[4] Dr. Young, in his Night Thoughts.

[5] See book the second, of Aristotle’s Metaphysics.

[6] Ennead vi. lib. vii.

[7] In his commentary on the 2d, 12th, and 13th books of Aristotle’s

Metaphysics, page 60. A Latin translation only of this invaluable work

is extant; but I have fortunately a copy in my possession, with the

version every where corrected by the learned Thomas Gale, and with

large extracts from the Greek.

[8] See Proclus on Plato’s Theology, p. 226.

[9] Ennead vi. lib. 6.

[10] In giving monadic number a subsistence in opinion, I have followed

the distribution of Proclus, in the conclusion of his comment on a

point; and, I think, not without sufficient reason. For since monadic

numbers are more immaterial than geometrical lines and figures, they

must have a more immaterial subsistence. But as they are correspondent

to matter, they cannot reside in the essential reasons of the soul;

nor can they subsist in the phantasy, because they are superior to

geometrical figures. It remains, therefore, that we must place them

between διάνοια or cogitation, and the phantasy; and this middle

situation is that of opinion. For cogitation, which Plato defines,

in his Sophista, to be an inward discourse, without voice, is an

energy of the rational soul, extending itself from propositions to

conclusions. And, according to Plato, in the same place, opinion is

the silent affirmation, or negation of διάνοια, or thought. Hence,

says he, “opinion is the conclusion of cogitation; but imagination,

the mutual mixture of sense and opinion.” So that opinion may, with

great propriety, be said to contain monadic number, to which it bears

the proportion of matter. And hence the reason is obvious, why the

Pythagoreans called the duad opinion.

[11]

Ἄτροπον, ἀκαμάτον Δεκάδα κλείουσιν μιν ἁγιὴν,

Ἀθάνατοί τε θεοὶ καὶ γηγενέεις ἃνθρωποι.

Syrian. in Meta. Aristot. p. 113. Gr.

i.e. (According to the Pythagoreans) “the immortal gods and earth-born

men, call the venerable decad, immutable and unwearied.”

[12] Αυτὸς μὲν Πυθαγόρας ἐν τῷ ἱερῷ λόγῳ διαῤῥηδην μορφῶν καὶ ἰδεῶν

κράντορα τὸν ἀριθμόν ἔλεγεν εἶναι.

Vid. Syrian. in Arist. Meta. p. 85. Gr.

[13] Φιλόλαος δέ, τῆς τῶν κοσμικὼν αἰωνίας διαμονῆς τὴν κρατιστεύουσαν

καὶ αὐτογειῆ συνοχὴν εἶναι ἀπεφήνατο τὸν ἀριθμόν.

Syrian. in eodem loco.

[14] Οἱ δὲ περὶ Ἴππασον ἀκουσματικοὶ, ἀριθμόν εἶπον παράδειγμα πρῶτον

κοσμοποιίας. Καὶ πάλιν κριτικὸν κοσμουργοῦ θεοῦ ὄργανον.

Jamb. in Nicomach. Arith. p. 11.

[15] In his Mathematical Lectures, page 48.

[16] In Arithmet. p. 23.

[17] In Aristot. Meta. p. 113. Gr. vel 59. b. Lat.

[18] For the tetrad contains all numbers within its nature, in the

manner of an exemplar; and hence it is, that in monadic numbers, 1, 2,

3, 4, are equal to ten.

[19] Notes to Letters on Mind, page 83.

[20] This bright light is no other than that of ideas themselves;

which, when it is once enkindled, or rather re-kindled in the soul,

becomes the general standard, and criterion of truth. He who possesses

this, is no longer the slave of opinion; puzzled with doubts, and lost

in the uncertainties of conjecture. Here the fountain of evidence is

alone to be found.--This is the true light, whose splendors can alone

dispel the darkness of ignorance, and procure for the soul undecaying

good, and substantial felicity. Of this I am certain, from my own

experience; and happy is he who acquires this invaluable treasure. But

let the reader beware of mixing the extravagancies of modern enthusiasm

with this exalted illumination. For this light is alone brought into

the mind by science, patient reflection, and unwearied meditation:

it is not produced by any violent agitation of spirits, or extasy of

imagination; for it is far superior to the energies of these: but it is

tranquil and steady, intellectual and divine. Avicenna, the Arabian,

was well acquainted with this light, as is evident from the beautiful

description he gives of it, in the elegant introduction of Ebn Tophail,

to the Life of Hai Ebn Yokdhan. “When a man’s desires (says he) are

considerably elevated, and he is competently well exercised in these

speculations, there will appear to him some small glimmerings of the

truth, as it were flashes of lightning, very delightful, which just

shine upon him, and then become extinct. Then the more he exercises

himself, the oftener will he perceive them, till at last he will

become so well acquainted with them, that they will occur to him

spontaneously, without any exercise at all; and then as soon as he

perceives any thing, he applies himself to the divine essence, so as to

retain some impression of it; then something occurs to him on a sudden,

whereby he begins to discern the _truth_ in every thing; till

through frequent exercise he at last attains to a perfect tranquillity;

and that which used to appear to him only by fits and starts, becomes

habitual, and that which was only a glimmering before, a constant

light; and he obtains a constant and steady knowledge.” He who desires

to know more concerning this, and a still brighter light, that arising

from an union with the supreme, must consult the eighth book of

Plotinus’ fifth Ennead, and the 7th and 9th of the sixth, and his book

on the Beautiful, of which I have published a translation.

[21] Lest the superficial reader should think this is nothing more than

declamation, let him attend to the following argument. If the soul

possesses another eye different from that of sense (and that she does

so, the sciences sufficiently evince), there must be, in the nature

of things, species accommodated to her perception, different from

feasible forms. For if our intellect speculates things which have no

real subsistence, such as Mr. Locke’s ideas, its condition must be much

more unhappy than that of the sensitive eye, since this is co-ordinated

to beings; but intellect would speculate nothing but illusions. Now, if

this be absurd, and if we possess an intellectual eye, which is endued

with a visive power, there must be forms correspondent and conjoined

with its vision; forms immoveable, indeed, by a corporeal motion, but

moved by an intellectual energy.

[22] The present section contains an illustration of almost all the

first book of Aristotle’s last Analytics. I have for the most part

followed the accurate and elegant paraphrase of Themistius, in the

execution of this design, as the learned reader will perceive: but I

have likewise everywhere added elucidations of my own, and endeavoured

to render this valuable work intelligible to the thinking mathematical

reader.

[23] See the twenty-eighth proposition of the first book of Euclid’s

Elements.

[24] We are informed by Simplicius, in his Commentary on Aristotle’s

third Category of Relation, “that though the quadrature of the

circle seems to have been unknown to Aristotle, yet, according to

Jamblichus, it was known to the Pythagoreans, as appears from the

sayings and demonstrations of Sextus Pythagoricus, who received (says

he) by succession, the art of demonstration; and after him Archimedes

succeeded, who invented the quadrature by a line, which is called

the line of Nicomedes. Likewise, Nicomedes attempted to square the

circle by a line, which is properly called τεταρτημόριον, or _the

quadrature_. And Apollonius, by a certain line, which he calls the

sister of the curve line, similar to a cockle, or tortoise, and which

is the same with the _quadratix_ of Nicomedes. Also Carpus wished

to square the circle, by a certain line, which he calls simply formed

from a twofold motion. And many others, according to Jamblichus, have

accomplished this undertaking in various ways.” Thus far Simplicius.

In like manner, Boethius, in his Commentary on the same part of

Aristotle’s Categories (p. 166.) observes, that the quadrature of

the circle was not discovered in Aristotle’s time, but was found out

afterwards; the demonstration of which (says he) because it is long,

must be omitted in this place. From hence it seems very probable,

that the ancient mathematicians applied themselves solely to squaring

the circle geometrically, without attempting to accomplish this by an

arithmetical calculation. Indeed, nothing can be more ungeometrical

than to expect, that if ever the circle be squared, the square to which

it is equal must be commensurable with other known rectilineal spaces;

for those who are skilled in geometry know that many lines and spaces

may be exhibited with the greatest accuracy, geometrically, though they

are incapable of being expressed arithmetically, without an infinite

series. Agreeable to this, Tacquet well observes (in lib. ii. Geom.

Pract. p. 87.) “Denique admonendi hic sunt, qui geometriæ, non satis

periti, sibi persuadent ad quadraturam necessarium esse, ut ratio lineæ

circularis ad rectam, aut circuli ad quadratum in numeris exhibeatur.

Is sane error valde crassus est, et indignus geometrâ, quamvis enim

irrationalis esset ea proportio, modo in rectis lineis exhibeatur,

reperta erat quadratura.” And that this quadrature is possible

geometrically, was not only the opinion of the above mentioned learned

and acute geometrician, but likewise of Wallis and Barrow; as may be

seen in the Mechanics of the former, p. 517 and in the Mathematical

Lectures of the latter, p. 194. But the following discovery will, I

hope, convince the liberal geometrical reader, that the quadrature of

the circle may be obtained by means of a circle and right-line only,

which we have no method of accomplishing by any invention of the

ancients or moderns. At least this method, if known to the ancients, is

now lost, and though it has been attempted by many of the moderns, it

has not been attended with success.

