Etymologies & Works 3

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:

'the voice of the trumpet roared', (Virgil, Aeneid 3,556): 'and the voices broken upon the shore'. For it is proper that the crags of the shore should resound, and (Aeneid 9,503): 'But the trumpet with its ringing bronze sang forth a terrible sound from afar'. Harmony is the modulation of the voice, and the concord of many sounds, or their fitting-together. Symphony is the tempering of modulation from low and high sounds that are in concord, whether in voice, or in breath, or in striking. For through it the higher and the lower voices are brought into concord, so that whoever should be dissonant from it offends the sense of hearing. Its contrary is diaphony, that is, discrepant or dissonant voices. Euphony is the sweetness of the voice. This is also called 'melos' from sweetness and from honey.

The interval (diastema) is a space of the voice fitted together from two or more sounds. The diesis is certain small spaces and gradations of modulation, inclining from one sound into another. The tone (tonus) is the high utterance of the voice. For it is the difference and quantity of harmony, which consists in the accent or tenor of the voice: of this the musicians have divided the kinds into fifteen parts, of which the hyperlydian is the last and highest, the hypodorian the lowest of all. Melody (cantus) is the inflection of the voice, for sound is direct; but sound precedes melody. Arsis is the raising of the voice, that is, the beginning. Thesis is the setting-down of the voice, that is, the end. Sweet voices are fine and thick, clear and high.

Clear (perspicuae) voices are those which are drawn out to a greater length, so that they continuously fill every place, like the blare of trumpets. Fine (subtiles) voices are those in which there is not much breath, such as belongs to infants, or women, or the sick, as in strings. For those strings which are the finest send forth fine and slender sounds. Full (pingues) voices are when much breath goes forth at once, such as belongs to men. The high (acuta) voice is thin and lofty, as we see in strings. The hard (dura) voice is that which sends forth its sounds violently, like thunder, like the sound of the anvil, whenever the hammer strikes upon hard iron. The rough (aspera) voice is hoarse, and one which is scattered through minute and dissimilar pulsings.

The dull (caeca) voice is that which, as soon as it has been sent forth, falls silent, and being choked off is by no means drawn out further, as is the case in earthenware vessels. The 'vinnola' is a soft and flexible voice. And it is called 'vinnola' from 'vinnus', that is, a curl softly bent. But the perfect voice is high, sweet and clear: high, so that it may suffice for the heights; clear, so that it may fill the ears; sweet, so that it may soothe the minds of the hearers. If any of these should be lacking, the voice is not perfect. XXI. ON THE SECOND DIVISION, WHICH IS CALLED ORGANIC. The second is the organic division, in those things which, being filled by the breath blowing back, are animated into the sound of a voice, such as trumpets, reeds, pipes, organs, pandoria, and instruments like these. 'Organum' is a general term for all vessels of music.

But that one to which bellows are applied, the Greeks call by another name. But that it should be called 'organum' is rather the common usage of the Greeks. The trumpet was first invented by the Tyrrhenians, of whom Virgil (Aeneid 8,526): 'And the Tyrrhenian blare of the trumpet bellowed through the sky'. Now it was employed not only in battles, but on all festal days, on account of the clearness of praise or of joy. Whence also it is said in the Psalter (81,4): 'Sound the trumpet at the beginning of the month, on the day of your notable solemnity'. For it had been commanded to the Jews that at the beginning of the new moon they should blare the trumpet, which they do even to this day. They report that the pipes (tibiae) were devised in Phrygia: these were indeed for a long time applied only to funerals, but soon also to the sacred rites of the heathen.

They think the pipes are called 'tibiae' because they were first made from the shin-bones of stags and the leg-bones of fawns; then by a misapplication those also came to be so called which were not made from leg-bones and bones. Hence also the 'tibicen' (piper), as it were the song of the pipes. 'Calamus' (reed) is the proper name of a tree, called from 'calere', that is, from pouring forth voices. Some think that the 'fistula' (panpipe) was invented by Mercury, others by Faunus, whom the Greeks call Pan. Some say it was by Idi, a shepherd of Agrigentum in Sicily. And the fistula is so called because it sends forth a voice. For FOS in Greek is voice, and STOLIA is called 'sent forth'. The sambuca is a species of the symphonies in music. For it is a kind of brittle wood, from which pipes too are constructed. The pandorius is named from its inventor. Concerning whom Virgil (Eclogues 2,32):

