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Historical Notes on Archimedes, Trigonometry,

and Algebra

- Fred Rickey
- USMA
- 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

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How many ways can you rearrange the pieces of the

stomachion?

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

The balancing act involves thin layers of the

cone, sphere and the cylinder.

x

2a

x

x

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

ArchimedesThe 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.

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What is a sine ?

- The Greeks used chords
- The Arabs used half-chords
- NB These are line segments, not numbers!

- Almagest
- Katz paper

Calculus Differentialis1727

- The Calculus of Finite Differences
- The differential Calculus in General
- Differentiation of Algebraic Functions
- 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 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

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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
- p is clearly irrational
- Value of p from de Lagny
- Note error in 113th decimal place
- scribam p
- W. W. Rouse Ball discovered (1894) the use of p

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

Editors 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.

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Abu Ja'far Muhammad ibn Musa al-Khwarizmi

- Lived c. 780 to c. 850
- Stamp issued September 6, 1983 in the Soviet

Union to commemorate the 1200th anniversary of

al-Khwarizmi'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 Quadratics

- Squares equal to roots
- x² 5x
- Squares equal to numbers
- x² 9
- Roots equal to numbers
- 4x 20
- Squares and roots equal to numbers
- x² 10x 39
- Squares and numbers equal to roots
- x² 21 10x
- Roots and numbers equal to squares
- 3x 4 x²

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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