詞源與文集 第 3 章 第 1/3 段

MATHEMATICA is called in Latin 'doctrinal science' (doctrinalis scientia), which considers abstract quantity. For abstract quantity is that which, separating it in the intellect from matter or from other accidents—such as even and odd, or others of this sort—we treat in mere reasoning. Of it there are four species: namely, Arithmetic, Music, Geometry, and Astronomy. Arithmetic is the discipline of numerable quantity in itself. Music is the discipline that speaks of the numbers which are found in sounds. Geometry is the discipline of magnitude and of forms. Astronomy is the discipline that contemplates the courses of the heavenly stars, and all their figures, and the dispositions of the stars.

These disciplines we now set forth a little more broadly, so that their principles may be fittingly displayed. I. ON THE NAME OF THE DISCIPLINE OF ARITHMETIC. Arithmetic is the discipline of numbers, for the Greeks call number ARITHMON. The writers of secular letters wished it to be the first among the mathematical disciplines, because in order to exist it needs no other discipline. But Music and Geometry and Astronomy, which follow, need its aid in order to be and to subsist. II. ON ITS AUTHORS. They report that among the Greeks Pythagoras first composed the discipline of number, and that it was afterwards more fully arranged by Nicomachus; which among the Latins first Apuleius, then Boethius, translated. III.

WHAT NUMBER IS. Number is a multitude constituted of units. For one is the seed of number, not number itself. Number gave the coin (nummus) its name, and from its frequent use bestowed upon it the word. 'One' (unus) draws its name from the Greek, for the Greeks say ENA for one; likewise two and three, which they call DUO and TRIA. But 'four' (quattuor) took its name from the square (quadrata) figure. 'Five' (quinque), however, received its word not according to nature but according to the pleasure of will, from him who assigned names to numbers. Six and seven come from the Greek; for in many names that have an aspiration in Greek, we put an S in place of the aspiration. Hence from EX comes 'six,' and from EPTA 'seven,' just as from herpillos comes the herb serpillum (wild thyme).

'Eight' (octo), moreover, comes by transference, just as with them and with us: so they say ENNEA, we novem; they DEKA, we decem. 'Ten' (decem) is so called from a Greek etymology, because they bind and join together the numbers lying below. For DESMOS is used by them for 'to join' or 'to bind.' Further, 'twenty' (viginti) is so called because they are ten twice-begotten, U being put for the letter B. 'Thirty' (triginta), because they are begotten from the ternary by the denary; and so up to ninety. 'A hundred' (centum) is named from cantho, which is a circle; 'two hundred' (ducenti) from 'two hundreds.' So too the rest up to a thousand. 'A thousand' (mille) is from multitude, whence also militia (soldiery), as if multitia; thence too milia (thousands), which the Greeks, with a letter changed, call a myriad. IV. WHAT NUMBERS AVAIL. The reckoning of numbers is not to be despised.

For in many passages of the Holy Scriptures how great a mystery numbers hold shines forth. For it was not said in vain in the praises of God (Wisdom 11:21): 'Thou hast ordered all things in measure and number and weight.' For the number six (senarius), which is perfect in its parts, declares by a certain signification of its number the perfection of the world. Likewise the forty days in which Moses and Elijah and the Lord himself fasted are not understood without a knowledge of numbers. So too other numbers exist in the sacred scriptures, whose figures none but those versed in the knowledge of this art are able to unravel.

It is granted to us also in some measure to subsist under the discipline of numbers: when by it we tell the hours, when we reckon the course of the months, when we recognize the span of the returning year. By number, indeed, we are instructed lest we be confounded. Take away number from all things, and all things perish. Remove computation from the age, and blind ignorance embraces all things; nor can he who knows not the method of calculation be distinguished from the other animals. V. ON THE FIRST DIVISION OF EVENS AND ODDS. Number is divided into evens and odds. The even number is divided into these: evenly even, evenly odd, and oddly even. The odd number is divided into these: first and simple, second and composite, third middling;

which in one manner is first and incomposite, but in another manner second and composite. An even number is one that can be divided into two equal parts, as 2, 4, and 8. An odd number is one that cannot be divided into equal parts, one middle either falling short or exceeding, as 3, 5, 7, 9, and the rest. An evenly even number is one that is evenly divided by an even number, until it arrives at the indivisible unit; for example, 64 has as its half 32, this again 16, 16 indeed 8, the eight 4, the four 2, the two 1, which is single and indivisible. Evenly odd is one that admits sectioning into equal parts, but its parts soon remain indissectible, as 6, 10, and 38, 50.

