Again, the god who is worshipped by the Adrotteni, did not less

conspicuously evince his great familiarity with this friend of the

gods; for, more than once, the god benevolently presented himself to

the sight of Proclus, frequenting his temple. And when he was doubtful,

and wished to know what god or gods resided and were worshipped in

that place, because the inhabitants were of different opinions in

this affair, some believing that the temple of Esculapius was there,

persuaded of this by various signs, as voices are certainly said to

be heard in that place, and a table is consecrated to the gods, and

oracles are given concerning the recovery of health, and those who

approach are miraculously delivered from the most grievous dangers

of life. Others, on the contrary, think that the Dioscuri reside in

that place, because there have been those who saw in the way tending

towards Adrotta, two young men of the most beautiful form, and riding

with great celerity on horseback; at the same time declaring, that

they hastened to the temple. They add besides, that the countenance of

these was indeed human, but that they immediately gave evidence of a

more divine presence; for when the men were arrived at the temple, the

above mentioned youths appeared to them, making no enquiry concerning

the affair, and occupied in the sacred concerns; but presently after,

withdrew themselves from their sight. As the philosopher, therefore,

was in doubt concerning these divinities, and did not discredit the

relations, having requested the gods who inhabited that place, that

they would condescend to manifest who they were; a god appeared to him

in a dream, and clearly spoke to him as follows: _What! Hast thou not

heard of Jamblichus teaching concerning, and celebrating those two,

Machaon and Podalirius?_ And besides this, the god condescended to

afford so great an instance of benevolence to this blessed man, that

he stood after the manner of those who bestow encomiums on others in

the theatres, and with a clear voice, and composed habit, extending his

right hand, did not hesitate to exclaim, (that I may relate the words

of the god,) _Proclus the ornament of the city_. And what could

possibly be a greater testimony of this blessed man’s friendship with

divinity? But he, indeed, in consequence of a most remarkable sympathy,

by which he was united with the gods, could never refrain from tears

when he mentioned this affair to us, and related the divine encomium of

himself.

Indeed, if I were willing to pursue every particular in like manner,

and to relate his familiarity with the Hermetic Pan, together with

the great benevolence and manifold assistance which that divinity

condescended to afford Proclus at Athens, and of that perfectly

singular kind of felicity which he obtained from the mother of the

gods, and in which he was accustomed especially to triumph and rejoice;

I should perhaps seem to many readers, to be rash in my assertions,

and to others, the author of things perfectly incredible. For many and

great were the daily instances of this goddess’s benevolence towards

him, in words and actions, which are both innumerable and unheard

of, and concerning which, I do not at present retain an accurate

remembrance. But if any one is desirous of knowing how great he was in

these, he must evolve his book concerning the mother of the gods, from

which he will understand, that it was not without divine assistance

he produced all the theology of that goddess into light; as likewise,

whatever is acted or affirmed concerning the same in the fables of

Athis, which he has explained after a philosophical manner, that

vulgar ears might afterwards cease to be disturbed, on perceiving the

lamentations and other obscure ceremonies with which her mysteries are

celebrated.

After having, therefore, run through the theurgic virtue of Proclus,

and its energies, and the happy circumstances which befell him,

respecting its operations, and having shewn, that he did not less excel

in every kind of virtue, and that he was a man such as mortals have

not beheld for a long period of time, it remains that we now bring our

discourse concerning him to a conclusion: for we are not now at the

beginning of our narration, nor does the half of the whole remain to

us, according to the proverb, but the whole itself is now perfect and

complete. Since having begun from the felicity of the philosopher,

and proceeded in its exemplification, our discourse now returns to

it again. For we have explained the great goods, and providential

exertions, which were granted to this most excellent man from the

gods, and have shewn their prompt attention to his prayers, the divine

visions which he enjoyed, and the help and solicitude which the gods

testified towards him. We have likewise explained his prosperous fate,

and propitious fortune, his country, parents, the good habit of his

body, his masters, friends, and other external advantages; each of

which we have shewn to have been far greater and more splendid in

him than in other men; and have likewise diligently enumerated such

things as cannot be reckoned amongst outward allurements, but entirely

depended on his will, such as are the upright and illustrious deeds of

his soul, according to universal virtue. And thus we have summarily

demonstrated that his soul arrived in reality at the summit of the most

consummate virtue, and was happily established in a perfect life, by

human and divine goods of every kind.

But that the lovers of more elegant studies way be able to conjecture

from the position of the stars under which he was born, that the

condition of his life, was by no means among the last or middling

classes, but ranked among the highest and most happy orders, we have

thought fit to expose in this place, the following scheme of his

nativity[68].

☉ 16° 26’ ♒︎

☽ 17 29 ♊︎

♄ 24 23 ♉︎

♃ 24 41 ♉︎

♂ 29 50 ♐︎

♀ -- 23 ♓︎

☿ 4 42 ♒︎

Horos. 8 19 ♈︎

Mid. }

Heaven. } 4 42 ♑︎

☋ or the head of the dragon.

24 33 ♏︎

The new moon preceding his birth.

8 51 ♒︎

But Proclus departed from this corporeal life, in the one hundred and

twenty-fourth year after the government of Julian, on the seventeenth

day of the Attic Munichion, or the April of the Romans. Nicagoras,

the junior, being at that time the Athenian archon. The dead body was

taken care of according to the Athenian rites, as he himself while

living had directed. And if that diligent attention which is due to

the deceased, was ever paid by any one, it was certainly bestowed by

this most blessed man. For he did not neglect any particular of the

accustomed ceremonies, but every year, on certain days, visited the

sepulchres of the Athenian heroes and philosophers, and besides this

sacrificed himself, and not through the medium of another, to the

manes of his friends and familiars. And having thus exhibited to each

a proper reverence and honour, proceeding into the Academy, he there,

in a certain place, pleased one by one the manes of his ancestors and

kindred; and shortly after, in another part of the same Academy,

he supplicated in common the souls of all philosophers. And these

ceremonies being ended, this most excellent man, having chosen a third

place, performed sacred rites to all the souls of deceased mankind. The

dead body of Proclus, therefore, being disposed of according to his

appointment, as we have said, was carried by his friends and buried

in the eastern part of the suburbs[69], near to Lycabetus, where also

the body of his preceptor Syrianus reposes, who, when he was alive,

had requested this of our philosopher, and, in consequence of this,

had taken care to procure a twofold receptacle in his sepulchre. But

when this most holy man was consulted how he would wish his funeral to

be conducted, lest there should be any thing indecent, or without a

proper decorum, he desired flutes, with which he was threatened in a

dream, and nothing besides. An epigram, consisting of four verses, is

inscribed on his tomb, which he composed himself, as follows:

I Proclus, here the debt of nature paid,

(My country Lycia) in the dust am laid;

Great Syrianus form’d my early youth,

And left me his successor in _the truth_.

One common tomb, our earthly part contains,

One place our kindred souls,--th’ ætherial plains.

Nor were prognostics wanting in the year prior to his decease, such

as an eclipse[70] of the sun, so great that night was produced in the

day-time: for the darkness was so profound, that the stars became

visible. And this happened when the sun was in Capricorn, in the

eastern centre. Besides, the writers of Ephemeride observe that there

will be another eclipse of the sun, at the conclusion of the next year.

But such like affections of the heavenly bodies are said to portend

events on the earth: to us indeed, the eclipse perspicuously signified

a privation and, as it were, defection of the illustrious luminary of

philosophy.

And thus much may suffice for the life of the philosopher. It now

remains for those who are willing to undertake the labour, to write

concerning his disciples and friends. For it appears that many resorted

to him from all parts; among which some were only his auditors, but

others firm adherents to his doctrine; and, on account of philosophy,

his familiars. I could likewise wish that some one, who is more

laborious than myself, would give a particular account of his writings.

For I have alone delivered these particulars of his life, that I might

satisfy my conscience, and might testify that I religiously reverence

the soul of Proclus, and his good dæmon. I shall, therefore, add

nothing concerning the philosopher’s writings, except that I have

always heard him prefer his commentaries on the Timæus of Plato beyond

all his works. He likewise vehemently approved of his commentaries on

the Theætetus. And he used frequently to say, that if he was endued

with despotic power, he would alone preserve, of all the writings of

the ancients, the Oracles and the Timæus. _For_, said he, _I

would abolish all the rest, and remove them from the present age,

because it happens that many are offended, who undertake to read them

rashly, and without proper institution_.

Concerning the Published Writings of PROCLUS.

Marinus having neglected to give us an account of the writings of

Proclus, I thought it would not be unacceptable to the reader, to

present him with a catalogue of his works which are still extant;

and the most perfect relation I am able of such as are unfortunately

lost. In the execution of this design, I shall follow, for the most

part, the accurate Fabricius in the order, and critical account of his

works; but shall not hesitate to dissent from him in deciding on their

philosophical merit: for it is very rare that philology and philosophy

are united in the same person, and coalesce in amicable conjunction.

* * * * *

_1. Four Hymns._ In the preceding life we are informed, by

Marinus, that Proclus composed many hymns, in which the divinities,

both of Greeks and Barbarians, were celebrated; but unfortunately there

are only four preserved, the first of which is to the Sun, the second

and third to Venus (which may be found in the Florilegium of Stobus, p.