In the circle _g o e f_, let _g o_ be the quadrantal arch, and the

right-line _g x_ its tangent. Then conceive that the central point

_a_ flows uniformly along the radius _a e_, infinitely produced; and

that it is endued with an uniform impulsive power. Let it likewise be

supposed, that during its flux, radii emanate from it on all sides,

which enlarge themselves in proportion to the distance of the point

_a_ from its first situation. This being admitted, conceive that the

point _a_ by its impulsive power, through the radii _a n_, _a m_, &c.

acting every where equally on the arch _g o_, impells it into its equal

tangent arch _g r_. And when, by its uniform motion along the infinite

line _a_ φ, it has at the same time arrived at _b_, the centre of the

arch _g r_, let it impel in a similar manner the arch _g r_, into its

equal tangent arch _g s_, by acting every where equally through radii

equal to _b r_. Now, if this be conceived to take place infinitely

(since a circular line is capable of infinite remission) the arch _g

o_ will at length be unbent into its equal, the tangent line _g x_;

and the extreme point _o_, will describe by such a motion of unbending

a circular line _o x_. For since the same cause, acting every where

similarly and equally, produces every where similar and equal effects;

and the arch _g o_, is every where equally remitted or unbent, it

will describe a line similar in every part. Now, on account of the

simplicity of the impulsive motion, such a line must either be straight

or circular; for there are only three lines every where similar, i. e.

the right and circular line, and the cylindric helix; but this last,

as Proclus well observes in his following Commentary on the fourth

definition, is not a simple line, because it is generated by two simple

motions, the rectilineal and circular. But the line which bounds more

than two equal tangent arches cannot be a right line, as is well known

to all geometricians; it is therefore a circular line. It is likewise

evident, that this arch _o x_ is concave towards the point _g_: for if

not, it would pass beyond the chord _o x_, which is absurd. And again,

no arch greater than the quadrant can be unbent by this motion: for any

one of the radii, as _a p_ beyond _g o_, has a tendency from, and not

to the tangent _g x_, which last is necessary to our hypothesis. Now if

we conceive another quadrantal arch of the circle _g o e f_, that is

_g y_, touching the former in _g_ to be unbent in the same manner, the

arch _x y_ shall be a continuation of the arch _x o_; for if _γ x κ_ be

drawn perpendicular to _x g_, as in the figure, it shall be a tangent

in _x_ to the equal arches _y x_, _x o_; because it cannot fall within

either, without making the sine of some one of the equal arches, equal

to the right-line _x g_, which would be absurd. And hence we may easily

infer, that the centre of the arch _y x o_, is in the tangent line _x

g_. Hence too, we have an easy method of finding a tangent right-line

equal to a quadrantal arch: for having the points _y_, _o_ given, it is

easy to find a third point, as _s_; and then the circle passing through

the three points _o_, _s_, _y_, shall cut off the tangent _x g_,

equal to the quadrantal arch _g o_. And the point _s_ may be speedily

obtained, by describing the arch _g s_ with a radius, having to the

radius _a g_ the proportion of 6 to 4; for then _g s_ is the sixth part

of its whole circle, and is equal to the arch _g o_. And thus, from

this hypothesis, which, I presume, may be as readily admitted as the

increments and decrements of lines in fluxions, the quadrature of the

circle may be geometrically obtained; for this is easily found, when a

right-line is discovered equal to the periphery of a circle. I am well

aware the algebraists will consider it as useless, because it cannot be

accommodated to the farrago of an arithmetical calculation; but I hope

the lovers of the ancient geometry will deem it deserving an accurate

investigation; and if they can find no paralogism in the reasoning,

will consider it as a legitimate demonstration.

[Illustration]

[Illustration]

[25] Axioms have a subsistence prior to that of magnitudes and

mathematical numbers, but subordinate to that of ideas; or, in other

words, they have a middle situation between essential and mathematical

magnitude. For of the reasons subsisting in soul, some are more simple

and universal, and have a greater ambit than others, and on this

account approach nearer to intellect, and are more manifest and known

than such as are more particular. But others are destitute of all

these, and receive their completion from more ancient reasons. Hence it

is necessary (since conceptions are then true, when they are consonant

with things themselves) that there should be some reason, in which the

axiom asserting, _if from equals you take away equals_, _&c._

is primarily inherent; and which is neither the reason of magnitude,

nor number, nor time, but contains all these, and every thing in which

this axiom is naturally inherent. Vide Syrian. in Arith. Meta. p. 48.

[26] Geometry, indeed, wishes to speculate the impartible reasons

of the soul, but since she cannot use intellections destitute of

imagination, she extends her discourses to imaginative forms, and

to figures endued with dimension, and by this means speculates

immaterial reasons in these; and when imagination is not sufficient

for this purpose, she proceeds even to external matter, in which she

describes the fair variety of her propositions. But, indeed, even then

the principal design of geometry is not to apprehend sensible and

external form, but that interior vital one, resident in the mirror of

imagination, which the exterior inanimate form imitates, as far as its

imperfect nature will admit. Nor yet is it her principal design to

be conversant with the imaginative form; but when, on account of the

imbecility of her intellection, she cannot receive a form destitute of

imagination, she speculates the immaterial reason in the purer form of

the phantasy; so that her principal employment is about universal and

immaterial forms. Syrian. in Arist. Meta. p. 49.

[27] Syrianus, in his excellent Commentary on Aristotle’s Metaphysics,

(which does not so much explain Aristotle, as defend the doctrine of

ideas, according to Plato, from the apparent if not real opposition of

Aristotle to their existence), informs us that it is the business of

wisdom, properly so called, to consider immaterial forms or essences,

and their essential accidents. By the method of resolution receiving

the principles of being; by a divisive and and definitive method,

considering the essences of all things; but by a demonstrative process,

concluding concerning the essential properties which substances

contain. Hence (says he) because intelligible essences are of the

most simple nature, they are neither capable of definition nor

demonstration, but are perceived by a simple vision and energy of

intellect alone. But middle essences, which are demonstrable, exist

according to their inherent properties: since, in the most simple

beings, nothing is inherent besides their being. On which account we

cannot say that _this_ is their essence, and _that_ something

else; and hence they are better than definition and demonstration.

But in universal reasons, considered by themselves, and adorning a

sensible nature, essential accidents supervene; and hence demonstration

is conversant with these. But in material species, individuals, and

sensibles, such things as are properly accidents are perceived by the

imagination, and are present and absent without the corruption of their

subjects. And these again being worse than demonstrable accidents, are

apprehended by signs, not indeed by a wise man, considered as wise, but

perhaps by physicians, natural philosophers, and all of this kind.

[28] See Note to Chap. i. Book i. of the ensuing Commentaries.

[29] Page 227.

[30] Page 250.

[31] Methodus hæc cum algebrâ speciosâ facilitate contendit, evidentiâ

vero et demonstrationum elegantiâ eam longe superare videtur: ut abunde

constabit, si quis conferat hanc Apollonii doctrinam _de Sectione

Rationis_ cum ejusdem Problematis Analysi Algebraicâ, quam exhibuit

clarissimus Wallisius, tom. ii. Operum Math. cap. liv. p. 220.

[32] Verum perpendendum est, aliud esse problema aliqualiter resolutum

dare, quod modis variis, plerumque fieri potest, aliud methodo

elegantissimâ ipsum efficere; Analysi brevissimâ et simul perspicuâ,

Synthesi concinnâ et minime operosâ.

[33] In his Mathematical Lectures, p. 44.

[34] Lib. iv.

[35] Lib. i. p. 30.

[36] In Theæteto.

[37] In his most excellent work on Abstinence, lib. i. p. 22, &c.

[38] See the Excerpta of Ficinus from Proclus, on the first Alcibiades

of Plato; his Latin version only of which is extant. Ficini Opera, tom.

ii.

[39] Marinus, the author of the ensuing life, was the disciple of

Proclus; and his successor in the Athenian school. His philosophical

writings were not very numerous, and have not been preserved. A

commentary ascribed to him, on Euclid’s data, is still extant; but

his most celebrated work, appears to have been, the present life of

his master. It is indeed in the original elegant and concise; and may

be considered as a very happy specimen of philosophical biography.

Every liberal mind must be charmed and elevated with the grandeur and

sublimity of character, with which Proclus is presented to our view. If

compared with modern philosophical heroes, he appears to be a being of

a superior order; and we look back with regret on the glorious period,

so well calculated for the growth of the philosophical genius, and the

encouragement of exalted merit. We find in his life, no traces of the

common frailties of depraved humanity; no instances of meanness, or

instability of conduct: but he is uniformly magnificent, and constantly

good. I am well aware that this account of him will be considered by

many as highly exaggerated; as the result of weak enthusiasm, blind

superstition, or gross deception: but this will never be the persuasion

of those, who know by experience what elevation of mind and purity of

life the Platonic philosophy is capable of procuring; and who truly

understand the divine truths contained in his works. And the testimony

of the multitude, who measure the merit of other men’s characters by

the baseness of their own, is surely not to be regarded. I only add,

that our Philosopher flourished 412 years after Christ, according to

the accurate chronology of Fabricius; and I would recommend those who

desire a variety of critical information concerning Proclus, to the

Prolegomena prefixed by that most learned man to his excellent Greek

and Latin edition of this work, printed at London in 1703.

[40] Plato in Phædro. Meminit et Plutarch. VIII. Sympos. Suidas in

μήτοι. Fabricius.

[41] For a full account of the distribution of the virtues according to

the Platonists, consult the sentences of Porphyry, and the Prolegomena

of Fabricius to this work.

[42] See the sixth book of his Republic, and the Epinomis.

[43] We are informed by Fabricius, that the Platonic Olympiodorus in

his MS. Commentary on the Alcibiades of Plato, divides the orders of

the Gods, into ὑπερκόσμιοι, or super-mundane, which are separate from

all connection with body; and into ἐγκόσμιοι, or mundane. And that

of these, some are οὐράνιοι, or celestial, others αἰθέριοι, or, or

etherial, or πύριοι, fiery, others ἀέριοι, or aerial, others ἔνυδροι,

or watry, others χθόνιοι, or earthly; and others ὑποταρτάριοι, or

subterranean. But among the terrestrial, some are κλιματάρχαι, or

governors of climates, others πολιοῦχοι, or rulers over cities, and

others lastly κατοικίδιοι, or governors of houses.

[44] This epithet is likewise ascribed by Onomacritus to the Moon, as

may be seen in his hymn to that deity; and the reason of which we have

given in our notes to that hymn.

[45] Divine visions, and extraordinary circumstances, may be fairly

allowed to happen to such exalted geniuses as Proclus; but deserve

ridicule when ascribed to the vulgar.