For as soon as you divide this number, you run into a number that you cannot cut. An oddly even number is one whose parts can indeed be divided, but they do not reach unity, as 24. For these, divided into half, make 12, and again into another half 6, then into another 3; and that section admits no further division, but before unity a limit is found which you cannot cut. Oddly odd is one that is oddly measured by an odd number, as 25, 49; which, since they are odd numbers, are divided also by odd parts, as seven times seven makes 49 and five times five makes 25. Of odd numbers, some are simple, others composite, others middling.

Simple are those which have no other part except unity alone, as the ternary has only a third, the quinary only a fifth, and the septenary only a seventh; for to these there is but a single part. Composite are those which are measured not only by unity but are also generated by another number, as 9, 15, and 21; for we say three times three, and seven times three, three times five, and five times five. Middling numbers are those which in one manner seem simple and incomposite, but in another manner also composite; as, for example, nine, when compared to 25, is first and incomposite, because it has no common number except the monad alone;

but if it be compared to fifteen, it is second and composite, because there is in it a common number besides the monad, that is, the ternary number; for three times three measures nine, and three times five measures fifteen. Likewise, of even numbers, some are superfluous, others deficient, others perfect. Superfluous are those whose parts drawn together exceed their own fullness, as, for example, twelve. For it has five parts: a twelfth, which is one; a sixth, which is two; a fourth, which is three; a third, which is four; a half, which is six. For one and two and three and four and six drawn together make 16, and far exceed twelve; and so likewise many others, as eighteen, and many such.

Deficient numbers are those which, computed by their parts, yield a lesser sum, as, for example, ten, whose parts are three: a tenth, which is one; a fifth, which is two; a half, which is five. For one and two and five drawn together make eight, far less than ten. Like this is eight, or many others which, reduced into their parts, fall short. A perfect number is one that is filled up by its own parts, as six; for it has three parts, a sixth, a third, and a half: its sixth is one, its third two, its half three. These parts drawn into a sum, that is, one and two and three together, consummate and perfect the same six.

There are, moreover, perfect numbers: within ten, 6; within a hundred, 28; within a thousand, 496. VI. ON THE SECOND DIVISION OF THE WHOLE OF NUMBER. Every number (1) is either considered in itself, (2) or in relation to something. (1) The former is divided thus: some are equal, others unequal. (2) The latter is divided thus: some are greater, others lesser. The greater are divided thus: multiples, superparticulars, superpartients, multiple superparticulars, and multiple superpartients. The lesser are divided thus: submultiples, subsuperparticulars, subsuperpartients, submultiple subsuperparticulars, and submultiple subsuperpartients. A number in itself is one that is stated without any relation, as 3, 4, 5,

6, and the rest alike. A number in relation to something is one that is compared relatively to others; as, for example, 4, when compared to 2, is called double [and multiple], 6 to 3, 8 to 4, 10 to 5; and again 3 to one triple, 6 to 2, 9 to 3, and the rest. Equal numbers are those which are equal in quantity, as, for example, 2 to 2, 3 to 3, 10 to 10, 100 to 100. Unequal numbers are those which, compared to one another, display inequality, as 3 to 2, 4 to 3, 5 to 4, 10 to 6; and universally a greater to a lesser, or a lesser to a greater, when so compared, is called unequal. A greater number is one that has within itself that lesser number to which it is compared, and something more;

as, for example, the quinary number is stronger than the ternary, in that the quinary number has within itself the ternary number and two other parts of it; and the rest are such. [A lesser number is one that is contained by the greater to which it is compared, together with some part of itself, as the ternary to the quinary; for it is contained by it together with two parts of itself.] A multiple number is one that has within itself a lesser number twice, or thrice, or four times, or manifoldly; as, for example, 2, when compared to one, is double; 3 to one, triple; 4 quadruple, and the rest. Contrariwise, a submultiple number is one that is contained within a multiple twice, or thrice, or four times, or manifoldly;

as, for example, one is contained by 2 twice, by 3 thrice, by 4 four times, by 5 five times, and by others manifoldly. A superparticular number is when the stronger contains within itself the inferior number with which it is compared, and likewise one part of it; as, for example, 3, when compared to 2, contains within itself 2 and one other, which is the middle part of two; 4, when compared to 3, contains in itself 3 and one other, which is the third part of three. Again, 5, when compared to 4, has in itself the quaternary number and one other, which is said to be the fourth part of the quaternary number; and the rest are such.

