The Quadvrium Thread [Archive]

Macrobius

06-27-2010, 01:31 AM

To begin, then, we start by quoting Boethius [repeating the quote from thread 54708 (http://www.thephora.net/forum/showthread.php?t=54708), as we have linked]:

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Among all the men of ancient authority who, following the lead of Pythagoras, have flourished in the purer reasoning of the mind, it is clearly obvious that hardly anyone has been able to reach the highest perfection of the disciplines of philosophy [B]unless the nobility of such wisdom was investigated by him in a certain four-part study, the quadrivium,[/B] which will hardly be hidden from those properly respectful of expertness. For this is the wisdom of things which are, and the perception of truth gives to these things their unchanging character.

[ ... ] arithmetic considers that multitude which exists of itself as an integral whole; the measures of musical modulation understand that multitude which exists in relation to some other; geometry offers the notion of stable magnitude; the skill of astronomical discipline explains the science of movable magnitude. If a searcher is lacking knowledge of these four sciences, he is not able to find the true; without this kind of thought, nothing of truth is rightly known. This is the knowledge of those things which truly are; it is their full understanding and comprehension. He who spurns these, the paths of wisdom, does not rightly philosophize. [ ... ]

[B]This, therefore, is the quadrivium by which we bring a superior mind from knowledge offered by the senses to the more certain things of the intellect.[/B] There are various steps and certain dimensions of progressing by which the mind is able to ascend so that by means of the eye of the mind, which (as Plato says) is composed of many corporeal eyes and is of higher dignity than they, truth can be investigated and beheld. This eye, I say, submerged and surrounded by the corporeal senses, is in turn illuminated by the disciplines of the quadrivium.
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Boethius did not invent this notion for himself. Indeed, the four arts comprising the Quadrivium, are named in a work of Plato. Here that matter is discussed, relating to Archytas, whom we will be meeting later:

[url]http://plato.stanford.edu/entries/archytas/[/url]

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He was the first to identify the group of four canonical sciences (logistic [arithmetic], geometry, astronomy and music), which would become known as the quadrivium in the middle ages.
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Which follows this aside:

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Archytas was most famous in antiquity and is most famous in the modern world for having sent a ship to rescue Plato from the tyrant of Syracuse, Dionysius II, in 361. In both the surviving ancient lives of Archytas (by Diogenes Laertius, VIII 79-83, and in the Suda) the first thing mentioned about him, after the name of his city-state and his father, is his rescue of Plato (A1 and A2). This story is told in greatest detail in the Seventh Letter ascribed to Plato. It has accordingly been typical to identify Archytas as “the friend of Plato” (Mathieu 1987). Archytas first met Plato over twenty years earlier, when Plato visited southern Italy and Sicily for the first time in 388/7, during his travels after the death of Socrates (Pl. [?], Ep. VII 324a, 326b-d; Cicero, Rep. I 10. 16; Philodemus, Acad. Ind. X 5-11; cf. D.L. III 6). Some scholars have seen Archytas as the “new model philosopher for Plato” (Vlastos 1991, 129), and he has been regarded as the archetype of Plato's philosopher-king (Guthrie 1962, 333). The actual situation appears to be considerably more complicated. The ancient evidence, apart from the Seventh Letter, presents the relationship between Archytas and Plato in diametrically opposed ways. One tradition does present Archytas as the Pythagorean master at whose feet Plato sat, after Socrates had died (e.g., Cicero, Rep. I 10.16), but another tradition makes Archytas the student of Plato, to whom he owed his fame and success in Tarentum (Demosthenes [?], Erotic Oration 44).

The Seventh Letter (http://en.wikipedia.org/wiki/Seventh_Letter_(Plato)) itself is of contested authenticity, although most scholars regard it either as the work of Plato himself or of a student of Plato who had considerable familiarity with Plato's involvement in events in Sicily (see e.g., Brisson 1987; Lloyd 1990; Schofield 2000).
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Our course, will, in a certain sense, be an extended commentary on the [i]Epinomis[/i] of Plato, since it is there that the Tradition of the quadrivium surfaces, as would be well known to Archytas. 'Epinomis' means after, or upon The Laws [Nomoi], which is the last of the acknowledged authentic works of Plato.

Here is the relevant passage -- and the following page or so.

[url]http://www.perseus.tufts.edu/hopper/text?doc=Perseus:text:1999.01.0180:text%3DEpin.:page%3D990[/url] [but the rest may be read with profit].

For those interested, a quick comparison with the books commonly studied by Egyptian priests may be found here: [url]http://www.sacred-texts.com/afr/stle/stle11.htm[/url]

It is clear the passage in the Epinomis is about the Arithmetic and Geometric mean, and the 'power' of multiplication. That is, if a number is represented by a set of pebbles in a straight line, and we add that many pebbles to itself -- arithmetic addition in one dimension -- then we obtain a 'line' of pebbles, or number, that is twice as long.

If, on the other hand, we start with our pebbles arranged in a square [it is clear we must have a square number of them!], say we have 4 x 4 or 16, then if we 'double' that, we no longer have a square, but a geometric shape of a different species -- a rectangle with 32 pebbles, and 32 is not a square number.

Yet, if we double the double, we *do* end up with a square -- 64 pebbles, which may be arranged in an 8x8 square, and that is indeed 'twice the size' on the edge of our 4 x 4.

That is, the 'power' of the second dimension [its propensity to produce like, given a function 2x] is different from the power of one dimension -- x + x = 2x always.

In short, doubling in the arithmetic sense, and doubling multiples, results in two different sorts of 'power', or of two different powers, in the plural as Plato would write it.

