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Euclid

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 up to the early twentieth century. For his work in the field, he is known as the father of geometry and is considered one of the great Greek mathematicians. Very little is known about the life of Euclid. Both the dates and places of his birth and death are unknown.

It is believed that he was educated at Plato’s academy in Athens and stayed there until he was invited by Ptolemy I to teach at his newly founded university in Alexandria. There, Euclid founded the school of mathematics and remained there for the rest of his life. As a teacher, he was probably one of the mentors to Archimedes. Personally, all accounts of Euclid describe him as a kind, fair, patient man who quickly helped and praised the works of others. However, this did not stop him from engaging in sarcasm.

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One story relates that one of his students complained that he had no use for any of the mathematics he was learning. Euclid quickly called to his slave to give the boy a coin because he must make gain out of what he learns. Another story relates that Ptolemy asked the mathematician if there was some easier way to learn geometry than by learning all the theorems. Euclid replied, There is no royal road to geometry and sent the king to study. Euclid’s fame comes from his writings, especially his masterpiece Elements. This 13 volume work is a compilation of Greek mathematics and geometry.

It is unknown how much if any of the work included in Elements is Euclid’s original work; many of the theorems found can be traced to previous thinkers including Euxodus, Thales, Hippocrates and Pythagoras. However, the format of Elements belongs to him alone. Each volume lists a number of definitions and postulates followed by theorems, which are followed by proofs using those definitions and postulates. Every statement was proven, no matter how obvious. Euclid chose his postulates carefully, picking only the most basic and self-evident propositions as the basis of his work. Before, rival schools each had a different set of postulates, some of which were very questionable.

This format helped standardize Greek mathematics. As for the subject matter, it ran the gamut of ancient thought. The subjects include: the transitive property, the Pythagorean theorem, algebraic identities, circles, tangents, plane geometry, the theory of proportions, prime numbers, perfect numbers, properties of positive integers, irrational numbers, 3-D figures, inscribed and circumscribed figures, LCD, GCM and the construction of regular solids. Especially noteworthy subjects include the method of exhaustion, which would be used by Archimedes in the invention of integral calculus, and the proof that the set of all prime numbers is infinite. Elements was translated into both Latin and Arabic and is the earliest similar work to survive, basically because it is far superior to anything previous. The first printed copy came out in 1482 and was the geometry textbook and logic primer by the 1700s.

During this period Euclid was highly respected as a mathematician and Elements was considered one of the greatest mathematical works of all time. The publication was used in schools up to 1903. Euclid also wrote many other works including Data, On Division, Phaenomena, Optics and the lost books Conics and Porisms. Today, Euclid has lost much of the godlike status he once held. In his time, many of his peers attacked him for being too thorough and including self-evident proofs, such as one side of a triangle cannot be longer than the sum of the other two sides.

Today, most mathematicians attack Euclid for the exact opposite reason that he was not thorough enough. In Elements, there are missing areas which were forced to be filled in by following mathematicians. In addition, several errors and questionable ideas have been found. The most glaring one deals with his fifth postulate, also known as the parallel postulate. The proposition states that for a straight line and a point not on the line, there is exactly one line that passes through the point parallel to the original line. Euclid was unable to prove this statement and needing it for his proofs, so he assumed it as true. Future mathematicians could not accept such a statement was unproveable and spent centuries looking for an answer.

Only with the onset of non- Euclidean geometry, that replaces the statement with postulates that assume different numbers of parallel lines, has the statement been generally accepted as necessary. However, despite these problems, Euclid holds the distinction of being one of the first persons to attempt to standardize mathematics and set it upon a foundation of proofs. His work acted as a springboard for future generations.

Euclid

Greek Mathematics
Centered on Geometry (Euclid)
The ancient Greeks have contributed much to the development of the Western World as we know it today. The Greeks questioned all and yearned for the answers to many of lifes questions. Their society revolved around learning, which allowed them to devote the majority of their time to enlightenment. In answering their questions, they developed systematic activities such as philosophy, psychology, astronomy, mathematics, and a great deal more. Socrates (469-399 BC) was an ancient Greek philosopher whose ideas mark the turning point in the history of knowledge and formal thought. Plato (428-347:348 BC) one of Socrates students founded the Academy. The Academy was key in spreading thought and knowledge because of its devotion to teaching the sciences. Aristotle (384-322 BC), Platos brightest student, founded Biology and is given credit for his accomplishments in varying fields. Out of all of the great Greek accomplishments which influence the world today, I chose the one which I believe is the most important, Euclidean Geometry and its effects.

