computation with roman numerals

7/28/2019 Computation With Roman Numerals

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Computa tion wi th Rom an Numer a lsM I C H A E L D E T L E F S E N , D O U G L A S K . E R L A N D S O N , J . C L A R K H E S T O N

C H A R L E S M . Y O U N GC o m m u n i c a t e d b y C . TRUESDELL

I t is w i d e l y h e l d t h a t c o m p u t a t i o n w i t h R o m a n n u m e r a l s i s d i ff i cu l t i f n o ti m p o s s i b l e ) B y p r e s e n t i n g s i m p le p r o c e d u r e s f o r a d d i n g a n d m u l t i p l y i n g w i t hR o m a n n u m e r a l s, w e s h o w t h a t t h i s c o m m o n i d e a is m i s t a k e n . W e a l s o s u g ge s tt h a t i t s a c c e p t a n c e a r i s e s f r o m t h e m i s t a k e n b e l i e f t h a t c o m p u t a t i o n s i n d i ff e r e ntn u m e r a l s y s t e m s w i ll m i r r o r o n e a n o t h e r , a b e l i e f w h i c h c a n b e e x p l a i n e d a sd e p e n d i n g u p o n a c o n f u s i o n o f n u m e r a l s w i th n u m b e r s .

I. AdditionT h e R o m a n a n d A r a b i c n u m e r a l s y s t e m s di ff er i n t h a t t h e la t t e r h a s s y m b o l s o f

d i f fe r e nt t y p e s f o r u n i t s ( ' 0 ' , ' 1 ', . . . . ' 9 ' ) a n d i n d i c a t e s p o w e r s o f t e n b y s y m b o lp o s i t i o n , w h i l e t h e f o r m e r h a s s y m b o l s o f di f f er e n t t y p e s f o r p o w e r s o f t en C I ' ,' X ' , "C ' , . .. )2 a n d i n d i c a t e s u n i t s b y s y m b o l i t e r a t i o n ? F o r t h i s r e a s o n , R o m a nn u m e r a l s a r e n u m e r a l - w i s e a d d i t i v e i n t h a t e ac h R o m a n n u m e r a l d e n o t e s t he s umo f t h e n u m b e r s d e n o t e d b y t he s i m p l e R o m a n n u m e r a l s 4 o c c u r r i n g w i t h i n it . F o re x a m p l e , ' C X L V I ' d e n o t e s t h e s u m o f C , X L , V , a n d I . T h a t R o m a n n u m e r a l sh a v e t hi s p r o p e r t y s u g ge s t s t h a t o n e c o u l d a d d w i t h t h e m b y s i m p l e c o n c a t e n a t i o n ,

Cf, e.g, K. MENNINGER,Num ber Symbols and Num ber Words,trans, by P. BRONEER (Cambridge,Mass.: M.I.T. Press, 1969), p. 298; C.M.TAISBAK,"'Roman Numerals and the Abacus", Classica etMediaevalia, 26 (1965), p. 153; and D.D.STgEBE, Elements o f Modern Arithmetic (Glenville, I1.: Scott,Freeman, and Co , 1971), p. 13.It should perhaps be observed that two obvious methods for computing with Roman numeralsare (1) translate the Roman numerals into Arabic, perform the appropriate Arabic computations,and trans late the result into Ro man numerals, and (2) tran slate the Roma n numerals into tally marks,perform the appropriate tally computations, and translate the result into Ro man numerals. W hat onewants, though, and what we provide, are simple and efficient procedures for computing with R omannumerals which are not parasiti c upon co mpu tation s in some other numeral system.z For the purpose of this point we ignore Roman numerals denoting five times some power often (i.e, 'V', 'L', ...) and the subtractive numerals (i.e., 'IV ', 'XL ', ...; 'I X' , 'XC ', ...), which make itit possible to indicate units economically.3 Sinc e he Roman system indicates powers of ten by the presence of numer als of the appro pria tetype (and not by numeral position), a sym bol for zero is not needed for writing Rom an numerals.A simple Roman numeral is either a numeral consisting of a single symbol or a subtractivenumeral.


