approach was considered to be speculative and "metaphysical" in contrast to the causal arguments supporting Newtonian mechanics. This opposition delayed full appreciation of the variational approach until the start of the 20\(^{th}\) century.


Johann Bernoulli (1667-1748) was a Swiss mathematician who was a student of Leibniz’s calculus, and sided with Leibniz in the Newton-Leibniz dispute over the credit for developing calculus. Also Bernoulli sided with the Descartes’ vortex theory of gravitation which delayed acceptance of Newton’s theory of gravitation in Europe. Bernoulli pioneered development of the calculus of variations by solving the problems of the catenary, the brachistochrone, and Fermat’s principle. The Bernoulli family is famous for its contributions to mathematics and science; Johann’s son Daniel played a significant role in the development of the well known Bernoulli Principle in hydrodynamics.


Pierre Louis Maupertuis (1698-1759) was a student of Johann Bernoulli and conceived the universal hypothesis that in nature there is a certain quantity called action which is minimized. Although this bold assumption correctly anticipates the development of the variational approach to classical mechanics, he obtained his hypothesis by an entirely incorrect method. He was a dilettante whose mathematical prowess was far behind the high standards of that time, and he could not establish satisfactorily the quantity to be minimized. His teleological argument was influenced by Fermat’s principle and the corpuscle theory of light that implied a close connection between optics and mechanics.

Leonhard Euler (1707-1783) was the preeminent Swiss mathematician of the 18th century and was a student of Johann Bernoulli. Euler developed, with full mathematical rigor, the calculus of variations following in the footsteps of Johann Bernoulli. Euler used variational calculus to solve minimum/maximum isoperimetric problems which had attracted and challenged the early developers of calculus, Newton, Leibniz, and Bernoulli. Euler also was the first to solve the rigid-body rotation problem using the three components of the angular velocity as kinematical variables. Euler became blind in both eyes by 1766 but that did not hinder his prolific output in mathematics due to his remarkable memory and mental capabilities. Euler’s contributions to mathematics are remarkable in quality and quantity; for example during 1775 he published 1Teleology is any philosophical account that holds that final causes exist in nature, meaning that – analogous to purposes found in human actions – nature inherently tends toward definite ends. 1.6. 19 CENTURY 5 one mathematical paper per week in spite of being blind. Euler implicitly implied the principle of least action using vis visa which is not the exact form explicitly developed by Lagrange. Jean le Rond d’Alembert (1717-1785) was a French mathematician and physicist who had the clever idea of extending use of the principle of virtual work from statics to dynamics. D’Alembert’s Principle rewrites the principle of virtual work in the form X  =1 (F − p˙ )r = 0 where the inertial reaction force p˙ is subtracted from the corresponding force F. This extension of the principle of virtual work applies equally to both statics and dynamics leading to a single variational principle.


Joseph Louis Lagrange (1736-1813) was an Italian mathematician who was a student of Leonhard Euler and his work paralleled that of Euler. In 1788 Lagrange published his monumental treatise on analytical mechanics entitled "Mécanique Analytique" which describes his new, immensely powerful, analytical technique that can solve any mechanical problem without resort to geometrical considerations. His theory only required the analytical form of the scalar quantities kinetic and potential energy. In the preface of his book he refers modestly to his extraordinary achievements with the statement "The reader will find no figures in the work. The methods which I set forth do not require either constructions or geometrical or mechanical reasonings: but only algebraic operations, subject to a regular and uniform rule of procedure." Lagrange also introduced the concept of undetermined multipliers to handle auxiliary conditions which plays a vital part of theoretical mechanics. William Hamilton, an outstanding figure in the analytical formulation of classical mechanics, called Lagrange the "Shakespeare of mathematics," on account of the extraordinary beauty, elegance, and depth of the Lagrangian methods. Lagrange also pioneered numerous significant contributions to mathematics. For example, Euler, Lagrange, and d’Alembert developed much of the mathematics of partial differential equations. Lagrange survived the French Revolution and, in spite of being a foreigner, Napoleon named Lagrange to the Legion of Honour and made him a Count of the Empire in 1808. Lagrange was honoured by being buried in the Pantheon.


Jean Baptiste Joseph Fourier (1768-1830) was a French mathematician and physicist who was a student of Lagrange. Fourier is most famous for the development of Fourier analysis which includes Fourier series, and Fourier transforms. His work has many applications to classical mechanics such as all forms of wave motion, signal processing, and solving for the eigenfunctions of linear equations.