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

1727

1. 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

Solve y k4d4y

dx4 0.

Factor 1 k4p4 0 :1 k p1 kp1 k2p2The solution is :

y xk C

xk D E Cosx

k F Sinx

k

• 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 BernoulliSeptember 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

Solve y k4d4y

dx4 0.

Factor 1 k4p4 0 :1 k p1 kp1 k2p2The solution is :

y xk C

xk D E Cosx

k F Sinx

k

• 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 113th

decimal place• “scribam π”• W. W. Rouse Ball

discovered (1894) the use of π in Wm Jones 1706.

• 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.