Early developments in trigonometry began with the ancient Egyptians and Babylonians who studied ratios of sides in similar triangles. Trigonometry advanced significantly through the works of Hipparchus, Euclid, Archimedes, and later Islamic mathematicians such as Al-Khwarizmi, who produced the first table of tangents. Nasir al-Din al-Tusi established spherical trigonometry as an independent field and listed the six cases of right triangles. Modern analytic trigonometry was established by Euler, while Brook Taylor derived general Taylor series for all six trigonometric functions.

1. HISTORY of GEOMETRY

2. Early Geometry
• Geometry is the branch of mathematics
that studies shapes and their relationships
to each other.
• 2000 BC – Scotland, Carved Stone Balls
exhibit a variety of symmetries including all
of the symmetries of Platonic solids.

3. Early Geometry
• 1800 BC – Moscow Mathematical Papyrus,
findings volume of a frustum

4. Early Geometry
• 1650 BC – Rhind Mathematical Papyrus,
copy of a lost scroll from around 1850 BC,
the scribe Ahmes presents one of the first
known approximate values of π at 3.16,
the first attempt at squaring the circle,
earliest known use of a sort of cotangent,
and knowledge of solving first order linear
equations

5. Indian Geometry
• people were using geometry to construct
elaborate ceremonial altars to the Hindu
gods throughout the Indian subcontinent
• instructions used to construct these alters
were recorded in a series of books called
the Sulba Sutras

6. Indian Geometry
• developed a method for calculating the
mathematical constant pi
• estimated the square root of two,
• wrote down the earliest known statement
of what would later come to be known as
the Pythagorean theorem

7. Indian Geometry
• describe ways to create various geometric
shapes with the same area

8. Chinese Geometry
• earliest evidence of a systematic
organization of regular shapes reproduced
on material objects
• Painted motifs displaying geometrical
patterns such as symmetrical
arrangements of triangles, lozenges, or
circles have been found on pottery pieces
unearthed at Banpo

9. Chinese Geometry
• Mo Jing - the oldest existent book on
geometry in China
• described various aspects of many fields
associated with physical science
• 'atomic' definition of the geometric point,
stating that a line is separated into parts

10. Chinese Geometry
• The Nine Chapters on the Mathematical
Art
– finding surface areas for squares and circles
– volumes of solids in various three-dimensional
shapes
– illustrated proof for the Pythagorean theorem
– properties of the right angle triangle and the
Pythagorean theorem
– calculated pi as 3.14159 using a 3072 sided
polygon

11. Chinese Geometry
• The Nine Chapters on the Mathematical
Art

12. Greek Geometry
• developed the idea of the "axiomatic
method"
• Thales of Miletus
– five geometric propositions for which he
wrote deductive proofs
• Euclid
– The Elements began with definitions of terms,
fundamental geometric principles (called
axioms or postulates), and general
quantitative principles

13. Greek Geometry
• Archimedes
– greatest of the Greek mathematicians
– developed methods very similar to the
coordinate systems of analytic geometry
– limiting process of integral calculus

14. Islamic Geometry
• establishment of the House of Wisdom in
Baghdad
• development of algebraic geometry
• Al-Mahani - conceived the idea of reducing
geometrical problems such as duplicating
the cube to problems in algebra
• Al-Karaji - completely freed algebra from
geometrical operations and replaced them
with the arithmetical type of operations

15. Islamic Geometry
• Thābit ibn Qurra - theorems in spherical
trigonometry, analytic geometry, and non-
Euclidean geometry
–one of the first reformers of the
Ptolemaic system
–mechanics he was a founder of statics
–arithmetical operations applied to ratios
of geometrical quantities
–generalization of the Pythagorean
theorem to all triangles

16. Islamic Geometry
• Ibrahim ibn Sinan - method of integration
more general than that of Archimedes
• al-Quhi - revival and continuation of Greek
higher geometry in the Islamic world
• Ibn al-Haytham - studied optics and
investigated the optical properties of
mirrors made from conic sections

17. Modern Geometry
• René Descartes and Pierre de Fermat -
creation of analytic geometry, or geometry
with coordinates and equations
• Girard Desargues - systematic study of
projective geometry
– the study of geometry without measurement,
just the study of how points align with each
other

18. Modern Geometry
• development of Calculus - beginning of a
new field of mathematics now called
analysis
• Non-Eudcledian Geometry
– Omar Khayyám - proof of properties of figures
in non-Euclidean geometries
– Gauss, Johann Bolyai, and Lobatchewsky -
develop a self-consistent geometry in which
the fifth Eucledian postulate was false

19. Modern Geometry
• Non-Eudcledian Geometry
– Bernhard Riemann - had applied methods of
calculus in a ground-breaking study of the
intrinsic geometry of all smooth surfaces

20. Modern Geometry
• Introduction of mathematical rigor
– separation of logical reasoning from intuitive
understanding of physical space
– Hilbert's axioms - David Hilbert
• Topology
– method of studying calculus- and analysis-
related concepts
– properties of more general figures (such as
connectedness and boundaries)

