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A Partial Answer
to an Inevitable Question
Mr. Turner, lately of the University of Vermont, is now at Heidelberg College. He is a graduate of the University of Toronto (B.A., 1940) and the University of Cincinnati (Ph.D., 1944). Before going to Vermont he taught at Bishop's College School, Lennoxville, Quebec, and headed the Latin Department of the McCallie School in Chattanooga. His special interest, of which this article is an illustration, is life among the Romans.
"How did the Romans work problems with their numerals? This question or one like it must be at least an annual experience in the life of most Latin teachers. Moreover, it is by no means a simple one to answer. First, information is not easy to find. In the notes attached to this article will be found only a small part of the relevant literature.1 Yet even of this handpicked fraction there is little that could be called readily accessible. Among the standard handbooks there are two groups which might be expected to contain the desired information, histories of mathematics and books on Roman private antiquities. But unfortunately, with one or two exceptions, the former are impressed chiefly by the Roman's lack of interest in theoretical mathematics, the latter are content with a few general statements in their accounts of Roman elementary education. Secondly, the source material, although it has rarely been adequately exploited, is still very far from satisfactory. As has been well pointed out before in this journal,2 it is just the simple everyday things about which it is hardest to find information. Thus in the field of mathematics, the important mathematicians concentrated on what seemed to them important topics; and so for knowledge of the fundamental operations — addition, subtraction, multiplication, division — we have to make use of late commentators and text book writers and of widelyscattered incidental references in classical literature.
Yet despite these difficulties the question is one to which an answer ought to be available, and not merely because it is one that troublesome schoolboys insist on asking. Practical calculation was a part of the Roman's life from his early youth; and as a result some p64knowledge of his methods will contribute not alone to our understanding of his daily life, but also to our appreciation of his literature.3
This calculation of the Roman with, or perhaps despite, his numerals might as a matter of fact proceed in more than one way. In the first place, although inferior in this respect to our so‑called Arabic numerals, the Roman's system of notation was somewhat more adaptable to use in written calculation than is generally assumed. He would, however, place greater reliance on two aids to reckoning, his hands, and the abacus or counting board.a It is in a consideration of these three possible methods that the answer to our question is to be found. In the discussion which follows, in spite of the fact that among the Romans the ability to handle duodecimal fractions was necessary for those with even an elementary training in mathematics, problems involving fractions have been omitted from consideration. This has been done in the interests of simplicity, and the reader should be quite content to have it so.4
Before we proceed to the examination of these alternative methods, it may be wise to point out that they do not in themselves tell the whole story, that all but the most primitive or completely mechanized forms of calculation depend only secondarily on such aids as these, primarily on tables of numerical relations which have been committed to memory and made almost second nature by constant drill. Such drill in practical mathematics was, according to Horace, especially characteristic of the Roman system of education. The poet favors us with a brief glimpse into a schoolroom where the subject of the drill, as it happens, not whole numbers, but the duodecimal fractions mentioned just above:
Let Albinus' son tell us: "If onetwelfth (uncia) is taken from fivetwelfths (quincunx), what is left?" You could have said, "Onethird (triens)." "Good, you will be able to preserve your estate; add a twelfth; what have you?" "Onehalf (semis)."5 
But for a more complete picture of Roman mathematical tables it is necessary to turn to the fifth century and a treatise by a Christian writer, Victorius of Aquitania (fl. 457 A.D.), whose Calculus contains very extensive and detailed tables which demonstrate not merely multiplication, addition, subtraction, but also the squares of whole numbers and mixed numbers and the relations of the duodecimal fractions and their subdivisions.6 This set of tables was presumably for consultation and therefore vastly more complicated than any but the most skilled calculators would be expected to master. Even so there is no reason that we should not assume for the educated Roman a knowledge, on more modest scale, of similar tables to assist him, whether his calculation was written or with the aid of his fingers or that of the abacus.
Although, as remarked above, calculation using the Roman numerals themselves is more practical than at first appears, their advantages are chiefly for recording: they are convenient; for large sums they are compact; they are rarely illegible; unlike the Latin sentence, they require less suspension of judgment on the part of the reader than does the current system. The simplicity of form of the individual symbols, incidentally, has insured their popularity through the centuries for use on monuments.
But for use in computation, they are inferior to the Arabic numerals. The great superiority of the latter is due to the fact that thanks to the use of the zero to indicate a blank space, each numeral obtains a part of its value from its position. Thus the value of the symbol 9 is 9 or 90 or 900 depending on whether or not it is followed by one or more spaces indicated by zeros or other numerals. This is admittedly an elementary observation, but like many other fundamental principles is so deeply embedded in our consciousness that for many of us its application has become automatic. This particular advantage the Arabic numerals share with the abacus; which is in fact sometimes used in schools in this country as a visual aid in the teaching of p65arithmetic to illustrate the principle of value by position.7
Even so, three at least of the operations could be performed without great difficulty, particularly if a little care were employed in grouping the numerals. Addition and subtraction would in fact be slightly easier than with Arabic numerals if corresponding symbols were placed together. In subtracting, for example, 126 from 378 the elimination of the symbols occurring in both numbers provides the answer.
