A Short Account of the History of Mathematics · W. W. Rouse Ball
See the article on Arithmetic by G. Peacock in the Encyclopaedia Metropolitana, vol. i, London, 1845; Arithmetical Books by A. De Morgan, London, 1847; and an article by P. Treutlein of Karlsruhe, in the Zeitschrift für Mathematik, 1877, vol. xxii, supplement, pp. 1--100.
We have seen in the last chapter that by the end of the thirteenth century the Arabic arithmetic had been fairly introduced into Europe and was practised by the side of the older arithmetic which was founded on the work of Boethius. It will be convenient to depart from the chronological arrangement and briefly to sum up the subsequent history of arithmetic, but I hope, by references in the next chapter to the inventions and improvements in arithmetic here described, that I shall be able to keep the order of events and discoveries clear.
The older arithmetic consisted of two parts: practical arithmetic or the art of calculation which was taught by means of the abacus and possibly the multiplication table; and theoretical arithmetic, by which was meant the ratios and properties of numbers taught according to Boethius--a knowledge of the latter being confined to professed mathematicians. The theoretical part of this system continued to be taught till the middle of the fifteenth century, and the practical part of it was used by the smaller tradesmen in England, Germany, and France till the beginning of the seventeenth century.
The new Arabian arithmetic was called algorism or the art of Alkarismi, to distinguish it from the old or Boethian arithmetic. The text-books on algorism commenced with the Arabic system of notation, and began by giving rules for addition, subtraction, multiplication, and division; the principles of proportion were then applied to various practical problems, and the books usually concluded with general rules for many of the common problems of commerce. Algorism was in fact a mercantile arithmetic, though at first it also included all that was then known as algebra.
Thus algebra has its origin in arithmetic; and to most people the term universal arithmetic, by which it was sometimes designated, conveys a more accurate impression of its objects and methods than the more elaborate definitions of modern mathematicians--certainly better than the definition of Sir William Hamilton as the science of pure time, or that of De Morgan as the calculus of succession. No doubt logically there is a marked distinction between arithmetic and algebra, for the former is the theory of discrete magnitude, while the latter is that of continuous magnitude; but a scientific distinction such as this is of comparatively recent origin, and the idea of continuity was not introduced into mathematics before the time of Kepler.
Of course the fundamental rules of this algorism were not at first strictly proved--that is the work of advanced thought--but until the middle of the seventeenth century there was some discussion of the principles involved; since then very few arithmeticians have attempted to justify or prove the processes used, or to do more than enunciate rules and illustrate their use by numerical examples.
I have alluded frequently to the Arabic system of numerical notation. I may therefore conveniently begin by a few notes on the history of the symbols now current.
Their origin is obscure and has been much disputed. On the whole it seems probable that the symbols for the numbers 4, 5, 6, 7, and 9 (and possibly 8 too) are derived from the initial letters of the corresponding words in the Indo-Bactrian alphabet in use in the north of India perhaps 150 years before Christ; that the symbols for the numbers 2 and 3 are derived respectively from two and three parallel penstrokes written cursively; and similarly that the symbol for the number 1 represents a single penstroke. Numerals of this type were in use in India before the end of the second century of our era. The origin of the symbol for zero is unknown; it is not impossible that it was originally a dot inserted to indicate a blank space, or it may represent a closed hand, but these are mere conjectures; there is reason to believe that it was introduced in India towards the close of the fifth century of our era, but the earliest writing now extant in which it occurs is assigned to the eighth century.
