Rozhanskaya1987 (1)

On a Mathematical Problem in al-Khaini's Book of the Balance of Wisdom M AR l A M R O Z H AN S K A Y A Institute for the History of Science and Technology USSR Academy of Science Staropanski Per. 1/5 103012 MOSCOWK-12, USSR

SECTION 1 BU'L-FATH 'Abd

al-RahmPn al-Mansiir al-KlGzini was an eminent scholar who lived in the first half of the 12th century.' He worked at Marw at the court of Sultan Sanjar and wrote in Arabic rather than in Persian. The compass of his scientific interests was extremely wide: it included astronomy and design of astronomical instruments, mechanics and mathematics. A few of his extant writings are a tract on astronomical instruments,z a monumental astronomical handbook with tables (zij) called al-Zq ~ l - S a n j a f ia, ~work evidently compiled under considerable influence of al-Birtini's Zij called alQiiniin al-Mas'iidi, and finally, the work which concerns us here, Kitiib Mizcn al-hikma, The book of the balance of wisdom. In the middle of the 19th century the Russian orientalist Khanikov discovered a manuscript of this work in Iran, and published part of it with an English translation and commentary.* German translations of some of the fragments from Khanikov's edition have also been published.5 Three manuscripts of al-KhPzini's treatise are now known: the "Khanikov manuscript" deposited at the Leningrad Saltykov-SchedrinPublic Library (Khanikov Collection, item no. 17) and two other manuscripts discovered in India (in Bombay and Hyderabad).The 1941Hyderabad edition of the Balance ofwisdom6was based on these two manuscripts preserved in India. Finally, a Russian translation of the complete text of the work based on all three manuscripts, together with a commentary, was published in Moscow in 1983.' The Balance of wisdom is a comprehensive explication of the problems of theoretical and practical statics discussed by contemporary scientists. It is composed just as modern scholarly monographs are: the author first reviewed everything done in this branch of science by earlier scholars, both Greek and Muslim, and then he presented his own findings. Al-KhPzini's book thus represents an entire stage in the history of statics: it enables its students to assess the achievements of the scholars in the medieval East who dealt with these problems.

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A characteristic trait of the Balance of wisdom is the use of mathematical methods for solving numerous mechanical problems. I shall consider one such problem on the selection of a minimal number of weights for weighing a certain load, in other words, the so-called "problem of weighing," well known in the history of mathematics. Al-Khgzini devotes the first chapter of Maqiifa 6 of his treatise to this problem. SECTION 2

Al-Khazini begins by explaining what he calls the "universally adopted method of selecting weights to weigh any arbitrary load. He says: 'At present, people follow the natural sequence of the orders of numbers," that is, the decimal system. Three numbers are chosen from each order: from the order of units, 1, 2, and 5; from the order of tens, 10, 20, and 50; from the order of hundreds, 100, 200, and 500 (Arabic text (al-Khazini 1941), 109). Thus, the standard set consisted of nine weights whose total added up to 888 units. Such a selection of weights, however, leads to some inconvenience. Placing all the weights only on one scale, it was indeed possible to weigh a load of, say, three units (since 3 = 1 2), but not those of four or nine units (since4 = 2 2 = 1 1 2 a n d 9 = 1 1 i- 2 5 =2 2 5). By using only one generally adopted set of weights and placing all the weights only on one scale, it was possible to weigh and, for that matter, only in one single manner, any loads from 1to 888 (for example, 6 = 5 1,8 = 5 i2 1,etc.), with only the following exceptions: those of 4 and 9 units among the first ten, 14 and 19 among the second ten, 24 and 29 among the third ten, . . . , 104 and 109,114 and 119, . . . , 204 and 209, . . . etc. To weigh these exceptional loads it was necessary to possess two universally adopted sets, i.e., 18 weights totalling 888 X 2 = 1,776 units. It was of course possible to distribute the weights on both scales of the balance. In this instance, one batch of weights was sufficient, but the procedure was no longer unique since one and the same load could have been weighed in many ways. For example, a load of 3 dirhams (1dirham = 4.23 grams) could be weighed in four ways, viz.:

+

+ +

+

+

+

+ +

+

+

1

+ 2 = 5 - 2 = 10 - 5 - 2 = 20 - 10 - 5 - 2.