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142 M. DETLEFSEN, D .K . ERLANDSON, J.C . HESTON & C. M, YOUNG

i f one ha d t r a ns f o r m a t ion r u l es w i th wh ich to p r oduc e a R om a n num e r a l f r om theresul t. The pro cedure we now p resent i s based on th is idea .

Our procedure has two stages, The f irst (s teps A and B) generates f rom two orm or e R om a n a dde nds a c onc a te na t ion o f s im ple R om a n nu m e r a l s ha v ing twoproper t ies:

(1) h ie ra rc h ic a l d i s t r ib u t io n : no s imple numera l s tands to the le ft of any s implenum e r a l d e no t ing a l a rge r num be r th a n i t de no te s .

(2) a d d i t i v e a d e q u a c y : t he sum o f the num be r s de n o te d by the s im ple num e r a l sin the conca tena t io n is the sum of the num bers denoted by the or ig ina l addends .The second s tage ( step C) produces the R om an num era l for the sum of the denota -t ions of the or ig ina l addends by applying t ransformat ions which prese rve theseproper t ies to the result o f s tage one,

S t a g e O n eA. C onc a te na te the R o m a n a dde nds , f la gg ing the r igh t -m os t s im ple num e r a l

of each.B. Regroup so tha t the resul t ing conca tena t ion i s h ie ra rchica l ly d is t r ibuted .

Eliminate f lags.S tep A s imply requi res tha t we wr ite Ro ma n adden ds in rows ra ther th an thecolumns w e use for Arabic add ends . W e wr i te Arab ic add ends in colum ns becausethis make s i t (psychologically) easier for us to exp loit the posi t io nal c harac ter o fArabic num era ls when we add. S ince Ro ma n n umera ls a re not pos i t iona l , the re isno rea son to wr i te Ro ma n adden ds in th is way. The device of f lagging preventsposs ib le ambigui t ie s . I t prevents one f rom reading, for example , ' IX ' where Tis the las t s imple numera l o f one ad den d and 'X ' i s the f ir s t s imple numera l of thenext , as the su btract ive n um eral for nine ( ' IX' ) . Step B el iminate s the need forf lags by h ie ra rchica l ly d is t r ibut ing the s imple nu mera ls in the resul t of step A,produc ing an addi t ive ly adequa te , h ie ra rchica l ly d is t r ibuted conca tena t ion ofs im ple R om a n num e r a l s. S t age two is de s igne d to p r oduc e a R o m a n num e r a l f r omthis conca tena t ion .

S t a g e T w oC. Ap ply these t ransfo rmat ions 5 wherever poss ib le to the resul t of s tep B( the order in which these t ransfo rmat ions a re appl ied i s i r re levant ) :

1 . i i i i i~ v2 . i i i i ~ iv3. iv iv --+ v i i i4 . v v ~ x

5 If N denotes he non-negative nteger n and R is the R oman numeral denoting r, we let r(R)~denote the R oman numeral for r. llY.E.g,, (I)x is ' X ', and (XIV)2 is" MCD'. (Those unfamiliarwith the


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C o m p u t a t i o n w i t h R o m a n N u m e r a l s

5. i x i x ---> x v i i i6 . i v i ~ v7. v iv --* i x8. ix iv ~ x i i i9. ix i ~ x

10. ix v ~ x iv

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square corners nota t ion will f ind i t expla ined in the sec t ion on " 'quas i - quota t ion" in W. V.D . QUINE,Mathematical Logic, rev. ed. [-Cambridge, Mass.: Ha rva rd Univ ersi ty Press, 1951], pp. 33-7.)

Transformation C8 is to be read, ~"F or e a ch non-ne ga t i ve i n t ege r n , i f t he c onc a t e na t i on c on t a i nsboth an o ccurrence of (IX) , and an occurrence of (IV) , , replace them w i th one occur rence of (X) ,and three occurrences of ( I ) ,, h ie ra rchica l ly d i s t r ibut ing them wi th in the conca tena t ion ," and theothe r t ransformat ions in l ike fashion.The use of these na tura l con vent ion s makes poss ib le not only the economica l exp ress ion of theset ransformat ion ru les (e .g . , C4 appl ied three t imes to the conca tena t ion "DDLLVV" would y ie ld'MCX' ) but a l so a surpr i s ingly compac t mul t ip l ica t ion table (see the next sec t ion) .

