A Short Account of the History of Mathematics · W. W. Rouse Ball
The subject of this chapter has been discussed by Cantor and by Hankel. See also the Philosophy of Arithmetic by John Leslie, second edition, Edinburgh, 1820. Besides these authorities the article on Arithmetic by George Peacock in the Encyclopaedia Metropolitana, Pure Sciences, London, 1845; E. B. Tylor's Primitive Culture, London, 1873; Les signes numéraux et l'arithmétique chez les peuples de l'antiquité by T. H. Martin, Rome, 1864; and Die Zahlzeichen by G. Friedlein, Erlangen, 1869, should be consulted.
I have in many places alluded to the Greek method of expressing numbers in writing, and I have thought it best to defer to this chapter the whole of what I wanted to say on the various systems of numerical notation which were displaced by the system introduced by the Arabs.
First, as to symbolism and language. The plan of indicating numbers by the digits of one or both hands is so natural that we find it in universal use among early races, and the members of all tribes now extant are able to indicate by signs numbers at least as high as ten: it is stated that in some languages the names for the first ten numbers are derived from the fingers used to denote them. For larger numbers we soon, however, reach a limit beyond which primitive man is unable to count, while as far as language goes it is well known that many tribes have no word for any number higher than ten, and some have no word for any number beyond four, all higher numbers being expressed by the words plenty or heap: in connection with this it is worth remarking that (as stated above) the Egyptians used the symbol for the word heap to denote an unknown quantity in algebra.
The number five is generally represented by the open hand, and it is said that in almost all languages the words five and hand are derived from the same root. It is possible that in early times men did not readily count beyond five, and things if more numerous were counted by multiples of it. It may be that the Roman symbol X for ten represents two "V"s, placed apex to apex, and, if so, this seems to point to a time when things were counted by fives. In connection with this it is worth noticing that both in Java and among the Aztecs a week consisted of five days.
The members of nearly all races of which we have now any knowledge seem, however, to have used the digits of both hands to represent numbers. They could thus count up to and including ten, and therefore were led to take ten as their radix of notation. In the English language, for example, all the words for numbers higher than ten are expressed on the decimal system: those for 11 and 12, which at first sight seem to be exceptions, being derived from Anglo-Saxon words for one and ten and two and ten respectively.
Some tribes seem to have gone further, and by making use of their toes were accustomed to count by multiples of twenty. The Aztecs, for example, are said to have done so. It may be noticed that we still count some things (for instance, sheep) by scores, the word score signifying a notch or scratch made on the completion of the twenty; while the French also talk of quatrevingts, as though at one time they counted things by multiples of twenty. I am not, however, sure whether the latter argument is worth anything, for I have an impression that I have seen the word octante in old French books; and there is no question that septante and nonante were at one time common words for seventy and ninety, and indeed they are still retained in some dialects.
The only tribes of whom I have read who did not count in terms either of five or of some multiple of five are the Bolans of West Africa who are said to have counted by multiples of seven, and the Maories who are said to have counted by multiples of eleven.
Up to ten it is comparatively easy to count, but primitive people find great difficulty in counting higher numbers; apparently at first this difficulty was only overcome by the method (still in use in South Africa) of getting two men, one to count the units up to ten on his fingers, and the other to count the number of groups of ten so formed. To us it is obvious that it is equally effectual to make a mark of some kind on the completion of each group of ten, but it is alleged that the members of many tribes never succeeded in counting numbers higher than ten unless by the aid of two men.
Most races who shewed any aptitude for civilization proceeded further and invented a way of representing numbers by means of pebbles or counters arranged in sets of ten; and this in its turn developed into the abacus or swan-pan. This instrument was in use among nations so widely separated as the Etruscans, Greeks, Egyptians, Hindoos, Chinese, and Mexicans; and was, it is believed, invented independently at several different centres. It is still in common use in Russia, China, and Japan.
1.
In its simplest form (see Figure 1) the abacus consists of a wooden board with a number of grooves cut in it, or of a table covered with sand in which grooves are made with the fingers. To represent a number, as many counters or pebbles are put on the first groove as there are units, as many on the second as there are tens, and so on. When by its aid a number of objects are counted, for each object a pebble is put on the first groove; and, as soon as there are ten pebbles there, they are taken off and one pebble put on the second groove; and so on. It was sometimes, as in the Aztec quipus, made with a number of parallel wires or strings stuck in a piece of wood on which beads could be threaded; and in that form is called a swan-pan. In the number represented in each of the instruments drawn on the next page there are seven thousands, three hundreds, no tens, and five units, that is, the number is 7305. Some races counted from left to right, others from right to left, but this is a mere matter of convention.