Devanagari (Indian) numerals, circ. 950. Bigr -2ex
Gobar Arabic numerals, circ. 1100 (?). Bigr
From a missal, circ. 1385, of German origin. Biggr
European (probably Italian) numerals, circ. 1400. Biggr
From the Mirrour of the World, printed by Caxton in 1480. Biggr
From a Scotch calendar for 1482, probably of French origin. Biggr
The numerals used in India in the eighth century and for a long time afterwards are termed Devanagari numerals, and their forms are shewn in the first line of the table given above. These forms were slightly modified by the eastern Arabs, and the resulting symbols were again slightly modified by the western Arabs or Moors. It is perhaps probable that at first the Spanish Arabs discarded the use of the symbol for zero, and only reinserted it when they found how inconvenient the omission proved. The symbols ultimately adopted by the Arabs are termed Gobar numerals, and an idea of the forms most commonly used may be gathered from those printed in the second line of the table given above. From Spain or Barbary the Gobar numerals passed into western Europe, and they occur on a Sicilian coin as early as 1138. The further evolution of the forms of the symbols to those with which we are familiar is indicated below by facsimiles of the numerals used at different times. All the sets of numerals here represented are written from left to right and in the order 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. From 1500 onwards the symbols employed are practically the same as those now in use.
The further evolution in the East of the Gobar numerals proceeded almost independently of European influence. There are minute differences in the forms used by various writers, and in some cases alternative forms; without, however, entering into these details we may say that the numerals they commonly employed finally took the form shewn above, but the symbol there given for 4 is at the present time generally written cursively.
Leaving now the history of the symbols I proceed to discuss their introduction into general use and the development of algoristic arithmetic. I have already explained how men of science, and particularly astronomers, had become acquainted with the Arabic system by the middle of the thirteenth century. The trade of Europe during the thirteenth and fourteenth centuries was mostly in Italian hands, and the obvious advantages of the algoristic system led to its general adoption in Italy for mercantile purposes. This change was not effected, however, without considerable opposition; thus, an edict was issued at Florence in 1299 forbidding bankers to use Arabic numerals, and in 1348 the authorities of the university of Padua directed that a list should be kept of books for sale with the prices marked "non per cifras sed per literas claras."
The rapid spread of the use of Arabic numerals and arithmetic through the rest of Europe seems to have been as largely due to the makers of almanacks and calendars as to merchants and men of science. These calendars had a wide circulation in medieval times. Some of them were composed with special reference to ecclesiastical purposes, and contained the dates of the different festivals and fasts of the church for a period of some seven or eight years in advance, as well as notes on church ritual. Nearly every monastery and church of any pretensions possessed one of these. Others were written specially for the use of astrologers and physicians, and some of them contained notes on various scientific subjects, especially medicine and astronomy. Such almanacks were not then uncommon, but, since it was only rarely that they found their way into any corporate library, specimens are now rather scarce. It was the fashion to use the Arabic symbols in ecclesiastical works; while their occurrence in all astronomical tables and their Oriental origin (which savoured of magic) secured their use in calendars intended for scientific purposes. Thus the symbols were generally employed in both kinds of almanacks, and there are but few specimens of calendars issued after the year 1300 in which an explanation of the Arabic numerals is not included. Towards the middle of the fourteenth century the rules of arithmetic de algorismo were also sometimes added, and by the year 1400 we may consider that the Arabic symbols were generally known throughout Europe, and were used in most scientific and astronomical works.
Outside Italy most merchants continued, however, to keep their accounts in Roman numerals till about 1550, and monasteries and colleges till about 1650; though in both cases it is probable that in and after the fifteenth century the processes of arithmetic were performed in the algoristic manner. Arabic numerals are used in the pagination of some books issued at Venice in 1471 and 1482. No instance of a date or number being written in Arabic numerals is known to occur in any English parish register or the court rolls of any English manor before the sixteenth century; but in the rent-roll of the St Andrews Chapter, Scotland, the Arabic numerals were used in 1490. The Arabic numerals were used in Constantinople by Planudes in the fourteenth century.