No te tha t , i f one were to lay down as the f i rst ru le "expan d a l l subt rac t ive numera ls ," our addi t ionprocedure would be much s imple r : only t ransformat ions C 1 , C2, and C4 would be needed.Note fur the r tha t a ca lcula tor has an e ffec tive proced ure for de te rm ining the Rom an numera l

denoted by ~(R)N ~, whenever ~(R)N ~ denotes a Ro ma n numera l tha t can be wri t ten wi th the s impleR om a n nume ra l s a t h i s c omma n d . He be g i ns by wr i ti ng t he s i mp l e R om a n nume ra l s a t h is c omm a ndin rows four column s wide , in o rde r of inc reas ing "s ize ':

Char t 0 : I IV V IXX XL L XCC C D D C MM . . . . . . . . .

He then cons t ruc ts fur the r cha r t s , one for each row o f Cha r t 0 :Ch art 1: ( I ) o (IV)o (V)o (IX)0

(I) ~ (IV)x (V)~ (IX)~( I)2 (w )2 (vh ( Ix )2( I )~ . . . . . . . . .

C h a rt 2 : . . . . . . . . . . . .(X)o (XL)o (L)o (XC)o(X)~ (XL)I (L)I (XC)I(X) 2 . . . . . . . . .

C h a r t 3 : . . . . . . . . . . . .(C) o (CD )o (D)0 (CM )o( C ) , . . . . . . . . .

C h a r t 4 : . . . . . . . . . . . .

( M) o . . . . . . . . .and so on. These charts give him a pro ced ure for finding the R om an num eral den ote d by ~-(R)t - ' i fr (R)N ~ denotes a s imple R om an numer a l a t h i s comm and : i t i s found in the pos i t ion of Char t 0 cor-resp ond ing to the posi t ion of r(R)N n on th e chart on which i t occurs. If r-(R)Nn denotes a non- s imple

numera l wri teable wi th the s imple Roman numera ls a t the ca lcula tor ' s command, he can a l so f indthe Rom an numera l i t denotes : i t would be the (h iera rchical ly d is t r ibuted) conca tena t io n of thes imple numera ls denot ed by r (A) N n . . . . r (K)N " , where A . . . . . K a re the s imple numera ls occu rr ingi n k


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144 M. DETLEFSEN,D. K. ERLANDSON,J.C. HESTON & C. M. YOUNGThese t ransformat ions y ie ld a Rom an numera l f rom the concatenat i on of

simple numer als resul ting from stage one. 6 Since that con cat ena ti on is additivelyadequate and the transf orma tion rules preserve addit ive adequacy, the Ro ma nnumera l result ing from stage two denote s the sum of the numbe rs deno ted by theoriginal addends.

Wor king t hrough an example wil l help to mak e this procedure clear . Supposeone wished to compute the Roman numeral for the sum of CXLVIII , CXLIV,and LXXII . Then:

Step A yieldsStep B yieldsStep C 6 (thrice) yields vSte p C 3 (twice) yieldsStep C 2 yields

C X L V I I I * C X L I V * L X X I IC C L X L X L X X V I V I I I I IC C L L L V V I I I IC C C L X I I I ICCCLXI__y_v.

And this is the Ro ma n num era l for the sum desired.If the rea der will work thr oug h this exam ple a nd a few of his own, he will see

that this procedure is simple and natural, though it might at first appear a bitcomplicated. With obvious shortcuts (e l iminating step A, applying transforma-tions as seems appropriate during conca tenat ion, and using a black board toavoid having to re-write) one can, with a little practice, add several numbersdictated in Ro ma n numera ls almost as quickly as they can be dictated. 8

6 A hierarchically distributed concatenation of simple Roman numerals is a Roman numeralit; and only i for every non-negative integer n, it contains:(a) at most three occurrences of (I).(b) at most one occurrence of (IV),(c) at most one occurrence of (V),(d) at most one occurrence of (IX),(e) no joint occurrences of (IV). and (I).(f/ no joint occurrences of (IV), and (V),(g) no joint occurrences of (IV), and (IX),(h) no joim occurrences of(IX), and (I),(i) no jo im occurrences of(IX)~ and (V)~.

