The Elements

The Elements is a treatise on geometry and mathematics written by the Greek mathematician Euclid (flourished 300 BCE).[1] The Elements is one of the most influential books ever written.[1] It set a standard for deductive reasoning and geometric instruction that persisted, practically unchanged, for more than 2,000 years.[1][2] It is sometimes said that, other than the Bible, the Elements is the most translated, published, and studied of all the books produced in the Western world.[1][2]

Euclid and His Background

Euclid (flourished c. 300 BCE, Alexandria, Egypt) was the most prominent mathematician of Greco-Roman antiquity, best known for his treatise on geometry, the Elements.[11] Of Euclid's life nothing is known except what the Greek philosopher Proclus (c. 410–485 CE) reports in his "summary" of famous Greek mathematicians.[11][5] According to him, Euclid taught at Alexandria in the time of Ptolemy I Soter, who reigned over Egypt from 323 to 285 BCE.[11]

Early Life and Education

He was born in around 325 BC, was probably educated in Plato's school in Athens, and he taught mathematics in Alexandria.[12][13] Though details of his early life remain largely unknown, he is believed to have studied at Plato's Academy before moving to Alexandria, where he became a key figure in the scholarly community under Ptolemy Soter.[14] He probably studied for a time at Plato's Academy in Athens but, by Euclid's time, Alexandria, under the patronage of the Ptolemies and with its prestigious and comprehensive Library, had already become a worthy rival to the great Academy.[15]

Professional Development

Euclid traveled to Alexandria and was appointed to the faculty of the Museum, the great research institution that was being organized under the patronage of Ptolemy Soter, who ruled Egypt from 323 to 283.[14] Around 300 BCE, he ran his own school in Alexandria, Egypt.[9] Euclid presumably became the librarian, or head, of the Museum at some point in his life.[14]

Key Influences and Mentors

Proclus, the last major Greek philosopher, who lived around 450 AD wrote: "Not much younger than these [pupils of Plato] is Euclid, who put together the 'Elements', arranging in order many of Eudoxus's theorems, perfecting many of Theaetetus's, and also bringing to irrefutable demonstration the things which had been only loosely proved by his predecessors."[5] Euclid compiled his Elements from a number of works of earlier men. Among these are Hippocrates of Chios (flourished c. 440 BCE), not to be confused with the physician Hippocrates of Cos (c. 460–375 BCE).[11]

Personal Life

Almost nothing is known of Euclid's life.[9] We do not know the years or places of his birth and death.[9] Euclid was apparently a kind, patient man, and did possess a sarcastic sense of humor.[18] Proclus supported his date for Euclid by writing "Ptolemy once asked Euclid if there was not a shorter road to geometry than through the Elements, and Euclid replied that there was no royal road to geometry."[11]

Structure and Content of the Elements

The Elements consists of thirteen books.[6] Euclid's Elements is divided into 13 "books", containing a total of 465 theorems (and 131 definitions).[26] The thirteen volumes of Euclid's "Elements" contains 465 formulas and proofs, described in a clear, logical style using only a compass and a straight edge.[4]

Foundational Framework

Euclid understood that building a logical and rigorous geometry (and mathematics) depends on the foundation—a foundation that Euclid began in Book I with 23 definitions (such as "A point is that which has no part" and "A line is a length without breadth"), five unproved assumptions that Euclid called postulates (now known as axioms), and five further unproved assumptions that he called common notions.[1]

#### The Five Postulates

The five postulates are: 1. To draw a straight line from any point to any point.[19] 2. To produce a finite straight line continuously in a straight line.[19] 3. To describe a circle with any center and radius.[19] 4. That all right angles equal one another.[19] 5. That, 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, meet on that side on which are the angles less than the two right angles.

#### The Five Common Notions

The five common notions are: 1. Things which equal the same thing also equal one another.[19] 2. If equals are added to equals, then the wholes are equal.[19] 3. If equals are subtracted from equals, then the remainders are equal.[19] 4. Things which coincide with one another equal one another.[19] 5. The whole is greater than the part.[19]

Organization by Books

The first four books focus on plane geometry, dealing with points, lines, angles, triangles, and circles. Book I introduces the basic definitions, postulates (axioms), and common notions, and it concludes with one of the most famous geometric propositions: the Pythagorean Theorem.[22]

Book 1 outlines the fundamental propositions of plane geometry, including the three cases in which triangles are congruent, various theorems involving parallel lines, the theorem regarding the sum of the angles in a triangle, and the Pythagorean theorem.[6] The subject of Book II has been called geometric algebra, because it states algebraic identities as theorems about equivalent geometric figures.[1]

The Elements consists of 13 different books, including book 3 which is about circles and their properties, and book 12 is about the relative volumes of cones.[3] Books V and VI explore the theory of proportion, particularly as applied to magnitudes, as well as similar figures. Book V provides the definition of proportionality and lays the groundwork for later mathematical concepts such as ratios and fractions.[22]

