(37) "The symmetrical solids were regarded by Pythagoras, and by the Greek thinkers after him, as of the greatest importance. To be perfectly symmetrical or regular, a solid must have an equal number of faces meeting at each of its angles, and these faces must be equal regular polygons, i. e., figures whose sides and angles are all equal. Pythagoras, perhaps, may be credited with the great discovery that there are only five such solids.* * *

(4) Earlier in the same work, Plutarch also notes: "For as the power of the triangle is expressive of the nature of Pluto, Bacchus, and Mars; and the properties of the square of Rhea, Venus, Ceres, Vesta, and Juno; of the Dodecahedron of Jupiter; so, as we are informed by Eudoxus, is the figure of fifty-six angles expressive of the nature of Typhon." Plutarch did not pretend to explain the inner significance of the symbols, but believed that the relationship which Pythagoras established between the geometrical solids and the gods was the result of images the great sage had seen in the Egyptian temples.

(5) Such, then, is the style of the example in arithmetic. And let the testimony of geometry be the tabernacle that was constructed, and the ark that was fashioned, - constructed in most regular proportions, and through divine ideas, by the gift of understanding, which leads us from things of sense to intellectual objects, or rather from these to holy things, and to the holy of holies. For the squares of wood indicate that the square form, producing fight angles, pervades all, and points out security. And the length of the structure was three hundred cubits, and the breadth fifty, and the height thirty; and above, the ark ends in a cubit, narrowing to a cubit from the broad base like a pyramid, the symbol of those who are purified and tested by fire. And this geometrical proportion has a place, for the transport of those holy abodes, whose differences are indicated by the differences of the numbers set down below.

(28) To the five symmetrical solids of the ancients is added the sphere (1), the most perfect of all created forms. The five Pythagorean solids are: the tetrahedron (2) with four equilateral triangles as faces; the cube (3) with six squares as faces; the octahedron (4) with eight equilateral triangles as faces; the icosahedron (5) with twenty equilateral triangles as faces; and the dodecahedron (6) with twelve regular pentagons as faces.

(3) "The Pythagoreans indeed go farther than this, and honour even numbers and geometrical diagrams with the names and titles of the gods. Thus they call the equilateral triangle head-born Minerva and Tritogenia, because it may be equally divided by three perpendiculars drawn from each of the angles. So the unit they term Apollo, as to the number two they have affixed the name of strife and audaciousness, and to that of three, justice. For, as doing an injury is an extreme on the one side, and suffering one is an extreme on the on the one side, and suffering in the middle between them. In like manner the number thirty-six, their Tetractys, or sacred Quaternion, being composed of the first four odd numbers added to the first four even ones, as is commonly reported, is looked upon by them as the most solemn oath they can take, and called Kosmos." (Isis and Osiris.)

(1) [Trismegistus] ’Tis in this way, Asclepius;—by mixing it, by means of subtle expositions, with divers sciences not easy to be grasped,—such as arithmetic, and music, and geometry. But Pure Philosophy, which doth depend on godly piety alone, should only so far occupy itself with other arts, that it may [know how to] appreciate the working out in numbers of the fore-appointed stations of the stars when they return, and of the course of their procession. Let her, moreover, know how to appreciate the Earth’s dimensions, its qualities and quantities, the Water’s depths, the strength of Fire, and the effects and nature of all these. [And so] let her give worship and give praise unto the Art and Mind of God.

(1) Of his wisdom, however, the commentaries written by the Pythagoreans afford, in short, the greatest indication; for they adhere to truth in every thing, and are more concise than all other compositions, so that they savour of the ancient elegance of style, and the conclusions are exquisitely deduced with divine science. They are also replete with the most condensed conceptions, and are in other respects various and diversified both in the form and the matter. At one and the same time likewise, they are transcendently excellent, and without any deficiency in the diction, and are in an eminent degree full of clear and indubitable arguments, accompanied with scientific demonstration, and as it is said, the most perfect syllogism; as he will find to be the case, who, proceeding in such paths as are fit, does not negligently peruse them.

