1.1. From the Publisher.

Professor Aaboe gives here the reader a feeling for the universality of important mathematics, putting each chosen topic into its proper setting, thus bringing out the continuity and cumulative nature of mathematical knowledge. The material he selects is mathematically elementary, yet exhibits the depth that is characteristic of truly great thought patterns in all ages. The success of this exposition is due to the author's unique approach to his subject. He wisely refrains from attempting a general survey of mathematics in antiquity, but selects, instead, a few representative items that he can treat in detail. He describes Babylonian mathematics as revealed from cuneiform texts discovered only recently, as well as more familiar topics developed by the Greeks. Although each chapter can be read as a separate unit, there are many connecting threads. Aaboe stays as close to the original texts as is comfortable for a modern reader, and the bibliography enables the interested student to delve more deeply into any aspect of ancient mathematics that catches his or her fancy.

1.2. From the Publisher (of the 1998 reprint).

Among other things, Aaboe shows us how the Babylonians did calculations, how Euclid proved that there are infinitely many primes, how Ptolemy constructed a trigonometric table in his Almagest, and how Archimedes trisected the angle. Some of the topics may be familiar to the reader, while others will seem surprising or be new.

1.3. Contents.

Introduction.

Chapter 1. Babylonian Mathematics.

Chapter 2. Early Greek Mathematics and Euclid's Construction of the Regular Pentagon.

Chapter 3. Three Samples of Archimedean Mathematics.

Chapter 4. Ptolemy's Construction of a Trigonometric Table.

Appendix

1.4. From the Introduction.

If a schoolboy suddenly finds himself transplanted to a new school in foreign parts, he is naturally puzzled by much of the curriculum. The study of languages and of subjects strongly depending on language, such as literature, changes radically from nation to nation, and some subjects, history for one, may even be interpreted differently in different parts of a single country. But in the sciences and in mathematics the boy will probably be quite at home; for, even though order and fashion of presenting details may vary from place to place, these subjects are essentially international.

But if we now imagine our schoolboy transported not only to a different place but also to a different age - say to Greece two thousand years ago, or Babylonia four thousand years ago - he would have to look hard to find anything that he could recognise as science, either in content or in method. What was called "physics" in Aristotle's day, with its discussions of the number of basic principles and of the nature of motion, we would classify as philosophy; and its connection with modern physics appears only after a careful study of the development of the physical sciences. Mathematics alone would now look familiar to our schoolboy: he could solve quadratic equations with his Babylonian fellows and perform geometrical constructions with the Greeks. This is not to say that he would see no differences, but they would be in form only, and not in content; the Babylonian number system was not the same as ours, but the Babylonian formula for solving quadratic equations is still in use.

The unique permanence and universality of mathematics, its independence of time and cultural setting, are direct consequences of its very nature. In Chapter 2 I shall say something about the structure of mathematical theories, so here I will content myself with drawing attention to only a few facets of our subject's singular character.

First I must mention that mathematics is cumulative; that is, it never loses territory, and its boundaries are ever moving outwards. This is in part a consequence of its absolute standards which ensure that what once is good mathematics will always be so and will remain part of the living body of mathematical knowledge. This steady growth offers a contrast to the progress of physics, to take but one example, which has been the victim, or rather beneficiary, of several radical revolutions. So, while Greek physics has only historical interest for a modern physicist, Greek mathematics is still good mathematics which is unavoidable for a modern mathematician. It was the English mathematician Littlewood. who said, with a donnish simile, that we should think of the Greek mathematicians not as clever schoolboys or "scholarship candidates," but as "Fellows of another college."

Another facet I must mention is the deductive character of mathematics: a mathematical theory progresses in an orderly, logical fashion from explicitly stated axioms. One consequence of this is that knowledge of a certain theorem implies, or should imply, knowledge of all the theorem's predecessors linking it to the axioms. A beginner must then begin at the beginning, and the beginning is often old in substance. I can illustrate this point with a biological dictum which, because of its curious phrasing, has stuck in my mind. It says that ontogeny recapitulates phylogeny, and it means that in the development of an individual we see, in swift review, the development of its entire species. If taken literally this dictum can lead, and has led, to all sorts of nonsense, but properly qualified it contains a truth. In the same modified sense it applies to the species of mathematicians. The embryonic development of a mathematician, that is, the education which leads him from the beginnings up to the research front of his day, indeed follows crudely the development of mathematics itself.

Thus, whether we want it or not, the past is very much with us in mathematics, and, whether he wants to or not, a mathematician must begin by studying what in substance is ancient mathematics, in whatever garb the mathematical fashion may dictate. Also, mathematicians are justly proud of the high antiquity of their subject: mathematics is so ancient a discipline that even the study of its history became a recognised field of scholarly endeavour long before most of the sciences. It is therefore particularly natural for students of mathematics to acquaint themselves with the history of their subject, and it is the purpose of the present little volume to help them do so.

