Review: A Manual of Greek Mathematics

Review: A Manual of Greek Mathematics

A Manual of Greek Mathematics by Thomas L. Heath

My rating: 4 of 5 stars

In the case of mathematics, it is the Greek contribution which it is most essential to know, for it was the Greeks who first made mathematics a science.


As a supplement to my interest in the history of science, I figured that I ought to take a look into the history of mathematics, since the two are quite intimately related. This naturally led me to the Greeks and to Sir Thomas L. Heath, who remains the most noteworthy translator, divulgator, and commentator in English eighty years after his death. This book is likely the best single volume you can get on the subject, as it covers all of the major mathematicians in some detail while giving a complete overview.

It is also reasonably accessible (“reasonably” being the operative word). Certainly it is no work of popular math in the modern sense; it is not pleasure reading, and Heath assumes a certain amount of knowledge on the reader’s part. A thorough knowledge of algebra and geometry is assumed, and a few words in ancient Greek are not translated. What is more, large sections of the book are essentially extended summaries and explications of Greek treatises, which makes them almost impossible to read without the original text alongside. Personally I would certainly have appreciated more spoon-feeding, as it was quite difficult for me to prevent my eyes from glazing over.

The book is divided primarily by subject-matter and secondarily by chronology. Heath introduces us to notation, fractions, and techniques of calculation, and then on to arithmetic. Geometry, of course, dominates the book, as it was the primarily form of Greek mathematical thought. Heath summarizes the contributions to geometry by Pythagoras and his followers, and the scattered mathematicians we know of in the years between Thales and Euclid. Once Euclid appears, he writes his famous Elements, which encapsulates the entire subject and which rendered many previous works obsolete. After Euclid we come to the divine Archimedes and the great Apollonius, who put the capstone on the tradition. Ptolemy (among others) made great advances in trigonometry, while Diophantus made strides in algebra (as well as inspired Fermat).

Heath’s account of these mathematicians is largely internal, meaning that he is focused on the growth of their ideas rather than anything external to the science. Reading this convinced me—as if further evidence was needed—that I do not have the moral fiber or intellectual temper to appreciate mathematics. Heath writes admiringly of the works of Euclid and Archimedes, finding them not only brilliant but beautiful. While I can normally appreciate the brilliance, the beauty normally escapes me. Ratios, volumes, lines, and equations simply do not make my heart beat.

Indeed, the questions that I find most fascinating are those that are hardly touched upon in this book. Most important, perhaps, is this: What aspect of a culture or a society is conducive to the development of pure mathematics? Though claims of Greek specialness or superiority seem antiqued at best nowadays, it is true that the Greeks made outstanding contributions to science and math; while the Roman contribution to those fields—at least on the theoretical side—is close to nil. The mathematics of Ancient Egypt amount to techniques for practical calculations. Admittedly, as Otto Neugebauer wrote about in his Exact Sciences of Antiquity, the Babylonians had quite advanced mathematics, allowing them to solve complex polynomials; they also had impressive tabulations of the heavenly motions.

Even so, it was the Greeks who created science and math in the modern sense, by focusing on generality. That is, rather than collect data or develop techniques for specific problems, the Greeks were intent on proving theorems that would hold in every case. This also characterizes their philosophy and science: a rigorous search after an absolute truth. This cultural orientation towards the truth in the most general, absolute form seems quite historically special. It arose in one fairly limited area, and lasted for only a few centuries. Most striking is the Greek disdain of the practical—something that runs from Pythagoras, through Plato, to Archimedes.

Of the top of my head, here are some possible factors for this cultural development. The Greek economy was based on slavery, so that citizens often could afford to disdain the practical. What is more, the Greek political model was based on the city-state—a small, close-knit community with limited expansionist aims and thus with limited need for great infrastructure or novel weapons. The relative lack of economic, political, or military pressure perhaps freed intellectuals to pursue wholly theoretical projects, with standards that arose from pure logic rather than necessity. Maybe this seems plausible; but I am sure many other societies fit this description, not just the Greeks. The development of culture is something that we do not fully understand, to say the least.

This has taken me quite far afield. In sum, this book is an excellent place to start—either by itself, or as a companion to the original Greek works—if you are interested in learning something about this astounding intellectual tradition. That the Greeks could get so far using geometry alone—that is, without variables or equations—is a testament to human genius and persistence.



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Review: The Oresteia

Review: The Oresteia

The Oresteia: Agamemnon, The Libation Bearers, The Eumenides by Aeschylus

My rating: 5 of 5 stars

The Greeks had an intoxicating culture, or at least it seems to us. All of the iniquities and superstitions of the ancient people have been buried or lost, leaving only the perfect skeletons of buildings and the greatest of their literary productions. As a result, they strike us as a race of superpeople. This trilogy certainly furthers this impression, for it is a perfect poetic representation of the birth of justice and ethics out of the primordial law of retaliation.

The most basic ethical principal is loyalty. We are born into a family, establish reciprocal relationships with friends, become a contributing member of a mutually supporting group, and so naturally feel bound to treat this network of people with the proper respect and kindness. But loyalty has several problems. First, one’s family, friends, and group are largely determined by chance—and who is to say that our family and friends are the most worthy? Second, loyalty does not extend outside a very limited group, and so does not preclude the horrid treatment of others. And, as the Greek plays show us, the bounds of loyalty can sometimes cross, putting us in a situation where we must be disloyal to at least one person.

This is the essential problem of Antigone, where the titular character must choose between loyalty to her city or to her dead brother, who betrayed the state. This is also the problem faced by Orestes, who must choose between avenging his father and treating his mother properly. In Sophocles’ play, the problem proves intractable, leading to yet another string of deaths. But Aeschylus shows that by submitting the bonds of loyalty to a higher, impartial court that we can resolve the contradictions and put an end to the endless series of mutual retaliations that loyalty can give rise to.

The rise of judicial procedures, and of concepts of ethics that extend beyond loyalty to fairness, was a crucial step in the rise of complex societies. Aeschylus has given us an immortal dramatization of this epochal step. But, of course, this play is more than a philosophical or historical exercise. It is a work of high drama and poetry, worthy to stand at the first ranks of literature for its aesthetic merit alone. The Greeks continue to enchant.



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