This science, therefore, concerning intelligible natures and the Gods, Pythagoras delivers in his writings from a supernal origin. Afterwards, he teaches the whole of physics, and unfolds completely ethical philosophy and logic. He likewise delivers all-various disciplines, and the most excellent sciences. And in short there is nothing pertaining to human knowledge which is not accurately discussed in these writings. If therefore it is acknowledged, that of the [Pythagoric] writings which are now in circulation, some were written by Pythagoras himself, but others consist of what he was heard to say, and on this account are anonymous, but are referred to Pythagoras as their author;—if this be the case, it is evident that he was abundantly skilled in all wisdom.

But it is said that he very much applied himself to geometry among the Egyptians. For with the Egyptians there are many geometrical problems; since it is necessary that from remote periods, and from the time of the Gods themselves, on account of the increments and decrements of the Nile, those that were skilful should have measured all the Egyptian land which they cultivated. Hence also geometry derived its name. Neither did they negligently investigate the theory of the celestial orbs, in which likewise Pythagoras was skilled. Moreover, all the theorems about lines appear to have been derived from thence. For it is said that what pertains to computation and numbers, was discovered in Phœnicia. For some persons refer the theorems about the celestial bodies to the Egyptians and Chaldeans in common.

It is said therefore, that Pythagoras having received and increased all these [theories,] imparted the sciences, and at the same time demonstrated them to his auditors with perspicuity and elegance. And he was the first indeed that denominated philosophy, and said that it was the desire, and as it were love of wisdom. But he defined wisdom to be the science of the truth which is in beings. And he said that beings are immaterial and eternal natures, and alone possess an efficacious power, such as incorporeal essences. But that the rest of things are only homonymously beings, and are so denominated through the participation of real beings, and such are corporeal and material forms, which are generated and corrupted, and never truly are.

And that wisdom is the science of things which are properly beings, but not of such as are homonymously so. For corporeal natures are neither the objects of science nor admit of a stable knowledge, since they are infinite and incomprehensible by science, and are as it were, non-beings, when compared with universals, and are incapable of being properly circumscribed by definition. It is impossible however to conceive that there should be science of things which are not naturally the objects of science. Hence it is not probable that there will be a desire of science which has no subsistence, but rather that desire will be extended to things which are properly beings, which exist with invariable permanency, and are always consubsistent with a true appellation.

For it happens that the perception of things which are homonymously beings, and which are never truly what they seem to be, follows the apprehension of real beings; just as the knowledge of particulars follows the science of universals. For he who knows universals properly, says Archytas, will also have a clear perception of the nature of particulars. Hence things which have an existence are not alone, nor only-begotten, nor simple, but they are seen to be various and multiform. For some of them are intelligible and incorporeal natures, and which are denominated beings; but others are corporeal and fall under the perception of sense, and by participation communicate with that which has a real existence. Concerning all these therefore, he delivered the most appropriate sciences, and left nothing [pertaining to them] uninvestigated.

He likewise unfolded to men those sciences which are common [ to all disciplines ,] as for instance the demonstrative, the definitive, and that which consists in dividing, as may be known from the Pythagoric commentaries. He was also accustomed to pour forth sentences resembling Oracles to his familiars in a symbolical manner, and which in the greatest brevity of words contained the most abundant and multifarious meaning, like the Pythian Apollo through certain oracles, or like nature herself through seeds small in bulk, the former exhibiting conceptions, and the latter effects, innumerable in multitude, and difficult to be understood. Of this kind is the sentence, The beginning is the half of the whole , which is an apothegm of Pythagoras himself.

But not only in the present hemistich, but in others of a similar nature, the most divine Pythagoras has concealed the sparks of truth; depositing as in a treasury for those who are capable of being enkindled by them, and with a certain brevity of diction, an extension of theory most ample and difficult to be comprehended, as in the following hemistich:

(4) 'He made each of these tripartite; and how these three beings become each of them tripartite, that learn from me now, my friend!

