This chapter is addressed to those who wish to read the original sources of Greco-Roman mathematics, either in the original languages or in modern translations. Hence, it focuses on the kinds of mathematics that was disseminated in treatises written by scholars who were members of a relatively small literary elite. This theoretical style of mathematics was not the only kind of mathematics practised in Greco-Roman antiquity, and, indeed, the total number authors of philosophical mathematics must have been dwarfed by the number of individuals who used practical mathematics in their daily work, and who passed on such mathematical skills to their sons, disciples, and apprentices. Nevertheless, the literary works produced by this self-selected group of individuals have elicited the admiration and study of mathematical scholars through the centuries, and have justly been regarded as one of the most important products of ancient scholarship.

There is some uncertainty among historians of science as to when the history of science first appeared. Unlike the old historical debate over the origin of mathematics or astronomy, the origin of the history of science has never been widely discussed or properly considered, and the interested reader will find a variety of starting points which reflect the professional preoccupations of historians. One is in the twentieth century with George Sarton, another in the nineteenth century with William Whewell, and yet another in the eighteenth century with Joseph Priestley. Thus, Helge Kragh regards Priestley’s The History and the Present State of Electricity (1767) as the best example of the history of science in the age of the Enlightenment, which ‘saw history as an instrument for progress in the battle against the old feudal order.

Ask yourself: what is a plant? You will probably answer that it is an organism able to photosynthesise chlorophyll. Depending on your level of knowledge in biology, your answer will be more or less elaborate. Now, ask a young child what a plant is, and their answer is likely to be very different. Their definition may centre on the notion of plant rootedness: a plant is something that is rooted to the ground and cannot move as a result.

Where does musical beauty come from? Can it be comprehended into rules or formulae? Since music was ubiquitous in the life of the ancient Greeks, it is no surprise that they posed these questions, thus triggering one of the most fascinating debates in their intellectual history.

Astronomy has often been called the oldest science, and some ancient Greek authors acknowledged a debt to Babylonian predecessors. Many of the earliest Greek philosophers are credited with astronomical observations, predictions, explanations, and discoveries. Yet, throughout much of Greco-Roman antiquity – and later – astronomy was regarded as a branch of mathematics, along with arithmetic, geometry, and harmonics. This view of astronomy – as a branch of mathematics – raises questions about what constituted astronomy, as well as its relationship to philosophy.

Greco-Roman meteorology will be described in four overlapping developments. In the archaic period, astro-meteorological calendars were written down, and one appears in Hesiod’s Works and Days; such calendars or almanacs originated thousands of years earlier in Mesopotamia. In the second development, also in the archaic period, the pioneers of prose writing began writing speculative naturalistic explanations of meteorological phenomena: Anaximander, followed by Heraclitus, Anaxagoras, and others.

The phenomena of life have special significance for us. Living things impress us in ways inanimate things couldn’t. This is because livings things do things. They act for the sake of some purpose, a purpose which moreover seems to be their very own. They instil in us the impression that there is something they are ‘up to’. This certainly seems to be the case with animals and, to a lesser degree, with plants and other growing things. Their goal-directed behaviours are presumably the reasons why living things are closer, more interesting – and sometimes also more repelling – to us than inanimate things: they have, one might say, certain interests they pursue. We can understand these interests and therefore interact with these pursuits in sometimes cooperative and sometimes inimical ways. This is why our attitudes towards animate things are so very different from our attitudes towards the inanimate.

One of Aristotle’s major contributions to natural science was his development of an idea that he called hulê’, which we translate into English as ‘matter’. The notion of matter seems quite ordinary to us today, but Aristotle’s idea was new at the time. The word hulê’ originally meant forest, brushwood, or cut wood and, aside from a single occurrence in Plato’s Philebus (54c), Aristotle is the first (extant) author to use the term in a general account of things in the natural world. Our notion of matter is a descendant of his, but we should be careful not to assume that he thought of it the way we do today.

The opposition between reason and experience is one of the legacies that we owe to ancient Greek philosophy and science, from which, via Latin, our terminology for it comes. The earliest plain statement of the distinction, if not in precisely these terms, is found in the Gorgias, which though set during Socrates’ lifetime in the fifth century BCE, was composed by Plato in the following century (501a; cf. 465a; Phdr. 270b)

Natural teleology, associated above all with Aristotle, and dealt with in a separate chapter of this volume, seeks to establish and elucidate the explanatory role of purposive structures and processes in the natural world, especially biological. Acorns exist for the sake of producing oak trees, eyelids for protecting the eyes. Natural teleology may or may not go on from there to seek the explanatory role of larger cosmic features, such as the shape and position of the earth. Much less does it need to ask – although it is not debarred from asking – the even bigger question, of how those structures and purposes came to be present in the first place. Thus it comes with no unavoidable theological implications.

A teleological explanation is an explanation in terms of an end or a purpose. So saying that ‘X came about for the sake of Y’ is a teleological account of X. It is a striking feature of ancient Greek philosophy that many thinkers accepted that the world should be explained in this way. However, before Aristotle, teleological explanations of the cosmos were generally based on the idea that it had been created by a divine intelligence. If an intelligent power made the world, then it makes sense that it did so with a purpose in mind, so grasping this purpose will help us understand the world. This is the pattern of teleological explanation that we find in the Presocratics and in Plato. However, with Aristotle teleology underwent a change: instead of thinking that the ends were explanatory because a mind had sought to bring them about, Aristotle took the ends to operate in natural beings independently of the efforts of any creative intelligence. Indeed, he thought that his predecessors had failed to understand what was distinctive of nature, namely, that its ends work from the inside of natural beings themselves.

One of the hallmarks of Herodotus’ Histories (fifth century BCE) is the evidence that he offers of first-hand accounts from people living in various places, with different customs and points of view. However, regarding an explanation of the seasonal flooding of the river Nile, he complained that he could get no information from Egyptian priests, or ordinary Egyptians.