Euclid (365-300 BC) is often considered synonymous with geometry. Euclids works have been so influential that they serve as the basis for most geometrical teachings for the past 2000 years. His works supercede all other works of its kind. Euclids interests in spatial knowledge lead him to detailed definitions, postulates, and axioms that are used today. Data is a collection of given measurements and postulates that Euclid collected. Data expresses that lines, angles, and ratios can be given in magnitude; rectilinear figures may be given in species or form; and points and lines may be given in position. Euclids 94 propositions state that when certain aspects of a figure are given, other aspects can be found by using concrete formulas. For example, proposition 66 states, If a triangle have one angle given, the area of the rectangle contained by the sides including the angle has to the area of that triangle a given ratio. Divisions of Figures consists of 36 propositions concerning the divisions of various figures into two or more equal parts in given ratios. Optics is an elaboration on Platonic thought stating that discrete rays cause vision, and that vision can be explained by geometry. Euclid states that, Things seen under a greater angle appear greater, and those under a lesser angle appear less, while those under equal angles appear equal. Euclid used this statement and his mathematical formulas to explain elusions in size comparison. Conics, Porisms, Psiedese, and Surface Loci are lost works attributed to Euclid. These four works are the link between elementary geometry, and higher mathematics. Catoptrica explains the theory of mirrors and brought about Euclids Elements of Music. Elements of Music is a brief excursion into the uses of mathematics in music and sound.
Euclids most important works are summarized in the Elements, which consists of 13 detailed books. Elements presents all of the Greek geometrical knowledge of Euclids day in a logical fashion. These books give us a little insight into Euclid and were designed and are used as learning tools. Including theorems and constructions of plane geometry, solid geometry theory of proportions, incommensurable, commensurable, number theory, and the basis for what is known as geometrical algebra. Proclus (Greek Philosopher) defined Elements as those theorem whose understanding leads to knowledge of the rest. Elements is a detailed explanation of geometric shapes, and measurements using the number theory. The impact of the Elements has been so great that translated forms are widely studied today. Since Euclid based his entire geometric study on points, straight lines, and circles, his work leaves three main geometrical questions open. The three famous problems left unsolved were squaring a circle, doubling the cube, and trisecting the angle. But the Greeks say other Greek philosophers later solved these unsolved mysteries. Euclidean Geometry was not elaborated upon greatly until 1667 when Girolamo Saccheri wrote Euclid Freed of Every Flaw. Girolamo Saccheri through his works started the basis for elliptical geometry (obtuse angles) and hyperbolic geometry (acute angles) which was a continuation on Euclids work eventually forming Non Euclidean Geometry.
Although a large part of mathematics can be attributed to Euclid, there are other Greek philosophers who have also contributed greatly to the study of mathematics. Pythagoras of Somos regarded numbers as sums of units. Pythaagoras is considered the father of irrational numbers, and the Pythagorean Theorem. Eudorus of Cnides solved Pathagorases dilemma of incommensurable magnitudes with the theory of proportion. Plato the teacher of many, considered geometry as the model of certain reasoning. Euclid during the 3rd century compiled and edited existing ideas. Pappus used Euclids writings as the basis for trigonometry, which is recorded in Almagest. Altogether the Greeks formalized geometry started the basis for modern trigonometry and set the grounds for the algebra of today, without all of the great mathematical contributions the world would be much different.
The mathematics ideas of ancient Greece are used in every aspect of life. The ideas of Greek mathematicians can be seen wherever you travel. From simple things such as buildings, to complex computers and engineering of all kinds, it is evident that their influence is ever present.
I am very impressed with the extent to which the ancient Greeks have influenced not only history but also our future. Euclidean Geometry and mathematics derived from it are used daily all over the world bringing order to the construction and understanding of almost everything.

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