The history of modern mercantile arithmetic in Europe begins then with its use by Italian merchants, and it is especially to the Florentine traders and writers that we owe its early development and improvement. It was they who invented the system of book-keeping by double entry. In this system every transaction is entered on the credit side in one ledger, and on the debtor side in another; thus, if cloth be sold to A, A's account is debited with the price, and the stock-book, containing the transactions in cloth, is credited with the amount sold. It was they, too, who arranged the problems to which arithmetic could be applied in different classes, such as rule of three, interest, profit and loss, &c. They also reduced the fundamental operations of arithmetic "to seven, in reverence," says Pacioli, "of the seven gifts of the Holy Spirit: namely, numeration, addition, subtraction, multiplication, division, raising to powers, and extraction of roots." Brahmagupta had enumerated twenty processes, besides eight subsidiary ones, and had stated that "a distinct and several knowledge of these" was "essential to all who wished to be calculators"; and, whatever may be thought of Pacioli's reason for the alteration, the consequent simplification of the elementary processes was satisfactory. It may be added that arithmetical schools were founded in various parts of Germany, especially in and after the fourteenth century, and did much towards familiarizing traders in northern and western Europe with commercial algoristic arithmetic.
The operations of algoristic arithmetic were at first very cumbersome. The chief improvements subsequently introduced into the early Italian algorism were (i) the simplification of the four fundamental processes; (ii) the introduction of signs for addition, subtraction, equality, and (though not so important) for multiplication and division; (iii) the invention of logarithms; and (iv) the use of decimals. I will consider these in succession.
(i) In addition and subtraction the Arabs usually worked from left to right. The modern plan of working from right to left is said to have been introduced by an Englishman named Garth, of whose life I can find no account. The old plan continued in partial use till about 1600; even now it would be more convenient in approximations where it is necessary to keep only a certain number of places of decimals.
The Indians and Arabs had several systems of multiplication. These were all somewhat laborious, and were made the more so as multiplication tables, if not unknown, were at any rate used but rarely. The operation was regarded as one of considerable difficulty, and the test of the accuracy of the result by "casting out the nines" was invented as a check on the correctness of the work. Various other systems of multiplication were subsequently employed in Italy, of which several examples are given by Pacioli and Tartaglia; and the use of the multiplication table--at least as far as 5 × 5 --became common. From this limited table the resulting product of the multiplication of all numbers up to 10 × 10 can be deduced by what was termed the regula ignavi. This is a statement of the identity (5+a)(5+b) = (5-a)(5-b)+10(a+b) . The rule was usually enunciated in the following form. Let the number five be represented by the open hand; the number six by the hand with one finger closed; the number seven by the hand with two fingers closed; the number eight by the hand with three fingers closed; and the number nine by the hand with four fingers closed. To multiply one number by another let the multiplier be represented by one hand, and the number multiplied by the other, according to the above convention. Then the required answer is the product of the number of fingers (counting the thumb as a finger) open in the one hand by the number of fingers open in the other together with ten times the total number of fingers closed. The system of multiplication now in use seems to have been first introduced at Florence.
1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 0 /2 & /4 & /6 & /8 & 1/0 & 1/2 & 1/4 & 1/6 & 1/8 & /0 /3 & /6 & /9 & 1/2 & 1/5 & 1/8 & 2/1 & 2/4 & 2/7 & /0 /4 & /8 & 1/2 & 1/6 & 2/0 & 2/4 & 2/8 & 3/2 & 3/6 & /0 /5 & 1/0 & 1/5 & 2/0 & 2/5 & 3/0 & 3/5 & 4/0 & 4/5 & /0 /6 & 1/2 & 1/8 & 2/4 & 3/0 & 3/6 & 4/2 & 4/8 & 5/4 & /0 /7 & 1/4 & 2/1 & 2/8 & 3/5 & 4/2 & 4/9 & 5/6 & 6/3 & /0 /8 & 1/6 & 2/4 & 3/2 & 4/0 & 4/8 & 5/6 & 6/4 & 7/2 & /0 /9 & 1/8 & 2/7 & 3/6 & 4/5 & 5/4 & 6/3 & 7/2 & 8/1 & /0
Figure 1.
7 1/4 2/1 2/8 3/5 4/2 4/9 5/6 6/3
Figure 2.