CCC  LXX  VIII 
C  XX  VI 
CC  L  II 
With Arabic numerals, on the other hand, knowledge of subtraction tables would be necessary for performing the operation. There is no reason again why a multiplication problem could not be carried out with Roman numerals in essentially the same way as with the symbols now in use. The reader can easily verify this either by doing a problem himself or by referring to the example given in Smith's History of Mathematics.8 He should remember in connection with this that, as is pointed out in Heath's History of Greek Mathematics, it is not the numerals we multiply, but the numbers they represent.9
The first of the alternatives to written calculation mentioned above is mental arithmetic aided by flexing of the fingers. Digital notation was especially prevalent in the Middle Ages, and it is from this period that our most complete descriptions come, but there is enough evidence, dating back as far as we may reasonably expect, to indicate a continuous tradition and to show that all, or almost all, Romans used and understood it.10 Plautus refers to one of his characters as doing sums with the fingers of his right hand, while the elder Pliny tells that a statue of Janus, sufficiently antique in his time that he ascribed its erection to King Numa, had its hands in the proper position to represent 365. Cicero, writing to Atticus, expected the latter to work out with the help of his fingers, as he read, a problem involving the difference between 12% compounded annually and 12% simple interest. Quintilian regarded the knowledge of the correct positions as essential for all persons with claims to literacy.11 And, in addition, a considerable number of other ancient authors, both pagan and Christian, show a personal acquaintance with finger counting and, what is even more significant, assume the same of their readers.12
For the precise nature of this method of expressing numbers we have to depend primarily, though not entirely, on evidence from the early Middle Ages. Its apparent universality is possibly the reason why the earliest written instructions which have come down to us, the Romana Computatio, are as late as the seventh century of our era.13 The Romana Computatio is a short pamphlet whose chief importance is that it served as the source for a better known, and considerably more literate, description by the Venerable Bede (673‑735 A.D.) which forms part of the first chapter of his De Temporum Ratione (725 A.D.). Both the complete work and the single chapter published separately, frequently accompanied by picturesque illustrative diagrams, were very widely distribute bound in the early Middle Ages, as is shown by the large number of surviving manuscripts.14 For most numbers these versions agree with a second mediaeval tradition represented by a late Greek writer on elementary mathematics, Nicolaus Rhabdas of Smyrna (fl. 1340 A.D.),15 and by this fact the probability is increased that they both are close to methods used in ancient times.
As stated above, the numerical values were expressed by positions of the fingers. The digits 1‑9 were represented by the use of the last three fingers of the left hand, 10 to 90 with the thumb and index finger of the same hand; for 100 to 900 the thumb and index of the right hand were used, in positions corresponding to those for the tens, and for 1000 to 9000 the remaining fingers of the same hand. By combining these gestures the two hands could represent any number up to 9999.
The evidence seems to be against the assumption that the ancient Greeks and Romans had a system of representing higher numbers by placing the hands with palm or back exposed against various parts of the body as described by Bede. There seems to be no ancient evidence of this and two of the manuscripts of Romana Computatio offer quite different instructions for 10,000 and above, although up to this point all agree with one another and with Bede. Furthermore Nicolaus offers no instructions beyond 9999. The ancient Persians are reported to have represented 10,000 in the same way as unity.17
What follows is a version of the pertinent passage from Bede with remarks containing information from earlier sources. These should be read in conjunction with the plate and with appropriate gestures on the part of the reader. I treat the subject in rather disproportionate detail because this is the first time so far as I know that all the evidence mentioned below has been used in conjunction, and because I believe that this knowledge could prove very useful to many of the readers of this journal.
"When therefore you say 1, bending the smallest finger on the left hand, you will place it on the middle joint of the palm." This should be compared with the instructions for 7. The representation of these numbers, as of 2 compared with 8, 3 with 9, is not consistent in illustrations found in the manuscripts or on the tesserae. The general distinction, however, seems to be that the fingers were bent over less far for the small numbers, to their own base or over to the place where the palm folds, farther for the higher numbers, towards the hollow of the palm (Nicolaus) or beyond towards its base. "When you say 2, you will bend the finger next the smallest (secundum a minimo) and place it in the same spot. "When you say 3 you will bend down the third (i.e. middle finger) likewise. "When you say 4, then you will raise the smallest finger. "When you say 5, you will straighten the finger next the smallest similarly. "When you say 6, you will raise the third likewise, keeping only (dumtaxat) the middle finger, which is called medicus (i.e. the ring finger), fixed in the middle of the palm." This is corroborated by Macrobius when he discusses the reason current in Egypt why this finger, the digitus medicus or medicalis, is used as the ring finger. After asserting its direct connection with the heart he cites the fact that when bent it expresses the number 6 qui est omnifariam plenus, perfectus atque divinus.18 "When you say 7, you will place the smallest only above the root of the palm, keeping the rest raised as you do so (interim)." See the comment on the instructions for representing 1. "Next to this, when you say 8, put the medicus. "When you say 9, you will lower the impudicus (i.e. the middle finger) from its position. "When you say 10, you will put the nail of the index on the middle joint of the thumb." This does not correspond exactly with earlier evidence. I discuss this at the end of the translation. "When you say 20, you will place the tip of the thumb between the middle joints of p68the index and impudicus." That is to say, the thumb is placed against the palm of the hand in such a way that its tip comes between the index and the finger next to it. According to Nicolaus the four fingers are straight, but bent slightly at the base towards the palm. This