BOOK IIIOn the Four Mathematical SciencesON ARITHMETIC
On the Four Mathematical Sciences
In examiningIsidore’sDe Arithmeticatwo peculiarities of the development of the subject should be borne in mind. In the first place, the predominant position among the mathematical sciences which Isidore claims for arithmetic was one acquired by it comparatively late. Owing perhaps to the awkwardness of the Greek notation of number[235]geometry had been developed first, and historically arithmetic was an off-shoot from geometry and borrowed its terminology largely from it.[236]It was not given an independent form until the time of Nicomachus (fl. 100 A.D.) whoseIntroductio Arithmeticawas “the first exhaustive work in which arithmetic was treated quite independently of geometry.”[237]Once it become independent, arithmetic, instead of geometry, came to be regarded as the fundamental mathematical science. The old tradition is reflected in Martianus Capella’s order of subjects, in which geometry is placed first and arithmetic second, while the newer tradition is seen in the order of Cassiodorus and Isidore, who both have passages also emphasizing the fundamental character of arithmetic.
The second peculiarity is one which will surprise the modern reader who is familiar with arithmetic as a utilitarian study. The ancientarithmeticahad nothing to do with the art of reckoning, which was calledlogistica.[238]The science and the art of numbers were completely divorced and the latter was excluded from the higher education as we have it in the seven liberal arts. Consequently we can expect nothing practical in Isidore’sDe Arithmetica. Nothing is said of methods of calculation, elementary or advanced, and, as a matter of course, nothing is to be found here on such topics as the use of the abacus[239]or the method of computing Easter, though the latter was the greatest mathematical problem of the time.
Isidore’s source in theDe Arithmeticawas Cassiodorus,[240]whom he copies with little change; while Cassiodorus’ work was apparently a bare abstract of Boethius’ translation of Nicomachus. Isidore’s account is of great brevity and contains a number of unexplained technical terms.
Preface.Mathematics is called in Latindoctrinalis scientia. It considers abstract quantity. For that is abstract quantity which we treat by reason alone, separating it by the intellect from the material or from other non-essentials, as for example, equal, unequal, or the like. And there are four sorts of mathematics, namely, arithmetic, geometry, music and astronomy. Arithmetic is the science of numerical quantity in itself. Geometry is the science of magnitude and forms.[241]Music is the science that treats of numbers that are found in sounds. Astronomy is the science that contemplates the courses of the heavenly bodies and their figures, and all the phenomena of the stars. These sciences we shall next describe at a little greater length in order that their significance may be fully shown.
Chapter 1. On the name of the science of arithmetic.
1. Arithmetic is the science of numbers. For the Greeks call number ἀριθμός. The writers of secular literature have decided that it is first among the mathematical sciences since it needs no other science for its own existence.
2. But music and geometry and astronomy, which follow, need its aid in order to be and exist.
Chapter 2. On the writers.
1. They say that Pythagoras was the first among the Greeks to write of the science of number, and that it was later described more fully by Nicomachus, whose work Apuleius first, and then Boethius, translated into Latin.
Chapter 3. What number is.
1. Number is multitude made up of units. For one is the seed of number but not number.Nummus(coin) gave its name tonumerus(number), and from being frequently used originated the word.
Unusderives its name from the Greek, for the Greeks callunusἕνα, likewiseduo,tria, which they call δύο and τρία.
2.Quattuortook its name from a square figure (figura quadrata).Quinque, however, received its name from one who gave the names to numbers not according to nature but according to whim.Sexandseptemcome from the Greek.
3. For in many names that are aspirated in Greek we usesinstead of the aspiration. We havesexfor ἑξ,septemfor ἕπτα, and also the wordserpillum(thyme) forherpillum.Octois borrowed without change; they have ἔννεα, wenovem; they δέκα, wedecem.
4.Decemis so-called from a Greek etymology, because it ties together and unites the numbers below it. For to tie together and unite is called among them δεσμεύειν.[242]
Chapter 4. What numbers signify.
1. The science of number must not be despised. For in many passages of the holy scriptures it is manifest what great mystery they contain. For it is not said in vain in the praises of God: “Omnia in mensura et numero et pondere fecisti.” For the senarius, which is perfect in respect to its parts,[243]declares the perfection of the universe by a certain meaning of its number. In like manner, too, the forty days which Moses and Elias and the Lord himself fasted, are not understood without an understanding of number.
3. So, too, other numbers appear in the holy scriptures whose natures none but experts in this art can wisely declare the meaning of. It is granted to us, too, to depend in some part upon the science of numbers, since we learn the hours by means of it, reckon the course of the months, and learn the time of the returning year. Through number, indeed, we are instructed in order not to be confounded. Take number from all things and all things perish. Take calculation from theworld and all is enveloped in dark ignorance, nor can he who does not know the way to reckon be distinguished from the rest of the animals.
Chapter 5. On the first division intoevenandodd.
1. Number is divided into even and odd. Even number is divided into the following: evenly even, evenly uneven, and unevenly even, and unevenly uneven.[244]Odd number is divided into the following: prime and uncompounded, compounded, and a third class which comes between (mediocris) which in a certain way is prime and uncompounded, but in another way secondary and compounded.
2. An even number is that which can be divided into two equal parts, as II, IV, VIII.[245]An odd number is that which cannot be divided into equal parts, there being one in the middle which is either too little or too much, as III, V, VII, IX, and so on.
3. Evenly even number is that which is divided equally into even number, until it comes to indivisible unity, as for example, LXIV has a half XXXII, this again XVI; XVI, VIII; VIII, IV; IV, II; II, I, which is single and indivisible.
4. Evenly uneven is that which admits of division into equal parts, but its parts soon remain indivisible, as VI, X, XVIII, XXX, and L, for presently, when you divide such a number, you run upon a number which you cannot halve.
5. Unevenly even number is that whose halves can be divided again, but do not go on to unity, as XXIV. For this number being divided in half makes XII, divided again VI, and again, III; and this part does not admit of further division, but before unity a limit is found which you cannot halve.
6. Unevenly uneven is that which is measured unevenly by an uneven number, as XXV, XLIX; which, being unevennumbers, are divided also by uneven factors, as, seven times seven, XLIX, and five times five, XXV. Of odd numbers some are prime, some compounded, some mean (mediocris).
7. Prime numbers are those which have no other factor except unity alone, as three has only a third, five only a fifth, seven only a seventh, for these have only one factor.
Compound numbers are they which are not only measured by unity, but are produced by another number, as IX, XV, XXI, XXV. For we say three times three are nine, and seven times three are XXI, and three times five are XV, and five times five are XXV.
8. Mean (mediocris) numbers are those which in a certain fashion seem prime and uncompounded and in another fashion secondary and compounded. For example, when IX is compared with XXV, it is prime and uncompounded, because it has no common factor except unity alone, but if it is compared with XV it is secondary and compounded, since there is in it a common factor in addition to unity, that is, III. Because three times three make nine, and three times five make fifteen.[246]
9. Likewise of even numbers some are excessive, others defective, others perfect.[247]Excessive are those whose factors being added together exceed its total, as for example, XII. For it has five factors: 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 being added together make XVI, which is far in excess of twelve....
10. Defective numbers are those which being reckoned by their factors make a less total, as for example, ten....
11. The perfect number is that which is equalled by its factors, as VI.... The perfect numbers are, under ten, VI; under a hundred, XXVIII; under a thousand, CCCCXCVI.
Chapter 6. On the second division of all number.
1. All number is considered either with reference to itself or in relation to something. The former is divided as follows: some are equal, as for example, two; others are unequal, as for example, three.[248]The latter is divided as follows: some are greater, some are less. The greater are divided as follows: intomultiplices(multiple),superparticulares,superpartientes,multiplices superparticulares,multiplices superpartientes. The less are divided as follows:Sub-multiplices(sub-multiple),sub-superparticulares,sub-superpartientes,sub-multiplices sub-superparticulares,sub-multiplices sub-superpartientes.
6. ... Thesuperparticularis numerusis when a greater number contains in itself a lesser number with which it is compared, and at the same time one part of it.
7. For example; III when compared with II contains in itself two and also one, which is the half of two. IV when compared with III, contains three and also one, which is the third of three. Likewise V, when compared with IV, contains the number four and also one, which is the fourth part of the said number four, and so on.
8. Thesuperpartiens numerusis that which contains the whole of a lesser number and in addition two parts of it, either thirds or fifths or other parts. For example, when V is compared with III, the number five contains three and in addition to this two parts of it.
Chapter 7. On the third division of all number.
1. Numbers are abstract or concrete. The latter are divided as follows: first, lineal; second, superficial; third, solid. Abstract number is that which is made up of abstract units. For example, III, IV, V, VI, and so on.
2. Concrete number is that which is made up of units that are not abstract, as for example, the number three, if it is understood of magnitude, whether line, superficies, or solid, is called concrete.
4. The number of superficies is that which is constituted not only by length but also by breadth, as triangular, square, pentangular, or circular numbers, and the rest that are contained in a plane surface or superficies.
5. The circular number, when it is multiplied by itself, beginning with itself, ends with itself. For example,Quinquies quini vicies quinque.
6. ... The spherical number is that which being multiplied by the circular number begins with itself and ends with itself; for example, five times five are twenty-five, and this circle being multiplied by itself makes a sphere, that is, five times XXV make CXXV.
Chapter 8. On the distinction between arithmetic, geometry, and music.
1. Between arithmetic, geometry and music there is a difference in finding the means. In arithmetic in the first place you find it in this way. You add the extremes and divide and find the half; as for example, suppose the extremes are VI and XII, you add them and they make XVIII. You divide and get IX, which is the mean of arithmetic (analogicum arithmeticae), since the mean is surpassed by the last by as many units as it surpasses the first. For IX surpasses VI by three units, and XII surpasses it by the same number.
2. According to geometry you find it this way. The extremes multiplied together make as much as the means multiplied, for example, VI and XII multiplied make LXXII; the means VIII and IX multiplied make the same.
3. According to music you find it in this way: The mean is exceeded by the last term by the part by which it exceeds the first term, as for example, VI is surpassed by VIII by two units, which is a third part, and by the same part the mean VIII is surpassed by the last term which is XII.
Chapter 9. That infinite numbers exist.
1. It is most certain that there are infinite numbers, since at whatever number you think an end must be made I say not only that it can be increased by the addition of one, but, however great it is, and however large a multitude it contains, by the very method and science of numbers it can not only be doubled but even multiplied.
2. Each number is limited by its own proper qualities, so that no one of them can be equal to any other. Therefore in relation to one another they are unequal and diverse, and the separate numbers are each finite, and all are infinite.
In spite of the high development of geometry among the Greeks it never took root as a pure science in the western Roman world,[249]and neither the various practical applications of its principles nor its use as a disciplinary educational subject sufficed to fasten thoughtful attention upon it; in consequence, it lost almost its entire content. As it appears in the four writers who treat of it in later Roman and early medieval times, Martianus Capella, Boethius,[250]Cassiodorus, and Isidore, it furnishes a striking commentary upon the intellectual conservatism that could retain without a suspicion of criticism a subject that was no longer anything but empty form.
The substance of Isidore’sDe Geometriacomes with little change from Cassiodorus. It is noteworthy that these two writers have nothing that does not go with the subject according to the modern conception of it, and do not follow the example of their predecessor Martianus Capella,[251]in whose account of the seven liberal arts the void caused by the loss of the proper content of geometry is filled with geography.
Book III, Chapter 10. On the inventors of geometry and its name.
1. The science of geometry is said to have been discovered first by the Egyptians, because when the Nile overflowed and all their lands were overspread with mud, its origin in the division of the land by lines and measurements gave the name to the art. And later, being carried further by the keenness of the philosophers, it measured the spaces of the sea, the heavens, and the air.
2. For, having their attention aroused, students began to search into the spaces of the heavens, after measuring the earth; how far the moon was from the earth, the sun itself from the moon, and how great a measure extended to the summit of the sky; and thus they laid off in numbers of stades with probable reason the very distances of the sky and the circuit of the earth.
3. But since this science arose from the measuring of the earth, it took its name also from its beginning. Forgeometriais so named from the earth and measuring. For the earth is called γῆ in Greek, and measuring, μέτρον. The art[253]of this science embraces lines, intervals, magnitudes, and figures, and in figures, dimensions and numbers.
Chapter 11. On the four-fold division of geometry.
1. The four-fold division of geometry is into plane figures, numerical magnitude, rational magnitude, and solid figures.
2. Plane figures are those which are contained by length and breadth. Numerical magnitude is that which can be divided by the numbers of arithmetic.
3. Rational magnitudes are those whose measures we can know, and irrational, those the amount of whose measurement is not known.
4. Solid figures are those that are contained by length, breadth, and thickness, which are five in number, according to Plato.
Chapter 12. On the figures of geometry.
1. The first of the figures on a plane surface is the circle, a figure that is plane, and has a circumference, in the middle of which is a point upon which everything converges (cuncta convergunt) which geometers call the center, and the Latins call the point of the circle.
2. A quadrilateral figure is one on a plane surface, and it is contained by four straight lines....
3. A sphere is a figure of rounded form equal in all its parts.
A cube is a solid figure which is contained by length, breadth, and thickness.
5. A cone (conon) is a solid figure which narrows from a broad base like the right-angled triangle.
6. A pyramid is a solid figure which narrows to a point from a broad base like fire. For fire in Greek is called πῦρ.
7. Just as all number is contained within ten so the outline of every figure is contained within the circle.
Chapter 13. On the first principles of geometry.
1. ... A point is that which has no part. A line is length without breadth. A straight line is one which lies evenly in respect to its points. A superficies is that which has length and breadth alone.
Chapter 14. On the numbers of geometry.
1. You search into the numbers of geometry as follows: the extremes being multiplied, amount to as much as the means multiplied; as for example, VI and XII being multiplied, make LXXII; the means VIII and IX being multiplied, amount to the same.
As an educational subject music is the oldest of those grouped under the heading of the seven liberal arts. In Plato’s time music and gymnastic were the staples of education, and the former term meant chiefly the study of poetry, with music in the proper sense of the word as a mere adjunct. As the different subjects, such as grammar, rhetoric, geometry, arithmetic, appeared in the curriculum, the field of music narrowed and it held a less commanding place. Conflicting points of view in regard to it appear to have arisen. The older educational tradition connected music with grammar and the other literary studies. On the other hand, the influence of the Pythagorean theory of number and of its application to music tended to dissociate grammar and music, and to place the latter in relation to the mathematical sciences. It has been noticed that among the older Roman writers from whom evidence on this matter can be drawn—Cicero, Varro, Seneca, Quintilian, and others—the association of music and grammar appears the natural one, while in the Roman writers of the second, third, and fourth centuries both traditions prevail, with an increasing preference for placing music among the mathematical sciences, where it finally found itself when the canon of the seven liberal arts was formed, and where it remained to the end of the middle ages.[254]
In Isidore little is to be found to justify the mathematical environment of music. It is true that at times he defines it as a mathematical science[255]and he insists on the musical view of the universe as a necessary complement to other views. “Without music,” he says, “there can be no perfect knowledge, for there is nothing without it. For even the universe itself is said to have been formed under the guidance of harmony.”[256]But, with the exception of a paragraph on the musical mean, his treatment is entirely taken up with the non-mathematical aspect of the subject, and the definition “music is the practical knowledge of melody”[257]is the one that more closely fits the occasion.
The treatment[258]of music is of about the same length asthat of arithmetic, and is devoted mainly to definitions of musical terms and brief descriptions of wind and stringed instruments. It appears that Isidore knew nothing of music in a technical sense.[259]
Book III, Chapter 15. On music and its name.
1. Music is the practical knowledge of melody, consisting of sound and song; and it is called music by derivation from the Muses. And the Muses were so-called ἀπὸ τοῦ μῶσθαι, that is, from inquiring, because it was by them, as the ancients had it, that the potency of songs and the melody of the voice were inquired into.
2. Since sound is a thing of sense it passes along into past time, and it is impressed on the memory. From this it was pretended by the poets that the Muses were the daughters of Jupiter and Memory. For unless sounds are held in the memory by man they perish, because they cannot be written.
Chapter 16. On its discoverers.
1. Moses says that the discoverer of the art of music was Jubal, who was of the family of Cain and lived before the flood. But the Greeks say that Pythagoras discovered the beginnings of this art from the sound of hammers and the striking of tense cords. Others assert that Linus of Thebes, and Zethus, and Amphion, were the first to win fame in the musical art.
2. After whose time this science in particular was gradually established and enlarged in many ways, and it was as disgraceful to be ignorant of music as of letters. And it had a placenot only at sacred rites, but at all ceremonies and in all things glad or sorrowful.
Chapter 17. On the power of music.
1. And without music there can be no perfect knowledge, for there is nothing without it. For even the universe itself is said to have been put together with a certain harmony of sounds, and the very heavens revolve under the guidance of harmony. Music rouses the emotions, it calls the senses to a different quality.
2. In battles, too, the music of the trumpet fires the warriors, and the more impetuous its loud sound the braver is the spirit for the fight. Also, song cheers the rowers. For the enduring of labors, too, music comforts the mind, and singing lightens weariness in solitary tasks.
3. Music calms overwrought minds also, as is read of David, who by his skill in playing rescued Saul from an unclean spirit. Even the very beasts and snakes, birds and dolphins, music calls to hear its notes. Moreover whatever we say or whatever emotions we feel within from the beating of our pulses, it is proven that they are brought into communion with the virtues through the musical rhythms of harmony.
Chapter 18. On the three parts of music.
1. There are three parts of music, namely,harmonica,rhythmica,metrica.Harmonicais that which distinguishes in sounds the high and the low.Rhythmicais that which inquires concerning the succession of words as to whether the sound fits them well or ill.
2.Metricais that which learns by approved method the measure of the different metres, as for example, the heroic, iambic, elegiac, and so on.
Chapter 19. On the triple division of music.
1. It is agreed that all sound which is the material of music is of three sorts. First isharmonica, which consists of vocal music; second isorganica, which is formed from the breath; third isrhythmica, which receives its numbers from the beat of the fingers.
2. For sound is produced either by the voice, coming through the throat; or by the breath, coming through the trumpet or tibia, for example; or by touch, as in the case of the cithara or anything else that gives a tuneful sound on being struck.
Chapter 20. On the first division of music which is calledharmonica.
1. The first division of music, which is calledharmonica, that is, modulation of the voice, has to do with comedians, tragedians, and choruses, and all who sing with the proper voice.[260]This [coming] from the spirit and the body makes motion, and out of motion, sound, out of which music is formed, which is called in man the voice.
2.Harmonicais the modulation of the voice and the concord or fitting together of very many sounds.
3.Symphoniais the managing of modulation so that high and low tones accord, whether in the voice or in wind or stringed instruments. Through this, higher and lower voices harmonize, so that whoever makes a dissonance from it offends the sense of hearing. The opposite of this isdiaphonia, that is, voices grating on one another or in dissonance.
7.Tonusis a high utterance of voice. For it is a difference and measure of harmony which depends on the stress and pitch of the voice. Musicians have divided its kinds into fifteen parts, of which the hyperlydian is the last and highest, the hypodorian the lowest of all.
8. Song is the modulation of the voice, for sound is unmodulated, and sound precedes song.
Chapter 21. On the second division, which is calledorganica.
1. The second division, organica, has to do with those [instruments] that, filled with currents of breath, are animated so as to sound like the voice, as for example, trumpets, reeds, Pan’s pipes, organs, the pandura, and instruments like these.[261]
Chapter 22. On the third division, which is calledrhythmica.
1. The third division isrhythmica, having to do with strings and instruments that are beaten, to which are assigned the different species of cithara, the drum, and the cymbal, the sistrum, acitabula of bronze and silver, and others of metallic stiffness that when struck return a pleasant tinkling sound, and the rest of this sort.[262]
2. The form of the cithara in the beginning is said to have been like the human breast, because as the voice was uttered from the breast so was music from the cithara, and it was so-called for the same reason. Forpectusis in the Doric language called κίθαρα.
Chapter 23. On the numbers of music.
1. You inquire into the numbers according to music as follows: setting down the extremes, as for example, VI and XII, you see by how many units VI is surpassed by XII, and it is by VI units; you square it; six times six make XXXVI. You add those first-mentioned extremes, VI and XII; together they make XVIII; you divide XXXVI by XVIII; two is the result. This you add to the smaller amount, VI namely; the result will be VIII and it will be the mean between VI and XII. Because VIII surpasses VI by two units, that is by a third of six, and VIII is surpassed by XII by four units, a third part [of twelve]. By what part, then, the mean surpasses, by the same is it surpassed.
2. Just as this proportion exists in the universe, being constituted by the revolving circles, so also in the microcosm—not to speak of the voice—it has such great power that man does not exist without harmony.[263]
The science of astronomy, in its history from the great period of Greece down to the dark ages, furnishes almost as complete a spectacle of decay as does geometry. It is quite certain “that Aristarchus taught the annual motion of the earth around the sun, and both he and Seleukus taught the diurnal rotation of the earth,”[264]but the general scientific development of the age was not sufficient to assimilate this advanced theory, and astronomers went back to a geocentric universe. Strange to say, the later rise of practical astronomy at Alexandria, and the development of pure mathematics, did not secure a return to the more advanced theory, the efforts of the later astronomers being devoted, not to a reconsideration of the fundamental theses of the subject, but to putting the geocentric theory on a secure mathematical basis. The greatest of these astronomers, Ptolemy (second century A.D.), left in hisSyntaxisa comprehensive summing up of mathematical astronomy.
Among the Romans no scientists arose to assimilate the results of the work of the Greeks, and sound ideas as to the form of the universe were rare even in the most intelligent circles. Since systematic observation was not practiced, and a knowledge of the higher mathematics did not exist among the Romans, their astronomy was a matter of tradition and authority. Therefore upon the acceptance of Christianity and the realization that there was a conflict between the Greek and the Hebrew cosmologies, it was a comparatively easy matter to accept the Scriptures instead of the secular writers as the source of authority.
In Isidore’s ideas on cosmology a curious inconsistency appears. On the one hand, he shows that he regards the words of the Scripture as the final authority, and he frequently gives expression to primitive notions in accord with the Hebrew cosmology. On the other hand, he displays a greater liberality than is shown by his predecessor, Cassiodorus, or by any other Christian writer in the Latin language up to his time, in borrowing from the pagan writers on astronomy. The explanation of this may be that it was a natural reaction from dogmatic narrowness, made possible for him by the favorable conditions offered by contemporary Spain; but the more probable supposition is that his natural vagueness of mind and lack of critical power enabled him to be much more liberal in effect than he in reality would have wished to be.[265]
Another feature of Isidore’sDe Astronomiathat deserves notice is his attitude toward the forbidden science of astrology.[266]He denies a fundamental assumption of the science, namely, that Mercury and Venus, for example, have as planets an influence analogous to their characters in mythology, and he asserts that the names of the planets and fixed stars, as used in astrology, have no validity. This was vigorous reasoning for the dark ages, and to all appearance it completely cut away the foundation of astrology.[267]Nevertheless Isidore believed that astrology had some truth—the magi who announced the birth of Christ were, he believed, astrologers—but this truth arose “outof a deadly alliance of men and bad angels.” His attitude, then, seems to be that astrologers may forecast the future, but that their ability to do so depends on the assistance of demons, and that the drawing up of nativities is merely a pretence to cloak this partnership.
Little is known of astronomy as a subject in the Roman schools. It no doubt formed part of the curriculum, but apparently no text-book was produced between the time of Varro and that of Martianus Capella. The three school treatises of late Roman and early medieval times, written by Capella, Cassiodorus, and Isidore, were all the work of educational encyclopedists from whom nothing of a scientific character could be expected.
Book III, Chapter 24. On the name of astronomy.
1. Astronomy is the law of the stars, and it traces with inquiring reason the courses of the heavenly bodies, and their figures, and the regular movements of the stars with reference to one another and to the earth.
Chapter 25. On its discoverers.
1. The Egyptians were the first to discover astronomy. And the Chaldeans first taught astrology and the observance of nativity. Moreover, Josephus asserts that Abraham taught astrology to the Egyptians. The Greeks, however, say that this art was first elaborated by Atlas, and therefore it was said that he held the heavens up.
2. Whoever was the discoverer, it was the movement of the heavens and his rational faculty that stirred him, and in the light of the succession of seasons, the observed and established courses of the stars, and the regularity of the intervals, he considered carefully certain dimensions and numbers, and getting a definite and distinct idea of them he wove them into order and discovered astrology.
Chapter 26. On its teachers.
1. In both Greek and Latin there are volumes written on astronomy by different writers. Of these Ptolemy[268]is considered chief among the Greeks. He also taught rules by which the courses of the stars may be discovered.[269]
Chapter 27. The difference between astronomy and astrology.
1. There is some difference between astronomy and astrology. For astronomy embraces the revolution of the heavens, the rise, setting, and motion of the heavenly bodies, and the origin of their names. Astrology, on the other hand, is in part natural, in part superstitious.
2. It is natural astrology when it describes the courses of the sun and the moon and the stars, and the regular succession of the seasons. Superstitious astrology is that which the mathematici follow, who prophesy by the stars, and who distribute the twelve signs of the heavens among the individual parts of the soul or body, and endeavor to predict the nativities and characters of men from the course of the stars.
Chapter 28. On the subject-matter of astronomy.
1. The subject-matter of astronomy is made up of many kinds. For it defines what the universe is, what the heavens, what the position and movement of the sphere, what the axis of the heavens and the poles, what are the climates of the heavens, what the courses of the sun and moon and stars, and so forth.
Chapter 29. On the universe and its name.
1.Mundus(the universe) is that which is made up of the heavens and earth and the sea and all the heavenly bodies. And it is calledmundusfor the reason that it is always inmotion. For no repose is granted to its elements.
Chapter 30. On the form of the universe.
1. The form of the universe is described as follows: as theuniverse rises toward the region of the north, so it slopes away toward the south; its head and face, as it were, is the east, and its back part the north.
Chapter 31. On the heavens and their name.
1. The philosophers have asserted that the heavens are round, in rapid motion, and made of fire, and that they are called by this name (coelum) because they have the forms of the stars fixed on them, like a dish with figures in relief (coelatum).
2. For God decked them with bright lights, and filled them with the glowing circles of the sun and moon, and adorned them with the glittering images of flashing stars.
Chapter 32. On the situation of the celestial sphere.
1. The sphere of the heavens is rounded and its center is the earth, equally shut in on every side. This sphere, they say, has neither beginning nor end, for the reason that being rounded like a circle it is not easily perceived where it begins or where it ends.
2. The philosophers have brought in the theory of seven heavens of the universe, that is, globes with planets moving harmoniously, and they assert that by their circles all things are bound together, and they think that these, being connected, and, as it were, fitted to one another, move backward and are borne with definite motions in contrary directions.
Chapter 33. On the motion of the same.
1. The sphere revolves on two axes, of which one is the northern, which never sets, and is called Boreas; the other is the southern, which is never seen, and is called Austronotius.
2. On these two poles the sphere of heaven moves, they say, and with its motion the stars fixed in it pass from the east all the way around to the west, theseptentrionesnear the point of rest describing smaller circles.
Chapter 34. On the course of the same sphere.
1. The sphere of heaven, [moving] from the east towards the west, turns once in a day and night, in the space of twenty-four hours, within which the sun completes his swift revolving course over the lands and under the earth.
Chapter 35. On the swiftness of the heavens.
1. With such swiftness is the sphere of heaven said to run, that if the stars did not run against its headlong course in order to delay it, it would destroy the universe.
Chapter 36. On the axis of the heavens.
1. The axis is a straight line north, which passes through the center of the globe of the sphere, and is called axis because the sphere revolves on it like a wheel, or it may be because the Wain is there.
Chapter 37. On the poles of the heavens.
1. The poles are little circles which run on the axis. Of these one is the northern which never sets and is called Boreas; the other is the southern which is never seen, and is called Austronotius.
Chapter 38. On thecardinesof the heavens.
1. Thecardinesof the heavens are the ends of the axis, and are calledcardines(hinges) because the heavens turn on them, or because they turn like the heart (cor).
Chapter 40. On the gates of the heavens.
1. There are two gates of the heavens, the east and the west. For by one the sun appears, by the other he retires.
Chapter 42. On the four parts of the heavens.
1. Theclimataof the heavens, that is, the tracts or parts, are four, of which the first part is the eastern, where some stars rise; the second, the western, where some stars set; the third, the northern, where the sun comes in the longer days; the fourth, the southern, where the sun comes in the time of the longer nights.
4. There are also otherclimataof the heavens, seven in number, as if seven lines from east to west, under which the manners of men are dissimilar, and animals of different species appear; they are named from certain famous places, of which the first is Meroe; the second, Siene; the third,Catachoras, that is Africa; the fourth, Rhodus; the fifth, Hellespontus; the sixth, Mesopontus; the seventh, Boristhenes.[270]
Chapter 43. On the hemispheres.
1. A hemisphere is half a sphere. The hemisphere above the earth is that part of the heavens the whole of which is seen by us; the hemisphere under the earth is that which cannot be seen as long as it is under the earth.
Chapter 44. On the five circles of the heavens.
1. There are five zones in the heavens, according to the differences of which certain parts of the earth are inhabitable, because of their moderate temperature, and certain parts are uninhabitable because of extremes of heat and cold. And these are called zones or circles for the reason that they exist on the circumference of the sphere.
2. The first of these circles is called the Arctic, because the constellations of the Arcti are visible enclosed within it; the second is called the summer tropic, because in this circle the sun makes summer in northern regions, and does not pass beyond it but immediately returns, and from this it is called tropic.
3. The third circle is called ἰσημερινὸς, which is equivalent toequinoctialisin Latin, for the reason that when the sun comes to this circle it makes equal day and night (for ἰσημερινὸς means in Latin day equal to the night) and by this circle the sphere is seen to be equally divided. The fourth circle is called Antarctic,[271]for the reason that it is opposite to the circle which we call Arctic.
4. The fifth circle is called the winter tropic (χειμερινὸς τροπικός), which in the Latin ishiemalisorbrumalis, because when the sun comes to this circle it makes winter for those who are in the north and summer for those who dwell in the parts of the south.
Chapter 47. On the size of the sun.
1. The size of the sun is greater than that of the earth and so from the moment when it rises it appears equally to east and west at the same time.[272]And as to its appearing to us about a cubit in width, it is necessary to reflect how far the sun is from the earth, which distance causes it to seem small to us.
Chapter 48. On the size of the moon.
1. The size of the moon also is said to be less than that of the sun. For since the sun is higher than the moon and still appears to us larger than the moon, if it should approach near to us it would be plainly seen to be much larger than the moon. Just as the sun is larger than the earth, so the earth is in some degree larger than the moon.
Chapter 49. On the nature of the sun.
1. The sun, being made of fire, heats to a whiter glow because of the excessive speed of its circular motion. And its fire, philosophers declare, is fed with water, and it receives the virtue of light and heat from an element opposed to it. Whence we see that it is often wet and dewy.
Chapter 50. On the motion of the sun.
1. They say that the sun has a motion of its own and does not turn with the universe. For if it remained fixed in the heavens all days and nights would be equal, but since we see that it will set to-morrow in a different place from where it set yesterday, it is plain that it has a motion of its own and does not move with the universe. For it accomplishes its yearly orbits by varying courses, on account of the changes of the seasons.
2. For going further to the south it makes winter, in order that the land may be enriched by winter rains and frosts. Approaching the north it restores the summer, in order that fruits may mature, and what is green in the damp weather may ripen in the heat.