It is easy to see that, if step C terminates, it terminates with a Roman numeral: C1 and C2 guaranteecondition (a), C3 guarantees condition (b) ... . and C 10 guarantees condition (i). That Step C doesindeed terminate follows from the fact that each transformation, each time it is applied, yields a simpleRoman numeral whose denotation is larger than the denotation of each simple numeral removedby that application of the transformation.Some systems of Roman numerals allow subtractive numerals other than those we allow in n. 2above (e.g., ' IC ' or 'IIX'). Allowing such numerals would require appropriate but obvious changesin the conditions under which a concatenation of simple Roman numerals is a Roman numeral andin our addition and multiplication procedures.We have underlined the simple numerals which result from each set of transformations.8 Just as the procedure for Arabic addition makes possible a procedure for Arabic subtraction,so our procedure for Roman addition makes possible a procedure for Roman subtraction (consistinglargely in cancellation facilitated by an analogue of borrowing)which, for reasons of space, we do notpresent. Similarly, our procedure for Roman multiplication in the next section makes possible a pro-cedure for Roman division.It might be thought that a weakr/ess of the Roman numeral system is that one requires arbitrarilymany symbols with which to denote arbitrarily large numbers. While this holds for the Roman system


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Compu tatio n with Roma n Numerals 145

II . Mult ip l ica t ionArabic multiplication takes advantage of the positional character of Arabic

numerals and the procedure for Arabic addition. In like manner, our procedurefor Roman multiplication exploits the numeral-wise additivity of Roman numeralsand our procedure for Roman addition.Suppose that a and b are positive integers, that A 1 . .. . A m are the simplenumerals (including repetitions, if any) in the Roman numeral for a, that B 1 .. .. Bnare the simple numerals in the Roman numeral for b, that each A i denotes the

integer ai, and that each Bj denotes the integer bj. Then by numeral-wise ad-ditivity,a . b= a i - j .~ i= l / \ j~ l /

And by the distributive law, a.b= ~ ~(a,.bj).i= l j= lSince each a i and bj is denoted by a simple Roman numeral that occurs in theRoman numerals for a and b, the right sides of the above equalities can be expressedusing only the signs for addition and multiplication, parentheses, and the simplenumerals occurring in the Roman numerals for a and b. With this multiplication

( I ) = ( I V ) m ( V ) = (IX)m(I)n ( I ) = + n . . . . . . . . . . . . . . . . . .( i v ) . ( IV ) m + . ( X V I ) m + . . . . . . . . . . . . .( v ) ( v ) = + , ( x x ) ~ , . . ( x x v ) = + . . . . . .( I X ) . ( I X )= + . ( X X X V I ) = + n ( X L V ) = + . ( L X X X I ) = + .

table:

the Roman numeral for a- b can be computed from the Roman numerals for aand b using our addition procedure. The multiplication table gives the Romannumerals for the products of the numbers denoted by all possible pairs of simpleRoman numerals. 9 For example, the table gives "MM' as the Roman numeralfor IV. D:' IV ' is (IV)o and ' D' is (V)2; according to the table the Roman numeralfor the product of the numbers they denote is (XX)2 + o; this is (XX)2, i.e., MM'.Multiplication with Roman numerals can thus be accomplished by successivelyapplying numeral-wise additivity, the distributive law, the multiplication table,and our addition procedure.Although this procedure can be cumbersome for numbers denoted by longRoman numerals, it works quite well for numbers denoted by short ones. Suppose,with which most of us are familiar, it is false for Roman type systems generally. Indeed, one can makedo with but four distinct symbols: 'I', "V', "X', and a bar. Using the symbols 'I', "IV', "V', and "IX"with n bars above the m to denote, respectively, 1 - 10", 4- 10", 5.10", and 9- 10", one can perspicac iouslyand uniquely mention any positive integer with but four distinct symbols.

9 Within the range of simple Roman numerals at a calculator's command. For the system de-scribed in the previous note, the table would be complete.

Note that, if one were to disallow subtractive numerals, this multiplication table would be evenmore compact: the second and fourth rows and columns could be eliminated.


