"Therefore, every intensity which can be acquired successively ought to be imagined by a straight line perpendicularly erected on some point of the space or subject of the intensible thing, e.g., a quality. For whatever ratio is found to exist between intensity and intensity, in relating intensities of the same kind, a similar ratio is found to exist between line and line, and vice versa." (Nicole Oresme, "Tractatus de configurationibus qualitatum et motuum" ["A treatise on the uniformity and difformity of intensities"], 1352) [definition of a functional relationship between two variables]
"Here, we call function of a variable magnitude, a quantity formed in whatever manner with that variable magnitude and constants." (Johann I Bernoulli, 1718)
"A function of a variable quantity is an analytic expression composed in any way whatsoever of the variable quantity and numbers or constant quantities. […] Functions are divided into algebraic and transcendental. The former are those made up from only algebraic operations, the latter are those which involve transcendental operations."(Leonhard Euler, "Introduction to Analysis of the Infinite", 1748)
“After exponential quantities the circular functions, sine and cosine, should be considered because they arise when imaginary quantities are involved in the exponential." (Leonhard Euler, ”Introductio in analysin infinitorum”, 1748)
"If some quantities so depend on other quantities that if the latter are changed the former undergoes change, then the former quantities are called functions of the latter. This definition applies rather widely and includes all ways in which one quantity could be determined by other. If, therefore, x denotes a variable quantity, then all quantities which depend upon x in any way, or are determined by it, are called functions of x." (Leonhard Euler, 1755)
"Those quantities that
depend on others in this way, namely, those that undergo a change when others
change, are called functions of these quantities. This definition applies
rather widely and includes all ways in which one quantity could be determined
by another."
"I take the word 'mapping' in the widest possible sense; any point of the spherical surface is represented on the plane by any desired rule, so that every point of the sphere corresponds to a specified point in the plane, and inversely." (Leonhard Euler, "On the representation of Spherical Surfaces onto the Plane", 1777)
"When we have grasped the spirit of the infinitesimal method, and have verified the exactness of its results either by the geometrical method of prime and ultimate ratios, or by the analytical method of derived functions, we may employ infinitely small quantities as a sure and valuable means of shortening and simplifying our proofs." (Joseph-Louis de Lagrange, "Mechanique Analytique", 1788)
"Every quantity whose value depends on one or more other quantities is called a function of these latter, whether one knows or is ignorant of what operations it is necessary to use to arrive from the latter to the first." (Sylvestre-François Lacroix, "Traité de calcul differéntiel et du calcul intégral", 1797-1798)
"Certain authors who seem to have perceived the weakness of this method assume virtually as an axiom that an equation has indeed roots, if not possible ones, then impossible roots. What they want to be understood under possible and impossible quantities, does not seem to be set forth sufficiently clearly at all. If possible quantities are to denote the same as real quantities, impossible ones the same as imaginaries: then that axiom can on no account be admitted but needs a proof necessarily." (Carl F Gauss, "New proof of the theorem that every algebraic rational integral function in one variable can be resolved into real factors of the first or the second degree", 1799)
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