100 great problems of elementary mathematics;: Their history by Heinrich Dorrie

By Heinrich Dorrie

"The assortment, drawn from mathematics, algebra, natural and algebraic geometry and astronomy, is very attention-grabbing and attractive." — Mathematical Gazette
This uncommonly attention-grabbing quantity covers a hundred of the main recognized ancient difficulties of trouble-free arithmetic. not just does the booklet endure witness to the intense ingenuity of a few of the best mathematical minds of historical past — Archimedes, Isaac Newton, Leonhard Euler, Augustin Cauchy, Pierre Fermat, Carl Friedrich Gauss, Gaspard Monge, Jakob Steiner, and so on — however it offers infrequent perception and idea to any reader, from highschool math scholar to expert mathematician. this is often certainly an strange and uniquely invaluable book.
The 100 difficulties are awarded in six different types: 26 arithmetical difficulties, 15 planimetric difficulties, 25 vintage difficulties pertaining to conic sections and cycloids, 10 stereometric difficulties, 12 nautical and astronomical difficulties, and 12 maxima and minima difficulties. as well as defining the issues and giving complete ideas and proofs, the writer recounts their origins and historical past and discusses personalities linked to them. frequently he provides now not the unique answer, yet one or less complicated or extra attention-grabbing demonstrations. in just or 3 circumstances does the answer think whatever greater than an information of theorems of simple arithmetic; accordingly, this can be a publication with a really extensive appeal.
Some of the main celebrated and exciting goods are: Archimedes' "Problema Bovinum," Euler's challenge of polygon department, Omar Khayyam's binomial growth, the Euler quantity, Newton's exponential sequence, the sine and cosine sequence, Mercator's logarithmic sequence, the Fermat-Euler leading quantity theorem, the Feuerbach circle, the tangency challenge of Apollonius, Archimedes' decision of pi, Pascal's hexagon theorem, Desargues' involution theorem, the 5 standard solids, the Mercator projection, the Kepler equation, selection of the location of a boat at sea, Lambert's comet challenge, and Steiner's ellipse, circle, and sphere problems.
This translation, ready particularly for Dover through David Antin, brings Dörrie's "Triumph der Mathematik" to the English-language viewers for the 1st time.

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Additional resources for 100 great problems of elementary mathematics;: Their history and solution

Sample text

Now each of the Rn n-membered products P includes (n – 1) paired multiplications of the form A · B. If we use f once as the multiplier in front of A, once as the multiplicand after A, once as the multiplier before B and once as the multiplicand after B, we thereby obtain from A · B four new paired products (f · A) · (B), (A · f) · (B), (A) · (f · B), and (A) · (B · f). Since these four arrangements of the factor f can be effected for each of the n – 1 paired subproducts of P, we obtain from P 4(n – 1) (n + l)-membered paired products.

Possibly, however, it may be just these little problems, which are, in their way, true jewels of mathematical miniature work, that will find the readiest readers and win new admirers for the queen of the sciences. As we have indicated already, a knowledge of higher analysis is not assumed. Consequently, the Taylor expansion could not be used for the treatment of the important infinite series. I hope nonetheless that the derivations we have given, particularly the striking derivation of the sine and cosine series, will please and will not be found unattractive even by mathematically sophisticated readers.

A product of this sort has the following appearance: in which the factor a is taken from the first α1 parentheses, the factor b from the next β1 parentheses, the factor a from the next α2 parentheses, etc. , n. If we set α1 + α2 + α3 + … equal to α and β1 + β2 + … equal to β the expression can be written in the simpler form Now the product P can generally be obtained in many other ways than the one described, for example, by taking a from the first a parentheses and b from the last β parentheses, or by taking b from the first β parentheses, and a from the last α parentheses, etc.

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