Unit I — Philosophy, the Quadrivium, the Odd Sieve & Perfect Numbers¶
Magnitude vs. Multitude, the Four Sister Sciences (Mathemata), the Odd Sieve of Eratosthenes & Euclid–Nicomachus Perfect Numbers (Book I, Chs. 1–7, 13, 16)
«ταῦτα γὰρ τὰ μαθήματα δοκοῦντι εἶμεν ἀδελφεά· περὶ γὰρ ἀδελφεὰ τὰ τοῦ ὄντος πρώτιστα δύο εἴδεα τὰν ἀναστροφὰν ἔχει.»
“For these sciences seem to be sisters; for they deal with sister subjects, the first two forms of being.” — Archytas of Tarentum (DK 47 B 1), quoted in Book I, Ch. 3.4 (Hoche p. 6)
Book I, Ch. 1 — Philosophy as Love of Wisdom & Immaterial Reality¶
English Translation by Martin Luther D’Ooge (1926) · Hoche pp. 1–3 · D’Ooge pp. 181–182
1. The ancients, who under the leadership of Pythagoras first made science systematic, defined philosophy as the love of wisdom. Indeed the name itself means this, and before Pythagoras all who had knowledge were called “wise” indiscriminately—a carpenter, for example, a cobbler, a helmsman, and in a word anyone who was versed in any art or handicraft. Pythagoras, however, restricting the title so as to apply to the knowledge and comprehension of reality, and calling the knowledge of the truth in this the only wisdom, naturally designated the desire and pursuit of this knowledge philosophy, as being desire for wisdom.
2. He is more worthy of credence than those who have given other definitions, since he makes plain the sense of the term and the thing defined. This wisdom he defined as the knowledge, or science, of the truth in real things, conceiving “science” to be a steadfast and firm apprehension of the underlying substance, and “real things” to be those which continue uniformly and the same in the universe and never depart even briefly from their existence; these real things would be things immaterial, by sharing in the substance of which everything else that exists under the same name and is so called is said to be “this particular thing,” and exists.
3. For bodily, material things are, to be sure, forever involved in continuous flow and change—in imitation of the nature and peculiar quality of that eternal matter and substance which was from the beginning, and which was throughout changeable and variable. The bodiless things, however, of which we conceive in connection with or together with matter, such as qualities, quantities, configurations, largeness, smallness, equalities, relations, actualities, dispositions, places, times, all those things, in short, whereby the qualities found in each body are comprehended—all these are of themselves immovable and unchangeable, visibility and change belonging to them only accidentally through their participation in and affection by the body to which they belong. Now it is with such things that wisdom is particularly concerned, and accidentally also with things that share in them, that is, bodies.
Book I, Ch. 2 — Eternal Being vs. Bodily Flux; Magnitude & Multitude¶
Hoche pp. 3–5 · D’Ooge pp. 182–183
1. Those things, however, are immaterial, eternal, without end, and it is in their nature to persist ever the same and unchanging, abiding by their own essential being, and each one of them is called real in the proper sense. But what are subject to generation and destruction, growth and diminution, all kinds of change and participation, are seen to vary continually, and while they are called real things, by the same term, so far as they partake of reality, by their own nature they are not truly real; for they do not abide even for the shortest moment in the same condition, but are always passing over into all sorts of changes, in the words of Plato's Timaeus (plato:timaeus:27d-36d): “What is that which always is, and has no becoming, and what is that which is always becoming, and never is? The one is apprehended by thought with reason, always existing the same, while the other is conjectured by opinion with unreasoning sense, becoming and perishing, but never really existing.”
4. Of things that are—both those properly so called and those that have the name by homonymy, that is, both intelligible and sensible things—some are unified and continuous, for example, an animal, the universe, a tree, and the like, which are properly and peculiarly called magnitudes (megethē); others are discontinuous, in a side-by-side arrangement, and, as it were, in heaps, which are called multitudes (plēthē), a flock, for instance, a populace, a heap, a chorus, and the like.
5. Wisdom, then, must be considered to be the knowledge of these two forms. Since, however, all multitude and all magnitude are by their own nature of necessity infinite—for multitude starts from a definite root and never ceases increasing; and magnitude, when division beginning with a limited whole is carried on, cannot bring the severing process to an end, but proceeds therefore infinitely—and since sciences are always sciences of limited things, and never of infinites, it is furthermore evident that a science dealing either with magnitude per se, or with multitude per se, could never be formulated, for each of them is limitless in itself, multitude in the direction of the more, and magnitude in the direction of the less. A science would arise to deal with something separated from each of them: with quantity (to poson), set off from multitude, and size (to pēlikon), set off from magnitude.
