By far the most developed science of ancient times was mathematics. We often hear of the external form of Euclidean mathematics, which supposedly derives everything from self-evident axioms and definition through incontestable demonstrations. All of this sounds like Greek mathematics would have been quite formal and removed from ordinary experience. Yet, when we look at e.g. Euclid's Elements, we quickly find that Greek mathematics was quite heavily linked to experience. Just think about some key proofs at the beginning of the first book: Euclid asks us to move one triangle on top of another triangle and thus to conclude that the two triangles coincide. This is a primary manner in which the similar magnitude of areas is seen – through quite concrete experiment. In fact, many of the more complex proofs dealing with areas work by cutting compared areas to pieces that we could then know to be similar by relying on some previous propositions – and ultimately on direct observation. This is in fact the manner in which Euclid proves the most famous theory of elementary geometry, the Pythagorean formula.

Although Euclid called his book Elements, an average modern reader would probably find only the first book filled with recognisable questions of elementary geometry: when are triangles similar, what is the sum of the angles in a triangle, how are the angles generated by parallel line related etc.? The second book already opens up avenues unfamiliar from geometry. The book apparently deals with rectangles, but the true topic of the book are certain equations of algebra. Ancient mathematicians had as yet no way to express complex calculations in any other way than through geometric means – squaring through actual squares, multiplication through rectangles. Thus, a simple task of squaring the sum of two magnitudes was represented as a square made out of a line combined of two smaller lines – and the result was then shown to consist of two smaller squares, with sides equal to the smaller lines, together with two rectangles formed by same two lines.

The need to express such equations explains also certain uses of circle by ancient mathematicians. Circle was, of course, important because it provided a way to copy a length of one line to another place. It was also a source for more intricate tools. At the very end of the third book of Elements, Euclid proves that no matter what sort of lines you draw, from a spot outside the circle, towards the circle and cutting it in two places, the rectangle formed by the first line between the spot and the first cutting place and the second line between the two cutting places is always of the same size – and if the line merely touches the circle, the square formed by the line between the spot and the touching place equals all these rectangles. This at first somewhat uninteresting result works in many proofs as a convenient trick e.g. to express some multiples as squares.

The fourth book of Elements continues the study of circles, and its topic is especially drawing figures of certain shape within and around circles. While at first this might seem rather unimportant pursuit, fit only for decorative purposes, it is actually of utmost importance at least in two cases. Firstly, it gives a convenient manner to divide the circumference of circle to equal parts – something which is especially useful for astronomical purposes. Secondly, it can be used to estimate the area of circle. The latter case deserves a more thorough explanation.

Elements is not precisely meant to provide for exact formulas for measurement. Instead, it gives exact means for measuring relations between different geometrical objects. In fact, the fifth book contains a theory of ratios between different magnitudes – that is, it tells us when a pair of number is related exactly in the same manner as another pair and when the pair has a larger or smaller ratio than the other pair. This is convenient in case of estimating magnitude of figures that one cannot express through simple algebraic means. For instance, in the sixth book Euclid can show that two circles have the exactly same ratio as squares on their diameters. Here Euclid is helped by what is called a method of exhaustion. By drawing polygons within and around the circles, he points out that their ratios provide limits between which the ratios of the circles must fall – and because every other ratio exceeds limits given by some of these polygons, the only alternative is the one already mentioned.

Elements is then a book on two-dimensional geometry, but it is also a book on arithmetics – or at least for certain arithmetical principles. Euclid doesn't go so far as to do exact calculations, but we do know from other sources that actual counting at antiquity was rather cumbersome. Instead, in the seventh, eighth and ninth book of Elements Euclid explicates such trivialities as ”odd times even is even”, but also goes through very intricate truths about number, for instance, that there is no highest prime number.

This theory of numbers provides also a ground for Euclid's theory of rational and irrational quantities, which is developed in the tenth book. The notion of ratios provides again a starting point: we just choose some magnitude as rational and then all magnitudes of same type that have the same ratio to the rational magnitude as a number has to another number are rational, while other magnitudes of same type are irrational. Since it is especially geometric magnitudes, or lines and areas, Euclid deals with, he makes the somewhat confusing choice that even all lines with rational squares should be called rational. Going through various relations between lines and areas, Euclid provided a crude classification of all irrationals, only to conclude that his classification fails to account for all irrationals.

The final three books of Elements extend the geometrical account to three-dimensional cases. What is most intriguing is perhaps the manner in which Euclid defines the various three-dimensional shapes through a method for their construction, for instance, when a circle is thought as moving around one of its diameters, a sphere is formed. The actual propositions of the last books are mostly rather uninspiring: we are shown in a similar manner to two-dimensional cases what are the relations of e.g. pyramids and rectangular solids with same bases and same height and we see that relations of cones and cylinders can then be determined through the method of exhaustion. The final book still closes with a true high note, when Euclid shows how to inscribe all the five Platonic shapes into balls and demonstrates that the sides of these shapes must be certain irrationals.

