11 quotes
If discovery takes place by the soul who keeps silent, how could the flow of language through the mouth be sufficient to bring to light the essential nature of what has been discovered?
Commentary on Plato’s Timaeus (2008)
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.
The Philosophical and Mathematical Commentaries of Proclus on the First Book of Euclid's Elements Vol. 2 (1789)
This, therefore, is mathematics: she reminds you of the invisible form of the soul; she gives life to her own discoveries; she awakens the mind and purifies the intellect; she brings light to our intrinsic ideas; she abolishes oblivion and ignorance which are ours by birth.
Proofs must vary. . . and be differentiated according to kinds of being concerned, since mathematics is a texture of all these strands and adapts its discourse to the whole range of things.
Those who have written the history of geometry have thus far carried the development of this science. Not much later than these is Euclid, who wrote the 'Elements,' arranged much of Eudoxus' work, completed much of Theaetetus's and brought to irrefragable proof propositions which had been less strictly proved by his predecessors.
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.
It arouses our innate knowledge. ..takes away the forgetfulness and ignorance [of our former existence] that we have from birth,. . . fills everything with divine reason, moves our souls towards Nous,. . . and through the discovery of pure Nous leads us to the blessed life.
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.
The Philosophical and Mathematical Commentaries of Proclus on the First Book of Euclid's Elements Vol. 1 (1788)
But Eudoxus the Cnidian, who was somewhat junior to Leon, and the companion of Plato, first of all rendered the multitude of those theorems which are called universals more abundant; and to three proportions added three others; and things relative to a section, which received their commencement from Plato, he diffused into a richer multitude, employing also resolutions in the prosecution of these.
The Philosophical and Mathematical Commentaries of Proclus on the First Book of Euclid's Elements Vol. 1 (1788)
For the circle is divisible into parts unlike in definition or notion, and so is each of the rectilineal figures; this is in fact the business of the writer of the Elements in his Divisions, where he divides given figures, in one case into like figures, and in another into unlike.
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.
The Philosophical and Mathematical Commentaries of Proclus on the First Book of Euclid's Elements Vol. 1 (1788)