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.

A BRIE

F HIS

TORY

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.

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Activity

Water “Proof”

Possible Shapes

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 .

=

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 ElementsAdopted Euclidean approachFormulated 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 and for every point P that does not lie on , there is exactly one line such that P lies on and ||.”

Elliptic Parallel Postulate: “For every line and for every point P that does not lie on , there is no line such that P lies on and ||.”

Hyperbolic Parallel Postulate: “For every line and for every point P that does not lie on , there are at least two lines and such that P lies on and and and are parallel to.”

Through a point not on a line, there is exactly one line parallel to the given line

PLAYFAIR’S FORM OF THE PARALLEL POSTULATE

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

C OX E T E R , H . S . M . N O N - E U C L I D E A N G E O M E T RY ( 1 9 6 1 ) . G R E AT B R I TA I N : U N I V E R S I T Y O F T O R O N T O P R E S S , P. V I I

H I S T O RY T E X T

EUCLIDEAN VS. NON-EUCLIDEAN

E U C L I D E A N ( P A R A B O L I C )

Only one where similar but not congruent triangles are possible

Sum of angles of a triangle is exactly 180°

Ratio of circumference of a circle to its diameter is exactly π

R I E M A N N I A N ( E L L I P T I C )

Sum of the angles of a triangle is less than 180°

Ratio of circumference of a circle to its diameter is greater than π and not constantL O B A C H E V S K I A N

( H Y P E R B O L I C )

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 15th and 16th centuries

Seen in the trigonometry of Europe and the Middle East

F I N A L T R I G N O T E F R O M " A H I S T O RY O F N O N -E U C L I D E A N G E O M E T RY: E V O LU T I O N O F T H E C O N C E P T

O F A G E O M E T R I C S PA C E " BY B . A . R O S E N F E L D . P U B L I S H E D 1 9 8 8 N E W YO R K : S P R I N G E R-V E R L A G

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

I S A L M I C I N F O F R O M " T H E C R E S T O F T H E P E A C O C K " BY G E O R G E G H E V E R G H E S E J O S E P H

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.