DE COMPUTO

647Augustine said concerning the four divisions of scripture: Four things are necessary in the Church of God: the divine Canon, in which the life to come is recounted and preached; History, in which the deeds of things done are recounted; Number, by which the facts of things to come and the divine solemnities are numbered out; Grammar, in which the knowledge of words is understood. Therefore there are four parts of Scripture: the divine Canon, History, Number, Grammar. And these divisions are, as it were, the four foundations of Scripture. Isidore says in praise of Reckoning: The reckoning of numbers is not to be despised. In many places of the Holy Scriptures it shines out how great a mystery it holds. For not without cause was it said in the praises of God: Thou hast made all things in measure and number and weight. For the number six, which is perfect in its parts, 648declares the perfection of the world by a certain signification of its own number. Likewise also the forty days, in which Moses and Elijah and the Lord himself fasted, are not understood without the knowledge of numbers. So also other numbers occur in the sacred Scriptures, whose figures none but those acquainted with the knowledge of this art can unravel. It is granted to us also, in some part, to stand under the discipline of numbers, when we discuss the courses of the months, when we recognize by number the span of the returning year. Indeed, it is by number that we are instructed, lest we be confounded. Take number away from all things, and all things perish. Remove reckoning from the world, and blind ignorance embraces all things. Nor can those who do not know the reckoning of calculation be distinguished from the other animals.

649ADISCIPLE: We desire to know, then, whence this reckoning of numbers first proceeded.

MASTER: From God, clearly, since all wisdom and knowledge is from the Lord God, from whom all things were made.

D. Tell me then, when was this reckoning first discovered?

M. From that time from which the creatures were made, that is, from the origin of the world. For then number first began; as we read in Genesis: And there was ves ere and morning, one day. Then, therefore, he spoke of night and day, when he said evening, that is, night; and morning, that is, day. But of number he spoke when he said, one day, and so a second, so a third, so a fourth, so a fifth, so a sixth, so a seventh. Again God spoke of number, when he said of the sun and moon: And let them be for signs and seasons and days and years. For who can understand days and seasons and years, except through number? Whence Boethius said: All things whatsoever that have been constructed from the first nature of things appear to be formed by the reckoning of numbers. For this was the principal exemplar in the mind of the Creator. For hence the multitude of the four 649Belements was borrowed; hence the changes of the seasons; hence the motion of the stars and the revolution of the heaven are understood. Likewise Boethius says: Properly by the very nature of numbers has every course of the stars and every astronomical reckoning been established. For thus we gather risings and settings; thus we keep track of the slowness and the swiftness of the wandering stars; thus we recognize the eclipses and the manifold variations of the moon.

D. This art, then, that is, number — what general name does it have?

M. Philosophy, clearly, since all wisdom is named philosophy.

D. What is philosophy? Of what tongue is it?

M. It is a Greek name, and is interpreted as love of knowledge. For Φίλος among the Greeks is called love in Latin. And Σοφία is interpreted as wisdom or knowledge.

D. How is philosophy defined?

M. Isidore defined it thus, saying: Philosophy is the knowledge of things human and divine. In this 649Cinstance it is shown that all wisdom, whether divine or human, is termed philosophy. Likewise another definition of philosophy: Philosophy is the probable knowledge of things divine and human, in so far as is possible. Likewise another definition of philosophy: Philosophy is the art of arts, and the discipline of disciplines.

D. How many divisions of philosophy are there?

M. Three, that is, physics, ethics, logic; that is, natural, moral, rational. Whence Isidore says: The kind of philosophy is threefold. One is natural, which in Greek is φυσικὴ, that is, physics, in which the investigation of nature is discussed; another is moral, which among the Greeks is called ἠθικὴ, that is, ethics, in which there is treatment of morals; and the third is rational, which by a Greek word is called λογικὴ, that is, logic, in which, by disputing, it is sought how truth itself may be found in the causes of things or in the morals of life. In physics, therefore, is engaged the cause of seeking; in ethics, the order of finding; in logic, the reckoning of understanding. Likewise, there are then three parts of philosophy, that is, three divisions of wisdom: Physics, 649DEthics, Logic. In these three kinds of philosophy also the divine utterances consist. For they are wont to discuss either nature, as in Genesis, and in Ecclesiastes, or in Proverbs, and in the Epistles of Paul the apostle, or in the Song of Songs, and in the Gospels, and in many other books; or morals; or the higher heavenly mysteries, that is, logic.

D. Who then discovered these three parts of philosophy?

M. Thales of Miletus, one of the seven wise men of the Greeks, discovered physics. Socrates discovered ethics. Plato the philosopher discovered logic.

D. To which part, then, of these three of philosophy does reckoning belong?

M. To physics, without doubt.

D. And how many divisions has physics?

M. Four, that is, arithmetic, geometry, music, astronomy.

D. What is arithmetic?

M. The art of numbering.

650AD. What is geometry?