would make it possible for the tip of the thumb to reach the middle joints of the two adjoining fingers. "When you say 30, you will join the nails of the index and thumb in a gentle embrace." St. Jerome, in a comment on the parable of the Sower which was quoted by Bede in the introduction to this passage, speaks of 30 as the number related to marriages because of the union of the fingers quasi molli se complexans osculo.19a As in the case of 10, there is conflicting earlier evidence which I discuss at the conclusion of the translation. "When you say 40, you will place the inner part of the thumb close to the side or back of the index finger, keeping both erect." Apuleius remarks that 40 is easier than the other numbers because it is indicated with the palm extended (palma porrecta).20 "When you say 50, you will hold the thumb against the palm with its outer joint bent like the Greek letter gamma." That is, the thumb, bent at right angles like capital gamma (Γ), will be placed against the fleshy "hill" at the base of the index. According to Quintilian previous writers on oratory warned against the inadvertent use of the gesture for 500, made flexo pollice, as being inelegant.21 This gesture differs from that for 50 only in that it is made with the right hand rather than the left. "When you say 60, with the thumb curved as above, you will encircle it carefully by bending the index finger forward." Jerome, continuing the number symbolism mentioned above in connection with the number 30, says that 60 is used in reference to widows quod in angusta et tribulatione sint positae — in the same way that the thumb is pressed down by the index finger.19b "When you say 70, with the index bent around as above, you will cover the cavity (superimplebis) by applying the thumb; the thumbnail will be upright across the middle joint of the index." Thus the thumb will not be under the index as for 60, but bent (artu curvato, Romana Computatio) and placed against the joint of the index at the side. "When you say 80, with the index bent around as above, you will fill the space under with the outstretched thumb; its nail will be fixed in the middle joint of the index." Thus the thumb is held up straight and the index wrapped around its end.22 "When you say 90, you will fix the nail of the bent index against the base of the erect thumb. "Up to this point (the numbers are represented) on the left hand, but (you will represent) 100 on the right, as 10 on the left." This shift to the right hand for 100 is well attested. Probably the earliest reference is a well known passage where Juvenal comments on the supposed good fortune of Nestor in being able at last to count his years with his right hand.23 "(You will indicate) 200 on the right as 20 on the left; "300 on the right as 30 on the left; "and the rest too in the same manner up to 900. "Likewise (you will represent) 1000 on the right as 1 on the left, "2000 on the right as 2 on the left, "3000 on the right as 3 on the left, and the rest up to 9000." 
At this point we shall leave Bede for the reasons stated at the beginning of this section and consider some earlier evidence which shows that despite the general continuity of tradition, a change had taken place in the way 10 and 30 were represented.
In a passage of his Apology which is usually described as obscure, Apuleius defends himself against the charge of marrying a woman aged 60. After proving that his wife, Pudentilla, is actually little more than 40 he goes on to comment on the improbability of a genuine error involving 40 and 60. Such similarly formed numbers as 30 and 10 would admit of such an error, but not 40 and 60 since 40 is so simple to form. "If you had said 30 years instead of 10, you might appear to have erred in the gestures involved in p69your computation — to have opened the fingers which you should have circled."24 The assumption that this must be made to conform to the instructions of Bede has led to unnecessary emendation. Although Bede tells us that 30 is formed by placing the tip of the index against that of the thumb, 10 by placing it against the thumb's joint, it is clear from the tesserae for 12, 13, and 15 that in the early centuries of our era the method of forming 10 was by means of a circle formed by joining the tips of the index and thumb. Apuleius' statement agrees with this in making 10 a closed circle, 30, apparently, an open circle. Unless the kisses of the married in the time of St. Jerome were notably lacking in intimacy the circle for 30 had closed by his date, and the tips of the fingers had gently come together.19c Possibly Jerome made a less clear distinction between the two numbers than did Bede, since he speaks of 100, the right hand gesture corresponding to 10, as symbolizing the corona virginitatis.
These differences are, however, minor, no more than should be expected, since over half a millennium separates Apuleius from Bede. The principles of Bede's system clearly date back at least to the first centuries of the Christian era, and we can be confident that in general his system goes back to quite early times.
These gestures were widely used for simple computation, as, for example, in a monument from Aesernia of L. Calidius Eroticus and Fannia Voluptas on which is a crude illustration of a traveller and an innkeeper (caupo) with hands extended reckoning the former's bill, accompanied by appropriate text which (with the omission of the epitaph proper) reads:
Copo, computemus. Habes vini ⊃ (sextarium) I, pane assem I, pulmentarium [i.e. food other than bread] asses II. Convenit. Puellam asses VIII. Et hoc convenit. Faenum mulo asses II. Iste mulus me ad factum dabit [That mule will be the ruin of me].25 
In a situation like this the fingers would register the successive sums in a series of additions, both helping in the calculation and preventing misunderstanding. Computare seems to have been the usual word for such reckoning as calculos ponere or some similar expression for the use of the abacus.26
More complex problems also were worked with the help of the fingers by skilled reckoners like Atticus or Regulus, the villain of so many of the younger Pliny's letters, problems of finance as in the instance mentioned earlier,27 or of astrology as in the celebrated incident where Regulus gained himself a legacy by working out a sick woman's horoscope and assuring her in the name of the stars that she would recover. The will changed, it was his good fortune to be speedily proved in error.28 The great advantages of finger reckoning were that the hands were always available whereas writing equipment or an abacus might not be, that all interested parties could observe and follow the stages of the computation. For the Middle Ages a third advantage could be noted: that at the mediaeval fairs all the traders, whatever their native tongue, understood the gestures.
