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The 18th century had sought a huge, immense progress in mathematics. As late as 17th century people still wrote algebraic equations with words. But by the end of 18th century we had

  • Mathematical analysis and theory of differential equations
  • Complex numbers fully formalized and the connection between trigonometric and hyperbolic functions discovered
  • Gamma and Zeta functions discovered and described
  • Calculus of variations
  • Theory of graphs
  • Probability theory and combinatorics

etc etc.

It seems that the most of mathematics that could be used in today's engineering applications was developed in 18th century. Neither 19th nor 20th century had saught such a major breakthrough despite the rise of industry, technology and appearance of computers.

I wonder what was the drive behind these successes in mathematics at a century when it would not be expected to find a lot of applications.

Was it indeed driven by the needs of industry, military and technological progress?

Or was it developed as a form of amusement because some nobility had a lot of spare time?

Or was it due to personality traits of some of government leaders who sponsored mathematics and made it a very prestigious profession of the time?

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    $\begingroup$ Galois theory ? $\endgroup$ – user2255 Sep 1 '15 at 17:46
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    $\begingroup$ More generally, group theory. One very very famous mathematician said in his lectures that 18 century was the age of Analysis, 19s of Group theory and 20s of cohomology. Somewhat controversial point of view:-) $\endgroup$ – Alexandre Eremenko Sep 2 '15 at 21:59
  • $\begingroup$ Because the master lived in that century. $\endgroup$ – timur Sep 28 '17 at 3:22
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One can easily name one main intrinsic reason: invention of Calculus in the very end of the previous century, and "invention of mathematical physics" by Newton. It happened in the very beginning of 18-s century that sufficiently many people suddenly realized that mathematics can effectively explain the world.

But of course, there were also outside reasons, like development of industry and capitalism. That is people were interested in the explanation of the world.

By "invention of physics" I mean rigorous formulation of principal laws of mechanics and explanation of Kepler laws, explanation of tides, explanation of the shape of the Earth. All this was shortly confirmed by measurements, and these discoveries made an enormous impression. For the first time it was evident to many people that mathematics can really explain the world.

On the other hand, the external reasons were also important: people ( also kings and governments) were really INTERESTED in these questions, they payed for expeditions to check the true shape of the Earth and built observatories to observe stars and planets. They were interested in supporting mathematicians and several well payed jobs appeared.

Laws of motions of stars and planets had one important application at that time: to navigation. And reliable navigation was important for overseas trade and functioning of the navies. Another very important application was military science (gunnery and fortification). True industrial applications of mathematics came a bit later.

But people were so very much impressed by physics and calculus that they immediately started applying mathematics literally everywhere, not always with success, initially. For example, D. Bernoulli developed his famous laws of hydrodynamics trying to model the blood flow in the blood vessels. This could not give interesting results in 18s century, but the laws of hydrodynamics were created. Similarly, Euler wrote a lot about interior ballistics and shipbuilding. It was too early for any success in such complicated things. But this shows how great was enthusiasm about applications of mathematics they had at that time.

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The progress in 18th century followed two important breakthroughs in 17th. Algebraic notation which was essentially modern appeared in Descartes's Geometry published already in 1637, and the calculus was invented by Newton and Leibniz by 1680s. Some methods of the calculus of variations were also developed in 17th century, in connection with the famous brachistochrone problem. If 17th century can be compared to breaking the wall the 18th was more of gathering of the stones, systematization and advancement of previously made discoveries. This progress accelerated in the 19th century and led to another major breakthrough at the end of it, comparable to that in the 17th century, creation of set theory by Cantor and establishment of logical foundations of modern mathematics by Weierstrass, Dedekind, Frege, Hilbert and others. Creation of new domains of mathematics that followed in the 20th century, mathematical logic, abstract algebra, algebraic geometry, algebraic topology, functional analysis, category theory, theory of computation, etc., is no less impressive if not more impressive than the progress in the 18th century.

As for reasons, it is hard to miss that both revolutions in 17th and 19th centuries were contemporary with revolutions in natural sciences, especially in physics, breakthroughs in technology, and major social and cultural changes (transitions from medieval to capitalist, and from national to global societies) that fueled them directly and indirectly. Some of it was intrinsic, many ancient mathematical works became available in Europe during Renaissance and stimulated great improvements in algebraic notation and methods, by Cardano and Viete especially, as well as ideas about tangents and volumes that led to calculus; some was technological, invention of the printing press made dissemination of scientific works much easier and broader, so more people could be pulled into doing mathematics and science; some was political, 17th and 18th centuries saw creation of government funded scientific societies like the Royal Society in England and Academy of Sciences in France; and some was economical, emerging capitalist trade and industry stimulated quantitative methods in mechanical engineering, cartography, chemistry, etc. that created steady supply of new problems and demand for new methods to solve them.

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    $\begingroup$ I disagree with one point of this answer: that 18s "was more of gathering of the stones, systematization and advancement of previously made discoveries". 18s century was the time of real breakthroughs in mathematics. What we teach in the universities to non-math students (scientists and engineers) was mostly invented in 18s century. $\endgroup$ – Alexandre Eremenko Sep 1 '15 at 22:53
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    $\begingroup$ I don't think establishig firm foundations of mathematics in 19th century can be called "accelerated progress". Nineteenth century is characterized mostly by justifying and rigorously strengthening the achievements of 18th century. There were not so many breakthroughs. $\endgroup$ – Anixx Sep 2 '15 at 1:24
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    $\begingroup$ @Anixx Galois theory, Riemannian geometry, Fourier analysis, Boolean algebra, non-Euclidean geometry come from 19th century, so does development of abstract algebraic notions by Dedekind, and of axiomatic methods by Pasch and Hilbert. For comparison, in 18th century analysis remained very formal in nature, and functions were essentially identified with closed formulas. Even complex numbers remained ghosts until Argand (1806). 18th century saw foremost advances in probability and differential equations (including calculus of variations), the rest required breakthroughs. $\endgroup$ – Conifold Sep 2 '15 at 2:49
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    $\begingroup$ @Anixx Yes, in formal manipulations. And although his are most bold and spectacular "generality of algebra" behind them goes back to Leibniz. hsm.stackexchange.com/questions/2035/… Theories of complex functions, elliptic functions and integrals in particular, analytic continuation, Riemann surfaces, etc. were based on geometric and topological intuitions, which did not come to light until the geometric interpretation of complex numbers was discovered. $\endgroup$ – Conifold Sep 2 '15 at 4:15
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    $\begingroup$ @Conifold: I would not compare establishing foundations in 19 century (Cantor, etc.) to the Newtonian revolution. Since that revolution, mathematics experiences fast and continuous growth and progress until now, and the achievements of 18, 19 and 20 century are inseparable. $\endgroup$ – Alexandre Eremenko Sep 2 '15 at 21:56

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