Book VII describes a method, antanaresis (now known as the Euclidean algorithm), for finding the greatest common divisor of two or more numbers; Book VIII examines numbers in continued proportions, now known as geometric sequences; and Book IX proves that there are an infinite number of primes.[1]

Book 10 attempts to classify incommensurable (i.e., irrational) magnitudes using the so-called "method of exhaustion", an ancient precursor to integration. Book 11 deals with the fundamental propositions of three-dimensional geometry. Book 12 calculates the relative volumes of cones, pyramids, cylinders, and spheres using the method of exhaustion. Finally, Book 13 investigates the five so-called Platonic solids.[10]

Mathematical Methodology and Innovation

The logical approach to mathematics is one of the factors that many people point to to explain the success of the Elements. Euclid used a systematic approach to the content within the Elements, where a set of axioms (a statement that is accepted to be true) are used to develop more detailed results and proofs.[3]

Euclid did not originate most of the ideas in The Elements. His contribution was fourfold: He collected important mathematical and geometric knowledge in one book. The Elements is a textbook rather than a reference book, so it does not cover everything that was known. He gave definitions, postulates, and axioms. He called axioms "common notions." He presented geometry as an axiomatic system: Every statement was either an axiom, a postulate, or was proven by clear logical steps from axioms and postulates. He gave some of his own original discoveries, such as the first known proof that there are infinitely many prime numbers.[9]

It set, for all time, the model for mathematical argument, following logical deductions from initial assumptions (which Euclid called "axioms" and "postulates") in order to establish proven theorems.[15]

Historical Impact and Transmission

Ancient Reception

Euclid's contemporaries considered his work final and authoritative; if more was to be said, it had to be as commentaries to the Elements.[1] In ancient times, commentaries were written by Heron of Alexandria (flourished 62 CE), Pappus of Alexandria (flourished c. 320 CE), Proclus, and Simplicius of Cilicia (flourished c. 530 CE).[1]

Islamic World

The immense impact of the Elements on Islamic mathematics is visible through the many translations into Arabic from the 9th century forward, three of which must be mentioned: two by al-Ḥajjāj ibn Yūsuf ibn Maṭar—first for the ʿAbbāsid caliph Hārūn al-Rashīd (ruled 786–809) and again for the caliph al-Maʾmūn (ruled 813–833)—and a third by Isḥāq ibn Ḥunayn (died 910), son of Ḥunayn ibn Isḥāq (808–873), which was revised by Thābit ibn Qurrah (c. 836–901) and again by Naṣīr al-Dīn al-Ṭūsī (1201–74).[1]

Medieval Europe

Euclid first became known in Europe through Latin translations of these versions. The first extant Latin translation of the Elements was made about 1120 by Adelard of Bath (flourished 12th century), who obtained a copy of an Arabic version in Spain, where he traveled while disguised as a Muslim student.[1]

Parts of the Elements also became an important component of the quadrivium, which was the classical curriculum taught in schools in Europe and the Middle East during the Middle Ages. The quadrivium included arithmetic, music, astronomy, and geometry. The geometry element consisted of the work in Euclid's Elements.[3]

Modern Era

The first English translation of the Elements was by Sir Henry Billingsley in 1570.[2] Nowhere was this more so than in the early modern period when the Elements was particularly visible, with nearly 200 printed editions appearing between 1500 and 1700.[8]

The impact of this activity on European mathematics cannot be exaggerated; the ideas and methods of Kepler, Pierre de Fermat (1601–65), René Descartes (1596–1650), and Isaac Newton (1642 [Old Style]–1727) were deeply rooted in, and inconceivable without, Euclid's Elements.[2]

Educational Legacy

This work has had a long-lasting influence in the field of geometry, with it being used widely up until the 20th Century, when other school textbooks became commonplace.[3] Even centuries after this, The Elements continued to play a role in education, with it forming the basis of almost all maths teaching until around 1900.[3]

The long lasting nature of The Elements must make Euclid the leading mathematics teacher of all time.[5] Euclidean geometry is still as valid today as it was 2,300 years ago, it is widely used in many disciplines, including art, architecture, science and engineering, to name but a few.[4]

Philosophical and Scientific Influence

Almost from the time of its writing, the Elements exerted a continuous and major influence on human affairs. It was the primary source of geometric reasoning, theorems, and methods at least until the advent of non-Euclidean geometry in the 19th century.[2]

Due to this spread of the Elements throughout the west, many other mathematicians and scientists studied and used The Elements for the basis of their work, including Sir Isaac Newton and Albert Einstein, who said it was "the miracle of a logical system".[3]

Euclid's Elements form one of the most beautiful and influential works of science in the history of humankind. Its beauty lies in its logical development of geometry and other branches of mathematics. It has influenced all branches of science but none so much as mathematics and the exact sciences.[7]