I have chosen not to attempt a survey of the history of mathematics from its beginnings to the present. Such a treatment, when confined to a reasonable length, is necessarily weak in mathematical detail and is meaningful only to those who are proficient enough in mathematics to supply depth to a shallow picture. Instead I have selected four episodes from the early history of mathematics and treated them in detail, with comments to convey some notion of their proper setting. As guiding principles for my choice of topics I have used first, that their mathematical content should be within reach of a student with knowledge of high school algebra and geometry. So I have excluded anything that has to do with limit processes and calculus (except the short and elegant argument that led Archimedes to his discovery of the volume and surface area of a sphere, and which I could not resist including). Further, I wanted my selections to be mathematically significant, representative of their periods and authors, and yet off the track beaten by popular histories of mathematics; I wanted them capable of independent treatment, yet having some themes and ideas in common.

Of course, such goals can only be approximated. The topics I came up with are, in order of their appearance in this book and also in chronological order, a presentation of Babylonian mathematics recovered from cuneiform texts only during the last half century; Euclid's construction of the regular pentagon from his Elements; three small samples of Archimedes' mathematics: his trisection of an angle, his construction of the regular heptagon, and his discovery of the volume and surface of a sphere; and, lastly, Greek trigonometry as it is presented by Ptolemy in his Almagest. I have endeavoured throughout to emphasise what the sources of our knowledge of ancient mathematics are, and in my presentation of the material I have tried to stay as close to the texts as is comfortable for a modern reader.

A recurrent theme in the selections from Greek mathematics is the problem of dividing the circle into a number of equal parts; Euclid achieves the division into five parts by compasses and straightedge alone, Archimedes must employ more complicated tools, and Ptolemy is interested in computing the length of the chord subtending a proper part of the circumference of a circle. The Babylonian Number System, which was the backbone of Babylonian mathematics, is adopted by Ptolemy as the only reasonable manner of expressing fractions (and is hence preserved in our subdivisions of degrees and hours). Babylonian influence may be detected in Euclid's formulation of quadratic equations, and though his method of solution differs on the surface from that of the Babylonians, there are similarities in the two approaches to the same problem. I shall leave it to the reader to discover other connecting threads between the four chapters, though each can be read separately.

Finally, I wish to make two apologetic and warning remarks about Greek names in the last three chapters. First, I have made no attempt at consistency in their spelling, but have simply written down what came naturally to my pen. If a reader should be interested in the proper Greek. form of a particular name, he can readily reconstruct it from my spelling; consistency would prohibit such time-honoured usages as Plato, Aristotle, and Euclid. Second, the number of names of Greek mathematicians and scholiasts who make but one or two insignificant appearances each in my tale is large, and it might well be argued that they were better omitted. But whenever I had the choice of writing, for example, "Stobaeus tells" or "we have it on ancient authority," I chose the former alternative, for I see no excuse for imprecision when precision is so easily attained. The reader who wishes to look up the reference is helped by my choice, and he who does not is not harmed. I have, however, not wanted to clutter up the pages of this book with any more detailed learned apparatus; at any rate, several of the works in the bibliography at the end contain exhaustive references.

There is great excitement in discovering the patterns of thought of great minds of the distant past, and in the mathematical sciences one can recognise when resonance is achieved with a much higher degree of certainty than anywhere else. It is a privilege to show others along paths first trodden so long ago, or, in a fine old phrase, to make the lips of the ancients move in their graves. There is, however, no real substitute for reading the old mathematicians themselves, and if this little book should induce some of its readers to do so, it will have served its purpose well.

1.5. Review by: Derek T Whiteside.
Journal of the History of Medicine and Allied Sciences 20 (2) (1965), 184-185.

In this slim paperbound volume the author presents selected highlights of Babylonian and Greek mathematics. In four chapters he deals with the structure and achievement of Babylonian arithmetic and geometry, and in particular the theory of Pythagorean triads implicit in the now-famous cuneiform tablet Plimpton 322; then passes quickly over the achievements of early Greek mathematics, stressing the novelty and rigour of its axiomatic approach to geometry and the logical penetration of Euclid's Elements, in which so much of its structure and content was incorporated. In the final two chapters he suggests the technical elaborations of later Greek mathematicians by concentrating on the contributions to geometry, trigonometry, and theoretical astronomy of Archimedes and Ptolemy. Outline solutions of problems interleaved in the text and a useful bibliography for the uninitiated reader conclude the book, while certain more detailed references are set as footnotes in the body of the work.

Professor Aaboe is an authority in his chosen field but his present book, number 13 in the New Mathematical Library (a series aimed dually at the high school student and the intelligent layman), is written in more didactic style. His approach, reminiscent - in a different context - of Toeplitz' classic Die Entwicklung der Infinitesimalrechnung, recreates the mathematical significance of his topics through a well-chosen notation which keeps close to the original and does not hide its difficulties merely by eliminating all but the non-technical. He is particularly adept at showing the continuity between successive stages of development: he traces, for example, growing knowledge of the theory of the regular polygon from Babylonian computation of the square's diagonal in terms of its side, through Euclid's work on the pentagon to Archimedes' construction of the regular heptagon, and then reveals how basic that theory was in Ptolemy's calculation of chord tables in the Almagest. The historical introduction and asides which weave together these mathematical strands are at once accurate and freshly written and the comparisons made - that, for example, between Archimedes and al-Kâshî - stimulating. There is little that is dull and uninformative, though perhaps the short section on Ptolemy's epicycloidal models of planetary motion is too brief and confused to be useful. A few other points are ones of personal taste: thus, I myself would have preferred to sketch Euclid's elegant comparison of the sides of the regular pentagon and decagon in Elements XIII, 10 rather than propose the modern algebraic reduction suggestion in solution to Problem 2.1. In general, the student cannot fail to be taken with the interest and quality of this introduction to ancient mathematics and the scholar will be impressed by its integrity and lack of deadwood and cant.