To summarize, according to al-KhHzini two essential faults were inherent in the standard set of weights. If weights were placed only on one scale of the balance, two sets would be needed to weigh any load from 1to 888; if, however, the weights could have been placed on either scale, the manner of weighing would not be unique. What was needed to eliminate these deficien-

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cies, al-Khiizini declared, was an entirely different principle for selecting the weights. He explains: If we want to choose the weights in accordance with the natural order of numbers, and to place them on one scale without opposinp them by weights on the other scale, it is necessary to select [them]in accordance with duplication. Let US choose the first [weight] equal to 1 [unit], the second [one equal to] 2, the third [one equal to] 4 . . . [He continues the enumeration to the tenth one equal to 512.1 The quantity of weights will exceed by unity their quantity in a standard set. If, however, we want to use a lesser [quantity of weights] this is achieved by means of opposing. [In this instance,] we choose the weights beginning with unity. We then multiply it [the unity] by three and continue in the same way. The first weight is equal to one 1i.e. to unity of weight], the second [one is equal to] 3, the third [one is equal to] 9 . . . [He continues the enumeration to the seventh one equal to 729.1 The quantity of these weights is less by two than that in a universally adopted set [text, 1091.

Thus, for the first case, when the weights are placed only on one scale of the balance, al-Khiizini suggests choosing not the standard set but rather a batch of ten weights selected so as to form a geometric progression with first term one and common ratio two. The quantity of weights would be one more than in the universal set, but, on the other hand, the batch enables one to weigh up any load up to P = 1,023 (this being the sum of the first ten terms of this progression). In the second instance, when the weights are distributed on both scales of the balance, the set consists only of seven weights (two less than in the standard batch) chosen so as to constitute a geometric progression with first term one and common ration three. The seven weights are then sufficient to weigh any load up to P = 1,093 units. In both cases the problem is thus reduced to finding the least quantity of weights sufficient to weigh any integer load less than or equal to 1,093units. The mathematical meaning of the problem is quite transparent. Represent, in the first instance, load P as

P

= ao i- alpl i- azpz

+ . . . iaspg,:gaipi < 1,023

where pi

=

1, 2, 22,

..,,

29

and ai = 0 or 1.

Then al-Khzzini's solution is tantamount to expressing numbers P in the binary number system. In the second case

P

= a.

+ alpl + azpz + . . . +

i-6 a6Pbr

& aipi < 1,097

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where pi

=

1, 3, 32,

. . . , 36 and ai

=

-I, 0, or I.

Here the solution is equivalent to representing numbers P in the ternary s ~ s t e m . ~ It is possible to consider both these instances as particular cases of representing integers n by an algebraic sum of different powers of a certain integer, i.e. of expressing natural number n by a sum of m (m < n) natural numbers. (In al-Khiizini’s first case m = 10; in his second case, m = 7.) Let us return, however, to al-Khazini’s objections to the generally adopted set of weights and his own criteria for choosing a set. Exactly this point characterizes the standard of rigor in formulating the problem. Al-Khiizini demands that only one set of weights be used, and, consequently, that in weighing any integral load up to and including a given value, not a single weight be used more than once. In present day terminology, this problem might be formulated as follows: to find such a subset M* of a set M of natural numbers that any element n in M could be expanded in a sum of m (m < n) elements belonging to M*. Besides, subset M* must consist of a minimal number of noncoincident elements. This is a problem well known in number theory. Al-Khazini remarks: ”If the demand formulated above is not met, this would (become) quite another problem belonging to another branch of mathematics” (text, 110). It may be assumed that by mentioning “another problem” al-Khiizini was probably thinking about finding the number of ways to weigh a certain load by means of a given set of weights, or about determining the number of expansions of a given integer into a sum of lesser integral numbers. His reference to “anotherbranch of mathematics”may mean that he had in mind methods peculiar to the solution of linear indeterminate equations such as had been applied by Indian mathematicians in the 5th and 6th centuries and by Muslim mathematicians in the 9th and 10th centuries.10

