Proclus

"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."

"To a given right line to apply a parallelogram equal to a given triangle in an angle which is equal to a given right lined angle. According to the Familiars of Eudemus, the inventions respecting the application, excess, and defect of spaces, is ancient and belongs to the Pythagoric muse. But junior mathematicians receiving names from these, transferred them to the lines which are called conic, because one of these they denominate a parabola, but the other an hyperbola, and the third an ellipsis; since, indeed these ancient and divine men, in the plane description of spaces on a terminated right line, regarded the things indicated by these appellations. For when a right line being proposed, you adapt a given space to the whole right line, then that space is said to be applied, but when you make the longitude of the space greater than that of the right line, then the space is said to exceed; but when less, so that some part of the right line is external to the described space, then the space is said to be deficient. And after this manner, Euclid, in the sixth book, mentions both excess and defect. But in the present problem he requires application..."

"But after these, Pythagoras changed that philosophy, which is conversant about geometry itself, into the form of a liberal doctrine, considering its principles in a more exalted manner; and investigating its theorems immaterially and intellectually; who likewise invented a treatise of such things as cannot be explained in geometry, and discovered the constitution of the mundane figures."

"If two right lines cut one another, they will form the angles at the vertex equal. ... This... is what the present theorem evinces, that when two right lines mutually cut each other, the vertical angles are equal. And it was first invented according to Eudemus by Thales..."

"For this, to draw a right line from every point, to every point, follows the definition, which says, that a line is the flux of a point, and a right line an indeclinable and inflexible flow."

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

"It is told that those who first brought out the irrationals from concealment into the open perished in shipwreck, to a man. For the unutterable and the formless must needs be concealed. And those who uncovered and touched this image of life were instantaneously destroyed and shall remain forever exposed to the play of the eternal waves."

"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."

"The term axiom was used by Proclus, but not by Euclid. He speaks, instead, of common notions—common either to all men or to all sciences."

"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 scholiast on Euclid, thought to be Proclus, says that Eudoxus practically invented the whole of Euclids fifth book."

"The mathematician speculates the causes of a certain sensible effect, without considering its actual existence; for the contemplation of universals excludes the knowledge of particulars; and he whose intellectual eye is fixed on that which is general and comprehensive, will think but little of that which is sensible and singular."