1.6. Review by: Carl B Boyer.
Science, New Series 144 (3619) (1964), 726.

The number 13 is very appropriate for this volume in the New Mathematical Library series. The excellent collection of which it is the most recent (and possibly the best) number is intended for extracurricular use by superior high school students, but it is also highly appropriate reading for laymen. The material in number 13 is pitched nicely at the level intended, with clarity of exposition equal to that of its predecessors in the series; and the volume is especially welcome, for its history is as accurate as its mathematics is understandable. All too often the little mathematical history that does make its way into works on the secondary school level is tarnished by half-truths or worse; but such is not the case here. Aaboe took his doctorate under Neugebauer, and he has read widely and deeply, with the result that in this book he has combined mathematics and history of comparable soundness, without a show of profundity. Elementary geometry and trigonometry suffice for comprehension of the themes undertaken, but the material is considerably removed from the routine topics characteristic of textbooks at this level.

The reader is properly warned in the title not to anticipate a systematic history for the author has adopted a "block-and-gap" approach in which a limited number of "episodes" are explored in some depth. In the first episode, Aaboe describes Babylonian mathematics, with particular reference to place-value notation and its use in algebraic and geometrical problems, including the solution of quadratic equations and the use of the Pythagorean theorem. The episode closes with a brief account of the Mesopotamian table of Pythagorean triads (Plimpton 322) and with a reminder that, apart from some geometry, the Egyptians "did not get past elementary arithmetic."

The author's second episode is "Early Greek mathematics and Euclid's construction of the regular pentagon." Aaboe is appropriately cautious about contributions traditionally ascribed to Thales and Pythagoras, and he emphasises the "critical reaction" that set in after Zeno had propounded his paradoxes and the existence of incommensurable line segments had been disclosed. The algebra that Greece had adopted from the Babylonians was reformulated in the geometric garb later definitively presented in Euclid's Elements. The solution of quadratic equations, for example, was now a problem in the "application of areas," rather than one of "finding a number.''

The third and least unified of the four episodes (and in some ways also the least successful) is entitled "Three samples of Archimedean mathematics." Here for the first time Aaboe allows biography and legend to obtrude into an otherwise mathematically oriented account. Following his brief summary of the life and principal works of Archimedes, the author focuses attention on the Syracusan's trisection of an angle, his construction of the regular heptagon and the application of his "mechanical method" in the discovery of the volume and surface area of a sphere. These aspects are well presented, but two questions come to mind in this connection: (i) Are the trisection and the heptagon (minor works which have come down through the Arabic) well adapted to the purposes of the series? (ii) Is not the section 3.3, on modern criteria of constructibility, something of an anachronism as far as this volume is concerned?

The last of the four episodes is a tightly woven summary of Greek trigonometry, as found especially in Ptolemy's Almagest. Methods used in the construction of tables and in applications to the solution of triangles are described in admirably clear detail. In laudably relating the material to earlier contributions? it is pointed out that Ptolemy's value for √2 is the very same sexagesimal - 1; 24,51,10 - that is found in an old Babylonian tablet. Although there is no reference to the fact that the law of cosines appears in Euclid or that certain trigonometric inequalities were known to Aristarchus and others, the reader is reminded that trigonometry did not originate with Ptolemy (about A.D. 150). Such methods go back at least to the time of Hipparchus (about 150 B.C.), and it is quite possible that they were known to Apollonius. Aaboe also closes with a caveat that deserves close attention:

It should be clear to anyone who has read this chapter that the commonly held notion that Greek mathematics is entirely geometrical is not quite correct. Greek mathematicians were perfectly capable of doing numerical work when they had to; indeed, one has to look far and wide to find Ptolemy's equal as a computer.

Intended as it is for student use (copies are available to elementary and secondary schools in the SMSG paperback series through the L W Singer Co., Syracuse, N.Y., at 90 cents, to the teacher or the school), the book includes a small number of well-graded problems and a short list of admirably selected and critically evaluated "Suggestions for further reading." However, a book as well written as this one will inevitably find its way into hands far more numerous than those of schoolboys. It is recommended reading for all who delight in a thoroughly authoritative account of just how certain aspects of mathematics came about. Modern notations and language are used, but explanations are kept as close to the thoughts of the original creators as is consistent with ready comprehension by a reader today. We are far removed in space and time from the ancient cities of Babylon, Alexandria, and Syracuse, but that the gap can be bridged through superb exposition is effectively demonstrated in this book. The little volume is as high in value as it is low in price.

