Accounting History - Roman Notation

History of Accounting > Roman Notation

As the Roman notation was adopted throughout Europe, and was almost exclusively employed during twelve or thirteen centuries, and is still for certain purposes (to distinguish chapter from verse, volume from page, &c.) current everywhere, it is expedient to state the rules to be observed in reading any number expressed in their manner.

(1) If to the right of any number another number is written which is equal to or less than the first, the value of the first must be increased by that of the second. Thus II VI XII LV DC MDCLXV 26 12 55 600 1665

(2) If to the left of any number another number is written which is less than the first, the value of the first must be diminished by that of the second. Thus

IV IX XL CD

4 9 40 400

(8) If a number is written between two others which are greater than it, it must be subtracted from the one on the right of it. Thus

XIV XIX LIX CXL MCM

14 19 59 140 1900

As regards the invention of the nine digits and the cipher, with the application to them of the principle of local value, it is not possible to speak with certainty. It is now, however, an accepted opinion that it is to the Hindoos we must ascribe this momentous improvement in arithmetical notation, but we have no evidence to enable us to say when the improvement was first made. Before the end of the ninth century the Hindoo figures were known to the Arabs, and before the end of the tenth they were in general use among them. By the eleventh century they had been introduced into Spain by the Moors, and they were known in Italy at the beginning of the thirteenth century. It has been conjectured that the commercial intercourse between Italy and the East would suffice to account for the introduction of these numerals; and it is certain that the first Italian who wrote about them (1202), Leonardo of Pisa, the son of Bonacci (Fibonacci), had travelled extensively in the East.

It is sometimes asserted that Gerbert, who was born at Aurillac, in Auvergne, and who was afterwards promoted to the bishoprics of Rheims and Ravenna, and finally became Pope under the title of Sylvester II., introduced into France a knowledge of the Arabic numerals in the latter part of the tenth century, but this is rather improbable. It is known that in early life Gerbert studied among the Saracens, and he is said to have written extensively on arithmetic and geometry.

When or how the Arabic notation came into England it is impossible to say with any approach to certainty. One of the modes in which persons who could read obtained a knowledge of it was from the ecclesiastical calendars, which were widely distributed in the fourteenth and fifteenth centuries.

After the introduction of the Hindoo numerals for the expression of integers, the next great improvement in arithmetical notation was the invention of decimal fractions. From the time of Ptolemy, and probably also before his time, it had been customary to divide the circle into 360 degrees (nolpai). Each of these degrees was divided into 60 equal parts called primes, each prime into 60 equal parts called seconds, each second into 60 equal parts called thirds, and so on. Our names, minutes and seconds, whether applied to angular magnitude or to time, are shortened forms of the expressions first minute parts (partes minutae primae) and second minute parts (partes minutae secundae). The notation adopted for degrees was a stroke written above the number of them and accents,’ ” ‘”, &c., for the different orders of sexagesimals.

The first indubitable appearance of decimal fractions occurs in the year 1525, in the extraction of a square root. Orontius Finaeus, a professor of mathematics in Paris, wishing to approximate to the square root of 10, affixes to it six ciphers, extracts the root in the usual way, and obtains the number 3162. Then taking 162 he multiplies it by 60, getting 9720, whence 9 primes are obtained by cutting off the three right-hand digits. Again, 720 x 60 = 43,200, whence 43 seconds; lastly, 200 x 60 = 12,000, whence 12 thirds. According to our mode of working this would stand—

3-162

60

9720

60

43-200

60

Thus ^/10 = 39? 43? 12?”. Though Finaeus expresses the root sexagesimally, yet he expressly states that in 3162 the 3 denotes units, the 1 one-tenth of a unit, the 6 six-tenths of onetenth of a unit, the 2 two-tenths of one-tenth of one-tenth of a unit. This is the germ of the doctrine of decimal fractions. The most notable development of it is found in Stevin’s Arithmetic, which was published in 1585 in French. It contained a small treatise, La Disme, “by the which we can operate with whole numbers without fractions.” This is not quite the modern view, namely, that by extension of the notation for integers, integers and fractions can be treated by the same rules, but it comes near to it. The number which we write 27’847 Stevin writes 27(0) 8(1) 4(2) 7(3), or

(0) (1) (2) (3)

when using it in operation 27 8 4 7. The following are other notations for decimal fractions which occur in books subsequent to Stevin’s time :

27 | 847 27 8? 4? 7?”

27 847?” 27 847(3)

27 | 847 thirds 27 ’8-4 7

27 847 27 | 847

The question of who introduced the point or comma to separate the integers from the fractions has been discussed by De Morgan, and he does not admit the claim that has been made by Peacock for Napier, the inventor of logarithms. Whether or not Napier habitually used the comma, there is at any rate one instance where it occurs in his Rabdologia, which was published in 1617; he gives a quotient as 1993,273 or 1993,2? 7? 3?”. This simplification of the notation for decimal fractions, obvious enough as it seems, did not become common till the middle of the seventeenth century.

It was only about a century ago that decimals were applied to metrological reform. The tables of the measures for length, area, capacity, weight in use in France were very irregular, and the French National Assembly in 1790 resolved to create a new system of measures, the sub-divisions of which should harmonise with the decimal system of numbers. The commission of scientific men to whom this reform was entrusted selected as a basis a length which should be the ten-millionth part of the distance between the North Pole and the Equator. This distance they called a metre, and to express the multiples of it they used as prefixes the Greek words, somewhat modified, for 10, 100, 1000 (deca-, hecto-, kilo-) ; to express the sub-multiples they used the Latin words for 10, 100, 1000 (deci-, centi-, milli-) in the sense of A, T

This is not the place in which to discuss the advantages or the disadvantages of the Metric System, but after the exposition which has been made of how a tolerably uniform system of numeration and a completely uniform system of notation have been gradually built up among civilised nations, it may be worth while to see what improvement awaits them in the distant future.

Herbert Spencer, in a pamphlet1 entitled ” Against the Metric System,” proposes as the radix of numeration the number ] 2. He says, ” This process of counting by groups and compound groups, tied together by names, is equally practicable with other groups than 10. We may form our numerical system by taking a group of 12, then 12 groups of 12, then 12 of these compound groups; and so on as before. … It needs only a small alteration in our method of numbering to make calculation by groups of 12 exactly similar to calculation by groups of 10; yielding just the same facilities as those now supposed to belong only to decimals. . . . To prevent confusion different names and different symbols would be needed for the digits, and to acquire familiarity with these, and with the resulting multiplication-table, would of course be troublesome: perhaps not more troublesome, however, than learning the present system of numeration and calculation as carried on in another language.” Spencer states that he thinks this system will not be adopted for generations. ” But it is not an unreasonable belief that further intellectual progress may bring the conviction that since a better system would facilitate both the thoughts and actions of men, and in so far diminish the friction of life throughout the future, the task of establishing it should be undertaken.”