Final Project

Math 4311

Lauren Samaniego

Origins of Geometry

Geometry literally translates to “Earth Measurement.” (1)

The first form of geometry was the drawing of straight lines and circles by man; the most important lines of geometry. (1)

Egyptians (c. 2000-500 B.C.)

Some of the first geometry experts were Ancient Egyptian land surveyors, called “arpedonapti,” which means those who tie knots. (1)

The tightening of ropes on the land to mark lines and circles was used as a method to survey land and is a method still used today. (1)

Egyptians- Moscow Mathematical Papyrus

The material in the Moscow Papyrus dates back to about 1850 B.C. (2)

The Moscow Papyrus is a collection of twenty-five practical problems regarding arithmetic and geometry. (3)

Seven of the twenty-five problems are geometry based and include computing the areas, surface area, and volume of various shapes. (3)

Egyptians- The Rhind Mathematical Papyrus

The Rhind Mathematical Papyrus is the primary source for Egyptian mathematics and dates back to around 1650 B.C. (2)

There are three parts of the Rhind Papyrus, the second part is a collection of geometry problems.

The geometry problems include:

Finding the volume of cylindrical and rectangular based granaries.

Finding the area of various shapes.

Finding the slopes of pyramids. (2)

Problem 50 deals with finding the area of a circular field with a specific diameter. In doing so an approximation of π is given as

π=256/81= 3+1/9+1/27+1/81≈ 3.1605 (2)(4)

Babylonians (c. 2000-500 B.C.)

Knowledge of Babylonian mathematics comes from about 400 clay tablets dating back to 1800 B.C. (5)

The tablets include problems and explanations on fractions, algebra, quadratic and cubic equations, and the Pythagorean theorem. (5)

The Babylonians understood the relationship between the sides of a right triangle and it is believed that Pythagoras developed his theorem by studying Babylonian geometry.

To calculate volume and area, it is found that the Babylonians used 3 as the value of pi, although on one tablet 3 and 1/8 was used as a value of pi. (5)

Greeks (c. 750-250 B.C.)

Greek geometry was influenced by other civilizations, particularly those Egyptian and Babylonian. (4)

The Greek culture has influenced many aspects of our modern life including our, government, universities, literature, alphabet, and mathematics, just to name a few. (4)

Thales of Miletus

Around 575 B.C. Thales of Miletus introduced Babylonian mathematics to Greece, where he used geometry to solve problems such as calculating the height of pyramids and the distance of ships form the shore. (6)

Pythagoras of Samos

Around 530 B.C. Pythagoras of Samos inhabits Croton and begins teaching mathematics, geometry, music, and his belief of reincarnation. The Pythagoreans worked with the properties of parallels, proportions of similar figures, and of course the Pythagorean theorem. (6)

Hippocrates of Chios

Hippocrates of Chios, a student of Pythagoras, wrote the first “Elements of Geometry” which included solutions to quadratic equations and illustrated some of the first methods of integration. Hippocrates also was the “first to show that the ratio of the areas of two circles equals the ratio of the squares of the circles’ radii. (6)

Euclid of Alexandria

Euclid is one of the most significant Greek geometers, most famously known for his work, the Elements. The Elements is a collection of mathematical theorems developed by various Greek mathematicians.

Euclid’s approach to geometry has had immeasurable influence on the teaching of geometry since his time.

Also, around 400 BC Euclid proposed five geometric postulates, four having proofs. The fifth postulate, also known as the "Parallel Postulate" was without proof and for the next few centuries mathematicians failed to provide one. (7)

SOURCES

1. Giusti, E. (n.d). The Origins of Geometry. Retrieved from http://php.math.unifi.it/archimede/archimede_NEW_inglese/curve curve_giusti/prima.php?id=1

2. Williams, S., (2008). Egyptian Mathematics Papyri. Retrieved from http://www.math.buffalo.edu/mad/Ancient-Africa/mad_ancient_egyptpapyrus.html#ahmes/rhind%20papyrus

3. Moscow Mathematical Papyrus. (8 March 2013) In Wikipedia, The Free Encyclopedia. Retrieved from http://en.wikipedia.org/wiki/Moscow_Mathematical_Papyrus

4. Burton, D.M. (2011). In S.K. Mattson (Ed.), The History of Mathematics. New York: The McGraw-Hill Companies.

5. Babylonian Mathematics. (21 July 2013). In Wikipedia, The Free Encyclopedia. Retrieved from http://en.wikipedia.org/wiki/Babylonian_mathematics#Geometry