[46] What glorious times! when it was considered as an extraordinary

circumstance for a teacher of rhetoric to treat a noble and wealthy

pupil as his domestic. When we compare them with the present, we can

only exclaim, _O tempora! O mores!_ Philosophy sunk in the ruins

of ancient Greece and Rome.

[47] Fabricius rightly observes, that this Olympiodorus is not the

same with the Philosopher of that name, whose learned commentaries, on

certain books of Plato, are extant in manuscript, in various libraries.

As in these, not only Proclus himself, but Damascius, who flourished

long after Proclus, is celebrated.

[48] Concerning the various mathematicians of this name, see Fabricius

in Bibliotheca Græca.

[49] The word in the original is λογικὰ, which Fabricius rightly

conjectures has in this place a more extensive signification than

either Logic, or Rhetoric: but I must beg leave to differ from that

great critic, in not translating it simply _philosophical_, as I

should rather imagine, Marinus intended to confine it to that part of

Aristotle’s works, which comprehends only logic, rhetoric, and poetry.

For the verb ἐξεμάνθανω, or _to learn_, which Marinus uses on

this occasion, cannot with propriety be applied to the more abstruse

writings of Aristotle.

[50] Hence Proclus was called, by way of eminence, διάδοχος Πλατωνικός,

or the Platonic Successor.

[51] Concerning Polletes, see Suidas; and for Melampodes, consult

Fabricius in Bibliotheca Græca.

[52] This Syrianus was indeed a most excellent philosopher, as we may

be convinced from his commentary on the metaphysics of Aristotle,

a Latin translation only of which, by one Hieronimus Bagolinus,

was published at Venice in 1558. The Greek is extant, according to

Fabricius, in many of the Italian libraries, and in the Johannean

library at Hamburg. According to Suidas, he writ a commentary on the

whole of Homer in six books; on Plato’s politics, in four books; and

on the consent of Orpheus, Pythagoras, and Plato, with the Chaldean

Oracles, in ten books. All these are unfortunately lost; and the

liberal few, are by this means deprived of treasures of wisdom, which

another philosophical age, in some distant revolution, is alone likely

to produce.

[53] Socrates, in the 6th book of Plato’s Republic, says, that from

great geniuses nothing of a middle kind must be expected; but either

great good, or great evil.

[54] The reader will please to take notice, that this great man is not

the same with Plutarch the biographer, whose works are so well known;

but an Athenian philosopher of a much later period.

[55] Aristotle’s philosophy, when compared with the discipline of

Plato is, I think, deservedly considered in this place as bearing the

relation of the proteleia to the epopteia in sacred mysteries. Now the

proteleia, or things previous to perfection, belong to the initiated,

and the mystics; the former of whom were introduced into some lighter

ceremonies only: but the mystics, were permitted to be present with

certain preliminary and lesser sacred concerns. On the other hand, the

epoptæ were admitted into the sanctuary of the greater sacred rites;

and became spectators of the symbols, and more interior ceremonies.

Aristotle indeed appears to be every where an enemy to the doctrine of

ideas, as understood by Plato; though they are doubtless the leading

stars of all true philosophy. However, the great excellence of his

works, considered as an introduction to the divine theology of Plato,

deserves the most unbounded commendation. Agreeable to this, Damascius

informs us that Isidorus the philosopher, “when he applied himself to

the more holy philosophy of Aristotle, and saw that he trusted more to

necessary reasons than to his own proper sense, yet did not entirely

employ a divine intellection, was but little solicitous about his

doctrine: but that when he had tasted of Plato’s conceptions, he no

longer deigned to behold him in the language of Pindar. But hoping he

should obtain his desired end, if he could penetrate into the sanctuary

of Plato’s mind, he directed to this purpose the whole course of his

application.” Photii Bibliotheca. p. 1034.

[56] _according to the oracle._

[57] Nothing is more celebrated by the ancients than that strict

friendship which subsisted among the Pythagoreans; to the exercise of

which they were accustomed to admonish each other, _not to divide

the god which they contained_, as Jamblichus relates, lib. i. c.

33. De Vita Pythagoræ. Indeed, true friendship can alone subsist in

souls, properly enlightened with genuine wisdom and virtue; for it then

becomes an union of intellects, and must consequently be immortal and

divine.

[58] Pythagoras, according to Damascius, said, that friendship was the

mother of all the political virtues.

[59] A genuine modern will doubtless consider the whole of Proclus’

religious conduct as ridiculously superstitious. And so, indeed,

at first sight, it appears; but he who has penetrated the depths

of ancient wisdom, will find in it more than meets the vulgar ear.

The religion of the Heathens, has indeed, for many centuries, been

the object of ridicule and contempt: yet the author of the present

work is not ashamed to own, that he is a perfect convert to it in

every particular, so far as it was understood and illustrated by

the Pythagoric and Platonic philosophers. Indeed the theology of

the ancient, as well as of the modern vulgar, was no doubt full of

absurdity; but that of the ancient philosophers, appears to be worthy

of the highest commendations, and the most assiduous cultivation.

However, the present prevailing opinions, forbid the defence of such

a system; for this must be the business of a more enlightened and

philosophic age. Besides, the author is not forgetful of Porphyry’s

destiny, whose polemical writings were suppressed by the decrees of

emperors; and whose arguments in defence of his religion were so

very futile and easy of solution, that, as St. Hierom informs us,

in his preface on Daniel, Eusebius answered him in twenty-five, and

Apollinaris in thirty volumes!

[60] See Proclus on Plato’s Politics, p. 399. Instit. Theolog. num.

196; and the extracts of Ficinus from Proclus’s commentary on the first

Alcibiades, p. 246. &c.

[61] Alluding to the beautiful description given of Ulysses, in the 3d

book of the Iliad, v. 222.

Καί ἔπεα νιφάδεσιν ἐοικότα χειμερίησιν.

Which is thus elegantly paraphrased by Mr. Pope.

But when he speaks, what elocution flows!

Soft as the fleeces of descending snows

The copious accents fall, with easy art;

Melting they fall, and sink into the heart! &c.

[62] Concerning Domninus, see Photius and Suidas from Damascius in his

Life of Isidorus.

[63] Nicephorus, in his commentary on Synesius de Insomniis, p. 562.

informs us, that the hecatic orb, is a golden sphere, which has a

sapphire stone included in its middle part, and through its whole

extremity, characters and various figures. He adds, that turning this

sphere round, they perform invocations, which they call Jyngæ. Thus

too, according to Suidas, the magician Julian of Chaldea, and Arnuphis

the Egyptian, brought down showers of rain, by a magical power. And by

an artifice of this kind, Empedocles was accustomed to restrain the

fury of the winds; on which account he was called ἀλεξάνεμος, or a

chaser of winds.

[64] No opinion is more celebrated, than that of the metempsychosis

of Pythagoras: but perhaps, no doctrine is more generally mistaken.

By most of the present day it is exploded as ridiculous; and the few

who retain some veneration for its founder, endeavour to destroy the

literal, and to confine it to an allegorical meaning. By some of the

ancients this mutation was limited to similar bodies: so that they

conceived the human soul might transmigrate into various human bodies,

but not into those of brutes; and this was the opinion of Hierocles, as

may be seen in his comment on the Golden Verses. But why may not the

human soul become connected with subordinate as well as with superior

lives, by a tendency of inclination? Do not similars love to be united;

and is there not in all kinds of life, something similar and common?

Hence, when the affections of the soul verge to a baser nature, while

connected with a human body, these affections, on the dissolution of

such a body, become enveloped as it were, in a brutal nature, and the

rational eye, in this case, clouded with perturbations, is oppressed by

the irrational energies of the brute, and surveys nothing but the dark

phantasms of a degraded imagination. But this doctrine is vindicated

by Proclus with his usual subtilty, in his admirable commentary on the

Timæus, lib. v. p. 329, as follows, “It is usual, says he, to enquire

how souls can descend into brute animals. And some, indeed, think

that there are certain similitudes of men to brutes, which they call

savage lives: for they by no means think it possible that the rational

essence can become the soul of a savage animal. On the contrary,

others allow it may be sent into brutes, because all souls are of

one and the same kind; so that they may become wolves and panthers,

and ichneumons. But true reason, indeed, asserts that the human soul

way be lodged in brutes, yet in such a manner, as that it may obtain

its own proper life, and that the degraded soul may, as it were, be

carried above it, and be bound to the baser nature, by a propensity and

similitude of affection. And that this is the only mode of insinuation,

we have proved by a multitude of reasons, in our commentaries on the

Phædrus. But if it is requisite to take notice, that this is the

opinion of Plato, we add, that in his politics, he says, that the soul

of Thersites assumed an ape, but not the body of an ape: and in the

Phædrus, that the soul descends into a savage life, but not into a

savage body; for life is conjoined with its proper soul. And in this

place he says it is changed into a brutal nature: for a brutal nature

is not a brutal body, but a brutal life.”

[65] Pericles Lydus, a Stoic philosopher.

[66] Vide Pausan. lib. i. Atticorum, cap. 21. et 20.

[67] He means the Christians.

[68] Proclus was born in the year of Christ 412, on the 6th of the Ides

of February. But, for the sake of the astrologers, I have subjoined

the following figure from the Prolegomena of Fabricius to this life:

and though I am not skilled in the art myself, I am persuaded, from

the arguments of Plotinus, that it contains many general truths; but

when made subservient to particulars, is liable to great inaccuracy

and error. In short, its evidence is wholly of a physiognomic nature;

for such is the admirable order and connection of things, that

throughout the universe, one thing is signified by another, and wholes

are after a manner contained in their parts. So that the language of

the obscure and profound Heraclitus is perfectly just, when he says,

“You must connect the perfect and the imperfect, the agreeing and

the disagreeing, the consonant and the dissonant, and out of one all

things, and out of all things one.”

Illustration: A Scheme of the situation of the Stars, such as it

was at Byzantium, when the philosopher Proclus was born.

[69] It was formerly the custom of almost all nations, to have their

burial places in the suburbs, and not in the city itself.

