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Additional info for Mathematical Recreations and Essays
Sample text
5, 6. Crelle’s Journal, 1832, vol. ix, pp. 390–393. Liouville’s Journal, 1841, vol. v, pp. 195–215, 276–9, 348–9. References to Kummer’s Memoirs are given in Smith’s Report to the British Association on the Theory of Numbers, London, 1860. 34 ARITHMETICAL RECREATIONS. [CH. I whether any number can be found to satisfy these conditions, but it seems unlikely, and it has been shown that there is no number less than 100 which does so. The proof is complicated and difficult, and there can be little doubt is based on considerations unknown to Fermat.
Suppose that the system is referred to rectangular axes Ox, Oy; and that OP , OM , M P make respectively angles θ, θ , θ with Ox. Hence, by projection on Oy and on Ox, we have Therefore l sin θ = l sin θ + l sin θ , l cos θ = l cos θ + l cos θ . n sin θ + sin θ , tan θ = n cos θ + cos θ where n = l /l . This result is true whatever be the value of n. But n may have any value (ex. gr. n = ∞, or n = 0), hence tan θ = tan θ = tan θ , which obviously is impossible. Seventh Fallacy. Here is a fallacious investigation, to which Mr Chartres first called my attention, of the value of π: it is founded on well-known quadratures.
27 numbers or elementary algebra as seemed worth reproducing. It will be noticed that the majority of them either are due to Bachet or were collected by him in his classical Probl`emes; but it should be added that besides the questions I have mentioned he enunciated, even if he did not always solve, some other problems of greater interest. One instance will suffice. Bachet’s Weights Problem* . Among the more difficult problems proposed by Bachet was the determination of the least number of weights which would serve to weigh any integral number of pounds from 1 lb.