6. University of St. Andrews, School of Mathematics and Statistics. (August 2001). A Mathematical Chronology. Retrieved from http://www-groups.dcs.st-and.ac.uk/~history/Chronology/full.html

7. Lanius, C. (1997). History of Geometry. Retrieved from http://math.rice.edu/~lanius/Geom/his.html

8. Snell's Law. (Updated 20 April 2009). In Encyclopedia Britannica online. Retrieved from http://www.britannica.com/EBchecked/topic/550450/Snells-law

9. University of St Andrews, School of Mathematics and Statistics. (August 1995). Girard Desargues. Retrieved from http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Desargues.html

10. Descriptive Geometry. (15 June 2013). In Wikipedia, The Free Encyclopedia. Retrieved from http://en.wikipedia.org/wiki/Descriptive_geometry

11. Synthetic Geometry. (29 July 2013). In Wikipedia, The Free Encyclopedia. Retrieved from http://en.wikipedia.org/wiki/Synthetic_geometry#History

12. Edgell, John J. (2006) Heronian Simplexes and Constructs:Teaching:Simplexes-Technology Connection. Retrieved February 20, 2011, from

<http://rfdz.phnoe.ac.at/fileadmin/Mathematik_Uploads/ACDCA/DESTIME2006/

DES_contribs/Edgell/edgell.pdf>

Aristarchus of Samos

Around 290 B.C. Aristarchus of Samos proposed a sun-centered universe and calculated the distance to the sun and moon using a geometric method. (6)

**2000 BC**

**1000 BC**

**The Development Over Time**

**1 AD**

Heron of Alexandria

Around 60 AD Heron of Alexandria writes Metrica which contains formulas for calculating area and volume. (6)

He is best known for his formula of the area of a triangle in terms of the lengths of it's sides.

T=√s(s-a)(s-b)(s-c), where s is the semiperimeter of the triangle and a,b,and c are the side lengths of the triangle.

s=(a+b+c)/2

Heron also had many inventions, one being the steam turbine. (12)

Liu Hui

Around 263 Liu Hui calculates the value of pi as 3.14159 using a 192 sided regular polygon. (6)

**500**

**1000**

Al-Khayyami

Around 1072 Al-Khayyami (aka. Omar Khayyam) wrote Treaties on Demonstration of Problems of Algebra in which Khayyami solves cubic equations with geometric solutions. (4)

Gerbert d' Aurillac

Gerbert was able to determine the sides of a right triangle whose hypotenuse a and area b^2. His solution being:

(1/2)(√(a^2+4b^2 )±√(a^2-4b^2 ))

Also, in Gerbert's Geometry, he determined the area of an equilateral triangle of side s to be (3s^2)/7.

(4)

Levi ben Gerson

Around 1342 Gerson writes De sinibus, chordis et arcubus, which includes a proof for the sine theorem for plane triangles. (6)

Ptolemy of Alexandria

Ptolemy develops a trigometric version of Pythagoas' theorem in his treatise, Almagest. The theorem states:

Let a convex quadrilateral ABCD be inscribed in a circle. Then the sum of the products of the two pairs of opposite sides equals the product of its two diagonals.

AD x BC + AB x CD = AC x BD

Willebrord van Royen Snell

Snell developed an optical law that describes the "relationship between the path taken by a ray of light in crossing the boundary or surface of separation between two contracting substance and the refractive index of each." (8)

**1500**

Girard Desargues

Desargues was the founder of projective geometry, which studies shapes projected on to a non-parallel plane with emphasis on conic sections and perspective. (9)

Gaspard Monge

Monge begins the study of descriptive geometry, which is the representation of 3-D objects in 2-D, around 1763.

(10)

Jakob Steiner

In 1824, Steiner contributes to projective geometry using a synthetic geometric approach which focuses on the use of "axioms, theorems and logical arguments to draw conclusions." (11)

Johann Carl Friedrich Gauss

In 1828 Gauss introduces differential geometry which draws on conclusions using differential and integral calculus. (6)

1946

Andre Weil publishes Foundations of Algebraic Geometry. (6)

**The Present**

Geometry has made a long journey from just lines and circles and it was the curiosity and determination of those before us that have brought geometry this far.

**THE END**

(4)