The History of
Geometry Concept
History of Mathematics
Pendidikan Matematika
Universitas Negeri Jakarta
Introduction
• The word “geometry” comes from Greek words. “Geo” means “earth” and
“metron” means “measurement.” At first, geometry was all about understanding how things fit together in space.
• Geometry is considered as the oldest branch of mathematics besides arithmetic, as initially geometry originated from practical necessity and the need to measure land (e.g., Egyptian and Babylonian: no axioms and rules, pragmatic geometry)
• Greek mathematicians brought a deductive reasoning to Geometry, finding out axioms and underlying principles governing geometry.
Egyptian Geometry
• They know how to calculate areas of rectangle, triangles, trapezoids, and circle.
• Problem 50 (Rhind Papyrus): “Example of a round field of diameter 9. What is the area? Take away 1/9 of the diameter; the remainder is 8. Multiply 8 times 8; it makes 64. Therefore, the area is 64.”
• The Egyptian scribe was using a procedure described by the formula A = (d − d/9)2.
• A comparison with the formula A = (π/4)d2shows that the Egyptian value for the constant π in the case of area was 256/81= 3.16049 ....
• Able to calculate: the volume of a cylinder; volume of a rectangular box; volume of a truncated pyramids; surface area of a hemisphere.
Babylonian Geometry
• They developed procedures for determining areas and volumes of various kinds of figures. They worked out algorithms to determine square root. The problems often related to agriculture or building.
• In determining areas of circles, the Babylonians took the circumference as the defining component of a circle. Thus, they gave two coefficients for the circle:
0;20 (= 1/3) for the diameter and 0;05 (= 1/12) for the area. The first coefficient means that the diameter is one-third of the circumference, while the second means that the area is one-twelfth of the square of the circumference.
Example 1:
Source: Katz, 2009, p. 15
Example 2:
Source: Katz, 2009, p. 25
Greek Geometry
• Thales of Miletus (6th century BCE), began the process of using deduction proof of geometry.
• He proposed some axioms that he believed to be mathematical truths:
A circle is bisected by any of its diameters
The base angles of an isosceles triangle are equal
When two straight lines cross, the opposing angles are equal An angle drawn in a semi-circle is a right angle
Two triangles with one equal side and two equal angles are congruent
Pythagoras (570 BC – 490 BC)
Euclid
Postulate : A statement, also known as an axiom, which is taken to be true
without proof.
Lemma : A short theorem used in proving a larger theorem
Theorem: is a statement that can be demonstrated to be true by accepted mathematical operations and
arguments. The process of showing a theorem to be correct is called a proof.
Elements
BOOK I Triangles, parallels, and area BOOK II Geometric algebra
BOOK III Circles
BOOK IV Constructions for inscribed and circumscribed figures BOOK V Theory of proportions
BOOK VI Similar figures and proportions BOOK VII Fundamentals of number theory
BOOK VIII Continued proportions in number theory BOOK IX Number theory
BOOK X Classification of incommensurables BOOK XI Solid geometry
BOOK XII Measurement of figures BOOK XIII Regular solids
Source: Katz, 2009, p. 53
Source: Katz, 2009, p. 54
Euclid: Book I and the Pythagorean Theorem
Source: Katz, 2009, p. 55
Euclid: Book II and Geometry Algebra
Euclid began Book II with a definition: Any rectangle is said to be contained by the two straight lines forming the right angle.
Euclid: Book III & IV: Circles and Pentagon Construction
Source: Katz, 2009, p. 67
Euclid: Book XI and Solid Geometry
Source: Katz, 2009, p. 84
Euclid: Book XII and the Method of Exhaustion
Source: Katz, 2009, p. 85
The central feature of Book XII, which distinguishes it from the other books of the Elements, is the use of a limiting process, generally known as the method of
exhaustion
Euclid: Book XIII and Polyhedra
Source: Katz, 2009, p. 87
The final book of the Elements, Book XIII, is devoted to the construction of the five regular polyhedra and their “comprehension” in a sphere.
Archimedes and Geometry
Chinese Geometry
• The Chinese geometry was generally practical and had no notion of an axiomatic system from which theorems could be derived.
• The Chinese developed numerous formulas for calculating the areas and volumes of geometrical figures, such as volume of parallelepipeds.
• The Nine Chapters also gives the correct formula for the volume of a pyramid.
• The Nine Chapters and other ancient Chinese documents assume known the Pythagorean Theorem.
• Calculations using similar triangles may often be thought of as “trigonometry”
Indian Geometry
Geometry in Islamic Golden Age
• Practical Geometry➔ Al-Khwarizmi: geometric demonstrations in algebra;
mensuration
• Geometrical Constructions ➔ Abu Kamil showed, using algebra, how to
construct an equilateral pentagon in a given square, each of whose sides is 10
• The Parallel Postulates ➔ the provability of Euclid’s fifth postulate
• Volumes and the Method of Exhaustion ➔ ibn al-Haytham proved that the volume of the solid formed by rotating the parabola x = ky2 around the line x = kb2
Postulate 5: The Parallel Postulate
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.
Omar Khayyam (1048 – 1131)
In Commentaries on the difficult postulates of Euclid's book Khayyam made a contribution
to non-euclidean geometry, although this was not his intention. In trying to prove the parallels
postulate he accidentally proved properties of figures in non-euclidean geometries.
Non-Euclidean Geometry
https://www.youtube.com/watch?v=i5goUkT1irw