Euclid The Elements “There is no royal road to Geometry.” c. 330 -275 B.C AKA – Father of Geometry Euclid told Ptolemy (the local ruler) that “There is no royal road to Geometry,” in response to Ptolemy’s request for a quick and easy knowledge of the subject

Development of Geometry Rhind papyrus – first written accounts Ahmes (Eqyptain scribe) – 3,4,5 right triangle & approximation of 𝜋 to four decimal places. Chinese – used the ruler, square, compass and level Greeks formalized & expanded the knowledge base of geometry. Thales (625-547 B.C.) – deductive proofs for several theorems Euclid (330-275 B.C.) – Elements, first textbook of geometry Names associated with the early development of Greek mathematics, beginning approximately 600 B.C., include Thales, Pythagoras, Archimedes, Appolonius, Diophantus, Eratosthenes, and Heron. Euclid collected, summarized, ordered, and verified the vast quantity of knowledge of geometry in his time.

Euclid of Alexandria Younger than the pupils of Plato, but older than Archimedes. Received mathematical education in Athens. The Arabs found that the name of Euclid, which they took to be compounded from ucli (key) an dis (measure) revealed the “key to geometry.” Greek philosophers posted on the doors of their school “Let no one come to our school who has not learned the Elements of Euclid.” Plato’s academy – all scholastic doors substituted the Elements for geometry. Asked to head the mathematics department in the University of Alexandria (in Egypt), which was the center of Greek learning.

Book One - Definitions A point is that which has no part. A line is breathless (never-ending) length. The extremities of a line are points. A straight line is a line which lies evenly with the points on itself.

Book One - Definitions A surface is that which has length and breadth (width) only. The extremities of a surface are lines. A plane surface is a surface which lies evenly with the straight lines on itself. A plane angle is the inclination to one another of two lines in a plane which meet one another and do not lie in a straight line. And when the lines containing the angle are straight, the angle is called rectilineal.

Book One - Definitions When a straight line set up on a straight line makes the adjacent angles equal to one another, each of the equal angles is right and the straight line standing on the other is called a perpendicular to that on which it stands. An obtuse angle is an angle greater than a right angle. An acute angle is an angle less than a right angle.

Book One - Definitions A boundary is that which is an extremity of anything. A figure is that which is contained by any boundary or boundaries. A circle is a plane figure contained by one line such that all the straight lines falling upon it from one point among those lying within the figure are equal to one another. And the point is called the center of the circle.

Book One - Definitions A diameter of the circle is any straight line drawn through the center and terminated in both directions by the circumference of the circle, and such a straight line also bisects the circle. A semicircle is the figure contained by the diameter and the circumference cut off by it. And the center of the semicircle is the same as that of the circle.

Book One - Definitions Rectilineal figures are those which are contained by straight lines, trilateral figures being those contained by three, quadrilateral those contained by four, and multilateral those contained by more than four straight lines. Of the trilateral figures, an equilateral triangle is that which has its three sides equal, an isosceles triangle that which has two of its sides equal, and a scalene triangle that which has three sides unequal. Further, of trilateral figures, a right-angled triangle is that which has a right angle, an obtuse-angled triangle that which has an obtuse angle, and an acute-angled triangle that which has three angles acute.

Book One - Definitions Of quadrilateral figures, a square is that which is both equilateral and right-angled; an oblong (rectangle) that which is right-angled but not equilateral; a rombus that which is equilateral but not right-angled; and a rhomboid that which has its opposite sides and angles equal to one another but is neither equilateral or right-angled. And let quadrilaterals other than these be called trapeziod. Parallel straight lines are straight lines which, being in the same plane and being produced indefinitely in both directions, do not meet one another in either direction.

Euclidean Geometry Axioms A statement that is held as true without any proof. These are often things that are obviously true Used to prove more complicated math theorems Axioms are self-evident assumptions, which are common to all branches of mathematics, while postulates are related to the particular mathematics.

Undefined Terms In Geometry the terms point, line & plane are described but not defined. Point – that which has no part, represented as a dot and has location but not size (meaning no dimensions).

Undefined Terms Line – infinite set of points, have a quality of “straightness” that is not defined but assumed. Plane – flat surface Parallel lines – two lines in a plane (or two planes) that never intersect.

Euclid’s Postulates A straight line segment can be drawn joining any two points. Any straight line segment can be extended indefinitely in a straight line. Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center.

Euclid’s Postulates (cont.) All right angles are congruent. If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough. This postulate is equivalent to what is known as the parallel postulate. Link explaining 5th postulate further.