2 & 9 & 8 & 5 /4 & 1/8 & 1/6 & 1/0 /6 & 2/7 & 2/4 & 1/5 /8 & 3/6 & 3/2 & 2/0 1/0 & 4/5 & 4/0 & 2/5 1/2 & 5/4 & 4/8 & 3/0 1/4 & 6/3 & 5/6 & 3/5 1/6 & 7/2 & 6/4 & 4/0 1/8 & 8/1 & 7/2 & 4/5
Figure 3.
The difficulty which all but professed mathematicians experienced in the multiplication of large numbers led to the invention of several mechanical ways of effecting the process. Of these the most celebrated is that of Napier's rods invented in 1617. In principle it is the same as a method which had been long in use both in India and Persia, and which has been described in the diaries of several travellers, and notably in the Travels of Sir John Chardin in Persia, London, 1686. To use the method a number of rectangular slips of bone, wood, metal, or cardboard are prepared, and each of them divided by cross lines into nine little squares, a slip being generally about three inches long and a third of an inch across. In the top square one of the digits is engraved, and the results of multiplying it by 2, 3, 4, 5, 6, 7, 8, and 9 are respectively entered in the eight lower squares; where the result is a number of two digits, the ten-digit is written above and to the left of the unit-digit and separated from it by a diagonal line. The slips are usually arranged in a box. Figure 1 above represents nine such slips side by side; figure 2 shews the seventh slip, which is supposed to be taken out of the box and put by itself. Suppose we wish to multiply 2985 by 317. The process as effected by the use of these slips is as follows. The slips headed 2, 9, 8, and 5 are taken out of the box and put side by side as shewn in figure 3 above. The result of multiplying 2985 by 7 may be written thus--
2 9 8 5
7
3 5
5 6
6 3
1 4
2 0 8 9 5
Now if the reader will look at the seventh line in figure 3, he will see that the upper and lower rows of figures are respectively 1653 and 4365; moreover, these are arranged by the diagonals so that roughly the 4 is under the 6, the 3 under the 5, and the 6 under the 3; thus
1 6 5 3
4 3 6 5
The addition of these two numbers gives the required result. Hence the result of multiplying by 7, 1, and 3 can be successively determined in this way, and the required answer (namely, the product of 2985 and 317) is then obtained by addition.
The whole process was written as follows:
822985
820895 000/7
82985 00/1
8955 0/3
946245
The modification introduced by Napier in his Rabdologia, published in 1617, consisted merely in replacing each slip by a prism with square ends, which he called "a rod," each lateral face being divided and marked in the same way as one of the slips above described. These rods not only economized space, but were easier to handle, and were arranged in such a way as to facilitate the operations required.
1.5pt
1 7 9 7 8 1 7 9 7 8 1 7 9 7 8
1 2 1 2
5 9 7 8 5 9 7 8
2 1 2 1
3 8 7 8 3 8 7 8
6 6
|*5 | | |*4-| |-|*4=| 3 8 1 8 3 8 1 8
3 2
6 1 8
5 6
5 8
1 6
|*5 | |*5 | |-|-|-|=|=| 4 2
|*5 | |*5 | |*5=| 4 7 2
4 7 2 4 7 2
4 7 2 backslash 4 backslash 7 backslash 2 4 7 2
0 0 3 0 3 8
|*5=| |*5=| |*5=| 5c0pt3exFigure 1. 1c 5c Figure 2. 1c 5c Figure 3.