'Pan was the first to teach the joining of several reeds with wax; Pan cares for the sheep and the masters of the sheep'. For among the heathen he was the pastoral god, who first fitted unequal reeds for song and composed them with diligent art. XXII. ON THE THIRD DIVISION, WHICH IS NAMED RHYTHMIC. The third is the rhythmic division, pertaining to strings and striking, to which are assigned the species of the various citharas, also the drum (tympanum), the cymbal, the sistrum, cups of bronze and of silver, or other things which, being struck by their metallic stiffness, give back a ringing sound with sweetness, and other things of this kind. The discoverer of the cithara and the psaltery is held to be Tubal, as was said before. But according to the opinion of the Greeks, the use of the cithara is believed to have been discovered by Apollo.

It is handed down that the form of the cithara was at first like the human breast, so that just as the voice comes forth from the breast, so from it the song might be produced, and that it was named for the same reason. For 'breast' in the Doric tongue is called KITHARA. But gradually more species of it came into being, such as psalteries, lyres, barbita, phoenices and pectides, and those which are called Indian and are struck by two at once. Likewise others and yet others, of square or of triangular shape. [4] The number of strings too was multiplied, and the kind was changed. But the ancients named the cithara 'fidicula' or 'fides', because its strings sound together as well as those between whom there is fidelity (fides) agree well. Now the ancient cithara was seven-stringed. Whence also Virgil (Aeneid 6,646):

'the seven distinctions of tones'. They are called 'distinctions' for this reason, that no string gives back a sound like that of the neighboring string. But there are seven strings for this reason, either because they fill up the whole compass of the voice, or because heaven sounds with seven motions. The strings (chordae) are named from the heart (cor), because just as the pulse of the heart is in the breast, so is the pulse of the string in the cithara. Mercury first devised these; and he too was the first to draw sound from strings. The psaltery, which is commonly called the 'canticum', is named from 'psallere' (to play), because to its sound the chorus responds by singing in concord. Now it is a likeness of the barbaric cithara in the manner of the letter DELTA;

but between the psaltery and the cithara there is this difference, that the psaltery has that hollow wood, from which the sound is given back, in the upper part, and the strings are struck below, and they sound from above. But the cithara has the hollow of the wood in the lower part. Now the Hebrews used a ten-stringed psaltery on account of the number of the Decalogue of the Law. The lyre is called APO TOU LUREIN, that is, from the variety of tones, because it produces diverse sounds. They say that the lyre was first invented by Mercury, in this manner. When the Nile, returning into its channels, had left behind various animals in the fields, a tortoise too was left behind. And when it had rotted, and its sinews had remained stretched within the shell, being struck by Mercury it gave a sound;

and after its likeness Mercury made the lyre and handed it over to Orpheus, who was most zealous in this pursuit. Whence it is thought that by the same art he drew to himself by the modulation of song not only wild beasts, but also rocks and woods. And on account of their love of the pursuit and their praise of song, the musicians in the fictions of their fables imagined that it was placed even among the stars. The drum (tympanum) is a skin or hide stretched over wood on one side. For it is the middle part of the symphonia, in the likeness of a sieve. And the drum is so called because it is 'middle' (medium), whence also a half-pearl is called a 'tympanum'; and it too, like the symphonia, is struck with a little rod. The cymbals are certain cups which, when struck, touch one another and make a sound.

They are called cymbals because they are struck together with dancing (ballematia); for when the Greeks say SUN (together), and BALA is dancing. The sistrum is named from its inventress. For Isis, the queen of the Egyptians, is proved to have invented this kind. Juvenal (13,93): 'Let Isis strike my eyes with her angry sistrum'. Hence women too strike this, because the inventress of this kind was a woman. Whence also among the Amazons the army of women was called to war by the sistrum. The little bell (tintinnabulum) has its name from the sound of its voice, just like the clapping of hands, the creaking of doors. The symphonia is commonly the name for a hollow piece of wood stretched with skin on both sides, which the musicians strike with little rods on this side and that, and there arises in it, from the concord of low and high, a most sweet song. XXIII. ON MUSICAL NUMBERS.

