These selected mathematical writings cover the years when the foundations were laid for the theory of numbers, analytic geometry, and the calculus. This book includes well-known works and also other more unusual selections. Introductory explanations put each excerpt in historical and mathematical context.
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A Source Book in Mathematics, 1200â"1800
By D. J. Struik
PRINCETON UNIVERSITY PRESSCopyright © 1986 Princeton University Press
All rights reserved.
The study of mathematics in medieval Latin Europe, after the eleventh century, was stimulated by Latin translations from the Greek and, especially, from the Arabic. They were prepared in those places where the contact between the Christian and the Islamic civilizations was the most intimate, notably in Sicily, Southern Italy, and Spain. Some of the most prolific translators were Adelard of Bath, Robert of Chester, and Gerhard of Cremona in the twelfth century and Johannes Campanus in the thirteenth. In this way Latin Europe became acquainted with the geometry of Euclid (c. 300 B.C.), originally composed in Greek (and previously known only through secondhand abstracts), and the arithmetic and algebra developed in the countries of Islam. Here the author with perhaps the greatest influence was Mohammed Al-Khwarizmi, who worked at Bagdad and wrote his Arabic in the early part of the ninth century. His name is preserved in our term "algorithm," and our word "algebra" is derived from the Arabic title of Mohammed's book on equations (see Chapter II).
It was through Islamic channels that Latin Europe became acquainted with our present decimal position system of numbers, based on the ten symbols which we now write 1, 2, 3, 4, 5, 6, 7, 8, 9, 0. Their gradual penetration into Europe may have come about, in ways only partially traceable, along the trade routes that connected the Christian and the Islamic worlds. It was also accomplished through scholarly manuscripts, among which those by Al-Khwarizmi and the learned merchant Leonardo of Pisa had a considerable reputation. Leonardo's Liber abaci (1202) begins with the introduction of the ten symbols and then develops an arithmetic based on these symbols, followed by a theory of equations.
With Fermat, in the first half of the seventeenth century, begins the study of the abstract theory of numbers. But it was not until a century later that a first-class mathematician accepted Fermat's challenge. Euler's work, starting around 1730, marks the beginning of the period of continual research in number theory. But for many years Euler worked almost alone in this field. Only in the second half of the eighteenth century was he joined by some others, such as Lagrange and Legendre. This phase can be said to have ended with the great texts by Legendre (1797–98) and Gauss (1801). The work of the young Gauss inaugurates a new phase in the history of number theory.
In the choice of the selections, especially those of Euler, we had to be fairly arbitrary. We decided to concentrate on power residues. The reader who wants to know more about eighteenth-century number theory should consult M. Cantor, Vorlesungen über Geschichte der Mathematik, vol. Ill (Teubner, Leipzig, 1898) and especially, in vol. IV (1908), the article by F. Cajori. Further details can be found in the three volumes of L. E. Dickson, History of the theory of numbers (Carnegie Institution, Washington, D.C., 1919–1923; 2d ed., 1934).
Among the problems solved in the Liber abaci one has acquired considerable fame. It may well have been invented by Leonardo, and with it we open our collection of texts.
1 LEONARDO OF PISA. THE RABBIT PROBLEM
In Leonardo of Pisa, also called Fibonacci, we meet the first outstanding mathematician of the Latin Middle Ages. He was a merchant of Pisa who traveled widely in the world of Islam, and took the opportunity of studying Arabic mathematical writings. His work is in the spirit of the Arabic mathematics of his day, but also reveals his own position as an independent thinker. Leonardo's Liber abaci (1202, revised 1228) circulated widely in manuscript, but was published only in 1857: Scritti di Leonardo Pisano (pubbl. da B. Boncompagni, Tipografia delle scienze matematiche e fisiche, Rome; 2 vols., 459 pp.). It is a voluminous compendium on arithmetic and its mercantile practice (even finger counting), the theory of linear, quadratic, and simultaneous sets of equations, square and cube roots. One of the principal features of the book is that, from the first page on, Leonardo introduces and uses the decimal position system. The first chapter opens with the sentence: "These are the nine figures of the Indians
9 8 7 6 5 4 3 2 1
With these nine figures, and with this sign 0 which in Arabic is called zephirum,1 any number can be written, as will below be demonstrated."
We confine ourselves here to presenting, in translation from the Latin, two interesting sections from the Liber abaci that may very well be original contributions. Since the first introduces paria coniculorum, we know it as the rabbit problem. It stands by itself (vol. I, 283–284), sandwiched in between other problems; the one before it deals with the so-called perfect numbers 6, 28, 496, ..., and the one after with the solution of a system of four linear equations with four unknowns.
How many pairs of rabbits can be bred from one pair in one year?