249. edit. Grotii,) and the fourth to the Muses. They are collected by

Fabricius, in the eighth volume of his Bibliotheca Græca; and are most

happy specimens of philosophical poetry. Indeed, they bear most evident

marks of a mind full of divine light, and agitated with the fury of

the Muses; and possess all that elegance of composition for which the

writings of Proclus are so remarkable. So that it is very strange

Gyraldus should ascribe them to a Hierophant of Laodicæa, of that name;

since, as Fabricius observes, Suidas mentions other writings of this

priest, but does not speak of him as the author of any hymns. And if he

had, these hymns breathe too much of the spirit and manner of Proclus,

to be the production of any other.

_2. Two Books concerning the useful Parts of Learning, contained

in Grammar._ Three books on this subject are mentioned by Suidas,

and four by Photius; but there are alone extant fragments of two,

in the Bibliotheca of Photius; which were published separate by

Andrew Schottus, at the end of the syntax of Apollonius Alexandrinus,

Francof. 1590. Hanov. 1615. quarto. It appears from these extracts,

that Proclus, in this work, not only treats of every kind of verse, but

also of the more celebrated poets. And hence Fabricius conjectures,

that the short life of Homer, which Leo Allatius published under the

name of Proclus, was taken from the first book of this Chrestomathia.

_3. Eighteen Arguments against the Christians._ The learned Cave,

was egregiously mistaken, in supposing that this work, as well as the

answer to it by Philoponus, is lost; not considering, probably, that

these arguments in defence of the world’s eternity are (as Fabricius

well observes) directly against one of the Christian dogmata, the

creation of the world. These arguments (except the first, which is

lost,) are preserved in the answer of Philoponus; the Greek edition of

which was published at Venice, in 1535, folio; and in Latin from the

version of Joh. Mahatius, at Lyons, 1557. folio. Simplicius, in his

commentaries on the first book of Aristotle de Cœlo, and elsewhere,

opposes this work of Philoponus; at the same time, suppressing

his name. The arguments are, in my opinion, extremely subtle, and

unanswerable; and Philoponus, in his refutation, every where discovers,

that he has by no means fathomed the depth of our philosopher.

_4. A Commentary on the Timæus of Plato, in five Books._ A

most admirable work, composed in the flower of his age, when he was

twenty-eight years old, according to Marinus. This invaluable treasure,

may be said to contain all the wisdom of the highest antiquity:

for Proclus every where cites the most celebrated interpreters on

the Timæus, such as Adrastus the Peripatetic; Ælianus and Albinus,

Platonists; Aristocles, Clearchus, Crantor, Dercyllides; and especially

Jamblichus; likewise Longinus, whom he often refutes; Numenius the

Pythagorean, and Origen, (different from Adamantius,) together with

Plotinus, Porphyry the Platonic, Ptolemy, Severus, and Theodorus

Asinæus; but he always prefers the explications of his master Syrianus,

before all the rest. Add too, that he every where conciliates the

Orphic writings and the Chaldean oracles with the theology of Plato.

And with respect to Orpheus in particular, we are indebted to these

sublime commentaries for the greatest part of the fragments of that

divine poet. He is perpetually on the wing, and rises gradually in

elevated conceptions, until his mind, like Homer’s discord, reaches the

heavens. His style is at the same time nervous and diffuse, accurate

and elegant. We are astonished with the magnificence of his metaphors,

delighted with the copious variety of his matter, and filled with

a divine light, by the sacred truths he unfolds. This great work,

however, unfortunately, scarcely explains a third part of the Timæus;

from whence it is probable, as Fabricius well observes, that several

books have been lost through the injury of time. It was published in

Greek, at Basil, in the year 1556; and is full of mistakes, as is

generally the case with the Basil editions of books, so that a more

correct edition is greatly to be desired, though at present not much

to be expected. On concluding my account of this inestimable work, I

find my indignation roused by the following words of Dr. Cudworth,

in his Intellectual System, p. 306. “Proclus (says he) had some

peculiar fancies and whims of his own, and was indeed a confounder

of the Platonic theology, and a mingler of much unintelligible stuff

with it.” I must confess, (and I am neither afraid, nor ashamed of

the declaration,) that I never found any thing in Proclus, but what

by patient thought, accompanied with a sincere and vehement thirst

after truth, I have been able to fathom. Had Dr. Cudworth been endued

with these requisites, he would doubtless have had equal success;

but without them, the sublimest truths will certainly appear to be

_unintelligible stuff_. Besides this consideration is not to be

omitted, that a modern priest makes a bad philosopher.

_5. On Plato’s Republic._ These commentaries, or rather fragments

of commentaries, are extant in Greek, at the end of Proclus on the

Timæus. Suidas mentions four books of Proclus on Plato’s politics; and

some of Proclus’s dissertations on these books, were found (according

to Fabricius) in the library of Lucas Holstenius. The chief design of

this work seems to be the unveiling the theological mysteries concealed

under the fables of Homer, and other divine poets; which Proclus has

accomplished (in my opinion) in a most wonderful manner. That Homer,

indeed, every where abounds with Egyptian learning, is obvious to

every one; but few are acquainted with the profound wisdom which his

fables conceal. The latent meaning of most of these is unfolded in

the present invaluable, though imperfect work; and he whose mind is

sufficiently enlightened by the ancient philosophy, to comprehend the

beauty of these illustrations, will receive an additional delight from

the study of Homer, which it is impossible to express. An epitome of

this work was published in Latin, by the learned Gesner, 1542. 8vo.

under the following title: Apologiæ quædam pro Homero, et Arte Poetica,

Fabularumque aliquot Enarrationes ex commentariis Procli Lycii Diadochi

philosophi Platonici in libros Platonis de Rep. in quibus plurimæ de

Diis Fabulæ non juxta grammaticorum vulgus historicè, physicè aut

ethicè tractantur, sed Theologicis, ut Gentiles loquntur, ex prima

Philosophia rationibus explanantur.

_6. On the first Alcibiades of Plato._ Ficinus translated parts

of this work into Latin, and published them under the title of Procli

de Anima ac dæmone, de Sacrificio et Magia, Venice 1497. and 1516.

fol. by Aldus; and in a more simple form at Lyons. Fabricius informs

us, that the manuscript commentary of Proclus in Greek, but scarcely

explaining the half part of the Alcibiades, is to be found in various

libraries of France, England, and Italy. Also at Lyons, among the books

of Isaac Vossius; and at Hamburgh in the Johannean library. From the

specimen given of this work by Ficinus, it appears, like all Proclus’s

philosophical writings, to be an invaluable treasury of wisdom; and

nothing certainly, reflects greater disgrace on a nation than suffering

such monuments of ancient learning and wisdom to lie concealed in

colleges, covered with dust, and never consulted.

_7. Six Books on Plato’s Theology._ A most divine work, in which

the philosopher collects into a system the theology dispersed in the

writings of Plato, and establishes it by invincible demonstrations. He

deduces, in a beautiful and connected series, all the divine orders,

from the retreats of ineffable unity; every where connects them by

proper mediums, and, after leading us through the long gradation of

principles, brings us back again to the original from whence they

flowed, and to which they constantly tend. The whole is uncommonly

profound and abstruse; and it was not before the third reading, that

I could fathom the depth it contains. Fabricius observes, “that it

is a subtle and learned work, but from which, you will sooner learn

the opinion of Syrianus and Proclus, concerning the deity and divine

concerns, than that of Plato. He adds, that it is usual with the

Platonists, even from Plotinus, to unite to the doctrine of Plato, a

thousand dogmata, foreign from his philosophy, as if Plato, though he

did not perceive after this manner, ought certainly so to perceive.”

When men mistake their abilities, they always act absurdly, and often

dangerously. As a laborious and accurate critic on philological

matters, Fabricius merits the highest commendation such attainments

can deserve; but when he leaves the beaten road in which nature

designed him to walk, and attempts the tractless paths of philosophy,

he perpetually stumbles, and often falls on the ground. The wings of

philology, like those of the swallow, were never destined for a lofty

flight:--it must be the eagle wing of genius, which can alone soar to

the sun of philosophy. The Greek and Latin edition of this valuable

work, was published at Hamburgh, by Æmilius Portus, 1618, folio.

_8. Theological Institutions_; or, as it may be called, _the

Elements of Theology_. This admirable work contains two hundred and

ten propositions, disposed in a scientific order, and supported by the

firmest demonstrations. They begin from super-essential unity, and

proceed gradually through all the beautiful and wonderful progressions

of divine causes, ending in the self-moving energies of soul. They

possess all the accuracy of Euclid, and all the subtilty and sublimity

necessary to a knowledge of the most profound theology; and may be

considered as bearing the same relation to the Pythagoric and Platonic

wisdom, as Euclid’s Elements, to the most abstruse geometry. Patricius,

the first Latin translator of this divine work, seems to have been very

sensible of the truth of this observation: for he every where carefully

distinguishes the propositions from their demonstrations; and adds

the word corollary to such consequencies as merit that appellation.

His edition was published at Ferraria, 1583. quarto, under the title

of Theological Elements. The Greek and Latin edition, is subjoined to

Proclus’s six books on Plato’s Theology, Hamburgh 1618. folio.

_9. Two Books concerning Motion._ This useful work, collected, as

Fabricius observes, from the third and following books of Aristotle’s

physics, was published in Greek at Basil, 1531, and with the Latin

version of one Justus Velsius, a physician, Basil, 1545. octavo. It

was likewise translated by Patricius, and is annexed to his version of

the Theological Institutions.