Now you inquire into numbers according to music thus. The extremes being set down, for example 6 and 12, you see by how many units 6 is exceeded by 12, and it is by 6 units: you carry it through the square, six times six make 36. You join those two first extremes, 6 to 12, and together they make 18. You divide 36 by 18, and it makes 2. These you join with the lesser sum, that is with 6, and they will be 8, and 8 will be the mean between 6 and 12. Wherefore 8 exceeds 6 by two units, that is, by a third of 6, and 8 is exceeded by 12 by four units, by a third portion. By whatever part therefore it exceeds, by the same it is exceeded.

But just as this ratio exists in the world from the revolving of the circles, so also in the microcosm it prevails so greatly beyond the voice, that without its perfection man himself, lacking the symphonies, does not stand consistent. By the perfection of this same music, meters too consist in arsis and thesis, that is, in raising and setting-down. XXIV. ON THE NAME OF ASTRONOMY. Astronomy is the law of the stars, which traverses by searching reason the courses of the constellations and the figures and dispositions of the stars in relation to one another and to the earth. XXV. ON ITS DISCOVERERS. The Egyptians were the first to discover astronomy. But the Chaldeans were the first to teach astrology and the observation of nativities. Josephus, moreover, asserts as author that Abraham instituted astrology among the Egyptians.

But the Greeks say that this art was first devised by Atlas, and for that reason he was said to have upheld the heaven. But whoever he may have been, stirred by the motion of the heaven and by the reasoning of his mind, he considered, through the changes of the seasons, through the fixed and defined courses of the stars, through the measured spaces of the intervals, certain dimensions and numbers, which by defining and distinguishing he wove into order, and so discovered astrology. XXVI. ON ITS ESTABLISHERS. Now in each language there are indeed volumes written on astronomy by various authors, among whom, however, Ptolemy, king of Alexandria, is held foremost among the Greeks: he also instituted the canons by which the courses of the stars may be found. XXVII. ON THE DIFFERENCE BETWEEN ASTRONOMY AND ASTROLOGY.

Now between astronomy and astrology there is some difference. For astronomy comprises the revolution of the heaven, the risings, settings and motions of the constellations, or from what cause they are so named. But astrology is partly natural, partly superstitious. It is natural, when it follows the courses of the sun and moon, or the fixed stations of the stars at set seasons. But that is superstitious which the astrologers (mathematici) follow, who take auguries from the stars, and who also assign the twelve signs of heaven to the several members of the soul or the body, and by the course of the constellations attempt to predict the nativities and characters of men. XXVIII. ON THE METHOD OF ASTRONOMY. The method of astronomy consists in very many ways.

For it defines what the world is, what the heaven is, what the position and course of the sphere is, what the axis of heaven and the poles are, what the climes of heaven are, what the courses of the sun and moon and of the stars are, and the rest. XXIX. ON THE WORLD AND ITS NAME. The world is that which is composed of heaven and earth and sea and all the constellations. Which is therefore called 'mundus' (world), because it is always in motion (motu); for no rest has been granted to its elements. XXX. ON THE FORM OF THE WORLD. The form of the world is shown thus. For just as the world is raised up toward the northern region, so it is inclined toward the southern. But its head and, as it were, its face is the eastern region, and the last part is the northern. XXXI. ON HEAVEN AND ITS NAME.

The philosophers said that heaven is round, revolving and burning; and it was called by this name (caelum) because, like a chased vessel (vas caelatum), it has the signs of the stars impressed upon it. For God distinguished it with bright lights, and filled it, namely, with the sun and the gleaming orb of the moon, and adorned it with the shining signs of the glittering stars. And this in Greek is called OURANOS, APO TOU ORASTHAI, that is, from seeing, because the air is transparent and purer for beholding. XXXII. ON THE POSITION OF THE CELESTIAL SPHERE. The sphere of heaven is a certain shape formed in the round, whose center is the earth, equally enclosed on all sides.

They say that this sphere has neither beginning nor end, for the reason that, being round like a circle, where it begins or where it leaves off is not easily comprehended. But the philosophers of the world introduced seven heavens, that is, the planets, of the harmonious motion of their globes, and they relate that all things are connected with their orbs; and these, joined to themselves and as it were inserted, they judge to be turned back, and to be borne in a course contrary to the rest. XXXIII. ON THE MOTION OF THIS SAME SPHERE. The motion of the sphere is revolved on two axes, of which one is the northern, which never sets, and is called Boreas; the other the southern, which is never seen, and is called Austronotius.