Chapter 51. What the sun does.
1. The rising sun brings the day, the setting sun the night; for day is the sun above the earth, night is the sun beneath the earth. From the sun come the hours; from the sun, when it rises, the day; from the sun, too, when it sets, the night; from the sun the months and years are numbered; from the sun come the changes of the seasons.
2. When it runs through the south it is nearer the earth; when it passes toward the north it is raised aloft. God has appointed for it different courses, places, and times for this reason, lest if it always remained in the same place all things should be consumed by its daily heat—just as Clement says: “It takes on different motions, by which the temperature of the air is moderated with a view to the seasons, and a regular order is observed in its seasonal changes and permutations. For when it ascends to the higher parts it tempers the spring, and when it comes to the summit of heaven it kindles the summer heats; descending again, it gives autumn its temperature. And when it returns to the lower circle it leaves to us the rigor of winter cold from the icy quarter of the heavens.”
Chapter 52. On the journey of the sun.
1. The eastern sun holds its way through the south, and after it comes to the west and has bathed itself in ocean, it passes by unknown ways beneath the earth, and again returns to the east.
Chapter 53. On the light of the moon.
1. Certain philosophers hold that the moon has a light of its own, that one part of its globe is bright and another dark, and that turning by degrees it assumes different shapes. Others, on the contrary, assert that the moon has no light of its own, but is illumined by the rays of the sun. And therefore it suffers an eclipse if the shadow of the earth is interposed between itself and the sun.
Chapter 56. On the motion of the moon.
1. The moon governs the times by alternately losing and recovering its light. It advances like the sun in an oblique,and not a vertical course, for this reason, that it may not be opposite the center of the earth and often suffer eclipse. For its orbit is near the earth. The waxing moon has its horns looking east; the waning, west; rightly, because it is going to set and lose its light.
Chapter 57. On the nearness of the moon to the earth.
1. The moon is nearer the earth than is the sun. Therefore having a narrow orbit it finishes its course more quickly. For it traverses in thirty days the journey the sun accomplishes in three hundred and sixty-five. Whence the ancients made the months depend on the moon, the years on the course of the sun.
Chapter 58. On the eclipse of the sun.
1. There is an eclipse of the sun as often as the thirtieth moon reaches the same line where the sun is passing, and, interposing itself, darkens the sun. For we see that the sun is eclipsed when the moon’s orb comes opposite to it.
Chapter 59. On the eclipse of the moon.
1. There is an eclipse of the moon as often as the moon runs into the shadow of the earth. For it is thought to have no light of its own but to be illumined by the sun, whence it suffers eclipse if the shadow of the earth comes between it and the sun. The fifteenth moon suffers this until it passes out from the center and shadow of the interposing earth and sees the sun and is seen by the sun.
Chapter 60. On the distinction betweenstella,sidus, andastrum.
1.Stellae,sidera, andastradiffer from one another. Forstellais any separate star.Sideraare made of very many stars, as Hyades, Pleiades.Astraare large stars as Orion, Bootes. But the writers confuse these names, puttingastraforstellaandstellaforsidera.[273]
Chapter 61. On the light of the stars.
1. Stars are said to have no light of their own, but to be lighted by the sun like the moon.
Chapter 62. On the position of the stars.
1. Stars are motionless, and being fixed are carried along by the heavens in perpetual course, and they do not set by day but are obscured by the brilliance of the sun.
Chapter 63. On the courses of the stars.
1. Stars either are borne along or have motion. Those are borne along which are fixed in the heavens and revolve with the heavens. Certain have motion, like the planets, that is, the wandering stars, which go through roaming courses, but with definite limitations.
Chapter 64. On the varying courses of the stars.
1. According as stars are carried on different orbits of the heavenly planets, certain ones rise earlier and set later, and certain rising later come to their setting earlier. Others rise together and do not set at the same time. But all in their own time revolve in a course of their own.
Chapter 65. On the distances of the stars.
1. Stars are at different distances from the earth and therefore, being of unequal brightness, they are more or less plain to the sight; many are larger than the bright ones which we see, but being further away they appear small to us.
Chapter 66. On the circular number of the stars.
1. There is a circular number of the stars by which it is said to be known in what time each and every star finishes its orbit, whether in longitude or latitude.[274]
2. For the moon is said to complete its orbit in eight years, Mercury in twenty, Lucifer in nine, the sun in nineteen, Pyrois in fifteen, Phaeton in twelve, Saturn in thirty. Whenthese are finished, they return to a repetition of their orbits through the same constellations and regions.
3. Certain stars being hindered by the rays of the sun become irregular, either retrograde or stationary, as the poet relates, saying: