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I was wondering if there is any book that chronicles the progress of Math over the centuries and also mentions about how/when applications of various theories were discovered/invented.

I have been trying to study university level math and I keep thinking about applications. Applications like principal component analysis, cryptography needed modern computing power. So before computers, how was all this abstract algebra used?

What kind of math was required for the manufacturing industry since the industrial revolution started (manufacturing heavy and precise equipment from steam engines to transistors...)?

EDIT: I saw a lot of such "history of math" books on amazon, but I am looking for something like "this technology is dependent on that math/science theory", preferably written in somewhat "math history for kids/dummies" style.

I had asked this on math.stackexchange.com some time back, got no suggestions and now it can't be migrated here. So, posting this as a new question. Please advise.

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  • $\begingroup$ I believe this is too broad and in fact can be divided in different questions. I particularly like: "What kind of math was required for the manufacturing industry since the industrial revolution started?". It'd be interesting to know in which way the industrial revolution influenced math and have a complete account on how it did on physics. $\endgroup$ – hjhjhj57 Jan 20 '15 at 10:06
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I do not know of such book, and will be glad if someone suggest it. But let me attempt a broad sketch of applications of Mathematics before 1900:-)

The first main application was astronomy, there is no doubt in this. It started in the Hellenistic Greece, and probably until the 18-th century this was the main application. Astronomy and (after Newton) celestial mechanics. Ability to predict the Moon motion and to predict the existence of new planets were probably the greatest "practical" achievements of mathematics until the middle 19-th century.

Serious applications to physics and engineering begin in 17-th century. With the invention of calculus, we see an explosive growth of applied mathematics. Bernoulli's and Euler tried to model literally everything: blood flow in the blood vessels, behavior of a ship at sea, behavior of a projectile inside the gun barrel (not only outside!) and so on. (It is questionable whether these early theories had any influence on the industry and technology. The problems they considered were too complicated for mathematics of that time). One early example, of successful applications of mathematics that I know besides astronomy is the design of clocks and watches.

Two entirely new playgrounds were opened in the beginning of 19-th century: electricity/magnetism and heat conduction. Here we already have examples of direct influence of mathematics on industry. I mean design of steam engines, design of rails and wires, and various machines, etc. One especially famous story is Lord Kelvin's mathematical contribution to the design of the first transatlantic cable. (He became a Lord because of this! This was around 1850-s) He also improved the compass for marine navigation using modern mathematical theories.

These are just few examples. But in general, one can say that since the beginning of 19-th century, mathematics becomes a true productive force. Necessary for engineering and industry (not only for astronomy and physics). Hydrodynamics was successively applied to ship construction in the end of 19-th century, though the first attempts start with Bernoulli and Euler.

You ask explicitly about algebra. Algebra had fewer applications before the modern period. One of them was to coding. They say this application begins with Vieta, the founder of algebraic notation, but I do not have precise references.

I did not mention applications of geometry to geodesy, and building construction, and this is perhaps as old as applications to astronomy.

It will indeed be interesting if someone writes a book along these lines. But the examples I mentioned are scattered in many books. One notable example is T. Korner's books "The pleasures of counting" and "Fourier Analysis".

A remark on computation before electronic computers. Of course, massive computations were always needed. One milestone was invention of logarithms in the early 17 century. This was not enough, and people were always working on the invention of computing machines, beginning from 17-th century. There were also "computing centers" where many people (called "computOrs") computed. Specialized tables for all kinds of computation were developed. There were quite advanced analog computing machines, like harmonic analizers (invented by the same Lord Kelvin for tide prediction). Of course, all this changed with the wide spread of cheap electronic computers, but this happened only very recently.

EDIT. I think, fdb's comment requires somewhat longer answer than fits in a comment. Indeed, Pythagoras lived in Italy, Thales in Turkey:-) and Euclid in Egypt. However I think it will be a ridiculous anachronism to call Thales "a Turk". All these people are commonly called Greeks, and there are at least some reasons for this.

Second, and more important. I use the words "mathematics" and "astronomy" in a narrow sense. (You may use them differently, I do not want to argue about words). All human cultures had some means of counting and measuring the amounts of grain, beer, and the areas of land plots. Many of them also had a tax code. From my point of view this is not "mathematics" yet. Mathematics (in the narrow sense), as far as I know was invented by some Greeks (according to the legend, either by Thales or Pythagoras) and the earliest documentary evidence of it comes to us from Hellenistic times/states. Namely from Alexandria, now in Egypt.

Astronomy was first practiced in Babylon (and we have well-documented prime sources of this) but there was no much mathematics yet. Real APPLICATIONS of mathematics to astronomy come from Hellenistic Greeks (who lived everywhere on the Mediterranean coast and islands.)

Eudoxus was a great mathematician, of course (judging by what is attributed to him) but his contribution to astronomy (comparable to the contribution of Aristarchus) is not really an application of mathematics to astronomy. Eudoxus's spheres (and Aristarchus determination of the ratio of distances to Sun and Moon) where nice, beautiful mathematical exercises, but they were not real "successful applications of mathematics to astronomy". Aristarchus proposal is not realistic, and it is clear that he did not even try to measure the quantities involved. If you do not believe me, try to determine the ratio of these distances with Aristarchus method. I tried. Really. That the planets do not actually move as Eudoxus described was probably clear even to is contemporaries, and certainly to anyone who did observations. So applications of mathematics to astronomy, to the best of my knowledge, begin with Apollonius and Hypparchus (who lived in Egypt:-)

Of course, astronomy in the narrow sense (as I understand it), does not include such facts that there are stars and planets, or names of constellations, or the fact that that Sun raises in the East and sets in the West, and so on. The facts that anyone discovers once she looks at the sky regularly for few months.

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    $\begingroup$ The job for humans who compute was sometimes called computer rather than computor. See the heading and text of the 19th century New York Times ad here: danwin.com/2013/02/…. $\endgroup$ – KCd Jan 21 '15 at 4:47
  • $\begingroup$ “It started in Hellenistic Greece….” I am not entirely sure what “Greece” means in this context (Is Alexandria in Greece?). But that is not the issue. The ancient Egyptians had at least a rudimentary form of algebra. Then there is the “Pythagorean” theorem (which may or may not have been first posited by Pythagoras). There is lots of geometry in Plato (e.g. in the Timaeus). And what about Eudoxus? All definitely pre-Hellenistic. $\endgroup$ – fdb Jan 21 '15 at 16:59
  • $\begingroup$ You are right:-) I should have said "Hellenistic states". The most important of them for mathematics was Ptolemy's state on the territory of current Egypt. $\endgroup$ – Alexandre Eremenko Jan 21 '15 at 22:45
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    $\begingroup$ Concerning pre-Hellenistic mathematics, on my opinion there were no serious applications to astronomy, but of course it is always risky to say who did something "for the first time". Anyway, I do not insist on this statement:-) I just know too little about pre-Hellenistic math and astronomy. $\endgroup$ – Alexandre Eremenko Jan 21 '15 at 22:48
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Some suggestions; not a single history, but some "pictures" :

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As a general and high level overview, I enjoyed Mathematics for the Million by Lancelot Hogben. It's not very in-depth or modern, but it does put many mathematical developments in their historical context.

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