The other method of working mathematical problems available to the ancient Roman would be the use of the counting board or abacus.29 That this was the standard instrument for any type of complex calculation is, I believe, a safe assumption, but an assumption based on surprisingly little direct evidence. The very word "abacus" which I have been using does not seem to have been found in this sense in classical Latin literature.30 The most common terms used are rather calculi (pebbles) for counters, tabula for the table; calculos ponere for undertaking a problem. The widespread metaphorical use of calculus31 is ample evidence of the universal and early use of the device, but precise evidence on its nature or on the way it was used is remarkably scarce.
The terms just mentioned seem to refer to an abacus which resembled mediaeval counting boards in that it consisted of a flat surface p70probably marked with lines on which calculi were placed in appropriate positions and in appropriate number to indicate the sum desired. These lines would be vertical in relation to the user and, if fractions are omitted, would have a value from right to left of 1, 10, 100 and so on as high as the maker desired. These values would be those which single counters would possess when placed on the lines. The lines might each be divided into a lower and an upper portion. In this case the value of a calculus on the upper portion would be five times that of one on the lower. A schematic rendering of such an abacus bearing the number 94826 would be as follows.
o  o  o  
(((I)))  ((I))  (I)  C  X  I 
o  o  o  o  o  
o  o  o  o  
o  o  o  
o  o 
As it happens no such abacus from Roman times is known to have survived, and if any counters exist, they have not been identified.32 There is a well preserved Greek abacus from Salamis •nearly 5½ by 2½ feet in size with spaces for representing several separate numbers.33a
The type of Roman abacus which has survived is a small portable instrument with movable beads which could be slid up and down slots. This type, which has some resemblance to the Chinese suanpan or Japanese soroban, is nowhere mentioned in Latin literature. The two surviving examples, in Rome and Paris, are both •about 5 by 3½ inches.33b It is from these that the illustrations in encyclopedias and handbooks are derived. It may be assumed that they are a late development. Unless they are much more convenient than the author's homemade copy, the surviving models are too small to be of much practical use.
The fact that we have little or no direct evidence on the use of the abacus before the mediaeval period is no reason for despairing regarding our ability to describe the operations. The nature of the instrument limits the possibility of any great variety in the performance of the simpler operations, addition and subtraction. Multiplication and division are more complex if not performed by the cumbersome means of multiple addition or subtraction, and there has been enough variety among other known methods to make dogmatism undesirable. In the case of multiplication, however, we have actual examples of how the Greeks worked specific problems with their alphabetical numerals. For reasons I state below, I have assumed their procedure to be the ancient method for abacus as well, and use as illustration an actual problem of Archimedes as worked out by a commentator.
Before I proceed, I should like to remind those who approach this subject as amateurs that the abacus in the hands of an expert is an instrument of remarkable efficiency. This point is well illustrated by a relatively recent incident. In the November 25, 1946 issue of Time there was reported the result of a public contest between a skilled Japanese abacus operator using a soroban, an instrument possibly descended from the Roman abacus, and an American private using an electrically operated calculating machine. Of the four divisions to the contest, the abacus won the addition, subtraction and division, losing only the multiplication.
I give below examples of addition, multiplication, division illustrating each step. Readers will follow these most satisfactorily if they will make themselves a simple line abacus by means of a few coins and a sheet of paper marked with units, tens, hundreds, thousands, and after performing the sample operations attempt similar problems of their own manufacture.
The problem:
4739 
1456 
6195 
p71 At the beginning of the operation the abacus will bear 4739 and have this appearance: 1.
o  o  
(I)  C  X  I 
o  o  o  o 
o  o  o  o 
o  o  o  
o  o 
The operation will proceed from right to left, from digits to fives and fives to tens. The stages of this operation would have this appearance.
2. Add 6 to the units column.
o  o  
(I)  C  X  I 
o  o  o  
o  o  o  
o  o  
o  o 
3. Add 5 to the tens column.
o  o  o  
(I)  C  X  I 
o  o  o  
o  o  o  
o  o  
o  o 
4. Add 4 to the hundreds column.
o  o  o  
(I)  C  X  I 
o  o  
o  
o  
o 
5. Add 1 to the thousands column.
o  o  o  
(I)  C  X  I 
o  o  o  
o  
o  
o 
The result reads 6195.
Subtraction would be the same process in reverse.
The opinion that in Roman times multiplication was merely a process of multiple addition may be dismissed. The word multiplicare (fold many times) indicates that originally it was such a process; later perhaps shortcuts were introduced, for example, doubling combined with addition until the desired result was reached; but by the time of Archimedes (287‑212 B.C.) and of Hero (date uncertain) the Greeks had evolved a method of written multiplication not unlike ours, except that the operation began with the highest power of 10 instead of with the digits; that is, it proceeded left to right instead of right to left. This, incidentally, has been standard practice in abacus reckoning from the time of our earliest recorded instructions. And as will be clear from the example below, the Greek method of multiplication can be adapted so easily to the abacus that it may well have originated in calculation on that instrument. It is rather difficult to believe that the Romans used a more primitive method than was readily available to them.
The primary difficulty in multiplication which starts with the highest power is to find the highest power contained in the product, or, in terms of abacus reckoning, the column in which to place the first counter of the product. There is a rather convenient rule for this, which, as expressed by Alfred Nagl,34 reads a+b‑1=c where a is the number of columns occupied by the multiplicand, b the number occupied by the p72multiplier and c the number occupied by the product. This rule was followed by the mediaeval abacists and was known to Archimedes.35 Its practical application is as follows: when 265 is multiplied by itself the number of spaces occupied by the product is 3+31=5. Therefore the first counter is placed in the 10,000 column, the fifth from the right.
