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The chapter studies nine long-known extant Chinese mathematical texts, and three recently excavated texts, all composed prior to the beginning of the Sui 隋 dynasty (581–618 ce). Most of these were compiled for use as school texts. They include problems on fractions, on proportions and extraction of square and cube roots, on simultaneous linear equations and computations of areas and volumes. Among the more advanced techniques deployed in these texts are computing the area of a circle, that is, obtaining certain approximate values of π; computing the volume of a pyramid; and computing the volume of a sphere.

The chapter studies ancient Chinese astronomy, which focused on computing and predicting the movements of the heavens (天 tian), the sun, moon, stars, and asterisms, which was the duty of the rulers, in order that the people be well-regulated. Heavenly bodies were allocated to terrestrial zones, especially 28 constellations roughly along the equator or the ecliptic, the seven stars of the Big Dipper (regarded as the carriage of heaven), and the five planets. Unusual celestial phenomena were recorded, such as solar eclipses, comets, and meteorites. The 盖天 gai tian theory (celestial dome theory), the 浑天 hun tian Theory (celestial sphere theory) and the 宣夜 xuan ye theory (infinite empty space theory) were the three primary theories of the structure of the heaven and the earth, in the Han dynasty (202 bce—220 ce). The earliest extant Chinese star catalogue of the whole sky was composed in the 1st century bce, and the definitive constellation system of 283 constellations, 1464 or 1465 stars was composed in the 3rd century ce.

The chapter explores Mesopotamian astrology and astronomy, which were not distinct as pseudo-science is from science, and were together a major part of cuneiform intellectual culture. The writings consist of scholarly compendia, observational records, and predictive ephemerides, mostly produced after ca700 bce, at the court in Assyria, and later, in the major temples of Babylonian cities. Heavenly phenomena (astral, planetary, and lunar) were objects of study both as signs of future events and as phenomena in their own right. The systematic study and recording of astral phenomena as signs began in the Old Babylonian period (2000–1600 bce), and continued well after the conquest by Alexander. Babylonian astronomy used mathematical models (based on numerical sequences) to calculate periodic phenomena. The astral sciences, including celestial and natal divination, that is, omens and horoscopes, were parts of a scholarly discipline sustained over two millennia by cuneiform scribes in both Assyria and Babylonia.

In its original Babylonian and Egyptian contexts, astrology was the interpretation of celestial signs and omens sent by the gods as warnings to rulers and the elite. Roman fondness for Stoicism fertilized the growth of astrology in the Greco-Roman world, which developed into a natural science, fully integrated with the prevailing cosmology. Astrology became popularized, and anyone who could afford some level of the service knew basic features of his natal chart. The chapter explains the various forms and purposes of judicial or divinatory astrology: “mundane” (heavenly effects on regions), “genethlialogical” (heavenly effects on a life from its birth or conception), “horary” (heavenly effects on the present moment), and “catarchic” (heavenly effects on the future). The chapter also provides an historical sketch of classical astrology, from Babylonian origins through the major surviving handbooks, and an elaborated ancient example of a natal chart (of the emperor Hadrian), its methods, and interpretation.

This chapter studies Egyptian astronomy based on the very few surviving texts. The Egyptian calendar was purely solar, unlike most ancient calendars. The oldest astronomical monuments from Egypt are the star clocks, mainly on the interior of coffin lids from the Eleventh and Twelfth Dynasties. They divide the year into 36 ten-day intervals (decades), each with 12 stars, to mark the hours of the night for religious purposes. The major text of Egyptian astronomy is the Book of Nut, the sky goddess, which describes the behavior of the sun, moon, and especially fixed stars, as well as shadow clocks and water clocks. The Egyptian constellations were fundamentally different from ours (based on Mesopotamian and Greek myths), with Osiris (our Orion), Seth (our Big Dipper), and Sirius playing a prominent role, plus the Ship, the Sheep, and the Two Tortoises. Late Egyptian astronomy borrows some techniques from Mesopotamian astronomy. In the Greco-Roman period, Egyptian astronomy borrows elements from Greco-Roman astronomy.

