By Heinrich Dorrie
"The assortment, drawn from mathematics, algebra, natural and algebraic geometry and astronomy, is awfully fascinating and attractive." — Mathematical Gazette
This uncommonly fascinating quantity covers a hundred of the main well-known old difficulties of easy arithmetic. not just does the e-book undergo 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 concept to any reader, from highschool math scholar to specialist mathematician. this can be certainly an strange and uniquely important book.
The 100 difficulties are awarded 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 heritage and discusses personalities linked to them. usually he provides no longer the unique resolution, yet one or easier or extra attention-grabbing demonstrations. in just or 3 cases does the answer think whatever greater than an information of theorems of straight forward arithmetic; consequently, this can be a ebook with a very huge 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 top quantity theorem, the Feuerbach circle, the tangency challenge of Apollonius, Archimedes' choice of pi, Pascal's hexagon theorem, Desargues' involution theorem, the 5 usual solids, the Mercator projection, the Kepler equation, choice of the placement of a boat at sea, Lambert's comet challenge, and Steiner's ellipse, circle, and sphere problems.
This translation, ready specifically for Dover through David Antin, brings Dörrie's "Triumph der Mathematik" to the English-language viewers for the 1st time.
Reprint of Triumph der Mathematik, 5th version.
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Additional info for 100 great problems of elementary mathematics: their history and solution
From the indicated values and formulas (2) and (5) it immediately follows that in general [The proof is by induction. We assume that (7) is true for all indices through n, so that E2 = Cl, E3 = C2, … En = Cn –1. According to (2) and (5) Since the right sides of the two last equations correspond member for member, it also follows that i. ] (6) and (7) give us Euler’s formula immediately: In conclusion we would like to give a slight simplification of Euler’s formula. It is and consequently where f = n – 2 is the number of triangles into which the n-gon can always be divided and k = 2n – 3 is the number of sides bounding these triangles.
We will let an equation such as R = s indicate that the Roman number indicated by the letter R possesses the same numerical value as the Arabic numeral corresponding to the letter s. Also, we will designate the days of the week Sunday, Monday, …, Saturday by the numerals 0, 1, 2, …, 6. Let the Sunday arrangement have the following order: From this, by adding r = R to each numeral, we obtain the arrangement for the rth weekday. Here every figure thus obtained that exceeds 7, such as perhaps c + r or D + R, will represent the girl who receives a number (c + r – 7 or D + R – 7), that is 7 below the figure and is subsequently converted into that number.
Thus, if r represents the number of the day of the week that is congruent to x – β (or y – b), then so that the girls x and y walk in the same row on weekday r. 2. Each girl x of the first group comes together exactly once with each girl X of the second group. ) can be congruent to only one of the seven differences A – a, A – α, B – b, B – β, C – c, C – γ, D – d. Let us assume X – x ≡ C – γ or X – C ≡ x – γ. If s = S is the weekday number that is congruent to X – C (or x – γ), then we have so that the girls X and x walk in the same row on weekday s.