All the things in the two-dimensional part of Elements are accomplished through circles and straight line. In later mathematics, this was made a sort of virtue, but it seems clear that at least not all ancient mathematicians did not put as strict limits to mathematics. This is shown even by Euclid's three-dimensional mathematics and its method of constructing three-dimensional shapes, but even more clearly by the ancient treatises of various types of curves, most interesting of which are possibly the conic sections, which get their names from being generated by planes cutting the cones in different angles. The use of conic section is, once again, not arbitrary, but important e.g. for expressing relations between one- and two-dimensional magnitudes.

Another interesting point in ancient mathematics is Arhcimedes' account of how to calculate various curvilinear areas and volumes. On some level, not much has changed from Euclidean account of finding relations that hold e.g. between circles. In many cases Archimedes even uses the same method of exhaustion Euclid had applied. For instance, when trying to determine the segment of a parabola, Archimedes shows that we can limit the area of the segment through sums of triangles, and by making the triangles smaller and smaller the limits become closer and closer to a certain area. But Archimedes had also another type of proof for such theories – he showed that when put to a lever the segment could be balanced with a triangle of a certain area, which was then equal to the segment. In other words, Archimedes used mechanical principles to determine an answer to a mathematical problem. As he himself testifies, this mostly just helped him to discover the areas, but he didn't accept it as a proper proof, which could be determined only through mathematical means.

Although Archimedes did use then mechanics as a tool in mathematics, more familiar is his application of mathematics in mechanics. There was considerable interest on mechanical questions even before Archimedes, as shown by a work on mechanical problems, attributed to Aristotle, but in a sense, in Archimedes the ancient tradition of mechanics culminated. Especially important in this context is the notion of a centre of gravity, that is, a place to which we can think that the whole weight of a body is concentrated. Through this notion Archimedes can prove a number of important mechanical principles, such as the lever law – the longer the stick the more weight you can lift. Note how close to practice the Archimedean practice of mechanics is – it is all about making machines, like levers and pulleys.

Even more important field of application for mathematics was the study of stars and their comings and goings. Here especially the study of circles and angles was of primary importance. Earth and the universe were pictured as a sphere within a sphere. By carefully noting the distances between places, measuring differences of shadows and the places of stars in the sky, one could get a reasonable model of the place of the Earth in relation to the universe. Then by following the changes in the positions of the stars, and especially of the strangest of these, that is, planets, and by assuming that the movements of all heavenly objects must be somehow based on circles, Ptolemy constructed a reasonably accurate model of planetary movement.

Ancient thinkers were not satisfied with mere modelling of planetary movement, but they also tried to explain this phenomenon. The basic notions were simplistically mechanical or then compared the movement of stars to movements of animal things. Whatever the case, the stars themselves and their movement was supposed to have an effect on Earthly things through their pressure on Earth's atmosphere. Such influences were also felt to be a justification for attempts to predict key events in human life.

In addition to these astrological considerations, study of stars was quite useful in making maps – one can use the positions of stars in the night sky for approximating one's position on the globe. Here ancient mathematical learning came in contact with ancient empirical knowledge. One cannot say that Greeks and Romans would not have appreciated empirical learning. Quite the contrary, as Strabo's geographical writings and Pliny's collection of all sorts of interesting factoids show.

What was lacking was such technological advances like telescope and microscope, which have allowed us to see farther and in more detail. Also, the lack of trustworthy and fast communications often made it impossible to verify the facts collected, which makes these empirical compilations into an often tantalising mix of truths and fairy tales. One can see that there was e.g. great technological expertise on various special subjects, such as metallurgy, but this expertise was dispersed to many different individuals – while one person could know everything of, say, gold and its behaviour, he might have no idea about forging iron.

While ancient culture did have empirically discovered information on many questions of nature, the role of theoretical foundation of such information was rather ambiguous, reflecting the conflicts between various philosophical schools. The complex interactions between theory and empiria and many viewpoints on these interactions can be well exemplified through the difference of ancient schools of medicine. We have methodists, who were affiliated with the Pyrrhonic school and who accepted only some rules of thumb that had appeared to work in previous occasions. We have empiricists, who accepted immediate appearances and experiences as a tool for suggesting treatment, but who denied the possibility of any knowledge on the hidden causes of various diseases. And there were various rationalist schools, who attempted to justify their procedures through some theoretical framework, although what was a correct theory of human body was a point hotly debated.

Probably the most famous of these researchers of medicine was Galen, whose position was somewhere between empirical and rational schools. Galen despised methodists, whom he thought as mere bunglers and frauds. He was quite sympathetic of empiricists and admitted that in many cases empirical observations were enough for discovering a good treatment. He endorsed the rationalist ideal of a science of medicine based on certain foundations, but only if the foundations were correct. His preferred theory of human body was quite Aristotelian: the constituents of human body were characterised by four qualities (hotness, dryness, coldness and wetness), and if some quality took a too forceful position in some part of the body, causing imbalance and disharmony, a proper treatment was to apply opposed quality.

