Bryn Mawr Classical Review 2002.09.34
Michael N. Fried, Sabetai Unguru, Apollonius of Perga's Conica: Text, Context, Subtext. Mnemosyne Supplement no. 222. Leiden: Brill, 2001. Pp. xii, 499. ISBN 9004119779.
Reviewed by Reviel Netz, Stanford
Word count: 3042 words
This book brings to completion the longstanding research project of the second author  delineating the geometrical character of Greek mathematics and showing that a historian of mathematics ought, first and foremost, to be a historian. It is also the first booklength publication by the first author, giving in detail many of his new interpretations of Apollonius' Conics. For both authors this book should be considered a clear success. It effectively makes the case for a geometrical interpretation of Greek mathematics  and will be considered the standard work on Apollonius.
The two leading concerns of the book  with historiography, and with Apollonius  are visible in its structure. The First Chapter as well as the final Chapter Nine discuss the major historiographical question: does Greek mathematics in its original form differ essentially from its transcription in algebraic expressions? The remaining chapters 28 form a systematic, though not exhaustive, survey of the structure and contents of the Conics. (Books IIII are studied in Chapter Two; Chapters Three to Six deal with Books IV to VII, respectively; Chapters Seven and Eight each deal with overall features of Apollonius' Conics).
The two leading concerns also form a natural whole. The historiographical question was first forcefully riased by Unguru in his now classic article from 1975.^{}1 The central notion in this debate is Geometrical Algebra, a concept first developed by Zeuthen in 1886.^{}2 Zeuthen suggested that much of Greek geometry makes sense not as geometry as such but as a geometrical way of presenting algebraical relations  'geometrical algebra'. Unguru claimed this was false. Zeuthen's main example was, as is obvious from his very title, Apollonius' Conics; thus it was incumbent upon Unguru to offer a study of this ancient work, showing how it could make sense without any algebra assumed as its background. Now this duty is discharged.
Still an interesting tension runs, I believe, between the general and particular aims of the book. This is worth noticing briefly as, indeed, the tension between the general and the particular is at the heart of the historiographical question addressed by the book.
What Unguru had criticized, first and foremost, was the view of mathematics as an ahistorical monolith. To a reader such as Zeuthen, it was of major importance that in Greek mathematical texts one could find statements equivalent to certain algebraic equations. Consider e.g. Euclid's Elements II.5, stating that 'If a straight line is cut into equal and unequal <segments>, the rectangle contained by the unequal segments of the whole, with the square on the <line> between the cuts, is equal to the square on the half.' This contains a truth which is equivalent, in some obvious sense, to the modern equation (a+b)(ab)+b^{2}=a^{2}. Thus the historian of mathematics was allowed, according to Zeuthen, to say that the equation (a+b)(ab)+b^{2}=a^{2} was known to Euclid. In a sense, Zeuthen was clearly right, and, as a matter of logic, histories of mathematics can be written where all results are reduced to a canonical form such as that of symbolic algebra. Many such histories have in fact been written but they seem to be historically inadequate for two reasons. First, in such a history, we can no longer see the essential originality of the first author in presenting a truth such as that contained in Euclid's Elements II.5 in the form (a+b)(ab)+b^{2}=a^{2}. Second, in such a history, we clearly lose something about Euclid  that he thought about lines and rectangles, not about unknowns and equations. The view of history as a monolith loses out on the real nature of change (what change can there be in a monolith?) as well as on the real nature of individual works (what individuality can be there in a monolith?).
To criticize Zeuthen is no easy thing  which brings us to the tension inherent in the project initiated by Unguru (1975). The fundamental argument made by the historicists is that it is always necessay, in history, to look at the individual characteristics of the object at hand: to transform different objects according to their mathematical equivalences is an act not of history, but of mathematics. However, it is also obvious that this, in itself, is not going to convince the ahistoricists. After all, merely to say that Euclid's formulations were not those of a modern mathematician would tell us very little, in itself. Few modern mathematicians write in Classical Greek, either: but we do not consider this linguistic difference of any significance for the history of mathematics. Some differences matter, and some not; some go to the essence, and some not. So we are back at the world of essences: we must look for such divisions in form that touch on the essence of a mathematical argument. And so the historian is almost inevitably led to something such as a study of the character of Greek mathematics. Instead of purist historical individualism, where each text is to be read on its own merits, we are lead back to a more philosophical, somewhat ahistorical exercise in defining something such as 'the Greek approach to mathematics'.
Here, then, is the resulting tension. One option of historicizing Apollonus' Conics would be to see this work as a unique document motivated by its own unique interests or, even more (especially given the nature of ancient writing) as a collection of unique documents (namely, each of the books) motivated, each, by their own unique interests. Another option would be to see this work as one specimen among many of 'The Greek way of doing mathematics'. Fried's and Unguru's book cautiously treads the line between the two options.
