It was not only the philosophers in antiquity who laid down theories that would reappear in the Early Modern Period, as that which we now term physics began to be created, several Ancient Greek mathematicians made contributions that would go on to play a significant role in that creation; most notable are Eudoxus of Cnidus (c. 408–c. 355 BCE), Euclid (fl. [maybe?] 300 BCE), Archimedes (c. 287–c. 212 BCE), Apollonius of Perga (c. 240–c. 190 BCE), Hipparchus (c. 190–c. 120 BCE), and Ptolemaios (c. 100–c. 170 CE). In what follows I shall be taking a brief look at the work of Euclid and Eudoxus, taking Euclid first, although Eudoxus predates him and will deal with the others in later episodes.

To begin I will deal with the maybe above in the life dates of Euclid, which doesn’t refer to the dates but to the man himself. For a man, who supposedly wrote the most successful textbook of any sort in the history of the world, he is remarkably elusive. There are four books attributed to him–the *Elements*, *Optics*, *Data*, *Phaenomena*–and that is quite literally all that we know about the man:

In 1966, French mathematician Jean Itard, introducing a reissue of François Peyrard’s translation of the

Elements, asserted that the book was the work not of an individual but of a group, a school, for which ‘Euclid’ was a collective name. (As it happens, a group of French mathematicians had been publishing under the name ‘Bourbaki’ since the 1930s, providing Itard with the relevant model for his claim about Euclid.) In the twenty-first century it is still sometimes said that theElementsis an accretive text for which there is no need to name an author.^{[1]}

The *Elements* is a sort of geometrical encyclopaedia that incorporates and systemises the work of several others and as such, one can understand Itard’s thought process, but I shall simply assume that there really was a man, who lived around 300 BCE in ancient Greece, was named Euclid and was the author of the book we know of as the *Elements*.

In what follows I shall only be dealing with the *Elements*. I shall cover the *Optics* in a separate episode over Greek optics. To quote Wikipedia: “*Data* (Greek: Δεδομένα, *Dedomena*) is a work by Euclid. It deals with the nature and implications of “given” information in geometrical problems.” The subject matter is closely related to the first four books of Euclid’s *Elements*. As such it need not concern us here, which is also true of *Phaenomena*, which is a non-technical introduction to astronomy.

It is almost impossible to overemphasise the significance and importance of the *Elements* as a textbook in European science over a period of more than two thousand years, only giving way to other texts in the nineteenth century, much to the annoyance of Lewis Carrol, a mathematics lecturer at Oxford University, who wrote a superb play, *Euclid and His Modern* *Rivals ^{[2]}*, criticising this development.

Euclid’s wasn’t the first *Elements* (Στοιχεῖα, *Stoicheia*) to be written by an ancient Greek mathematician. The earliest known was written by Hippocrates of Chios (c. 470–410 BCE) of which only one fragment is known to exist embedded in the work of Simplicius of Cilicia (c. 480–c. 560 CE). Other *Elements* were supposedly written by Leon (fl. 370 BCE), Theudius (4^{th} C BCE) and Hermotimus of Colophon (4^{th} C BCE) all three mentioned by Proclus (412–485 CE). It can be assumed that just as Ptolemaios’ *Mathēmatikē Syntaxis*because it was so superior to all previous astronomy textbooks made them obsolete and they simply disappeared. Now known only through later hearsay or because Ptolemaios mentions them, so Euclid’s *Elements* made all previous geometry textbooks obsolete, and they too disappeared.

Because it was the only major mathematics textbook to survive antiquity. The opening books formed the geometry part of the quadrivium, the basic mathematical education on the medieval Latin schools and later the undergraduate curriculum on the medieval universities. This meant that, at least within science, if, over the centuries, scholars were doing mathematics they were mostly doing Euclidean geometry. This applies to John Philoponus in the sixth century, to the Oxford Calculatores and the Paris physicists in the fourteenth century, to Tartaglia, Benedetti, Copernicus, and Tyco Brahe in the sixteenth century, and even to Kepler and Galileo in the seventeenth century. The Euclidean element of Newton’s *Principia* is even greater, but I’ll deal with that in more detail later.

I would need a whole blog series to explain Euclid’s *Elements* in detail. It has thirteen books and Thomas Heath’s three volume, annotated, English translation runs to more than fourteen hundred pages. However, I will briefly cover some of the salient points that help to explain its longevity.

The first aspect of the Elements is its logical structure. In general, the mathematical propositions within it have proofs. This is new. If you look at the earlier mathematics of Mesopotamia, Egypt, or India there are no proofs. The propositions in these various early forms of mathematics are demonstrated and explained by worked examples. When you have a problem of this type, then you solve it in this way. To use a modern term the mathematics was algorithmic. This changes with Euclid, here mathematical propositions have proofs utilising deductive logic. Perhaps the most well-known example being his proof of the so-called Pythagorean Theorem, which Euclid does not attribute to Pythagoras.

More significant is the logical structure of the entire book. Book 1 opens with a set of basic self-evidently true concepts that require no proof on which the whole structure of the book is erected using step by step logical deduction, adding new definitions where necessary. The entire book has a coherent logical structure. If you accept the basics, then everything else follows logically and must therefore also be accepted. Caveat, critical examination in the nineteenth century, in particular by Moritz Pash (1843–1930) and David Hilbert (1862–1943), showed that many of Euclid’s definitions were inadequate and some of his logic suffered from lacuna but as the historian of mathematics W. W. Rouse Ball pointed out “the fact that for two thousand years [the *Elements*] was the usual text-book on the subject raises a strong presumption that it is not unsuitable for that purpose.”^{[3]}

Of interest, is the fact that the *Elements* is the most obvious work from antiquity that fulfils Aristotle’s fundament epistemological concept: Starting from self-evident premises that require no proof one uses a chain of deductive logic until one arrives at empirically observed facts. This is ironical as Aristotle was fundamentally anti-mathematics because did not consider mathematical object as related in anyway to the real world. Some Aristotle fans argue that he was not anti-mathematics exactly because his epistemology was borrowed from mathematics.

