Elementary

100 great problems of elementary mathematics: their history by Heinrich Dorrie PDF

By Heinrich Dorrie

ISBN-10: 0486318478

ISBN-13: 9780486318479

ISBN-10: 0486613488

ISBN-13: 9780486613482

"The assortment, drawn from mathematics, algebra, natural and algebraic geometry and astronomy, is very fascinating and attractive." — Mathematical Gazette

This uncommonly attention-grabbing quantity covers a hundred of the main well-known historic difficulties of common arithmetic. not just does the ebook endure witness to the intense ingenuity of a few of the best mathematical minds of background — Archimedes, Isaac Newton, Leonhard Euler, Augustin Cauchy, Pierre Fermat, Carl Friedrich Gauss, Gaspard Monge, Jakob Steiner, etc — however it presents infrequent perception and suggestion to any reader, from highschool math scholar to specialist mathematician. this can be certainly an strange and uniquely worthy book.
The 100 difficulties are offered in six different types: 26 arithmetical difficulties, 15 planimetric difficulties, 25 vintage difficulties touching on 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 suggestions and proofs, the writer recounts their origins and background and discusses personalities linked to them. usually he provides no longer the unique answer, yet one or easier or extra attention-grabbing demonstrations. in just or 3 cases does the answer think something greater than an information of theorems of trouble-free arithmetic; for that reason, it is a ebook with an incredibly extensive appeal.
Some of the main celebrated and fascinating goods are: Archimedes' "Problema Bovinum," Euler's challenge of polygon department, Omar Khayyam's binomial enlargement, the Euler quantity, Newton's exponential sequence, the sine and cosine sequence, Mercator's logarithmic sequence, the Fermat-Euler top quantity theorem, the Feuerbach circle, the tangency challenge of Apollonius, Archimedes' selection of pi, Pascal's hexagon theorem, Desargues' involution theorem, the 5 usual solids, the Mercator projection, the Kepler equation, decision of the placement of a boat at sea, Lambert's comet challenge, and Steiner's ellipse, circle, and sphere problems.
This translation, ready particularly for Dover by means of David Antin, brings Dörrie's "Triumph der Mathematik" to the English-language viewers for the 1st time.

Reprint of Triumph der Mathematik, 5th variation.

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But this method soon becomes impossible as the number of angles is increased. In 1758 Segner, to whom Euler had communicated the first seven division numbers 1, 2, 5, 14, 42, 132, 429, established a recurrence formula for En (Novi Commentarii Academiae Petropolitanae pro annis 1758 et 1759, vol. VII) which we will begin by deriving. Let the angles of any convex polygon of n angles be 1, 2, 3, …, n. For every possible division En of the polygon of n angles we may take the side nl as the base line of a triangle the apex of which is situated at one of the angles 2, 3, 4, …, n – 1 in accordance with the division selected.

CATALAN’S PROBLEM has the form: How many different ways can a product of n different factors be calculated by pairs? We say that a product is calculated by pairs when it is always only two factors that are multiplied together and when the product arising from such a “paired” multiplication is used as one factor in the continuation of the calculation. Calculation by pairs of the product 3 · 4 · 5 · 7, for example, is carried out in the following manner: 3 · 5 = 15, 4 · 15 = 60, 7 · 60 = 420. ” {[(a · b) · C] · [(d · e) · (f · g)]} · {(h · i) · k} is therefore a paired product of the ten factors a to k.

No man sits next to his wife 2. One man sits next to his own wife (namely when 3. Two men sit next to their own wives (when Mμ = Mv or Mμ = Mv + 1 and at the same time Xn = M1 that is, when in our arrangement the order M1F1 occurs). Thus, we must consider other seating arrangements in addition to the one prescribed in the problem. In the following we will distinguish between three types of arrangements: arrangements A, B, and C. An A-arrangement will be one in which no man sits next to his wife.

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100 great problems of elementary mathematics: their history and solution by Heinrich Dorrie


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