Glen Van Brummelen

This survey of the movements of astronomical tables within medieval Islam provides a backdrop for the remaining articles in this volume. We discuss the types of document that contained astronomical tables, and some of the analytical tools that have produced discoveries with respect to the transmission of tables from one author or subculture to another. We conclude with a summary of table transmission: from origins with the appropriation of traditions of astronomical tables taken from Greece and India, through the establishment of a Ptolemaic tradition in the 9th century and a uniquely Islamic expansion of methods and concepts starting in the 10th, to important sub-traditions in al- Andalus and the Maghrib.

Preface.- Introduction: The Birth and Growth of a Community by Amy Shell-Gellasch.- History or Heritage? An Important Distinction in Mathematics and for Mathematics Education, by Ivor Grattan-Guinness.- Ptolemy's Mathematical Models and their Meaning, by Alexander Jones.- Mathematics, Instruments and Navigation, 1600-1800, by Jim Bennett.- Was Newton's Calculus a Dead End? The Continental Influence of Maclaurin's Treatise of Fluxions, byJudith V. Grabiner.- The Mathematics and Science of Leonhard Euler (1707-1783), by Ruediger Thiele.- Mathematics in Canada before 1945: A Preliminary Survey by Thomas Archibald and Louis Charbonneau.- The Emergence of the American Mathematical Research Community, by Karen Hunger Parshall.- 19th Century Logic Between Philosophy and Mathematics, by Volker Peckhaus.- The Battle for Cantorian Set Theory, by Joseph W. Dauben.- Hilbert and his Twenty-Four Problems, by Ruediger Thiele.- Turing and the Origins of AI, by Stuart Shanker.- Mathemati...

In 1551, Georg Rheticus published a compact set of tables that effectively completed the set of all six trigonometric functions. However, his work was not widespread, and may not have been known to Francesco Maurolico when he published a secant table in 1558. Before the end of the century, several authors argued whether Maurolico had borrowed the notion of the secant, and his table, from Rheticus. We present Maurolico’s text on his table (named the tabula benefica in a nod to Regiomontanus’s tangent table, the tabula foecunda), as well as a translation and analysis. Finally, we demonstrate that Maurolico’s table is too accurate to have derived from Rheticus’s work, absolving him of the historical accusation.

‘ … and beyond, to complex things’ first considers the Taylor series for the exponential function. One of the most famous, yet enigmatic, numbers in mathematics, e is an irrational number equal to 2.718281828. … Exponential functions deal with the phenomena of growth and decay. As calculus was starting to become established, curious parallels between the apparently disparate worlds of trigonometry and exponential functions were starting to appear. Imaginary numbers, Euler’s formula, and Euler’s identity are discussed along with the Argand diagram, De Moivre’s formula, hyperbolic trigonometric functions, and the catenary curve. Imaginary numbers are now at the heart of science and technology, and are used in the study of electromagnetic waves, cellular and wireless technologies, and fluid dynamics.

AbstractGiovanni Bianchini’s fifteenth-century Tabulae primi mobilis is a collection of 50 pages of canons and 100 pages of tables of spherical astronomy and mathematical astrology, beginning with a treatment of the conversion of stellar coordinates from ecliptic to equatorial. His new method corrects a long-standing error made by a number of his antecedents, and with his tables the computations are much more efficient than in Ptolemy’s Almagest. The completely novel structure of Bianchini’s tables, here and in his Tabulae magistrales, was taken over by Regiomontanus in the latter’s Tabulae directionum. One of the tables Regiomontanus imported from Bianchini contains the first appearance of the tangent function in Latin Europe, which both used as an auxiliary quantity for the calculation of stellar coordinates.

In the Almagest and the Planetary hypotheses, Claudius Ptolemy (fl. c. A.D. 150) devised sophisticated geometrical models to predict the positions of the seven 'planets': Moon, Mercury, Venus, Sun, Mars, Jupiter and Saturn. These models contain a number of subtleties and special properties devised to account for peculiarities of the motions of these bodies. Some of these properties either caused Ptolemy certain difficulties or allowed him to make certain simplifications. The planets are moving objects, but teachers of the history of astronomy (like Ptolemy himself) have had to attempt to demonstrate the nature of the models and their features using static images. The increasing graphical power of the computer removes this limitation of visual representation. The geometer's sketchpad.' a dynamic drawing program whose principles are governed by geometric relationships rather than artistic considerations, is capable of drawing animations of figures given the relationships between the geometric objects. This tool is ideally suited to representing Ptolemy's geometricallydefined planetary models. I have compiled a set of animations within The geometer's sketchpad to aid teachers and students of the history of astronomy in understanding Ptolemy's models. With a free demonstration version of this program, available from the publishers (Key Curriculum Press) via the Internet, asers of reasonably recent-vintage Windows or Macintosh computers may run my animations on their computers. My files may be downloaded from my Web site: http://www.kingsu.ab.ca/-glenlhome.htm.Alternatively, send me a 3.5" floppy disk at The King's University College, 9125-50 Street, Edmonton, Alberta, Canada T6B 2H3. The animations make it easy to demonstrate a number of the fine points of Ptolemaic astronomy. By simple click-and-drag mouse operations, the parameters of the model may be changed. Various important angles may be highlighted or removed from the screen by clicking "Show" and "Hide" buttons. The values of these angles are displayed and continuously updated on-screen as the animation progresses. Three animation speeds should aid users of slower computers. The paths of the planets and other points in the model may be shown. Finally, the animations can be controlled manually by dragging a point back and forth along a line at the bottom of the screen. Several sample uses appear in the illustrations.? In Figure 1, both the eccentric and the epicyclic solar model are displayed (in different colours; the epicyclic model