_10. An Hypotyposis, or Information concerning Astronomical

Hypotheses._ This work, which Fabricius observes is a compendium of

Ptolemy’s Almagest, was published in Greek, at Basil, 1540. quarto;

and in Latin by George Valla, folio, 1541. A part of this work, which

treats of the use of the astrolabe, Fabricius informs us, is extant in

manuscript, in various libraries. The same accurate critic likewise

observes, that a small treatise, inscribed Uranodromus, is extant,

under the name of Proclus, in some libraries, as in that of Vindobona,

and of Oxford, among the Barrocian volumes. The comprehensive variety

of Proclus’s genius equally demands our admiration and applause.

_11. A small Treatise concerning the Sphere, or Celestial

Circles._ This little work is an accurate and elegant introduction

to astronomy; and is almost wholly taken from the Isagoge of Geminus

Rhodius, on the phænomena. The best editions are the Greek and Latin

one published at Paris in 1553, quarto; and that of Bainbridge,

professor of astronomy at Oxford, London 1620. quarto.

_12. A Paraphrase in four Books, on the Quadripartite of Ptolemy._

This elegant work must, I should imagine, be an invaluable treasure to

the lovers of astrology. It was first published in Greek by Melancthon;

and afterwards in Greek and Latin by Leo Allatius, at Lyons Bat. 1654.

octavo.

_13. Four Books, on the first Book of Euclid’s Elements._ For an

account of this work, see the introduction, and the following sheets,

in which it speaks for itself, in an English dress.

_14. A Commentary on Hesiod’s Works and Days._ This work contains

a valuable moral explanation of this great poet’s meaning; and

Fabricius justly observes, that he is often assaulted without occasion,

by the petulant jeers of that vain man Joh. Tzetzes. The best edition

of this work is that of Daniel Heinsius, Lugd. Bat. 1603. quarto.

_15._ Fabricius informs us, that in some manuscripts, as in the

Vindobonensian and Barrocian, a small treatise is usually ascribed

to Proclus, entitled _Epistolic Characters_; and is prefixed to

the Epistles of Phalaris and Brutis, and published under the name of

Libanius, in Greek, with the version of Casp. Stibilinus, Commelin.

1597, octavo. But it is doubtful whether Proclus is the genuine author

of this work: from the title, I should suppose the contrary. And thus

much for an account of those writings of Proclus which have escaped

the ravages of time, and have been fortunately exposed to public

inspection: it now remains that we relate such inestimable works of

this philosopher, as are yet preserved in shameful concealment; or are

utterly lost in the ruins of antiquity.

Concerning the Unpublished Writings of PROCLUS.

16. _On the Alcibiades of Plato._ See num. 6.

17. _On Plato’s Politics._ See num. 5.

18. _On Plato’s Parmenides._ A commentary, in seven books; the

last of which was not completed by Proclus, but by Damascius. From

occasional fragments, which have been published of this commentary, it

appears to be a most divine work; and indeed it cannot be otherwise, if

we consider it as the production of one of the greatest philosophers,

on the most sublime and profound of all Plato’s Dialogues. It is

dedicated to Asclepiodotus, a physician and philosopher, and is

not only extant in Greek MS. in the library of the German emperor,

according to Lambecius, lib. vii. p. 41. but also in Latin, from the

unpublished version of one Antonius Hermannus Gogava, as the same

Lambecius informs us, p. 41. Four books of this work are extant in

Greek, in the Bodleian library at Oxford; and it is much to be lamented

that Thomson did not publish these, instead of his trifling edition of

the Parmenides. Fabricius likewise informs us, that Livius Galantes

mentions his having found six of these books in some of the Italian

libraries. They are also extant in the Medicæan library of the great

Etruscan commander.

_19. On the Cratylus of Plato._ We have already observed, in the

dissertation on the Orphic theology, p. 105. what a great treasure of

ancient mythology, must be contained in this work; but there is little

hope of its ever emerging from the obscurity of public libraries. It

is extant in Greek, not only in the Italian libraries, but also among

the manuscript books of Isaac Vossius.

_20. Uranodromus._ See above, num. 10.

_21. Ten Doubts concerning Providence, in one Book._ Philoponus

mentions this work, in his second book against Proclus on the eternity

of the world; and a Latin version of it is extant by one William de

Morbeka, in the Johannean library of Hamburgh. Extracts from this

translation are preserved by Fabricius, in his Greek Library; and they

are in every respect worthy of the genius of Proclus.

_22. Concerning Providence and Fate, and that which is in our Power,

one Book._ This work is dedicated to one Theodorus, a mechanist; and

is extant in the Latin translation of the same Morbeka, in the Greek

Library of Fabricius. The translation is for the most part barbarous,

but is, however, sufficiently legible to discover that it is a most

valuable treatise, replete with the usual elegance, subtilty, and

sublimity of our philosopher.

_23. Concerning the Hypostasis, or Subsistence of Evil._ This

book is extant in Latin, in the Johannean library; and fragments of

it are preserved by Fabricius, in his Greek library. It is to be

regretted, that Fabricius did not preserve the whole in that excellent

philological work.

Concerning the Lost Writings of PROCLUS.

_24. On the Speech of Diotima, in Plato’s Banquet, concerning the

Subsistence of the Beautiful._ Fabricius informs us, that this

work is distributed into many books; and Holstenius observes, that it

is mentioned in a certain scholium of the Medicean copy of Proclus’s

commentaries on Plato’s politics; but it is unfortunately no where

extant.

_25. On the Philebus of Plato_; as may be inferred from the

narration of Damascius in Photius, p. 550; and Suidas in Marinus.

For Damascius relates, that Marinus having composed a commentary on

this dialogue, on shewing it to Isidorus for his approbation, that

philosopher observed, _that those of his master were sufficient_;

which words Fabricius, with great propriety, applies to the

commentaries of Proclus on the Philebus.

_26. On the Theætetus of Plato._ This work is praised by Marinus,

in the last chapter of the preceding life; and no doubt with great

propriety: for this abstruse and sublime dialogue would naturally call

forth all the divine fire and elegance of our philosopher.

_27. Commentaries on the Enneads of Plotinus._ This work is

mentioned by Gyraldus, in his second dialogue on ancient poets; by

Ficinus on Plotinus; by Philip Labbeus, in his account of MS. books,

p. 286; and in the notes of Bullialdus to Theo of Smyrna, p. 224. But

also in a certain note prefixed to an ancient manuscript of Jamblichus,

on the Egyptian mysteries, to this effect: “The philosopher Proclus,

commenting on the Enneads of the great Plotinus, says, that it is the

divine Jamblichus who answers the epistle of Porphyry.” This note is

in Greek, in the original, and is (in my opinion) of itself sufficient

to prove that such a work was once extant, though now unfortunately

lost. How much the want of these commentaries is to be regretted, must

be deeply felt by every lover of the Platonic philosophy. For the

unequalled profundity, and divine mysteries, contained in the writings

of Plotinus, could never be more happily illustrated than by the

irradiations of such a genius as Proclus.

_28. Lectures on Aristotle’s Book_ Περὶ Ερμηνείας, _or concerning

Interpretation_. This work, it seems, was never published; but Ammonius

Hermeas, the disciple of Proclus, has inserted in his valuable

commentary on this book all that he could retain in his memory of

Proclus’s lectures.

_29. Hymns_, not a few, see num. 1.

_30. Chrestomathea._ See num. 3.

_31. On the Mother of the Gods_, one book, mentioned by Marinus,

in the preceding Life.

_32. On the Theology of Orpheus._ This work is mentioned by

Marinus, in the preceding Life, and by Suidas; and its loss must be

particularly regretted by all the lovers of recondite theology.

_33. Ten Books, on the Chaldean Oracles._ This most valuable work

is mentioned by Marinus, in the preceding Life, and by Proclus himself

on Plato’s Politics, p. 359. It was doubtless not extant at the time

when Psellus and Pletho undertook the illustration of a few of these

oracles: at least the inconsiderable merit of their commentaries,

strongly favours this supposition.

_34. A Commentary on the whole of Homer._ Suidas. A specimen

of the great value of this work may be seen in our philosopher’s

commentaries on Plato’s republic. The works of Homer are not only the

great fountain of poetry, but likewise of philosophy; and are no less

admirable for inspiring the fury of the Muses than for containing the

mysteries of the most recondite theology.

_35. Concerning the Gods, according to Homer._ Had this work been

preserved, we should doubtless have been furnished with a defence of

the heathen religion, which would have silenced the ignorant clamours

of its opponents.

_36. The Symphony or Concord of Orpheus, Pythagoras, and Plato._

Suidas. Proclus, in his published writings, is every where studious of

reconciling the doctrines of these great men, and is always successful

in this undertaking. Indeed, the same divine genius seems to have

irradiated and inspired these wonderful heroes, but in different

ways: in Orpheus it was accompanied with the fire of the Muses; in

Pythagoras it shone through the mysterious veil of numbers; and in

Plato, combining the preceding modes, it was seen enshrined in awful

majesty of thought, clothed with the graces of poetical diction, and

resplendent with ineffable light.

_37. Two Books on the Theurgic Discipline._ Suidas. How much

Proclus excelled in this art, may be seen in the preceding Life.

_38. Concerning the Oppositions of Aristotle to Plato’s Timæus._

This work is mentioned by Proclus in the 3d book of his commentary

on the Timæus, p. 226. and seems to have escaped the notice of the

accurate Fabricius. Aristotle may, no doubt, in many particulars

be reconciled with Plato; but it is also certain, that in some he

is perfectly dissonant. And thus much for the Life and Writings of

Proclus.

