Module 6

• Geometry comes from two Greek words geo and metron which mean earth and measurement, respectively.
• Several early civilizations studied areas, lengths and volumes and their discoveries were the beginnings of Geometry.
• Before Euclid, in the 1850s, 400 clay tablets were unearthed containing Babylonian mathematics written in cuneiform script.
• Before Euclid

  The Babylonians gave us the formulas for finding areas and volumes of

  some geometric figures like circles and cylinders.

  • The Circumference of the circle was known to be three times the

  diameter. C = 3d = 3(2r) = 6r

  • The area was one-twelfth of the square of the circumference.

  A =1/12(6r)^2 = 3r2

  Hence, the Babylonians approximated the value of π to be 3.
• Before Euclid

  Egyptian civilization also gave us the formula in finding the area of areas and volumes. These formulas were essential in the construction of their famous pyramids and the determination of food supply. These formulas are recorded in the famous Ahmes Papyrus (also known as the Rhind Mathematical Papyrus).
• Rhind Papyrus was named after Alexander Henry Rhind, a Scottish antiquaria, who purchased the papyrus in 1858 in Luzor, Egypt.
• Before Euclid

  • One of the most significant contributions of the Greeks was the discovery of irrational numbers like √ 2, π, and the famous golden ratio φ, also known as the divine proportion.
  • How do we get the Golden Ratio? Divide any line segment (say one with unit length 1) into two parts, in the following • The ratio of the whole (1) to the longer part x equals the ratio of the longer to the shorter part, that is 1/x = x /1 − x
• Before Euclid

  value of golden proportion 1.618 . . .

  • This value is what we call the golden ratio, and is commonly denoted by φ
  • Rectangles having the ratio of the length and its width to be φ are called in the literature as golden rectangles, and these are considered by many as a rectangle with the most pleasing proportions.
• Before Euclid

  Erathosthenes was known to be the first mathematician to calculate the

  circumference of the earth. His calculation was indeed remarkable as the

  value he obtained was very much close to the value we have these days

  which was obtained by using modern calculations.

  The equatorial circumference of Earth is about 24, 901 miles (40, 075 km).

  However, from pole-to-pole — the meridional circumference — Earth is

  only 24, 860 miles (40, 008 km) around. This shape, caused by the

  flattening at the poles, is called an oblate spheroid.\
• Before Euclid

  The Greek mathematician Pythagoras gave us the concept that given a right triangle with lengths of sides a and b; and, length of side opposite the right angle is c, then a^2 + b^2 = c^2

  • This concept is known as the Pythagorean Theorem.
• Before Euclid

  • Archimedes of Syracuse was another Greek mathematician who contributed a handful of geometric concepts like the volumes of irregular shapes. He was also known for deriving an accurate approximation of the value of π using the method of exhaustion developed by Eudoxus of Cnidus.
  • Method of exhaustion – method of finding area of a shape by inscribing inside it a sequence of polygons whose areas converge to the area of the containing shape
• Euclidean Geometry

  • Euclid of Alexandria was one of the most famous Greek mathematician.
  • In his textbook The Elements, Euclid described a mathematical system known as Euclidean Geometry. Euclidean geometry is the oldest geometry.
  • Though many of the results mentioned by Euclid were stated by earlier mathematicians, he was the first to show how these results would fit into a comprehensive deductive and logical mathematical system.
• Geometry deals with points and set of points.
• There are concepts in geometry which are called Undefined Terms. These are points, lines and planes.
• Euclid Geometry

  Axioms Euclid formulated the following axioms to develop a structure mathematical system:

  1. Things that are equal to the same thing are also equal.
  2. If equals are added to equals, then the whole are equal.
  3. If equals are subtracted from equal, then the remainders are equal.
  4. Things that coincide with one another are equal to one another.
  5. The whole is greater than the part.
• Euclidean Postulates


  Statements or propositions that abstractly defined Euclid's axiomatic system
• There were five postulates that truly governed Euclid's system
• Postulate 1