1.7. Review by: Kurt Vogel.
Mathematical Reviews MR0159739 (28 #2956).

In four chapters (1) Babylonian mathematics (including position system, quadratic equations, √2), (2) that of the Greeks before and during Euclid (especially lunulae, irrationality of √2, elements, prime number, pentagon), (3) Archimedes (including spirals, angle trisection, heptagons, the sphere in methodology) and (4) Ptolemy (Almagest, chord chart, epicycle model), which also provide insights into further development (non-Euclidean geometry, constructability of polygons, trigonometry in later times), the author describes the achievements of antiquity in laying the foundation and expanding mathematical knowledge using selected, clearly presented examples that particularly characterise the individual periods and are accessible to school algebra and geometry. The book, which also introduces some new things with its numerous illustrations (e.g. the Mesopotamian school room, the "Tomahawk" angle trisection), is the 13th volume of the School Mathematics Study Group prompted for the "New Mathematical Library" and is certainly suitable for arousing interest in the history of mathematics and for conveying to the student the fundamental difference in attitudes towards mathematical matters. A number of tasks (along with solutions) and references to further literature also serve this purpose. The title of the book itself is perhaps too undemanding: there are no interpolations that the author offers, no "Epeisodia" that come along "in the way", but the anonymous Babylonians, Euclid, Archimedes and Ptolemy are among the main protagonists of mathematical events.

1.8. Review by: Clifford A Long.
The Mathematics Teacher 59 (3) (1966), 287-288.

The purpose of this little book, which I found to be fascinating and informative, is to help students of mathematics acquaint themselves with the history of their subject. Instead of at tempting a survey of the history of mathematics, Dr Aaboe "... selected four episodes from the early history of mathematics and treated them in detail, with comments to convey some notion of their proper setting." In so doing, he has written a book which contains depth in the mathematics of the episodes, but which is within reach of a student with knowledge of high school algebra and geometry. His selections meet his requirements that they "... be mathematically significant, representative of their periods and authors, ... capable of independent treatment, yet having some themes and ideas in common." The episodes have corresponding chapter headings:

Babylonian Mathematics

Early Greek Mathematics and Euclid's Construction of the Regular Pentagon

Three Samples of Archimedean Mathematics

Ptolemy's Construction of a Trigonometric Table

In Chapter One, the Babylonian number system is considered, compared to our own, and applied in arithmetical computations. Selections from three texts are presented, followed by a fine summary of the nature and content of Babylonian mathematics. The figures and the photographs are particularly well chosen throughout the book, and are particularly so in the first chapter.

Brief mention is made in Chapter Two of the work of Thales, Pythagoras, Hippocrates, Zeno, and Eudoxos, followed by a longer section on Euclid and the Elements. The eighteen-page section on Euclid's construction of the regular pentagon contains Euclid's theorems and proofs of which lead to the desired end.

After some commentary on Archimedes' life, death, and works, Dr Aaboe presents the work of Archimedes concerning trisection of an angle, construction of the regular heptagon, and finding the volume and surface of a sphere ac cording to The Method. (The first three chapters of this book served as a valuable reference in a short honours seminar entitled "Modern Minds in Ancient Bodies" led by Dr Irvin Brune and the reviewer.)

Ptolemy, his table of chords, and its uses made up the episode of Chapter Four. This section will be more fully appreciated by a reader with a knowledge of trigonometry.

The reviewer recalls one error. On page 40, the ratio should be $\large\frac{1}{2}\normalsize$, rather than 2. The problems sprinkled here and there keep the reader on his toes as to the mathematics involved.

Any student of mathematics (from high school on up) not already well versed in the history of mathematics should find this book a welcome addition to his library. Even those who tend to avoid texts on the history of mathematics should enjoy this volume and get acquainted with the history in spite of themselves. Dr Aaboe is to be congratulated for writing this very readable little volume.

2.1. From the Publisher.

Before streets were brightly illuminated at night, astronomy was accessible to everyone and was a matter of great importance: for divination; for setting appropriate dates for planting, harvest, and festivals; for regulating lives. Phenomena in the heavens are still of great importance to many, and much of the lore of astronomy and astrology dates back to the earliest days of civilization. The astronomy of the ancients is thus of interest not only as history but also as the basis for much of what is known or believed about the heavens today. Because phenomena in the heavens are less familiar today than in earlier eras, this book begins with a brief description of what one can see in the sky on dark nights with the naked eye. It then turns to the astronomy of the Babylonians, who named many of our constellations, who are responsible for many of the fundamental insights of early astronomy, and who married mathematics to astronomy to make it an exact science. A chapter on Greek astronomy discusses various models of planetary motion, showing that the cycles and epicycles used by the Greeks have their modern counterparts in the computations used to compute the ephemerides listed in the Nautical Almanac. The book then turns to a detailed discussion of Ptolemy's cosmology, the first to include quantitative models in an integral way. Though the Ptolemaic system is now often dismissed as unsound and inefficient, it is in fact a logically pleasing structure which, for more than a millennium, provided a framework for educated people throughout the Christian and Moslem worlds to think about the universe.

2.2. From the Preface.

More years ago than I can easily count, I published a small book entitled Episodes from the Early History of Mathematics (New Mathematical Library Vol. 13). Here I discussed in some detail a selection of subjects from Babylonian and Greek mathematics that could be fully mastered by someone with a background in high school mathematics.