COMMENTARIES

OF

PROCLUS.

BOOK I.

CHAP. 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 evidently

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)[71] 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 an 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.

CHAP. II.

_Concerning the common Principles of Beings, and of the Mathematical

Essence,[72] 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. But _bound_ 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 hand 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.

CHAP. III.

_What the common Theorems are of the Mathematical Essences._

But as we have contemplated the common principles of things, which are

diffused through all the mathematical genera, after the same manner

we must consider those common and simple theorems, originating from

one science, which contains all mathematical knowledge in one. And we

must investigate how they are capable of according with all numbers,

magnitudes and motions. But of this kind are all considerations

respecting proportions, compositions, divisions, conversions, and

alternate changes: also the speculation of every kind of reasons,

multiplex, super-particular, super-partient, and the opposite to these:

together with the common and universal considerations respecting equal

and unequal, not as conversant in figures, or numbers, or motions,

but so far as each of these possesses a common nature essentially,

and affords a more simple knowledge of itself. But beauty and order

are also common to all the mathematical disciplines, together with

a passage from things more known, to such as are sought for, and

a transition from these to those which are called resolutions and

compositions. Besides, a similitude and dissimilitude of reasons are

by no means absent from the mathematical genera: for we call some

figures similar, and others dissimilar; and the same with respect

to numbers. And again, all the considerations which regard powers,

agree in like manner to all the mathematical disciplines, as well the

powers themselves, as things subject to their dominion: which, indeed,

Socrates, in the Republic, dedicates to the Muses, speaking things

arduous and sublime, because he had embraced things common to all

mathematical reasons, in terminated limits, and had determined them in

given numbers, in which the measures both of abundance and sterility

appear.

CHAP. IV.

_How these Common Properties subsist, and by what Science they are

considered._

But it is requisite to believe, that these common properties do not

primarily subsist in many and divided forms, nor originate from things

many and the last: but we ought to place them as things preceding

in a certain simplicity and excellence. For the knowledge of these

antecedes many knowledges, and supplies them with principles; and

the multitude of sciences subsist about this, and are referred to

it as their source. Thus the geometrician affirms, that when four

magnitudes are proportional, they shall be alternately proportional;

and he demonstrates this from principles peculiar to his science, and

which the arithmetician never uses. In like manner, the arithmetician

affirms, that when four numbers are proportional, they shall be so

alternately: and this he evinces from the proper principles of his

science. For who is he that knows alternate ratio considered by

itself, whether it subsists in magnitudes or in numbers? And the

division of composite magnitudes or numbers, and in like manner, the

composition of such as are divided? For surely it cannot be said that

there are sciences and cognitions of things divisible: but that we

have no science of things destitute of matter, and which are assigned

a more intellectual contemplation; for the knowledge of these is by

a much greater priority science, and from these the common reasons

of many sciences are derived. And there is a gradual ascent in

cognitions from things more particular to more universal, till we

revert to the science of that which _is_, considered _as it

is_, abstracted from all secondary properties. For this sublime

science does not think it suitable to its dignity, to contemplate

the common properties which are essentially inherent in numbers, and

are common to all quantities; but it contemplates the one, and firm

essence of all the things which _are_. Hence, it is the most

capacious of all sciences, and from this all the rest assume their

own peculiar principles. For the superior sciences always afford the

first suppositions of demonstrations to such as are subordinate. But

that which is the most perfect of all the sciences, distributes from

itself principles to all the rest, to some indeed, such as are more

universal, but to others, such as are more particular. Hence, Socrates,

in the Theætetus, mingling the jocose with the serious, compares the

sciences which reside in us to doves: but he says they fly away, some

in flocks, but others separate from one another. For such, indeed,

as are more common and more capacious, comprehend in themselves many

such as are more particular: but such as being distributed into forms,

touch things subject to knowledge, are distant from one another, and

can by no means be copulated together, since they are excited by

different primary principles. One science, therefore, precedes all

sciences and disciplines, since it knows the common properties which

pervade through all the genera of beings, and supplies principles to

all the mathematical sciences. And thus far our doctrine concerning

dialectic[73] is terminated.

CHAP. V.

_What the Instrument is, which judges of the Mathematical Genera and

Species._

Let us now consider what that instrument is[74], adapted to the

judgment of mathematical concerns; and let us appoint Plato as our

guide in this affair, who, in his Republic, divides cognitions

separately from such things as are the objects of knowledge; and

distributes cognitions in conjunction with things subject to knowledge.

For of the things which are, some he ranks among intelligibles, and

others among sensibles. And of intelligibles, some are again pure

intelligibles, and others subject to cogitation. And of sensibles,

some are purely sensibles, but others conjectural. To intelligibles,

indeed, which are the first of the four genera, he assigns an

intelligible knowledge; but to those which are subject to cogitation,

he attributes thought: to sensibles, faith; but to conjecturals, a

conjectural or assimilatory power. And he shews, that the assimilatory

power has the same proportion to sense as thought to intelligence. For

the conjectural power knows the spectres of sensible forms, while they

are beheld in water and other bodies, which perspicuously represent

their image: since, by their situation in water, they are after a

manner, allotted the last seat in the gradations of forms, and truly

become the resemblances of resemblances. In like manner, thought

beholds the images of intelligibles in a degraded state, fallen from

primary simple and indivisible forms, into multitude and division.

Hence, a knowledge of this kind, depends on other more ancient

hypotheses; but intelligence arrives at that principle which is no

longer supposed. If then, mathematical concerns are neither allotted an

essence separate from all division and variety, nor that nature which

is apprehended by sense, which is obnoxious to many mutations, and is

in every proportion divisible, it must be manifest to every one, that

they are essentially subject to cogitation: but cogitation presides

over these as an instrument adapted to judgment, in the same manner as

sense to sensibles, and the assimilatory power to conjecturals. From

whence, indeed, Socrates determines that the knowledge of these is more

obscure than the first science, but is more evident than the impulsive

apprehension of opinion. For in this the mathematical sciences are

inferior to intelligence, because they contemplate that which is

evolved, and is endued with a power of progression; but they are

superior to opinion, by that stability of reasons which they contain,

and which cannot be confuted. And they originate from supposition,

through a diminution of the first science; but they contain forms

independent of matter, from their possessing a knowledge more perfect

than that of sensibles. We have therefore determined an instrument

adapted to the judgment of all mathematical concerns, i. e. cogitation,

according to the mind of Plato; which places itself indeed above

opinion, but is excelled by intelligence.

CHAP. VI.

_Concerning the Essence of Mathematical Genera and Species[75]._

It now remains, that we consider what subsistence or essence ought

to be assigned to mathematical genera and species? Whether we must

deduce their origin and subsistence from sensible objects, or from

abstraction, or from a collection of such things as are dispersed by

parts into one common definition; or must allow them an existence

prior to that of sensibles, as Plato affirms, and as the progression

of universal being demonstrates? First then, if we affirm that

mathematical species are composed from sensibles; whilst the soul

from material triangles or circles, forms in herself the trigonic, or

circular species, by a kind of secondary generation; I would ask from

whence is derived the great certainty and accuracy of definitions?

For it must either proceed from sensibles, or from the soul herself.

But from sensibles is impossible, for these, in a continual flow of

generation and decay, do not for a moment retain an exact sameness

of being; and consequently fall far short of the exactness contained

in the definitions themselves. It must therefore proceed from the

soul, which, by her immaterial nature, procures perfection from the

imperfect, accurate subtilty from that which is neither accurate nor

subtle, and rekindles the light of ideas from the obscure and unreal

objects of sense.

For where shall we find, amongst sensible objects, an indivisible

nature, such as that of a point, or a line without the dimension

of breadth, or a superficies without depth, or the ever constant

proportion of sides, and exact rectitude of angles? For my part, I

cannot see where, since all divisible natures are thus mixed and

confused together, nothing sincere, nothing free from its contrary,

but things every where yielding to separation, as well such as are

removed by distance of place, as those which are united together.

How then shall we obtain this durable essence for these immoveable

natures from the ever fluctuating forms of sense? For whatever derives

its existence from moveable beings, must of necessity be mutable and

frail. And how shall we gain this perfect accuracy for the stable

species, from the inaccurate and imperfect? For whatever is the cause

of a conception, always immutable, is itself much more stable than its

effect. We must therefore admit the soul to be the generator of these

mathematical species and reasons. But if she contains them in herself,

as first exemplars, she gives them an essential being, so that the

generations are nothing else than propagations of species, which had

a prior subsistence in herself: and thus we shall speak agreeably to

the sentiments of Plato, and discover the true essence of mathematical

entities. But if the soul, though she neither possesses nor received

the mathematical reasons prior to the energies of sense, yet fabricates

this admirable immaterial building, and generates this fair series of

speculations; how can she discern whether her productions are stable

and constant, or things which the winds may dissipate, and phantoms

rather than realities? What standard can she apply as the measure of

their truth? Or how, since she is destitute of their essence, can she

generate such a variety of reasons? For from such an hypothesis, we

make their subsistence fortuitous, not tending to any scientific bound.

Mathematical species are therefore the genuine offspring of the soul:

nor does she derive from sensible objects the definitions she frames,

but rather the first are propagated from the second; they are the

energies of soul, which, as it were, pregnant with forms, delivers her

immaterial progeny into the dark and fluctuating regions of matter, as

evidences of the permanent duration of her species.

