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Who was Euclid?
Euclid of Alexandria, author of the most successful mathematics textbook ever written and with the exception of Autolycus' Sphere, is author of the oldest Greek mathematical treatise extant. More than half of Euclid's works have been completely lost, others surviving the ages only through translations and interpretations. Little is known about his life, yet his name and writings have managed to continue being a major influence on geometry for two millennia.
Much of what is believed to be known about Euclid comes from Proclus's Eudemian Summary. Proclus states that he was a contemporary of King Ptolemy I Soter. (305 - 285 B.C.E.) Some students of his work have asserted that he was in fact, a disciple of the Platonic school and had studied with students of Plato, possibly in Athens at the Academy. Euclid was older than Archimedes (287 - 212 B.C.E.) and Eratosthenes and younger than Plato. It is uncertain, but Euclid was probably a Greek who had traveled to the city of Alexandria to learn and teach. Euclid and Demetrius Phalereus were invited to open the mathematical school and to take charge of the library, at the Museum and Library at Alexandria. Euclid thus compiled the Elements as an elementary mathematics textbook covering arithmetic (number theory), synthetic geometry and the theories of proportion and of irrational lines. He probably wrote the Elements around 320 B.C.E., thus leaving him around the age of 40. No actual original copies exist from his time; the earliest copy in existence is dated from 888 A.D. Some fragments of the Elements have been found on potsherds discovered in Egypt from around 225 B.C.E. and pieces of papyrus dated from 100 B.C.E. The former containing notes on two propositions from Book XIII, and the latter having inscribed parts of Book II.
The Elements is a work contained in thirteen books. The first six books are on the topic of plane geometry, including a proportion theory for all geometrical magnitudes and a theory of similar plane figures. Books VII-IX are on number theory, Book X covers commensurability and incommensurability. Books XI and XII explore three-dimensional geometric objects, and Books XIII constructs the five regular polyhedra.
The Elements opens with definitions for the point, lines, surfaces, angles, etc., and lists five postulates and common notions. From Book I, the first three propositions deal with the constructions of an equilateral triangle and line segments equal to given ones. Proposition four, covers the side-angle-side criterion of triangle congruency. Book I ends with the Pythagorean Theorem and its converse.
Book II covers relationships between rectangles and squares, of which Euclid begins with the definition, "Any rectangle is said to be contained by the two straight lines forming the right angle." He never multiplies two lengths together, but does multiply lengths by positive integers. Many theorems in Book II can be easily restated as algebraic identities.
Book Three, Four, Five and Six
Book III covers properties of circles, the most fundamental curved figure. The Greeks considered this to be the most perfect of plane figures. Book IV contains the construction of a regular hexagon and a regular fifteen-gon inside a circle. The most difficult construction is that of a regular pentagon.
Book V's main concept is that of equal ratios (proportion). It is thought the Pythagoreans knew most of the results from Book VI.
Books Seven through Nine
Book VII-IX deals with the elementary theory of numbers. It appears that most of the compilation of Book VII was due to Theaetetus. Book VIII deals with sequences, or numbers that are continued in proportion. Much is this book is believed to be from Archytas (5th century B.C.E.) Book IX looks upon prime numbers being unbounded and finding perfect numbers.
Book Ten through Thirteen
Book X, arguably the most difficult of the Elements, is mainly credited to the work of Theaetetus. Book XI-XIII focus on problems of solid geometry. Book XI contains some redundant constructions, while Book XII uses what is known as the method of exhaustion, which was developed by Eudoxus Book XIII is dedicated to constructing the five regular polyhedra.
Book Fourteen and Fifteen
Additional books, Book XIV, not of Euclid's writings, contains 8 propositions and is a supplement to Book XIII. It is thought that this was contributed by Hypsicles in the second century B.C.E. Book XV is also not of Euclid's writing but probably added by John Damascianos of Damascus.
You can see the complete collections of Euclid's elements by clicking here.