A man has one pair of rabbits at a certain place entirely surrounded by a wall. We wish to know how many pairs can be bred from it in one year, if the nature of these rabbits is such that they breed every month one other pair and begin to breed in the second month after their birth. Let the first pair breed a pair in the first month, then duplicate it and there will be 2 pairs in a month. From these pairs one, namely the first, breeds a pair in the second month, and thus there are 3 pairs in the second month. From these in one month two will become pregnant, so that in the third month 2 pairs of rabbits will be born. Thus there are 5 pairs in this month. From these in the same month 3 will be pregnant, so that in the fourth month there will be 8 pairs. From these pairs 5 will breed 5 other pairs, which added to the 8 pairs gives 13 pairs in the fifth month, from which 5 pairs (which were bred in that same month) will not conceive in that month, but the other 8 will be pregnant. Thus there will be 21 pairs in the sixth month. When we add to these the 13 pairs that are bred in the 7th month, then there will be in that month 34 pairs ... [and so on, 55, 89, 144, 233, 377, ...]. Finally there will be 377. And this number of pairs has been born from the first-mentioned pair at the given place in one year. You can see in the margin how we have done this, namely by combining the first number with the second, hence 1 and 2, and the second with the third, and the third with the fourth ... At last we combine the 10th with the 11th, hence 144 with 233, and we have the sum of the above-mentioned rabbits, namely 377, and in this way you can do it for the case of infinite numbers of months.
Here is a section (I, 24) in which Leonardo introduces a kind of continued fraction, which he writes eca/fdb, or in our notation:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Below some line [branchlet, virgula] let there be 2, 6, 10, and above the 2 be 1, above 6 be 5, and above 10 be 7, which appears as 1/2 5/6 7/10. The 7 above 10 at the head of the line represents seven-tenths and the 5 above 6 denotes five-sixths of the decimal part and 1 above 2 denotes half of the sixth of the decimal part ... We shall say that fractions which are on a branch [virga] will be in grades, so that the first grade of them is the fraction which is at the head of the branch to the right. The second grade is the fraction that follows towards the left. For instance, in the branch above, hence in 1/2 5/6 7/10, the 7/10 are in the first grade of this branch, 5/6 in the second one ...
2 RECORDE. ELEMENTARY ARITHMETIC
With the growing interest in mercantile reckoning and the spread of printing, the number of textbooks of elementary arithmetic increased rapidly from the latter half of the fifteenth century onward. Their character has been described in L. C. Karpinski, The history of arithmetic (Rand McNally, Chicago, 1925) and in Smith, History of mathematics, II, chaps. 1–3, with many illustrations. Smith, Source booh, 1–12, has an English translation of a section of the so-called Treviso arithmetic of 1478 (the first printed arithmetic). Typical of all is their introduction to the art of reckoning with the aid of the decimal position system, with digits almost or exactly the same as those we use. Many books have chapters on finger reckoning and on the use of counters for computation on an abacus. Notations for addition, subtraction, multiplication, and division still vary, though the use of + and — for addition and subtraction is fairly common. As an example we present here, in facsimile, some pages of the first arithmetic printed in the English language, The ground of artes by Robert Recorde (c. 1510-1558). Recorde, a Cambridge M.D. and physician to Edward YI and Mary Tudor, wrote several books on mathematics and astronomy that were long in use in England. The ground of artes, first published in London between 1540 and 1542 (the oldest extant edition has the date 1543), was regularly reprinted and reedited; there exists an edition of 1699.
In the pages that we reproduce (Figs. 1, 2, 3) we see how Recorde performed division in Arabic numerals, and how he taught addition by means of counters, which have long been in use and are still popular in Russia, Japan, and China. In the United States they are used by Chinese laundrymen and restaurant workers, and on baby pens. Recorde used the + and — signs, and in his algebra, The whetstone of witte (London, 1557), he introduced our sign for equality:
I will sette as I doe often in woorke use, a paire of paralleles, or Gemowe lines of one lengthe, thus: [??] bicause noe.2. thynges, can be moare equalle.
In his use of the strange word "Gemowe" we see an example of Recorde's attempt to substitute English technical words for the current Latin ones. Stevin tried the same in Dutch, Kepler in German. The only one who was partly successful was Stevin. "Gemowe," also "gemew," means twin (French gémeaux, Latin gemini).
On Recorde see F. M. Clarke, "New light on Robert Recorde," I sis 8 (1926), 50–70; see also L. D. Patterson, Isis 42 (1951), 208–218.
For those who find it difficult to read the text, we transcribe here, in slightly modernized form, page 84v, beginning with the third line. It is a discussion between M, the master, and S, the scholar:
S. So is it, and also more certainer, for such as I am, that might quickly err in multiplying, especially being smally practised therein. M. Then prove in some brief example whether you can do it, and so will we make an end. S. I would divide 38468 by 24, therefore first I set the table thus. Then set I the two sums of division thus. And over the divisor I find 38, which I seek in the table, and find it not, therefore take I the next beneath it, which the table has, and that is 24, the divisor itself, against which is set 1, which I take for the quotient, which I set in his place. And now I need not to multiply the divisor by it, but only to withdraw the divisor out of the 38 that is over it, and so remains 14, as thus.
