Proclus Quotes (412-485)

“Arithmetic is indeed more accurate than Geometry, for its Principles respect Simplicity. Unity implies no Position, a Point does, and a Point requiring Position is the Principle of Geometry, Unity of Arithmetic.”

“Eratosthenes declares that it is no longer necessary to inquire as to the cause of the overflow of the Nile, since we know definitely that men have come to the sources of the Nile and have observed the rains there.”

“Pythagoras transformed the study of geometry into a liberal education, examining the principles of the science from the beginning and probing the theorems in an immaterial and intellectual manner: he it was who discovered the theory of irrationals [or 'proportions'] and the construction of the cosmic figures.”

“[Thales] first went to Egypt and hence introduced this study [geometry] into Greece. He discovered many propositions himself, and instructed his successors in the principles underlying many others, his method of attack being in some cases more general [i.e. more theoretical or scientific], in others more empirical [...more in the nature of simple inspection or observation].”

“He is verbose and dull, but luckily he has preserved for us quotations from other and better authorities.”

“What Science can be more accurate than Geometry? What Science can afford Principles more evident, more certain, yea I will add, more simple than Geometrical Axioms, or exercises a more strictly accurate Logic in drawing its Conclusions? But Aristotle and Proclus affirm that Unity (they had more rightly said Numbers) the Principle of Arithmetic, is more simple than a Point which is the Principle of Geometry, or rather of Magnitude. Because a Point implies Position, but Unity does not. A Point, says Aristotle, and Unity are not to be divided, as Quantity: Unity requires no Position, a Point does. But this Comparison of a Point in Geometry with Unity in Arithmetic is of all the most unsufferable, and derives the worst Consequences upon Mathematical Learning.”

“It is well known that the commentary of Proclus on Eucl. Book I is one of the two main sources of information as to the history of Greek geometry which we possess, the other being the Collection of Pappus.”

“The Porisms. Our only source of information about the nature and contents of the Porisms is Pappus. ...With Pappus's account of Porisms must be compared the passages of Proclus on the same subject. ...Proclus's definition... agees well enough with the first, the 'older', definition of Pappus. A porism occupies a place between a theorem and a problem; it deals with something already existing, as a theorem does, but has to find it (e.g. the centre of a circle) and, as a certain operation is therefore necessary, it partakes to that extent of the nature of a problem, which requires us to construct or produce something not previously existing. ...all the positive information which we have about the nature of a porism and the contents of Euclid's Porisms ...is obscure and leaves great scope for speculation and controversy; naturally, therefore, the problem of restoring the Porisms has had a great fascination for distinguished mathematicians ever since the revival of learning. But it has proved beyond them all.”

“Of his surviving works, the Commentary, which treats Book I of Euclid's Elements, is the most valuable. Proclus apparently intended to discuss more of the Elements, but there is no evidence that he ever did so.”

“According to the account of Proclus (Book II. c. 4 ), Pythagoras was the first who gave to Geometry the form of a deductive science, by shewing the connexion of the geometrical truths then known, and their dependence on certain first principles.”

“It is also problematical whether Proclus could have ever written such a clear, sober, and concise piece of work. His predominant interest in any subject, even mathematics, is always the epistemological aspect of it. He must ever inquire into the how and the why of the knowledge relevant to that subject, and its kind or kinds; and such speculation is apt with him to intrude into the discussion of even a definition or proposition. Moreover Proclus can never forego theologizing in the Pythagorean vein. Mathematical forms are for him but veils concealing from the vulgar gaze divine things. Thus right angles are symbols of virtue, or images of perfection and invariable energy, of limitation, intellectual finitude, and the like, and are ascribed to the Gods which proceed into the universe as the authors of the invariable providence of inferiors, whereas acute and obtuse angles are symbols of vice, or images of unceasing progression, division, partition, and infinity, and are ascribed to the Gods who give progression, motion, and a variety of powers. This epistemological interest and this tendency to symbolism are entirely lacking in our commentary; and another trait peculiar to Proclus is also absent, namely, his inordinate pedantry, his fondness of quoting all kinds of opinions from all sorts of ancient thinkers and of citing these by name with pedagogical finicalness. Obviously the author of our commentary had a philosophical turn of mind, but he was a temperate thinker compared with Proclus.”

“A full history of Greek geometry and astronomy during this period written by Eudemus, a pupil of Aristotle, has been lost. It was well known to Proclus, who, in his commentaries on Euclid, gives a brief account of it. This abstract constitutes our most reliable information. We shall quote it frequently under the name of Eudemian Summary.”