If multiplication was considered difficult, division was at first regarded as a feat which could be performed only by skilled mathematicians. The method commonly employed by the Arabs and Persians for the division of one number by another will be sufficiently illustrated by a concrete instance. Suppose we require to divide 17978 by 472. A sheet of paper is divided into as many vertical columns as there are figures in the number to be divided. The number to be divided is written at the top and the divisor at the bottom; the first digit of each number being placed at the left-hand side of the paper. Then, taking the left-hand column, 4 will go into 1 no times, hence the first figure in the dividend is 0, which is written under the last figure of the divisor. This is represented in figure 1 above. Next (see figure 2) rewrite the 472 immediately above its former position, but shifted one place to the right, and cancel the old figures. Then 4 will go into 17 four times; but, as on trial it is found that 4 is too big for the first digit of the dividend, 3 is selected; 3 is therefore written below the last digit of the divisor and next to the digit of the dividend last found. The process of multiplying the divisor by 3 and subtracting from the number to be divided is indicated in figure 2, and shews that the remainder is 3818. A similar process is then repeated, that is, 472 is divided into 3818, shewing that the quotient is 38 and the remainder 42. This is represented in figure 3, which shews the whole operation.
The method described above never found much favour in Italy. The present system was in use there as early as the beginning of the fourteenth century, but the method generally employed was that known as the galley or scratch system. The following example from Tartaglia, in which it is required to divide 1330 by 84, will serve to illustrate this method: the arithmetic given by Tartaglia is shewn below, where numbers in thin type are supposed to be scratched out in the course of the work.
0 7
4 9
0 5 9 0
1 3 3 0 ( 15
8 4 4
8
The process is as follows. First write the 84 beneath the 1330, as indicated below, then 84 will go into 133 once, hence the first figure in the quotient is 1. Now 1 × 8 = 8 , which subtracted from 13 leaves 5. Write this above the 13, and cancel the 13 and the 8, and we have as the result of the first step
5
1 3 3 0 ( 1
8 4
Next, 1 × 4 = 4 , which subtracted from 53 leaves 49. Insert the 49, and cancel the 53 and the 4, and we have as the next step
4
5 9
1 3 3 0 ( 1
8 4
which shews a remainder 490.
We have now to divide 490 by 84. Hence the next figure in the quotient will be 5, and re-writing the divisor we have
4
5 9
1 3 3 0 ( 15
8 4 4
8
Then 5 × 8 = 40 , and this subtracted from 49 leaves 9. Insert the 9, and cancel the 49 and the 8, and we have the following result
4 9
5 9
1 3 3 0 ( 15
8 4 4
8
Next 5 × 4 = 20 , and this subtracted from 90 leaves 70. Insert the 70, and cancel the 90 and the 4, and the final result, shewing a remainder 70, is
7
4 9
5 9 0
1 3 3 0 ( 15
8 4 4
8
The three extra zeros inserted in Tartaglia's work are unnecessary, but they do not affect the result, as it is evident that a figure in the dividend may be shifted one or more places up in the same vertical column if it be convenient to do so.
The medieval writers were acquainted with the method now in use, but considered the scratch method more simple. In some cases the latter is very clumsy, as may be illustrated by the following example taken from Pacioli. The object is to divide 23400 by 100. The result is obtained thus
0
0 4 0
0 3 4 0 0
2 3 4 0 0 ( 234
1 0 0 0 0
1 0 0
1
The galley method was used in India, and the Italians may have derived it thence. In Italy it became obsolete somewhere about 1600; but it continued in partial use for at least another century in other countries. I should add that Napier's rods can be, and sometimes were used to obtain the result of dividing one number by another.
(ii) The signs + and - to indicate addition and subtraction occur in Widman's arithmetic published in 1489, but were first brought into general notice, at any rate as symbols of operation, by Stifel in 1544. They occur, however, in a work by G. V. Hoecke, published at Antwerp in 1514. I believe I am correct in saying that Vieta in 1591 was the first well-known writer who used these signs consistently throughout his work, and that it was not until the beginning of the seventeenth century that they became recognized as well-known symbols. The sign = to denote equality was introduced by Record in 1557.