While Galenian medicine thus offered a comprehensive account of the biological make-up of human body, rhetoric was at least advertised as a comprehensive account of the cultural side of human beings. On the superficial level, the topic of the rhetoric is much more restricted – the production of speeches, and especially of speeches meant to be used in a court setting. Thus, the central aim of ancient rhetoric was to show how to invent good arguments for one's position, how to arrange these arguments into a coherent whole, how to express this whole in a good style, how to memorise the written speech and how to present it in a compelling manner.

Yet, writers like Cicero and Quintilian were eager to suggest that rhetoric was something more, namely the lost half of philosophy. That is, they pictured a time, when philosophical study of truths and especially of truths concerning good human life and rhetorical expertise of expressing those truths were an undivided unity. A conclusion they both draw was that a good rhetorician should also be a good person and know all the intricacies of human life.

An important part in rhetoric, Quintilian suggested was then the education of a future rhetoric. As it should be evident from his wide definition of rhetoric, Quntilian plans a careful curriculum for his imaginary students who wish to learn ,not just the tricks of the trade, but also the rudiments for becoming an ideal speaker. It appears that an ideal speaker should learn as much a possible from every topic – even if her skills are to be used in the courthouse, it is a definite possibility that some case would require expertise knowledge of some special topic. In addition to specialised knowledge, the rhetorician-to-be should also learn history, since past events often play part in modern cases. A special place in the curriculum is given to literature, which at the same time teaches the students the basics of good language, but also suggests various examples of human behaviour.

Quintilian's school of rhetoric incorporates then two important pieces of humanist learning: history and literature, both of which were important fields of study in antiquity. Study of history in ancient Greece and Rome was in a sense something quite different from academic history of our time. The ancient historians had progressed not much beyond mere chronicling of events. Often their standards of criticism were quite suspect. As a case in point, we might raise the historian of philosophy, Diogenes Laërtius, whose work is at worst a rambling collection of anecdotes from the lives of philosophers, without a proper explanation of their philosophies. The main historical guide line of Diogenes is the notion of one philosopher being a student of another, which allows him to create two lines of thinkers, but fails in explaining how the ideas of earlier philosophers led to ideas of future philosophers.

As for study of different types of literature, a scholarly criticism was quite developed. As an example, we might pick out the investigation of works of Aristotle, ancient commentaries on which could have filled libraries. An important example of this tradition is Alexander of Aphrodisias, who apparently set out to explain all of Aristotle's extant works, although only a handful of his commentaries have been passed down to us. And explain he does. Following through Aristotle's account of different types of deductions, Alexander goes carefully through every reasoning and every example Aristotle uses, expanding Aristotle's sometimes rather summarised sentences.

In some cases, Alexander picks on details that Aristotle probably did not mean as important. Why did Aristotle choose to present the three different figures of syllogisms in a certain order, Alexander asks. Pretty clearly the first figure is most important, because it is the only one allowing us to prove universal and affirmative propositions, required in scientific reasoning. Alexander takes this line of thought further and suggests that second figure, leading only to negative conclusions, is second best, because it can be used in philosophical debates, meant for refuting the ideas of opponents – and the third figure is then worst, leading only to particular conclusions and thus useful only in sophistical reasoning, in which propositions applying merely for particular cases are deceptively presented as holding in general.

At times, Alexander defends Aristotle against later critics. For instance, some Stoics had claimed that Aristotle went wrong in assuming that one cannot deduce impossibilities out of possibilities, because a possible statement ”Dion is dead” implied a statement ”he is dead” (”he” referring to Dion), but since the Greek term for ”he” could be used only of living persons, the implied statement was always an impossibility. Alexander noted that, if Stoics were correct in their grammar, then the first statement simply did not imply the latter. Furthermore, he pointed out that since Stoics believed in eternal recurrence of all things, death was always only a temporary state and one could refer to a dead person with a pronoun.

Alexander also opposed Stoics on their understanding of negation. Stoics had suggested that ”Socrates is not white” is not the negation of ”Socrates is white”, because both of these presuppose that Socrates exist, while ”It is not so that Socrates is white” does not, making it the sought-out negation. Alexander pointed out that a proper name did not necessarily refer to an existing entity, like in a sentence ”My future son will be called Eric”, which made Stoic assumption doubtful.

At other times, Alexander tried to show that certain innovations of later logicians were already accounted for by Aristotle. Thus, he attempts to show that a deduction consisting of nothing else but conditionals (”if A then B, if B then C, thus, if A then C”) must still follow syllogistic rules, because the premisses must have been ultimately deduced syllogistically.

Sometimes Alexander went clearly too far in his attempt to speak for Aristotle. A good example is latter's attempt to add necessity and possibility in his logic. Aristotle's efforts become quickly a hopeless muddle, when he distinguished many different meanings of possibility or contingency and then himself confused these various meanings from time to time. Alexander's valiant effort to make sense out of it manages just to muddle the issue even further.

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