The language of analysis: the function as the basic object
A reform of mathematical analysis after which the basic object becomes the function, rather than the curve or the geometric magnitude. A function is defined as an analytic expression composed of a variable and of constants.
The reform was carried out in Introductio in analysin infinitorum (Lausanne, 1748) by the Swiss mathematician Leonhard Euler. He moved the trigonometric quantities out of the line segments tied to the circle and into analytic functions, and in the same book fixed the notation e for the base of natural logarithms.
After this the calculus of Newton and Leibniz could be taught and applied as a ready language, not as a set of devices belonging to its inventors.
Why this date
The date is that of the first publication: Lausanne, 1748, two volumes (Eneström E101 and E102). The definition of function stands in book I, chapter 1, §4. The notation e is introduced in chapter 7, the trigonometric quantities are treated in chapter 8.
What it grew from
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Differential and integral calculus
improved · Confidence: high
The calculus of Newton and Leibniz existed as a set of devices, and it was the Introductio of 1748 that assembled it into a system in which the function became the explicit and central object.
Link sources
- Carl B. Boyer, The Foremost Textbook of Modern Times, 1950, American Mathematical Monthly 58 (4), 223-226 (1951) Verified 20 Sep 2026
yet it was the Introductio which in 1748 fashioned these into the third member of the triumvirate
- Israel Kleiner, Evolution of the Function Concept: A Brief Survey, The College Mathematics Journal 20 (4), 282-300 (1989) Verified 20 Sep 2026
Euler's Introductio was the first work in which the concept of function plays an explicit and central role
- Introduction to the Analysis of Infinities, Encyclopaedia Britannica (2024) Verified 24 Sep 2026
he made the notion of function the central organizing concept of analysis
"Verified" means that on that day the quotation was found at the address given, word for word. No date means we have not been able to open the source yet, not that the quotation is wrong.
- Carl B. Boyer, The Foremost Textbook of Modern Times, 1950, American Mathematical Monthly 58 (4), 223-226 (1951) Verified 20 Sep 2026
Node sources
- Leonhard Euler, Introductio in analysin infinitorum, liber I, cap. I, §4 (1748) Verified 20 Sep 2026
Functio quantitatis variabilis est expressio analytica quomodocunque composita ex illa quantitate variabili et numeris seu quantitatibus constantibus.
- Israel Kleiner, Evolution of the Function Concept: A Brief Survey, The College Mathematics Journal 20 (4), 282-300 (1989) Verified 20 Sep 2026
Euler's Introductio was the first work in which the concept of function plays an explicit and central role
- H N Jahnke (ed.), A History of Analysis, AMS, MacTutor, The function concept (2003) Verified 20 Sep 2026
were regarded as lines connected with the circle rather than functions
- Introduction to the Analysis of Infinities, Encyclopaedia Britannica (2024) Verified 24 Sep 2026
he made the notion of function the central organizing concept of analysis
"Verified" means that on that day the quotation was found at the address given, word for word. No date means we have not been able to open the source yet, not that the quotation is wrong.