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The Theoretic Arithmetic

of the Pythagoreans

Thomas Taylor

Extract - from Book III, chapters 3 & 4


 On the monad.

 The monad, as we learn from the extracts preserved by Photius from Nicomachus, was called by the Pythagoreans intellect, male and female, God, and in a certain respect matter. They also said, that it in reality mingled all things, is the recipient and capacious of all things, is Chaos, confusion, commixtion, obscurity, darkness, a chasm, Tartarus, Styx, and horror, and void of mixture, a subterranean profundity, Lethe, a rigid virgin, and Atlas. It is likewise called by them the axis, the Sun, and Pyralios, Morpho, the tower of Jupiter, and spermatic reason. Apollo likewise, the prophet, and ambiguous.

 With respect to the first of these appellations, intellect, the reason why the Pythagoreans thus denominated the monad will be evident if it is considered that forms or ideas were called by them numbers, and that as the monad contains in itself the cause of numbers, so intellect is the source of all ideas. As the monad too comprehends in itself the multitude which it produces, and with which it accords, so intellect comprehends in itself all the forms which proceed from it, and with which it is coordinate. But they appear to have called the monad male and female, from containing in itself causally, the odd and the even, the former of which corresponds to the male, and the latter to the female, or according to the anonymous author in Theologumenis Arithmeticis, it was so called as being the seed of all things. Hence we are informed by Theo of Smyrna, (p. 30) that Aristotle in his treatise called Pythagoric, said, "that the one partakes of both these natures; for being added to the odd it makes the even number, and to the even, the odd number, which it could not do, if it did not participate of both these." And he further informs us that Archytas also was of the same opinion. As God too is the cause of all multitude, the reason is obvious why they called the monad God. They also very properly denominated it in a certain respect, and not wholly matter, from its similitude to divinity. For God is the first, and matter the last of things, and each subsists by a negation of all things. Hence matter is said to be dissimilarly similar to divinity. It is similar, so far as it alone subsists by a negation of all things. But it is dissimilarly similar, because divinity is better and beyond all things, but matter is worse than, and below all things.

 Again, when they said that the monad in reality mingled and is the recipient of all things, this likewise was asserted by them from the analogy of the monad to deity; for all things are mingled by and comprehended in the ineffable nature of divinity. But they called it chaos from resembling the infinite, for chaos, according to Pythagoras, is analogous to infinity, in the same manner as ether is said by him to correspond to bound. And bound and infinity are the two great principles of things immediately posterior to the ineffable. For the same reason they called it a chasm. But they called it confusion, commixture, obscurity, and darkness, because in the ineffable principle of things of which it is the image, all things are profoundly one without any separation or distinction, as being all things prior to all, and in consequence of being involved in unfathomable depths, are concealed in unknown obscurity and darkness. Hence, as we are informed by Damascius in his admirable MS. treatise Concerning Principles, the Egyptians asserted nothing of the first principle of things, but celebrated it as a thrice unknown darkness transcending all intellectual perception. As Tartarus too subsists at the extremity of the universe, in a descending series, it is dissimilarly similar to the ineffable which is the extremity of things in an ascending series. But when they called the monad Styx, it was in consequence of looking to its immutable nature. For Styx, according to its first subsistence, is the cause by which divine natures retain an immutable sameness of essence; for this is the occult meaning of the fabulous assertion, that the Gods swear by Styx, viz. they continue through this invariably the same. They appear to have called the monad, horror, from considering that the ineffable is perfectly unknown, and unconnected with our nature; for the perception of any sensible thing of this kind, is attended with terror. But they denominated it void of mixture, from the simplicity of the nature of the ineffable and a subterranean profundity from its unfathomable depths, which are beyond all knowledge. As with respect to the ineffable likewise, knowledge, as Damascius beautifully observes, is refunded into ignorance, the monad, which is the image of it, is very appropriately called Lethe or oblivion. But from the purity of its nature, the monad is denominated a rigid virgin. And it is called Atlas, because the ineffable supports, connects, and separates all things; for of this the fabulous pillars of Atlas are an image.