[70] This eclipse happened, according to Fabricius, in A. C. 484. 19

Cal. Feb. at sun-rise.

[71] All the ancient theologists, among whom Plato holds a

distinguished rank, affirmed that the soul was of a certain middle

nature and condition between intelligibles and sensibles: agreeable

to which doctrine, Plotinus divinely asserts that she is placed in

the horizon, or in the boundary and isthmus, as it were, of eternal

and mortal natures; and hence, according to the Magi, she is similar

to the moon, one of whose parts is lucid, but the other dark. Now the

soul, in consequence of this middle condition, must necessarily be the

receptacle of all middle energies, both vital and gnostic; so that her

knowledge is inferior to the indivisible simplicity of intellectual

comprehension, but superior to the impulsive perceptions of sense.

Hence the mathematical genera and species reside in her essence, as

in their proper and natural region; for they are entirely of a middle

nature, as Proclus proves in this and the sixth following chapter.

But this doctrine of Plato’s, originally derived from Brontinus and

Archytas, is thus elegantly explained by that philosopher, in the

concluding part of the sixth book of his Republic. “Socrates, know

then, they are, as we say, two (the Good itself, and the Sun,) and

that the one reigns over the intelligible world, but the other over

the visible, not to say the heavens, lest I should deceive you by the

name. You comprehend then, these two orders of things, I mean the

visible and the intelligible?--Glauco. I do.--Socrates. Continue this

division then, as if it were a line divided into two unequal segments;

and each part again, i. e. the sensible and intelligible, divided

after a similar manner, and you will have evidence and obscurity

placed by each other. In the visible segment, indeed, one part will

contain images. But I call images, in the first place, shadows;

afterwards, the resemblances of things appearing in water, and in

dense, smooth, and lucid bodies, and every thing of this kind, if you

apprehend me?--Glauco. I apprehend you.--Socrates. Now conceive that

the other section comprehends the things, of which these images are

nothing more than similitudes, such as the animals around us, together

with plants, and whatever is the work of nature and art.--Glauco. I

conceive it.--Socrates. Do you consider this section then, as divided

into true and false? And that the hypothesis of opinion is to the

knowledge of science, as a resemblance to its original?--Glauco. I

do, very readily.--Socrates. Now then, consider how the section of

the intelligible is to be divided.--Glauco. How?--Socrates. Thus:

one segment is that which the soul enquires after, using the former

divisions as images, and compelled to proceed from hypotheses, not

to the principle, but to the conclusion. The other is that which

employs the cogitative power of the soul, as she proceeds from

an hypothesis to a principle no longer supposed, and, neglecting

images, advances through their _obscurity_ into _the light of_ ideas

themselves.--Glauco. I do not, in this, sufficiently understand

you.--Socrates. But again, for you will more easily understand me from

what has been already premised. I think you are not ignorant, that

those who are conversant in geometry, arithmetic, and the like, suppose

even and odd, together with various figures, and the three species of

angles, and other things similar to these, according to each method of

proceeding. Now, having established these, as hypotheses sufficiently

known, they conceive that no reason is to be required for their

position: but beginning from these, they descend through the rest, and

arrive at last, at the object of their investigation.--Glauco. This

I know perfectly well.--Socrates. This also you know, that they use

visible forms, and make them the subject of their discourse, at the

same time not directing their intellect to the perception of these,

but to the _originals_ they resemble; I mean the square itself, and

the diameter itself; and not to the figures they delineate. And thus,

other forms, which are represented by shadows and images in water,

are employed by them, merely as resemblances, while they strive to

behold that which can be seen by cogitation alone.--Glauco. You speak

the truth.--Socrates. This is what I called above a species of the

intelligible, in the investigation of which, the soul was compelled to

use hypotheses; not ascending to the principle, as incapable of rising

above hypotheses, but using the images formed from inferior objects,

to a similitude of such as are superior, and which are so conceived

and distinguished by opinion, as if they perspicuously contributed to

the knowledge of things themselves.--Glauco. I understand indeed, that

you are speaking of the circumstances which take place in geometry,

and her kindred arts.--Socrates. Understand now, that by the other

section of the intelligible, I mean that which reason herself reaches,

by her power of demonstrating, when no longer esteeming hypotheses for

principles, but receiving them in reality for hypotheses, she uses

them as so many steps and handles in her ascent, until she arrives at

that which is no longer hypothetical, the principle of the universe;

and afterwards descending, holding by ideas which adhere to the

principle, she arrives at the conclusion, employing nothing sensible

in her progress, but proceeding through ideas, and in these at last

terminating her descent.--Glauco. I understand you, but not so well

as I desire: for you seem to me to propose a great undertaking. You

endeavour, indeed, to determine that the portion of true being and

intelligible, which we speculate by the science of demonstration, is

more evident than the discoveries made by the sciences called arts;

because in the first hypotheses are principles, and their masters are

compelled to employ the eye of cogitation, and not the perceptions

of the senses. Yet, because they do not ascend to the principle,

but investigate from hypotheses, they seem to you not to have

intelligence concerning these, though they are intelligible, through

the light of the principle. But you seem to me to call the habit of

reasoning on geometrical and the like concerns, cogitation, rather

than intelligence, as if cogitation held the middle situation between

opinion and intellect.--Socrates. You understand me sufficiently well.

And again: with these four proportions take these four corresponding

affections of the soul: with the highest intelligence; with the second

cogitation; against the third set opinion; and against the fourth

assimilation, or imagination. Besides this, establish them in the

order of alternate proportion, so that they may partake of evidence,

in the same manner as their corresponding objects participate of

reality.” I have taken the liberty of translating this fine passage

differently from both Petvin and Spens; because they have neglected

to give the proper meaning of the word διάνοια, or cogitation, the

former translating it _mind_, and the _eye of the mind_, and by

this means confounding it with intellect; and the latter calling it

_understanding_. But it is certain that Plato, in this place, ranks

intellect as the first, on account of the superior evidence of its

perceptions; in the next place, cogitation; in the third, opinion;

and in the fourth, imagination. However, the reader will please to

remember, that by διάνοια, or cogitation, in the present work, is

understood that power of the soul which reasons from premises to

conclusions, and whose syllogistic energy, on active subjects, is

called prudence; and on such as are speculative, science. But for

farther information concerning its nature, see the dissertation

prefixed to this work, and the following fifth chapter.

[72] These two principles, _bound_ and _infinite_, will

doubtless be considered by the unthinking part of mankind, as

nothing more than general terms, and not as the most real of beings.

However, an accurate contemplation of the universe, will convince

every _truly_ philosophic mind of their reality. For the heavens

themselves, by the coherence and order of their parts, evince their

participation of bound. But by their prolific powers, and the

unceasing revolutions of the orbs they contain, they demonstrate their

participation of _infinity_. And the finite and perpetually

abiding forms with which the world is replete, bear a similitude _to

bound_: while, on the contrary, the variety of particulars, their

never-ceasing mutation, and the connection of more and less in the

communion of forms, represents an image of _infinity_. Add too,

that every natural species, by its form is similar to _bound_;

but by its matter, to _infinity_. For these two, form and matter,

depend on _bound_ and _infinity_, and are their ultimate

progressions. And each of these, indeed, participates of unity; but

form is the measure and bound of matter, and is more _one_. But

matter is in capacity all things, because it subsists by an emanation

from the first capacity, or the _infinite itself_.

[73] Of human disciplines, those alone deserve to be called sciences

which use no hypotheses, which resolve things into their principles,

which are conversant with true being, and elevate us to ideas

themselves. Dialectic is wholly of this kind (I mean the dialectic of

Plato); for this alone uses no suppositions, but, neglecting shadows

and images, raises us, by a sublime investigation, to the principle

of the universe; and on this account, deserves to be called the

very apex of disciplines. But we must not imagine, that by the word

dialectic here, is meant logic, or any part of logic, or that method

of disputation, by which we fabricate probable reasons; but we must

conceive it as signifying a discipline, endued with the greatest

acuteness; neglecting all hypotheses, truly soaring to primary causes,

and ultimately reposing in their contemplation. Plotinus has given us

most happy specimens of this method, in his books _on the genera of

being_.

[74] See note to the first chapter.

[75] I would particularly recommend this chapter to modern

mathematicians, most of whom, I am afraid, have never considered

whether or not the subjects of their speculation have any real

subsistence: though it is surely an enquiry worthy the earnest

attention of every liberal mind. For if the objects of mathematical

investigation are merely imaginary, I mean the point without parts,

the line without breadth, &c. the science, founded on these false

principles, must of course be entirely delusive. Indeed, an absolutely

true conclusion, can never flow from an erroneous principle, as from

its cause: as the stream must always participate of its source. I

mean such a conclusion as is demonstrated by the proper cause, πλὴν

οὐ διότι, ἀλλ’ ὅτι, says Aristotle, in his first Analytics; that is,

a syllogism from false principles will not prove the _why_, but

only simply _that it is_: indeed it can only simply prove _that

it is_, to him who admits the false propositions; because he who

allows the premises, cannot deny the conclusion, when the syllogism is

properly constructed. Thus we way syllogize in the first figure,

Every thing white, is an animal:

Every bird is white:

Therefore, Every bird is an animal.

And the conclusion will be true, though the major and minor terms are

false; but then these terms are not the causes of the conclusion, and

we have an inference without a proof. In like manner, if mathematical

species are delusive and fictitious, the conclusions deduced from them

as principles, are merely hypothetical, and not demonstrative.

[76] Aristotle, in his last Analytics. The reader will please to

observe, that the whole force of this nervous, accurate, and elegant

reasoning, is directed against Aristotle; who seems unfortunately to

have considered, with the moderns, that mathematical species subsist

in the soul, by an abstraction from sensibles. See the preceding

Dissertation.

[77] Viz. 1, 2, 4, 8, 3, 9, 27. Concerning which, see lib. iii. of

Proclus’s excellent Commentary on the Timæus.