They say that the sphere of heaven is moved by these two poles, and that with its motion the constellations fixed in it go around from east to west, while those near the pole in the north complete shorter circuits near the pivot. XXXIV. ON THE COURSE OF THIS SAME SPHERE. The sphere of heaven is turned from east and west once in a day and night, in the space of twenty-four hours, in which the sun, by its revolving, completes its course above the lands and below the lands. XXXV. ON THE SWIFTNESS OF HEAVEN. The sphere of heaven is said to run with such swiftness that, unless the stars, which delay it, ran against its headlong course, it would cause the ruin of the world. XXXVI. ON THE AXIS OF HEAVEN. The axis is the straight line of the north, which extends through the middle ball of the sphere;

and it is called axis because on it the sphere is turned like a wheel, or because there is the Wain (plaustrum) there. XXXVII. ON THE CELESTIAL POLES. The poles are circles which run around the axis. Of these one is the northern, which never sets, and is called Boreas; the other the southern, which is never seen, and is called Austronotius; and they are called 'poli' because they are the cycles of the axes, from the use of the wains, being named, namely, from 'polishing' (poliendo); but the northern pole is always seen, the Austronotius never, because the right-hand parts of heaven are higher, the parts of the South being pressed down. XXXVIII. ON THE CARDINAL POINTS OF HEAVEN. The cardinal points of heaven are the extreme parts of the axis. And they are called 'cardines' because heaven is turned through them, or because they are revolved like the heart (cor). XXXIX. ON THE VAULTS OF HEAVEN.

The vaults (convexa) of heaven are its extremities, so called from their curvature, as in this line: 'as often as damp night closes in the vaulted heaven'. For the vaulted (convexum) is curved, as it were turned about or inclined, and bent in the manner of a circle. XL. ON THE GATES OF HEAVEN. The gates of heaven are two, the east and the west. For by one gate the sun comes forth, by the other it withdraws itself. XLI. ON THE TWIN FACE OF HEAVEN. The face or head of heaven is the eastern region, the last part is the northern. Concerning which Lucan (4,106): 'Thus the lowest part of the world lies, which the snowy Zone and perpetual winters press down'. XLII. ON THE FOUR PARTS OF HEAVEN. The climes of heaven, that is, its regions or parts, are four, of which the first part is the eastern, whence some stars rise.

The second is the western, where some stars set for us. The third is the northern, where the sun arrives on the longer days. The fourth is the southern, where the sun arrives on the longer nights. Now the East (Oriens) is named from the rising (exortus) of the sun. The West (Occidens), because it makes the day fall (occidere) and perish. For it hides the light from the world and brings on darkness. The North (Septentrio) is named from the seven stars (septem stellae) of the axis, which, revolving in it, are wheeled about. This is properly also called the summit (vertex), because it is turned (vertitur). But the South (Meridies) is so named either because there the sun makes the midday (medius dies), as it were 'medidies', or because then the aether shines more purely. For 'merum' means pure.

There are also seven other climes of heaven, as it were seven lines from east to west, under which the characters of men are unlike and animals specifically diverse are born, which are named from certain famous places; of which the first is Meroe, the second Syene, the third Catachoras, that is, Africa, the fourth Rhodes, the fifth the Hellespont, the sixth Mesopontus, the seventh the Borysthenes. XLIII. ON THE HEMISPHERES. The hemisphere is the half part of the sphere. The hemisphere above the earth is that part of heaven which is wholly seen by us; the hemisphere below the earth is that which cannot be seen, as long as it shall be below the earth. XLIV. ON THE FIVE CIRCLES OF HEAVEN.

The zones of heaven are five, by whose distinctions some parts are inhabited by reason of their temperateness, while some exist uninhabitable through the vastness of cold or of heat. And they are therefore called zones or circles, because they exist in the circumduction of the sphere. Of these the first circle is called ARKTIKOS for this reason, that within it the signs of the Bears (Arctoi) are seen enclosed. The second circle, THERINOS, which is called TROPIKOS, because in that circle the sun, making summer in the regions of the North, does not pass beyond that circle, but immediately turns back; and from that it is called TROPIKOS. The third circle is EMERINOS, which by the Latins is therefore called the equinoctial, because the sun, when it has come to that orbit, makes the equinox.

