That a formal science like algebra, the creation of our abstract thought, should thus, in a sense, dictate the laws of its own being, is very remarka… - George Ballard Mathews

The invention of the symbol ≡ by Gauss affords a striking example of the advantage which may be derived from an appropriate notation, and marks an epoch in the development of the science of arithmetic.

It may fairly be said that the germs of the modern algebra of linear substitutions and concomitants are to be found in the fifth section of the Disquisitiones Arithmeticae; and inversely, every advance in the algebraic theory of forms is an acquisition to the arithmetical theory.

Strictly speaking, the theory of numbers has nothing to do with negative, or fractional, or irrational quantities, as such. No theorem which cannot be expressed without reference to these notions is purely arithmetical: and no proof of an arithmetical theorem, can be considered finally satisfactory if it intrinsically depends upon extraneous analytical theories.