[78] Plato frequently, both in the Meno and elsewhere, shews that

science is Reminiscence; and I think not without the strongest

reason. For since the soul is immaterial, as we have demonstrated

in the dissertation to this work, she must be truly immortal, i. e.

both _a parte ante_, & _a parte post_. That she must be

eternal, indeed, with respect to futurity, if immaterial, is admitted

by all; and we may prove, with Aristotle, in his first book de Cœlo,

that she is immortal, likewise _a parte ante_, as follows.

Every thing without generation, is incorruptible, and every thing

incorruptible, is without generation: for that which is without

generation, has a necessity of existing infinitely _a parte ante_

(from the hypothesis); and therefore, if it possesses a capacity of

being destroyed, since there is no greater reason why it should be

corrupted now, rather than in some former period, it is endued with

a capacity of being destroyed and ceasing to be, in every instant of

infinite time, in which it necessarily is. In like manner, that which

is incorruptible, has a necessity of existing infinitely _a parte

post_; therefore, if it possesses a capacity of being generated,

since there is no greater reason why it should be generated now rather

than afterwards, it possesses a capacity of being generated, in every

instant of time, in which it necessarily is. If then the soul is

essentially immortal, with respect to the past and future circulations

of time; and if she is replete with forms or ideas of every kind, as

we have proved in the dissertation, she must, from her circulating

nature, have been for ever conversant in alternately possessing and

losing the knowledge of these. Now, the recovery of this knowledge by

science, is called by Plato, reminiscence; and is nothing more than

a renewed contemplation of those divine forms, so familiar to the

soul, before she became involved in the dark vestment of an earthly

body. So that we may say, with the elegant Maximus Tyrus, (Disser.

28.) “Reminiscence is similar to that which happens to the corporeal

eye, which, though always endued with a power of vision, yet darkness

sometimes obstructs its passage, and averts it from the perception of

things. Art therefore, approaches, which though it does not give to the

eye the power of vision, yet removes its impediments, and affords a

free egress to its rays. Conceive now, that our rational soul is such a

power of perceiving, which sees and knows the nature of beings. To this

the common calamity of bodies happens, that darkness spreading round

it, hurries away its aspect, blunts its sharpness, and extinguishes its

proper light. Afterwards, the art of reason approaches, which, like

a physician, does not bring or afford it a new science, but rouses

that which it possesses, though very slender, confused, and unsteady.”

Hence, since the soul, by her immersion in body, is in a dormant state,

until she is roused by science to an exertion of her latent energies;

and yet even previous to this awakening, since she contains the

vivid sparks, as it were, of all knowledge, which only require to be

ventilated by the wings of learning, in order to rekindle the light of

ideas, she may be said in this case to know all things as in a dream,

and to be ignorant of them with respect to vigilant perceptions. Hence

too, we may infer that time does not antecede our essential knowledge

of forms, because we possess it from eternity: but it precedes our

knowledge with respect to a production of these reasons into perfect

energy. I only add, that I would recommend the liberal English reader,

to Mr. Sydenham’s excellent translation of Plato’s Meno, where he

will find a familiar and elegant demonstration of the doctrine of

Reminiscence.

[79] Concerning this valuable work, entitled ΙΕΡΟ‘Σ ΛΟΓΟ’Σ, see the

Bibliotheca Græca of Fabricius, vol. i. p. 118 and 462, and in the

commentary of Syrianus on Aristotle’s metaphysics, p. 7, 71, 83, and

108, the reader will find some curious extracts from this celebrated

discourse; particularly in p. 83. Syrianus informs us, “that he who

consults this work will find all the orders both of Monads and Numbers,

without neglecting one, fully celebrated (ὐμνουμένας.)” There is no

doubt, but that Pythagoras and his disciples concealed the sublimest

truths, under the symbols of numbers; of which he who reads and

understands the writings of the Platonists will be fully convinced.

Hence Proclus, in the third book of his excellent commentary on the

Timæus, observes, “that Plato employed mathematical terms for the sake

of mystery and concealment, as certain veils, by which the penetralia

of truth might be secluded from vulgar inspection, just as the

theologists made fables, but the Pythagoreans symbols, subservient to

the same purpose: for in images we may speculate their exemplars, and

the former afford us the means of access to the latter.”

[80] Concerning this Geometric Number, in the 8th book of Plato’s

Republic, than which Cicero affirms there is nothing more obscure, see

the notes of Bullialdus to Theo. p. 292.

[81] I am sorry to say, that this part of the enemies to pure geometry

and arithmetic, are at the present time very numerous; conceptions

of utility in these sciences, extending no farther than the sordid

purposes of a mere animal life. But surely, if intellect is a part of

our composition, and the noblest part too, there must be an object of

its contemplation; and this, which is no other than truth in the most

exalted sense, must be the most noble and useful subject of speculation

to every rational being.

[82] In the 13th book of his Metaphysics, cap. iii.

[83] In. I. De Partib. Animalium, et in primo Ethic. cap. iii.

[84] See more concerning this in the Dissertation.

[85] Since number is prior to magnitude, the demonstrations of

arithmetic must be more intellectual, but those of geometry more

accommodated to the rational power. And when either arithmetic or

geometry is applied to sensible concerns, the demonstrations, from the

nature of the subjects, must participate of the obscurity of opinion.

If this is the case, a true mathematician will value those parts of his

science most, which participate most of evidence; and will consider

them as degraded, when applied to the common purposes of life.

[86] This division of the mathematical science, according to the

Pythagoreans, which is nearly coincident with that of Plato, is blamed

by Dr. Barrow in his Mathematical Lectures, p. 15. as being confined

within too narrow limits: and the reason he assigns for so partial a

division, is, “because, in Plato’s time, others were either not yet

invented, or not sufficiently cultivated, or at least were not yet

received into the number of the mathematical sciences.” But I must

beg leave to differ from this most illustrious mathematician in this

affair; and to assert that the reason of so confined a distribution

(as it is conceived by the moderns) arose from the exalted conceptions

these wise men entertained of the mathematical sciences, which they

considered as so many preludes to the knowledge of divinity, when

properly pursued; but they reckoned them degraded and perverted, when

they became mixed with sensible objects, and were applied to the common

purposes of life.

[87] That is, a right and circular line.

[88] I am afraid there are few in the present day, who do not consider

tactics as one of the most principal parts of mathematics; and who

would not fail to cite, in defence of their opinions, that great

reformer of philosophy, _as he is called_, Lord Bacon, commending

pursuits which come home to men’s businesses and bosoms. Indeed, if

what is lowest in the true order of things, and best administers to the

vilest part of human nature, is to have the preference, their opinion

is right, and Lord Bacon is a _philosopher_!

[89] By this is to be understood the art new called Perspective: from

whence it is evident that this art was not unknown to the ancients,

though it is questioned by the moderns.

[90] From hence it appears, that it is doubtful whether Plato is the

author of the dialogue called Epinomis; and I think it may with great

propriety be questioned. For though it bears evident marks of high

antiquity, and is replete with genuine wisdom, it does not seem to be

perfectly after Plato’s manner; nor to contain that great depth of

thought with which the writings of this philosopher abound. Fabricius

(in his Bibliotheca Græca, lib. i. p. 27.) wonders that Suidas should

ascribe this work to a philosopher who distributed Plato’s laws into

twelve books, because it was an usual opinion; from whence it seems,

that accurate critic had not attended to the present passage.

[91] This proximate conjunction of the mathematical sciences, which

Proclus considers as subordinate to dialectic, seems to differ from

that vertex of science in this, that the former merely embraces the

principles of all science, but the latter comprehends the universal

genera of being, and speculates the principle of all.

[92] In the Meno.

[93] This is certainly the true or philosophical employment of the

mathematical science; for by this means we shall be enabled to

ascend from sense to intellect, and rekindle in the soul that divine

light of truth, which, previous to such an energy, was buried in the

obscurity of a corporeal nature. But by a contrary process, I mean,

by applying mathematical speculations, to experimental purposes, we

shall blind the liberal eye of the soul, and leave nothing in its stead

but the darkness of corporeal vision, and the phantoms of a degraded

imagination.

[94] The design of the present chapter is to prove that the figures

which are the subjects of geometric speculation, do not subsist in

external and sensible matter, but in the receptacle of imagination,

or the matter of the phantasy. And this our philosopher proves with

his usual elegance, subtilty, and depth. Indeed, it must be evident

to every attentive observer, that sensible figures fall far short

of that accuracy and perfection which are required in geometrical

definitions: for there is no sensible circle perfectly round, since

the point from which it is described is not without parts; and, as

Vossius well observes, (de Mathem. p. 4.) there is not any sphere in

the nature of things, that only touches in a point, for with some

part of its superficies it always touches the subjected plane in a

line, as Aristotle shews Protagoras to have objected against the

geometricians. Nor must we say, with that great mathematician Dr.

Barrow, in his Mathematical Lectures, page 76, “that all imaginable

geometrical figures, are really inherent in every particle of matter,

in the utmost perfection, though not apparent to sense; just as the

effigies of Cæsar lies hid in the unhewn marble, and is no new thing

made by the statuary, but only is discovered and brought to sight by

his workmanship, i. e. by removing the parts of matter by which it is

overshadowed and involved. Which made Michael Angelus, the most famous

carver, say, _that sculpture was nothing but a purgation from things

superfluous. For take all that is superfluous_, (says he) _from

the wood or stone, and the rest will be the figure you intend_. So,

if the hand of an angel (at least the power of God) should think fit to

polish any particle of matter, without vacuity, a spherical superficies

would appear to the eyes, of a figure exactly round; not as created

anew, but as unveiled and laid open from the disguises and covers of

its circumjacent matter.” For this would be giving a perfection to

sensible matter, which it is naturally incapable of receiving: since

external body is essentially full of pores and irregularities, which

must eternally prevent its receiving the accuracy of geometrical body,

though polished by the hand of an angel. Besides, what polishing

would ever produce a point without parts, and a line without breadth?

