Euclid – ‘Father Of Geometry’

An Introduction• Euclid ,sometimes called Euclid of

Alexandria , was a Greek Mathematician, often referred to as the "Father of Geometry". He was active in Alexandria during the reign of Ptolemy I (323–283 BC). His Elements is one of the most influential works in the history of Mathematician, serving as the main textbook for teaching Mathematics (especially Geometry) from the time of its publication until the late 19th or early 20th century.

Euclid’s Elements

• In the Elements, Euclid deduced the principles of what is now called Euclidean Geometry from the small set of axioms. Euclid wrote works on:-

• Number Theory• Perspectives•  Conic Sections• Spherical Geometry etc. 

Number theory

• Euclid although is best known for its geometric results, the Elements also includes number theory. It considers the connection between perfect numbers and mersenne primes( Euclid-Euler theorem), the infinitude prime numbers, Euclid’s lemma on factorization (which leads to the fundamental theorem of arithmetic  on uniqueness of prime factorization), and the Euclidean algorithm for finding the greatest common divisor of two numbers.

Geometry• The geometrical system described in

the Elements was long known simply as Geometry, and was considered to be the only geometry possible. Today, however, that system is often referred to as Euclidean Geometry .

Geometry Works

• The concepts include : Dimension, Angles, Curve, Diagonals, Parallel, Perpendicular, Vertex, Symmetry, Similarity, Congruence etc.

• His works are also done on Zero/One-Dimensional, Two-Dimensional, Triangles and its Properties, Quadrilateral, Parallelogram, Three-Dimensional, Circles, Four Dimensional (Tesseract) and Volume, Perimeter and Area.

Euclid’s Perspectives

• Optics is the earliest surviving Greek treatise on perspective. In its definitions Euclid follows the Platonic tradition that vision is caused by discrete rays which emanate from the eye.

• Euclid relates the apparent size of an object to its distance from the eye and investigates the apparent shapes of cylinders and cones when viewed from different angles.

• He proved that for any two unequal magnitudes, there is a point from which the two appear equal.

Perspectives - Definitions• Things seen under a greater angle appear

greater, and those under a lesser angle less, while those under equal angles appear equal."

• A point is that which has no part.• A line has a breathless length.• The extremities of lines are points. • A straight line lies equally with respect to

the points on itself. • One can draw a straight line from any point

to any point.• Things equal to the same thing are also

equal to one another.• A straight line lies equally with respect to

the points on itself.

 

Perspectives - Definitions• A surface is that which has length and

breadth only.• The edges of a surface are lines.• A plane surface is a surface which lies

evenly with the straight lines on itself.• When the lines containing angle are

straight, the angle is called rectilinear.• And when the lines containing angle are

straight, the angle is called rectilinear.• When a straight line standing 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.

Perspectives - Definitions

• An obtuse angle is an angle greater than a right angle.

• An acute angle is an angle less than a right angle.

• A boundary is that which is an extremity of anything.

• A figure is that which is contained by any boundary or boundaries.

• 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.

• Rectilinear 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 trilateral figures, an equilateral triangle is that which has its three sides equal, an isosceles triangle that which has two of its sides alone equal, and a scalene triangle that which has its three sides unequal.

Perspectives - Definitions

• Of quadrilateral figures, a square is that which is both equilateral and right-angled; an oblong that which is right-angled but not equilateral; a rhombus 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 nor right-angled. And let quadrilaterals other than these be called trapezoid.

• 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.

Perspectives - Definitions

Perspectives-Notions

• Things which equal the same thing also equal one another.

• If equals are added to equals, then the wholes are equal.

• If equals are subtracted from equals, then the remainders are equal.

• Things which coincide with one another equal one another.

• The whole is greater than the part.