The Bernoulli family

As already noted in section 10.3the early part of the 18th century saw with the work of Antoine parent and alexis clairaut , the beginnings of solid analytic geometry. The first half of 18th century also saw the significant work of girolamo sacceri the fore runner of non Euclidean geometry, this work was considered in section 5.7 it will be recalled that lambert’s work in this area occurred toward of the end of the century and that the actual discovery of non Euclidean geometry by lobachevsky, janos bolyai and gauss took place in the early 19th century. But the bulk of the mathematics of the 18th found not until well into the 19th century that mathematical research generally emancipated itself from this view point.
The principal contributions to mathematics in the 18th century were made by members of the Bernoulli  family , Abraham de Moivre, Brook Taylor, Colin Maclaurin, Leonhard Euler, Alexis Claude Clairaut, jean, Johann, Joseph, Gaspard
One of the most distinguished families in the history of mathematics and science is the Bernoulli family of Switzerland which , from the late 17th century on has produced a remarkable number of capable mathematicians and scientist. The family record starts with the two brothers jakob Bernoulli (1654-1705) and Johann Bernoulli (1667-1748), some of whose mathematical accomplishments have already been mentioned in our book. These two man gave up earlier vocational interest and became mathematicians  when Leibniz’ 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 Jacob occupied the mathematical at basel University . Johan 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 on almost constant exchange of ideas with Leibniz and with each other.
The derivation in both rectangular and polar coordinates of the formula for the radius of curvature of a plane curve, the study of catenery curve with extensions to strings of variables density and strings under the action of the central force. The study of a number of other higher plane curves, the discovery of the so-called isochrones-or curve along with a body     will fall with uniform vertical velocity (it turned out to be semicubical parabola with fall with vertical cusptangent), the determination of the form taken by an elastic rod fixed at one end and carrying a weight at the other, the form assumed by a flexible rectangular sheet having to opposite edges held horizontally fixed 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 isoperimetric figures (planar closed path of given spesies and fixed perimeter which include a maximum area), and was thus one of the first mathematicians to work in calculus of variations. He was also (10.5) one of the early students of mathematical probalility. His book in this field the ars Conjectandi was posthumously published in 1713. There are several things in mathematics which now bear Jacob’s Bernoulli name: among these are the Bernoulli distributions and Bernoulli theorem of statistics and problability theorem. The Bernoulli equation met by every students of first course in differential equation, the Bernoulli’s number  and bernoulli’s polynomials of number theory interest and the lemniscale of Bernoulli encountered in any first course in calculus. In jakob Bernoulli solution to the problem of isochrone curve which published in the acta eruditorum in 1690 , we meet for the first time the word integral in calculus sense. Leibniz called the integral in calculus “calculus summatorius” ; in 1696 leibniz and johann Bernoulli agreed to call it calculus integralis. Jakob Bernoulli stuck by the way the equiangular spiral reproduces itself under a variety of transformations and asked in imitations of Archimedes, that such a spiral be engraved on his tombstone, along with the inscription “Eadem mutate eresurgo” ( I shall arise the same, though changed)
Johann Bernoulli was an even more prolific contributor to mathematics 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. As we have seen (in section 11-10) it was his material that the marquis de I’Hospital (1661-1704), under a curious financial agreement with johann, assembled in 1696 in to the first calculus textbook . it was in this  way that the familiar method of evaluating the indeterminate form 0/0 become incorrectly kown, in later calculus text, as I’Hospital 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, rectifications of curves and quadrature of areas by series, analytical trigonometry, the exponential calculus, and other subjects.  One of his more noted pieces of work in his contributions to the problem of the branchystochrone – the determination of the curve of quickest descent of the weighted particle moving between two given points in a gravitational field; the curved turned out to be an arc of an appropriate cycloid curve. This problem was also discussed by Jakob Bernoulli. The cycloid curved 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 from what initial point of the curves it starts.This problem , which was more generally discussed by johann, euler , and lagrange, had earlier be solve by Huygens (1673) and newton(1687) and applied by Huygens in the constructions of pendulum clocks.
(10.9)