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146 M. DETLEFSEN, D. K . ERLANDSON, J. C. HESTON & C. M. YOUNG

for example, we wish to com pute the R om an numera l for the produ c t of XLVIand XIV. Then:Num eral-wise addit ivi ty yie ldsThe distributive law yieldsThe multiplication table yieldsStep A (addition) yieldsStep B yieldsStep C 6 yieldsStep C4 yieldsStep C 2 yields

(XL + V + I)- (X + IV)( X L - X ) + ( X L - I V ) + ( V . X ) +(V . IV ) + ( I . X ) + ( I . IV)C D + C L X + L + X X + X + I VC D * C L X * L * X X * X * I VC D C L L X X X X I V_DLLXXXXIVD_C_CXXXXIVD C X L I V

And this is the Ro m an numeral for the produ ct desired. As with the addit ion p roce-dure , obvio us shortcuts (with t raining, one can often go direct ly to step B of theaddit ion p rocedure) make this mult ipl icat ion proce dure m uch quicker in practicethan it might appear.

III. Roman Numerals and the AbacusThe in t imate connec t ion be tween Rom an numera ls and the abacus has of ten

been noted. 1 I t may be described by saying that the Ro m an num eral for a num berdescribes the sta te of an abacu s when i t represents that nu mber, the nu mbe r o foccurrences of each simple numeral correspon ding to the num ber of countersat play in the appro pria te column s of an abacus w hen i t represents that number.11

In l ight of this , it should com e as no surprise that our addit ion and mult iplica-t ion procedures paralle l qu i te c losely the procedu res for addi t ion and m ult ipl icat ionon an abacus. Indeed, our proce dures are a kind of penci l and paper a bacus: themain difference between our procedures an d the correspon ding abacus proceduresis that an abacus operator concatenates step by step, applying t ransformationswhen appropria te , while we concatenate a l l a t once and apply t ransformationsto the result . The suggested shortcuts in o ur procedu res w ould b ring the two setsof techniques in line.

IV. Numbers and NumeralsIn view of the re la t ive simplic i ty of our computat ional procedures and theirconnect ion with com putat ion al techniques on the abacus, the comm on conv ict ion1o CS MENNINGER, op. cir., p. 298, an d TAISBAK,op. cir., pp. 158-160. TAISBAKalso c ites A. NAGL'S

artic le on the abacus in PAULY-WIsSOWA, Real-Encyclopaedie , suppl. I I I , col . 11.11 T he n um b e r o f oc c u r r e nc e s o f "I" i n a R om a n n um e r a l w ou ld c o r r e s pond to t he num be r o f

c oun te r s a t p l a y i n t he l ow e r po r t i on o f t he r i gh t - m os t c o lum n o f a n a b a c us w h e n i t r e p r ese n t s t hesa m e num be r ; t he num be r o f oc c u r r e nc e s o f "V ' , t o t he num be r a t p l a y i n t he uppe r po r t i o n o f t ha tc o lu m n; a n d so f o rth . F o r t h e pu r pose o f t h is po in t , one m u s t t h ink o f sub t r a c t i ve num e r a l s l i ke ' I V 'a s sho r tha n d f o r a n a pp r op r i a t e num be r o f a pp r op r i a t e non - sub t r a c t i ve num e r a l s , bu t T AIS BA K op. cir.,pp . 158 -160 ) obse r ve s t ha t sub t r a c t i ve num e r a l s m a y ha ve a r i s e n f r om a n a ba c us m a ne uv e r : a na ba c us ope r a to r a dds f ou r by d r opp ing o ne c oun te r i n t he l ow e r po r t i on o f t he r i gh t - m os t a ba c usc o lum n a nd r a i sing one c oun te r i n t he uppe r po r t i on , i f t he se ope r a t i ons a r e pos s ib l e .


