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« Reply #15 on: February 06, 2009, 01:18:26 pm » 

but it is the power * of a certain figure. And as heat is incisive, so cold has a connective property. And as the former subsists according to sharpness of angles and tenuity of sides, so, on the contrary, the latter subsists according to obtuseness of angles and thickness of sides. Hence, the former power is contrary to the latter, the figures themselves not being contrary, but the powers inherent in the figures. The argument, however, requires a figure, not in reality contrary, but adapted to a contrary power. Such figures, therefore, as have obtuse angles and thick sides, have powers contrary to the pyramid, and are connective of bodies. But such figures are the elements of three bodies. Hence all things that congregate, congregate through impulsion; but fire alone, as we have observed, has a separating power. †
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Aristotle adds a fifteenth argument, after all that has been said, objecting to magnitude, and shewing that the Pythagoreans make the power of cold a cause, as consisting of great parts, because it compresses and does not pass through pores, as is indicated by what Plato says in the Timæus about cold. * Proclus, however, in opposition
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to this, observes as follows: "We do not determine the elements of simple bodies by magnitude alone, but also by thinness and thickness, by sharpness and facility of motion, and by immobility and difficulty of motion, which give variety to forms, and cause things which have the same form, not to differ by magnitude alone. For the magnitude of planes makes the largeness or smallness of parts in bodies; since the parts of them are called elements. Thus, the pyramids of fire, of which fire consists, are the parts of fire, and octaedra are the parts of air. For the octaedron is greater than the pyramid, both being generated from an equal triangle. But the composition, together with so great a multitude, make the acute and the obtuse. For more or fewer triangles coming together, an angle, either acute or obtuse, is generated; an acute angle, indeed, from a less, but an obtuse from a greater multitude. But the characteristic property of the planes produces facility or difficulty of motion; these planes existing in a compact state, through similitude, but being prepared for tendency
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through dissimilitude. Large pyramids, therefore, do not belong to things which refrigerate, but to the larger parts of fire; just as larger octaedra belong to the larger parts of air, and larger icosaedra to larger parts of water. For from this cause waters are thin and thick, and airs are attenuated and gross; since it is evident that these are determined by quantity."

Footnotes 11:* In order to understand what is said by Proclus in answer to the objections of Aristotle, it is requisite to relate, from Simplicius, the hypothesis of the Pythagoreans and Plato, respecting the composition of the elements from the five regular bodies. "They supposed two primogenial rightangled triangles, the one isosceles, but the other scalene, having the greater side the double in length of the less, and which they call a semitriangle, because it is the half of the equilateral triangle, which is bisected by a perpendicular from the vertex to the base. And from the isosceles triangle, which Timæus calls a semisquare, four such having their right angles conjoined in one centre, a square is formed. But the union of six such triangles † having eight angles, p. 19 forms a cube, which is the element of earth. The semitriangle, however, constitutes the pyramid, the octaedron, and the icosaedron, which are distributed to tire, air, and water. And the pyramid, indeed, consists of four equilateral triangles, each of which composes six semitriangles. But the octaedron consists of eight equilateral triangles, and fortyeight semitriangles; and the icosaedron is formed from twenty equilateral triangles, but one hundred and twenty semitriangles. Hence, these three, deriving their composition from, one element, viz. the semi triangle, are naturally adapted, according to the Pythagoreans and Plato, to be changed into each other; but earth, as deriving its composition from another triangle specifically different, can neither be resolved into the other three bodies, nor be composed from them."
11:† Viz. of six squares, or six times four isosceles triangles, whose right angles are conjoined in one centre.
17:* In planes this can only be accomplished by the equilateral triangle, the square, and the hexagon; viz. by six equilateral triangles, four squares, and three hexagons. But in solids, the pyramid and cube alone can fill the place, which is about one point. Of the first part of this admirable theorem, which is also mentioned, with the praise it deserves, by Proclus in his Commentary on the First Book of Euclid, the following demonstration is given by Tacquet.—In order that any regular figures frequently repeated may fill space, viz. may form one continued superficies, it is requisite that the angles of many figures of that species composed about one point make four right angles; for so many exist about one point as is evident from Coroll. 3. Prop. 13. of the First Book of Euclid. Thus, for instance, that equilateral triangles may fill place, it is requisite that some angles of such triangles composed about one point should make four right angles. But 6 equilateral triangles make 4 right angles; for 1 makes 2/3 of one right angle, and therefore 6 make 12/3 of 1 right, i.e. 4 right angles. The 4 angles of a square, also, as is evident, make 4 right angles; and this is likewise the case with the 3 angles of a hexagon. For one makes 4/3 of 1 right, and consequently 3 make 12/3 of 1 right, that is, again 4 right. But that no other figure can effect this, will clearly appear, if, its angle being found, it is multiplied by ally number; for the angles will always be less than, or exceed, 4 right angles.
28:* It is well observed by Simplicius, (De Cœlo, p. 142,) "that Plato and the Pythagoreans by a plane denoted something more simple than a body, atoms being evidently bodies; that they assigned commensuration and a demiurgic analogy [ i.e. active and fabricative powers] to their figures, which Democritus did not to his atoms; and that they differed from him in their arrangement of earth."
28:† Simplicius here remarks, "that it may be doubted, how the powers which are in figures, being contrary, the figures themselves will not be contrary; for powers are adapted to the p. 29 things by which they are possessed. Perhaps, therefore, he H. e. Proclus] calls the four figures, the pyramid and the other regular bodies, which not being contrary, their powers are contrary; since their powers are not according to their figures. For neither the thick nor the thin, neither that which has large nor that which has small parts, neither that which is moved with difficulty nor that which is easily moved, are the differences of figure. Perhaps, too, neither are acuteness nor obtuseness of angles simply the differences of figure, since neither is an angle simply a figure. If, therefore, the dispositions of the hot and the cold, which are contrary, are effected according to these contrarieties, no absurdity will ensue. Hence the proposition which says, that things which are determined by figures are not contrary, requires a certain circumscription. For they are not contrary according to figures, yet they are not prevented from having contraries. If, however, some one should insist, that contrarieties are according to figures, it is necessary to recollect that Aristotle in this treatise says, that there is also in figures a certain contrariety."
29:* What Plato says on this subject in the Timæus, is as follows: "The moist parts of bodies larger than our humid parts, entering into our bodies, expel the smaller parts; but not being able to penetrate into their receptacles, coagulate our moisture, and cause it through equability to pass from an anomalous and agitated state, into one immovable and collected. p. 30 But that which is collected together contrary to nature, naturally opposes such a condition, and endeavours by repulsion to recall itself into a contrary situation. In this contest and agitation, a trembling and numbness takes place; and all this passion, together with that which produces it, is denominated cold."