Again, if we collect mathematical reasons from externals, why

are not demonstrations composed from sensibles, better than the

demonstrations of universal and simple species? For we say, in order

to the investigation of any thing sought, that the principles and

propositions, should be allied to the conclusions. If then, particulars

are the causes of universals, and sensibles the sources of reasoning,

why does the boundary of demonstration always refer to that which is

more universal, and not to that which is partial and particular? And

how can we prove that the essence of intelligibles is more allied to

demonstration than the essence of sensibles? For thus they speak[76]:

his knowledge is not legitimate, who demonstrates that the isosceles,

the equilateral, or the scalene triangle, have angles equal to two

right; but he possesses science, properly so called, who demonstrates

this of every triangle simply, or of triangle itself. And again,

that universals, for the purpose of demonstration, are superior to

particulars; that demonstrations concern things more universal; but

that the principles from which demonstrations are composed, have a

priority of existence, and a precedency in nature to singulars, and

are the causes of the propositions they prove. It is very remote,

therefore, from the nature of Apodictical sciences, that from converse

with things of posterior origin, and from the dark perceptions of

sense, they should falsely collect their indubitable propositions.

I add farther, that they who affirm this, make the soul of a baser

nature than the material species themselves. For if matter derives from

nature beings essential, and participating a high degree of entity and

evidence; but the soul, by a posterior energy, receives these from

sensible objects, and fashions in herself resemblances and images

of posterior origin, contemplating vile essences, and abstracting

from matter, the forms inseparable from its nature; do they not make

the soul more obscure and indigent than matter itself? For matter

is the receptacle of forms materialized, as the soul is of species

immaterialized. But in this case, matter would be the place of primary

beings, and the soul of such as are secondary and subordinate: matter

and its forms obtaining the lead in being, and existing as the sources

of the subsistence of immaterial forms. Lastly, the material forms

would have an essential existence, the others only an intentional

denomination. How then can the soul, which is the first participant of

intellect, and an intellective essence, and which derives from thence

consummate knowledge, and a plenitude of life, become the receptacle

of the most obscure species, the lowest in the order of things, and

participating the most imperfect existence. But this opinion, which has

been sufficiently exploded by others, needs no farther confutation.

If then, mathematical species do not subsist by material abstraction,

nor by a collection of those common properties inherent in individuals;

nor are at all, in their origin, posterior to sensibles, nor derived

in any manner from them: it is necessary that the soul should either

deduce them from herself, or from intellect; or lastly, from herself

and intellect united. But if from herself alone, Whence do the

images of intellectual species arise; whence do they derive their

middle nature, linking, as it were, the divisible and indivisible

essence together; if they do not participate the fullness of entity

from primary essences? Lastly, how, upon this hypothesis, are the

first exemplars, paradigms, or ideas, which subsist in intellect,

the principles of universals? But if they are derived from intellect

alone into the soul, how can the soul remain self-operative, and

self-motive, if her inherent reasons flow from an external source, and

are regulated by its operations? And in what respect does the soul

differ from matter, which is all things in mere dormant capacity,

but generates nothing appertaining to material species? It remains,

therefore, that the soul deduces these species from herself, and

intellect; and that she is the absolute consummation of the forms which

originate from intellectual exemplars, but which are allotted from

themselves a transition to permanent being. The soul, therefore, is by

no means to be compared to a smooth tablet, void of all reasons; but

she is an ever-written tablet, herself inscribing the characters in

herself, of which she derives an eternal plenitude from intellect.

For soul is a certain subordinate intellect, revolving round an

intellect prior to herself, formed to its image, and participating its

divine irradiations. If then, this superior intellect is all things

intellectually, soul will all things animally; if the first exists as

the exemplar, soul will be as its image; if as contracted and united

in itself, soul as divisible and expanded. And this is what Plato

understood, when in his Timæus, he composes the soul of the world

from all things, dividing her according to harmonical reasons, and

analogies; assigning to her the first principles effective of figures,

I mean the right and circular line, and giving an intellectual motion

to her inherent circles. All mathematical species, therefore, have a

primary subsistence in the soul: so that, before sensible numbers,

there are to be found in her inmost recesses, self-moving numbers;

vital figures, prior to the apparent, ideal proportions of harmony

previous to concordant sounds; and invisible orbs, prior to the bodies

which revolve in a circle. So that soul is the prolific abundance of

all these, and is another ornament producing herself, and produced

from a proper principle, filling herself with life, and at the same

time filled from the demiurgus of the universe, is an incorporeal

and indistant manner. When, therefore, she produces and unfolds

her latent reasons, she then detects every science and virtue. The

essence of soul then consists in these species, nor must we suppose

her inherent numbers to be a multitude of units, nor her archytipal

ideas of divisible forms to be corporeal: but we must conceive all

these as subsisting ever vitally, and intellectually, as the exemplars

of apparent numbers, figures, reasons and motions. And here we must

follow the doctrine of Timæus, who derives the origin, and consummates

the fabric of the soul, from mathematical forms, and reposes in her

nature the causes of every thing which exists. For the seven bounding

terms[77], comprehending the principles of all numbers, lines,

planes and solids, pre-exist in soul according to cause. And again,

the principles of figures are placed in her essence, according to a

demiurgical power. And lastly, the first of all motions, which embraces

every other motion in its comprehensive ambit, is co-existent with

soul. For the principle of every thing which is moved is a circle, and

the circular motion. The mathematical reasons, therefore, which fully

consummate the soul, are essential, and self-moving: and the soul, by

her cogitative power, diffusing, propagating, and evolving these, from

her profound recesses, constitutes all the fair variety of mathematical

sciences. Nor will she ever cease to generate, and waken into energy,

succeeding species, while she divests her indivisible reasons of

their intellectual simplicity. For she previously received all

things, after a primary manner; and according to her infinite power,

from pre-existent principles, deduces a beautiful series of various

speculations.

CHAP. VII.

_What the Employments and Powers are of the Mathematical Science,

and how far they extend themselves in their Energies._

But, after contemplating the essence of mathematical forms, it is

necessary we should recur to that one master-science of these, which

we have shewn is prior to a multitude of others, and that we should

contemplate what its employment is, what are its powers, and how far

it advances in its energies. The employment, therefore, of the whole

mathematical science, possessing, as we have before said, the power

of cogitation, must not be placed so high as that of intelligence;

which is firmly seated in its own stable essence, is perfect, is

contained by itself, and in itself continually verges. Nor must it be

situated so low as that of opinion and sense, since these cognitions

dwell upon external concerns, energize upon them, and do not possess

the causes of the objects of their knowledge. But the mathematical

science, receives its commencement, indeed, extrinsically from[78]

recollection, but ends in the most intimate reasons, residing in the

depths of the soul; and is excited, indeed, from things posterior,

but arrives by gradual advances at the principal essence of forms.

Nor is its energy immoveable, like that of intelligence, nor is it

affected with local motion and alteration, like sense, but it revolves

with a vital energy, and runs through the ornament of incorporeal

reasons, sometimes advancing from principles to such things as are

perfected by principles, but at other times yielding in a retrograde

progression from conclusions to their forming principles: and sometimes

proceeding from things previously known, to such as are the subject

of investigation: but at other times, from things placed in the

question, to such as precede in cognition. Besides, it does not excel

all inquisition, as if it were perfect from itself, like intellect,

nor is it perfected from others, like sense, but it proceeds by

enquiry to invention, and ascends from the imperfect to perfection.

But it likewise possesses twofold powers, one kind of these deducing

principles into multitude, and generating the different paths of

contemplation: but the other endued with a power of collecting many

transitions into proper suppositions. For since it proposes to itself

as principles, as well unity, and multitude, as bound and infinite,

and such things as are subject to its comprehension, are allotted a

middle order, between forms indivisible and every way divisible; with

great propriety (I think) the gnostic powers of the whole science of

these are essentially twofold. One species indeed, hastens to union,

and contracts the expansion of multitude: but the other possesses

a power of distinguishing things simple into such as are various,

more universals into more particulars, and reasons digested in their

principle, into things secondary and multifariously multiplied from

these principles. For rising higher from its commencement it penetrates

even to such things as are the perfections of sensible concerns,

is joined with nature, and demonstrates many things together with

natural science. Since ascending from inferiors, it accedes in a

certain respect proximate to intellectual knowledge, and touches the

contemplation of things primary and divine. And hence, in the limits

which flow from its essence, it produces the whole mechanic, optic,

and catoptric speculation, together with many other sciences which are

inwoven and entangled with sensible concerns, and which operate through

their assistance. Besides, in its ascensions from corporeal natures,

it derives intelligences indivisible and destitute of matter: and with

these it perfects its divisible apprehensions, those cognitions which

subsist in progressions, and its own genera and forms: it likewise

indicates the truth respecting the gods themselves, and in its peculiar

treatises exhibits a contemplation of the things which _are_. And

thus much concerning the employment and powers of the Mathematical

Science.

CHAP. VIII.