The chapter shows how the texts of early Byzantine alchemy transformed the alchemical tradition. This period is characterized by a generation of “commentators” tied to the Neoplatonic milieu. Their writings, designed primarily to clarify the ideas of the previous generations, represent the most advanced stage of ancient alchemical theory. In the fifth century, authors external to alchemy explicitly speak of alchemy as a contemporary practice to produce gold from other metals. Around the seventh century, the corpus of alchemical texts began to be assembled as an anthology of extracts. The object of the research was agents of transformations of matter. The cause of the transformation is an active principle that acts by dissolution: “divine water” (or sulfur water), mercury, “chrysocolla” (gold solder), or raw sulfur. Mercury is at once the dyeing agent and the prime metallic matter, understood as the common substrate of the transformations and the principle of liquidity.

The chapter maps out the various forms and modes of Greek geographical writing and thinking in the early Byzantine period. Geography in Byzantium means transmission of the ancient Greeks’ knowledge and the handling of this material by certain scholars, via copying manuscripts, making extracts from the old writings, or writing separate commentaries. The two ancient writers that most influenced geographical ideas of the Byzantines were Strabo and Ptolemy. Besides geographical treatises, such as those by Marcian, Protagoras, Hierocles, and Stephen, many historical works contained extensive geographical excursuses (for example, Philostorgius), and reports of ambassadors often included a geographical description of the land(s) they visited. Christians of the school of Antioch tended to deny the spherical earth theory (Theophilus of Antioch, Diodorus of Tarsus, Theodorus of Mopsuestia, Kosmas of Alexandria). Itineraries and traveler’s accounts provide a glimpse of how people traveled. The Peutinger map displays the Roman roads around 435 ce.

The contribution of the Byzantine medical encyclopedists (ca 370 ce–ca 650 ce) to the preservation of medical knowledge still awaits its full scholarly assessment. This chapter highlights the achievements of Oribasius of Pergamum, Aëtius of Amida, Alexander of Tralleis, and Paul of Aegina. In the shadow of Galen’s legacy, these writers preserve, organize, and explicate the diverse body of medical knowledge, available to them. Relying on Galen but also on medical writers from the sixth century bce onwards, Oribasius composes his Compilations as a comprehensive source of medical authority. Aëtius produces a collection of 16 books covering the full medical spectrum from pharmacology to diagnostics to pathology. To it, Alexander adds a collection of 12 books, entitled Therapeutics, with corollary treatises On Fevers and On Intestinal Worms. Paul compiles the Epitome of Medicine, featuring all medical branches.

The chapter discusses ancient Greek dietetic theory and practice; for Greek physicians and patients, diet was one of the primary means of medical intervention. Diet was treated along with exercise, as well as sleep, massage, drugs, and sex. The many authors of the Hippocratic differed about diet, and Galen favors his own advice. We have mere scraps from Hellenistic authors such as Diocles of Carystus, Archestratus, Mnesitheus of Athens, Diphilus of Siphnos, and Phylotimus. The late antique Latin cookbook “Apicius” (likely named for the first-century ce gourmand), promotes a style of cookery that survived into the mediaeval world: blending numerous seasonings and the characteristic balance between sweet and sour, still redolent in the traditional agrodolce sauces of Italian cuisine. Theories of diet and digestion varied enormously as well, Hippocrates claiming concoction, Erasistratus thinking of mechanical grinding and dispersion, and others deciding upon distribution, the more dominant theory in late antiquity.

This chapter re-examines the historiography of Greco-Egyptian alchemy. The author challenges the representation of alchemy as a superficial amalgam (or “syncretism”) of Greek philosophical theories. He argues for an indigenous Egyptian development of the art, rooted distantly in the artisanal traditions of the temples, and stresses the originality of alchemical matter theory, which drew creatively on Greek philosophical (especially Aristotelian) concepts. The alchemists were also innovative technicians, developing new methods and apparatus in pursuit of the goal of transmutation. These theoretical and technical innovations contradict the commonplace image of alchemy as an irrational pseudo-science. The esotericism of alchemy, far from undermining its rational development, served to stimulate research, since the secrets of the ancient masters could only be decoded in the light of laboratory experience. Alchemy does not provide evidence for a (supposed) decline into mystical obscurantism in the late ancient world.