This problem, the multiplication of 265 by itself, has been selected because it is one of the problems found worked out in full in Eutocius' commentary on Archimedes.36 Hero's calculations37 involved the use of fractions and are too complex for our purposes.38a Eutocius' calculation, transcribed into the Arabic system, has this appearance.
265  
265  
40000  12000  1000 
12000  3600  300 
1000  300  25 
70225 
The fact that the operation begins with 200 times 200 rather than 5 times 5 is what makes it necessary to write out nine rather than three steps. If performed on the abacus the operation would be simpler since the process of addition would be continuous, not reserved until the end.
1. 40,000 (200 × 200)
((I))  (I)  C  X  I 
o  
o  
o  
o 
2. 40,000 + 12,000 (200 × 60)
o  
((I))  (I)  C  X  I 
o  
o 
3. 52,000 + 1,000 (200 × 5)
o  
((I))  (I)  C  X  I 
o  
o  
o 
4. 53,000 + 12,000 (60 × 200)
o  o  
((I))  (I)  C  X  I 
o  
5. 65,000 + 3,600 (60 × 60)
o  o  o  
((I))  (I)  C  X  I 
o  o  o  
o  
o  
6. 68,600 + 300 (60 × 5)
o  o  o  
((I))  (I)  C  X  I 
o  o  o  
o  o  
o  o  
o 
7. 68,900 + 1,000 (5 × 200)
o  o  o  
((I))  (I)  C  X  I 
o  o  o  
o  o  
o  o  
o  o 
8. 69,900 + 300 (5 × 60)
o  
((I))  (I)  C  X  I 
o  o  
o  o 
9. 70,200 + 25 (5 × 5)
o  o  
((I))  (I)  C  X  I 
o  o  o  
o  o  o 
p73 The multiplier and multiplicand could be carried in the head or recorded elsewhere, either on paper or wax or on a section of the abacus marked for that purpose, or the multiplier could be placed on the board and the pebbles for 200, 60 and 5 could be removed as their turn to multiply came.
For division it is necessary to resort more to conjecture, since there is no surviving Greek example of simple division, only one involving degrees, minutes and seconds.39 The following is an example of what is probably the simplest form of division on the abacus. It cannot, of course, be maintained that the Romans divided in exactly this manner, but it shows one way in which the operation could have been performed.38b Once more the sample operation as performed with Arabic numerals is the same as that on the abacus. The problem is 35 divided into 808. The operation begins with the dividend on the abacus and proceeds by a process of subtraction, the first number subtracted is 350 rather than 35, and the quotient is established as being at least 10. This fact could be remembered, recorded elsewhere — on the abacus or on the fingers of the left hand.
1.  808 
35  
2.  458 
35  
3.  108 
35  
4.  73 
35  
5.  38 
35  
6.  3 
1. The dividend (808)
o  o  
C  X  I  
o  o  
o  o  
o  o 
2. quotient is at least 10
o  o  
C  X  I  
o  o  
o  o  
o  o  
o 
3. quotient is 20
o  
C  X  I  
o  o  
o  
o 
4. quotient is 21
o  
C  X  I  
o  o  
o  o  
o 
5. quotient is 22
o  
C  X  I  
o  o  
o  o  
o  o 
6. quotient is 23, remainder left on abacus
C  X  I  
o  
o  
o 
A person with even a small amount of experience p74could shorten this process considerably by guessing the highest multiple of 35 which is smaller than 80 and the highest which is smaller than 108, as is required in our so‑called long division.
At the beginning of this article I suggested that part of the importance of this subject was that elementary calculation as part of the Roman's everyday life had become part of his literature. This statement should not be left unsupported, especially since not all places where a knowledge of fingerreckoning or the abacus brings illumination to a Latin text are as easily recognized as the one mentioned or discussed above. I shall cite three examples from Catullus; Martial, Symphosius and Alcuin, the full significance of which remained long unrecognized.
The appeal that begins with these words has long been famous as one of the great love poems of all times.40 But, despite the energy which has been invested so unstintingly in the elucidation of Catullus' verse, it was as recently as 1941 that the form taken by the arithmetic was explained in print, when Professor Levy of Hunter College pointed out that Catullus was thinking in terms of abacus reckoning.41 After urging Lesbia that they take advantage of the brief time of life allotted them and disregard the rumores senum severiorum, Catullus continues:
Da mi basia mille, deinde centum, Dein mille altera, dein secunda centum, Deinde usque altera mille, deinde centum. Dein, cum milia multa fecerimus, Conturbabimus illa, ne sciamus, Aut ne quis malus invidere possit, Cum tantum sciat esse basiorum. 
There should be no need of having to show in detail that Catullus is here setting imaginary counters on thousand and hundred columns of an imaginary abacus. In the arithmetic of love what elsewhere might be a handicap is a distinct advantage: a vigorous shake of the board and the counters are scattered, the record irretrievably lost, beyond the recall of those persons who wish the lovers ill.
This epigram is on the occasion of the 62nd birthday of one Cotta whose health has been so perfect that he can afford to direct towards his doctor acquaintances his middle finger, the impudicus.
Sexagesima, Marciane, messis Acta est et, puto, iam secunda Cottae, Nec se taedia lectuli calentis Expertum meminit die vel uno. Ostendit digitum sed inpudicum, Alconti Dasioque Symmachoque. 
