### Struik: A Concise History of Mathematics: The Eighteenth Century

#### by Carson Reynolds

Mathematics of the 18th Century focused on calculus and its application to mechanics. The major figures were:

- Leibniz (1646-1716)
- Jakob and Johann Bernoulli (1667-1748)
- Euler (1707-1783)
- Lagrange (1736-1813)
- Laplace (1749-1827)

Related to these leading figures was the work of Clairaut, D’Alembert, and Maupertuis, more closely associated with the French enlightenment. There were also the Swiss mathematicians Labert and Daniel Bernoulli. Scientific work centered around academies and the courts of enlightened despots. Much of the applications of math during this period were military and astronomical.

The Bernoulli family (of Basel, Switzerland) made several notable contributions to math. After Jakob struck up a correspondence with Leibniz, he and his brother Johann they made many contributions to calculus and solutions to ODEs. Jakob contributed the use of polar coordinates, the study of the cantenary, the lemniscate (1694), and the logarithmic spiral. He also found the isochrome, and isoperimetric figures, leading to a problem in the calculus of variations. He also did work on probability, contributing the theorem of Bernoulli on binomial distributions and Bernoulli numbers.

Johann Bernoulli is credited with the invention of the calculus of variations because of his brachystochrone problem which is solved by the cycloid. He also worked with his brother to find the equation of the geodesics on a surface. Johann’s two sons, Nicolaus and Daniel also contributed to mathematics. Nicholaus is associated with the “St. Petersburg problem” while Daniel is better known for his *Hydrodynamica* (1783). Daniel also provided a kinetic theory of gases and vibrating strings. While Johann worked on ODEs, Daniel worked on PDEs.

Leonhard Euler (also from Basel) studied under Johann. Euler is considered one of the most productive mathematicians of all time, despite losing one eye in 1735, and the other in 1766. One estimate puts his published works at 886. Euler contributed to every field of mathematics of his day, in several codifying the presentation into a nearly final form. For instance, our present style of trigonometry is drawn from his *Introductio in analysin infinitorum* (1748). This text covers a wide variety of subjects: infinite series, curves with the aid of equations, number theory, algebraic theory of elimination, and Zeta functions. It is considered the first text on analytic geometry. In Euler’s *Institutiones calculi differentialis* (1755) and *Institutiones calculi integralis* (1768-1774) we find calculus, a theory of differential equations, Taylor’s theorem, Euler’s summation formula, Eulerian integrals. His presentation of diff eqs (distinguishing between linear, exact, and homogeneous) is still our model for texts on the subjects. His *Mechanica, sive motus scientia analytice exposita* (1736) applied analytical methods to Newton’s dynamics. *Theoria motus corporum solidorum seu rigidorum* (1765) followed with a mechanics of solid bodies using analytical methods. *Vollstandige Anleittung zur Algebra* (1770) served to as a model for later texts on Algebra, provides theory for cubic and biquadratic equations and proves Fermat’s Last Theorem for the limit case of n=3,4. *Methodus inveniendi lineas curvas maximi minimive propietate gaudentes* (1744) was the first text on the calculus of variations, also providing Euler’s Equations, discovery of catenoid and right helicoid as minimal surfaces. A theorem connecting the number of vertices, edges, and faces of a closed polyhedron is shown. The line of Euler, obiform curbves, Euler’s constant were also provided. Graph theory might be traced to his paper on the seven bridges of Konigsberg. He contributed the law of reciprocity for quadratic residues to number theory. To astronomy he contributed *Theoria motus planetarum et cometarum* (1774), a treatise on celestial mechanics looked at the three body problem and the problem of finding longitude. Euler published works on hydraulics, ship construction, artillery, *Dioptrica* or rays moving through a system of lenses, music, and popular philosophy. Euler’s influence on later mathematicians can be seen in their use and our similarity to his notation.

Euler did have his weaknesses. His handling of infinite processes was careless, and primitive. He lacked tests of convergence or divergence modern math relies on. His presentation of calculus was also mired in the notion of different orders of zeros. The foundation of calculus remained controversial, prompting some like Guido Grandi (known for his study of rosaces) to think of infinite series in a mystical sense. D’Alembert in the *Encyclopedie* tried to a different foundation. He made use of infinities of different orders, but some felt his analysis, in which secant becomes tangent, did not escape Zeno’s paradox. In Berkeley’s *The Analyst* he attacked Newton’s fluxions and infinitesimals. Berkeley used the confusion and difficulties to strengthen his idealist philosophy. John Landen, who worked on a theory of elliptic integrals, tried to overcome calculus’ problems in a different manner. His derivation in *Residual Analysis* (1764) avoided infitesimals altogether. In more complex function, his method involved infinite series, which is somewhat similar to Lagrange’s algebraic methods.