3 STEVIN. DECIMAL FRACTIONS
The introduction of decimal fractions as a common computational practice can be dated back to the Flemish pamphlet De Thiende, published at Leyden in 1585, together with a French translation, La Disme, by the Flemish mathematician Simon Stevin (1548-1620), then settled in the Northern Netherlands. It is true that decimal fractions were used by the Chinese many centuries before Stevin and that the Persian astronomer Al-Kashi used both decimal and sexagesimal fractions with great ease in his Key to arithmetic (Samarkand, early fifteenth century). It is also true that Renaissance mathematicians such as Christoff Rudolff (first half sixteenth century) occasionally used decimal fractions, in different types of notation. But the common use of decimal fractions, at any rate in European mathematics, can be directly traced to De Thiende, especially after John Napier (see p. 13) had modified Stevin's notation into the present one with the decimal point or comma.
Stevin's notation strikes us as clumsy, showing an unnecessary relation to the notation of sexagesimal fractions. However, for beginners in the difficult arts of multiplication and division, his method may have had a certain advantage. See further the introduction to the edition of De Thiende in The principal works of Simon Stevin, IIA (Swets-Zeitlinger, Amsterdam, 1958), 373-385. We take from this edition the English translation, based on that of Richard Norton and published in 1608. Another English translation, by V. Sanford, can be found in Smith, Source book, 20–34.
THE FIRST PART
Of the Definitions of the Dime.
THE FIRST DEFINITION
Dime is a kind of arithmetic, invented by the tenth progression, consisting in characters of ciphers, whereby a certain number is described and by which also all accounts which happen in human affairs are dispatched by whole numbers, without fractions or broken numbers.
Explication. Let the certain number be one thousand one hundred and eleven, described by the characters of ciphers thus 1111, in which it appears that each 1 is the 10th part of his precedent character 1; likewise in 2378 each unity of 8 is the tenth of each unity of 7, and so of all the others. But because it is convenient that the things whereof we would speak have names, and that this manner of computation is found by the consideration of such tenth or dime progression, that is that it consists therein entirely, as shall hereafter appear, we call this treatise fitly by the name of Dime, whereby all accounts happening in the affairs of man may be wrought and effected without fractions or broken numbers, as hereafter appears.
THE SECOND DEFINITION
Every number propounded is called COMMENCEMENT, whose sign is thus [??].
Explication. By example, a certain number is propounded of three hundred sixty-four: we call them the 364 commencements, described thus 364[??], and so of all other like.
THE THIRD DEFINITION
And each tenth part of the unity of the COMMENCEMENT we call the PRIME, whose sign is thus (1), and each tenth part of the unity of the prime we call the SECOND, whose sign is (2), and so of the other: each tenth part of the unity of the precedent sign, always in order one further.
Explication. As 3(1) 7(2) 5(3) 9(4), that is to say: 3 primes, 7 seconds, 5 thirds, 9 fourths, and so proceeding infinitely, but to speak of their value, you may note that according to this definition the said numbers are 3/10, 7/100, 8/1000, 9/10000, together 3759/10000, and likewise 8(0) 9(1) 3(2) 7(3) are worth 8, 9/10, 2/100, 7/1000, together 8 937/10000, and so of other like. Also you may understand that in this dime we use no fractions, and that the multitude of signs, except (0), never exceed 9, as for example not 7(1) 12(2), but in their place 8(1) 2(2), for they value as much.
THE FOURTH DEFINITION
The numbers of the second and third definitions beforegoing are generally called DIME NUMBERS.
The End of the Definitions
THE SECOND PART OF THE DIME.
Of the Operation or Practice.
THE FIRST PROPOSITION: OF ADDITION
Dime numbers being given, how to add them to find their sum.
Excerpted from A Source Book in Mathematics, 1200â"1800 by D. J. Struik. Copyright © 1986 Princeton University Press. Excerpted by permission of PRINCETON UNIVERSITY PRESS.
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Table of Contents
- FrontMatter, pg. i
- ERRATA ET ADDENDA, pg. vii
- PREFACE, pg. ix
- CONTENTS, pg. xi
- ABBREVIATIONS OF TITLES, pg. xiv
- CHAPTER I ARITHMETIC, pg. 1
- CHAPTER II. ALGEBRA, pg. 55
- CHAPTER III. GEOMETRY, pg. 133
- CHAPTER IV. ANALYSIS BEFORE NEWTON AND LEIBNIZ, pg. 188
- CHAPTER V. NEWTON, LEIBNIZ, AND THEIR SCHOOL, pg. 270
- INDEX, pg. 421