The chapter surveys Greek mathematics and astronomy, as far as it can be known from works before circa 300 bce. Key sources are the now-fragmentary histories of astronomy and of geometry composed by Eudemus of Rhodes, a student of Aristotle. Eudemus focused on “first discoverers” of theorems or procedures. The role of deductive mathematical proof in Greek mathematics is central, derives from the agonistic character of Greek culture, and probably largely displaced earlier more practical or procedural mathematics. The main lines of mathematical investigation that survive concerned geometry and also arithmetic and number theory. Many of these early mathematicians were also astronomers. The main lines of astronomical investigation concerned the motions of the sun, moon, and planets, about which a variety of observations were made, and for which a variety of models were constructed.

This chapter studies ancient Egyptian medicine, which for nearly two millennia before the Greek conquest had combined both rational and irrational procedures. For Egyptians, magic was a divine force that, together with the creative word, could turn concepts into reality. Visible conditions were generally treated according to the perceived cause (such as bone setting or simple surgery); however, for fever, where the cause remained hidden, magic was used. The author makes extensive use of funerary evidence (mummies and tomb goods), since very few texts survive, and depictions of physical deformity or disease are restricted to portrayals of lower class people. The preserved texts, including especially the Ebers Papyrus and the Edwin Smith Papyrus, present fragmentary information about physiology and offer case studies that prescribe surgical, pharmaceutical, or magical treatments. Effective remedies identified in the Egyptian pharmacopoeia include laxatives, antacids, anti-diarrheals, and antiseptics.

The chapter gives an account Epicurus’ natural philosophy and his attitude to the sciences. Epicurus’ mission was to liberate people from the fear of death and the gods, and science was subordinate to that project, practiced to show that nature acts without divine intervention. He was skeptical about mathematics, due to his commitment to atomism, and about astronomy, because knowledge should be based on clear foundations unavailable for deciding issues such as the planets’ sizes. Sensation dictates there are two constituents of reality: bodies, directly attested by sense perception; and the void, the space where bodies exist and move. Infinite atoms move through space, forming countless worlds (kosmoi), which at some point will again fall apart into their constituent atoms. Epicurus considers naturalistic explanations of phenomena to show they are not divine. But his philosophy of nature insists upon natural causes (as opposed to geometrical models), is consistently materialist and mechanistic, and is thus anti-teleological.

Galen was part of the urban, Hellenic, leisured class and culture that produced the Second Sophistic. In the regimen he prescribes for a healthy way of life, and in his stories about his patients, he shows allegiance to the masculine, intellectual, and aristocratic values of the gymnasium, contrasted with the harsh deprivation of the peasant’s countryside. He identified with the class of pepaideumenoi, and positioned medicine among the “liberal arts.” He wrote widely on ethics, logic, and language, though his views on Atticism are complicated. Galen privileged classical writers (the palaioi) over more recent ones, and Hippocrates and Plato were especially central to his intellectual identity. Public demonstrations (epideixeis) and more informal debates were important in his professional life. Galen’s ambivalent position in the Roman aristocracy—a well-connected part of the imperial project, committed to the idea of Hellenic superiority—also locates him in the Second Sophistic.

The chapter treats the life and work of the physician Galen; his own works are the primary source for his life. He established a comprehensive system of medical practice based on a systematic theoretical foundation, which included all the basic medical sciences, as well as the relevant areas of philosophy, and endured with minor alterations until the mid-nineteenth century. He made important anatomical discoveries and carried out illuminating physiological experiments. He insisted that doctors must have a solid grounding in philosophy, and intervened in the debate between the contending medical sects of his era. Galen valued the works of Hippocrates and Plato and attempted to ground his work in their ideas. Galen adhered to the theory that four qualities (hot/cold and wet/dry) composed the body; he advocated teleology as an explanatory principle for all phenomena, and he understood the soul to have three parts as in Plato’s Timaeus.