When one finds a geometric mean, one solves a problem of equal *ratios* -- A : B :: C : D (A is to B as C is to D), or in modern notation, A/B = C/D -- the two ratios, [i]logoi[/i], are equal. If the geometric mean is unknown, we are finding A/X = X/D, that is seeking to find just one number, X, which is the same as B=C, that serves as a geometric mean to A and D.

In solving a 'double proportion', then one finds a magnitude, or two magnitudes, such that A:X :: X:Y :: Y:B, and finds a two-step proportion, rather than one-step one. If X = Y it will, in fact, go as before, but that is not the general case.

This observation of Plato, or a student of his, in the Epinomis, is the foundation of the Quadrivium, and the source for its other discoveries, as we shall elucidate. Using Mathematics for Power, and the Powers thus obtained, is our subject matter. 'Let none ignorant of Geometry enter herein', as the sign over the Academy read.

Quoting, then the passage, starting after the passage on the orbits of the heavenly bodies:

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But to avoid repeating again and again the same things on the same subjects [990c] in our discussion, the other courses of these bodies that we have previously described are not easily understood: we must rather prepare our faculties, such as they may possibly be, for these matters; and so one must teach the pupil many things beforehand, and continually strive hard to habituate him in childhood and youth. And therefore there will be need of studies: the most important and first is of numbers in themselves; not of those which are corporeal, but of the whole origin of the odd and the even, and the greatness of their influence on the nature of reality. [990d] When he has learnt these things, there comes next after these what they call by the very ridiculous name of geometry,2 when it proves to be a manifest likening3 of numbers not like one another by nature in respect of the province of planes; and this will be clearly seen by him who is able to understand it to be a marvel not of human, but of divine origin. And then, after that, the numbers thrice increased and like to the solid nature, and those again which have been made unlike, he likens by another art, namely, that which [990e] its adepts called stereometry; and a divine and marvellous thing it is to those who envisage it and reflect, how the whole of nature is impressed with species and class according to each analogy, as power and its opposite4 continually turn

[991a] upon the double. Thus the first analogy is of the double in point of number, passing from one to two in order of counting, and that which is according to power is double; that which passes to the solid and tangible is likewise again double, having proceeded from one to eight; but that of the double has a mean, as much more than the less as it is less than the greater, while its other mean1 exceeds and is exceeded by the same portion of the extremes themselves. Between six and [991b] twelve comes the whole-and-a-half（9=6+3）and whole-and-a-third（8=6+2）: turning between these very two, to one side or the other, this power（9）assigned to men an accordant and proportioned use for the purpose of rhythm and harmony in their pastimes, and has been assigned to the blessed dance of the Muses.2
In this way then let all these things come to pass, and so let them be. But as to their crowning point, we must go to divine generation and therewith the fairest and divinest nature of visible things, so far as God granted the vision of it to men; a vision that none of us may ever boast of having received at his leisure [991c] without the conditions here laid down. And besides these requirements, one must refer the particular thing to its generic form in our various discussions, questioning and disproving what has been wrongly stated; for it3 is rightly found to be altogether the finest and first of tests for the use of men, while any that pretend to be tests, without being so, are the vainest of all labors. And further, we must mark the exactness of time, how exactly it completes all the processes of the heavens, in order that he who is convinced of [991d] the truth of the statement which has been made—that the soul is at once older and more divine than the body—might believe it a most admirable and satisfactory saying that all things are full of gods, and that we have never been disregarded in the least through any forgetfulness or neglect in our superiors. And our view about all such matters must be that, if one conceives of each of them aright, it turns out a great boon to him who receives it in a proper way; but failing this, he had better always call it God. The way is this— [991e] for it is necessary to explain it thus far: every diagram, and system of number, and every combination of harmony, and the agreement of the revolution of the stars must be made manifest as one through all4 to him who learns in the proper way, and will be made manifest if, as we say, a man learns aright by keeping his gaze on unity;

[992a] for it will be manifest to us, as we reflect, that there is one bond naturally uniting all these things: but if one goes about it in some other way, one must call it Fortune, as we also put it. For never, without these lessons, will any nature be happy in our cities: no, this is the way, this the nurture, these the studies, whether difficult or easy, this the path to pursue: to neglect the gods is not permissible, when it has been made manifest that the fame of them, stated in proper terms, hits the mark. [992b] And the man who has acquired all these things in this manner is he whom I account the most truly wisest: of him I also assert, both in jest and in earnest, that when one of his like completes his allotted span at death, I would say if he still be dead, he will not partake any more of the various sensations then as he does now, but having alone partaken of a single lot and having become one out of many,1 will be happy and at the same time most wise and blessed, whether one has a blessed life in continents or in islands; and that such a man will partake [992c] always of the like fortune, and whether his life is spent in a public or in a private practice of these studies he will get the same treatment, in just the same manner, from the gods. And what we said at the beginning, and stands now also unchanged as a really true statement, that it is not possible for men to be completely blessed and happy, except a few, has been correctly spoken. For as many as are divine and temperate also, and partakers of virtue as a whole in their nature, [992d] and have acquired besides all that pertains to blessed study—and this we have explained—are the only persons by whom all the spiritual gifts are fully obtained and held. Those then who have thus worked through all these tasks we speak of privately, and publicly establish by law, as the men to whom, when they have attained the fullness of seniority, the highest offices should be entrusted, while the rest should follow their lead, giving praise to all gods and goddesses; and we should most rightly invite the Nocturnal Council to this wisdom, when we have duly distinguished and approved [992e] all its members.
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