Returning to Newton, not only is his Principia written in Euclidean geometry and not as often falsely assumed in the then new analysis that he helped substantially to create but the whole structure of the work mirrors the same epistemology as used by Euclid in the *Elements*. This form of epistemology, generally known today as the axiomatic method, is now well-known and widely used in many branches of science and mathematics. Two well known examples in the basics of mathematics are the Peano Axioms for the natural numbers, and the Zermelo-Fraenkel Axioms for set theory.

A short overview of the contents of the thirteen books:

Book I starts with five postulates, including the infamous parallel postulate, and five common notions continuing to cover basics of plane geometry ending with his proof of proposition 47, the Pythagorean theorem and proposition 48, its corollary.

Book II introduces the so-called geometric algebra i.e., presenting and solving algebraic propositions geometrically. It covers a lot of propositions about squares and rectangles, including the general solution of the quadratic equation. Euclid in the reason we use geometrical names quadratic, cubic etc. to refer to algebraic terms.

Book III moves onto circles.

Book IV construction of incircles and circumcircles of triangles and the construction of regular polygons with 4, 5, 6, and 15 sides.

Book V changes gear and we learn about the theory of proportion of magnitudes.

Book VI extends the theory of proportions to plane geometry and the construction of similar figures.

Book VII Elementary number theory, including the Euclidean algorithm for finding the greatest common divider and the lowest common multiple.

Book VIII deals with geometric series

Book IX is the application of Books VII and VIII and includes Proposition 20, Euclid’s elegant proof of the infinity of primes.

Book X deal with incommensurable magnitudes i.e., lines that cannot be used to measure each other, a geometrical presentation of irrational numbers. For example, the diagonal of a unit square, which has length √2, cannot be measured by the side.

Book XI applies Book VI to solid figures.

Book XII determines the volumes of cones, pyramids and cylinders using the method of exhaustion, showing, for example, that the volume of a cone is a third of the volume of the corresponding cylinder, and the volume of a sphere is proportional to the cube of its radius.

Book XIII Constructs and discusses the properties of the five regular Platonic Solids.

Having sketched the Elements, we can now turn our attention to Eudoxus of Cnidus (c. 408–c. 355 BCE) and the reason why we deal with him after Euclid, although he lived earlier. As we saw earlier in the episodes about Plato and Aristotle Eudoxus was the creator of the homocentric spheres model for modelling the orbits of the planets as viewed from the Earth, including a mechanism to recreate retrograde motion. However, Eudoxus was not just an astronomer, he was also an excellent mathematician, who can be counted alongside Archimedes as one of the best Ancient Greek mathematicians. Starting with Book V, much of the latter part of the Elements rests on mathematics created by him.

The theory of proportions, a powerful tool in the geometrical presentation of the Elements, and its subsequent application in other books is due to Eudoxus. Eudoxus’ theory of proportions provides a rigorous definition for the real numbers; a definition that inspired Richard Dedekind in developing his theory of Dedkind cuts. Eudoxus introduced the concept of incommensurable lengths as a replacement for working with irrational numbers.

The mathematics most closely associated with Eudoxus is the method of exhaustion, an early form of integration. It is claimed that the method was first propagated by Antiphon who used inscribed polygons with increasing numbers of sides to approximate the area of a circle. As is so often the case we don’t actually known who exactly this Antiphon was. He is mostly identified with Antiphon of Rhamnus (480–411 BCE) an orator but the identification is highly questionable. Eudoxus took over the theory and made it rigorous. In his hands and later those of Archimedes it proved a powerful tool, in particular for determining areas and volumes. The method of exhaustion was used to prove the following proposition in Book XII of the Elements:

Proposition 2: The area of circles is proportional to the square of their diameters.

Proposition 5: The volumes of two tetrahedra of the same height are proportional to the areas of their triangular bases.

Proposition 10: The volume of a cone is a third of the volume of the corresponding cylinder which has the same base and height.

Proposition 11: The volume of a cone (or cylinder) of the same height is proportional to the area of the base.

Proposition 12: The volume of a cone (or cylinder) that is similar to another is proportional to the cube of the ratio of the diameters of the bases.

Proposition 18: The volume of a sphere is proportional to the cube of its diameter.

Today, the names Euclid and *The Elements* are for most people just something to do with Ancient Greek mathematics, but, as already emphasised above, no other book in the history of humanity has had such a powerful influence on the discipline of mathematics and also those disciplines that used mathematics such as physics.

^{[1]} Benjamin Wardhaugh, *The Book of Wonders*: *The Many Lives of Euclid’s Elements*, William Collins, 2020 ppb. p. 304 Wardhaugh’s book is an excellent guide to the more than 2000-year history of the *Elements*, you can read my review here.

^{[2]} Lewis Carroll, *Euclid and His Modern Rivals*, Dover Books, NY, 1973

^{[3]} W. W. Rouse Ball, A short Account of the History of Mathematics, MacMillan, 6^{th} ed., 1915, p. 55