For EMERINOS in Latin means day and night, by which circle the half part of the sphere is seen to be established. The fourth circle is called ANTARKTIKOS because it is contrary to the circle which we name ARKTIKON. The fifth circle is CHEIMERINOS TROPIKOS, which by the Latins is called the winter or brumal circle, for this reason, that the sun, when it has come to that circle, makes winter for those who are toward the North, and summer for those who dwell in the parts of the South. XLV. ON THE ZODIAC CIRCLE. Now the Zodiac is a circle, which consists of the angles of five lines, and of one line. XLVI. ON THE WHITE CIRCLE. The Milky (lacteus) circle is a way which is seen in the sphere, so called from its whiteness (candor), because it is white.

Some say that it is the way by which the sun travels around, and that it shines thus from the passing of its own splendor. XLVII. ON THE MAGNITUDE OF THE SUN. The magnitude of the sun is greater than the earth's, whence also at the same moment, when it rises, it appears equally both to the east and to the west at once. But as for the fact that it seems to us as if a cubit in size, one ought to consider how great a distance the sun is from the lands, a length which makes it seem small to us. XLVIII. ON THE MAGNITUDE OF THE MOON. The magnitude of the moon too is reported to be less than the sun's. For while the sun is higher than the moon, and yet appears to us greater than the moon, then if it had approached near to us, it would be beheld much greater than the moon. But just as the sun is greater than the earth, so the earth is greater than the moon by some quantity.

49. ON THE NATURE OF THE SUN. Since the sun is fiery, it grows still hotter from the excessive motion of its revolution. The philosophers say that its fire is nourished by water, and that from the contrary element it receives the power of light and heat. Hence we often see it moist and dewy. 50. ON THE COURSE OF THE SUN. The sun moves by its own power and is not turned about together with the firmament. For if it remained fixed in the sky, all days and nights would be equal; but since we see that it will set tomorrow in one place, and set yesterday in another, it is clear that it moves by its own power and is not turned about with the world. For it completes its yearly circuits over unequal spaces on account of the changes of the seasons. Rising, it makes the day; setting, it brings on the night.

For going farther toward the south it makes winter, so that the earth grows rich with wintry moisture and frosts. Drawing nearer to the north it restores summer, so that crops are hardened to ripeness, and things which in moist places are unripe, being warmed, grow mellow. 51. ON THE EFFECT OF THE SUN. The rising sun makes the day, the setting sun brings on the night; for day is the sun above the lands, night is the sun beneath the lands. For from it come the hours; from it the day, when it has risen; from it also the night, when it has set; from it the months and years are numbered; from it the changes of the seasons come about. And when it runs through the south, it is nearer the earth; but when it is toward the north, it is raised on high.

For this reason God appointed for it different places and times of its course, lest, while it always lingered in the same places, it should consume them with its daily heat; but, as Clement says: 'It receives various courses, by which the temperature of the air is dispensed according to the seasons, and the order of successions and changes is preserved. For while it ascends to the higher regions, it tempers the spring; when it has come to the highest, it kindles the heats of summer; withdrawing again, it restores the mildness of autumn. But when it returns to the lower circle, from the icy fabric of heaven it leaves us the harshness of winter's cold.' (There follows a circular figure, which has in its middle the center of the world, and around it the stations of the sun inscribed thus: here the sunrise at the Lord's Nativity;

the sixth hour of the day; the setting at the Lord's Nativity; the setting at the equinox; the setting of the sun at John's Nativity; ever midnight; the rising of the sun at John's Nativity; here the rising of the sun at the equinox.) 52. ON THE JOURNEY OF THE SUN. The rising sun makes its journey through the south. After it has come to its setting and has dipped itself in the Ocean, it goes by unknown ways beneath the earth and runs back again to the east. 53. ON THE LIGHT OF THE MOON. Some philosophers say that the moon has its own light, and that one part of its globe is luminous but the other dark, [thus: (a figure follows)] and that by gradually turning itself it produces various shapes. Others, on the contrary, say that the moon does not have its own light, but is illuminated by the rays of the sun.