The Roman abaci seem to have been rather more elaborate. They contained two marginal grooves or wires, one with four beads to facilitate the addition of fractions whose denominators were four, and one with twelve beads for fractions whose denominators were twelve: but otherwise they do not differ in principle from those described above. They were commonly made to represent numbers up to 100,000,000. The Greek abaci were similar to the Roman ones. The Greeks and Romans used their abaci as boards on which they played a game something like backgammon.
2.
In the Russian tschotü (Figure 2) the instrument is improved by having the wires set in a rectangular frame, and ten (or nine) beads are permanently threaded on each of the wires, the wires being considerably longer than is necessary to hold them. If the frame be held horizontal, and all the beads be towards one side, say the lower side of the frame, it is possible to represent any number by pushing towards the other or upper side as many beads on the first wire as there are units in the number, as many beads on the second wire as there are tens in the number, and so on. Calculations can be made somewhat more rapidly if the five beads on each wire next to the upper side be coloured differently to those next to the lower side, and they can be still further facilitated if the first, second, , ninth counters in each column be respectively marked with symbols for the numbers 1, 2, , 9. Gerbert is said to have introduced the use of such marks, called apices, towards the close of the tenth century.
3.
Figure 3 represents the form of swan-pan or saroban in common use in China and Japan. There the development is carried one step further, and five beads on each wire are replaced by a single bead of a different form or on a different division, but apices are not used. I am told that an expert Japanese can, by the aid of a swan-pan, add numbers as rapidly as they can be read out to him. It will be noticed that the instrument represented in Figure 3 is made so that two numbers can be expressed at the same time on it.
The use of the abacus in addition and subtraction is evident. It can be used also in multiplication and division; rules for these processes, illustrated by examples, are given in various old works on arithmetic.
The abacus obviously presents a concrete way of representing a number in the decimal system of notation, that is, by means of the local value of the digits. Unfortunately the method of writing numbers developed on different lines, and it was not until about the thirteenth century of our era, when a symbol zero used in conjunction with nine other symbols was introduced, that a corresponding notation in writing was adopted in Europe.
Next, as to the means of representing numbers in writing. In general we may say that in the earliest times a number was (if represented by a sign and not a word) indicated by the requisite number of strokes. Thus in an inscription from Tralles in Caria of the date 398 b.c. the phrase seventh year is represented by eteos | | | | | | | . These strokes may have been mere marks; or perhaps they originally represented fingers, since in the Egyptian hieroglyphics the symbols for the numbers 1, 2, 3, are one, two, and three fingers respectively, though in the later hieratic writing these symbols had become reduced to straight lines. Additional symbols for 10 and 100 were soon introduced: and the oldest extant Egyptian and Phoenician writings repeat the symbol for unity as many times (up to 9) as was necessary, and then repeat the symbol for ten as many times (up to 9) as was necessary, and so on. No specimens of Greek numeration of a similar kind are in existence, but there is every reason to believe the testimony of Iamblichus who asserts that this was the method by which the Greeks first expressed numbers in writing.
This way of representing numbers remained in current use throughout Roman history; and for greater brevity they or the Etruscans added separate signs for 5, 50, &c. The Roman symbols are generally merely the initial letters of the names of the numbers; thus c stood for centum or 100, m for mille or 1000. The symbol v for 5 seems to have originally represented an open palm with the thumb extended. The symbols l for 50 and d for 500 are said to represent the upper halves of the symbols used in early times for c and m. The subtractive forms like iv for iiii are probably of a later origin.
Similarly in Attica five was denoted by Pi , the first letter of p'ente, or sometimes by Gamma ; ten by Delta , the initial letter of d'eka; a hundred by Eta for <ekat'on; a thousand by Chi for q'ilioi; while 50 was represented by a Delta written inside a Pi ; and so on. These Attic symbols continued to be used for inscriptions and formal documents until a late date.
