Euclid – Biography, Geometry Treatises, Accomplishments, & Elements

Euclid, also known as the Father of Geometry, was considered one of the greatest mathematicians of Greek times. He etched his name into the annals of history by authoring a mathematical textbook titled the Elements. The book, which contains a host of groundbreaking mathematical theorems and proofs in geometric algebra and among others, is undoubtedly the most famous math textbook of all time.

But who really was Euclid, and what were his major contributions to mathematics?

World History Edu delves straight into the early life, career, major treatise, and accomplishments by Euclid, the author of the Elements

Quick facts about Euclid

Born – late 4^th century BC (around c. 320 BC)

Place of birth – Alexandria, Egypt

Died – mid-3rd century BC (around 265 BC)

Math subjects – Geometry, Algebra

Other areas – Conic sections, number theory, mathematical rigour, spherical geometry, and perspective

Other names – The Founder of Geometry, The Author of Elements

Most known for – being the founder of geometry

Famous for – Elements, one of the most famous mathematical textbooks of all time

Influenced – Johannes Kepler, Pierre de Fermat, René Descartes, Isaac Newton , Nicolaus Copernicus, Galileo Galilei, and Albert Einstein, among others

Biography

Greek mathematician Euclid was born in the latter part of the 4^th century BC (most likely around 320 BC) in the City of Alexandria in Egypt. The Ptolemaic Era (i.e. Greco-Roman era) had just began, considering the fact that Alexander the Great’s most trusted companion general Ptolemy I Soter was ruler of Egypt (r. c.305-285 BC).

Euclid’s childhood and education benefited tremendously from the growth of Alexandria as the intellectual hub of the world at the time.

Aside from the above few facts, not very much is known about Euclid’s upbringing and personal life. The little that we know of this genius mathematician comes from Proclus Lycius (commonly called Proclus), a Greek philosopher who lived about eight centuries after Euclid’s death. Proclus talks about Euclid and his works in A Commentary on the First Book of Euclid’s “Elements”.

According to Greek philosopher and author Proclus, Euclid took up a teaching job in Alexandria following his studies.

It is been said that he arrived in Alexandria about a decade after the city was founded by Alexander the Great. That means that he arrived around 322 BC. His time in Alexandria most likely saw him visit places like the temple Serapeum of Alexandria and the Library of Alexandria.

There were some Arabian authors in the medieval era that claimed that Euclid’s place of birth was in the town of Tyre, a city in Lebanon. It must be noted that there exists no evidence to this day to support such claim.

The Ptolemaic era mathematician was influenced by mathematicians from the era of Plato, including Leodamas of Thasos, Theaetetus, and Archytas of Taras. Euclid also studied the works of Philip of Opus, one of the pupils of Plato. Other mathematicians (primarily Pythagorean mathematicians) and philosophers that influenced Euclid are Neoclides and Eudoxus of Cnidus.

It is commonly accepted that Euclid died around 270 BC. His place of death was most likely in Alexandria, Egypt.

Greek mathematicians and scholars that Euclid influenced

It’s been stated that Euclid lived during the reign of Ptolemy I. This is supported by the fact that Greek mathematician and inventor Archimedes makes mention of Euclid. Supporters of this notion also state that Euclid wrote his works before Archimedes wrote his.

According to Greek mathematician Pappus of Alexandria, the Greek geometer and astronomer Apollonius of Perga (c. 240 BC – c. 190 BC) interacted with a number of pupils of Euclid in Alexandria.

Ptolemy I Soter’s quest for a “royal road” to geometry

Such was the prominence of Euclid the Geometer that Ptolemy I Soter sought the mathematician once, inquiring for shorter route to geometry than through the Elements. At which point, Euclid responded that geometry has no royal road.

Euclid and Archimedes

Archimedes, the famed mathematician and inventor

Was Euclid older than Archimedes of Syracuse (287-212 BC)? The verdict is still out on that one. Considering the fact that a number of sources claim that Archimedes studied under/along with a host of successors of Euclid, it is probably likely that Archimedes was born after Euclid.

Meaning of Euclid

The name “Euclid” is actually the Anglicized version of the Greek name Εὐκλείδης. That name means “glorious” or “celebrated”.

What is Euclidean geometry?