Hence it also suffers eclipse, if the shadow of the earth comes between it and the sun. [For the sun is higher than that place. Thus it happens that, when it is beneath the sun, it shines on its upper part, but on its lower part, which it has toward the earth, it is dark.] 54. ON THE PHASES OF THE MOON. The first shape of the moon is two-horned, thus (a figure follows). The second is 'cut' [crescent], thus (a figure follows). The third is half, thus (a figure follows). The fourth is full, thus (a figure follows). The fifth is again half [from the greater], thus (a figure follows). The sixth is again 'cut,' thus (a figure follows). The seventh is two-horned, thus (a figure follows). The seven-and-a-half and the twenty-two-and-a-half are the midpoints in its orbit (a figure follows). The rest are in proportion. 55.

ON THE MOON'S CONJUNCTION [the interlunar period]. The moon's interlunium is that time between the waning and the newborn moon. It is the thirtieth day, on which the moon does not shine. It cannot then be seen, because being joined to the sun it is darkened; but at the same moment, being reborn and gradually withdrawing from it, it is seen. 56. ON THE COURSE OF THE MOON. The moon regulates the monthly periods by the alternations of light lost and regained. It therefore advances in an oblique course and not a straight one like the sun, namely lest it fall upon the center of the earth and frequently suffer eclipse. For its circle is near the earth. Waxing, it looks toward the east with its horns; waning, toward the west: rightly, because it is about to set and lose its light. 57. ON THE NEARNESS OF THE MOON TO THE EARTH. The moon is nearer to the lands than the sun.

Therefore it completes its course more swiftly in a shorter orbit. For the journey which the sun completes in three hundred and sixty-five days, the moon runs through in thirty days. Hence the ancients reckoned the months by the moon, but the years by the course of the sun. 58. ON THE ECLIPSE OF THE SUN. An eclipse of the sun occurs whenever the thirtieth-day moon comes to the same line on which the sun is borne, and, casting itself before it, darkens the sun. For the sun seems to us to fail while the orb of the moon is set against it. 59. ON THE ECLIPSE OF THE MOON. An eclipse of the moon occurs whenever the moon runs into the shadow of the earth. For it is thought not to have its own light, but to be illuminated by the sun, whence it suffers eclipse if the shadow of the earth comes between it and the sun.

The moon suffers this on the fifteenth day, up until the time when it comes out of the center and shadow of the obstructing earth, and sees the sun, or is seen by the sun. 60. ON THE DIFFERENCE BETWEEN STELLAE, SIDERA, AND ASTRA. Stars (stellae), constellations (sidera), and great stars (astra) differ among themselves. For a stella is any single star. But sidera are formed of many stars, like the Hyades and the Pleiades. Astra, moreover, are great stars, like Orion and Bootes. But writers confound these names, and put astra for stellae and stellae for sidera. 61. ON THE LIGHT OF THE STARS. The stars are said not to have their own light, but to be illuminated by the sun, like the moon also. 62. ON THE POSITION OF THE STARS.

The stars are immovable and, fixed with the heaven, are carried by perpetual motion; nor do they fall during the day, but are darkened by the splendor of the sun. 63. ON THE COURSE OF THE STARS. The heavenly bodies are either borne along or moved. Those are 'borne along' which are fixed to the heaven and revolve with the heaven. But certain ones are 'moved,' like the planets, that is the wandering stars, which accomplish their roving courses yet by a fixed rule. 64. ON THE VARIED COURSE OF THE STARS. Because the stars are borne through the different orbits of the heavenly planets, some, having risen more swiftly, set later; some, having risen more slowly, come more quickly to their setting; others rise together yet do not set at the same time; but all return in their own time to their proper course. 65. ON THE INTERVALS OF THE STARS.

The stars stand apart from the earth at various intervals from one another, and therefore appear more or less to our eyes with differing brightness. For many are greater than those we see as conspicuous, but being placed farther off, they seem small to us. 66. ON THE ORBITAL PERIOD OF THE STARS. The orbital number of the stars is that by which is said to be known in how much time each star runs through its circle, whether in longitude or in latitude. For the Moon is said to complete its circle in a whole year, Mercury in twenty years, Lucifer (the Morning Star) in nine years, the Sun in nineteen years, Vesper (the Evening Star) in fifteen years, Phaethon (Jupiter) in twelve years, Saturn in thirty years. When these are completed, they return to the beginning of their circle at the same signs and parts.