 Besides these appellations, the Pythagoreans also called the monad Apollo, as we are informed by Plutarch and Plotinus, from its privation of multitude. They likewise denominated it Prometheus, according to the anonymous author of Theologum. Arithmet. because it in no respect runs to the anterior part; for there is nothing beyond the ineffable. The same author, likewise, informs us that they called it "essence, the cause of truth, the simple paradigm, the order of symphony; in the greater and the less, the equal; in intension and remission, the middle; in multitude the moderate; in time the present now. And besides these, a ship, a chariot, a friend, life, and felicity." For as The One is all things prior to all, it is preeminently the most excellent of things, but this, according to The One, i.e. without departing from the ineffable simplicity of its nature. They also denominated it form, because, as Simplicius observes (in Phys. lib. 1.) form circumscribes and bounds every thing to which it accedes. But they called it Proteus, as we are informed by the above mentioned anonymous author, in consequence of comprehending the peculiarities of all things in itself. They also denominated it Jupiter, because what The One, or the ineffable principle of things is to all the orders of the Gods, that Jupiter is to all the divine orders posterior to him, as is beautifully observed by Proclus in Theol. Plat. lib. 5. They likewise called it Mnemosyne the mother of the Muses, because as the Muses generate all the variety of reasons with which the world is replete, and are the causes of the perfection of the universe, Mnemosyne will be analogous to The One which is the source of all multitude. It may also be said that as Mnemosyne is memory, and memory is stability of knowledge, the monad is thus denominated, as being the image of The One which is the stable root of all knowledge, and of all things. But they called it Vesta, or the fire in the centre of the earth, which as Simplicius observes (de Caelo lib. 2) possesses a fabricative power, nourishes the whole earth from the middle, and excites whatever in it is of a frigid nature. So that as a producing centre it is analogous to The One. On this appellation, there is the following remarkable passage in the before mentioned anonymous author. "In addition to these things, also, they say, that a certain fiery cube of the nature of unity, is situated about the middle of the four elements, the middle position of which Homer also knew, when he says,

As far beneath the unseen region hurl'd,
As earth is distant from the etherial world.

Empedocles, Parmenides, and nearly most of the ancient wise men, appear to accord in these things with the Pythagoreans; for they say that the monadic nature, after the manner of Vesta, is established in the middle, and on this account preserves that seat in equilibrium."

 In the last place, they called the monad multinominal, as we are informed by Hesychius; and this with the greatest propriety, because the ineffable one, of which the monad is the image, is, as we have observed, all things prior to all.



On the duad.

 The duad was called by the Pythagoreans, as we learn from Nicomachus, "audacity, matter, the cause of dissimilitude, and the interval between multitude and the monad. This alone produces equality from composition and mixture, on which account also it is equal. But it is likewise unequal, defect, and abundance, and is alone unfigured, indefinite, and infinite. It is also alone the principle and cause of the even, yet is neither evenlyeven, nor unevenlyeven, nor evenlyodd. But many of these things are proximate to the physical peculiarity of the duad. It is likewise the fountain of all symphony, and among the Muses is Erato. It is also harmony, patience, and a root, though not in a certain respect in energy. It is power too, the feet of Ida abounding with fountains, a summit and Phanes. It is also Justice, and Isis, Nature and Rhea, the mother of Jupiter, and the fountain of distribution. It was likewise called by them Phrygia, Lydia, Dindymene, Ceres, and Eleusinia, Diana and Cupid, Dictynna, Aeria, Asteria, Disamos, and Esto. Also Venus, Dione, Mychaea, Cytherea, ignorance, ignobility, falsehood, difference, indistinction, strife, dissension, Fate and Death."

 Prior, however, to a development of these appellations, it will be requisite to observe concerning the duad, that the Pythagoreans, before they evinced the multitude subsists in intelligibles, necessarily investigated the cause of the multitude which is there, and found that among the genera of being it is difference, which subsists according to nonbeing; but that in causes most eminently the first, it is the indefinite duad, which, says Syrianus, Pythagoras in the Sacred Discourse calls Chaos, and which he associates with intellect; for he assigns this appellation to the monad, which is the first of the two great principles after the ineffable one.

 This duad is indeed every where the cause of multitude, so far as it produces things fromthe one with their proper differences. But so far as it is a principle, there is also in the several orders of beings a proper monad; and a duad connate to this found, and which generates a number accommodated to itself.

 Every number too, subsists from these two principles, the monad and duad; but the odd number is rather characterized by the property of the monad, but the even by the property of the duad. In angles, also, the right angle subsists rather according to the monad, but the acute and obtuse, according to the indefinite duad, in which exuberance and defect are most apparent. Of figures, likewise, those which are characterized by equality, sameness, and similitude, have greater relation to the monad; but those in which inequality, difference, and dissimilitude are predominant, are more allied to the duad. In short, every figure subsists from these two principles: for the sphere, circle, equilateral triangle, square and cube, participate of the duad by their quantity, and their possession of interval. And again, beams of timber, altars, scalene triangles, and oblong figures accord with the monad, from which they receive their form.

