By D. J. Struik
These chosen mathematical writings hide the years while the rules have been laid for the idea of numbers, analytic geometry, and the calculus.
Originally released in 1986.
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Extra resources for A Source Book in Mathematics, 1200-1800
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]. , or ψ, φ, θ, etc. And those that are between two lines that run from the top downward are called cells of the same perpendicular rank, such as the cells Ο, ψ, A, D, etc.
16 I ARITHMETIC I For we observe that a moving point is declared more or less swift, according as it is seen to be borne over a greater or less space in equal times. Hence the ratio of the spaces traversed is necessarily the same as that of the velocities. , is that of the distances TS, IS, 2S, 3S, 4S, etc. , respectively. 26. The logarithm of a given sine is that number which has increased arith metically with the same velocity throughout as that with which radius began to decrease geometrically, and in the same time as radius has decreased to the given sine.
45. 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.