My own work, particularly since then, however, has largely been concerned with ancient mathematical astronomy, especially Babylonian arithmetical lunar and planetary theories. I had the great good fortune in 1963 to get access to a large collection of unstudied, relevant clay tablets in the British Museum, so quite naturally I began thinking about writing an astronomical companion to the mathematical volume. I was well aware, though, that it would be quite a different sort of enterprise.

When I wrote the former volume, I could take for granted that my readers would have some familiarity with the elements of the subject whose early history was my concern, for nearly everyone has seen some basic arithmetic, algebra, and geometry.

It is, however, far otherwise for astronomy. If students take any astronomy at all, they may learn of the evolution of stars, and even of things that were certainly deemed unknowable in my youth, such as what Mars's surface and the far side of the moon look like in detail, but they remain in most cases woefully ignorant of what you can expect to see when you look at the sky with the naked eye, intelligently and with curiosity. I write this from experience. I first became aware of the state of astronomical enlightenment when, many years ago, out of curiosity I took a vote, yes or no, on a few questions from elementary spherical astronomy in a mathematics class of students selected for their excellence.

I first asked if the sun rises and sets. After the students' Copernican scruples were stilled, and they were sure I knew they knew that it was really the earth's rotation that made it so appear, they voted yes, the sun rises and sets. Next I asked the same question about the moon and, after some mutterings about phases, they voted yes again, as they did for the planets, for, as they reasoned, "planet" means "wandering star." But for the fixed stars their answer was a quick and unanimous no, for they are fixed. [The correct answer is that some rise and set, while others never set, and the rest never rise. The two exceptions are at the equator, where all stars rise and set, and at the poles, where a star is either always above or always below the horizon. Thus, the students would have been nearly right if they had lived at either pole, but they did not.] My final question was whether one can see the moon in the daytime; again they voted no unanimously. This was disturbing, for the moon obliged by being plainly visible, and in broad daylight, through the lecture room's window, which I duly pointed out to them to their astonishment.

It was thus clear that an introduction to naked-eye astronomy would be necessary in a book on early astronomy; I have given one here in Chapter 0. I tried to keep it purely descriptive, uninfluenced by modern knowledge. A case in point is the retrogradation of planets. This phenomenon - that the planets, in their slow eastward motion among the fixed stars, come to a halt, reverse direction, and come to a halt again before continuing their eastward travel, is usually introduced in terms of the Copernican, heliocentric system. The argument involves the changing directions of lines of sight from a moving earth to a moving planet, and it is difficult, as I know only too well, to make students visualise, in this fashion, the phenomenon this explanation is supposed to explain. This is not strange, for planetary retrogression depends on the observer's being on the earth and so is best and easiest accounted for in a geocentric system.

In the astronomical introduction I simply describe the way the sun, moon, stars, and planets appear to behave to anyone who has the time, patience, will, and wit to observe and remember. Later in the book I discuss the various mathematical models, arithmetical and geometrical, that were devised in antiquity to account for the observed behaviour of these bodies, particularly the planets.

I need not, however, introduce the mathematical techniques ancient astronomers used - Babylonian arithmetic and Greek geometry and trigonometry - for I already treated them in my previous little book.

In the following I concentrate on planetary theory and try to avoid the moon as much as possible. Here I am reminded of a story about Ernest Brown (1866-1938), professor of mathematics at Yale University. Brown devoted his life to the study of the moon's motion, and he published his lunar tables in three folio volumes in 1919. Toward the end of his life he was inanely asked what he could say about lunar theory. His answer was heartfelt, "It is very difficult." And so it has always been, ever since its elegant beginning in Mesopotamia some 2500 years ago, and I could find no place for it within the limits I had set for this little book.

Nor shall I consider early attempts at accounting for the planets' motion in latitude. These motions were referred to the mean sun, so their descriptions remained unduly complicated until Kepler saw that the planes of the planetary orbits pass through the true sun (it is in this connection he wrote that Copernicus was unaware of his own riches).

In Chapter 1 I introduce Babylonian arithmetical astronomy, the earliest, and highly successful, attempt at giving a quantitative account of a well-defined class of natural phenomena. I also mention the preserved observational records and hint at how the theories could have been derived from such material. [The elegant English mathematician G H Hardy (1877-1947) was particularly devoted to the theory of numbers - the domain of mathematics dealing with the properties of whole numbers because of its purity in the sense that it found no application outside mathematics. He would have been greatly amused to learn that it was precisely number theory that, in the hands of the Babylonian astronomers, became the first branch of mathematics to be used to make a natural science exact.]

When I turn to Greek geometrical models for planetary behaviour in Chapter 2, I can no longer afford to cite observations or to deal with how the quantitative models' parameters were derived from them. It is not that I deem these matters unimportant - quite the contrary - but such discussions would stray too far from my main purpose, which is to describe the various geometrical models for the planets and show how they work. Further, I am particularly interested both in demonstrating that epicyclic planetary models are not just ad-hoc devices for mimicking how a planet seems to behave, but are good descriptions of how a planet in fact moves relative to the earth, and in identifying their various components with their counterparts in the solar system. Indeed, a well-read medieval astronomer who considered a sun-centred system of planetary orbits could immediately derive its dimensions in astronomical units (one astronomical unit equals the mean distance from the earth to the sun) from the parameters of Ptolemaic epicyclic models, as we know Copernicus did (a note in his hand of this simple calculation is preserved in Uppsala University's library).