 Again, the Pythagoreans, says Syrianus, considered accidents, and saw that the same principles had an analogous subsistence in these; and that they had their proper monad and duad; the former being the cause of identity to them, and the latter of difference and multitude. In natural reasons also, or productive seminal principles, they placed effective causes. There is therefore, in nature, one productive principle generative of all colours, and another which is indeed primarily perfected from this, but which produces together with it the multitude and diversity of colours: and these are the monad and duad of colours. In other accidents also, which are perfected through natural reasons, there will be found a monad and duad analogous to these.

 Having premised thus much, let us direct our attention in the next place, to the appellations of the duad. With respect to the appellation audacity, therefore, we are informed by the anonymous author that the duad was so called "because it first separated itself from the monad." For as the descent of the soul into body, and her abandoning an intellectual and divine life, for an irrational and mortal condition of being may be called audacity, as being in a certain respect an improper boldness, so with reference to the transcendant excellence of the monad, a departure from it as the paternal profundity and the adytum of godnourished silence, as it is called in the Chaldean oracles, may be said, metaphorically speaking, to be an audacious undertaking. But the duad was called matter, as being indefinite, and the cause of bulk and division, as Simplicius observes in his comment on the Physics. And it is the cause of dissimilitude, as being in its first subsistence theinfinite , from which dissimilitude is suspended, in the same manner as similitude is suspended from bound. But it is the interval between multitude and the monad, because it is yet perfect multitude, but is as it were parturient with it, and almost unfolding it into light. Of this we see an image in the duad of arithmetic. For as Proclus beautifully observes in his Comment on the 20th &c. definition of the first book of Euclid's Elements: "The duad is the medium between unity and number. For unity, by addition, produces more than by multiplication; but number, on the contrary, is more increased by multiplication than by addition; and the duad, whether multiplied into, or compounded with itself, produces an equal quantity." The duad was also called equal, because, says the anonymous author, "two and two are equal to twice two:" that is, the addition of two to itself, is equal to the multiplication of it by itself. But it is unequal, defect and abundance, as the same author observes, according to the conception of matter. For he adds, the Pythagoreans call matter homonymously with this, the indefinite duad, because so far as pertains to itself, it is deprived of morphe, form, and a certain definition, and is defined and bounded by reason and art. It is likewise alone unfigured, because, as the anonymous writer observes, "From the triangle and the triad polygonous figures proceed in energy, ad infinitum; from the monad all figures subsist at once according to power; but from two things, whether they are right lines, or angles, a rightlined figure can never be composed." But the duad was called indefinite and infinite, because in its first subsistence it is infinity, and therefore has no appropriate bound. And as we have shown that it was called equal, it is not wonderful that it was denominated the cause of the even, and consequently as the cause of it, was said to be neither evenlyeven, nor unevenly-even, nor evenly-odd.

 Again, it was called the fountain of all symphony, because the symphony diapason, which is most harmonic, is formed from a duple ratio. But is was denominated Erato, "because, says the anonymous author, it attracts to itself through love the accession of the monad, as of form, and thus generates the remaining effects." As the fountain of all symphony, it is evident why it was called harmony. But it was denominated patience, because, says the anonymous writer, it is the first multitude that sustains or endures separation, viz. a separation from the adytum of the monad. It is also a root, though not in a certain respect in energy, because it is the mother of number with which it is parturient, but is not number in perfect energy. It was likewise denominated power, because the first infinity is the first power. But it is the feet of Ida abounding with fountains, because it is the root of the region of ideas, or an intelligible essence. For the foot of a mountain is the same as the root of it; and mount Ida, as Proclus observes in his Apology for Homer, signifies the region of ideas. It is also called Phanes, or intelligible intellect, as being the occult power of it. But it may be said generally, that it is Justice, Isis, Nature, Rhea, &c. because, as being of a feminine nature, it is the fountain of all the divinities that are of a female characteristic. It likewise appears that it was called Cupid for the reason above assigned, for its being denominated Erato, viz. from desiring the accession of the monad. But it is ignorance, from its subsistence as infinity, about which there is an allperfect ignorance. And it is ignobility, falsehood, difference, &c. as being the leader of the worse coordination of things.

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