On Numbers: On Prime Numbers (1800-1899)

"The problem of distinguishing prime numbers from composite numbers and of resolving the latter into their prime factors is known to be one of the most important and useful in arithmetic. […] The dignity of the science itself seems to require that every possible means be explored for the solution of a problem so elegant and so celebrated.” (Carl Friedrich Gauss, "Disquisitiones Arithmeticae”, 1801)

“The difference of two square numbers is always a product, and divisible both by the sum and by the difference of the roots of those two squares; consequently the difference of two squares can never be a prime number.” (Leonhard Euler, “Elements of Algebra”, 1810)

"It is greatly to be lamented that this virtue of the real numbers [the ordinary integers], to be decomposable into prime factors, always the same ones [...] does not also belong to the complex numbers [complex integers]; were this the case, the whole theory [...] could easily be brought to its conclusion. For this reason, the complex numbers we have been considering seem imperfect, and one may well ask whether one ought not to look for another kind which would preserve the analogy with the real numbers with respect to such a fundamental property." (Ernst E Kummer, 1844)

"It is possible to express the laws of thermodynamics in the form of independent principles, deduced by induction from the facts of observation and experiment, without reference to any hypothesis as to the occult molecular operations with which the sensible phenomena may be conceived to be connected; and that course will be followed in the body of the present treatise. But, in giving a brief historical sketch of the progress of thermodynamics, the progress of the hypothesis of thermic molecular motions cannot be wholly separated from that of the purely inductive theory." (William J M Rankine, "A Manual of the Steam Engine and Other Prime Movers", 1859)

"It always seems to me absurd to speak of a complete proof, or of a theorem being rigorously demonstrated. An incomplete proof is no proof, and a mathematical truth not rigorously demonstrated is not demonstrated at all." (James J Sylvester, "On certain inequalities related to prime numbers", Nature Vol. 38, 1888)

"I think it would be desirable that this form of word [mathematics] should be reserved for the applications of the science, and that we should use mathematic in the singular to denote the science itself, in the same way as we speak of logic, rhetoric, or" (own sister to algebra) music." (James J Sylvester, Collected Mathematical Papers, 1869)

"One can divide the entire circumference of the lemniscate into m equal parts by ruler and compass alone if m is of the form 2^n or a prime of the form 2^n + 1, or if m is a product of numbers of these two kinds. This theorem, as one sees, is exactly the same as the theorem of Gauss for the circle." (Niels H Abel, "Euvres completes de Niels Henrik Abel", 1881)

"It always seems to me absurd to speak of a complete proof, or of a theorem being rigorously demonstrated. An incomplete proof is no proof, and a mathematical truth not rigorously demonstrated is not demonstrated at all." (James J Sylvester, On certain inequalities related to prime numbers", Nature Vol. 38, 1888)

"Mature knowledge regards logical clearness as of prime importance: only logically clear images does it test as to correctness; only correct images does it compare as to appropriateness. By pressure of circumstances the process is often reversed. Images are found to be suitable for a certain purpose; are next tested as to their correctness ; and only in the last place purged of implied contradictions." (Heinrich Hertz, "The Principles of Mechanics Presented in a New Form", 1894)

"The analogy between the results of the theory of algebraic functions of one variable and those of the theory of algebraic numbers suggested to me many years ago the idea of replacing the decomposition of algebraic numbers, with the help of ideal prime factors, by a more convenient procedure that fully corresponds to the expansion of an algebraic function in power series in the neighborhood of an arbitrary point." (Richard Dedekind, "New foundations of the theory of algebraic numbers", 1899)