Euclidean geometry encompasses the various theorems that Euclid and other geometers of his era deduced from a number of axioms. Many of those axioms are found in Euclid’s groundbreaking geometry textbook the elements. Euclidean geometry uses deductive reasoning to attain propositions (theorems) from the axioms. The methods used in Euclidean geometry played a crucial role in advancing logic, a host of areas in mathematics, and many other disciplines in modern science.

In the two millennia that followed the death of Euclid, the geometry that he and his colleagues conceived – i.e. Euclidean geometry – was the standard in the study of geometry and mathematics. Geometry was simply synonymous to Euclidean; therefore the adjective “Euclidean” was a sort of tautology in that period.

From the 19^th century onward, mathematicians and scientists alike have conceived of a number of non-Euclidean geometries, including analytic geometry (also known as coordinate geometry), hyperbolic geometry and elliptic geometry.

The Elements by Euclid of Alexandria

Prior to Euclid’s very comprehensive math textbook the Elements, students of math and science usually used the compilation by Greek mathematician Theudius. Therefore, Theudius’ works, along with works by the likes of Hippocrates of Chios, had tremendous influence on Euclid.

It is a common misconception that Euclid’s the Elements covers only geometry. That will be true for the first four books – Books I through to Book IV. However, reading beyond Book IV will expose the reader to mathematical concepts outside of geometry. Euclid paid close attention to establishing a very strong foundation of sound geometry so as to help the reader gain better understanding of the concepts in the subsequent books.

How many books are there in Euclid’s Elements?

Euclid’s the Elements comprises 13 books (Book I to Book XIII) written during the Ptolemaic Era, most likely around 300 BC. The books capture many concepts, propositions, constructions and mathematical proofs in not just Euclidean geometry, but also basic number theory and incommensurable lines.

In Book I of the Elements, Euclid exposes the reader to basic concepts and definitions including definitions of a point and a line. This is followed up by simple shapes like triangles, rectangles and parallelograms. The book also contains a number of axioms (postulates) and notions. It concludes with the famous Pythagorean theorem conceived by Pythagoras of Samos, an ancient Ionian Greek philosopher and mathematician.

It’s long been held that Euclid’s Elements bears semblance to a compilation of major works by mathematicians that came before Euclid. However, the verdict is still out on which part of the book is his own idea or adaptations from other mathematicians. In no way do those assertions take anything away from brilliance of those books; they were truly masterpieces.

Book II of Euclid’s Elements

In Book II of Euclid’s the Elements, the mathematician comes out with a number of algebraic identities. Those theorems cover geometric figures as well as the equality of rectangles and squares. Book II, a geometric algebraic book, covers lines and how when a line is divided into two parts can produce ratio of a larger segment to the smaller segment equal to the ratio of the original line to the larger segment. The works of Pythagoras (c. 571-495 BC) inspired quite a lot of propositions in Book I and Book II.

Book II also talks about the Pythagorean Theorem and its applications in arbitrary triangles.

Books III-V of Euclid’s Elements

Book III of the Elements dives into the properties of circles. It draws heavily on the works of Hippocrates of Chios (c. 470-410 BC) in discussing tangents and inscribed angles.

Book IV sheds light on regular polygons, particularly pentagon, and their construction.

Book V moves away from plane geometry and introduces theory of ratios and proportions. Euclid picked a lot from Eudoxus of Cnidus (c. 395 – c. 342 BC). According to Proclus, Book V of the Elements can be read independently of the rest of the first four books. Book V, which incorporates a lot of irrational numbers, basically serves as a foundation for a geometric theory of numbers.

Books VI – XIII

In Book VI of the Elements, Euclid compiles extant literature on the theory of ratios to plane geometry, triangles and parallelograms. The book also contains quadratic problems using geometric means.

Books VII to IX develops theorems under number theory. Book VII begins with definitions of terms like unity, even, odd, and prime numbers. Euclid also highlights the properties of the positive integers. In Book VII, a method known as antanresis (also known as Euclidean algorithm) can be found.

For Books X and XIII, Euclid borrows a lot of work from Theaetetus (c. 417-369 BC), a Pythagorean mathematician.

Book X, which is famed for being an inspiration to the scientist Johannes Kepler, is said to form about 25% of the Elements. The book’s theorems on incommensurable lines and areas influenced Kepler’s work on cosmological model.