(527) Yes, that is what we assert. Yet anybody who has the least acquaintance with geometry will not deny that such a conception of the science is in flat contradiction to the ordinary language of geometricians. How so? They have in view practice only, and are always speaking, in a narrow and ridiculous manner, of squaring and extending and applying and the like—they confuse the necessities of geometry with those of daily life; whereas knowledge is the real object of the whole science. Certainly, he said. Then must not a further admission be made? What admission? That the knowledge at which geometry aims is knowledge of the eternal, and not of aught perishing and transient. That, he replied, may be readily allowed, and is true. Then, my noble friend, geometry will draw the soul towards truth, and create the spirit of philosophy, and raise up that which is now unhappily allowed to fall down. Nothing will be more likely to have such an effect. Then nothing should be more sternly laid down than that the inhabitants of your fair city should by all means learn geometry. Moreover the science has indirect effects, which are not small. Of what kind? he said. There are the military advantages of which you spoke, I said; and in all departments of knowledge, as experience proves, any one who has studied geometry is infinitely quicker of apprehension than one who has not. Yes indeed, he said, there is an infinite difference between them. Then shall we propose this as a second branch of knowledge which our youth will study? Let us do so, he replied.

(16) "The design of a temple depends on symmetry, the principles of which must be most carefully observed by the architect. They are due to proportion, in ἀναλογία. Proportion is a correspondence among the measures of the members of an entire work, and of the whole to a certain part selected as standard. From this result the principles of symmetry. Without symmetry and proportion there can be no principles in the design of any temple; that is, if there is no precise relation between its members, as in the case of those of a well shaped man. For the human body is so designed by nature that the face, from the chin to the top of the forehead and the lowest roots of the hair, is a tenth part of the whole height; the open hand from the wrist to the tip of the middle finger is just the same; the head from the chin to the crown is an eighth, and with the neck and shoulder from the top of the breast to the lowest roots of the hair is a sixth; from the middle of the breast to the summit of the crown is a fourth. If we take the height of the face itself, the distance from the bottom of the chin to the under side of the nostrils [and from that point] to a line between the eyebrows is the same; from there to the lowest roots of the hair is also a third, comprising the forehead. The length of the foot is one sixth of the height of the body; of the forearm, one fourth; and the breadth of the breast is also one fourth. The other members, too, have their own symmetrical proportions, and it was by employing them that the famous painters and sculptors of antiquity attained to great and endless renown."

(4) Each one of the three feet, when it is placed on the ground, is as much as a flock (gird) of a thousand sheep comes under when they repose together; and each pastern is so great in its circuit that a thousand men with a thousand horses may pass inside.

(2) All things accord in number:

which he very frequently uttered to all his disciples. Or again, Friendship is equality; equality is friendship . Or in the word cosmos , i. e. the world ; or by Jupiter, in the word philosophy , or in the so much celebrated word tetractys . All these and many other inventions of the like kind, were devised by Pythagoras for the benefit and amendment of his associates; and they were considered by those that understood them to be so venerable, and so much the progeny of divine inspiration, that the following was adopted as an oath by those that dwelt together in the common auditory:

(38) The square base of the Pyramid is a constant reminder that the House of Wisdom is firmly founded upon Nature and her immutable laws. "The Gnostics," writes Albert Pike, "claimed that the whole edifice of their science rested on a square whose angles were: Σιγη, Silence; Βυθος, Profundity; Νους, Intelligence; and Αληθεια Truth." (See Morals and Dogma.) The sides of the Great Pyramid face the four cardinal angles, the latter signifying according to Eliphas Levi the extremities of heat and cold (south and north) and the extremities of light and darkness (east and west). The base of the Pyramid further represents the four material elements or substances from the combinations of which the quaternary body of man is formed. From each side of the square there rises a triangle, typifying the threefold divine being enthroned within every quaternary material nature. If each base line be considered a square from which ascends a threefold spiritual power, then the sum of the lines of the four faces (12) and the four hypothetical squares (16) constituting the base is 28, the sacred number of the lower world. If this be added to the three septenaries composing the sun (21), it equals 49, the square of 7 and the number of the universe.