For though body may be reduced to the greatest exility, it will not

by this means ever pass into an incorporeal nature, and desert its

triple dimension. Since external matter, therefore, is by no means the

receptacle of geometrical figures, they must necessarily reside in the

catoptric matter of the phantasy, where they subsist with an accuracy

sufficient for the energies of this science. It is true, indeed, that

even in the purer matter of imagination, the point does not appear

perfectly impartible, nor the line without latitude: but then the

magnitude of the point, and the breadth of the line is indefinite, and

they are, at the same time, unattended with the qualities of body, and

exhibit to the eye of thought, magnitude alone. Hence, the figures in

the phantasy, are the proper recipients of that universal, which is

the object of geometrical speculation, and represent, as in a mirror,

the participated subsistence of those vital and immaterial forms which

essentially reside in the soul.

[95] This division is elegantly explained by Ammonius, (in Porphyr. p.

12.) as follows, “Conceive a seal-ring, which has the image of some

particular person, for instance, of Achilles, engraved in its seal, and

let there be many portions of wax, which are impressed by the ring.

Afterwards conceive that some one approaches, and perceives all the

portions of wax, stamped with the impression of this one ring, and

keeps the impression of the ring in his mind: the seal engraved in the

ring, represents the universal, prior to the many: the impression in

the portions of wax, the universal in the many: but that which remains

in the intelligence of the beholder, may be called the universal,

after and posterior to the many. The same must we conceive in genera

and species. For that best and most excellent artificer of the world,

possesses within himself the forms and exemplars of all things: so that

in the fabrication of man, he looks back upon the form of man resident

in his essence, and fashions all the rest according to its exemplar.

But if any one should oppose this doctrine, and assert that the forms

of things do not reside with their artificer, let him attend to the

following arguments. The artificer either knows, or is ignorant of that

which he produces: but he who is ignorant will never produce any thing.

For who will attempt to do that, which he is ignorant how to perform?

since he cannot act from an irrational power like nature, whose

operations are not attended with animadversion. But if he produces

any thing by a certain reason, he must possess a knowledge of every

thing which he produces. If, therefore, it is not impious to assert,

that the operations of the Deity, like those of men, are attended with

knowledge, it is evident that the forms of things must reside in his

essence: but forms are in the demiurgus, like the seal in the ring;

and these forms are said to be prior to the many, and separated from

matter. But the species man, is contained in each particular man, like

the impression of the seal in the wax, and is said to subsist in the

many, without a separation from matter. And when we behold particular

men, and perceive the same form and effigy in each, that form seared

in our soul, is said to be after the many, and to have a posterior

generation: just as we observed in him, who beheld many seals impressed

in the wax from one and the same ring. And this one, posterior to the

many, may be separated from body, when it is conceived as not inherent

in body, but in the soul: but is incapable of a real separation from

its subject.” We must here, however, observe, that when Ammonius speaks

of the knowledge of the Deity, it must be conceived as far superior

to ours. For he possesses a nature more true than all essence, and a

perception clearer than all knowledge. And as he produced all things

by his unity, so by an ineffable unity of apprehension, he knows the

universality of things.

[96] In lib. vii. Metaphys. 35 & 39.

[97] In lib. iii. de Anima, tex. 20.

[98] That is, geometry first speculates the circle delineated on

paper, or in the dust: but by the medium of the circular figure in

the phantasy, contemplates the circle resident in cogitation; and by

that universal, or circular reason, participated in the circle of the

phantasy, frames its demonstrations.

[99] In his first Analytics, t. 42. See the Dissertation to this work.

[100] Such as the proportion of the diagonal of a square to its side;

and that of the diameter of a circle, to the periphery.

[101] The gnomons, from which square numbers are produced, are odd

numbers in a natural series from unity, i. e. 1, 3, 5, 7, 9, 11, &c.

for these, added to each other continually, produce square numbers

_ad infinitum_. But these gnomons continually decrease from the

highest, and are at length terminated by indivisible unity.

[102] This doctrine of ineffable quantities, or such whose proportion

cannot be expressed, is largely and accurately discussed by Euclid, in

the tenth book of his Elements: but its study is neglected by modern

mathematicians, _because it is of no use_, that is, because it

contributes to nothing mechanical.

[103] This proposition is the 11th of the second book: at least,

the method of dividing a line into extreme and mean proportion, is

immediately deduced from it; which is done by Euclid, in the 30th, of

the sixth book. Thus, Euclid shews (11. 2.) how to divide the line (A

G B)[diagram] A B, so that the rectangle under the whole A B, and

the segment G B, may be equal to the square made from A G: for when

this is done, it follows, that as A B is to A G, so is A G to G B; as

is well known. But this proposition, as Dr. Barrow observes, cannot be

explained by numbers; because there is not any number which can be so

divided, that the product from the whole into one part, may be equal to

the square from the other part.

[104] All polygonous figures, may, it is well known, be resolved

into triangles; and this is no less true of polygonous numbers,

as the following observations evince. All number originates from

indivisible unity, which corresponds to a point: and it is either

linear, corresponding to a line; or superficial, which corresponds to

a superficies; or solid, which imitates a geometrical solid. After

unity, therefore, the first of linear numbers is the duad; just as

every finite line is allotted two extremities. The triad is the first

of superficial numbers; as the triangle of geometrical figures. And

the tetrad, is the first of solids; because a triangular pyramid, is

the first among solid numbers, as well as among solid figures. As,

therefore, the monad is assimilated to the point, so the duad to the

line, the triad to the superficies, and the tetrad to the solid. Now,

of superficial numbers, some are triangles, others squares, others

pentagons, hexagons, heptagons, &c. Triangular numbers are generated

from the continual addition of numbers in a natural series, beginning

from unity. Thus, if the numbers 1, 2, 3, 4, 5, &c. be added to each

other continually, they will produce the triangular numbers 1, 3, 6,

10, 15, &c. and if every triangular number be added to its preceding

number, it will produce a square number. Thus 3 added to 1 makes 4; 6

added to 3 is equals 9; 10 added to 6 is equal to 16; and so of the

rest. Pentagons, are produced from the junction of triangular and

square numbers, as follows. Let there be a series of triangular numbers

1, 3, 6, 10, 15, &c.

And of squares 1, 4, 9, 16, 25, &c.

Then the second square number, added to the first triangle, will

produce the first pentagon from unity, i.e. 5. The third square added

to the second triangle, will produce the second pentagon, i.e. 12;

and so of the rest, by a similar addition. In like manner, the second

pentagon, added to the first triangle, will form the first hexagon from

unity; the third pentagon and the second triangle, will form the second

hexagon, &c. And, by a similar proceeding, all the other polygons may

be obtained.

[105] Intellections are universally correspondent to their objects,

and participate of evidence or the contrary, in proportion as their

subjects are lucid or obscure. Hence, Porphyry, in his sentences,

justly observes, that “we do not understand in a similar manner with

all the powers of the soul, but according to the particular essence

of each. For with the intellect we understand intellectually; and

with the soul, rationally: our knowledge of plants is according to a

seminal conception; our understanding of bodies is imaginative; and our

intellection of the divinely solitary principle of the universe, who is

above all things, is in a manner superior to intellectual perception,

and by a super-essential energy.” Ἀφορμαὶ πρὸς τὰ Νοητὰ, (10.) So that,

in consequence of this reasoning, the speculations of geometry are then

most true, when most abstracted from sensible and material natures.

[106] See Plutarch, in the life of Marcellus.

[107] In lib. i. de Cælo, tex. 22. et lib. i. Meteo. cap. 3. Aristotle

was called demoniacal by the Platonic philosophers, in consequence of

the encomium bestowed on him by his master, Plato, “That he was the

dæmon of nature.” Indeed, his great knowledge in things subject to

the dominion of nature, well deserved this encomium; and the epithet

_divine_, has been universally ascribed to Plato, from his

profound knowledge of the intelligible world.

[108] Εἰς νοῦν, is wanting in the original, but is supplied by the

excellent translation of Barocius.

[109] Ἀλόγων, in the printed Greek, which Fabricius, in his Bibliotheca

Græca, vol. i. page 385, is of opinion, should be read ἀναλόγων; but

I have rendered the word according to the translation of Barocius,

who is likely to have obtained the true reading, from the variety of

manuscripts which he consulted.

[110] The quadrature of the Lunula is as follows.

[Illustration]

Let A B C be a right-angled triangle, and B A C a semi-circle on the

diameter B C: B N A a semi-circle described on the diameter A B; A M

C a semi-circle described on the diameter A C. Then the semi-circle B

A C is equal to the semi-circle B N A, and A M C together: (because

circles are to each other as the squares of their diameters, 31, 6.)

If, therefore, you take away the two spaces B A, A C common on both

sides, there will remain the two lunulas B N A, A M C, bounded on both

sides with circular lines, equal to the right-angled triangle B A C.

And if the line B A, be equal to the line A C, and you let fall a

perpendicular to the hypotenuse B C, the triangle B A O will be equal

to the lunular space B N A, and the triangle C O A will be equal to

the lunula C M A. Those who are curious, may see a long account of an

attempt of Hippocrates to square the circle, by the invention of the

lunulas, in Simplicius on Aristotle’s Physics, lib. i.

[111] So Barocius reads, but Fabricius Μεδμᾶιος.

[112] i. e. The five regular bodies, the pyramid, cube, octaedron,

dodecaedron and icosaedron; concerning which, and their application to

the theory of the universe, see Kepler’s admirable work, De Harmonia

Mundi.

[113] It may be doubted whether the optics and catoptrics, ascribed to

Euclid in the editions of his works are genuine: for Savil, and Dr.

Gregory, think them scarcely worthy so great a man.

[114] There are two excellent editions of this work, one by Meibomius,

in his collection of ancient authors on harmony; and the other by Dr.

Gregory, in his collection of Euclid’s works.

[115] This work is most probably lost. See Dr. Gregory’s Euclid.