Book XI talks about the intersections of planes, lines and parallelepipeds; while Book XII uses Eudoxus’s method of exhaustion to secure proofs in areas of circles and the squares of their diameters, and the volumes of spheres and the cubes of their diameters

The final book, Book XIII, captures five regular Platonic solids, which are pyramid, cube, octahedron, dodechedron, and icosahedron.

More facts about the Elements

Some notable mathematicians that provided commentary on the Elements include Heron of Alexandria, Pappus of Alexandria, Proclus, and Simplicius of Cilica.

Theon of Alexandria, the father of Hypatia, edited the Elements. His edition is considered the most widely accepted edition, which in turn was used for subsequent translations into Latin and Arabic up until 1808.

Francois Peyrard, a French mathematician, discovered an earlier edition of the Elements in the Vatican. That edition was older than Theon’s edition.

In 1505, Bartolomeo Zamberti made the first direct translation of the Elements from Greek to Latin without an Arabic intermediary. The edition was published in Vienna.

Sir Henry Billingsley is credited with making the first English translation of Euclid’s the Elements. Billingsley, who was Lord Mayor of London and a renowned merchant, made the translation in 1570.

Euclid’s influence on Islamic scholars and mathematicians

The Arab world is said to have gotten hold of Euclid’s works, particularly the Elements, around the mid-8^th century CE. They received the textbook from the Byzantines.

Under the leadership of Harun al Rashid (also known as Harun the Just) – fifth Abbasid Caliph – Euclid’s the Elements was translated into Arabic. From then onward, the book Arabic scholarship by storm as more and more translations were made.

Euclid’s the Elements in Europe

For a some period, the Elements was lost to European scholars and mathematicians. Europeans rediscovered Euclid through Latin translations of those Arabic versions in the 12^th century CE. The first Latin translation was made around 1120 CE. The translation was made by the English natural philosopher Adelard of Bath. Adelard is said to have gotten the Arabic version in Spain.

The first printed edition of the Elements was made in 1482. The edition was based on Campanus of Novara’s 1260 edition. Since then the textbook has had more than thousand different editions.

Abraham Lincoln, the 16th president of the United States, is said to have kept a copy of the Euclid’s the Elements in his bag. Growing up in Springfield, Illinois, Lincoln studied the textbook late at night. He later praised the book for making him a better lawyer.

Impact of the Elements

Euclid’s the Elements has had a tremendous impact on the society since it was released over two millennia ago. It has been the primary source material for geometric reasoning, theorems, and methods until the coming of non-Euclidean geometry in the 19^th century.

Elements is probably the world’s oldest known textbook that uses deductive reasoning to tackle mathematics. The books played a crucial role in advancing logic and many disciplines in modern science.

These and many other reasons are why the book remained a standard mathematics book in the West and beyond up until the late 19^th century and even the early 20^th century. Such was the success of Elements that for over two millennia that many consider it the most influential academic textbook ever. It is generally considered the second-most translated, published, and studied book of all time in the West, coming in behind the Bible.

Elements is also second only to the Bible in terms of the number of editions published since the advent of the printing press in the mid-15^th century. It’s been noted that every educated person before the 20^th century most likely read the Elements.

In addition to influencing the likes of Sir Isaac Newton, Nicolaus Copernicus, Galileo, and Albert Einstein, Euclid’s the Elements influenced English philosophers like Thomas Hobbes, Alfred North Whitehead, and Bertrand Russell; and Dutch philosopher Baruch Spinoza.

Famous physicist Albert Einstein credits Euclid’s the Elements for having a tremendous impact on him as a child. He is said to have called the geometry textbook the “holy little geometry book”.

Euclid and Eukleides of Megara

During the medieval era, there was some bit of confusion by scholars and translators. Eclid was confused with Greek philosopher Eukleides of Megara. Eukleides the philosopher (also known as Megarensis) undoubtedly lived about a century before the birth of Euclid. That means that Eukleides was a contemporary of famous Greek philosopher Plato.

More Euclid Facts

Euclid is sometimes called Euclid of Alexandria in order to distinguish him from Euclid of Megara, the Socratic philosopher.

The European Space Agency (ESA) named one of its spacecrafts the Euclid Spacecraft.

Historical accuracy of stories about Euclid

Owing to the scanty nature of the biographical information about Euclid, there have been some unsubstantiated stories that Euclid was not a historical figure; proponents of the theory claim that the works attributed to Euclid were written by mathematicians who used the pseudonym Euclid from Euclid of Megara, a Socratic philosopher.