EULER, LEONHARD. - INTRODUCING "THE EULER CONSTANT" AND "THE FUNCTION NOTATION"

De Linea celerrimi descensus in Medio qvocunqve resistente. (On the curve of fastest descent in whatever resistent medium). (+) De progressionibus harmonicis observationes. (On harmonic progressions). (+) De infinitis curvis iusdem generis seu methodus inveniendi aequationes pro infinitis curvis eiusdem generis. (On infinite(ly many) curves of the same type, that is, a method of finding equations for infinite(ly many) curves of the same type). (+) Additamentum ad dissertationem de infinitis curvis eiusdem generis. (Addendum to the dissertation on infinite(ly many) curves of the same type).

(Petropoli, St. Petersburg, Typis Academiae, 1740). 4to. No wrappers. In: "Classes Prima continens Mathematica. Commentarii Academiae Scientiarum Imperialis Petropolitanae", Tomus VII ad Annum 1735, &..... Euler's papers: pp. 135-149, 150-161, 174-183 a. 184-200 and 2 engraved plates. Clean and fine.

First printing of 4 importent early papers by Euler. Enestroem: E42, E43, E44 a. E45.

E42: This is Euler's second paper on the "Brachistochrone problem".
E43. Here Euler introduces THE EULER CONSTANT. "The Euler-Mascheroni constant (also called Euler's constant) is a mathematical constant recurring in analysis and number theory, usually denoted by the lowercase Greek letter gamma."
E44: "This is an extensive paper that develops a method for finding a family of curves arising from the constant of integration of dz = Pdx, which is treated as the second variable; the rudiments of partial differentiation are presented, and there is an extensive survey of homogeneous functions centred around what is now know as Euler's Theorem for such functions. The origins of this paper would seem to be Proposition 15 of Vol. 2 of the Mechanica, relating to families of tautochronous curves, where an integration relying on Euler's Theorem is required." (Ian Bruce).
E45: Here Euler introduces the FUNCTION NOTATION f(x). "This is an equally extensive paper that continues the development of methods for finding a family of curves arising from the constant of integration of dz = Pdx, which is treated as the second variable. A method is developed for finding the modular equation for the first order equation that is extended to cover a number of cases; this in turn is extended to second and higher orders. The method involves finding suitable functions to integrate, starting from a part of the modular equation that is integrable, so that the whole equation is of this form. This paper is noteworthy in addition as it seems to be the first in which the function notation, albeit in a slightly different form from the modern meaning, is introduce. I have not been able to check all the equations at this stage."(Ian Bruce).
This section also contains DANIEL BERNOULLI: Demonstrationes theorematum svorum de oscillationibus corporum filo flexili connexorum et catenae verticaliter suspensae. Pp. 162-173.

Order-nr.: 50926