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Historical Notes on Archimedes, Trigonometry, and Algebra

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A 10th century codex is sold for $2,000,000 to an anonymous buyer ... Almagest. Katz paper. Calculus Differentialis. 1727. The Calculus of Finite Differences ... – PowerPoint PPT presentation

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Title: Historical Notes on Archimedes, Trigonometry, and Algebra


1
Historical Notes on Archimedes, Trigonometry,
and Algebra
  • Fred Rickey
  • USMA
  • USMAPS, 29 October 2008

2
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

3
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4
How many ways can you rearrange the pieces of the
stomachion?
http//www.math.ucsd.edu/fan/stomach/
5
The balancing act involves thin layers of the
cone, sphere and the cylinder.
x
2a
x
x
http//mthwww.uwc.edu/uwmc-math/pmartin/
6
ArchimedesThe Palimpsest Project
  • http//www.archimedespalimpsest.org
  • Many great pictures here of the restoration of
    the palimpsest.

7
  • 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.

8
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9
What is a sine ?
  • The Greeks used chords
  • The Arabs used half-chords
  • NB These are line segments, not numbers!

10
  • Almagest
  • Katz paper

11
Calculus Differentialis1727
  • The Calculus of Finite Differences
  • The differential Calculus in General
  • Differentiation of Algebraic Functions
  • Differentiation of Logarithmic and Exponential
    Quantities

12
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

13
  • Daniel Bernoulli to Euler, May 4, 1735
  • The DE arises in a problem about vibrations on an
    elastic band.
  • This matter is very slippery.

14
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.

15
Euler creates trig functions in 1739
16
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17
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18
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
19
Chapter 1 Functions
  • A change of Ontology
  • Study functions
  • not curves

20
VIII. Trig Functions
21
  • 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

22
  • 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

23
  • Linear Differential Equations with constant
    coefficients
  • De Integratione Aequationum Differentialium
    altiorum graduum
  • 1743
  • E62

24
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.

25
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26
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.

27
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

28
  • Latin translation, beginning with "Dixit
    algorizmi"
  • His name is the origin of our word algorithm

29
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

30
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31
Abstraction makes mathematics easier !
  • The introduction of zero
  • The coefficients include negative reals
  • a x2 b x c 0

32
Questions
  • On what I have just said
  • On any topic
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