By D. J. Struik
These chosen mathematical writings disguise the years while the rules have been laid for the speculation of numbers, analytic geometry, and the calculus.
Originally released in 1986.
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Additional info for A Source Book in Mathematics, 1200-1800
For instance, the cell F, that is, the number of the cell F, is equal to cell C plus cell E, and so the others. From this many consequences can be drawn. Here are the most important ones, where I consider the triangles whose generator is unity, but what can be said about them will also apply to the others. FIRST CONSEQUENCE In every arithmetic triangle all the cells of the first parallel rank and of the first perpendicular rank are equal to the generator. Indeed, by the construction of the triangle, every cell is equal to the cell which precedes it in its perpendicular rank plus the cell that precedes it in its parallel rank.
Corollary 1. In a similar way it can be shown that, when λ < (ρ — 1)/4, we never can have Λ > (ρ — 1)/5 and thus we have also here λ = (ρ — 1)/5 or λ < (ρ - 1)/5. 46. Corollary 2. And in general, if it is known that A < (p — 1)/», then one proves in the same way that we cannot have λ > (ρ — 1 )/(n + 1), therefore we must have λ = (ρ — 1)/(w + 1) or λ < (ρ — 1)(n + 1). 47. Corollary 3. Wherefrom it appears that the number of all numbers that cannot be residues is either = 0, or = λ, or = 2A or any multiple of A; for if there are more than ηλ of such numbers, then, if any at all, A new ones are added to them, so as to make their number = (η + 1)A; and if this does not yet comprise all the nonresidues, then at once A new ones are added.
Needham, Science and civilisation in China, III (Cambridge University Press, New York, 1959), 135. 2 Russian translation by B. A. Rozenfel'd (Gos. 3, footnote 1. 3 Smith, History of mathematics, II, 508-512. 9 (Girard). 22 I I ARITHMETIC Z 2 3 4 JL 6 4/1 /4 5 /l /5 /15 /35 A&\ /26 7] 8, 9] 10 /10 /20\ /35 /56 7777X7 νy / // [/26 Fig. 1 And. joining in this way all the division points which have the same indices I form with them as many triangles and bases. I draw through every one of the division points lines parallel to the sides, and these by their intersections form small squares which I call cells [cellules].