Certain heavenly bodies, hindered by the rays of the sun, become anomalous, or retrograde, or stationary, as the poet also records, saying (Lucan 10.201): 'The Sun divides the seasons of the age; it changes day into night, and with its powerful rays forbids the stars to go, and by its measure delays their wandering courses.' 67. ON THE PLANETS. Certain stars are therefore called planets, that is, wanderers, because they run through the whole world with varied motion. Hence, because they wander, they are called retrograde, or become anomalous, that is, when they add and subtract portions. But when they only subtract, they are called retrograde; and they make a station when they stand still. 68. ON THE PRECESSION AND FORWARD ADVANCE OF THE STARS.

The precession or forward advance of the stars is when a star seems to carry out its motion and goes forward somewhat beyond its custom. 69. ON THE WITHDRAWAL OR RETROGRADATION OF THE STARS. The withdrawal or retrogradation of the stars is that in which a star, while it carries out its motion, seems at the same time to be moved backward as well. 70. ON THE STATION OF THE STARS. The station of the stars is that by which, although a star always moves, yet in certain places they seem to stand still. 71. ON THE NAMES OF THE STARS, AND FROM WHAT CAUSES THEY RECEIVED THEIR NAMES. The Sun (Sol) is so called because it alone (solus) appears, all the constellations being darkened by its brilliance. The Moon (Luna) is called as if Lucina, with the middle syllable removed. Of her Virgil says (Ecl. 4.10): 'Chaste Lucina, be favorable.'

It took its name by derivation from the light of the sun, because it receives light from it and gives back what it has received. The stars (stellae) are so called from 'standing' (stando), because being fixed they always stand in the sky and do not fall. For what we see as it were gliding down from the sky are not stars, but little fires fallen from the ether; which come about when a wind, seeking the higher regions, draws the ethereal fire with it, which by its trailing imitates falling stars. For the stars cannot fall: they are immovable, as has been said, and are borne fixed with the heaven. Sidera are so called because sailors, by observing them, direct their plan for their course, lest by deceptive waves or winds they be carried astray.

But certain stars are called 'signs' (signa) because sailors observe them in steering their oars, contemplating their sharpness and brilliance, by which things the future state of the sky is shown. And indeed all men attend to them for foreseeing the qualities of the weather through summer and winter and the mildness of spring. For by their rising or setting at fixed stations they signify the quality of the seasons. The first of the signs is Arcton, which, fixed on the axis, is turned about with seven stars revolving upon itself. The name is Greek, which in Latin is called 'ursa' (bear); and because it turns in the manner of a wagon (plaustrum), our people called it Septentrio. For triones are properly plowing oxen, so called because they tread the earth (terant), as if 'teriones.'

The nearness of the axis causes the Septentriones (the Bears) not to set, because they are on it. Arctophylax is so called because it follows Arcton, that is, the Helice Bear. The same they also called Bootes, because it clings to the wagon: a sign notable for its many stars, among which is Arcturus. Arcturus is a star placed behind the tail of the Greater Bear in the constellation of Bootes. Whence it is called Arcturus, as if 'Arktou oura' (the Bear's tail), because it is placed at the breast of Bootes. It rises in the autumn season. Orion shines in the south before the footsteps of Taurus, and is called Orion from 'urina,' that is, from the inundation of waters. For rising in the time of winter, it troubles the sea and the lands with waters and storms.

The Latins call this one 'Iugula,' because it is armed like a sword, and terrible and most brilliant in the light of its stars; in which, if all these shine, fair weather is portended; if their edge is darkened, a storm is seen to threaten. The Hyades are so called 'apo tou hyein,' that is, from moisture and rains. For rain in Greek is called 'hyetos.' Indeed by their rising they produce rains. Whence the Latins also called them 'suculae' (little sows), because when they are born, signs of rains are shown. Of them Virgil says (Aen. 1.744): 'Arcturus and the rainy Hyades.' They are seven on the forehead of Taurus, and rise in the spring season. The Pleiades are named from plurality, because the Greeks call plurality 'apo tou pleiston.' For they are seven stars before the knees of Taurus;