Historical Notes on Archimedes, Trigonometry, and Algebra Fred Rickey USMA USMAPS, 29 October 2008.

Historical Notes on Archimedes,

Trigonometry, and Algebra

Fred RickeyUSMA

USMAPS, 29 October 2008

October 29, 1998

• A 10th century codex is sold for $2,000,000 to an anonymous buyer

• It contains the unique copies of two works by Archimedes: – The Stomachion– The Method

• The Walters Art Museum in Baltimore is conserving the manuscript

http://www.math.ucsd.edu/~fan/stomach/

How many ways can you rearrange the pieces of the stomachion?

The balancing act involves thin layers of the cone, sphere and the cylinder.

x

x

x2a

http://mthwww.uwc.edu/uwmc-math/pmartin/

Archimedes:The Palimpsest Project

• http://www.archimedespalimpsest.org

• Many great pictures here of the restoration of the palimpsest.

At 2pm on October 29th, 2008, ten years after the Archimedes Palimpsest was purchased by the present owner, the core data generated by the project to conserve, image and study the manuscript, will be released on the web.

What is a sine ?

• The Greeks used chords

• The Arabs used half-chords

• NB: These are line segments, not numbers!

• Almagest

• Katz paper

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

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

• Linear Differential Equations with constant coefficients

• De Integratione Aequationum Differentialium altiorum graduum

• 1743• E62

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.

Abu Ja'far Muhammad ibn Mūsā al-Khwārizmī

• Lived c. 780 to c. 850• Stamp issued

September 6, 1983 in the Soviet Union to commemorate the 1200th anniversary of al-Khwārizmī's birth.

Kitab al-jabr wa l-muqabala

• The book of restoration and balancing

• “what is easiest and most useful in arithmetic”

• Origin of our word algebra

• Latin translation, beginning with "Dixit algorizmi"

• His name is the origin of our word “algorithm”

Six Types of Quadratics1. Squares equal to roots

• x² = 5x2. Squares equal to numbers

• x² = 93. Roots equal to numbers

• 4x = 204. Squares and roots equal to numbers

• x² + 10x = 395. Squares and numbers equal to roots

• x² + 21 = 10x6. Roots and numbers equal to squares

• 3x + 4 = x²

x2 10x 39

x2 10x 52 39 25x 52 82

x 5 8

x 3

Abstraction makes mathematics easier !

• The introduction of zero

• The coefficients include negative reals

• a x2 + b x + c = 0

Questions

• On what I have just said

• On any topic