_Concerning the Utility of the Mathematical Science._

But let us now consider the utility of this Science, which extends

itself from the most principal to the last cognitions. Timæus,

therefore, calls the knowledge of the mathematical disciplines the

path of erudition, because, indeed, it has the same proportion to

universal science, and the first philosophy, which learning has to

virtue. For this last frames our soul to a perfect life, by the

possession of worthy manners; but the former prepares our cogitation,

and the divine eye of our soul to an elevation from the obscurity of

sensible information. Hence, Socrates in the Republic, says, “That

the eye of the soul, which is darkened and buried by other studies,

can by the mathematical disciplines alone be invigorated, and again

excited to the contemplation of that which _is_, and transferred

from resemblances to real beings, from an obscure light to that

light which has the power of intelligence, and from a cave, and

those bonds which exist in it as the authors of generation, and from

material impediments be able to rise to an incorporeal and indivisible

essence. For the beauty and order of mathematical reasons, and the

firmness and stability of the contemplations they afford, conjoins

us with intelligible objects, and perfectly determines us in their

essences; which perpetually remain the same, ever shining with divine

beauty, and preserving a mutual order without end. But Socrates, in

the Phædrus, delivers to us three characters who are elevated from

sense, because they fill up and accomplish the primary life of the

soul, i. e. the philosopher, the lover, and the musician. But the

beginning and path of elevation to the lover, is a progression from

apparent beauty, using as excitations the middle forms of beautiful

objects. But to the musician, who is allotted the third seat, the

way consists in a transition from sensible to invisible harmonies,

and to the reasons existing in these. So that to the one, sight is

the instrument of reminiscence, and to the other, hearing. But to

him who is by nature a philosopher, from whence and by what means is

reminiscence the prelude of intellectual knowledge, and an excitation

to that which truly is, and to truth itself? For this character also,

on account of its imperfection, requires a proper principle: for it is

allotted a natural virtue, an imperfect eye, and a degraded manner.

It must therefore be excited from itself; and he who is of such a

nature, rejoices in that which _is_. But to the philosopher,

says Plotinus, the mathematical disciplines must be exhibited, that

they may accustom him to an incorporeal nature, and that afterwards

using these as figures, he may be led to dialectic reasons, and to

the contemplation of all the things which _are_. And thus it is

manifest, from hence, that the mathematics are of the greatest utility

to philosophy. But it is requisite that we should be more explicit, and

mention the several particulars to which they conduce, and evince that

they prepare the intellectual apprehensions of theology. For whatever

to imperfect natures appears difficult and arduous in obtaining the

true knowledge of the gods, the mathematical reasons render, by their

images, credible, manifest, and certain. Thus, in numbers, they

indicate the significations of super-essential properties, but they

evince the powers of intellectual figures, in those figures which

fall under cogitation. Hence it is, that Plato, by mathematical forms

teaches us many and admirable sentences concerning the gods, and the

philosophy of the Pythagoreans, using these as veils, conceals from

vulgar inspection the discipline of divine sentences. For such is the

whole of the _Sacred and Divine Discourse_[79], that of Philolaus

in his Bacchics, and the universal method of the Pythagoric narration

concerning the Gods. But it especially refers to the contemplation of

nature, since it discloses the order of those reasons by which the

universe is fabricated, and that proportion which binds, as Timæus

says, whatever the world contains, in union and consent; besides, it

conciliates in amity things mutually opposing each other, and gives

convenience and consent to things mutually disagreeing, and exhibits

to our view simple and primary elements, from which the universe is

composed, on every side comprehended by commensurability and equality,

because it receives convenient figures in its proportions, and numbers

proper to every production, and finds out their revolutions and

renovations, by which we are enabled to reason concerning the best

origin, and the contrary dissolution of particulars. In consequence

of this, as it appears to me, Timæus discloses the contemplation

concerning the nature of the universe, by mathematical names, adorns

the origin of the elements with numbers and figures, referring to

these their powers, passions, and energies; and esteeming as well

the acuteness as the obtuseness of angles, the levity of sides, or

contrary powers, and their multitude and paucity to be the cause of the

all-various mutation of the elements. But why may we not say, that it

profits much, and in an admirable manner, to that philosophy which is

called Politic, as well by measuring the times of actions as affording

the various revolutions of the universe, and numbers convenient to

things rising into being; I mean the assimilating, and authors of

dissimilitude, the prolific too and the perfect, and the contraries

to these; together with orderly and elegant ministers of life, and

inelegance; and finally, such numbers as procure fertility and

sterility. Which, indeed, the speech of the Muses in the Republic[80]

evinces, placing the universal Geometric Number as the author of better

and more debased generations, and as the cause of the indissoluble

perseverance of good manners, and of the mutation of the best Republics

into such as are remote from reason, and are given to affections. For

it is sufficiently evident, that it belongs to the whole mathematical

discipline to deliver the science of this number which is called

geometrical, and not to one particular science, such as arithmetic, or

geometry: since the reasons or proportions of abundance and sterility,

permeate through all the mathematical disciplines. Again, it is the

means of our institution in moral philosophy which it brings to its

ultimate perfection, and gives order and an elegant life to our

manners. Besides this, it delivers to us figures, and modulations

and motions convenient to virtue, by which the Athenian guest wishes

those to be instituted and perfected, who are destined to pursue

moral virtue from their early youth. Add too, that it places before

our view the reasons of virtues, in one manner, indeed, in numbers,

in another in figures, but differently in musical symphonies; and

lastly, it indicates the excess and defect of vices, by which we are

enabled to moderate and adorn our manners. Hence it is, that Socrates,

in the Gorgias, accusing Calicles of an inordinate and intemperate

life, says to him, “You neglect geometry and geometric equality:” but,

in the Republic, he finds out the proportion of tyrannic pleasure to

a royal interval, according to a plane and solid generation. But we

shall learn what great utility is derived to other sciences and arts

from the mathematical science, when we consider that it adds order and

perfection to contemplative arts; I mean rhetoric, and all such as

consist in discourse. But it proposes to the poetic arts, the reasons

of poems in the place of an example, because it presides over the

measures existing in these. But to the active arts it determines action

and motion, by its own abiding and immoveable forms. For all arts, as

Socrates says, in the Philebus, require arithmetic, mensuration, and

statics, either in all, or in some of their operations. But all these

are contained in the discourses of the mathematical science, and are

terminated according to their diversity. For from this science the

divisions of numbers, and the variety of dimensions, and the difference

of weights are known. The utility, therefore, of the whole mathematical

science to philosophy itself, and to other sciences and arts, may be

from hence known to intelligent hearers.”

CHAP. IX.

_A Solution of an Objection raised by some against the Utility of the

Mathematical Sciences._

But some, who are prone to contradiction through those who wish to

subvert geometry, endeavour to destroy the dignity of this science.

One part, indeed, depriving it of ornament and good, because it

does not discourse on these. But another part[81] affirming that

sensible experiments are more useful than the universal objects of its

speculation; I mean, that Geodesia (for instance,) or the mensuration

of the earth, is preferable to geometry, and vulgar arithmetic to

that arithmetic which is conversant with theorems alone: and that

nautical astrology is more useful than that which teaches universally,

abstracted from any application to sensible concerns. For we are not,

say they, made rich by our knowledge of riches, but by using them;

nor are we happy by the merely understanding felicity, but by living

happily. Hence we must confess that those mathematical sciences,

which are conversant with cognition, do not profit human life, and

confer to action, but those only which are engaged in exercise. For

those who are ignorant of the reasons of things, but are exercised

in particular and sensible experiments, are in every respect more

excellent, for the purposes of human life, than those who are employed

in contemplation alone. Against objections then, of this kind, we

shall reply, by shewing the beauty of the mathematical disciplines

from those arguments by which Aristotle endeavours to persuade us. We

must therefore confess that there are three things which especially

cause beauty, both in bodies and souls; I mean, order, convenience, and

determination. Since corporeal baseness, indeed, arises from material

inordination, deformity, and inconvenience, and from the dominion

of the indefinite in the composite body. But the baseness of the

soul originates from its irrationality, and inordinate motion, and

from its being in a state of discord with reason, and not receiving

from thence its proper limitation. Hence, beauty exists even in

contraries, by means of order, convenience and determination. But we

may behold these in a more eminent degree in the mathematical science;

order, indeed, in the perpetual exhibition of things posterior and

more various, from such as are primary and more simple; for things

subsequent are always annexed to their precedents, the latter ranking

as principles, and the former as the first suppositions of things

consequent: but convenience is evinced in the mutual consonance of

things demonstrated, and in the relation of all of them to intellect,

since intellect is the common measure of all science, from which it

receives its principles, and to which it converts the learner: but

determination is perceived in its perpetually abiding and immoveable

reasons, for the objects of its knowledge are not, at times, subject

to variation, like those of opinion and sense, but present themselves

for ever the same, and are bounded by intellectual forms. If such

then, are the principal requisites of beauty, it is evident, that

in these sciences that illustrious ornament and gracefulness is

found. For how is it possible this should not be the case with a

science receiving a supernal illumination from intellect, to which

it continually advances, hastening to transfer us from the obscure

light of sensible information? With respect to the second objection,

we think it proper to judge of its utility, without regarding the

conveniencies and necessities of human life. For otherwise, we must

confess that contemplative virtue is also useless, which separates

itself from human concerns, which it is very little desirous to look

down upon and understand. Indeed Socrates, in the Theætetus, affirming

this concerning noblemen endued with the prophetic power, says, “that

it withdraws them from all regard to human life, and raises their

thoughts, properly liberated, from all necessity and use, to the very

summit of all true being.” The mathematical science, therefore, must

be considered as desirable for its own sake, and for the contemplation

it affords, and not on account of the utility it administers to human

concerns. But if it is necessary to refer the utility it produces to

something different from itself, it must be referred to intellectual

knowledge. For it leads us to this, and prepares the eye of the

soul for the knowledge of universals, removing and obliterating the

impediments arising from the senses, and from corporeal involution.