A major concern of the authors is to show the unity of the Conics as informed by a single project. However, their instinct is to historicize, to find the specific. Thus the bookbased chapters (Chapters 36 on Apollonius' Conics IVVIII)  which are in a real way the main content of the book  tend away from the picture of the Conics as a unified whole. Each of the four books discussed in detail is shown to have its own defining, sometimes surprising characteristics: the role of the opposite sections in Book IV; the inspiration of neusis problems in Book V; the question of the 'shape' of a conic section in Book VI; a line serving as an analogic 'image' of other line (the homologue) in Book VII. Such specific characteristics  over and above the different subject matter  make each book a unique work, and it is clear that Apollonius had put effort precisely into giving each of these books its own individuality. Apollonius' books do not spill over to each other the books are the result not of mere division into papyrus rolls, but of their being genuine separate entities. Fried and Unguru very effectively display the individuality of each of these four books but they keep wishing to deny this individuality and to stress, instead, the continuities between the books. Thus the role of the opposite sections in Book IV is cited as evidence that the double conical surface (a special tool introduced by Apollonius in Book I to define conic sections) is still effective in the later books. The inspiration of neusis problems in Book V, is seen to underline the character of the book as based on problems of fitting lines in space, making it continuous with the preceding books (independently argued to have similar interests). Similarly, the interest in the 'shape' of conic sections in Book VI is once again seen in the light of what it says for the Conics as a study of spatial objects. Finally, the homologue  a very special phenomenon very effectively marking Book VII off the rest of the books  is interpreted by Fried and Unguru mainly in the context of Apollonius' overall use of mathematical analogy. Finally, all such strands  the continuous reference to the double conic surface, the insistence on fitting lines in space, treating conic sections as spatial objects, and the use of analogy  all point to the geometrical character of the Conics. The double conic surface is after all a spatial object, and it is less natural to describe it as an algebraic equation. Fitting lines in space and dealing with spatial objects is clearly the business of geometry. And as for analogy as discussed by Fried and Unguru, this is once again spatial in character: it involves looking at shapes (primarily, conic sections) through the spatial metaphor of some other shape (primarily, circles). I emphasize that the above brief description does no justice to a discussion of the highest quality, extraordinarily subtle and attentive to mathematical detail. (It should be said, however, that the authors pay no attention to the manuscript evidence, and generally speaking raise and answer such questions as can be discussed with little philological detail.)
In their basic argument, the authors are, I believe, correct, so that this book does indeed offer an account of Apollonius' Conics, showing it as a piece of geometry rather than geometrical algebra. My one, somewhat unfair concern is that, showing that Apollonius' Conics is a piece of geometry became, for Fried and Unguru, more important than simply giving an account of Apollonius' Conics. Apollonius, after all, did not write his Conics to refute Zeuthen. He was indeed a geometer first and foremost and so one can detect traces of his geometrical character everywhere in his books. But he wrote his various books to achieve, each, a separate goal. Fried and Unguru come close to offering a textually close hermeneutic analysis of the manystranded project Apollonius was actually engaged in, but, throughout, they foreground not this positive account but the more limited and negative project of showing what Apollonius did not do.
I am not sure that there was a single goal unifying Apollonius' Conics. Naturally, then, I am even more skeptical as to whether there was a single goal unifying Greek mathematics as a whole. Fried and Unguru, largely following Jacob Klein, seem to suggest there was one.^{}3 It might perhaps be rephrased as "the study of the qualitative properties of 'natural' spatial objects" (where by 'natural' we mean something such as those geometrical objects that arise straightforwardly by thinking about obvious objects such as lines and circles in their possible combinations, movements, etc.  this is obviously a wider concept than just 'ruler and compass constructions'). This would be a single goal where the entirety of Greek mathematics can be seen as a series of contributions produced to this unique aim, each author making progress with different objects and different properties. The entire discipline is an exercise in the cartography of a given mathematical realm, the various authors distinguishing themselves by mapping different parts of the realm. Apollonius, then, appears as a supreme mapmaker producing a remarkable survey of the qualitative properties of conic sections.
This then is one possible understanding of the practice of Greek mathematicians. Following Knorr, we can outline an alternative account where Greek mathematicians are seen to be interested mainly in local problemsolving.^{}4 Thus the goal is to offer a new, surprising way of solving some well known problems (famous in this regard are the three classical problems: finding four lines in continuous proportion, trisecting an angle, and squaring the circle), or indeed to invent new problems, most spectacularly in Archimedes' project of extending the squaring of the circle into an entire range of curvilinear objects he had rectified. The main interest then is not in some allencompassing map of the world of geometrical objects but, almost the contrary, in isolated vignettes of this or that object. The cognitive goal is not understanding but, almost the contrary, surprise.
This then is a possible opposition (of course, somewhat exaggerated) between two interpretations of Greek mathematics, the Klein interpretation and the Knorr interpretation. Both are historical in character and call for a contextual understanding of mathematical texts. The Klein interpretation sees a unity to the Greek mathematical enterprise as a whole; the Knorr interpretation sees a single direction to the overall enterprise but is otherwise much more focused on the details of separate works taken on their own.
