## Essai sur la théorie des nombres.

Paris: Duprat, 1797-1798.

A fine copy of the first book entirely dedicated to number theory. The work contains Legendre’s discovery of the law of quadratic reciprocity, which Gauss referred to as the ‘golden theorem’ and for which he published six proofs in his *Disquisitiones arithmeticae* (1801).

❧Norman 1325; Parkinson *Breakthroughs* 231.

“The theory of numbers in the eighteenth century remained a series of disconnected results. The most important works in the subject were Euler’s *Anleitung zur Algebra* (1770) and Legendre’s *Essai sur la théorie des nombres* (1798).” (Kline). “Legendre was one of the most prominent mathematicians of Europe in the 19th Century… His texts were very influential. In 1798 he published his *Theory of Numbers*, the first book devoted exclusively to number theory. It underwent several editions, but was soon to be superseded by Gauss’ *Disquisitiones arithmeticae*

“Many of Legendre’s results were found independently by Gauss, and seriouspriority disputes arose between them. Legendre’s proofs, moreover, left much tobe desired, even by mid-18th-century standards. For example, he discovered thelaw of quadratic reciprocity, unaware of Euler’s prior discovery, and gave a proofbased on what he viewed as a self-evident fact, namely the existence of infinitelymany primes in any arithmetic progression *an + b (n = 1, 2, 3, …*, with *a* and *b* relatively prime). This was a very difficult result, proved subsequently by Dirichlet using deep methods of analysis. Legendre was chagrined when Gauss, who gave a rigorous proof, claimed the result as his own. In connection with this law, Legendre introduced the useful and celebrated *Legendre symbol* *(a/p)* (it does *not* denote division), with *p* an odd prime and *a* an integer not divisible by *p : (a/p) = 1* if *x ^{2} ≡ a* (mod

*p*) is solvable and

*(a/p)=-1*if it is not. In terms of this symbol, the law of quadratic reciprocity can be stated succinctly as

*(p/p)(q/p) = (-1)*” (Kleiner,

^{(p-1)(q-1)/4}*Excursions in the History of Mathematics*, pp. 17-18).

4to (258 x 193 mm ), pp [i-v] vixxii [2:errata] 472, [56:tables], [2:publisher's catalogue], contemporary and unrestored French gilt-ruled calf, all edges colored in red, red leather spine label, some light wear and rubbing, a piece of leather scratched from the rear board, book plate presenting the book as a prize, three interesting manuscript leaves with mathematical contents laid in, fine and clean throughout.

Item #3414

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Price:
$4,000.00
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