... That meant one of three things: spend sleepless nights with pencil and paper working your way through the drudgery of ... The hero of our story, an early 15th-century Iranian astronomer, probably spent many long nights with his numbers. ... 4 The proof of the relation cj = 2 + cl 41 ...

chose a favourite photograph from their respective missions. The astronauts also contributed a few sentences about the photographs, ranging from factual accounts to compelling reflections on space flight. The editors, all NASA staffers, selected several more images from each mission and wrote brief descriptions of the missions’ participants and main objectives, giving the reader or, more accurately, the viewer an introduction to the Apollo missions’ history and large photographic archive. The collection presents the Apollo missions as an extraordinary and very human endeavour. An astronaut, spacecraft, or their traces (footprints and tracks) appears in almost every photograph. Very few shots show the lunar surface without signs of humanity’s arrival. The number of shots of the Earth is also remarkable with more than half of the astronauts selecting a photograph that includes it. This frequency attests to the profundity of the view as well the distinctiveness of the journey. Discourse around space exploration is suffused with frontier rhetoric, and the Moon photographs invite comparisons to other visual records of human exploration. As daring as these other journeys were, as much as explorers might have longed for the comforts of home, they spent little time documenting the view over their shoulders and concentrated instead on the unfamiliar worlds before them. The Apollo astronauts, however, looked back. My personal favourite in this book is a photgraph of a photograph, a colourful family snapshot wrapped in a plastic lying in the grey lunar dust. With every other human carefully protected by a spacesuit, eyes hidden behind reflective visors, these figures seem incredibly vulnerable. Although blurred, we can see their eyes. Astronaut Charles Duke writes that he wanted his family to be part of his flight to the Moon. His decision to bring and leave this family photograph attests to the totemic power of images, to the equation we make between a representation and the original it portrays. The photograph of his family was a means to bring them to the Moon. In similar fashion, the camera-toting astronauts brought the Moon to us.

In the Almagest and the Planetary hypotheses, Claudius Ptolemy (fl. c. A.D. 150) devised sophisticated geometrical models to predict the positions of the seven 'planets': Moon, Mercury, Venus, Sun, Mars, Jupiter and Saturn. These models contain a number of subtleties and special properties devised to account for peculiarities of the motions of these bodies. Some of these properties either caused Ptolemy certain difficulties or allowed him to make certain simplifications. The planets are moving objects, but teachers of the history of astronomy (like Ptolemy himself) have had to attempt to demonstrate the nature of the models and their features using static images. The increasing graphical power of the computer removes this limitation of visual representation. The geometer's sketchpad.' a dynamic drawing program whose principles are governed by geometric relationships rather than artistic considerations, is capable of drawing animations of figures given the relationships between the geometric objects. This tool is ideally suited to representing Ptolemy's geometricallydefined planetary models. I have compiled a set of animations within The geometer's sketchpad to aid teachers and students of the history of astronomy in understanding Ptolemy's models. With a free demonstration version of this program, available from the publishers (Key Curriculum Press) via the Internet, asers of reasonably recent-vintage Windows or Macintosh computers may run my animations on their computers. My files may be downloaded from my Web site: http://www.kingsu.ab.ca/-glenlhome.htm.Alternatively, send me a 3.5" floppy disk at The King's University College, 9125-50 Street, Edmonton, Alberta, Canada T6B 2H3. The animations make it easy to demonstrate a number of the fine points of Ptolemaic astronomy. By simple click-and-drag mouse operations, the parameters of the model may be changed. Various important angles may be highlighted or removed from the screen by clicking "Show" and "Hide" buttons. The values of these angles are displayed and continuously updated on-screen as the animation progresses. Three animation speeds should aid users of slower computers. The paths of the planets and other points in the model may be shown. Finally, the animations can be controlled manually by dragging a point back and forth along a line at the bottom of the screen. Several sample uses appear in the illustrations.? In Figure 1, both the eccentric and the epicyclic solar model are displayed (in different colours; the epicyclic model

This chapter discusses the ancient approach to trigonometry, beginning with Hipparchus of Rhodes, the founder of trigonometry. It reconstructs when and where Hipparchus must have lived by taking into account the observations that he made as an astronomer and the references his successors made to him. It then considers the theorems of Menelaus of Alexandria, whose book Sphaerica completely reinvented the mathematical study of the sphere. In particular, it describes Menelaus's Theorem, which became the standard tool of spherical astronomy for the next 900 years. It also examines Abū Sahl al-Kūhī's use of the Menelaus theorems to solve the problem of rising times of arcs of the ecliptic.

The 10th-century mathematician Abū Sahl al-Kūhī, one of the best geometers of medieval Islam, wrote several treatises on the first three books of Euclid's Elements. We present an edition and translation of al-Kūhī's revision of Book I of the Elements, in which he altered the book's focus to the theorems and rearranged the propositions. The most dramatic of the changes