In 2012 the Union of German Academies of Sciences and Humanities announced the funding of a 25-year Academy Project of research into the reception and study of medieval Arabic and Latin versions of Ptolemaic works. Known as Ptolemaeus Arabus et Latinus (PAL), its primary aim is “to make the Ptolemaic corpus on the science of the stars available to researchers” (p. 3). To achieve this simply stated goal the organizers have created what is probably the largest collaborative, current project in the history of astronomy. Interested scholars may follow the activities of PAL on its website: https:// ptolemaeusbadwe.de. This volume represents results of the first triennial PAL conference, offering a deliberately broad coverage of Ptolemaic astronomy and astrology in Latin, Arabic and Greek. After a useful overview of each of its fourteen contributions, the book begins with four papers on the Greek and Near Eastern Traditions. In his lead-in paper, “The Ancient Ptolemy,” Alexander Jones seeks to present Ptolemy “in the round” and begins by asking which “Ptolemaic” works Ptolemy wrote. Reviewing the evidence on the doubtful works, he exploits the searchability of the digital Thesaurus Linguae Graecae (TLG) to identify traits of Ptolemy’s writing that allow a secure test of his authorship. One result of this is a strong argument for accepting the previously doubtful philosophical work, Criterion, as a work of Ptolemy. After considering the relative order in which Ptolemy composed his works, Jones presents two views of the Ptolemaic canon: a view by discipline and another by prevailing themes. For example, both the Almagest and the Harmonics, although in different disciplines, are “deeply concerned with applying sense perception . . .and reason” to create models underlying observed phenomena (p. 28). Jones concludes by emphasizing Ptolemy’s concern with “mathematically defined modes of representation of the cosmos” (p. 29). To Jones’s examples one might add the Analemma, the subject of Nathan Sidoli’s “Mathematical Methods in Ptolemy’s Analemma.” The analemma was an ancient method of geometrically representing a sphere on a plane, a problem Ptolemy faced in his Planispherium and Geography. Sidoli carefully explains the method, based on the division of the celestial sphere into eight octants by three mutually perpendicular great circles all passing through the observer’s position at the center of that sphere. These three 1081975 JHA0010.1177/00218286221081975Journal for the History of AstronomyEssay Review Essay Review2022

Originally published in 1996, this book contains a translation and study of Euclid's Phaenomena, a work which once formed part of the mathematical training of astronomers from Central Asia to Western Europe. Included is an introduction that sets Euclid's geometry of the celestial sphere, and its application to the astronomy of his day, into its historical context for readers not already familiar with it. So no knowledge of astronomy or advanced mathematics is necessary for an understanding of the work. The book shows mathematical astronomy shortly before the invention of trigonometry, which allowed the calculation of exact results and the subsequent composition of Ptolemy's Almagest. This work and the (roughly) contemporaneous treatises of Autolycus and Aristarchos form a corpus of the oldest extant works on mathematical astronomy. Together with Euclid's Optics one has the beginnings of the history of science as an application of mathematics.

ASTRONOMY IN ISLAM Islamic Astronomy and Geography. David A. King (Variorum Collected Studies Series, CS1009; AshgatePublishing, Franham, Surrey, 2012). Pp. xlii + 376. £90. ISBN 978-1-4094-4201-1.The papers reprinted in this welcome addition to the series of three previous collections of David King's papers further witness to the breadth and depth of his researches on medieval Islamic astronomy, as well as the extent to which that discipline was responsive to the concerns of Islamic religion and culture.King has divided the papers into five groups, the first of which (General) begins with a 1996 overview of problems and achievements in Islamic astronomy. The two following papers deal with Islamic astronomical instruments (2004) and illustrations in astronomical manuscripts (1995). In the former, inscriptions to context, King related a number of cases in which physical features (such as inscription) and (sometimes) separate treatises have given us information about the purpose of these instruments and the context in which they were produced. One such instrument is an astrolabe the design of whose rete and stars displayed thereon are identical with that of the sole surviving Byzantine astrolabe. It predates all known early Islamic astrolabes; the inscriptions tell us that the instrument was reworked (and slightly muddled) by an Ottoman craftsman almost a thousand years after it was made. In the past decades instruments and illustrations in scientific works have received careful scholarly attention and King has been a leading contributor to the study of both.The second section, Regional studies, treats astronomy in Fatimid and Mamluk Egypt, as well as astronomy in the medieval Maghrib. The treatment of astronomy in the medieval Maghrib has received much attention in past decades, as King points out, thanks to the efforts of the Barcelona school under the leadership of Julio Samso. However King has to be considered the principal authority on mathematics in Fatimid and Mamluk Egypt, and his paper, Aspects of Fatimid astronomy: From hard-core mathematical astronomy to architectural orientations in Cairo, connects, as do so many of his works, the exact sciences in Islam with the wider context of Islamic culture. In this case King shows how the effects of the work of the great Cairene astronomer, Ibn Yunus (d. 1009), and his successors were felt for centuries in such seemingly unlikely matters as the orientation of ventilators in Cairo.Much of King's work has been devoted to the study of astronomical tables, and the third section of this book, Mathematical astrology, contains one paper, tracing the Arabic tradition of two tables by the second-century Antiochian astronomer Vettius Valens. These tables ostensibly predicted a person's length of life based on the longitude of the ecliptic rising at the moment of birth, subject to the bound determined by half the maximum length of daylight at the place of birth. Although such use of exact science may seem nonsensical to a scientific 'modem' it seems appropriate to modify a remark that King makes in a later paper (VIII, p. …

