Euclid

Euclid is one of the most influential and best read mathematician of all time.His prize work, Elements, was the textbook of elementary geometry and logic upto the early twentieth century. For his work in the field, he is known as thefather of geometry and is considered one of the great Greek mathematicians. Verylittle is known about the life of Euclid. Both the dates and places of his birthand death are unknown. It is believed that he was educated at Plato’s academy inAthens and stayed there until he was invited by Ptolemy I to teach at his newlyfounded university in Alexandria. There, Euclid founded the school ofmathematics and remained there for the rest of his life.

As a teacher, he wasprobably one of the mentors to Archimedes. Personally, all accounts of Eucliddescribe him as a kind, fair, patient man who quickly helped and praised theworks of others. However, this did not stop him from engaging in sarcasm. Onestory relates that one of his students complained that he had no use for any ofthe mathematics he was learning.

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Euclid quickly called to his slave to give theboy a coin because “he must make gain out of what he learns.” Anotherstory relates that Ptolemy asked the mathematician if there was some easier wayto learn geometry than by learning all the theorems. Euclid replied, “Thereis no royal road to geometry” and sent the king to study. Euclid’s famecomes from his writings, especially his masterpiece Elements. This 13 volumework is a compilation of Greek mathematics and geometry. It is unknown how muchif any of the work included in Elements is Euclid’s original work; many of thetheorems found can be traced to previous thinkers including Euxodus, Thales,Hippocrates and Pythagoras. However, the format of Elements belongs to himalone.

Each volume lists a number of definitions and postulates followed bytheorems, which are followed by proofs using those definitions and postulates.Every statement was proven, no matter how obvious. Euclid chose his postulatescarefully, picking only the most basic and self-evident propositions as thebasis of his work. Before, rival schools each had a different set of postulates,some of which were very questionable. This format helped standardize Greekmathematics.

As for the subject matter, it ran the gamut of ancient thought. Thesubjects include: the transitive property, the Pythagorean theorem, algebraicidentities, circles, tangents, plane geometry, the theory of proportions, primenumbers, perfect numbers, properties of positive integers, irrational numbers,3-D figures, inscribed and circumscribed figures, LCD, GCM and the constructionof regular solids. Especially noteworthy subjects include the method ofexhaustion, which would be used by Archimedes in the invention of integralcalculus, and the proof that the set of all prime numbers is infinite. Elementswas translated into both Latin and Arabic and is the earliest similar work tosurvive, basically because it is far superior to anything previous. The firstprinted copy came out in 1482 and was the geometry textbook and logic primer bythe 1700s.

During this period Euclid was highly respected as a mathematician andElements was considered one of the greatest mathematical works of all time. Thepublication was used in schools up to 1903. Euclid also wrote many other worksincluding Data, On Division, Phaenomena, Optics and the lost books Conics andPorisms. Today, Euclid has lost much of the godlike status he once held. In histime, many of his peers attacked him for being too thorough and includingself-evident proofs, such as one side of a triangle cannot be longer than thesum of the other two sides. Today, most mathematicians attack Euclid for theexact opposite reason that he was not thorough enough. In Elements, there aremissing areas which were forced to be filled in by following mathematicians. Inaddition, several errors and questionable ideas have been found.

The mostglaring one deals with his fifth postulate, also known as the parallelpostulate. The proposition states that for a straight line and a point not onthe line, there is exactly one line that passes through the point parallel tothe original line. Euclid was unable to prove this statement and needing it forhis proofs, so he assumed it as true. Future mathematicians could not acceptsuch a statement was unproveable and spent centuries looking for an answer. Onlywith the onset of non- Euclidean geometry, that replaces the statement withpostulates that assume different numbers of parallel lines, has the statementbeen generally accepted as necessary. However, despite these problems, Euclidholds the distinction of being one of the first persons to attempt tostandardize mathematics and set it upon a foundation of proofs.

His work actedas a springboard for future generations.Mathematics