A superpartient number is one that contains within itself the whole inferior number, and above this two, or three, or four, or five, or other parts of it; as, for example, 5, when compared to 3, the quinary number has in itself the ternary and above this two other parts of it; 7, when compared to 4, has in itself 4 and three other parts of it; 9, when compared to 5, has in itself 5 and four other parts of it. A subsuperpartient number is one that is contained in a superpartient number together with some parts of itself, two or three or more; as, for example, 3 is contained by 5 with two other parts of itself; 5 by 9 with four parts of itself.

A subsuperparticular number is a lesser one that is contained in a stronger number together with one other part of itself, whether a half, or a third, or a fourth, or a fifth; as, for example, 2 to 3, 3 to 4, 4 to 5, and the rest. A multiple superparticular number is one that, when compared to a number inferior to itself, contains within itself the whole inferior number manifoldly together with some part of it; as, for example, 5, when compared to 2, contains within itself twice 2, namely 4, and one part of it; 9, when compared to 4, contains within itself twice 4, namely 8, and one part of it.

[A submultiple subsuperparticular number is one that, when compared to a number stronger than itself, is contained by it manifoldly together with one other part of itself; as, for example, 2, when compared to 5, is contained by it twice together with one part of itself.] A multiple superpartient number is one that, when compared to a number inferior to itself, contains it manifoldly together with other parts of it; as, for example, 8, when compared to 3, contains within itself twice 3 together with two other parts of it; 14, when compared to 6, contains within itself twice 6 together with two other parts of it; [16, when compared to 7, contains it twice together with two other parts of it;

21, when compared to 9, contains within itself twice 9 together with three other parts of it]. A submultiple superpartient number is one that, when compared to a number stronger than itself, is contained by it manifoldly together with some parts of itself; as, for example, 3, when compared to 8, is contained twice with two parts of itself; 4, when compared to 11, is contained twice with three parts of itself. VII. ON THE THIRD DIVISION OF THE WHOLE OF NUMBER. Numbers (1) are either discrete, (2) or continuous. The latter is divided thus: (1) lineal, (2) superficial, (3) solid. A discrete number is one that is composed of discrete units (monads), as, for example, 3, 4, 5, 6, and the rest. A continuous number is one that is composed of joined units;

as, for example, when the ternary number is understood in magnitude, that is, in a line, or in space, or in a solid, it is called continuous; likewise the quaternary and quinary numbers. A lineal number is one that, beginning from a monad, is written lineally to infinity. Whence alpha is set for the designation of lines, since among the Greeks this letter signifies 'one' (here follows a figure). A superficial number is one that is contained not only by length but also by breadth, as the triangular, square, pentagonal, or circular numbers, and the rest, which are always contained in a plane foot, that is, in a surface. A triangular number is thus (a figure follows). A square number is thus (a figure follows). A pentagonal, thus (a figure follows).

A circular number is thus, one that, when multiplied by itself, beginning from itself is turned back to itself, as, for example, five times five makes 25, thus (a figure follows). A solid number is one that is contained by length and breadth and height, as are the pyramids, which rise up in the manner of a flame, thus (a figure follows). A cube, as are dice, thus (a figure follows). Spheres, which have equal roundness on every side, thus (a figure follows). A spherical number, moreover, is one that, multiplied from a circular number, begins from itself and turns back into itself. Five times five is 25; this circle, when multiplied into itself, makes a sphere, that is, five times 25 makes 125. VIII. ON THE DIFFERENCE OF ARITHMETIC, GEOMETRY, AND MUSIC.

Between Arithmetic and Geometry and Music there is this difference, in how you find the mean. In Arithmetic you first seek thus: you join the extremes, and divide, and produce the mean; for example, let the extremes be 6 and 12, you join them and they make 18; you divide the sum and produce 9, which is the arithmetical analogy, so that by however many units the mean surpasses the first, by these it is surpassed by the extreme. For 9 surpasses 6 by three units, and by these it is surpassed by 12. But according to geometry you seek thus: the extremes multiplied together produce as much as the means multiplied together; for example, 6 and 12 multiplied make 72, and the means 8 and 9 multiplied make just as much. According to music, thus:

By whatever part the mean surpasses the first, by that same part the mean is surpassed by the extreme. For example, 6 and 8 (with 12 as the extreme): they surpass by two parts, and by that same middle part which is the mean, 8, it is surpassed by the last, 12. IX. HOW MANY INFINITE NUMBERS THERE ARE. That numbers are infinite is most certain, since at whatever number you suppose an end should be made, that very same number—I do not say can be increased by adding one, but, however great it be, and however vast a multitude it contain—can, by the very reasoning and knowledge of numbers, not only be doubled but even multiplied. And so each number is bounded by its own properties, so that none of them can be equal to any other whatsoever.