SECTION 3

The problem of weighing obviously originated in the East in remote antiquity. Its sources are traceable in Indian civilization. During excavations at Mohenjo-daro and Harappa, archaeologists have discovered a large quantity of weights of various denominations. They testify that the system of weights was then based on duplication. The small weights found there were of 1,2, 4, . . . , 64 units.ll It is still unknown whether Sumerian and Babylonian mathematicians included the problem of weighing, or problems similar thereto, within the compass of their inquiries. A comprehensive study of the immense metrolog-

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ical data discovered during excavations would indeed help to answer this question, the more so as archaeological findings convincingly testify in favor of sufficiently close ties having existed between the Indian and the Sumerian civilizations.12 The sources of the problem of weighing can also be traced in the method of multiplying integers in Egyptian mathematics. The relevant trick consisted in expanding one of the factors into a sum of terms of the type 2k, where k are odd integers.13 Here too, the study of metrological evidence would provide additional information. Plato, when describing the system of the world in his Timaeus (35c-36a), represented integers by sums of powers of numbers 2 and 3. He characterized the relations between orbits of the seven celestial bodies (the Moon, the Sun, and the five planets then known) by numbers 1,2,3,4,8,9, and 27, the terms of two geometric progressions, 1, 2, 22, 23, and 1, 3, 32,33.1* This fact can be regarded as evidence in favor of Plato’s cosmological conceptions being based upon Pythagorean sources. And of course the great influence of Egyptian and Greek mathematics on the making of this science in the medieval East is generally known. Finally, I note that a table which included powers of numbers 2 and 3 is contained in a medieval Chinese mathematical treatise. l5 Obviously, the problem of weighing was widely disseminated over the medieval Near and Middle East and al-Khiizini was not the first author to include it in his work. According to our present knowledge, the first mention of this problem dates back to the second half of the 11th century. The Iranian scholar Muhammad ibn Ayyiib al-Tabari formulated it among many others in his Persian treatise MiftZih al-mu‘ZimalZit, Key to (commercial] deals.16The set of weights in his version of the problem contained ten pieces, namely those of 1,3, 3 2 , . . , , 3 9 units. Thus, the batch enabled one to weigh all loads up to and including 1,000 units. The second source now known to mention the problem of weighing is the Balance of wisdom itself. Leonardo Pisano (Leonard of Pisa) (1180-1240) in his Liber abaci appears to have been the first to consider the problem in Western Eur0pe.1~He formulated it as follows: Determine four weights capable of measuring any load up to and including 40 units supposing that the weights could be placed either on one or on both scales of the balance. Leonardo’s answer was: 1,3, 9, and 27. This is al-Khiizini’s second case, with m = 4. Leonardo, however, generalized the problem, thus making the next step in its solution. In addition he considered the case when the load might weigh more than 40 units. Then, he observed, it is possible to include a fifth weight of 81 = 3 4 units so that the new set would become capable of weighing loads up to and including 121 units, etc. ad infinitum. Thus Leonardo actually formulated and solved alKhazini’s second instance for any n.18

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The problem of weighing is contained in Chapter 12 of the Liber abaci along with other problems, such as the summation of arithmetic and geometric and recurrent (Fibonacci) series. Some of these definitely originated in the East, such as the traditional problem on the summation of different powers of 7, one version of which had existed in ancient Egypt. According to one point of view, the Egyptian problem might have found its way (directly or via antique sources) into Arabic mathematical literature and thence into the Liber abaci.19 At any rate, konardo‘s sources could have hardly included the Balance of wisdom since it was unknown in medieval Europe. Leonardo’s contemporary, Jordanus Nemorarius, wrote a treatise called Demonstratio Jordani de algorismo.20 It was devoted to the substantiation of mathematical operations on integers. The contribution included a theorem tantamount to the inequality 9

