Download A Brief History of Geometry

THE BEGINNING
Geometry was created out of necessity by the ancient Egyptian s [1,4].
Flooding of the Nile River would cause some of the tenants of the Pharaoh’s land to
lose crops, so the tenants would refuse to pay “rent” for the flooded land.
The Egyptians then figured out how to measure the area to adjust the “rent” [1].
Picture 1: Flooding of the Nile.
EGYPTIAN GEOMETRY
Approximated area and volumes of different shapes including circles, hemispheres,
and cylinders.
Example: Approximation for the area of a circle was “Take the diameter of the circle,
remove the ninth part of it, and find the area of the square with the resulting side
length.” Translates into
[2]
Challenged by finding the volume of a truncated square pyramid.
BABYLONIAN GEOMETRY
Geometry mostly used for measurement.
Appeared to know the Pythagorean Theorem.
 The sliding ladder problem
 Plimpton 322 tablet
Had formulae to find areas and volumes of various common shapes.
Used “cut and paste” geometry where they would rearrange squares
and rectangles to find the solutions to quadratic equations.
THE MIGRATION TO “GREECE”
Simply being called a Greek mathematician, does not
make one from Greece [2].
Thales (624-547 B.C.) was said to have learned math
from the Egyptians and Babylonians, and bring his
findings to the Grecian Empire [1,2,3,4].
Thales is credited with 5 theorems [Encyclopedia]
THALES’S THEOREMS
1. A circle is bisected by the diameter.
2. Angles in a triangle opposite two sides of equal length are equal (Isosceles
Triangle Theorem).
3. Opposite angles formed by intersecting straight lines are equal (Vertical angles
are congruent).
4. The angle inscribed in a semicircle is a right angle.
5. A triangle is determined if its base and the two angles at the base are given (ASA
Triangle Congruence Theorem).
PYTHAGORAS (569-475 B.C.)
Founder of the Pythagorean Brotherhood.
Pythagoras is credited with the Pythagorean Theorem learned in middle school.
It is quite possible that Pythagoras did not create the Pythagorean Theorem.
HIPPOCRATES
Hippocrates (470-410 B.C.) was the
predecessor of Euclid, and wrote an
“Elements” of Geometry [3,E].
He worked with the “squaring of the circle”
problem.
SQUARING THE CIRCLE
“For if a parallelogram can be found equal to any rectilinear figure, it is worth
inquiring whether it is possible to prove that a rectilinear figure is equal to a
circular area” [1].
In essence, the problem is to find a square that has the same area as a circle.
Indirectly leads to an exploration of .
𝐴1 : 𝑟12 = 𝐴2 : 𝑟22
𝐴: 𝑟 2 = 𝜋
𝐴 = 𝜋𝑟 2
IN COMES EUCLID
Euclid (325-265 B.C.) was the author of
“The Elements” which was a gathering
of the work of many of his predecessors
such as Thales, Pythagoras, and
Hippocrates [3].
It is quite possible that Euclid modeled his
book after Hippocrates [3].
THE FIVE POSTULATES OF GEOMETRY
In “The Elements” Euclid proposed 5 postulates (things assumed true) [1].
1. “Let it be postulated to draw a straight line from any point to any point, and”
2. “to produce a limited straight line in a straight line,”
3. “to describe a circle with any center and distance,”
4. “that all right angles are equal to each other.”
5. “That if a straight line falling on two straight lines makes the interior angles on the
same side less than two right angles, the two straight lines, if produced
indefinitely, meet on that side on which are the angles less than the two right
angles.” (parallel postulate)
ARAB GEOMETRY
First Greek text translated was Elements
Adopted Euclidean approach
Formulated precise theorems and proofs based on Euclid’s style
Focused on Euclid’s fifth postulate (parallel postulate)
Art, decoration
 Repetitions of basic motif
 Mathematical because not all shapes can cover a plane surface through repetition
 Symmetry
 Two styles: Muqarnas, Arabesque
ARAB ART
PARALLEL POSTULATES
Multiple parallel postulates exist, and are used to define different types of geometry.
Euclidean Parallel Postulate: “For every line 𝑙 and for every point P that does not lie
on 𝑙, there is exactly one line 𝑚 such that P lies on 𝑚 and 𝑚 ||𝑙.”
Elliptic Parallel Postulate: “For every line 𝑙 and for every point P that does not lie on 𝑙,
there is no line 𝑚 such that P lies on 𝑚 and 𝑚 ||𝑙.”
Hyperbolic Parallel Postulate: “For every line 𝑙 and for every point P that does not lie
on 𝑙, there are at least two lines 𝑚 and 𝑛 such that P lies on 𝑚 and 𝑛 and 𝑚 and
𝑛 are parallel to 𝑙.”