This, if a clumsy, is a perfectly intelligible system; but the Greeks at some time in the third century before Christ abandoned it for one which offers no special advantages in denoting a given number, while it makes all the operations of arithmetic exceedingly difficult. In this, which is known from the place where it was introduced as the Alexandrian system, the numbers from 1 to 9 are represented by the first nine letters of the alphabet; the tens from 10 to 90 by the next nine letters; and the hundreds from 100 to 900 by the next nine letters. To do this the Greeks wanted 27 letters, and as their alphabet contained only 24, they reinserted two letters (the digamma and koppa) which had formerly been in it but had become obsolete, and introduced at the end another symbol taken from the Phoenician alphabet. Thus the ten letters alpha to iota stood respectively for the numbers from 1 to 10; the next eight letters for the multiples of 10 from 20 to 90; and the last nine letters for 100, 200, etc., up to 900. Intermediate numbers like 11 were represented as the sum of 10 and 1, that is, by the symbol iotaalpha' . This afforded a notation for all numbers up to 999; and by a system of suffixes and indices it was extended so as to represent numbers up to 100,000,000.
There is no doubt that at first the results were obtained by the use of the abacus or some similar mechanical method, and that the signs were only employed to record the result; the idea of operating with the symbols themselves in order to obtain the results is of a later growth, and is one with which the Greeks never became familiar. The non-progressive character of Greek arithmetic may be partly due to their unlucky adoption of the Alexandrian system which caused them for most practical purposes to rely on the abacus, and to supplement it by a table of multiplications which was learnt by heart. The results of the multiplication or division of numbers other than those in the multiplication table might have been obtained by the use of the abacus, but in fact they were generally got by repeated additions and subtractions. Thus, as late as 944, a certain mathematician who in the course of his work wants to multiply 400 by 5 finds the result by addition. The same writer, when he wants to divide 6152 by 15, tries all the multiples of 15 until he gets to 6000, this gives him 400 and a remainder 152; he then begins again with all the multiples of 15 until he gets to 150, and this gives him 10 and a remainder 2. Hence the answer is 410 with a remainder 2.
A few mathematicians, however, such as Hero of Alexandria, Theon, and Eutocius, multiplied and divided in what is essentially the same way as we do. Thus to multiply 18 by 13 they proceeded as follows:--
iota + iota &= (iota + )(iota + ) & 13×18&=(10+3) (10+8) &=iota(iota + ) + (iota + )& &=10 (10+8) +3 (10+8) &= + + + & &=100+80+30+24 &= & &=234
I suspect that the last step, in which they had to add four numbers together, was obtained by the aid of the abacus.
These, however, were men of exceptional genius, and we must recollect that for all ordinary purposes the art of calculation was performed only by the use of the abacus and the multiplication table, while the term arithmetic was confined to the theories of ratio, proportion, and of numbers.
All the systems here described were more or less clumsy, and they have been displaced among civilized races by the Arabic system in which there are ten digits or symbols, namely, nine for the first nine numbers and another for zero. In this system an integral number is denoted by a succession of digits, each digit representing the product of that digit and a power of ten, and the number being equal to the sum of these products. Thus, by means of the local value attached to nine symbols and a symbol for zero, any number in the decimal scale of notation can be expressed. The history of the development of the science of arithmetic with this notation will be considered below in chapter xi.
SECOND PERIODMathematics of the Middle Ages and Renaissancemiddle
This period begins about the sixth century, and may be said to end with the invention of analytical geometry and of the infinitesimal calculus. The characteristic feature of this period is the creation or development of modern arithmetic, algebra, and trigonometry.
In this period I consider first, in chapter viii, the rise of learning in Western Europe, and the mathematics of the middle ages. Next, in chapter ix, I discuss the nature and history of Hindoo and Arabian mathematics, and in chapter x their introduction into Europe. Then, in chapter xi, I trace the subsequent progress of arithmetic to the year 1637. Next, in chapter xii, I treat of the general history of mathematics during the renaissance, from the invention of printing to the beginning of the seventeenth century, say, from 1450 to 1637; this contains an account of the commencement of the modern treatment of arithmetic, algebra, and trigonometry. Lastly, in chapter xiii, I consider the revival of interest in mechanics, experimental methods, and pure geometry which marks the last few years of this period, and serves as a connecting link between the mathematics of the renaissance and the mathematics of modern times.
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End of part 10 of 28
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