Elements of Euclid: The Father of Geometry

Euclid was a Greek mathematician who was also called Euclid of Alexandria, the founder of geometry and the father of geometry. Euclid’s Elements, a collection of geometrical theorems, is his most significant work which has become part of the history of mathematics. Euclid’s geometry has always been part of mathematics textbooks ever since he published it. Euclid concluded the theorems of what is now called Euclidean geometry in Euclid’s Elements from a small set of axioms. Euclid’s contributions include works on number theory, perspective, mathematical rigour, conic sections, number theory and spherical geometry. 

Euclid’s Biography

Not much is known about the life of Euclid, as not much information about his existence has survived through history. Very little is known about the place and situation of his birth and death. 

It is believed that Euclid was born around mid 4th century BC in a town called Tyre which is a city presently in Lebanon. It is assumed that Euclid arrived at Alexandria, Egypt, about ten years after the arrival of Alexander the Great, which is around circa 322 BC.

Some of the historical references that have survived about Euclid are written by Proclus, a philosophical commentator who lived about eight centuries after Euclid. He confirmed the existence of Euclid as he briefly introduced the mathematician in his book “Commentary on the Elements”. Proclus also believed that Euclid probably lived during the time of Ptolemy I.  Euclid is also mentioned by Archimedes, a Greek mathematician and physicist, as “the author of the Elements”. 

Euclid died in circa 270 BC in Alexandria, Egypt.  

Euclid’s Geometry

Euclidean geometry, developed by Euclid, is a mathematical system. Euclidean geometry studies solid figures and planes on the foundation of theorems and axioms. He describes this in his book of geometry, Euclid’s Elements. Euclid’s method consists in believing a small set of axioms and deducing many other theorems from these. 

An axiom is like an assumption. It is a statement that is accepted to be true and to be a starting point for further reasoning, logic and arguments. 

Most of the results by Euclid have been expressed earlier. Although Euclid was the first to collect and organise and classify these propositions into a proper analytical system in which every result is proved from axioms and theorems that have been previously proved.

Euclid’s geometry was the only form of geometry available for a long period of time, and geometry only meant “Euclid’s geometry” until mathematicians like René Descartes introduced analytical geometry that uses coordinates instead of axioms to express geometric concepts like algebraic expressions. Hence, there are two forms of geometry Euclidean geometry and analytical geometry. 

Euclidean geometry is the common geometry taught in secondary schools. 

Elements of Euclid

Euclid’s Elements is a collection of definitions, postulates, theorems and constructions and also contains mathematical proofs of the propositions. The books also cover plane and solid Euclidean geometry, elementary number theory, and incommensurable lines. Elements are the oldest existent large-scale reasoned treatment of mathematics. It has been instrumental in the development of logic and modern science. 

Euclid’s geometry always starts with certain fundamental foundations. He began Elements with certain unclear phrases, such as “a point is something which has no part” and “a line is a length that does not have a breadth.” Moving from these terms, he further defined notions such as angles, triangles, circles, and varied other polygons and figures. For example, an angle was described as the inclination of two straight lines, and a circle was a plane figure consisting of all points that have a fixed distance from a given centre.

Euclid suggested five common concepts as a basis for logical deductions, such as “things equal to the same thing are equal” and five unprovable but instinctive principles known as axioms. The axioms stated in modern phrases are as follows:

1. Given two points, there is a straight line that joins them.
2. A straight line segment can be prolonged indefinitely.
3. A circle can be constructed when a point for its centre and a distance for its radius are given.
4. All right angles are equal.
5. 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, will meet on that side on which the angles are less than the two right angles.

In Euclid’s renowned work, the Elements, the only tools utilised for geometrical constructions were the ruler and the compass—a condition included in elementary Euclidean geometry to this day.

Facts About Euclid

• Euclid taught mathematics as a profession and also founded the Alexandrian School of Mathematics
• Many forms of mathematics are named after him, including Euclidean Geometry, Euclidean Number and Euclidean Algorithm
• Manuscripts of his most famous work ‘Euclid’s Elements’ were made in both Latin and Arabic languages
• He made huge contributions to the understanding of prime numbers, their behaviour, factorisation and divisors
• Until the 19th century, only Euclid’s work was considered ‘geometry’ as no other type of geometry was introduced

Conclusion

Euclid was a Greek mathematician who is also referred to as the father of geometry as his works on geometry have been used in the field of mathematics for the past 2000 years, and whose contributions to mathematics, geometry, in particular, have influenced modern mathematics since his publications. His most famous work is Euclid’s Elements. Euclidean geometry is based on axioms and theorems. Euclidean geometry is clearly explained in his book Elements.