The poem continues to the effect that only the years of good health should count towards the total: Non est vivere, sed valere vita est.
Commentaries have faithfully pointed out that the gentlemen mentioned in the last quoted verse are doctors and by means of cross references42 have given the reader the means of discovering the meaning of the insulting gesture in the preceding line; but apparently no one has thought of looking for anything in the early lines to give special appropriateness to the gesture described. Those who have completed the editing of more than five books of Martial may perhaps be forgiven for feeling that such a gratuitous insult is to be considered its own justification. I believe, however, that the poem is a trifle more clever than appears on the surface.
The reader may already have undertaken to reproduce the 62 of Cotta's age on his hands. If he has, he has bent his thumb and hooked his index finger over it to form 60 and bent the last two fingers of his hand to form 2, leaving one finger extended, the middle one known as the impudicus. Thus in passing his 62nd harvest Cotta, merely by showing his age to his medical acquaintances, has indicated his opinion of them and their tribe.b
In the 96th position among the riddles of Symphosius is a numerical riddle entitled p106De VIII tollas VII et remanet VI, believed by editors to be a late insertion. The text reads:
Nunc mihi iam credas, fieri quod posse negatur, Octo tenes manibus, sed me monstrante magistro Sublatis septem reliqui tibi sex remanebunt. 
The same riddle is to be found on page 172, line 30 of Beeson's Mediaeval Latin Primer, where Alcuin puts to his pupil Pippin the problem, "Vidi hominem octo in manu tenentem et de octonis rapuit septem, et remanserunt sex." Pippin answers succinctly, "Pueri in schola hoc sciunt." The answer to this has been presented independently by M. Froehner for Symphosius and by professor Eva Sanford for Alcuin.43 It is, as is not unusual in such cases, simple enough when explained. If a person with his two last fingers lowered for 8 raises the little finger, the one which by itself indicates 7, he now has his hand in the proper position for 6.
1 In addition to the works on special topics mentioned throughout the notes, the following will be found worth consulting:
Smith, D. E., History of Mathematics (Boston, 1925), Vol. 2, Special Topics of Elementary Mathematics, definitely the best available introduction to the history of elementary mathematics; it is well illustrated and to be recommended for browsing.
Friedlein, Gottfried, Die Zahlzeichen und das elementare rechnen der Griechen und Römer und des Christlichen Abendlandes vom 7. bis 13. Jahrhundert (Erlangen, 1869), still the most comprehensive work on the subject.
Smith, D. E., Mathematics (Our Debt to Greece and Rome, no. 36) (Boston, 1923), well worth reading for its many interesting bits of information.
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2 Edwards, Walter A., "What we don't know about the Romans," The Classical Journal, 41 (1947‑48), 325‑329.
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3 See passages mentioned and discussed throughout this article, especially at the conclusion.
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4 A problem from Frontinus, De Aquaeductibus, 26, will serve to illustrate the complexities sometimes encountered in the use of duodecimal fractions. For the calculations which follow I am indebted to Friedlein, op. cit., 91. Frontinus makes the statement that the unciae modulus (diameter 1⅓ digiti) has a capacity of 1+⅛ + 3/288 + (⅔ · 1/288) quinaria. One quinarium is the capacity of the quinarius modulus (diameter 1¼ digiti). The procedure which must be followed to reach the above conclusion is (a) to find the squares of 1⅓ and 1¼ and (b) to divide the square of the second into that of the first.
(a) Squares:
1⅓ × 1⅓ = 1 + ⅓º + ⅓ + 1/9 = 16/9 = 256/144
1¼ × 1¼ = 1 + ¼º + ¼ + 1/16 = 25/16 = 225/144
(b) Division:
256/225
= 1 31/225
= 1 248/(225 × 8)
= 1 (225 + 23)/(225 × 8)
= 1⅛ + 23/(225 × 8)
= 1⅛ + 828/(225 × 288)
= 1⅛ + (675 + 153)/(225 × 288)
= 1⅛ + 3/288 + 153/(225 × 288)
= 1⅛ + 3/288 + (⅔ · 1/288)
The reason ⅛ is selected is that 8 × 31 is the first product of 31 higher than 225. 1/288 is the scripulum, one of the named duodecimal fractions.
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5 Horace, Ars Poetica, 326‑330.
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6 Published by Friedlein, Gottfried, "Der Calculus des Victorius," Zeitschrift für Mathematik und Physik, 16 (1871), 42‑79. A brief description of the contents of the section illustrating multiplication will give an idea of the scope of the rest. These consist of 49 tables showing the results when the digits (1‑9), the tens (10‑90), the hundreds (100‑900) and 1000, as well as fractions going as low as 1/144 (dimidia sescle) are multiplied by each of the numbers from 2 to 50.
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7 Spitzer, F. H., "The abacus in the teaching of arithmetic," Elementary School Journal, 42 (1942), 448‑451.
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8 Vol. 2, 107.
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9 Heath, Thomas, A History of Greek Mathematics (Oxford, 1921), Vol. 1, 38.
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10 Some of the more accessible among the many articles written on this subject are: Bechtel, Edward A., "Finger counting among the Romans in the Fourth Century," CP 4 (1909), 25‑31; Richardson, Leon J., "Digital Reckoning among the Ancients," American Mathematical Monthly 23 (1915), 7‑13; Sanford, Eva Matthews, "De Loquela Digitorum," CJ 23 (1927‑28), 588‑593; Smith, D. E., History of Mathematics, Vol. 2, 196‑201; id., Article "Finger Notation," Encyclopedia Americana, 1943 ed.
