Euclid

(325 BC- 265 BC)

Euclid of Alexandria is the most prominent mathematician of antiquity best known for his treatise on
mathematics *The Elements*. The long lasting nature of The Elements must make Euclid the leading
mathematics teacher of all time. 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.

Little is known of Euclid's life except that he taught at Alexandria in Egypt.

According to Proclus (410-485 A.D.) in his Commentary on the
*First Book of Euclid's Elements*, Euclid came after the first pupils of Plato and lived during the
reign of Ptolemy I (306-283 B.C.). Pappus of Alexandria (fl. c. 320 A.D.) in his Collection
states that Apollonius of Perga (262-190 B.C.) studied for a long while in that city under the
pupils of Euclid. Thus it is generally accepted that Euclid flourished at Alexandria in around
300 B.C. and established a mathematical school there. Proclus also says that Euclid
"belonged to the persuasion of Plato,'' but there exists some doubt as to whether Euclid
could truly be called a Platonist. During the middle ages, Euclid was often identified as
Euclid of Megara, due to a confusion with the Socratic philosopher of around 400 B.C.

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. 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.

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.

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 also wrote many other works including *Data*, *On Division of Figures*, *Phaenomena*, *Optics and the lost books Conics and Porisms*.

All survive in the original Greek except Divisions, which is partially preserved in Arabic. These works all follow the basic logical structure of the Elements, having definitions and rigorously proved propositions.

The *Data* is closely related to the first four books of the Elements. It opens with definitions of
the different senses in which things are said to be ``given.'' Thus lines, angles, and ratios may
be given in magnitude, rectilinear figures may be given in species or given in form, points and
lines may be given in position, and so on. These definitions are followed by 94 propositions
which state that when certain aspects of a figure are given, other aspects are given.

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 the] triangle a given ratio.'' Pappus lists this work among those in the Treasury of Analysis; in fact, the propositions in the Data may be considered elementary exercises in analysis which supplement the theorems and problems found in the Elements. The Data is also considered important in the development of algebra. The so-called geometrical algebra of the Greeks is addressed in the discussion of Book II of the Elements.

*On Divisions of Figures* survives only in Arabic translation, although not a direct one. In its
present form, it consists of 36 propositions concerning the division of various figures into two
or more equal parts or parts in given ratios. These divisions may be into like figures -- a
triangle into two triangles, for example - or unlike figures - a triangle into a triangle and a
quadrilateral, say. The figures so divided include triangles, parallelograms, trapezia,
quadrilaterals, circles, and figures bounded by an arc of a circle and two straight lines which
form a given angle. The proofs of only four propositions have survived.

Two of these are the 19th: ``To divide a given triangle into two equal parts by a line which passes through a point situated in the interior of the triangle,'' and the 29th: ``To draw in a given circle two parallel lines cutting off a certain fraction from the circle.'' This work is similar to the Divisions of Figures by Heron of Alexandria, writing in perhaps the third century A.D., except that Heron supplements his discussion with numerical calculations.

Euclid's *Phaenomena* is a tract on sphaeric, the study of spherical geometry for the purpose
of explaining planetary motions. It survives in Greek and is quite similar to On the Moving
Sphere, by Autolycus of Pitane, who flourished around 310 B.C. However, the propositions of
Autolycus are more abstract than those of Euclid, who uses the convenient astronomical terms
horizon and circle of the zodiac in his presentation. The former work also exhibits the
``classical'' Greek form - discussed in section III below - found in all of Euclid's treatises,
demonstrating that this style of presentation was not original with Euclid, but was established
before his time. It is in the Phaenomena that Euclid first makes the observation that an
ellipse may be obtained from cutting a cylinder.

Euclid's *Optics* is the earliest surviving Greek treatise on perspective. In its definitions Euclid
follows the Platonic tradition that vision is caused by discrete rays which emanate from the
eye. One important definition is the fourth: "Things seen under a greater angle appear
greater, and those under a lesser angle less, while those under equal angles appear equal.''
In the 36 propositions which follow, Euclid relates the apparent size of an object to its
distance from the eye and investigates the apparent shapes of cylinders and cones when
viewed from different angles. Proposition 45 is interesting, proving that for any two unequal
magnitudes, there is a point from which the two appear equal. Pappus believed such results to
be important in astronomy and included Euclid's Optics, along with the previous work,
Phaenomena, in the Little Astronomy, a compendium of smaller works to be studied before the
Syntaxis (Almagest) of Claudius Ptolemy.

Of the lost treatises attributed to Euclid, four were unquestionably his works: *Conics*, *Porisms*,
*Pseudaria*, and *Surface Loci*. Euclid's *Conics* predated by a half-century the famous work by
Apollonius on the same subject. However, Euclid's treatment was most likely a compilation of
previously known information - much like the Elements - and was probably not very original. In
fact, according to Pappus, Euclid gave credit to Aristaeus, a contemporary, for his discoveries
in the conics.

Pappus further states that "Apollonius, having completed Euclid's four books of conics and added four others, handed down eight volumes of conics.'' The contents of Euclid's treatise is therefore regarded to have been quite similar to the first three or four books of Apollonius's work. The Conics of Apollonius quickly supplanted the former work, and by the time of Pappus, Euclid's work was already lost, while that of Aristaeus was still extant.

Both Pappus and Proclus attribute to Euclid a three-book work called Porisms, which contained 171 theorems and 38 lemmas. A porism may be a corollary, something which follows easily from a proved proposition, or it may mean a type of proposition intermediate between a theorem - a statement of the properties of a given thing - and a problem - the actual construction or bringing into existence of something.

Proclus gives as an example of the latter meaning finding the center of a circle (Proposition I, Book III of the Elements); the center already exists, but it must be found. A third meaning is given by Pappus: ``A porism is that which falls short of a locus-theorem in respect of its hypothesis.'' A locus (plural, loci) is a set of points all of which obey a certain property.

Theorems, problems, and porisms will be discussed further in the third section of this paper. There have been several attempts to reconstruct the Porisms, but controversy still rages over the mere meaning of the title, making discussion of content difficult. It is generally agreed, however, that the work was in the realm of higher mathematics.

Pappus thought it important enough to be included in the Treasury of Analysis. It has been suggested that the entire work was a by-product of Euclid's investigations into conic sections, making the propositions contained therein porisms in the first sense given above.

Proclus describes a work of Euclid's called Pseudaria, or Book of Fallacies, in which he showed beginners how to avoid errors in reasoning by ``setting the true beside the false and adapting his refutations of error to the seductions we may encounter.'' It is clear from Proclus's description that the work remained in the realm of elementary geometry, but nothing else is known about it.

The last of Euclid's works to be included in the*Treasury of Analysis* was *Surface Loci*. It is not
known whether this title referred to loci on surfaces or to loci which were themselves
surfaces. Pappus offers two lemmas to this work - thus they are not part of Euclid's original
text - one of which is the focus-directrix property of conic sections. The other appears to
give loci which are cones or cylinders; but this interpretation is based on a slight rewording of
the lemma to make it mathematically sound. In any event, the evidence would appear to favor
the interpretation that the loci were themselves surfaces. It is conjectured that some of
the loci may have been more complex quadric surfaces (paraboloids, hyperboloids, prolate
spheroids); however, none of Euclid's original two-book text survives to confirm such a
hypothesis. Euclid