21. 20th Century Geometry
• Algebraic Geometry
– study of curves and surfaces over finite fields
– André Weil, Alexander Grothendieck, and
Jean-Pierre Serre
• Finite Geometry
– study of spaces with only finitely many points
– found applications in coding theory and
cryptography

22. 20th Century Geometry
• Computational Geometry / Digital
Geometry
– geometric algorithms, discrete
representations of geometric data

23. HISTORY of TRIGONOMETRY

24. Development and Early Trigonometry
• Trigonometry
– came from the Greek words, trigonon
(triangle) and metron (measure)
• Ancient Egyptians and Babylonians -
theorems on the ratios of the sides of
similar triangles
• Babylonian astronomers - detailed records
on the rising and setting of stars, the
motion of the planets, and the solar and
lunar eclipses

25. Development and Early Trigonometry
• Egyptians - used a primitive form of
trigonometry for building pyramids in the
2nd millennium BC.
• Ancient Greek and Hellenistic
mathematicians made use of the chord.
– A chord's perpendicular bisector passes
through the center of the circle and bisects
the angle

26. Development and Early Trigonometry
• Euclid and Archimedes
– propositions twelve and thirteen of Book Two
of the Elements are the laws of cosines for
obtuse and acute angles
– Theorems on the lengths of chords are
applications of the law of sines
– theorem on broken chords is equivalent to
formulas for sines of sums and differences of
angles

27. Development and Early Trigonometry
• Hipparchus of Nicaea
– "the father of trigonometry"
– compilation of first trigonometric table
• Aristarchus of Samos - "On the Sizes and
Distances of the Sun and Moon" (ca. 260
BC)

28. Development and Early Trigonometry
• Menelaus of Alexandria (ca. 100 AD) -
"Sphaerica"
–In Book I, he established a basis for
spherical triangles analogous to the
Euclidean basis for plane triangles.
–Htwo spherical triangles are congruent if
corresponding angles are equal
–the sum of the angles of a spherical
triangle is greater than 180°

29. Development and Early Trigonometry
• Menelaus of Alexandria (ca. 100 AD) -
"Sphaerica"
–Book II of Sphaerica applies spherical
geometry to astronomy.
–Book III contains the "theorem of
Menelaus".
–"rule of six quantities"

30. • given any line that transverses (crosses)
the three sides of a triangle (one of them
will have to be extended), six segments
are cut off on the sides. The product of
three non-adjacent segments is equal to
the product of the other three

32. Development and Early Trigonometry
• Pythagoras discovered many of the
properties of what would become
trigonometric functions.

33. Development and Early Trigonometry
• Siddhantas
– defined the sine as the modern relationship
between half an angle and half a chord
– defining the cosine, versine, and inverse sine
• Bhaskara I
– produced a formula for calculating the sine of
an acute angle without the use of a table
– approximation formula for sin(x)

34. Development and Early Trigonometry

35. Development and Early Trigonometry
• Madhava
– developed the concepts of the power series
and Taylor series
– produced the power series expansions of
sine, cosine, tangent, and arctangent
– power series of π and the angle, radius,
diameter, and circumference of a circle in
terms of trigonometric functions

36. Development and Early Trigonometry
• Muhammad ibn Mūsā al-Khwārizmī
– produced accurate sine and cosine tables,
and the first table of tangents
– pioneer in spherical trigonometry
• Habash al-Hasib al-Marwazi
– produced the first table of cotangents
• Muhammad ibn Jābir al-Harrānī al-Battānī
– discovered the reciprocal functions of secant
and cosecant
– produced the first table of cosecants for each
degree from 1° to 90°

37. Development and Early Trigonometry
• l-Jayyani
– wrote The book of unknown arcs of a sphere
– contains formulae for right-handed triangles,
– the general law of sines
– the solution of a spherical triangle by means
of the polar triangle

38. Development and Early Trigonometry
• method of triangulation
– first developed by Muslim mathematicians
– applied to practical uses
• Omar Khayyám
– solved cubic equations using approximate
numerical solutions found by interpolation in
trigonometric tables

39. Development and Early Trigonometry
• Nasīr al-Dīn al-Tūsī
– first to treat trigonometry as a mathematical
discipline independent from astronomy
– he developed spherical trigonometry into its
present form
– listed the six distinct cases of a right-angled
triangle in spherical trigonometry

40. Development and Early Trigonometry
• Nasīr al-Dīn al-Tūsī
– first to treat trigonometry as a mathematical
discipline independent from astronomy
– he developed spherical trigonometry into its
present form
– listed the six distinct cases of a right-angled
triangle in spherical trigonometry
• Jamshīd al-Kāshī
– provided the first explicit statement of the law
of cosines in a form suitable for triangulation

41. Development and Early Trigonometry
• Levi ben Gershon (Gersonides)
– wrote On Sines, Chords and Arcs
• "toleta de marteloio"
– used by sailors in the Mediterranean Sea
during the 14th-15th Centuries to calculate
navigation courses
• Leonhard Euler
– Introductio in analysin infinitorum
– mostly responsible for establishing the
analytic treatment of trigonometric functions in
Europe

42. Development and Early Trigonometry
• Brook Taylor
– general Taylor series
– gave the series expansions and
approximations for all six trigonometric
functions