Book I, Ch. 3 — The Four Sister Sciences (Mathemata / Quadrivium)¶
Hoche pp. 5–9 · D’Ooge pp. 183–187
1. Again, to start afresh, since of quantity one kind is viewed by itself, having no relation to anything else, as ‘even,’ ‘odd,’ ‘perfect,’ and the like, and the other is relative to something else and is conceived of together with its relationship to another thing, like ‘double,’ ‘greater,’ ‘smaller,’ ‘half,’ ‘one and one-half times,’ ‘one and one-third times,’ and so forth, it is clear that two scientific methods will lay hold of and deal with the whole investigation of quantity: Arithmetic, absolute quantity, and Music, relative quantity.
2. And once more, inasmuch as part of size (to pēlikon) is in a state of rest and stability, and another part in motion and revolution, two other sciences in the same way will accurately treat of size: Geometry the part that abides and is at rest, Astronomy (sphairikē) that which moves and revolves.
3. Without the aid of these, then, it is not possible to deal accurately with the forms of being nor to discover the truth in things, knowledge of which is wisdom, and evidently not even to philosophize properly, for “just as painting contributes to the menial arts toward correctness of theory, so in truth lines, numbers, harmonic intervals, and the revolutions of circles bear aid to the learning of the doctrines of wisdom,” says the Pythagorean Androcydes.
4. Likewise Archytas of Tarentum, at the beginning of his Harmonics (archytas:dk47b1), says the same thing in about these words: “It seems that those who know have discerned well, and it is not at all strange that they think correctly concerning the nature of particular things; for having judged well concerning the nature of the whole they were likely also to see well how things are in detail. Thus they have handed down to us clear knowledge of the speed of the stars, and their risings and settings, and of geometry, numbers, and not least of spherics and music; for these sciences seem to be sisters, since they deal with sister subjects, the first two forms of being.”
5. Plato, too, at the end of the thirteenth book of the Laws—to which title some give the name Philosopher, because he investigates and defines in it what sort of man the real philosopher should be, and others Epinomis (plato:epinomis:991d-992a)—expresses it thus: “Every diagram, system of numbers, every combination of harmony, and the single harmonious system of the revolution of the stars must be made manifest as one through all to him who learns in the proper way; and it will become manifest if, as we say, a man learns rightly by keeping his eyes on unity; for it will be seen by those who reflect that there is one natural bond uniting all these things.”
6. Look, then, if these four methods do not resemble bridges (gephyrai) or ladders (klimakes), which carry our minds across from things apprehended by sense and opinion to those comprehended by the mind and understanding, and from those material, physical things, our foster-brethren known to us from childhood, to the things with which we are unacquainted, foreign to our senses, but in their immateriality and eternity more akin to our souls, and above all to the reason which is in our souls, just as Plato sets forth through Socrates in the Republic (plato:rep:vii-525a-534e).
Book I, Chs. 4–5 — Ontological Priority of Arithmetic as Mother of the Sciences¶
Hoche pp. 9–12 · D’Ooge pp. 187–189
I.4.1–2. Which then of these four methods must we first learn? Evidently the one which naturally exists before them all, is superior and takes the place of origin and root, and, as it were, of mother (mētēr) to the others. And this is Arithmetic, not solely because we said that it existed before all the others in the mind of the creating God like some universal and exemplary plan, relying upon which as a design and archetypal example the Creator of the universe sets in order his material creations and makes them attain to their proper ends; but also because it is naturally prior in birth, inasmuch as it abolishes other sciences with itself, but is not abolished together with them (synanairei men, ou synanaireitai de).
I.4.3–4. For example, ‘animal’ is naturally prior to ‘man,’ for abolish ‘animal’ and ‘man’ is abolished; if ‘man’ be abolished, it does not follow that ‘animal’ is abolished at the same time. So with Geometry and Arithmetic: if Geometry exists, Arithmetic must also needs be implied, for it is with the help of the latter that we can speak of triangle, quadrilateral, octahedron, icosahedron, double, eightfold, one and one-half times, or anything else of the sort which is used as a term by geometry, and such things cannot be conceived of without the numbers that are implied with each one. Yet three, four, eight, and the rest might exist even though there were no figures!
I.5.1–2. The same is true of Music: not only because the absolute exists prior to the relative—as ‘great’ to ‘greater’ and ‘man’ to ‘father’—so that the musical consonances, diatessaron (\(4:3\)), diapente (\(3:2\)), and diapason (\(2:1\)), stand to one another in exactly the arithmetical ratios epitritos, hemiolios, and diplasiōn, which exist in pure number prior to sound; and Astronomy in turn depends upon both Geometry (for spherical motion presupposes stationary sphere and circle) and Music (for the celestial revolutions move in harmonic intervals).
Book I, Chs. 6–7 — Number in the Divine Mind; Definition of Number, Even, and Odd¶
Hoche pp. 12–14 · D’Ooge pp. 189–192
I.6.1. All that has by nature with systematic method been arranged in the universe seems both in part and as a whole to have been determined and ordered in accordance with number, by the forethought and the mind of Him that created all things; for the pattern was fixed, like a preliminary sketch, by the domination of number preëxistent in the mind of the world-creating God, number conceptual only and immaterial in every way, but at the same time the true and the eternal essence, so that with reference to it, as to an artistic plan, should be created all these things—time, motion, the heavens, the stars, all sorts of revolutions.