This survey reviews research in four areas of the history of Greek mathematics: (1) methods in Greek mathematics (the axiomatic method, the method of analysis, and geometric algebra); (2) proportion and the theory of irrationals (controversies over the origins of the theory of incommensurables); (3) Archimedes (aspects of controversies over his life and works); and (4) Greek mathematical methods (including discussion of Ptolemy's work, connections between Greek and Indian mathematics, the significance of Greek mathematical papyri, Arabic texts, and even archaeological investigations of scientific instruments).

This collection of papers by an eminent authority on the history of mathematics and astronomy in medieval Andalusia and the Maghreb will be a welcome addition to the library of all scholars with a serious interest in this area. This volume supplements a previous one by the author in the same series, Islamic Astronomy and Medieval Spain, which dealt only with Andalusian and Alfonsine materials. The present collection divides its attention almost equally between al-Andalus (the Iberian peninsula at the time of Muslim rule there) and the Maghreb (northwest Africa), particularly the astronomical handbooks (zijes) produced in this region during the thirteenth and fourteenth centuries that reflect Andalusian material, a topic that the author identifies as ‘the main object of our research during the last fifteen years’. Completing this picture are the author's study of the criticism and gradual abandonment of that tradition during the course of the fourteenth century and its replacement by an Eastern tradition during the latter part of that period.

In der Antike wurde – spätestens vom 4. Jahrhundert vor Christus an – das wissenschaftliche Teilgebiet, Kreisbögen und Winkel auf einer Kugelfläche zu berechnen, als „Sphärik“ bezeichnet. In den Anwendungen ging es dabei entweder um die Himmelskugel oder um die Erde; die erste war eine Kugel, von der man dachte, dass an ihr die Fixsterne befestigt sind und dass sie einen solch großen Radius hat, dass die Erde im Verhältnis dazu nicht mehr als ein Punkt ist. Ihr Radius war dennoch endlich und konnte für mathematische Zwecke als Einheit verwendet werden. In der Geometrie der Oberfläche einer Kugel entsprechen den Geraden in einer Ebene die Großkreise, das sind die Schnittlinien der Kugeloberfläche mit irgendeiner Ebene, die durch den Mittelpunkt geht. Ebenso wichtig sind die Parallelkreise, die sich als Schnittlinien der Kugeloberfläche mit einer Ebene ergeben, die nicht durch den Mittelpunkt geht.

Muslimische Mathematiker waren die ersten Menschen, die Zahlen so schrieben, wie wir es auch heute tun. Und während wir die Erben der Griechen in der Geometrie sind, ist unsere Arithmetik ein Teil des Vermächtnisses der muslimischen Welt, auch wenn es indische Mathematiker waren, vermutlich einige Jahrhunderte vor dem Aufstieg der islamischen Kultur, die damit begannen, ein Zahlensystem mit folgenden zwei Eigenschaften zu verwenden: 1.  Die Zahlen von eins bis neun werden durch neun Ziffern dargestellt, die sich alle leicht mit ein oder zwei Strichen schreiben lassen. 2.  Die Ziffer ganz rechts in einer Zahl zählt die Einer und ein Eintrag an einer beliebigen Stelle hat den zehnfachen Wert der rechts daneben liegenden Stelle. So zählt die Ziffer an zweiter Stelle die Zehner, die an der dritten Stelle die Hunderter (das entspricht zehn Zehnern) und so weiter. Ein besonderes Zeichen, die Null, wird dazu benutzt, anzuzeigen, dass eine Stelle nicht besetzt ist. Diese beiden Eigenschaften beschreiben das heutige System für die Notation der ganzen Zahlen und wir können das eingangs Gesagte dahingehend zusammenfassen, dass die Inder die ersten waren, die ein zifferngestütztes, dezimales Stellenwertsystem verwendeten. „Zifferngestützt“ bedeutet, dass die ersten neun Zahlen durch neun Ziffern dargestellt werden und nicht, wie bei den Ägyptern und Babyloniern, durch die Häufung von Strichen. Dezimal bedeutet, dass die Basis zehn ist. Die Hindus erweiterten dieses System jedoch nicht, um Teile eines Ganzen durch Dezimalbrüche darzustellen. Weil es die Muslime waren, die dies als Erste taten, waren sie die ersten Menschen, die Zahlen so darstellten, wie wir es tun. Völlig zu Recht heißt dieses System indo-arabisch.