[116] All this is shewn by Proclus in the following Commentaries; and

is surely most admirable and worthy the investigation of every liberal

mind; but I am afraid modern mathematicians very little regard such

knowledge, because it cannot be applied to practical and mechanical

purposes.

[117] This work is unfortunately lost.

[118] Because this is true only in isosceles and equilateral triangles.

[119] This follows from the 32d proposition of the first book of

Euclid; and is demonstrated by Dr. Barrow, in his scholium to that

proposition.

[120] The method of constructing these is shewn by our philosopher,

in his comment on the first proposition, as will appear in the second

volume of this work.

[121] The reader will please to observe, that the definitions are,

indeed, hypotheses, according to the doctrine of Plato, as may be seen

in the note to chap, i. book I. of this work.

[122] In his last Analytics. See the preceding Dissertation.

[123] That part of this work enclosed within the brackets, is wanting

in the original; which I have restored from the excellent version of

Barocius. The philosophical reader, therefore, of the original, who may

not have Barocius in his possession, will, I hope, be pleased, to see

so great a vacancy supplied; especially, as it contains the beginning

of the commentary on the definition of a point.

[124] I do not find this ænigma among the Pythagoric symbols which are

extant; so that it is probably no where mentioned but in the present

work. And I am sorry to add, that a _figure and three oboli_, in

too much the general cry of the present times.

[125] The present Comment, and indeed most of the following, eminently

evinces the truth of Kepler’s observation, in his excellent work,

_De Harmonia Mundi_, p. 118. For, speaking of our author’s

composition in the present work, which he every where admires and

defends, he remarks as follows, “oratio fluit ipsi torrentis instar,

ripas inundans, et cæca dubitationum vada gurgitesque occultans, dum

mens plena majestatis tantarum rerum, luctatur in angustiis linguæ, et

conclusio nunquam sibi ipsi verborum copiâ satisfaciens, propositionum

simplicitatem excedit.” But Kepler was skilled in the Platonic

philosophy, and appears to have been no less acquainted with the great

depth of our author’s mind than with the magnificence and sublimity of

his language. Perhaps Kepler is the only instance among the moderns,

of the philosophical and mathematical genius being united in the same

person.

[126] That is, the reason of a triangular figure (for instance) in the

phantasy, or triangle itself, is superior to the triangular nature

participated in that figure.

[127] In the tenth book of his Republic.

[128] See the Hymn to the Mother of the Gods, in my translation of the

Orphic Initiations.

[129] The philosopher here seems to contradict what he asserts in the

end of his comment on the 13th Definition: for there he asserts, that

the circle is a certain plane space. Perhaps he may be reconciled, by

considering, that as the circle subsists most according to bound, when

we speculate its essence in this respect we may define it according

to the circumference, which is the cause of its bound. But when we

consider it as participating of infinity also, though not in so eminent

a degree, and view it from its emanations from the centre as well as in

its regressions, we may define it a plane space.

[130] That is, the essential one of the soul is the mother of number;

but that which subsists in opinion is nothing more than the receptacle

of the former; just as matter is the seat of all forms. For a farther

account of the subsistence of numbers, see the first section of the

preceding Dissertation.

[131] That is, number composed from units.

[132] This sentence within the brackets, is wholly omitted in the

printed Greek.

[133] In i. De Cælo.

[134] This sentence within the brackets, which is very imperfect in the

Greek, I have supplied from the excellent translation of Barocius. In

the Greek there is nothing more than λὲγω δὲ ἑνὸν τῂν γραμμὴν δυαδός

πρὸς τὸ στερεόν.

[135] In the Greek, γὰρ ἡ μονὰς ἐκεῖ πρῶτον, ὅπου πατρικὴ μονάς ἐστι

φησὶ τὸ λόγιον. The latter part only of this oracle, is to be found

in all the printed editions of the Zoroastrian oracles; though it is

wonderful how this omission could escape the notice of so may able

critics, and learned men. It seems probable, from hence, that it is

only to be found perfect in the present work.

[136] The word τανάη, is omitted in the Greek.

[137] This and the following problems, are the 1st, 22d, and 12th

propositions of the first book. But in the two last, instead of the

word ἄπειρος or infinite, which is the term employed by Euclid, Mr.

Simson, in his edition of the Elements, uses the word unlimited. But

it is no unusual thing with this great geometrician, to alter the

words of Euclid, when they convey a philosophical meaning; as we shall

plainly evince in the course of these Commentaries. He certainly

deserves the greatest praise for his zealous attachment to the ancient

geometry: but he would (in my opinion) have deserved still more, had

he been acquainted with the Greek philosophy; and fathomed the depth

of Proclus; for then he would never have attempted to restore Euclid’s

Elements, by depriving them of some very considerable beauties.

[138] This is doubtless the reason why the proportion between a right

and circular line, cannot be exactly obtained in numbers; for on this

hypothesis, they must be incommensurable quantities; because the one

contains property essentially different from the other.

[139]

[Illustration]

The cornicular angle is that which is made from the periphery of a

circle and its tangent; that is, the angle comprehended by the arch

L A, and the right line F A, which Euclid in (16. 3.) proves to be

less than any right-lined angle. And from this admirable proposition

it follows, by a legitimate consequence, that any quantity may

be continually and infinitely increased, but another infinitely

diminished; and yet the augment of the first, how great soever it

may be, shall always be less than the decrement of the second: which

Cardan demonstrates as follows. Let there be proposed an angle of

contact B A E, and an acute angle H G I. Now if there be other lesser

circles described A C, A D, the angle of contact will be evidently

increased. And if between the right lines G H, G I, there fall other

right lines G K, G L, the acute angle shall be continually diminished:

yet the angle of contact, however increased, is always less than the

acute angle, however diminished. Sir Isaac Newton likewise observes,

in his Treatise on Fluxions, that there are angles of contact made

by other curve lines, and their tangents infinitely less than those

made by a circle and right line; all which is demonstrably certain:

yet, such is the force of prejudice, that Mr. Simson is of opinion,

with Vieta, that this part of the 16th proposition is adulterated; and

that the space made by a circular line and its tangent, is no angle.

At least his words, in the note upon this proposition, will bear such

a construction. Peletarius was likewise of the same opinion; but is

elaborately confuted by the excellent Clavius, as may be seen in his

comment on this proposition. But all the difficulties and paradoxes in

this affair, may be easily solved and admitted, if we consider, with

our philosopher, that the essence of an angle does not subsist in ether

quantity, quality, or inclination, taken singly, but in the aggregate

of them all. For if we regard the inclination of a circular line to

its tangent, we shall find it possess the property, by which Euclid

defines an angle: if we respect its participation of quantity, we shall

find it capable of being augmented and diminished; and if we regard

it as possessing a peculiar quality, we shall account for its being

incommensurable with every right-lined angle. See the Comment on the

8th Definition.

[140] In i. De Cælo.

[141] It is from this cylindric spiral that the screw is formed.

[142]

[Illustration]

The present very obscure passage, may be explained by the following

figure. Let A B C, be a right angle, and D E the line to be moved,

which is bisected in G. Now, conceive it to be moved along the lines A

B, B C, in such a manner, that the point D may always remain in A B,

and the point E in B C. Then, when the line D E, is in the situations

_d e_, _δ ε_, the point G, shall be in _g, γ_, and these

points G, _g, γ_, shall be in a circle. And any other point F in

the line D E, will, at the same time, describe an ellipsis; the greater

axis being in the line A B, when the point F is between D and G; and in

the line B C, when the point F is between G and E.

[143] That is, the soul of the world.

[144] In Timæo.

[145] The ellipsis.

[146] The cissoid. For the properties of this curve, see Dr. Wallis’s

treatise on the cycloid, p. 81.

[147] The conchoid.

[148] Thus, a right line, when considered as the side of a

parallelogram, moving circularly, generates a cylindrical superficies:

when moving circularly, as the side of a triangle, a conical surface;

and so in other lines, the produced superficies varying according to

the different positions of their generative lines.

[149] Inv ii. De Rep.

[150] In multis locis.

[151] This definition is the same with that which Mr. Simson has

adopted instead of Euclid’s, expressed in different words: for he says,

“a plane superficies is that in which any two points being taken, the

straight line between them lies wholly in that superficies.” But he

does not mention to whom he was indebted for the definition; and this,

doubtless, because he considered _it was not worth while to relate

the trifles of Proclus at full length_: for these are his own words,

in his note to proposition 7, book i. Nor has he informed us in what

respect Euclid’s definition is _indistinct_.

[152] In the Greek ἐννοιὰς, but it should doubtless be read εἰκόνας,

_images_, as in the translation of Barocius.

[153] Mr. Simson, in his note on this definition, supposes it to

be the addition of some less skilful editor; on which account, and

because it is quite useless (in his opinion) he distinguishes it from

the rest by inverted double commas. But it is surely strange that the

definition of angle in general should be accounted useless, and the

work of an unskilful geometrician. Such an assertion may, indeed, be

very suitable to a professor of experimental philosophy, who considers

the useful as inseparable from practice; but is by no means becoming

a restorer of the liberal geometry of the ancients. Besides, Mr.

Simson seems continually to forget that Euclid was of the Platonic

sect; and consequently was a philosopher as well as a mathematician.

I only add, that the commentary on the present definition is, in my

opinion, remarkably subtle and accurate, and well deserves the profound

attention of the greatest geometricians.

[154] For a philosophical discussion of the nature of quality and

quantity, consult the Commentaries of Ammonius, and Simplicius on

Aristotle’s Categories, Plotinus on the genera of beings, and Mr.

Harris’s Philosophical Arrangements.

[155] That is, the ellipsis.

[156] That is, they are either right, acute, or obtuse.

[157] This oracle is not mentioned by any of the collectors of the

Zoroastrian oracles.

[158] This, indeed, must always be the case with those geometricians,

who are not at the same time, philosophers; a conjunction no less

valuable than rare. Hence, from their ignorance of principles and

intellectual concerns, when any contemplative enquiry is proposed,

they immediately ask, in what its utility consists; considering every

thing as superfluous, which does not contribute to the solution of some

practical problem.

