Historical Notes on Archimedes, Trigonometry, and Algebra

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Historical Notes on Archimedes, Trigonometry,
and Algebra

• Fred Rickey
• USMA
• USMAPS, 29 October 2008

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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/
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The balancing act involves thin layers of the
cone, sphere and the cylinder.
x
2a
x
x
http//mthwww.uwc.edu/uwmc-math/pmartin/
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ArchimedesThe Palimpsest Project

• http//www.archimedespalimpsest.org
• Many great pictures here of the restoration of
  the palimpsest.

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• 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!

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• Almagest
• Katz paper

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Calculus Differentialis1727

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

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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

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

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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.

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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
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Chapter 1 Functions

• A change of Ontology
• Study functions
• not curves

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VIII. Trig Functions
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• 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

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• 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

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

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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.

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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

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

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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

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Questions

• On what I have just said
• On any topic