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C o m p u t a t i o n w i th R o m a n N u m e r a l s 1 47

that computation with Roman numerals is difficult if not impossible is in needof explanation. An obvious (but uninteresting) explanation would be that no onethought of techniques that would work. A more interesting explanation, though,is suggested by a consideration of what seems to be the only argument for theconviction; that argument is due to MENNINGER. Discussing the importance of theintroduction of Arabic numerals into Europe (an importance we would not, ofcourse, deny), he writes:

But though at first glance one merely notices the greater brevity brought aboutby the new numerals, a second glance lets one see a little deeper: with the newdigits we can now for the first time make computations!With this remark we finally realize that writing numerals and making compu-tations are two entirely different things; up to now we have generally hadnothing to say about computations, although we have thoroughly discussedspoken numbers and written numerals. But didn't people make calculationswith Roman numerals? No, they did not! The fairly simple multiplication:

325- 47 in Roman numerals CCCXXV XLVII2275 MMCCLXXV

13000 KMMM15275 KWCCLXXV

looks clearly impossible to the uninitiated reader. 12MENNINGER'S claim that persons did not in fact make calculations with Romannumerals may well be true. (If so, the existence of our procedures shows that thefault lay in the persons, not in their numeral system.) But MENNINGER also holdsthat computation with Roman numerals is itself not possible. Using a translationof an Arabic computa tion into Roman numerals, he argues from the "impossiblelook" of the translated computation to this conclusion. MENNINGER'S argument,note, requires the assumption that computation with Roman numerals is possibleonly if Arabic and Roman computations mirror one another: without this idea,the impossibility of MENNINGER'S translated computation would be irrelevantto the impossibility of Roman computation itself. As a comparison between ourprocedures for Roman computation and the standard procedures for Arabiccomputation will show, this assumption is false. But one could be led to thinkit true by a confusion of numerals with numbers. If (making this confusion) onethought that when one is computing with Arabic numerals one is really manipulat-ing numbers, it would be natural for one to suppose that numbers must be mani-pulated in the same manner regardless of the numeral system one is using, and thusto suppose that the intermediate stages of Roman computations, if such computa-tions are possible, will parallel the intermediate stages of Arabic computations. ,3

12 MENNINGER, o p . c i t . , p. 298. W e have taken the l ibe r ty of changi ng MENNINGER'S Ro ma nnum e r a l s f o r fi ve t ho usa nd a nd t e n t hou sa nd to "W ' a nd "K ' , r e spec t ive ly .

~3 F ur th e r ev idence t ha t MENNINGER ma y be co nfus in g num era l s wi th nu mb ers i s h is use of thee xp r e s s ion " spo ke n nu m b e r s " i n t he pa s sa ge quo te d a bove . S ee a l so T AlS BA K,o p . c i r . , p. 148, whereon ly a f ew l ine s a f t er d i s t i ngu i sh ing be tw e e n num e r a l s a nd num be r s he p r oc ee ds t o c on f use t he m .


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148 M. DETLEFSEN, D. K. ERLANDSON, J. C. HESTON & C. M. YOUNGOnce one makes this supposition, an argument of the sort presented by MENNIN-GER will be persuasive, if no t comp elling.

Computation techniques, though, involve the manipulation of numerals, notof numbers. The standard procedures for Arabic com put atio n and those presentedhere for Roman computation are both techniques for computing numerals fornumbers from other designators o f those num bers, giving a calculator the numeralshe needs to write sums and products. Arabic computation, for example, wouldyield the numeral "714' from the designator "102-7', while Roman computationwould yield the numeral 'DCCXIV' from the designator 'CII-VII' . In l ight ofthis and of the significant differences between the Roman and Arabic numeralsystems, there is no reason to suppose that R oma n compu tation is possible onlyif Rom an and Arabic com putation s mirror one another. 14

Department of PhilosophyUniversity of Minnesota

DuluthDepartment of Philosophy

University of NebraskaLincoln

Department of PhilosophyOakland UniversityRochester, Michigan

an dDepartment of PhilosophyUniversit y o f Pittsburgh

(R eceived October 31, 1975)14 The authors wish to thank Professors ROBERT CUMMINS and DALE GOTTLIEBfor suggesting

that we devise techniques for comput ing with R om an numerals. We are also grateful to several personsfor their helpful comments on earlier versions of this paper, in particular, to Professors JAMESBOGENand ALDEN PIXLEYand to the editor of the Archive.

The final version of this paper was prepared by YOUNGduring the tenure of a 1975-6 MellonPostdoctoral Fellow ship at the Universi ty of Pittsburgh.

computation with roman numerals

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