Therefore they are both unlike one another and diverse, and each single one is finite, yet all together are infinite. X. ON THE DISCOVERERS OF GEOMETRY AND ITS NAME. The discipline of geometry is said to have been first discovered by the Egyptians, because, when the Nile flooded and everyone's holdings were overlaid with mud, the practice of dividing the land by lines and measures gave the art its name. Thereafter, advancing further through the acumen of the wise, it measures the spaces of sea and sky and air. For, spurred on by zeal, after the measuring of the earth they thus began to inquire into the spaces of the heavens as well:

namely, at how great an interval the moon stands from the earth, at how great a distance the sun itself is from the moon, and to what measure it extends all the way up to the summit of heaven; and thus by a plausible method they distinguished these very intervals of the sky and the circuit of the celestial sphere by the number of stades. But since this discipline began from the measuring of the earth, it kept its name from its very beginning. For geometry is named from earth and from measure. For earth in Greek is called GE, and METRA is measure. The art of this discipline contains within itself lines, intervals, magnitudes and figures, and in figures dimensions and numbers. XI. ON THE FOURFOLD DIVISION OF GEOMETRY. The division of geometry is fourfold: into the plane, into numerable magnitude, into rational magnitude, and into solid figures.

Plane figures are those which are bounded by length and breadth; according to Plato these are five in number. Numerable magnitude is that which can be divided by the numbers of arithmetic. Rational magnitudes are those whose measure we can know, but irrational are those whose quantity of measure is not held as known. XII. ON THE FIGURES OF GEOMETRY. Solid figures are those which are bounded by length, breadth and height, such as the cube, of which there are five species in the plane. Of these the first is the circle, a plane figure, which is called 'drawn around' (circumducta); in whose midst is a point, toward which all things converge, which they call the center of geometry, and which the Latins name the point of the circle (a figure follows). The quadrilateral figure is a square in the plane;

which lies under four straight lines, thus (a figure follows). The 'dianatheton grammon' is a plane figure, thus (a figure follows). The orthogonium, that is, the right-angled figure, is a plane figure. For it is a triangle and has a right angle (a figure follows). The isopleuros is a plane figure, straight and set below (a figure follows). The sphere is a figure formed into a round shape, equal in all its parts (a figure follows). The cube is a properly solid figure, which is bounded by length, breadth and height (a figure follows). The cylinder is a square figure, having a semicircle above (a figure follows). The cone (conon) is a figure which ends from broad into narrow, like the orthogonium (a figure follows). The pyramid is a figure which rises up in the manner of fire from broad to a point;

for fire among the Greeks is called PUR (a figure follows). But just as every number is below ten, so within this circle is enclosed the compass of all figures (a figure follows). Now the first figure of this art is the point, which has no part. The second is the line, length without breadth. A straight line is one which lies evenly with respect to its own points. But a surface is that which has lengths and breadths only. And the boundaries of a surface are lines, whose forms have therefore not been placed among the ten figures above, because they are found among them. XIII. ON THE NUMBERS OF GEOMETRY. You inquire into numbers according to Geometry thus. For its extremes, when multiplied, produce as much as the means when doubled:

for example, 6 and 12 multiplied make 72, and the means 8 and 9 multiplied make just as much. [XIV. EXPOSITION OF THE FIGURES WRITTEN BELOW. Another reckoning in the motion of the stars is likewise gathered under eight figures: namely, whether they are diametrical, or square, or trigonal, or hexagonal, or asyndeta, or conjoined, or 'circumferens', that is, 'superferens' or 'superfertur'. They are diametrical when five signs lie between them. Tetragonal, when two. Hexagonal, when one. Asyndeton, when none. Conjoined, when they are in the same portion. 'Superferens', when one comes upon another or performs an act. 'Superfertur', when it goes before. Trigonal, when three lie between as means. Likewise, according to another reckoning there are eight differences, that is:

sign, degrees, boundaries, conjunction, whether in retrograde or in direct course, latitude and longitude. The reckoning of the inner form. In this place such a question might arise. Since in the order of number 8 comes before 9, here he has set 9 first, because in the reckoning of arithmetic or geometry 8 is more than 9. For 8 is a cube, that is, a solid, that is, a body beyond which one can find nothing more. But 9 is a surface, that is, a thing which is not full, but lacks perfection. Here two cubes, that is, two solidities, are found in this manner. Six is the first perfect number; for it is divided by equal numbers thus: a sixth by one (as); into a third by twos (dupondii); three times two, six; into a half, that is, twice three, six.