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11 Plautus, Miles Gloriosus, 204‑206; Pliny, Hist. Nat. 34, 33 (cf. Macrobius, Saturnalia 1, 9, 10); Suidas s.v. Ἰᾶνος; Cicero, Ad Att. 5, 21, 12‑13; Quintilian, Instit. Orat. 1, 10, 35.
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12 For references to passages other than those mentioned in the course of this article the reader should consult the TLL s.v. digitus I, A, 2, c as well as the articles of Bechtel and Richardson mentioned above. Greek references from Herodotus and later, in addition p107to Roman, have been collected in Sittl, Carl, Die Gebärden der Griechen und Römer (Leipzig, 1890), 252‑259.
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13 Jones, Charles W., Bedae Pseudepigrapha: Scientific Writings Falsely Attributed to Bede (Ithaca, N. Y., 1939), 106‑108; cf. 53‑54. This edition of the pamphlet is based on a collation of two early MSS and a third as represented by the text in Migne, Patrologia Latina, 129, 1349‑50. This collation reveals two widely diverging traditions, which differ not merely in form, but in substance as well. An examination of as many as possible of the five remaining manuscripts listed by Jones will be necessary before we have a dependable text (or texts).
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14 Jones, Charles W., Bedae Opera De Temporibus (Cambridge, Mass., 1943), 179‑180; cf. 329‑331. This is the first truly critical edition of this work of Bede, based on complete collation of ten selected manuscripts and partial collation or examination of 94 others. Fortyfive of these were written between 70 and 100 years after its composition.
Reproductions of manuscript illustrations may be found in Burnam, John M., Palaeographica Iberica (Paris, 1912‑25), 3, Plates XLI‑XLIV (1‑90,000); 2, Plate XXIX (100,000‑1,000,000) from Madrid, Biblioteca Nacional A16 (19) (ca. 1130 A.D.) (The figure for 2000 has been copied in Smith, History of Mathematics, 2, 197 and in Encyc. Brit., 14th ed., s.v. Finger Numerals); Cordoliani, A., Bibliothèque de l'École des chartes, 103 (1942), Plate II (opp. p65), showing 30‑500 from Paris, Bibliothèque Nationale, MSS Lat. 7418, apparently a copy of the Madrid MS; Hermann, H. J., Die deutschen röm. Hss. (Leipzig, 1926), Plate V (reference from Jones, op. cit., 330), facsimile of Vienna MSS 12600 fol. 23 showing positions for higher numbers. Smith, op. cit., 198‑201 illustrates three other mediaeval systems. These illustrations are reproduced in part in Encyc. Brit., 14th ed., s.v. Numeral. See also Wüstemann, E. F., Neue Jahrbücher für Philologie und Pädagogik, Suppl.band, 15 (1849), 2, 511‑515 and Plates I‑IV which seem to be based on illustrations from J. Aventinus, Abacus atque vetustissima veterum latinorum per digitos manusque numerandi . . . consuetudo (Nuremberg, 1522) which contains the first printed text of this chapter of Bede.
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15 Tannery, Paul, "Notice sur les deux lettres arithmétiques de Nicolas Rhabdas," Notices et extraits des mss. de la Bibliothèque Nationale, 36 (1886), pt. 1, 121‑252 (reprinted in Tannery, Mémoires scientifiques, IV, Sciences exactes chez les Byzantins [Toulouse and Paris, 1920], 61‑198. The date of Nicolaus is established conclusively from his second letter; see PaulyWissowaKroll, s.v. Ῥαβδᾶς. A description of his system may be found in Heath, op. cit., vol. 2, 551‑552.
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16 Fröhner, Wilhelm, "Le comput digital," Annuaire de la société Française de numismatique et d'archéologie, 8 (1884), 232‑238 and Plate IV (wrongly labelled III). An illustration showing the obverse of the tesserae representing 8 and 9 is to be found in Bailey, Cyril, ed., The Legacy of Rome (Oxford, 1923), 296. These are in the British Museum, but not certainly from Pompeii as stated in Bailey. Fröhner illustrates the reverse of the former of these as bearing IIIV which he interprets as a retrograde VIII. I am, however, informed by R. A. Higgins of the British Museum that it actually reads INV, but that the crosspiece on the N is perhaps a little thinner than the other lines and could be a modern addition. Both, as Mr. Higgins has kindly informed me, are 2.8 cm in diameter (the corresponding measurements of our 25 and 50 cent pieces are approximately 2.4 and 3.0 cm respectively).
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17 Suidas, s.v. Ἀρβαζάκιος.
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18 Macrobius, Sat. VII.13.10. The tessera reproduced by Fröhner has all fingers erect, but the index finger separated as far as possible from its neighbour. As Fröhner points out, this must be a mistake since, according to all other evidence, the fingers left erect do not count, and therefore we have no number at all represented. The term medicus or medicalis digitus seems to have come from the practice on the part of doctors of using this finger in conjunction with the thumb to lift pinches of various materials; see Niedermann, Max, "Sprachliche Bemerkungen zu Marcellus Empiricus de medicamentis," Festgabe Hugo Blümner (Zurich, 1914), 329‑330.