M. The measuring of the earth.

D. What is music?

M. Modulation.

D. What is astronomy?

M. The law of the stars.

D. Of these four divisions of physics, that is, of natural knowledge, which is to be learned first?

M. Arithmetic beyond doubt, which holds, as it were, the portion of a beginning and of a mother toward the rest. For arithmetic is prior to all, since God, the founder of this worldly mass, had it first as the exemplar of his own reasoning: out of it he established all things, that is, through numbers he established and ordered all the elements.

D. Who first discovered number among the Hebrews and the Egyptians?

M. Abraham first discovered number among the Hebrews, then Moses; and Abraham handed on this knowledge of number to the Egyptians, and taught them: then 650BJosephus.

D. Who first had this knowledge of number among the Greeks and the Latins?

M. Pythagoras among the Greeks, Apuleius and Boethius among the Latins; whence Isidore says: They say that among the Greeks Pythagoras first wrote down the discipline of number, and that thereafter it was more fully set in order by Nicomachus; which among the Latins Apuleius and then Boethius translated.

D. How is number named among the Hebrews, and the Chaldeans, and the Syrians, and the Macedonians?

M. It is called Nonnan among the Hebrews, the Chaldeans, and the Syrians. Among the Macedonians, calculus. The calculator took its name from calculi, that is, small pebbles, which the ancients, holding in their hand, composed number with. Among other Greeks, cyclus, or rhythmus; among the Egyptians, latercus, whence laterculus is said as a diminutive, and rima, whence they are called rimarii.

D. That name which is called number — is it simple 650Cor compound?

M. Compound, namely from two corrupted words. For Nume comes from that noun which is nummus (coin), while rus is derived from that noun which is rivus (stream). For from the stream of coins, that is, from the multitude of the revenue which was paid to kings or emperors, number took its name. Whence Isidore says: Nummus gave its name to numerus, and from its frequent use bestowed the word.

D. Number — whence did it take its name?

M. From Numeria. Hence Augustine says: Number is named from a certain goddess Numeria, whose priests used to make offering with their faces turned backwards, and after the gifts had been offered went away backwards.

D. How is number defined?

M. Isidore defined it, saying: Number is a multitude constituted out of units. For we say that one is the seed of number, not a number. Likewise another says: One is not a number, but from it numbers grow. Yet Donatus also set down one as a number, saying: 650DThe number is singular, as, This master. Likewise Augustine says: Number is the gathering together of a single body and voice and signification. Likewise, number (numerus) is so called from numbering, as was said above, from the goddess Numeria, whom the ancients used to say was the goddess of number; or number is so called as though it were the stream of coins. For the ancients, not knowing how to count, used to number their times and their days by pebbles — white ones in prosperity, black ones in adversity. Whence Persius says: Number this day, Macrinus, with a better pebble.

D. How are the numbers named from one up to a thousand, and myriads?

M. They are named thus:--One, two, three, four, five, six, seven, eight, nine, ten;--eleven, twelve, thirteen, fourteen, fifteen, sixteen or six-and-ten, seventeen or ten-and-seven, eighteen, nineteen, twenty;--twenty-one, twenty-two, twenty-three, twenty-four, twenty-651Afive, twenty-six, twenty-seven, twenty-eight, twenty-nine, thirty;--thirty-one, thirty-two, thirty-three, thirty-four, thirty-five, thirty-six, thirty-seven, thirty-eight, thirty-nine, forty;--forty-one, forty-two, forty-three, forty-four, forty-five, forty-six, forty-seven, forty-eight, forty-nine, fifty;--fifty-one, quiquaginta two, fifty-three, fifty-four, fifty-five, fifty-six, fifty-seven, fifty-eight, fifty-nine, sixty;--sixty-one, sixty-two, sixty-three, sixty-four, sixty-five, sixty-six, sixty-seven, sixty-eight, sixty-nine, seventy;--seventy-one, seventy-two, seventy-three, seventy-four, seventy-five, seventy-six, seventy-seven, seventy-eight, seventy-nine, eighty;--eighty-one, eighty-two, eighty-three, eighty-four, 651Beighty-five, eighty-six, eighty-seven, eighty-eight, eighty-nine, ninety;--ninety-one, ninety-two, etc.;--a hundred, a hundred and one, a hundred and two, etc.;--two hundred, three hundred, four hundred, five hundred, six hundred, seven hundred, eight hundred, nine hundred, a thousand;--two thousand, three thousand, four thousand, five thousand, six thousand, seven thousand, eight thousand, nine thousand, ten thousand.--Thus it is established that ten multiplied ten times makes a hundred. A hundred multiplied ten times makes a thousand. A thousand multiplied ten times makes a myriad. So twenty makes two myriads; and thirty, three myriads; and forty, four myriads; and fifty, five myriads; and sixty, six myriads; and seventy, seven myriads; and eighty, eight myriads; and ninety, nine myriads; and a hundred, ten myriads; twice a hundred, twenty myriads; three times a hundred, thirty myriads; four times a hundred, forty myriads; five times 651Ca hundred, fifty myriads; six times a hundred, sixty myriads; seven times a hundred, seventy myriads; eight times a hundred, eighty myriads; nine times a hundred, ninety myriads; ten times a hundred thousands, a hundred myriads.

D. These names of number which we have spoken — by what marks are they signified among the Latins?

M. By these seven marks: that is, I V X L D C M.

D. How do these marks signify and multiply numbers?

M. Either they signify simply, alone, or in composition.

D. Some with others, or multiplied of themselves?

M. They signify simply, as I is one. V alone, five. X alone, ten. L by itself, fifty. C by itself, signifies only a hundred. D alone, five hundred. M alone, or with a stroke above, signifies a thousand.

D. But how, when multiplied, do they generate number of themselves?

M. As I doubled thus, II, signifies two; tripled, 651DIII, three; quadrupled, IIII, four.

D. Why does I signify one?

M. What is more fitting than that that letter, which is the smallest of the letters in its character, should signify that number which is the smallest among numbers, that is, one? And no mark among the Latins is multiplied more than four times over. Then after I follows V, which signifies five?

D. Why does V signify five?

M. Because it is the fifth vowel in the order of the vowels; and as it is in the order of the vowels, so it is in number. Hence it is said of the fifth letter, marked with five strokes; of the letter X, ten. Nor is V doubled, but with a following V, an I is placed in the number. If one I, there become 652Asix; if two, VII, seven; if three, VIII, eight; if four, VIIII, nine: it grows no more. Then follows X, which signifies ten.

D. Who shows that X signifies ten?

M. Isidore says: The names of the letters among the Greeks both compose words and make numbers. But the Latins do not compose numbers to letters, except the letter X, which in its figure signifies a cross, and in number denotes ten. With X following, I is multiplied up to XIIII, which denotes fourteen. Then follows V after X and there becomes XV. With V following, an I multiplied four times, thus XVIIII, there becomes nineteen.

D. What mark signifies twenty?

M. The letter X doubled. For it has twice ten, thus XX; they are therefore called twenty, because begotten twice ten. And thus XXX, thirty, begotten thrice ten.

D. By what kind of mark is thirty signified?

M. By X multiplied thrice. And X is not multiplied more than three times over.

D. Forty — by what kind of mark is it signified?

652BM. By X and L.

D. Forty — why is it so called? M. That is, begotten four times ten. (For every numeral letter signifying less, if it precedes, takes away from the one following only so much as it has in itself; as I, II, III, IV for IIII, V, VI, VII, VIII, VIIII for IX, X, XX, XXX, XL for XXXX, as some will have it, L, LX, LXX, LXXX, LXL or XC for LXXXX, C, CC, CCC, CD, for CCCC, D for CCCCC, DC, DCC, DCCC, CM for DCCCC, M.)

D. What mark signifies fifty?

M. The letter L.

D. Why?

M. Because the chief number in the law of the Lord is the fiftieth. For under that number fell the Jubilee, that is, in the fiftieth year. And it was fitting that that number which holds the primacy in the old law, that is, the fiftieth, should be signified by that letter L, which is the first in that name which is lex (law). But sixty by L and X; seventy, 652Cby L and two X's; eighty, by L and three X's.

D. What mark signifies ninety?

M. Two marks signify ninety, either an X between two L's, as LXL, or an X before C, as XC.

D. What mark signifies a hundred?

M. The letter C.

D. Why?

M. Because it is the first in that name which is centum (a hundred). For it was often the practice among the ancients to signify names by the first letter.

D. Whence is it called centum?

M. From cantus (song), that is, a circle, or from a choir of singers.

For it was the custom among the ancients to have a hundred men in a choir of singers. A hundred is therefore called from cantus (song) or from a circle. And C doubled thus, CC, signifies two hundred; tripled, CCC, three hundred; quadrupled, CCCC, four hundred.

D. What mark signifies five hundred?

652DM. The letter D itself.

D. Why?

M. It was fitting that, the number of the prior letter, that is, C, being finished, that letter which follows in the order of the letters, namely D, should hold the following number, that is, five hundred. Then C following D, thus DC, six hundred. Two C's following D, thus DCC, seven hundred. C drawn thrice, D following, thus DCCC, becomes eight hundred. C multiplied four times, D following, thus DCCCC, signifies nine hundred. But a thousand is marked by the mark M (as we have already said), since it is the first letter in this word mille (a thousand). And a thousand proceeds from a multitude, and reaches up to a hundred hundred-thousands.