A crucial point in the demonstration is the transformation from a heliocentric to a geocentric coordinate system. This ought to be simple enough, but I have found it extraordinarily difficult to make people visualise the same phenomenon in the two systems. One sticking point is to make them realise that even the nearest fixed star is so far away that the directions to it from the sun and from the earth are the same (except for the annual parallax, which is so small that it was not observed until the 19th century).

The fixed-star sphere therefore looks the same whether viewed from the earth or from the sun. For an observer on the earth, the sun will seem to travel in a near-circular orbit, one revolution a year relative to the background of the fixed stars. The planets, in turn, will revolve in their orbits around the moving sun. This arrangement, the "Tychonic system" as Tycho Brahe himself modestly called it, is the one earth-bound astronomers observe in and the one the Naval Observatory uses for compiling the Nautical Almanac (mariners, after all, are on the watery surface of the earth and do not care about what things look like from the sun).

The Copernican and Tychonic arrangements are geometrically equivalent, for either implies the other and yields the same directions and distances from one body to another. However, if you insist that the origin of your coordinate system - the earth for Ptolemy and Brahe, the mean sun for Copernicus (he has the true sun itself travel in a small circle around the mean sun) is "at rest in the centre of the universe," you do not, of course, have dynamical equivalence between the two systems, and the Copernican arrangement in Kepler's version, focused on the true sun, became the basis of Newton's mechanical treatment of the solar system.

In the Almagest Ptolemy does not give his planetary models absolute size: He measures a model's dimensions in units, each or which is one-sixtieth or the deferent's radius, because he only wants them to yield directions to the planets. So far they are perfectly compatible with the Tychonic system and, in fact, if we scaled the Ptolemaic models properly, they would also give us the distances to the planets correctly.

However, in his Planetary Hypotheses Ptolemy constructs his cosmological scheme, the Ptolemaic system, or snugly nested spherical shells, all centred on the earth, each containing the model of one planet, and with no wasted space. Here he commits himself to the dimensions of the structure in terrestrial units, and now agreement with the Tychonic system is no longer possible. I discuss these things in Chapter 3.

Though I originally intended to treat only ancient topics, I could not help including in Chapter 2 a few remarks on later modifications or Ptolemy's models at the hands of medieval Islamic scholars. These revisions were mostly of three kinds: improvements of parameters; much-needed corrections of serious flaws in his lunar theory; and attempts at replacing his philosophically objectionable equant with combinations of philosophically correct uniform circular motions that would work almost as well.

I end Chapter 2 with a few remarks about Copernicus and Brahe. Of Brahe's many achievements I only mention the Tychonic system, for its arrangement is, as said, compatible with the Almagest's planetary models.

In my comments on Copernicus's work, I concentrate on just two aspects. First, I try to make clear precisely which problem was resolved by the heliocentric hypothesis. The general literature is often vague on this point, suggesting, for example, a desire for higher accuracy, or a dislike of epicycles, as a motivation for the new system. Both suggestions are wrong. In fact, the motivation lay in a desire to get rid of the awkward questions raised by the sun's curious role in the Ptolemaic planetary models: For an inner planet the deferent's radius to the epicycle's centre always points toward the mean sun, while for the outer planets the radii from the epicycles' centres to the planets are all parallel and point in the same direction as that from the earth to the mean sun. This strange role was difficult to explain, but it becomes an immediate consequence of placing the mean sun in the middle of the planets' paths, as we shall see. (The Tychonic arrangement has the same virtue.)

Second, I discuss the fine-structure of Copernicus's planetary models. His ideal is clearly to have each planet move uniformly in a circular path - not, alas, around its centre, but around an equant point in analogy to Ptolemy's deferent. However, he has committed himself to the exclusive use of uniform circular motions and manages, by superposition of several of these, to make his planets move uniformly around his ideal equant points exactly, not approximately - while their resulting paths are nearly, but not quite, circular. I shall point out that these arrangements are precisely what we have found in the works of some of his Islamic predecessors.

In Chapter 3 I present, in some detail, Ptolemy's cosmology, the first cosmological scheme to include quantitative models in an integral way. It was long called the Ptolemaic system, even though Ptolemy's authorship of it was established only a few decades ago, when my then-colleague at Yale, Bernard Goldstein, found Ptolemy's own description of it in an Arabic translation of a lost part of the Greek original of his Planetary Hypotheses. Though the Ptolemaic system is now most often talked about in slighting terms, it is a logically pleasing structure that was, after all, the basis for how educated people thought about the universe for nearly a millennium and a half.

In the final chapter I show, from a more modern point of view, why Ptolemy's equant is so efficient. I take this opportunity to sketch how Kepler proceeded in order to find the longitude of a planet that moves according to his laws, and then I go on to analyse how a planet seems to behave when observed from the empty focus of its elliptical orbit, the one not occupied by the sun. It turns out that its angular motion is uniform but for terms involving second and higher powers of the ellipse's eccentricity, so the empty focus plays the role of an equant point and, for small eccentricities, ellipses are nearly circular. The combination of two eccentric circular motions with equants yields a planetary model very close to Ptolemy's.

The result of this analysis is not to be found in the more recent general literature, but it is far from new. In fact, in his Principia Isaac Newton addressed the problem of a planet's angular motion around the empty focus of its orbit and reached nearly the same result as the one I derive in Chapter 4.

In the course of this derivation I transgressed the limits I originally imposed on myself, for I could not help including an integral or two. However, having set the rules myself, I felt free to break them, and a few integrals never hurt anyone.

At one time or another, I have dealt with most of the above, very idiosyncratic selection of topics in courses and seminars at Yale University. Additionally, in March of 1988. I was pleased to be invited to give an Honours course at the University of Pittsburgh: in my lectures there I also presented some of this material. I am grateful to the students in both places who, by their questions, made what I thought and wrote clearer.

Furthermore, I wish to thank my colleagues, Dr John Britton and Professors Bernard Goldstein and Alexander Jones, for reading my manuscript and for helpful criticism and suggestions.

Finally, I must acknowledge my indebtedness and gratitude to Miss Izabela Zbikowska of the Polish Academy of Science and Yale University. Without her constant help during the last three years, this little book, begun so long ago, would still be unfinished.

2.3. Review by: Stephen C McCluskey.
The British Journal for the History of Science 38 (3) (2005), 356-358.

The distinguished scholar and teacher Asger Aaboe has now presented us with a historical and mathematical introduction to early astronomy to complement his Episodes from the Early History of Mathematics (1964). There is little social context in this book; it is history of astronomical ideas that expects the reader to be comfortable with basic mathematics. When, despite his intention to avoid them, a few integrals slip into the last chapter, Aaboe remarks that 'a few integrals never hurt anyone'.

Aaboe begins with an introduction to observational astronomy, in which he provides a full explanation of the principal astronomical phenomena, while deliberately avoiding any explanations of the causes of those phenomena. He then defines the circles, coordinate systems and the various years and months (tropical, sidereal, anomalistic and draconitic) used to record those phenomena quantitatively.

The longest chapter in the book provides a detailed survey of Babylonian astronomy (Aaboe's forte), beginning with a discussion of the cuneiform texts and the difficulties attending their decipherment and interpretation in the nineteenth and twentieth centuries. By way of introducing the content and methods of Babylonian astronomy, he describes the major categories of astronomical texts: omen texts; astronomical diaries with their related goal-year texts; and the mathematical texts, including lunar and planetary ephemerides and the related procedure texts. He notes that the highly precise theoretical schemes found in the ephemerides must be derived from the crude observations found in the astronomical diaries, and that historians have established possible techniques for such derivations.

Discussion of the techniques of planetary ephemerides leads Aaboe to consider the complex roles of period relationships and number theory in Babylonian planetary theory, and in our understanding of it. Although he treats lunar theory in somewhat less detail, he provides the clearest presentation I have ever seen of the relationships among the various factors appearing in the columns of the ephemerides and how they contributed to the accuracy of Babylonian predictions of the time of syzygy and the day of the visible new moon.

The Greek tradition of geometric astronomy is next, from its origins with Eudoxus' hippopede through the development of epicyclic astronomy. Again, Aaboe's strength lies in the clear and uncompromising mathematical elaboration of the geometrical models involved, demonstrating the purely qualitative nature of Eudoxus' model and the transformations that show the equivalence of Ptolemaic and modern astronomy. After Ptolemy he turns to his Arab and Latin interpreters, pointing out the striking mathematical similarity of Copernicus's models to the modified Ptolemaic models of Nasir al-Din al-Tusi, Ibn al-Shatir and Qutb al-Din al-Shirazi.

With the arrival of Copernicus, Brahe and Kepler, Aaboe turns his attention to cosmological considerations of the structure and dimensions of the universe. Here he returns to Ptolemy, this time to the cosmological model of the Planetary Hypotheses. Once again we are presented with a mathematical analysis of the Ptolemaic system and its relations to the underlying observational data, frequently related to Aristotelian cosmological principles.

In the final chapter Aaboe turns to a discussion of the mathematical properties of Keplerian ellipses and the nature of Kepler's solution to the technical problems involved, ending with an analysis of how closely Keplerian motion is approximated by uniform circular motion around the empty focus of the ellipse. Aaboe does not mention that Kepler tried and abandoned such an equant model. His purpose here is to show that Ptolemy 'was on the right track' when he abandoned uniform motion around the centre and introduced the philosophically unacceptable equant. The successes of ancient astronomy in creating mathematical models of the appearances of the heavenly bodies, rather than the achievements of its later progeny, remain the focus of Aaboe's excellent study.

One minor point of confusion for the book's intended audience appears in its Introduction, where azimuth is defined as 'usually counted from the south'. That convention is now almost entirely restricted to the more formal analyses of spherical astronomy, while many readers will already be familiar with the measurement of azimuth from the north, which is the dominant usage in hiking, navigation and even astronomy. Such minor caveats aside, this is an ideal book for those of us who want a clear introduction to early astronomical concepts, and for our students.

2.4. Review by: Benno van Dalen.
Mathematical Reviews MR1933640 (2003i:01003).

This nice little booklet is a companion volume to the author's Episodes from the early history of mathematics (1964). It presents a general introduction to naked-eye astronomy as well as descriptions of the main characteristics of the planetary models that were in use up to the time of Kepler. The five chapters can be read independently.

Chapter 0 explains in a very clear fashion the basic concepts of spherical and planetary astronomy, including, among other things, the three main coordinate systems and the synodic phenomena of the planets, namely first and last visibility, the stationary points, and conjunction and opposition.

Chapter 1 first discusses the various types of Babylonian astronomical sources that are extant on clay tablets from Babylon and Uruk. It then presents examples of the two mathematical systems that were utilised for expressing the times between successive synodic phenomena of the same type, namely ``step functions'' (System A) with various constant values on different parts of the ecliptic and ``zigzag functions'' (System B) that increase and decrease linearly between two limits.

Chapter 2 describes the Greek geocentric, geometrical planetary models consisting of epicycles and eccentres, which were expounded by Ptolemy in his Almagest. The author explains how these correctly represent the heliocentric motions of the planets with respect to the earth and briefly discusses the trigonometric corrections to the linear mean motions that Ptolemy applied. He ends with a short treatment of some Islamic alternative models that sought to maintain the Aristotelian principle of uniform circular motion and that were later adopted by Copernicus.

Chapter 3 deals with cosmological aspects of Ptolemy's planetary models, including cosmical dimensions and the physical arrangement of the planetary orbs. Chapter 4, finally, describes Kepler's new astronomy with general properties of ellipses, the area law, and the Kepler equation. The author points out how Ptolemy and Kepler used different solutions for the same problems in devising their planetary models.

The book under review is a very handy introduction to early astronomy for any reader with some background in mathematics. It contains various beautiful illustrations and the hand-drawn figures are nearly always very clear.

2.5. Review by: John P Britton.
Aestimatio 3 (2006), 106-107.

This slim, elegantly written volume by Asger Aaboe might have been more accurately titled, 'Highlights of Planetary Theory from Babylon to Kepler', since that is in fact its subject. Lunar theory is mentioned in passing, but mainly to explain its absence. The book begins with an introductory description, characterised as 'Chapter 0', of the principal phenomena of naked eye astronomy necessary to understand what follows.

Chapter 1 describes the arithmetical models and methods employed by astronomical scribes from Babylon and Uruk to depict the dates and positions of the planets' main synodic phenomena - appearances, disappearances, stations, and (for outer planets) oppositions. Especially noteworthy, since they are not published elsewhere, are the reconstruction of Jupiter's daily motion in Table 5, and the illustration in Figure 4 of the interrelations of the functions comprising Lunar System A. This is an area of the author's particular expertise; and his account, which emphasises the crucial role of period relations in Babylonian theory, is uncommonly readable as well as authoritative.

In many respects the crux of the book is chapter 2, which surveys the kinematic models depicting planetary motions, from their qualitative origins in the homocentric spheres of Eudoxus, through Ptolemy's first simplified and then detailed quantitative models, to the improvements introduced by Islamic astronomers, and finally to the transformations of these models by Copernicus and Tycho Brahe. This ambitious survey focuses on the geometrical relationships of the several models, and combines clear yet rigorous descriptions with novel and uniquely instructive illustrations of the fine details of Ptolemy's equant models and their modifications by al-Tusi, Qutb al-Din, and Ibn al-Shatir. Of particular note is the author's discussion of efforts to circumvent the philosophically distasteful equant motion. In all, the chapter comprises an admirably clear survey of the essential elements and evolution of kinematic planetary theory.

Two points are emphasised here and subsequently. The first is that Ptolemy's models yield planetary positions - where the planet is if you look for it - as well as any later models until Kepler's. The second is that while Copernicus' transformation of Ptolemy has the singular advantage of establishing the relative distances of the planets, it has no sensible advantage over the Tychonic system, which - as depicted in the splendid frontispiece to Riccioli's Almagestum novum [1651] - was preferred to Copernicus' by contemporary astronomers, perhaps because it is how we actually see the planets from the Earth.

After extended immersion in the mathematical details of kinematic planetary theory, chapter 3 presents a less mathematically demanding description of the fine structure of the Ptolemaic cosmological system of nested spheres, a system much maligned in conventional commentaries, but which, as the author notes, 'prevailed for nearly a millennium and a half in the West, and for longer in the Near East'. The account illustrates the internal consistency of this system with clarity and economy, adding little-known details about the recovery of its textual underpinnings in Ptolemy's Planetary Hypotheses.

In the fourth and last chapter, the author leads the reader gently (only two integrals) through the properties of 'Kepler Motion [viewed from] from Either Focus', to show that, but for second order differences, the angular motion of a planet moving about the Sun in one focus of an ellipse is very nearly uniform about the empty focus. This paves the way for an original answer to the question of whether the vector sum of the eccentricities of Earth and planet can be extended rigorously to a vector sum of eccentricity plus equant, as implied in Ptolemy's models. The answer is presented exquisitely in Figure 15 towards the end of the chapter.

In short, Aaboe's book provides a clear, authoritative, and frequently original introduction to the principal elements of mathematical planetary theory before Newton, which experts will find rewarding and novices accessible. It deserves and will repay a wider audience, despite its unfortunate mispricing by its publisher.