As therefore we call the whole of purgative virtue useful, or the

contrary, not regarding the use of the sensible life, but of that

which is contemplative, so indeed it is requisite to refer the end

of mathematics to intellect, and universal wisdom. Hence its energy

is worthy our study, both on its own account, and on account of an

intellectual life. But it appears, as Aristotle[82] says, that this

science is desirable of itself to its votaries, because though no

reward is proposed to its enquirers, yet the mathematical contemplation

receives, in a small time, an abundant increase. Besides, this is

farther evident from hence, that all men are willingly employed in

its pursuit, and wish to dwell on its speculations, omitting every

other concern; even those who have, with their lips, as it were, but

just touched its utility. And hence it follows, that they who despise

the knowledge of the mathematical disciplines, have very little

tasted of the pleasures they contain. The mathematics, therefore,

are not to be despised because their speculative parts do not

immediately confer to human utility, (for the ultimate limits of its

progressions, and whatever operates with matter, consider a use of

this kind;) but on the contrary we should admire its immateriality,

and the good which it contains, considered by itself alone. For when

mankind were entirely disengaged from the care of necessary concerns,

they converted themselves to the investigation of the mathematical

disciplines; and this, indeed, with the greatest propriety. Since

affairs familiar to human life in its most imperfect state, and which

are immediately connected with its origin, first of all employed the

studies of mankind: but, in the second place, those concerns succeeded

which separate the soul from generation, and restore its memory of

that which _IS_. After this manner, then, we are engaged in

necessaries, before things honourable for their own sakes, on account

of their intrinsic dignity and worth; and in things related to sense,

before such as are apprehended by the nobler energies of mind. For

every origin and life of the soul which is converted into herself, is

naturally adapted to proceed from the imperfect to the perfect. And

thus much against those who despise the mathematical science.”

CHAP. X.

_A Solution of another Objection of certain Platonists, against the

Utility of the Mathematical Sciences._

But, perhaps, some of our own family will here rise up against us,

and, proposing Plato as a witness, will endeavour to provoke ruder

understandings into a contemptuous disregard of the mathematical

disciplines. For they will say, that this philosopher entirely

excludes (in his Republic) the mathematical knowledge from the choir

of the sciences, and that he accuses it as being ignorant of its own

principles, that its very principle is to itself unknown, and its ends

and mediums composed from things of which it is ignorant. To these

objections they may likewise add whatever other reproaches are there

urged by Socrates against this contemplation. In answer then, to the

objections of our friends, we shall recall into their memory, that

Plato himself perspicuously asserts the mathematical science to be the

purgation of the soul, and that it is endued with a power of leading

it on high; because, like the Homeric Minerva, it removes the darkness

of a sensible nature from the intellectual light of thought, which

is better worth saving than ten thousand corporeal eyes, and which

not only participates of a mercurial gift, (preserving us from the

incantations and delusions of this material abode, which is similar

to the fascinating realms of Circe,) but also of the more divine arts

of Minerva. He likewise every where calls it by the name of science,

and asserts that it is the cause of the greatest felicity to those

who are exercised in its contemplation. But I will briefly explain

why, in the Republic he takes from it the surname of science: for my

present discourse is addressed to the learned. Plato, indeed, in most

places, calls all the knowledge (as I may say) of universals by the

name of science, opposing it in a division to sense which apprehends

only particulars, whether such a mode of cognition is accomplished

by art or experience. And in this sense, as it appears to me in

the _Civil_ Dialogue, and in the Sophista, he seems to use the

name of science; placing likewise the illustrious Sophistic science,

which Socrates in the Gorgias, says, is a certain experience: also,

the adulatory, and many others, which are experiences, but not true

sciences. But again, dividing this knowledge of universals into

that which knows causes, and into that which understands without a

cause, he thinks that the one should be called science, but the other

experience. And hence, to arts he sometimes attributes the name of

science, but to experience never. For how (says he in the Banquet)

can a thing which possesses no reason be science? All knowledge,

therefore, which contains the reason and cause of the things known,

is a certain science. Again, therefore, he divides this science which

is endued with a power from the cause of knowing, by the peculiarity

of its subjects, and he places one, conjectural of things divisible;

but the other of such as subsist by themselves, and are ever knowable

after the same manner. And according to this division he separates from

science, medicine, and every faculty which is conversant with material

concerns. But he calls mathematical knowledge, and whatever possesses

a power of contemplating eternal objects, by the name of science.

Lastly, dividing this science, which we distinguished from arts, he

considers one part as void of supposition; but the other as flowing

from supposition. And that the one which is void of supposition, has

a power of knowing universals: that it rises to good, and the supreme

cause of all; and that it considers good as the end of its elevation:

but that the other, which previously fabricates for itself definite

and determinate principles, from which it evinces things consequent to

such principles, does not tend to the principle, but to the conclusion.

And hence he asserts, that mathematical knowledge, because it makes

use of supposition, falls short of that science which is without

supposition, and is perfect. For there is one true science, by means

of which we are disposed to know all the things _which are_, and

from which also principles emerge to all sciences; to some, indeed,

constituted more proximately, but to others more remotely. We must

not say, therefore, that Plato expels mathematical knowledge from the

number of the sciences, but that he asserts it to be the second from

that one science, which possesses the supreme seat of all: nor must we

affirm, that he accuses it as ignorant of its own principles, but that

receiving these from the master science dialectic, and possessing them

without any demonstration, it demonstrates from these its consequent

propositions. For, indeed, he sometimes allows the soul, which is

constituted from mathematical reasons, to be the principle of motion:

and sometimes he affirms, that it receives its motion from genera

which are subject to intelligence. And these variations accord among

themselves. For to such things as are moved by another, the soul is a

certain cause of motion, but it is not the cause of every motion. After

the same manner, the mathematical science is indeed the second from the

first of all sciences, and, with reference to it, imperfect: but it

is, nevertheless, a science, not as being free from supposition, but

as knowing the peculiar reasons resident in the soul, and as bringing

the causes of conclusions, and containing the reason of such things as

are subject to its knowledge. And thus much for the opinion of Plato

respecting mathematics.

CHAP. XI.

But let us now consider what are the things which may be required of

a mathematician, and how any one may rightly judge concerning his

distinguishing peculiarities. For[83] Aristotle indeed, says, that

he who is simply learned in all disciplines, is adapted to judge of

all: but that he who is alone skilled in the mathematical sciences,

can alone determine concerning the magnitude of reasons inherent in

these. It is requisite, therefore, that we should previously assume

the terms of judging, and that we should know, in the first place,

in what things it is proper to demonstrate generally, and in what to

regard the peculiarities of singulars. For many of the same properties

reside in things differing in species, as two right angles in all

triangles: but many have indeed the same predicament, yet differ

in their individuals in a common species, as similitude in figures

and numbers. But one demonstration is not to be sought for by the

mathematician in these, for the principles of figures and numbers are

not the same, but differ in their subject genus. And if the essential

accident is one, the demonstration will also be one[84]: for the

possession of two right angles is the same in all triangles, and that

general something to which this pertains is the same in all, I mean

triangle, and a triangular reason. In the same manner, likewise, the

possession of external angles to four right ones, not only pertains to

triangles, but also to all right-lined figures; and the demonstration,

so far as they are right-lined, agrees in all. For every reason brings

with it, at the same time, a certain property and passion, of which all

participate through that reason, whether triangular, or rectilinear, or

universally figure. But the second limit by which a mathematician is

to be judged, is, if he demonstrates according to his subject-matter,

and renders necessary reasons, and such as cannot be confuted, but are

at the same time neither probable, nor replenished with a similitude

of truth. For, says Aristotle, it is just the same to require

demonstrations from a rhetorician, and to assent to a mathematician

disputing probably; since every one, endued with science and art, ought

to render reasons adapted to the subjects of his investigation. In

like manner also, Plato in the Timæus, requires credible reasons of

the natural philosopher, as one who is employed in the resemblances

of truth: but of him who discourses concerning intelligibles, and a

stable essence, he demands reasons which can neither be confuted nor

moved. For subjects every where cause a difference in sciences and

arts, since, if some of them are immoveable, others are conversant

with motion; and some are more simple, but others more composite;

and some are intelligibles, but others sensibles. Hence we must

not require the same certainty from every part of the mathematical

science. For if one part, after a manner, borders upon sensibles, but

another part is the knowledge of intelligible subjects, they cannot

both be equally certain, but one must inherit a higher degree of

evidence than the other. And hence it is, that we call arithmetic more

certain than the science of harmony. Nor must we think it just that

mathematics and other sciences should use the same demonstrations;

for their subjects afford them no small variety. In the third place,

we must affirm, that he who rightly judges mathematical reasons, must

consider sameness and difference, what subsists by itself, and what is

accidental, what proportion is, and every consideration of a similar

kind. For almost all errors of this sort happen to those who think

they demonstrate mathematically, when at the same time they by no

means demonstrate, since they either demonstrate the same thing as if

different in each species, or that which is different as if it were

the same: or when they regard that which is accidental, as if it were

an essential property; or that which subsists by itself, as if it were

accidental. For instance, when they endeavour to demonstrate that the

circumference of a circle is more beautiful than a right line, or an

equilateral than an isosceles triangle. For the determination of these

does not belong to the mathematician, but to the first philosopher

alone. Lastly, in the fourth place, we must affirm, that since the

mathematical science obtains a middle situation between intelligibles

and sensibles, and exhibits in itself many images of divine concerns,

and many exemplars of natural reasons, we may behold in it three kinds

of demonstration[85], one approaching nearer to intellect, the second

more accommodated to cogitation, and the third bordering on opinion.

