By D. J. Struik

ISBN-10: 0691610517

ISBN-13: 9780691610511

These chosen mathematical writings disguise 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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**Additional resources for A Source Book in Mathematics, 1200-1800**

**Example text**

1+ 1 (in base k ' + I). p + a):(B + A) = 3:2. These consequences can all be found in the translation of Pascal's paper in Smith, Source book, pp. 74-75. 26 I ARITHMETIC I For example, let it be proposed to find the number of the cell ξ of the fifth perpendicular rank and of the third parallel rank. Having taken all the numbers that precede the index of the perpendicular rank 5, that is, 1, 2, 3, 4, take as many natural numbers beginning with the index of the parallel rank 3, that is, 3, 4, 5, 6.

This process can be repeated. It was from these problems by Permat that Euler, in the paper of 1732/33, started his research on the "Pell" equation. i Xi 0 0 8 EULEB. POWER RESIDUES Here follow some contributions of Leonhard Euler (1707-1783) to the theory of numbers. Euler, born in Basel, Switzerland, studied with Johann Bernoulli, was from 1727 to 1741 associated with the Imperial Academy in Saint Petersburg, from 1741 to 1766 with the Royal Academy in Berlin (at the time of Frederick II, "the Great"), and from 1766 to his death again with the Saint Petersburg Academy (at the time of Catherine II, "the Great").

8. 6. 32 I ARITHMETIC I residues appear in the same order. 9. There are no more than ρ — 1 different residues, and 1 is always among them. The congruence is always modulo p. The algorithm of paragraphs 37-46 is the same as that used later to prove that the order of a subgroup is a divisor of the order of the group. Euler's case is that of cyclical groups. 37. Theorem 10. If the number of different residues resulting from the division of the powers 1, a, a2, a3, a4, a5, etc. by the prime number ρ is smaller than ρ — 1, then there will be at least as many numbers that are nonresidues as there are residues.

### A Source Book in Mathematics, 1200-1800 by D. J. Struik

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