I.7.1–3. Number (arithmos) is limited multitude, or a combination of monads, or a flow of quantity made up of units; and the first division of number is even (artios) and odd (perittos). The even is that which can be divided into two equal parts without a unit intervening in the middle, and the odd is that which cannot be divided into two equal parts because of the aforesaid intervention of a unit. According to the ancient Pythagorean definition, an even number is that which admits of division into the greatest and the smallest parts at the same operation—greatest in size (halves) and smallest in multitude (two, the minimum dyad)—while the odd, when divided into two parts, leaves one part even and the other odd.
Book I, Ch. 13 — Prime Numbers & the Odd Sieve (Koskinon) of Eratosthenes¶
Hoche pp. 29–36 · D’Ooge pp. 204–207
I.13.1–3. Among the odd numbers themselves, three classes are distinguished: first, the prime and incomposite (prōtos kai asynthetos), which receives no other measure save the universal monad alone, as 3, 5, 7, \(11, 13, 17, 19, 23, 29, 31\); second, the secondary and composite (deuteros kai synthetos), which is odd and measured by another odd number besides the unit, such as \(9, 15, 21, 25, 27, 33, 35, 39\) (for example 15 is measured both by \(3\) and by \(5\)); and third, that which is secondary and composite in itself, yet relatively prime (pros allēlous prōtos) to another, such as \(9\) in relation to \(25\), since they share no common measure except the monad.
I.13.3–8 (The Sieve of Eratosthenes). The method of obtaining these numbers is called by Eratosthenes the Sieve (koskinon), because we take the odd numbers mingled together and indiscriminate and out of them by this device, as if by some instrument or sieve, we separate the prime and incomposite by themselves, and the secondary and composite by themselves, and find the mixed class separately:
Set forth all the odd numbers in succession from three in as long a series as possible: \(3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, \dots\) Beginning with the first, 3, examine which numbers in the series it can measure, passing over two intervening odd numbers at a time (stride 3 along the odd array), and you will find it measures \(9, 15, 21, 27, 33, \dots\) Beginning next with the second prime, 5, pass over four intervening odd numbers at a time (stride 5 along the odd array), marking \(15, 25, 35, \dots\) Beginning with the third, 7, pass over six odd numbers at a time to mark \(21, 35, 49, \dots\) Those that escape every mark are prime and incomposite; those marked once by a prime multiplying itself (\(9 = 3^2, 25 = 5^2, 49 = 7^2\)) have a single proper odd measure; and those marked by two or more strides (like 15 marked by both \(3\) and \(5\)) are measured by distinct odd factors.
Book I, Ch. 16 — Superabundant, Deficient & Perfect Numbers (6, 28, 496, 8128)¶
Hoche pp. 39–44 · D’Ooge pp. 209–212
I.16.1–2. Among the simple even numbers, some are superabundant (perissoteleis, like \(12\) and \(24\), whose proper aliquot parts exceed the whole, as a beast with ten tongues or too many fingers), some are deficient (ellipeis, like \(8\) and \(14\), whose aliquot parts fall short of the whole, like the one-eyed Cyclops), and between these two extremes—like virtue and musical harmony between excess and defect—stand the perfect (teleioi) numbers, which are equal to their own parts.
I.16.3. Just as the fair and admirable are rare and easily counted, while the ugly and bad are prolific, so superabundant and deficient numbers are found in great multitude and disorder, whereas perfect numbers are rare and arranged with fitting order: there is only one among the units, 6 (\(1 + 2 + 3 = 6\)); only one in the tens, 28 (\(1 + 2 + 4 + 7 + 14 = 28\)); one in the hundreds, 496; and one in the thousands, below ten thousand, 8128—and their endings alternate in 6 and 8.
I.16.4 (Nicomachus’s Rule of Generation; cf. Euclid IX.36). There is an elegant and unfailing method of producing them, neither omitting any perfect number nor admitting any that is not perfect (euclid:elements:vii-ix): Set forth the double series from the monad, \(1, 2, 4, 8, 16, 32, 64, 128, \dots\), and at each step add the terms successively together. Whenever the cumulative sum (\(2^k - 1\)) turns out to be prime and incomposite, multiply this prime sum by the last power of two added (\(2^{k-1}(2^k - 1)\)), and the product will invariably be perfect:
\(1 + 2 = 3\) (prime) \(\implies 3 \times 2 =\) 6
\(1 + 2 + 4 = 7\) (prime) \(\implies 7 \times 4 =\) 28
\(1 + 2 + 4 + 8 = 15\) (composite, \(3 \times 5\), passed over!)
\(1 + 2 + 4 + 8 + 16 = 31\) (prime) \(\implies 31 \times 16 = \mathbf{496}\)
\(1 + \dots + 32 = 63\) (composite, \(7 \times 9\), passed over!)
\(1 + \dots + 64 = 127\) (prime) \(\implies 127 \times 64 = \mathbf{8128}\)