(iii) The invention of logarithms, without which many of the numerical calculations which have constantly to be made would be practically impossible, was due to Napier of Merchiston. The first public announcement of the discovery was made in his Mirifici Logarithmorum Canonis Descriptio, published in 1614, and of which an English translation was issued in the following year; but he had privately communicated a summary of his results to Tycho Brahe as early as 1594. In this work Napier explains the nature of logarithms by a comparison between corresponding terms of an arithmetical and geometrical progression. He illustrates their use, and gives tables of the logarithms of the sines and tangents of all angles in the first quadrant, for differences of every minute, calculated to seven places of decimals. His definition of the logarithm of a quantity n was what we should now express by 10^7log e(10^7/n) . This work is the more interesting to us as it is the first valuable contribution to the progress of mathematics which was made by any British writer. The method by which the logarithms were calculated was explained in the Constructio, a posthumous work issued in 1619: it seems to have been very laborious, and depended either on direct involution and evolution, or on the formation of geometrical means. The method by finding the approximate value of a convergent series was introduced by Newton, Cotes, and Euler. Napier had determined to change the base to one which was a power of 10, but died before he could effect it.
The rapid recognition throughout Europe of the advantages of using logarithms in practical calculations was mainly due to Briggs, who was one of the earliest to recognize the value of Napier's invention. Briggs at once realized that the base to which Napier's logarithms were calculated was inconvenient; he accordingly visited Napier in 1616, and urged the change to a decimal base, which was recognized by Napier as an improvement. On his return Briggs immediately set to work to calculate tables to a decimal base, and in 1617 he brought out a table of logarithms of the numbers from 1 to 1000 calculated to fourteen places of decimals.
It would seem that J. Bürgi, independently of Napier, had constructed before 1611 a table of antilogarithms of a series of natural numbers: this was published in 1620. In the same year a table of the logarithms, to seven places of decimals, of the sines and tangents of angles in the first quadrant was brought out by Edmund Gunter, one of the Gresham lecturers. Four years later the latter mathematician introduced a "line of numbers," which provided a mechanical method for finding the product of two numbers: this was the precursor of the slide-rule, first described by Oughtred in 1632. In 1624, Briggs published tables of the logarithms of some additional numbers and of various trigonometrical functions. His logarithms of the natural numbers are equal to those to the base 10 when multiplied by 10^8 , and of the sines of angles to those to the base 10 when multiplied by 10^12 . The calculation of the logarithms of 70,000 numbers which had been omitted by Briggs from his tables of 1624 was performed by Adrian Vlacq and published in 1628: with this addition the table gave the logarithms of all numbers from 1 to 101,000.
The Arithmetica Logarithmica of Briggs and Vlacq are substantially the same as the existing tables: parts have at different times been recalculated, but no tables of an equal range and fulness entirely founded on fresh computations have been published since. These tables were supplemented by Briggs's Trigonometrica Britannica, which contains tables not only of the logarithms of the trigonometrical functions, but also of their natural values: it was published posthumously in 1633. A table of logarithms to the base e of the numbers from 1 to 1000 and of the sines, tangents, and secants of angles in the first quadrant was published by John Speidell at London as early as 1619, but of course these were not so useful in practical calculations as those to the base 10. By 1630 tables of logarithms were in general use.
(iv) The introduction of the decimal notation for fractions is due to Pitiscus, in whose Tables, 1608 and 1612, it appears; it was employed in 1616 in the English translation of Napier's Descriptio, and occurs in the logarithmic Tables published by Briggs in 1617, after which date its use may be taken to be established. The idea was not new, for Stevinus had in 1585 used a somewhat similar notation, writing a number such as 25·379 either in the form 25, 3' 7" 9"' , or in the form 25 0 3 1 7 2 9 3. This latter notation was also used by Napier in 1617 in his essay on Rods, and by Rudolff. These writers employed it only as a concise way of stating results, and made no use of it as an operative form; probably Briggs did more than any other writer to establish its use as an operative form. The subject is one of much interest, and a considerable body of literature has grown up about it. Some of the facts are in dispute, and the above statement must only be taken to represent my general conclusions. The reader interested in the subject may consult the Napier Tercentenary Volume issued by the Edinburgh Royal Society in 1915. Before the sixteenth century fractions were commonly written in the sexagesimal notation.
· · ·
End of part 14 of 28
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