Why Euler Created Trigonometric Functions V. Frederick Rickey USMA, West Point NJ-MAA March 31, 2007

What is a sine ? The Greeks used chords The Arabs used half-chords NB: These are line segments, not numbers!

Calculus Differentialis The Calculus of Finite Differences 2.The differential Calculus in General 3.Differentiation of Algebraic Functions 4.Differentiation of Logarithmic and Exponential Quantities

Draft on Differential Calculus, 1827 Euler defines functions and then divides them into two classes: –Algebraic –Transcendental The only transcendental functions are logarithms and exponentials Euler gives a differential calculus of these functions NB: no trigonometry

Daniel Bernoulli to Euler, May 4, 1735 The DE arises in a problem about vibrations on an elastic band. “This matter is very slippery.”

Euler to Johann Bernoulli September 15, 1739 after treating this problem in many ways, I happened on my solution entirely unexpectedly; before that I had no suspicion that the solution of algebraic equations had so much importance in this matter.

Euler creates trig functions in 1739

Linear Differential Equations with constant coefficients De Integratione Aequationum Differentialium altiorum graduum 1743 E62

Often I have considered the fact that most of the difficulties which block the progress of students trying to learn analysis stem from this: that although they understand little of ordinary algebra, still they attempt this more subtle art. From the preface of the Introductio

Chapter 1: Functions A change of Ontology: Study functions not curves

VIII. Trig Functions

He showed a new algorithm which he found for circular quantities, for which its introduction provided for an entire revolution in the science of calculations, and after having found the utility in the calculus of sine, for which he is truly the author... Eulogy by Nicolas Fuss, 1783

Sinus totus = 1 π is “clearly” irrational Value of π from de Lagny Note error in 113 th decimal place “scribam π” W. W. Rouse Ball discovered (1894) the use of π in W m Jones Arcs not angles Notation: sin. A. z

Editor’s introduction in 1754 there occurs in analysis a very important type of transcendental quantity, namely the sine... which demands a special calculus, which the celebrated author of this dissertation is able rightly to claim all for himself.