+ 9 . 10 + 9

*

102

+ ... + 9.10

n-1<

10”’

which could be interpreted as a version of the problem of weighing, or as representing a given integer in the decimal number system. N. Chuquet (15th century) and L. Pacioli (14451-15141) considered the same version as konardo, but with m = 5. In the 16th century, N. Tartaglia (1499-1557), M. Stifel (14861-15671) and Gemma Frisius (in 1540)treated both cases of the problem, taking m = 6.21Other authors followed suit and in 1612 Bachet de Meziriac (1581-1638) was the first to publish a printed work on the problem and it is now called after him.22 Bachet formulated his problem in this way: Estant proposCe telle quantifb qu’on voudra pesant un nombre de livres depuis 1 iusques a 40 inclusivement (sans toutefois adrnettre les fractions) on dernande cornbien de pois pour les rnoins il faudroit employer li cet effect. [It is required to weigh all integral loads from 1to 40 livres. What would be the least quantity of weights necessary for the operation, and how should they be chosen?]

In the 18th century Euler offered a rigorous solution to the problem. He considered it as a particular case of partitioning numbers into sums (partitio nurnerorum), i.e. of expressing natural numbers n as sums of m (m < n) natural numbers.23He expressly devoted a series of memoirs to this more general problem. His investigations are known to have been started from a problem proposed to him in 1740 by the Berlin mathematician Philippe Naude (1684-1745). The problem was this: In how many ways can an integer be expressed as a sum of two, three, four, and more numbers? Euler gave its analytical solution the same year. Essentially, this is a more general case of alKhZzini’s “other problem.” Euler adduced both the problem of weighing and the ”other problem” as examples.

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Before Euler, however, not a single author amongst those I have mentioned above formulated the problem with the same degree of rigor as al-Khiizini did in the 12th century. ACKNOWLEDGMENT This paper was translated into English by Dr. O.B. Sheynin. NOTES 1. On al-Khazini, see the article by R. E. Hall in DSB. 2. On this, see Sayili. 3. O n this, see Kennedy, 129, no. 27. 4. See Khanikov. 5. Wiedemann and Ibel. 6. Listed as al-Khazini 1941. 7. See al-Khlzini 1983. 8. I find it remarkable that al-Khlzini uses here the term al-rnuqibala. Mathematicians in the medieval East used the same term to denote an algebraic operation of transferring terms containing the unknown onto one side of the equation and free terms onto the other side. 9. This method of presenting numbers in the ternary system differs from the modern method only insofar as numbers - 1 , O and 1, were used rather than 0,1 and 2. Simple transformations are sufficient to reduce one representation into the other and vice vena. Of course, this remark does not mean that al-Khazini really wanted to record numbers in the ternary system. Indeed, the mathematical equivalence of two specific problems does not in any way imply their historical identity. 10. On this, see, for example, Youschkevitsch, 143-47 and 221-23. 11. It is worth mentioning that ancient builders of these settlements have used bricks with the ratios of their dimensions being 1:2:4 or 1:3:9. See in addition Volodarsky, 13. 12. Knorozov, 4-15. 13. Vygodsky, 59-60. 14. Rozhansky, 257 15. Berezkina, 25. 16. Tropfke, 125, 635. 17. Leonardo, 297-98. 18. It is, however, possible that al-Khlzini knew the solution even for a larger number of terms. In this instance he simply compares his set with the standard batch used for weighing loads up to and including 888 units. Maybe for this reason he cut off both his series. Leonardo might have become acquainted with the problem of weighing either during his travels in the East or while studying mathematical literature written in Arabic. He is known to have transacted commercial deals in the East where he acquired knowledge of the working principle of balances and of the practice of weighing. 19. Vygodsky, 59-65. 20. See Enestrom and Matvievskaya. 21. See Knobloch for details. 22. See Bachet. 23. See Euler.

