(being continued  from 14/02/17)


……The first who methodically applied the principles of mathematics to mechanics: who imparted an organic motion to a geometric figure, by the section of the semi-cylinder seeking two means that would be proportional, in order to double the cube.” [p.178]

“Archytas of Tarentum, son of Mnesagoras, or Hestius, according to Aristoxenus, also a Pythagorean … (an alleged contemporary/communicator with Plato). … With the exception of the mathematical fragments and a few others, the fragments of Archytas are not considered genuine….. [p.177]

(cf., Sir Thomas Heath, A History of Greek Mathematics, Vol. 1., 246-49., and also Evans. G. Valens, The Number of Things, Methuen & Co. Ltd, London, 1965.)


1. There are necessarily two principles of beings: the one contains the series of beings organized, and finished; the other, contains unordered and unfinished beings. That one which is susceptible of being expressed, but speech, and which can be explained, embraces both beings, and determines and organizes the non-being.

For every time that it approaches the things of becoming, it orders them, and measures them, and makes them participate in the essence and form of the universe. On the contrary, the series of beings which escapes speech and reason, injures ordered things, and destroys those which aspire to essence and being; whenever it approaches them, it assimilates them to its own nature.

But since there are two principles of things of an opposite character, the one the principle of good, and the other the principle of evil, there are therefor also two reasons, the one of beneficent nature, the other of maleficent nature.

That is why the things that owe existence to art, and also those which owe it to nature, must above all participate in these two principles: form and substance.

The form is the cause of essence; substance is the substrate which receives the form. Neither can substance alone participate in form, by itself; nor can form by itself apply itself to substance; there must therefore exist another cause which moves the substance of things, and forms them. This cause is primary, as regards substance, and the most excellent of all. Its most suitable name is God.

There are therefore three principles; the substance is the matter, the moved; the essence is what you might call the art, and that to which the substance is brought by the mover. But since the mover contains forces which are self-contrary, those of simple bodies, and as the contraries are in need of a principle harmonizing and unifying them, it must necessarily receive its efficacious virtues and proportions from numbers, and all that is manifested in numbers and geometric forms, virtues and proportions capable of binding and uniting into form the contraries that exist in the substance of things. For, by itself, substance is formless; only after having been moved towards form does it become formed and receive the rational relations of order. Likewise, if movement exists, besides the thing moved, there must exist a prime mover; there must therefore be three principles: the substance pf things, the form, and the principle that moves itself, and which by its power is the first; not only must this principle be an intelligence, it must be above intelligence, and we call it God.

Evidently, the relation of equality applies to the being which can be defined by language and reason. The relation of inequality applies to the irrational being’ and cannot be fixed by language; it is substance, and that is why all begetting and destruction take place in substance and do no occur without it. [p.179]

2. In Short, the philosophers began only by so to speak contrary principles; but above these elements they knew another superior one, as is testified to by Philolaus, who says God has produced, and realized the Limited and Unlimited, and shown that at the Limit is attached the whole series which has a greater affinity with the One, and to the Unlimited, the series that is below. Thus, above these two principles they have posited a unifying cause, superior to everything; which, according to Archnetus, is the cause before cause, and, according to Philolaus, the universal principle. [p.179]

3. A. Which One are you referring to? The supreme One. or the infinitely small One that you can find in parts? The Pythagoreans distinguish between the One and the Monad, as says Archytas: the One and the Monad have natural affinity, yet they differ.

B. Archytas and Philolaus indiscriminately call the One a monad, and Monad a One. The majority, however, add to the name Monad, the distinction of first Monad, for there is a monad which is not the first, and which is posterior to the Monad itself, and to the One.

C. Pythagoras said the human soul was a tetragon with right angles. Archytas, on the contrary, instead of defining the soul by the tetragon, did so by a circle, because the soul is a self-mover, and consequently, the prime mover, and it is a sphere or a circle.

D. Plato and Archytas and the other Pythagoreans claim that there are three parts in the soul: reason, courage, and desires. [p.179]

4 The beginning of knowledge of beings is in the things that produce themselves. Of these some are intelligible, and other sensible; the former are immovable, the latter are moved. The criterion of intelligible things is the world; that of sensible things is sensation.

Of the things that do not manifest in things themselves, some are science, the other, opinion; science is immovable, opinion is movable,

We must, besides, admit these three things; the subject that judges., the object that is judged, and the rule by which that object is judged. What judges is the mind, or sensation; what is judged is the logos, or rational essence; the rule of judgement is the act itself which occurs in being, whether intelligible or sensible. the mind id judge of essence, whether it tends towards an intelligible being or a sensible one. When reason seeks intelligible things, it tends towards an intelligible element; when it seeks things of sense, it tends towards their element.

Hence come those false graphic representations in figures and numbers seen in geometry, those researches in causes and probably ends, whose object are beings subject to becoming, and moral acts, in physiology or politics. It is while tending toward the intelligible element that reason recognized that harmony is in the double relation [the octave] but sensation alone attests that this double relation is concordant. In mechanics, the object of science is figures, numbers, proportions – namely, rational proportions; the effects are perceived by sensation, for you can neither study nor know them outside of the matter or movement. In short, it is impossible to know the reason of an individual thing, unless you have preliminarily by the mind grasped the essence of the individual thing; the knowledge of the existence, and of quality, belongs to reason and sensation; to reason, whenever we effect a thing’s demonstration by a syllogism whose conclusion is inevitable; to sensation, when the latter is the criterion of a thing’s essence. [p.180] 

(to be continued)

About sooteris kyritsis

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