Arithmetic Quotes - page 6
That fondness for science, ... that affability and condescension which God shows to the learned, that promptitude with which he protects and supports them in the elucidation of obscurities and in the removal of difficulties, has encouraged me to compose a short work on calculating by al-jabr and al-muqabala, confining it to what is easiest and most useful in arithmetic.
Muhammad ibn Mūsā al-Khwārizmī
It is significant that we owe the first explicit formulation of the principle of recurrence to the genius of Blaise Pascal... Pascal stated the principle in a tract called The Arithmetic Triangle which appeared in 1654. Yet... the gist of the tract was contained in the correspondence between Pascal and Fermat regarding a problem in gambling, the same correspondence which is now regarded as the nucleus from which developed the theory of probabilities.
It surely is a fitting subject for mystic contemplation that the principle of reasoning by recurrence, which is so basic in pure mathematics, and the theory of probabilities, which is the basis of all deductive sciences, were both conceived while devising a scheme for the division of stakes in an unfinished match of two gamblers.
Blaise Pascal
It is probable... that the greatest inventor in science was never able to do more than to accelerate the progress of discovery, and to anticipate what time, "the author of authors," would have gradually brought to light. Though logarithms had not been invented by Napier, they would have been discovered in the progress of the algebraic analysis, when the arithmetic of powers and exponents, both integral and fractional, came to be fully understood. The idea of considering all numbers, as powers of one given number, would then have readily occurred, and the doctrine of series would have greatly facilitated the calculations which it was necessary to undertake. Napier had none of these advantages, and they were all supplied by the resources of his own mind. Indeed, as there never was any invention for which the state of knowledge had less prepared the way, there never was any where more merit fell to the share of the inventor.
John Napier
My father, in my earliest childhood, taught me the rudiments of arithmetic, and about that time made me acquainted with the arcana; whence he had come by this learning I know not. This was about my ninth year. Shortly after, he instructed me in the elements of the astronomy of Arabia, meanwhile trying to instill in me some system of theory for memorizing, for I had been poorly endowed with the ability to remember. After I was twelve years old he taught me the first six books of Euclid, but in such a manner that he expended no effort on such parts as I was able to understand by myself.
This is the knowledge I was able to acquire and learn without any elementary schooling...
Gerolamo Cardano
Wallis did not become interested in mathematics till the age of thirty-one, but devoted himself to the subject for the rest of his life. One of the earliest and most important books on algebra ever written in English was his treatise published in 1685. It contains a brief historical sketch of the subject which is unfortunately not entirely accurate, but his treatment of the theory and practice of arithmetic and algebra has made the book a standard work for reference ever since.
John Wallis
[E]arly analytic geometers-Descartes in particular-did not accept that geometry could be based on numbers or algebra. Perhaps the first to take the idea of arithmetizing geometry seriously was Wallis... [(1657) Mathesis universalis. Opera 1, 11-228.] Chs. XXIII and XXV, gave the first arithmetic treatment of Euclid's Books II and V, and he had earlier given purely algebraic treatment of conic sections [(1655) De sectionibus conicus. Opera 1, 291-364.]. He initially derived equations from classical definitions by sections of the cone but then proceeded to derive their properties from the equations, "without the embranglings of the cone," as he put it.
John Wallis
Not that I at all disallow the use of reasoning upon experiments, or the endeavouring to discern as early as we can the confederations, and differences, and tendencies of things: for such an absolute suspension of the exercise of reasoning were exceeding troublesome, if not impossible. And, as in that rule of arithmetic, which is commonly called regula falsi by proceeding upon a conjecturally-supposed number, as if it were that, which we inquire after, we are wont to come to the knowledge of the true number sought for; so in physiology it is sometimes conducive to the discovery of truth, to permit the understanding to make an hypothesis, in order to the explication of this or that difficulty, that by examining how far the phænomena are, or are not, capable of being solved by that hypothesis, the understanding may, even by its own errors, be instructed.
Robert Boyle
All relations are either qualitative or quantitative. Qualitative relations can be considered by themselves without regard to quantity. The algebra of such enquiries may be called logical algebra, of which a fine example is given by Boole.
Quantitative relations may also be considered by themselves without regard to quality. They belong to arithmetic, and the corresponding algebra is the common or arithmetical algebra.
In all other algebras both relations must be combined, and the algebra must conform to the character of the relations.
Benjamin Peirce