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BIBLIOGRAPHY Berezkina, E. I. Mathematika drevnego Kitaya (= Mathematics in ancient China). Moscow: 1980 Nauka. DSB Dictionary of scientific biography, 14 vols. and 2 supp. vols., New York: Charles Scribner's Sons, 1970-80. Enestrom, G. Uber die Demonstratio Iordani de algorismo. Biblioteca mathematica, F. 3,7: 1906-7 24-37. Euler, L. lntroductio in analysin infinitorum, especially Cap. 16: De partitione 1748 numerorum, 313-38.Lausanne: M.M. Bousquet. Ibel, Th. Die Waage im Attertum und Mittefalter, Inaug. Diss. Univ., Erlangen. 1908 Kennedy, E. S. A survey of Islamic astronomical tables. Transactions of the American 1956 Philosophical Society, N.S. 46:2: 123-77. Khanikoff, N. (Khanikov) Analysis and extracts of Kit& miziin al-kikrna (Book of the balance of 1859 wisdom), written by al-KhPzini in the twelfth century. Journal o f the American Oriental Society, 6: 1-128. al-KhPzini, 'Abd al-Rahman KitZb Miz'iin al-kikma (= The book of the batance of wisdom). Hyderabad: Osmania Oriental Publications Bureau, 1941. Kniga vessov mudrosti (= The book of the balance ofwisdom), In Nauchnoye nasleaktvo (= Scientific heritage), vol. 6:lz istoriyiphysiko - mathematicheskih nauk na srednevekovom Vostoke (= Essays on the history of the physical and mathematical sciences in the medieual East), 15-140.Moscow: Nauka, 1983. Knobloch, E. Zur Uberlieferung des Bachetischen Gewichtsproblems. Sudhofs Archiv, 57:2: 1973 142-51. Knorozov, Y. V. Klassifikatsiya proto-indiyskih nadpissey (= Classification of proto-Indian in1975 scriptions), In Soobscheniya ob isuchenii proto - indiyskih textov (= Report on the study of Proto-Indian texts), 4-15. Moscow: Nauka. Leonard0 Pisano Scritti maternatico del secolo decimo term. Rome: Publicati da Baldassare Bon1857 compagni, vol. 1 (Liber abaci). Matvievskaya, G. I? 1971 Razvitiye ucheniya o chisle v Europe do XVIl veka (= Development of the doctrine of number in Europe up to the 77th century). Tashkent: Fan. Rozhansky, I. D. Razvitiye estestvoznaniya v epoku antichnosti (= Development o f natural science 1979 during antiquity). Moscow: Nauka. Sayili, A. Al-KhHzin7s treatise on astronomical instruments. Ankara hiversitesi Dil ve 1956 Tarih - Cografya Faciiltesi Dergesi, 14: 15-19.

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Geschichte der Elementarmathematik, Bd. 1, Arithmetik und Algebra. 4. Aufl. Berlin-New York: Walter de Gruyter.

Volodarsky, A. I. Ocherkipo istoriisrednevekovoy indiyskoy mathematiki (= Essays on the his1977 tory of medieval Indian mathematics). Moscow: Nauka. Vygodsky, M. Y. Arithmetika i algebra v drevnem mire (= Arithmetic and algebra in the ancient n.d. world), Moscow. Wiedemann, E. Aufsutze zur arabischen Wissenschaftsgeschichte,2 vols. Hildesheim and New 1970 York: G. Olms. Vol. I, especially 'mer die Bestimmung der spezifischen Gewichte," I, 240-57; 'fJber die Bestimmung der Zusammensetzung von der Legierungen,"vol. II,57-68, and 'Qber die Verbreitung der Bestimmungen der spezifischen Gewichte nach al-BErtini," vol. 11, 1-4. Youschkevitch, A. P. Istoria mathematiki v sredniye veka (= History of mathematics in the Middle 1961 Ages). Moscow: Physmatgis.