For it is requisite that demonstrations should differ according to the

varieties of problems, and receive a division correspondent to the

genera of beings, since the mathematical science is connected with all

these, and adapts its reasons to the universality of things. And thus

much for a discussion of the subject proposed.

CHAP. XII.

_What and how many the Species of the whole Mathematical Science are,

according to the Opinion of the Pythagoreans._

But after these considerations, it is requisite to determine concerning

the parts of the mathematical science, what, and how many they are.

For it is just, after speculating its whole and entire genus, to

consider the differences of its more particular sciences, according

to their species. The Pythagoreans[86], therefore, thought that the

whole mathematical science should receive a fourfold distribution,

attributing one of its parts to the how-many, but the other to the

how-much; and they assigned to each of these parts a twofold division.

For they said, that discrete quantity, or the how-many, either subsists

by itself, or must be considered with relation to some other; but that

continued quantity, or the how-much, is either stable or in motion.

Hence they affirmed, that arithmetic contemplates that discrete

quantity which subsists by itself, but music that which is related

to another; and that geometry considers continued quantity so far as

it is immoveable; but spherics contemplates continued quantity as

moving from itself, in consequence of its union with a self-motive

nature. They affirmed besides, that these two sciences, discrete and

continued quantity, did not consider either magnitude or multitude

absolutely, but that alone which in each of these is definite from the

participation of bound. For sciences alone speculate the definite,

rejecting as vain the comprehension of infinite quantity. But when

these wise men assigned this distribution, we must not suppose they

understood that discrete quantity which is found in sensible natures,

nor that continued quantity which subsists about the fluctuating

order of bodies. For, I think, the contemplation of these pertains

to the natural and not to the mathematical science. But because the

demiurgus of the universe, employed the union, division, and identity

of general natures, together with difference, station, and motion, for

the purpose of completing the essence of the soul, and composed it from

these genera, as Timæus informs us, we must affirm, that cogitation,

abiding according to its diversity, its division of reasons, and its

multitude, and understanding itself to be both one and many, proposes

indeed to itself, and produces numbers, together with an arithmetical

knowledge of these: but it provides for itself music according to an

union of its multitude, and a communication and junction with itself;

and hence it is that arithmetic excels music in antiquity; since,

according to the narration of Plato, the demiurgus first divided the

soul, and afterwards collected it in harmonical proportions. Again,

thought establishing its energy according to the stability which it

contains, draws from its inmost retreats geometry, together with one

essential figure, and the demiurgical principles of all figures[87]:

but, according to its inherent motion, it produces the spherical

science. For it is moved also by circles, but abides perpetually the

same from the causes of circles. Hence, likewise, geometry precedes

spherics, in the same manner as station is prior to motion. But because

cogitation itself produces these sciences, not by looking back upon

its convolution of forms, endued with an infinite power, but upon the

inclosure of _bound_ according to its definite genera; hence they

say, that the mathematical sciences take away infinite from multitude

and magnitude, and are only conversant about finite quantity. Indeed,

intellect has placed in cogitation all the principles both of multitude

and magnitude. For since it wholly consists, with reference to itself,

of similar parts, and is one and indivisible, and again divisible,

educing the ornament of forms, it participates of _bound_ and

_infinite_, from intelligible essences themselves. But it

understands, indeed, from its participation of bound, and generates

vital energies, and various reasons from the nature of infinite.

The intellections, therefore, of thought, constitute these sciences

according to the _bound_ which they contain, and not according to

an infinity of life; since they bring with them an image of intellect,

but not of life. Such then is the opinion of the Pythagoreans, and the

division of the four mathematical sciences.

CHAP. XIII.

_Another Division of the Mathematical Science, according to

Geminus._

Again, some think (among whom is Geminus) that the mathematical science

is to be divided in a different manner from the preceding. And they

consider that one of its parts is conversant with intelligibles only,

but the other with sensibles, upon which it borders; denominating as

intelligibles whatever inspections the soul rouses into energy by

herself, when separating herself from material forms. And of that which

is conversant with intelligibles they establish two, by far the first

and most principal parts, arithmetic and geometry: but of that which

unfolds its office and employment in sensibles, they appoint six parts,

mechanics, astrology, optics, geodæsia, canonics, and logistics, or

the art of reckoning. But they do not think that the military art, or

tactics, should be called any one part of mathematics, according to

the opinion of some[88]; but they consider it as using at one time

the art of reckoning, as in the numbering of legions; but at another

time geodæsia, as in dividing and measuring the spaces filled by a

field of camps. As, say they, neither the art of writing, nor the

art of healing, are any part of mathematics, though frequently both

the historian and physician use mathematical theorems. This is the

case with historians indeed, when relating the situation of climates,

or collecting the magnitudes and dimensions of cities, or their

compass and circuit: but with physicians, when elucidating by ways of

this kind, many things in their art. For Hippocrates himself shews

the utility derived to medicine from astrology, and almost all who

speak of opportune times and places. By the same reason he also, who

accommodates his work to tactics, uses indeed mathematical theorems,

yet is not on this account a mathematician, although he is sometimes

willing that a numerous camp should exhibit a very small multitude,

and forms his army according to a circular figure; but sometimes in a

quadrangular, quinquangular, or some other multangular figure, when

he desires it to appear numerous. But since these are the species of

the whole mathematical science, geometry is again divided into the

contemplation of planes, and the dimension of solids, which is called

stereometry. For there is not any peculiar treatise about points and

lines, because no figure can be produced from these without planes

or solids. For geometry treats of nothing else in every one of its

parts, than that it may constitute either planes or solids: or that

when constituted, it may compare and divide them among themselves.

In like manner, arithmetic is distributed into the contemplation of

linear, plane, and solid numbers. For it considers the species of

numbers separate from sensible connections, proceeding from unity,

and the origin of plane numbers; I mean of the similar, dissimilar,

and solid, even to the third increase. But geodæsia, and the art

of reckoning, are divided similarly to arithmetic and geometry, as

they do not discourse concerning intelligible numbers or figures,

but of such as are sensible alone. For neither is it the office of

geodesia to measure the cylinder or the cone, but material masses

as if they were cones, and wells as if they were cylinders. Neither

does it accomplish this purpose by intelligible right lines, but by

such as are sensible, sometimes indeed by a more certain means, as

by the solar rays: but at other times by grosser ones, as by a line

and perpendicular. In like manner, the reckoner does not survey the

passions of numbers by themselves, but as they are resident in sensible

objects. From whence he also imposes a name upon these derived from

the things which he reckons, calling them μηλίαι, & φιαλίται. Besides

this, he does not, admit of any least, like the arithmetician, who

receives that minimum, as a genus of relation. For some one man is

considered by him as the measure of the whole multitude of men, as

unity also is the common measure of all numbers. Again, optics and

canonics are produced from geometry and arithmetic. And optics uses

the visual rays which are constituted by the rays of the eyes, as

lines and angles. But it is divided into that which is properly called

optics (because it renders the cause of these appearances, which are

accustomed to present themselves to us different from their reality, on

account of the different situations and distances of visible objects,

as the coincidence of parallel lines, or the appearance of quadrangles

as if they were circles); and into universal catoptrics, which is

conversant about various and manifold refractions, and is connected

with imaginative or conjectural knowledge: as also into that which

is called sciography[89], or the delineation of shadows, which shews

how appearances in images may seem neither inelegant nor deformed, on

account of the distances and altitudes of the things designed. But

canonics (music) or the regular art, considers the apparent reasons

of harmonies, finding out the sections of rules, every where using

the assistance of sense, and, as Plato says, seeming to prefer the

testimony of the ears to intellect itself. But to the parts we have

hitherto enumerated, mechanics must he added, as it is a certain part

of the whole science, and of the knowledge of sensible objects, and

of things united with matter. But under this exists the art effective

of instruments, which is called (ὀργανoποιητικὴ) I mean of those

instruments proper for the purposes of war: such, indeed, as Archimedes

is reported to have constructed, resisting the besiegers of sea and

land; and that which is effective of miracles, and which is called

(θαυματοποιητικὴ.) One part of this constructs with the greatest

artifice pneumatic engines, such as Ctesibius and Heron fabricated: but

another operates with weights, the motion of which is reckoned to be

the cause of inequilibrity; but their station of equilibrity, as Timæus

also has determined: and again, another part imitates animate foldings

and motions by strings and ropes. Again, under mechanics is placed

the knowledge of equilibriums, and of such instruments as are called

centroponderants: also (σφαιροποιία) or the art effective of spheres,

imitating the celestial revolutions, such as Archimedes fabricated;

and lastly, every thing endued with a power of moving matter. But the

last of all is astrology, which treats of the mundane motions, of the

magnitudes of the celestial bodies, their figures and illuminations,

their distances from the earth, and every thing of this kind; assuming

many things indeed to itself from sense, but communicating much with

the natural speculation. One part of this is gnomonics, which is

exercised in settling the dimension of hororary gnomons: but the other

is metheoroscopics, which finds out the differences of elevations, and

the distances of the stars, and also teaches many other and various

astrological theorems. The third part is dioptrics, which ascertains by

dioptric instruments of this kind the distances of the sun and moon,

and of the five other stars. And such is the account of the parts of

the mathematical science, delivered by the ancients, and transmitted to

our memory by the informing hand of time.