The principal contributions to Mathematics in the eighteen century were made by members of the Bernoulli  family, Abraham De Moivre, Brook Taylor, Colin Maclaurin, Leonhard Euler, Alexis Claude Clairaut, Jean-Le-Rond d’Alembert, Johann Heinrich Lambert, Joseph Louis Lagrange, Pierre-Simon Laplace, Adrien-Marie Legendre, Gaspard Monge and Lazare Carnot. It will be observed that the bulk of the mathematics of these men found its genesis and its goal in the applications of the calculus to the fields of mechanics and astronomy. It was not until well into the nineteenth century that mathematical research generally emancipated itself from this viewpoint. Now we will describe the remarkable Bernoulli family.

One of the most distinguished families in the history of mathematics and science is the Bernoulli family from Switzerland, which, from the late seventeenth century on, produced an unusual number of capable mathematicians and scientists. The family record starts with the two brothers, Jakob Bernoulli (1654-1705) and Johann Bernoulli (1667-1748). These two men gave up earlier vocational interests and became mathematicians when Leibniz’s papers began to appear in the Acta eruditorum. They were among the first mathematicians to realize the surprising power of the calculus and to apply the tool to a great diversity of problems. From 1687 until his death, Jakob occupied the mathematics chair at Basel University. Johann, in 1697, became a professor at Groningen University, and then, on Jakob’s death in 1705, succeeded his brother in the chair at Basel University, to remain there for the rest of his life. The two brothers, often bitter rivals, maintained an almost constant exchange of ideas with Leibniz and with each other.
Among Jakob Bernoulli’s contributions to mathematics are the early use of polar coordinates, the derivation in both rectangular and polar coordinates of the formula for the radius of curvature of a plane curve, the study of the catenary curve with extensions to strings of variable density and strings under the action of a central force, the study of a number of other higher plane curves, the discovery of the so-called isochrone-or curve along which a body will fall with uniform vertical velocity (it turned out to be a semicubical parabola with a vertical cusptangent), the determination of the form taken by an elastic rod fixed at one end and carrying a weight at the other, and the form assumed by a flexible rectangular sheet having two opposite edges held horizontally at the same height and loaded with a heavy liquid, and the shape of a rectangular sail filled with wind. He also proposed and discussed the problem of isometric figures (planar closed paths of given species and fixed perimeter which include a maximum area), and was thus one of the first mathematicians to work in the calculus of variations. He was also one of the early students of mathematical probability; His book in this field, Ars conjectandi, was posthumously published in 1713.

There are now several things in mathematics that now bear Jakob Bernoulli’s name. Among these are the Bernoulli distribution and Bernoulli theorem of statistics and probability theory, the Bernoulli equation met by every student of a first course in differential equations, the Bernoulli numbers and Bernoulli polynomials of number theory interest, and the lemniscate of Bernoulli encountered in any first course in the calculus. In Jakob Bernoulli’s solution to the problem of the isochrone curve, which was published in the Acta eruditorum in 1690, we meet for the first time the word “integral” in a calculus sense. Leibniz had called the integral calculus calculus summatorius; in 1696 Leibniz and Johann Bernoulli agreed to call it calculus integralis. Jakob Bernoulli was struck by the way the equiangular spiral reproduces itself under a variety of transformations and asked, in imitation of Archimedes, that such a spiral be engraved on his tombstone, along with the inscription “Eadem mutata resurgo” (“I shall arise the same though changed”).

Johann Bernoulli was an even more prolific contributor to mathematics than was his brother Jakob. Though he was a jealous and cantankerous man, he was one of the most successful teachers of his time. He greatly enriched the calculus and was very influential in making the power of the new subject appreciated in continental Europe. It was his material that the Marquis de L’Hospital (1661-1704), under a curious financial agreement with Johann, assembled in 1696 into the first calculus textbook. It was in this way that the familiar method of evaluating the indeterminate form 0/0 became incorrectly known, in later calculus texts, as L’Hospital’s rule. Johann Bernoulli wrote on a wide variety of topics, including optical phenomena connected with reflection and refraction, the determination of the orthogonal trajectories of families of curves, rectification of curves and quadrature of areas by series, analytical trigonometry, the exponential calculus and other subjects.

One of his more noted pieces of work is his contribution to the problem of the brachystochrone-the determination of the curve of quickest descent of a weighted particle moving between two given points in a gravitational field; the curve turned out to be an arc of an appropriate cycloid curve. The problem was also discussed by Jakob Bernoulli. The cycloid curve is also the solution to the problem of the tautochrone-the determination of the curve along which a weighted particle will arrive at a given point of the curve in the same time interval no matter form what initial point of the curve it starts. The latter problem, which was more generally discussed by Johann Bernoulli, Euler and Lagrange, had earlier been solved by Huygens(1673) and Newton(1687), and applied by Huygens in the construction of pendulum clocks.

Johann Bernoulli had three sons, Nicolaus (1695-1726), Daniel (1700-1782) and Johann II (1710-1790), all of whom won renown as eighteenth century mathematicians and scientists. Nicolaus, who showed great promise in the field of mathematics, was called to the St. Petersburg Academy, where he unfortunately died by drowning, only eight months later. He wrote on curves, differential equations and probability. A problem in probability, which he proposed from St. Petersburg, later became known as the Petersburg paradox. The problem was investigated by Nicolaus’ brother Daniel, who succeeded Nicolaus at St. Petersburg. Daniel returned to Basal seven years later.

Daniel was the most famous of Johann’s three sons, and devoted most of his energies to probability, astronomy, physics and hydrodynamics. In probability he devised the concept of moral expectation, and in his Hydrodynamica, of 1738, appears the principle of hydrodynamics that bears his name in all present-day elementary physics texts. He wrote on tides, established the kinetic theory of gases, studied the vibrating string and pioneered in partial differential equations. Johann II, the youngest of the three sons, studied law but spent his later years as a professor of mathematics at he University of Basel. He was particularly interested in the mathematical theory of heat and light.

There was another eighteenth-century Nicolaus Bernoulli (1687-1759), a nephew of Jakob and Johann, who achieved some fame in mathematics. This Nicolaus held, for a time, the chair of mathematics at Padua once filled by Galileo. He wrote extensively on geometry and differential equations. Later in life he taught logic and law. Johann Bernoulli II had a son Johann III (1744-1807) who, like his father, studied law but then turned to mathematics. When barely nineteen years old, he was called to the Berlin Academy. He wrote on astronomy, the doctrine of chance, recurring decimals and intermediate equations. Lesser Bernoulli descendants are Daniel II (1751-1834) and Jakob II (1759-1789), two other sons of Johann II, Christoph (1782-1863), a son of Daniel II and Johann Gustav (1811-1863), a son of Christoph.