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Proclus' Commentary on
the First Book of Euclid's Elements

Translated by Thomas Taylor

From Taylor's Preface:

  The design of the present work is to bring us acquainted with the nature and end of Mathematics in general, and of Geometry in particular: and in the execution of this design our Author has displayed an uncommon elegance of composition, and a most valuable store of recondite learning.  He is not content with every where unfolding the full, and most accurate meaning of Euclid; but he continually rises in his discourse, and leads us into the depths of the Pythagoric and Platonic philosophy.  We are surprised to find a use in Geometry, which at present it is by no means suspected to afford.  For who would conceive that it is the genuine passage to true theology, and the vestibule of divinity?  This, indeed, is by no means the case when it is studied for lucre, and applied to mechanical purposes; for then the soul is neither elevated nor enlightened, but degraded and filled with material darkness.  Hence these Commentaries are alone valuable to the liberal part of mankind, who look beyond sense for certainty; and who prefer things desirable for their own sakes, before such as minister to the necessities of life.



  Chapter I


 On the Middle Nature of the Mathematical Essence.

    It is necessary that the mathematical essence should neither be separated from the first nor last genera of things, nor from that which obtains a simplicity of essence; but that it should obtain a middle situation between substances destitute of parts, simple, incomposite and indivisible, and such as are subject to partition, and are terminated in manifold compositions and various divisions.  For since that which subsists in its inherent reasons remains perpetually the same, is firm and durable, and cannot be confuted, it evidentaly declares it is superior to the forms existing in matter.  But that power of progression which apprehends, and which besides uses the dimensions of subjects, and prepares different conclusions from different principles, gives it an order inferior to that nature which is allotted an indivisible essence, perfectly constituted in itself.  Hence (as it appears to me) Plato also divides the knowledge of things which are, into first, middle, and last substances. And to indivisible natures, indeed, he attributes an intelligence, which, in a collective manner, and by a certain simple power, divides the objects of intellectual perception; so that being divested of matter, and endued with the greatest purity, it apprehends things themselves, by a certain unifying perception, and excels the other kinds of knowledge.  But to divisible essences, and such as are allotted the lowest nature, and to all sensible beings, he attributes opinion, which obtains an obscure and imperfect truth.  But to middle essences (and such are mathematical forms), and to things inferior to an indivisible and superior to a divisible nature, he attributes cogitation.  For this, indeed, is inferior to intellect, and the supreme science dialectic; but is more perfect than opinion, and more certain and pure.  For it advances by a discursive procession, expands the indivisibility of intellect, and unfolds that which was involved in the unity of intellectual apprehension: but it collects things which are divided, and brings them back to mind.  Hence, as knowledges differ among themselves, so the objects of knowledge are distinguished by nature.  So that intelligible essences having a uniform subsistence, evidently excel all others.  But sensibles are entirely excelled by primary essences: and mathematical natures, and whatever falls under cogitation, are allotted a middle order: for they are excelled by the division of intelligibles; but because destitute of matter, they are superior to sensible natures; and by a certain simple power, they are excelled by the first; but by a certain reason are more exalted than the last.  Hence they possess notions of an intellectual essence, which are more manifest than sensibles, but which are, at the same time, only the images of an intellectual nature; and they imitate divisibly the indivisible, and, in a multiform manner, the uniform exemplars of things.  And, that I may sum up the whole in a few words, they are placed in the vestibules or entrances of primary forms, and disclose their indivisible and prolific subsistence collected into one, but they do not yet excel the division and composition of reasons, and an essence accommodated to the obscurity of images; nor are they capable of passing beyond the various notions of the soul, endued with a discursive power, and of adhering to intellections perfectly simple, and purified from all material imperfection.  After this manner then, is the middle nature of mathematical genera and forms to be understood; as filling up the medium between essences entirely indivisible, and such as are divisible about matter.


 Chapter II

  Concerning the common Principles of Beings, and of the Mathematical Essence, bound and infinite.

  But it is necessary that, considering the principles of the whole mathematical essence, we should return to those general principles, which pervade through and produce all things from themselves, I mean bound and infinite.  For from these two after that cause of one, which can neither be explained, nor entirely comprehended, every other thing, as well as the nature of the mathematical disciplines, is constituted.  In the former, indeed, producing all things collectively and separately; but in these proceeding in a convenient measure, and receiving a progression in a becoming order; and in some, subsisting among primary, but in others among middle, and in others again among posterior natures.  For intelligible genera, by their simplicity of power, are the first participants of bound and infinite: because, on account of their union and identity, and their firm and stable existence, they are perfected by bound: but on account of their division into multitude, their copious power of generation, and their divine diversity and progression, they obtain the nature of infinite.  But mathematical genera originate, indeed, from bound and infinite, yet not from primary, intelligible, and occult principles only; but also from those principles which proceed from the first to a secondary order, and which are sufficient to produce the middle ornaments of beings, and the variety which is alternately found in their natures.  Hence, in these also, the reasons and proportions advance to infinity, but are restrained and confined by that which is the cause of bound.  For number rising from the retreats of unity, receives an incessant increase, but that which is received as it stops in its progression, is always finite.  Magnitude also suffers an infinite division, yet all the parts which are divided are bounded, and the particles of the whole exist finite in energy.  So that without the being of infinity, all magnitudes would be commensurable, and no one would be found but what might either be explained by words, or comprehended by reason (in which indeed geometrical subjects appear to differ from such as are arithmetical;) and numbers would be very little able to evince the prolific power of unity, and all the multiplex and super-particular proportions which they contain.  For every number changes its proportion, looking back upon, and diligently enquiring after unity, and a reason prior to itself.  Butbound being taken away, the commensurability and communication of reasons, and one and the same perpetual essence of forms, together with equality, and whatever regards a better co-ordination, would never appear in mathematical anticipations: nor would there be any science of these; nor any firm and certain comprehensions.  Hence then, as all other genera of beings require these two principles, so likewise the mathematical essences.  But such things as are last in the order of beings, which subsist in matter, and are formed by the plastic band of nature, are manifestly seen to enjoy these two principles essentially.  Infinite as the subject seat of their forms; but bound as that which invests them with reasons, figures, and forms.  And hence it is manifest that mathematical essences have the same pre-existent principles with all the other genera of beings.


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