You will find another which you may thus divide by equal numbers, which agrees with the matter proposed. Between the first in the order, that is, 10, which on account of the first perfect number, multiplying with the first series six times nine gives 54; nine times six gives 54. And it is recognized that the material is not without reason found to have had so many parts by the two, of which it has one in such an order: 1, 2, 3, 4, 9, 8, and the others together 27.] XV. ON MUSIC AND ITS NAME. Music is the skill of modulation, consisting in sound and song. And it is called Music by derivation from the Muses. But the Muses are named APO TOU MASAI, that is, from inquiring, because through them, as the ancients would have it, the power of songs and the modulation of the voice were sought out.

Because their sound, being a sensible thing, flows past into time that is gone, and is impressed upon the memory. Hence it was fabled by the poets that the Muses were the daughters of Jove and of Memory. For unless sounds are held by man in memory, they perish, because they cannot be written. XVI. ON ITS DISCOVERERS. Moses says that the discoverer of the art of music was Tubal, who was of the stock of Cain before the flood. But the Greeks say that Pythagoras discovered the beginnings of this art from the sound of hammers and from the striking of stretched strings. Others report that Linus the Theban, and Zethus and Amphion, were first to become renowned in the art of music. After them this discipline was gradually set in order and increased in many ways, and it was as disgraceful to be ignorant of music as of letters.

Now it was employed not only in sacred rites, but also in all solemn occasions, and in all joyful or rather sorrowful affairs. For as in divine veneration there were hymns, so at weddings there were the songs of Hymen, and at funerals dirges and laments were sung to the pipes. But at banquets the lyre or the cithara was passed around, and a festal kind of songs was arranged for each of those reclining. XVII. WHAT MUSIC CAN DO. And so without Music no discipline can be perfect, for there is nothing without it. For even the world itself is said to have been composed with a certain harmony of sounds, and the very heaven to revolve under the modulation of harmony. Music moves the affections, and stirs the senses into a different disposition.

In battles too the concord of the trumpet kindles the fighters, and the more vehement the blast has been, the braver becomes the spirit for the contest. Indeed song urges on the rowers, and music soothes the mind for the enduring of toils too, and the modulation of the voice consoles the weariness of each several labor. Music also calms excited minds, as one reads of David, who by the art of modulation delivered Saul from the unclean spirit. Music also draws the very beasts, and even serpents, birds and dolphins, to the hearing of its modulation. But also whatever we speak, or however we are stirred inwardly by the pulsings of our veins, is shown to be bound up through musical rhythms with the powers of harmony. XVIII. ON THE THREE PARTS OF MUSIC.

The parts of Music are three, that is, harmonic, rhythmic, and metric. The harmonic is that which distinguishes the high and the low in sounds. The rhythmic is that which examines the succession of words, whether the sound coheres well or ill. The metric is that which recognizes by a plausible method the measure of the various meters, as for example the heroic, the iambic, the elegiac, and the rest. XIX. ON THE THREEFOLD DIVISION OF MUSIC. Now with respect to every sound, which is the material of songs, its nature is agreed to be threefold. The first is the harmonic, which consists of the singing of voices. The second is the organic, which consists of blowing. The third is the rhythmic, which takes its measures from the striking of the fingers.

For a sound is produced either by the voice, as through the throat, or by breath, as through the trumpet or pipe, or by striking, as through the cithara, or through anything else which is tuneful when struck. XX. ON THE FIRST DIVISION OF MUSIC WHICH IS CALLED HARMONIC. The first division of Music, which is called harmonic, that is, the modulation of the voice, pertains to comic actors, tragic actors, or choruses, or to all who sing with their own voice. This produces motion from soul and body, and from motion sound, from which Music is gathered, which in man is called the voice. The voice is air struck by the breath, whence also words (verba) are named. But properly the voice belongs to men, or to irrational animate beings. For in other things sound is called 'voice' abusively, not properly, as: