The Beginnings of Greek Mathematics
When this book was first published, more than five years ago, I added an appendix on How the Pythagoreans discovered Proposition 11.5 of the 'Elements'. I hoped that this appendix, although different in some ways from the rest of the book, would serve to illustrate the kind of research which needs to be undertaken, if we are to acquire a new understanding of the historical development of Greek mathematics. It should perhaps be mentioned that this book is not intended to be an introduction to Greek mathematics for the general reader; its aim is to bring the problems associated with the early history of deductive science to the attention of classical scholars, and historians and philos ophers of science. I should like to conclude by thanking my translator, Mr. A. M. Ungar, who worked hard to produce something more than a mechanical translation. Much of his work was carried out during the year which I spent at Stanford as a fellow of the Center for Advanced Study in the Behavioral Sciences. This enabled me to supervise the work of transla tion as it progressed. I am happy to express my gratitude to the Center for providing me with this opportunity. Arpad Szabo NOTE ON REFERENCES The following books are frequently referred to in the notes. Unless otherwise stated, the editions are those given below. Burkert, W. Weisheit und Wissensclzaft, Studien zu Pythagoras, Philo laos und Platon, Nuremberg 1962.
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algebra ancient Ao'yo Archytas argument Aristotle arithmetic assertion axioms Becker Book VII canon commensurable concept conjecture consonances construction definition denote dialectic diastema didarrjfia discovery discussed dvdXoyov dynamis Eleatic Elements end points Euclid Euclid's Elements example existence explain expression fact fifth fourth Furthermore geometrical geometrical algebra Greek mathematics Hence Hippocrates of Chios historical hypothesis indirect proof investigation irrationals kind later length line segments linear incommensurability logoi magnitudes mathe mathematical term mathematicians matics measure mentioned monochord musical interval numerical ratios octave Oenopides original Parmenides passage Plato postulates pre-Euclidean principles problem Proclus propositions proved Pythagoreans question rectangle refer section of string seems sense side and diagonal similar plane numbers Socrates square numbers straight line successive subtraction T. L. Heath Tannery Theaetetus Theodorus theorem theory of music theory of proportions tion translation triangle unit van der Waerden verb Waerden word Zeno Zeno's