[159] Concerning the soul’s descent into body, see lib. ix., Ennead

iv. of Plotinus; and for the method by which she may again return to

her pristine felicity, study the first book of Porphyry’s Treatise on

Abstinence.

[160] This Definition too, is marked by Mr. Simson with inverted

commas, as a symbol of its being interpolated. But for what reason

I know not, unless because it is useless, that is, because it is

_philosophical_!

[161] That is, the various species of forms, with which the four

elements are replete.

[162] That is, the circle.

[163] An admirer of the moderns, and their pursuits, will doubtless

consider all this as the relics of heathen superstition and ignorance;

and will think, perhaps, he makes a great concession in admitting

the existence of one supreme god, without acknowledging a multitude

of deities subordinate to the first. For what the ancients can urge

in defence of this obsolete opinion, I must beg leave to refer the

reader to the dissertation prefixed to my translation of Orpheus; in

addition to which let him attend to the following considerations. Is it

possible that the machinery of the gods in Homer could be so beautiful,

if such beings had no existence? Or can any thing be beautiful which

is destitute of all reality? Do not things universally please in

proportion as they resemble reality? Perhaps it will be answered,

that the reverse of this is true, and that fiction more generally

pleases than truth, as is evident from the great avidity with which

romances are perused. To this I reply, that fiction itself ceases to be

pleasing, when it supposes absolute impossibilities: for the existence

of genii and fairies cannot be proved impossible; and these compose all

the marvellous of romance. This observation is verified in Spencer’s

Fairy Queen: for his allegories, in which the passions are personified,

are tedious and unpleasant, because they are not disguised under the

appearance of reality: while the magic of Circe, the bower of Calypso,

the rocks of Scylla and Charybdis, and the melody of the Syrens, in the

Odyssey of Homer, though nothing but allegories, universally enchant

and delight, because they are covered with the semblance of truth.

It is on this account that Mikon’s battles in heaven are barbarous

and ridiculous in the extreme; for every one sees the impossibility

of supposing gun-powder and cannons in the celestial regions: the

machinery is forced and unnatural, contains no elegance of fancy, and

is not replete with any mystical information. On the contrary, Homer’s

machinery is natural and possible, is full of dignity and elegance, and

is pregnant with the sublimest truths; it delights and enobles the mind

of the reader, astonishes him with its magnificence and propriety, and

animates him with the fury of poetic inspiration. And this, because it

is possible and true.

[164] The sentence within the brackets is omitted in the Greek.

[165] That is, the circular form proceeds from _bound_, but

right-lined figures from _infinity_.

[166] That is, the number three.

[167] In Timæo.

[168] πρὸς ὃ, or, _to which_, is wanting in the original, and in

all the published collections of the Zoroastrian oracles.

[169] That is Jupiter, who is called triadic, because he proceeds from

Saturn and Rhea; and because his government is participated by Neptune

and Pluto, for each of these is called Jupiter by Orpheus.

[170] This sentence, within the brackets, is omitted in the printed

Greek.

[171]

[Illustration: Fig. I.]

[Illustration: Fig. II.]

Thus let a part A E B cut off by the diameter A B (fig. I.) of the

circle A E B D be placed on the other part A D B, as in fig. II.. Then,

if it is not equal to the other part, either A E B will fall within A D

B, or A D B within A E B: but in either case, C E will be equal to C D,

which is absurd.

[172] This objection is urged by Philoponus, in his book against

Proclus on the eternity of the world; but not, in my opinion, with

any success. See also Simplicius, in his third digression against

Philoponus, in his commentary on the 8th book of Aristotle’s Physics.

[173] This definition is no where extant but in the commentaries of

Proclus. Instead of it, in almost all the printed editions of Euclid,

the following is substituted. _A segment of a circle is the figure

contained by a diameter, and the part of the circumference cut off by

the diameter._ This Mr. Simson has marked with commas, as a symbol

of its being interpolated: but he has taken no notice of the different

reading in the commentaries of Proclus. And what is still more

remarkable, this variation is not noticed by any editor of Euclid’s

Elements, either ancient or modern.

[174] As in every hyperbola.

[175] The Platonic reader must doubtless be pleased to find that Euclid

was deeply skilled in the philosophy of Plato, as Proclus every where

evinces. Indeed, the great accuracy, and elegant distribution of these

Elements, sufficiently prove the truth of this assertion. And it is no

inconsiderable testimony in favour of the Platonic philosophy, that its

assistance enabled Euclid to produce such an admirable work.

[176] Concerning these crowns, or annular spaces, consult the great

work of that very subtle and elegant mathematician Tacquet, entitled

_Cylindrica et Annularia_.

[177] In the preceding tenth commentary.

[178] This in consequence of every triangle possessing angles alone

equal to two right.

[179] This too, follows from the same cause as above.

[180] Thus the following figure A B D C has four sides, and but three

angles.

[Illustration]

[181] The Greek in this place is very erroneous, which I have restored

from the version of Barocius.

[182] For the Greek word ῥόμβος is derived from the verb ῥέμβω, which

signifies to have a circumvolute motion.

[183] See the Orphic Hymns of Onomacritus to these deities; my

translation of which I must recommend to the English reader, because

there is no other.

[184] These twelve divinities, of which Jupiter is the head, are,

_Jupiter_, _Neptune_, _Vulcan_, _Vesta_, _Minerva_, _Mars_, _Ceres_,

_Juno_, _Diana_, _Mercury_, _Venus_, and _Apollo_. The first triad of

these is demiurgic, the second comprehends guardian deities, the third

is vivific, or zoogonic, and the fourth contains elevating gods. But,

for a particular theological account of these divinities, study Proclus

on Plato’s Theology, and you will find their nature unfolded, in page

403, of that admirable work.

[185] For it is easy to conceive a cylindric spiral described about a

right-line, so as to preserve an equal distance from it in every part;

and in this case the spiral and right-line will never coincide though

infinitely produced.

[186]

[Illustration]

As the conchoid is a curve but little known, I have subjoined the

following account of its generation and principal property. In

any given right line A P, call P the pole, A the vertex, and any

intermediate point C the centre of the conchoid: likewise, conceive

an infinite right line C H, which is called a rule, perpendicular to

A P. Then, if the right line A _p_ continued at _p_ as much

as is necessary, is conceived to be so turned about the abiding pole

_p_, that the point C may perpetually remain in the right line C

H, the point A will describe the curve A _o_, which the ancients

called a conchoid.

In this curve it is manifest (on account of the right line P O, cutting

the rule in H that the point _o_ will never arrive at rule C H;

but because _h_ O is perpetually equal to C A, and the angle of

section is continually more acute, the distance of the point O from C

H will at length be less than any given distance, and consequently the

right line C H will be an asymptote to the curve A O.

When the pole is at P, so that P C is equal to C A, the conchoid A O

described by the revolution of P A, is called a primary conchoid, and

those described from the poles _p_, and _π_, or the curves

A _o_, A _ω_, secondary conchoids; and these are either

contracted or protracted, as the eccentricity P C, is greater or less

than the generative radius C A, which is called the altitude of the

curve.

Now, from the nature of the conchoid, it may be easily inferred, that

not only the exterior conchoid A _ω_ will never coincide with

the right line C H, but this is likewise true of the conchoids A O, A

_o_; and by infinitely extending the right-line A _π_, an

infinite number of conchoids may be described between the exterior

conchoid A _ω_, and the line C H, no one of which shall ever

coincide with the asymptote C H. And this paradoxical property of

the conchoid which has not been observed by any mathematician, is a

legitimate consequence of the infinite divisibility of quantity. Not,

indeed, that quantity admits of an actual division in infinitum, for

this is absurd and impossible; but it is endued with an unwearied

capacity of division, and a power of being diffused into multitude,

which can never be exhausted. And this infinite capacity which it

possesses arises from its participation of the indefinite duad; the

source of boundless diffusion, and innumerable multitude.

[Illustration]

But this singular property is not confined to the conchoid, but is

found in the following curve. Conceive that the right line A C which is

perpendicular to the indefinite line X Y, is equal to the quadrantal

arch H D, described from the centre C, with the radius C D: then from

the same centre C, with the several distances C E, C F, C G, describe

the arches E _l_, F _n_, G _p_, each of which must be conceived equal

to the first arch H D, and so on infinitely. Now, if the points H, _k_,

_l_, _n_, _p_, be joined, they will form a curve line, approaching

continually nearer to the right-line A B (parallel to C Y) but never

effecting a perfect coincidence. This will be evident from considering

that each of the sines of the arches H D, _l_ E, _n_ F, &c. being less

than its respective arch, must also be less than the right-line A C,

and consequently can never coincide with the right-line A B.

But if other arches D _i_, E _m_, F _o_, &c. each of them equal to

the right-line A C, and described from one centre, tangents to the

former arches H D, _l_ E, _n_ F, &c. be supposed; it is evident that

the points H, _i_, _m_, _o_, &c. being joined, will form a curve line,

which shall pass beyond the former curve, and converge still nearer

to the line A B, without a possibility of ever becoming coincident:

for since the arches D _i_, E _m_, F _o_, &c. have less curvature than

the former arches, but are equal to them in length, it is evident that

they will be subtended by longer lines, and yet can never touch the

right-line A B. In like manner, if other tangent arches be drawn to the

former, and so on infinitely, with the same conditions, an infinite

number of curve-lines will be formed, each of them passing between

H _p_ and A B, and continually diverging from the latter, without a

possibility of ever coinciding with the former. This curve, which I

invented some years since, I suspect to be a parabola; but I have not

yet had opportunity to determine it with certainty.

Transcriber’s Notes:

1. Obvious printers’, punctuation and spelling errors have been

corrected silently.

2. Where hyphenation is in doubt, it has been retained as in the

original.

3. Some hyphenated and non-hyphenated versions of the same words have

been retained as in the original.

4. The errata have been soilently corrected.

5. Italics are shown as _xxx_.