(1) After this we must narrate how, when he had admitted certain persons to be his disciples, he distributed them into different classes according to their respective merits. For it was not fit that all of them should equally participate of the same things, as they were naturally dissimilar; nor was it indeed right that some should participate of all the most honorable auditions, but others of none, or should not at all partake of them. For this would be uncommunicative and unjust. While therefore he imparted a convenient portion of his discourses to each, he benefited as much as possible all of them, and preserved the proportion of justice, by making each a partaker of the auditions according to his desert.

Hence, in conformity to this method, he called some of them Pythagoreans, but others Pythagorists; just as we denominate some men Attics, but others Atticists. Having therefore thus aptly divided their names, some of them he considered to be genuine, but he ordained that others should show themselves to be the emulators of these. He ordered therefore that with the Pythagoreans possessions should be shared in common, and that they should always live together; but that each of the others should possess his own property apart from the rest, and that assembling together in the same place, they should mutually be at leisure for the same pursuits. And thus each of these modes was derived from Pythagoras, and transmitted to his successors.

Again, there were also with the Pythagoreans two forms of philosophy; for there were likewise two genera of those that pursued it, the Acusmatici, and the Mathematici. Of these however the Mathematici are acknowledged to be Pythagoreans by the rest; but the Mathematici do not admit that the Acusmatici are so, or that they derived their instruction from Pythagoras, but from Hippasus. And with respect to Hippasus, some say that he was a Crotonian, but others a Metapontine. But the philosophy of the Acusmatici consists in auditions unaccompanied with demonstrations and a reasoning process; because it merely orders a thing to be done in a certain way, and that they should endeavour to preserve such other things as were said by him, as so many divine dogmas.

They however profess that they will not speak of them, and that they are not to be spoken of; but they conceive those of their sect to be the best furnished with wisdom, who retained what they had heard more than others. But all these auditions are divided into three species. For some of them indeed signify what a thing is; others what it especially is; but others, what ought, or what ought not, to be done. The auditions therefore which signify what a thing is, are such as, What are the islands of the blessed? The sun and moon. What is the oracle at Delphi? The tetractys. What is harmony? That in which the Syrens subsist . But the auditions which signify what a thing especially is, are such as, What is the most just thing?

To sacrifice. What is the wisest thing? Number. But the next to this in wisdom, is that which gives names to things. What is the wisest of the things that are with us, [i. e. which pertain to human concerns]? Medicine. What is the most beautiful? Harmony. What is the most powerful? Mental decision. What is the most excellent? Felicity. What is that which is most truly asserted? That men are depraved. Hence they say that Pythagoras praised the Salaminian poet Hippodomas, because he sings:

(6) Pythagoras thereupon discovered that the first and fourth strings when sounded together produced the harmonic interval of the octave, for doubling the weight had the same effect as halving the string. The tension of the first string being twice that of the fourth string, their ratio was said to be 2:1, or duple. By similar experimentation he ascertained that the first and third string produced the harmony of the diapente, or the interval of the fifth. The tension of the first string being half again as much as that of the third string, their ratio was said to be 3:2, or sesquialter. Likewise the second and fourth strings, having the same ratio as the first and third strings, yielded a diapente harmony. Continuing his investigation, Pythagoras discovered that the first and second strings produced the harmony of the diatessaron, or the interval of the third; and the tension of the first string being a third greater than that of the second string, their ratio was said to be 4:3, or sesquitercian. The third and fourth strings, having the same ratio as the first and second strings, produced another harmony of the diatessaron. According to Iamblichus, the second and third strings had the ratio of 8:9, or epogdoan.

(42) Iamblichus gathered thirty-nine of the symbolic sayings of Pythagoras and interpreted them. These have been translated from the Greek by Thomas Taylor. Aphorismic statement was one of the favorite methods of instruction used in the Pythagorean university of Crotona. Ten of the most representative of these aphorisms are reproduced below with a brief elucidation of their concealed meanings.