The chapter looks at Greek and Roman surgical instruments. The survival of Greco-Roman surgical instruments falls into two divisions: tools available in Hippocratic times (fifth to fourth century bce), and instruments at the disposal of surgeons, mostly Greek, from the late Republic through the Empire (first century bce to fifth century ce). From the former, most survivals are cupping vessels from graves. The texts suggest the Hippocratic physician often created his tool on the spot or had a tool prepared for an immediate need, whereas most of an Imperial surgeon’s repertoire consisted of instruments professionally made and sold by smiths. The various kinds of instruments are described, explained, and illustrated: cupping vessels, scalpels, phlebotomes (for phlebotomy), lithotomes (for bladder stones), needles, probes, cauteries, hooks, forceps, saws, drills, chisels, files, levers, tubes, douches, specula, and abortives.

The chapter discusses the mathematics and astronomy of “Late Antiquity” (the early Byzantine period). The period was one of intensive innovation, transformation, and change, involving appropriation and assimilation, and also involved fierce cultural and doctrinal competition between various allegiances. The mathematical or astronomical works of this period are consistently original in their attempt to consolidate mathematical and astronomical practice for literate and/or philosophical education. Cultural competition and emulation existed between various justifications of mathematics and astronomy. These cultural choices were furthermore justified in the terms of wider domains of knowledge and intellectual activities. The period also displays a deep love of traditional knowledge, taken as an almost unavoidable reference point. Fourthly, mathematics and astronomy in Late Antiquity followed a wide range of stylistic patterns. Last, the diversity of works must be seen to include anonymous corpora, such as the pseudo-Heronian metrology, or the scholia on mathematical and astronomical texts, etc.

This chapter focuses on four texts in Hellenistic astrologia, which covers both astrology and astronomy. Its guiding thesis is that understanding ancient astrologia should not be restricted to the mathematics deployed but should include its contexts, such as its relations to other intellectual disciplines and to broader philosophical and cultural concerns about how to live one’s life. The texts studied are Diodorus Siculus, Bibliotheca 1–2 (ca 50 bce), Vitruvius, De architectura 9 (ca 25 bce), Geminus, Introductio astronomiae (ca 50 bce), and Pliny, Naturalis historia 2 (77 ce). What each includes is found to depend on the readers addressed and how its author sought to persuade or inform them. Hellenistic astrologia is a work in progress during the two centuries about the millennium, with different writers urging in different contexts divergent views of what astrologia should be and why it is important.

Geography as a scholarly discipline originated in the period from the 4th century BCE through the 1st century CE. Ephorus in the 340s BCE was the first to write about it in detail, and the slightly later explorations of Pytheas to the North Atlantic and Baltic, and Alexander the Great to India, provided data that allowed Eratosthenes of Cyrene, after ca 250 BCE, to write the first scholarly treatise on geography, even inventing the term. Other Hellenistic authors added to the topic, especially Hipparchus and Polybius. Fantasy geography—mythical places at the end of the known world—also developed, but such works could contain valuable information. The culmination of Hellenistic geographical thought was the 17-book treatise of Strabo of Amaseia, completed ca 25 CE. It is the most complete ancient work on geography, and the source for most of the previous history of geography, including numerous quotations from works now lost.

The article analyses the forms and texts of Hellenistic mathematics. Three stylistic codes were employed in these texts, each related to a specific mathematical content: the demonstrative, the procedural, and the algorithmic code. Greek mathematical works, such as those by Apollonius of Perge, were not diffused through official channels, and instead were usually sent to some addressee, and were frequently preceded by a prefatory epistle, so each “edition” was actually a single copy, to be diffused by the recipient to relevant people. The prefatory letters of Archimedes’ works show that he regularly circulated lists of open problems and conjectures, and then their solutions. Mathematical texts were canonical, and elicited a variety of activities, especially the writing of commentaries, perhaps first by Heron of Alexandria. Late antique authors, such as Diophantus, Pappus, and Serenus, transform this canon.