Mathematical activity in France also continued to blossom. Newton was introduced to French culture in Voltaire’s *Lettres sur les Anglais* (1734) and translated into French by Mmme. Du Chatelet (1759). Mathematician divided themselves into Cartesian and Newtonian camps. The Cartesian school believed the earth to elongate at the poles, while the Newtonians thought it flat. Cassini provided astrological evidence in favor of the Cartesian hypothesis, which was refuted by Pierre de Maupertuis. Maupertuis was later ridiculed for a proof of God’s existence, but in the same work presented the basis of the principle of least action. Euler restated the principle of least action, later to be adopted by Lagrange and Hamilton. Alexis Claude Clairaut, a contemporary of Maupertuis, made a first attempt to deal with the analytical and differential geometry of space curves. He also published in 1743 *Theorie de la figure de la terre* which dealt with equilibrium of fluids and attraction of ellipsoids of revolution. Following this he published *Theorie de la lune* (1752) which dealt with the three body problem. Clairaut also made contributions to the theory of line integrals and differential equations. Clairaut’s name is preserved in Clairaut’s equation and Clairaut’s theorem.

D’Alembert was the leading mathematician of the Encyclopedists. He is known for D’Alembert’s principle which reduces the dynamics of solid bodies to statics. His work focused on hydrodynamics, aerodynamics, and the three body problem. He collaborated with Daniel Bernoulli in 1747 on a theory of vibrating strings. D’Alembert also anticipated Fourier in the construction of any function using trigonometric series, with which Euler took issue. D’Alembert also did work on the fundamental theory of algebra and probabilistic theory. *The Doctrine of Chances* (1716) by de Moivre, introduced his namesake theorem but also the normal pdf. Georges Louis Leclerc showed in 1777 the first instance of geometrical probability. The needle problem and the *probabilite des judgements* were prominent topics during enlightenment mathematics, and the work of the Marquis de Condorcet.

De Moivre, Stirling, and Landen were representatives of English 18th century math. The growth of English math was retarded by slow replacement of Newton’s notation with Leibniz’s. The Leading English-speaking mathematician was Colin Maclaurin, a disciple of Newton. He worked on cubes of second and higher order, and the attraction of ellipsoids. Several of his theorems are of important in projective geometry and the theory of plane curves. His *Geometria organica* (1720) presented what is now known as Cramer’s paradox. Maclaurin also defended Newton against Berkeley in his *Treatise of Fluxions* (1742) which contained Maclaurin’s series, previously appearing in *Methodus incrementorum* (1715) by Brook Taylor, and more commonly known as Taylor’s series.

Lagrange’s early work was on the calculus of variations, putting it on a purely analytical basis. He applied Euler’s principle of least action to problems in dynamics. He also did work on lunar theory and the problem of determining latitude. Lagrange found some of the first particular solutions to the 3-body problem. Lagrange’s theorem states that it is possible to start 3-bodies in such a manner that their orbits are similar ellipses describable in the same time (1772). He developed methods for separating the real roots of algebraic equations, and approximating them with continued fractions. Lagrange looked at the properties of higher degree equations compared to those of a degree Mecanique analytique, appearing 100 years after Newton’s *Principia* applied analysis to points and rigid bodies. By use of D’Alembert’s principle, he arrived at the Lagrangian form. His work was a triumph of pure analysis over geometry, not containing a single figure.

Laplace was best known for two great works: *Theorie analytique des probabilites* (1812) and *Mecanique celeste* (1799-1825). His celestial mechanics was the culmination of the work of Newton, Clairaut, D’Alembert, Euler, Lagrange, and Laplace’s own. It dealt with lunar theory, the 3-body problem, perturbations of the planets, and the stability of the solar system. It also provided potential theory and Laplace’s equation. Lagrange provided a negative definition of probabilities by postulating equally likely events. Laplace also provided the Laplace transform and reworked Thomas Bayes’ theory of inverse probabilities.

There was a sort of pessimism that followed the triumph of Laplace, a feeling that all that could be done had been done. Of course Guass and Poincare were to show how little truth there was in that. This brings Quine’s circle to mind: the more things we know, the more related things we don’t know.

I wanted to know who was involved in the invention of the binomial theorem. Where there many persons or just one person.