From the Introduction to Arithmetic, by Nicomachus of Gerasa (ca. 100 AD), trans. by Martin Luther D'Ooge, Macmillan, 1926 (pp. 181-190).
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.
He is more worthy of credence than those who have given other definitions, since he makes clear 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.
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 has been from the beginning, and which was all changeable and variable throughout. The bodiless things, however, of which we conceive in connection with or together with matter, such as qualities, quantities, configurations, largeness, smallness, equality, relations, actualities, dispositions, places, times, all those things, in a word, whereby the qualities found in each body are comprehended — all these are of themselves immovable and unchangeable, but accidentally they share in and partake of the affections of the body to which they belong. Now it is with such things that 'wisdom' is particularly concerned, but accidentally also with things that share in them, that is, bodies. 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 involved in birth 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 as the former, so far as they partake of them, they are not actually real
by their own nature; for they do not abide for even the shortest moment in the
same condition, but are always passing over in all sorts of changes. To quote
the word of Timaeus, in Plato, "What is that which always is, and has no
birth, and what is that which is always becoming but never is? The one is apprehended
by the mental processes, with reasoning, and is ever the same; the other can
be guessed at by opinion in company with unreasoning sense, a thing which becomes
and passes away, but never really is."
Therefore, if we crave for the goal that is worthy and fitting for man, namely, happiness of life — and this is accomplished by philosophy alone and by nothing else, and philosophy, as I said, means for us desire for wisdom, and wisdom the science of the truth in things, and of things some are properly so called, others merely share the name — it is reasonable and most necessary to distinguish and systematize the accidental qualities of things.
Things, then, both those properly so called and those that simply have the name, are some of them unified and continuous, for example, an animal, the universe, a tree, and the like, which are properly and peculiarly called 'magnitudes'; others are discontinuous, in a side-by-side arrangement, and, as it were, in heaps, which are called 'multitudes,' a flock, for instance, a people, a heap, a chorus, and the like. 2
Wisdom, then, must be considered to be the knowledge of these two forms. Since, however, all multitude and 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 dividing process to an end, but proceeds therefore to infinity 3 — and since sciences are always sciences of limited things, and never of infinites, it is accordingly 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, however, would arise to deal with something separated from each of them, with quantity, set off from multitude, and size, set off from magnitude. 4
Again, to start afresh,
since of quantity one kind is viewed by itself, having no relation to anything
else, as 'even,' 'odd,' 'perfect,' 5
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
[... I]n Plato's Republic, when the interlocutor of Socrates appears to bring certain plausible reasons to bear upon the mathematical sciences, to show that they are useful to human life, arithmetic for reckoning, distributions, contributions, exchanges and partnerships, geometry for sieges, the founding of cities and sanctuaries, and the partition of land, music for festivals, entertainment, and the worship of the gods, and the doctrine of the spheres, or astronomy, for farming, navigation and other undertaking, revealing beforehand the proper procedure and suitable season, Socrates, reproaching him, says: "You amuse me, because you seem to fear that these are useless studies that I recommend; but that is very difficult, nay, impossible. For the eye of the soul, blinded and buried by other pursuits, is rekindled and aroused again by these and these alone, and it is better that this be saved than thousands of bodily eyes, for by it alone is the truth of the universe beheld."
Which then of these four
methods 6 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 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. For example, 'animal'
is naturally antecedent to 'man,' for abolish 'animal' and 'man' is abolished;
but if 'man' be abolished, it no longer follows that 'animal' is abolished at
the same time. And again, 'man' is antecedent to 'schoolteacher,' but if 'schoolteacher'
is nonexistent, it is still possible for 'man' to be. Thus since it has the
property of abolishing the other ideas with itself, it is likewise the older.
Conversely, that is called younger and posterior which implies the other thing with itself, but is not implied by it, like 'musician,' for this always implies 'man.' Again, take 'horse'; 'animal' is always implied along with 'horse,' but not the reverse; for if 'animal' exists, it is not necessary that 'horse' should exist, nor if 'man' exists, must 'musician' also be implied.
So it is with the foregoing sciences; if geometry exists, arithmetic must also needs be implied, for it is with the help of this latter that we can speak of triangle, quadrilateral, octahedron, icosahedron, double, eightfold, or 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. For how can 'triple' exist, or be spoken of, unless the number 3 exists beforehand, or 'eightfold' without 8? But on the contrary, 3, 4, and the rest might be without the figures existing to which they give names. Hence arithmetic abolishes geometry along with itself, but is not abolished by it, and while it is implied by geometry, it does not itself imply geometry.
And once more is this
true in the case of music; not only because the absolute is prior to the relative,
as 'great' to 'greater' and 'rich' to 'richer' and 'man' to 'father,' but also
because the musical harmonies, diatessaron, diapente, and diapason 7,
are named for numbers; similarly all of their harmonic ratios are arithmetical
ones, for the diatessaron is the ratio of 4:3, the diapente that of 3:2, and
the diapason the double ratio; and the most perfect, the di-diapason, is the
More evidently still, astronomy attains through arithmetic the investigations that pertain to it, not alone because it is later than geometry in origin — for motion naturally comes after rest — nor because the motions of the stars have a perfectly melodious harmony, but also because risings, settings, progressions, retrogressions, increases, and all sorts of phases are governed by numerical cycles and quantities.
So then we have rightly undertaken first the systematic treatment of this, as the science naturally prior, more honorable, and more venerable, and, as it were, mother and nurse of the rest; and here we will take our start for the sake of clearness.
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 preexistent
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.
It must needs be, then, that scientific number, being set over such things as these, should be harmoniously constituted, in accordance with itself; not by any other but by itself. Everything that is harmoniously constituted is knit together out of opposites and, of course, out of real things; for neither can nonexistent things be set in harmony, nor can things that exist, but are like one another, nor yet things that are different, but have no relation to one another.
Of such things, therefore, scientific number consists; for the most fundamental species in it are two, embracing the essence of quantity, different from one another and not of a wholly different genus, odd and even, and they are reciprocally woven into harmony with each other, inseparably and uniformly, by a wonderful and divine Nature, as straightway we shall see.
Number is limited multitude
or a combination of units or a flow of quantity made up of units; and the first
division of number is even and odd.
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.
Now this is the definition after the ordinary conception; by the Pythagorean doctrine, however, the even number is that which admits of division into the greatest and the smallest parts at the same operation, greatest in size and smallest in quantity, in accordance with the natural contrariety of these two genera; and the odd is that which does not allow this to be done to it, but is divided into two unequal parts.
In still another way, by the ancient definition, the even is that which can be divided alike into two equal and two unequal parts, except that the dyad, which in its elementary form, admits but one division, that into equal parts; and in any division whatsoever it brings to light only one species of number, however it may be divided, independent of the other. The odd is a number which in any division whatsoever, which is necessarily a division into unequal parts, shows both the two species of numbers together, never without intermixture one with another, but always in one another's company.
By the definition in terms of each other, the odd is that which differs by a unit from the even in either direction, that is, toward the greater or the less, and the even is that which differs by a unit from the odd, that is, is greater by a unit or less by a unit.
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