PLAYFAIR’S FORM OF THE PARALLEL POSTULATE
Through a point not on a line, there is exactly one line parallel to
the given line
SACCHERI’S NEGATION
Through a point not on a line, either
 there are no lines parallel to the given line, or
 there is more than one line parallel to the given line
RIEMANNIAN GEOMETRY
Considered a system where Saccheri’s 1st conjecture is true
“extended indefinitely”
 Saccheri interprets as “infinitely long”
 Riemann says, “not necessarily” (case of circles)
Created and published (1854) a system using postulates 1-4 and part 1 of parallel
negation with no contradictions
RIEMANNIAN GEOMETRY
Plane = sphere
Lines = great circles (circles that divide the sphere into two equal parts)
Great circles are the biggest possible circles drawn on the sphere, thus the
least curvature and thus the shortest distance
LOBACHEVSKIAN GEOMETRY
A geometric system where the Parallel Postulate is replaced with part 2 of Saccheri’s
negation
Carl Friedrich Gauss (German) explored but did not publish
Nicolai Lobachevsky (Russian) published first in 1829
Janos Bolyai (Hungarian) published in 1832
The joint conclusion: The substitution produces a system with no contradictions. Thus
the Parallel Postulate cannot be proven by postulates I - IV.
CLASSIFICATION OF NON-EUCLIDEAN GEOMETRIES
In 1871, Felix Klein assigned these three geometries their more commonly known
names:
 Euclidean = parabolic
 Lobachevskian = hyperbolic
 Riemannian = elliptic
Klein also linked the new geometries to new algebraic ideas
COXETER, H.S.M. NON-EUCLIDEAN GEOMETRY (1961). GREAT BRITAIN: UNIVERSIT Y OF TORONTO PRESS,
P. VII
HISTORY TEXT
EUCLIDEAN VS. NON-EUCLIDEAN
EUCLIDEAN (PARABOLIC)
RIEMANNIAN (ELLIPTIC)
Only one where similar but not
congruent triangles are
possible
Sum of the angles of a triangle
is less than 180°
Sum of angles of a triangle is
exactly 180°
Ratio of circumference of a
circle to its diameter is
exactly π
Ratio of circumference of a
circle to its diameter is
greater than π and not
constant
LOBACHEVSKIAN (HYPERBOLIC)
Sum of the angles of a triangle is less
than 180°
Ratio of circumference of a circle to its
diameter is less than π and not
constant
BABYLONIAN ASTRONOMY
Early studies of astronomy were more predictive than explanatory.
Computational schemes to determine the next time and location of a celestial event.
This is where the sexagesimal system was formed.
SPHERICAL GEOMETRY
Geometry of the surface of a sphere
Came about from the Greek’s interest in astronomy
Borrowed ideas from the Babylonians
 Use of numbers to measure angles
 Sexagesimal system
Propelled the development of trigonometry in the 15 th
and 16th centuries
Seen in the trigonometry of Europe and the Middle East
FINAL TRIG NOTE FROM "A HISTORY OF NON-EUCLIDEAN
GEOMETRY: EVOLUTION OF THE CONCEPT OF A GEOMETRIC
SPACE" BY B.A.ROSENFELD. PUBLISHED 1988 NEW YORK:
SPRINGER-VERLAG
TRIGONOMETRY
Hipparchus of Rhodes (190-120 B.C.):
Attempted to describe celestial movement by relating angles to chords.
Believed to have constructed a table of chords with radius 3438 (see Sketch 18)
Claudius Ptolemy (85-165 A.D.) :
Wrote the book Almagest in which he proved theorems about chords (see Sketch
18)
He also worked with “spherical triangles” to solve basic astronomy problems
[Encyclopedia]
Also created a table of chords.
TRIGONOMETRY
Indians introduced idea of half-chord which led to the modern day idea of sine
Islamic culture
 Introduced the six basic trigonometric functions
 Derived the sine rule and other trigonometric identities
 Constructed trigonometric tables
ISALMIC INFO FROM "THE CREST OF THE PEACOCK" BY GEORGE
GHEVERGHESE JOSEPH
INDIAN GEOMETRY
Western philosophy on geometry a mixture
 Greek geometry
 Traditional measuring and surveying
 Metaphysical speculation
Study of mathematics propelled by interest in astronomy
Created the idea of “half-chord” or sine into trigonometry
 Built upon Greek trig
Extended Ptolemy’s approximation techniques to obtain sophisticated formulas for
approximate computation [2]
REFERENCES
1. Artmann, B. (1999). Euclid - the creation of mathematics. New York: Springer.
2. Berlinghoff, W. P., & Gouvêa, F. Q. (2004). Math through the ages: a gentle
history for teachers and others (Expanded ed.). Washington, DC: Mathematical
Association of America.
3. Heath., T. (n.d.). History of Geometry. Geometry Algorithm Home. Retrieved April
16, 2012, from http://softsurfer.com/history.htm
4. Heilbron, J. L. (1998). Geometry civilized: history, culture, and technique. Oxford:
Clarendon Press ;.
5. Venema, G. (2006). The foundations of geometry. Upper Saddle River, N.J.:
Pearson Prentice Hall.