The two interpretations lead to different accounts of the conic sections themselves. According to the Klein interpretation, the conic sections are primarily spatial bodies created from the cut of a cone, and their study is always focused on this question: what happens when you take a cone and pass a plane through it? This is a question concerning the mapping of the qualitative features of natural geometrical objects. According to the Knorr interpretation, a different role is seen for conic sections. Among the properties of conic sections are some remarkable proportions, typically involving both squares and lines. Thus conic sections may be used in problem solving. By allowing a conic section to pass through given points according to given parameters, one can force desired proportions between lines and squares, e.g. so as to transform a given ratio involving squares into a ratio involving lines, etc. Thus the conic sections are helpful in geometrical manipulation inside individual problems. Seen from this perspective, the main interest in developing a theory of conic sections is in widening the set of useful proportions known to hold in conic sections so as to have more tools available for the solution of problems.
Note that if we follow the Knorr interpretation and think of conic sections primarily not as spatial, geometrical objects, but as arenas for proportion, we have reached an understanding of them which is indeed not quite so different from that of geometrical algebra. Of course, the proportions themselves have a strict geometrical meaning, but the conic section itself becomes not unlike a locus specifying a quantitative property, the hallmark of the algebraical interpretation of the conic sections.
Which of the two interpretations fits Apollonius' work? This question translates into a very specific question which, famously, Apollonius makes very difficult to answer. This is the question of the meaning of Apollonius' definition of the conic sections. In fact, Apollonius does not clearly define the conic sections but, instead, allows them to emerge out of the mathematical discussion itself. In Book I, propositions 1113, Apollonius introduces the conic sections by showing, for each, that if a cone is cut in a certain way by a plane, then the resulting curve has a certain fundamental proportion property; having shown that this property holds, Apollonius then says that the result is a parabola, ellipse, hyperbola. In other words, Apollonius has each conic section characterized twice, both by the protasis of the conditional (a cone is cut in a certain way by a plane), as well as by the apodosis of the conditional (the resulting curve has a certain fundamental proportion property). The protasis fits the conic sections as understood in the Klein interpretation, the apodosis fits the conic sections as understood in the Knorr interpretation.
A central theme of Fried and Unguru's book is that, ultimately, the definition is wholly based on the protasis: it is purely the geometrical cut that defines the conic section, and the fundamental proportion property is a mere epiphenomenona that happens to arise, coincidentally, at the proposition where Apollonius chose to introduce the conic sections. Fried and Unguru bring many arguments to this thesis, which is indeed plausible. If we were to press Apollonius to answer directly the question just what a conic section was, what would his reply be? A likely answer could well be 'a line produced by cutting a cone'. Yet it must also be significant that we would have to press Apollonius to elicit this reply. Apparently he did not care so much about this question, just what a conic section was. Knorr might say that Apollonius was busy studying not the ontology of conic sections, but their geometry. The evidence of I.1113, finally, goes both ways: perhaps it could be made to show that Apollonius upheld a geometrical ontology of the conic section; perhaps it shows that Apollonius was indifferent to ontology. Once again, then, I find that Fried and Unguru may be right about the facts but may be wrong about the emphasis. A geometrical ontology can be found in Apollonius but only at the price of forcing upon him an ontological interest he did not share.
Fried and Unguru foreground the ontological question of the nature of the geometrical object, which may have been much more at the background for Apollonius himself. Readers of the book may do best, perhaps, to concentrate on what Fried and Unguru leave to the background of their argument: the detailed texture of Apollonius' thought in the individual books. I wish to bring attention in particular to a theme running through the book: the role of analogy in mathematical thinking. Throughout, Fried and Unguru point out how the circle served for Apollonius as a model on the basis of which, by analogy, he had developed his studies of the conic sections. For Fried and Unguru this is yet another argument for the geometrical character of the conic sections but I believe this discussion of analogical thinking in mathematics is of great importance in its own terms. The role of nondeductive thinking in mathematics is yet very little understood, and there is especially a lack in historical studies showing such nondeductive thinking in action. In Fried and Unguru's book, such an extended study is to be found. For this reason alone, it deserves a wide readership among historians and philosophers of mathematics.
One could even suggest the following. I describe, in Fried and Unguru's book, a tension between attention to the particular of each book, and the quest for a general theme unifying the book. Ascribing the tension to Fried and Unguru makes it appear as a criticism. But what if the tension can be found in Apollonius himself? This seems a plausible suggestion, that when setting out to write the Conics, Apollonius had aimed to produce a treatise that was, simultaneously, inspired by a single spirit and distinctive in each of its constituent books. He had largely succeeded, and so, inspired by him, did Fried and Unguru. Their book, finally, is a worthy homage to one of the most remarkable and difficult masterpieces of western mathematics.
Notes:
1. Unguru, S. 1975. "On the Need to Rewrite the History of Greek Mathematics", (Archive for the History of Exact Sciences 15) 67114. 2. Zeuthen, H.G. 1886. Die Lehere von den Kegelschnitten im Alterum. Kopenhagen. 3. Klein, J. 19346/1968. Greek Mathematical Thought and the Origins of Algebra. Cambridge MA. 4. Knorr, W.R. 1986. The Ancient Tradition of Geometric Problems. Boston.