  A straight line can be drawn from any point to any point
• Postulate 2


  A finite straight line can be produced continuously in a straight line
• Postulate 3


  A circle may be drawn with any point as center and any distance as radius
• Postulate 4


  All right angles are equal to one another
• Postulate 5


  If a transversal falls on two lines in such a way that the interior angle on one side of the transversal are less than two right angles, then the lines meet on that side on which the angles are less than two right angles
• A postulate is synonymous with the term axiom.
• The 5th Postulate is also called the Parallel Postulate as it

  characterizes what happens to a pair of parallel lines.
• Euclidean Geometry

  • It was Playfair (19th Century) who gave us the most popular version of the Euclid’s 5th postulate. Now called Playfair’s Axiom, the following is equivalent to Euclid’s 5th Postulate: “Through a point P not on a line l, there exists exactly one line passing through a point P parallel to l”.
• (Euclidean Geometry)

  Congruent Triangle - Two Euclidean triangles are congruent if and only if they have the same size and shape.

  • Same size and shape means that corresponding sides are of the same length and angles are equal.
• (Euclidean Geometry)

  Congruence Criteria for Two Euclidean Triangles

  Given △ABC and △XYZ

  • SSS Congruence: Three corresponding sides are congruent.
  • SAS Congruence: Two corresponding sides and the angle between them are congruent
  • ASA Congruence: Two corresponding angles and the side they include are congruent.
• (Euclidean Geometry)

  • Similar Triangles - Two triangles are similar if they have the same shape but can be different sizes.
  • Three corresponding angles of similar triangles are equal.
  • Congruent triangles are said to be similar triangles. Similar triangles need not be congruent.
• Non-Euclidean Geometry

  • Playfair’s Axiom - Through a point P not on a line l, there exists exactly one line passing through a point P parallel to l.
• Non-Euclidean Geometry


  A type of geometry that differs from Euclidean geometry
• When early development of non-Euclidean geometry began


  Early 19th century
• Early developers of non-Euclidean geometry


  • Nikolai Ivanovich Lobachevsky (Russian mathematician and geometer)
  • Janos Bolyai (Hungarian mathematician)
  • Johann Carl Friedrich Gauss (German mathematician)
• Early development of non-Euclidean geometry


  1. Lobachevsky published his work in 1829

  2. Bolyai published his work in 1832
• Lobachevsky and Bolyai were the main contributors to Hyperbolic geometry
• (Non-Euclidean Geometry)

  • Hyperbolic Geometry

  By assuming the negation of Euclid’s 5th Postulate, we get a new geometry called Hyperbolic Geometry.

  A model for this geometry can be seen by considering not the usual plane, but a circular disk. The model of this geometry conceived by the great universal mathematician Henri Poincar´e is now called the Poincare Disk of ´ Hyperbolic Geometry.
• (Non-Euclidean Geometry)

  Hyperbolic Geometry

  • Points - usual points inside the disk (but not lying on the boundary).
  • Lines - either diameters (line segments passing through the center of the disk but excluding points on the boundary) or arcs of circles that intersect the disk at right angles.
• (Non-Euclidean Geometry)

  Characteristic Postulate of Hyperbolic Geometry - Through a point P not on a line l, there exists at least two lines passing through point P parallel to a line l.
• (Non-Euclidean Geometry)

  Consequences of the Characteristic Postulate:

  1. The sum of the angles of a (hyperbolic) triangles are less than 180o . 2. Similar triangles are congruent
• Non-Euclidean Geometry

  Let us consider the first question: What if there is no line through P parallel to l?

  • This is a geometry where there are no parallel line, where all lines intersects.
  • The easiest way to visualize this is to think of the globe.
  • This geometry was studied deeply by the German mathematician George Friedrich Bernhard Riemann (1826 – 1866)
• (Non-Euclidean Geometry)

  Elliptic Geometry

  • Points - points on the surface of the globe.
  • Lines (great circles) - the circles of maximum diameter on the surface of the globe (such as the equator, or any longitude – these are the great circles passing through the North and South Pole).