(2) The temperance also of those men, and how Pythagoras taught this virtue, may be learnt from what Hippobotus and Neanthes narrate of Myllias and Timycha who were Pythagoreans. For they say that Dionysius the tyrant could not obtain the friendship of any one of the Pythagoreans, though he did every thing to accomplish his purpose; for they had observed, and carefully avoided his monarchical disposition. He sent therefore to the Pythagoreans, a troop of thirty soldiers, under the command of Eurymenes the Syracusan, who was the brother of Dion, in order that by treachery their accustomed migration from Tarentum to Metapontum, might be opportunely effected for his purpose. For it was usual with them to change their abode at different seasons of the year, and they chose such places as were adapted to this migration.

In Phalæ therefore, a craggy part of Tarentum, through which the Pythagoreans must necessarily pass in their journey, Eurymenes insidiously concealed his troop, and when the Pythagoreans, expecting no such thing, came to that place about the middle of the day, the soldiers rushed upon them with shouts, after the manner of robbers. But the Pythagoreans being disturbed and terrified at an attack so unexpected, and at the superior number of their enemies (for the whole number of the Pythagoreans was but ten), and considering also that they must be taken captive, as they were without arms, and had to contend with men who were variously armed,—they found that their only safety was in flight, and they did not conceive that this was foreign to virtue.

For they knew that fortitude, according to the decision of right reason, is the science of things which are to be avoided and endured. And this they now obtained. For those who were with Eurymenes, being heavy-armed, would have abandoned the pursuit of the Pythagoreans, if the latter in their flight had not arrived at a certain field sown with beans, and which were in a sufficiently florishing condition. Not being willing therefore to violate the dogma which ordered them not to touch beans, they stood still, and from necessity attacked their pursuers with stones and sticks, and whatever else they happened to meet with, till they had slain some, and wounded many of them. All the Pythagoreans however, were at length slain by the spearmen, nor would any one of them suffer himself to be taken captive, but preferred death to this, conformably to the mandates of their sect.

(15) Endeavour not to conceal your errors by words, but to remedy them by reproofs. Pythagoras. Stob. p. 146.

It is not so difficult to err, as not to reprove him who errs. Pythagoras. Stob. p. 147.

(37) Your attention is called to the fact that each circle in the symbol is called to and blended with the one on either side of it. Accordingly in the circular extent of each circle there is to be found FOUR different spaces or regions, as follows: (1) Its own unblended space or region; (2) the space or region in which its own space or region is blended with that of one of the neighboring circles, which constitutes a shield-shaped space; (3) the space or region in which its own space or region is blended with that of the other neighboring circle, constituting a shield-shaped space; and (4) the space or region in the very centre of the symbol, in which the space or region of each circle is blended with that of both of the other two—thus producing a Triune Region. This arrangement, again, furnishes us with SEVEN distinct regions, as follows (giving each circle the name of a letter, as A, B, or C, respectively) I. Circle A; II. Circle B; III. Circle C; IV. Space A-B; V. Space A-C; VI. Space B-C and finally VII. Region A-B-C, at the centre. There are thus three unblended areas; also three blended areas of two elements; and finally one blended area of three elements; the latter combining within itself all three elements in equal proportion. Let him who wishes for the Light solve this Riddle of the Symbol!

(526) And indeed, you will not easily find a more difficult study, and not many as difficult. You will not. And, for all these reasons, arithmetic is a kind of knowledge in which the best natures should be trained, and which must not be given up. I agree. Let this then be made one of our subjects of education. And next, shall we enquire whether the kindred science also concerns us? You mean geometry? Exactly so. Clearly, he said, we are concerned with that part of geometry which relates to war; for in pitching a camp, or taking up a position, or closing or extending the lines of an army, or any other military manoeuvre, whether in actual battle or on a march, it will make all the difference whether a general is or is not a geometrician. Yes, I said, but for that purpose a very little of either geometry or calculation will be enough; the question relates rather to the greater and more advanced part of geometry— whether that tends in any degree to make more easy the vision of the idea of good; and thither, as I was saying, all things tend which compel the soul to turn her gaze towards that place, where is the full perfection of being, which she ought, by all means, to behold. True, he said. Then if geometry compels us to view being, it concerns us; if becoming only, it does not concern us?