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19a b c Adv. Iovinianum, 1, 3 (Migne, 23, 223); quoted in substantially the same words, Ep. 49 (48), 2 (Corp. Vind., 54, 353‑354); mentioned Ep. 123, 9 (Migne, 22, 1052) and Comm. in Evang. Matthaei, 2, 13, 23 (Migne 26, 92). See Bechtel, Loc. cit. (note 10). For 60 add Cassiodorus, In Psalm. 60 concl.
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20 Apologia, 89. Nicolaus' instructions are slightly different in that he speaks of the thumb as held in the same position but bent like capital gamma.
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22 Here Nicolaus' instructions are that the thumb should be laid across the palm to the base of the middle finger and the index hooked over its lower part.
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23 10, 248‑249; cf. Jerome, adv. Iovinianum, 1, 3 (and other references in note 19); Augustine, Serm. 175, 1; 251, 7; Tractatus in Iohannis Evangelium, 122, 7; Sidonius, Ep. 99, 16; Cassiodorus, in Psalm. 100 concl.; Cassianus, Conlatio, 24, 26, 7 (Corp. Vind. 13, 707).
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24 Apologia, 89, si triginta annos pro decem dixisses posses videri computationis gestu errasse, quos circulare debueris digitos aperuisse. Aperuisse is the reading of the later MSS, the two early readings quoted by Helm in the Teubner edition are adperisse and apisse. The suggestion, unsupported by outside evidence, that the gestures for these numbers might have been somewhat different in Apuleius' Africa and Bede's England, was made by H. E. Butler in Classical Review, 25 (1911), 72‑73.
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25 Dessau, Inscr. Lat. Sel., 7478 (CIL IX, 2689). The precise interpretation of the idiom used in the last remark is doubtful, but we may be fairly confident that a comic effect was intended. The relief was in the Louvre at the beginning of the present century after a mysterious disappearance from Naples and has been reproduced in various pictorial dictionaries, e.g. Baumeister, A., Denkmäler des Klassischen Altertums (Munich and Leipzig, 1885‑88), fig. 2373; Schreiber, Atlas of Classical Antiquities (London and New York, 1895), LXII, 12. It should be noted that the figures seem to have their p108right hands extended, rather than their left as we should expect.
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26 See Bechtel, loc. cit. and the TLL s.v. calculus, II, A, 2; s.v. computo, I, A, 1; s.v. digiti, I, A, 2, c.
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29 The most thorough discussion of the ancient abacus is Nagl, Alfred, "Die Rechentafel der Alten," Sitzungsberichte der Kais. Akademie der Wissenschaften in Wien, Phil.hist. Klasse, 177 (1914), abh. 5 (also published separately) to which the reader is referred for evidence for statements made throughout this section, unless otherwise directed. See also Nagl's article in PaulyWissowaKroll, s.v. abacus (9) (Supplementband, 3, 4‑13) and, for the history of the abacus, Smith, History of Mathematics, Vol. 2, 156‑196.
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30 All the examples of the word listed in the Thesaurus, s.v. abacus have other and more natural interpretations.
Thayer's Note: The word abacus does in fact seem to have been in use to mean a countingtable. The Abacus article in Smith's Dictionary of Greek and Roman Antiquities cites Pers. Sat. I.131, as does Lewis & Short, which further cites the postclassical Apul. Apol. 284.
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31 Thesaurus, s.v. calculus.
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32 This is less strange in the light of the Greek that almost no identifiable examples of the counting tables in wide use in Europe down to the beginning of the 17th century have survived although many of the counters or jetons are in existence. See on this subject Barnard, F. P., The Castingcounter and the Countingboard (Oxford, 1916).
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33a b Nagl, "Rechentafel," 15‑18. Smith, op. cit., 162‑168. Smith's identification of the abacus pictured on page 167 as from the British Museum is apparently wrong; for I am informed that there is no Roman abacus in their collection.
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34 See either article cited in note 29.
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35 Psammites, 3, 6.
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36 Eutocii Comm. in Dimensionem Circuli in Archimedes, Ed. J. L. Heiberg, Vol. 3, 272‑273.
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37 Examples of these problems with explanation may be found in Heath, A History of Greek Mathematics, Vol. 1, 57‑60; Gow, James, A Short History of Greek Mathematics (Cambridge, England, 1884), 49‑53.
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38a b Examples of multiplication and division involving fractions may be found in Nagl's article in PaulyWissowa (note 29) or (in greater detail) in Friedlein, op. cit., 89‑90.
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39 In Theon of Alexandria's commentary on the Almagest of Ptolemy, edited by Rome, A., Studi e Testi, 72, 461, 1‑462, 17 (reprinted with translation in Selections Illustrating the History of Greek Mathematics, ed. Ivor Thomas [Loeb Classical Library] [London, 1939], Vol. 1, 50‑53).
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41 Levy, Harry L., "Catullus, 5, 7‑11 and the Abacus," AJP 62 (1941), 222‑224.
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42 E.g., Martial, 2, 28; Juvenal, 10, 53; Priapea, 56.
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a the abacus: our author will discuss it in considerable detail below; but if you find that with this webpage you've bumbled onto far more detail than you want or can use, a much simpler summary is provided by the article in Smith's Dictionary of Greek and Roman Antiquities.
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b It's an act of some bravery; every Roman with the slightest medical knowledge knew that Cotta, entering his 63rd year, was on the cusp of the great climacteric, the most dangerous year of a person's life according to ancient medicine.
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