ÿWPCL ûÿ2BJ|xÕ!Ð x ÐÐÐüð ä ØÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÐЊ‚ÐÈÐÁ`ÁTabarÀ…À © p. !ÕÐ °x ÐÐа¤˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐЊ‚ÐÈÐ Ã ÃÁàÁNOTES AND QUERIES ON PERSIAN MATERIAL ÁàÁIN ELEMENTARY AND RECREATIONAL MATHEMATICSÄ Ä ÁàÁà ÃDAVID SINGMASTERÄ Ä ÁàÁ87 Rodenhurst Road, London, SW4 8AF, UK ÁàÁTel/fax: 020©8674 3676; email: ZINGMAST @ LSBU.AC.UK ÁàÁLast updated on ØD1 3 4. DØ ÁÁThis is mostly based on the entries for Mohammed ibn AyyÀÁÀb TabarÀ…À in the following. ÁÁThis was originally a letter to a Persian graduate student who was going to work on TabarÀ…À and did not read German, so I give translations of the German text. I have now deleted the material copied from my à ÃSourcesÄ Ä. ÁÁJohannes Tropfke, revised by Kurt Vogel, Karin Reich and Helmuth Gericke. à ÃGeschichte der Elementarmathematik.Ä Ä 4th ed., Vol. 1: Arithmetik und Algebra. De Gruyter, Berlin, 1980. [The 1st ed. was De Gruyter?, Leipzig, 1902, 2 vols. 2nd ed., De Gruyter, Berlin & Leipzig, 1921©1924, 7 vols. 3rd ed., De Gruyter, Berlin & Leipzig, 1930-1940, vols. 1©4 (the MSS of the remaining volumes were lost in 1945).] ÁÁThere is considerably more in Tropfke than I recalled, though some entries are very minor. ÁÁFor more information, see the cited sections of my à ÃSources in Recreational Mathematics. An Annotated Bibliography.Ä Ä ÁÁà Ã7.H.7.ÁÁDIGGING PART OF A WELL.Ä Ä ÁÁà Ã7.E. ÁÁMONKEY AND COCONUTS PROBLEMSÄ Ä ÁÁà Ã10.A.ÁÁOVERTAKING AND MEETING PROBLEMSÄ Ä ÁÁà Ã10.H.ÁÁSNAIL CLIMBING OUT OF WELLÄ Ä ÁÁà Ã7.AK.ÁÁLAZY WORKERÄ Ä ÁÁà Ã7.R.1.ÁÁMEN FIND A PURSE AND 'BLOOM' OF THYMARIDESÄ Ä ÁÁà Ã7.R.2.ÁÁ"IF I HAD 1/3 OF YOUR MONEY, I COULD BUY THE HORSE"Ä Ä ÁÁà Ã7.R.ÁÁ"IF I HAD ONE FROM YOU, I'D HAVE TWICE YOU"Ä Ä ÁÁà Ã7.P.1.ÁÁHUNDRED FOWLS AND OTHER LINEAR PROBLEMSÄ Ä ÁÁà Ã6.BF.2.ÁÁSLIDING SPEAR = LEANING REEDÄ Ä ÁÁà Ã7.L.3.ÁÁ1 + 3 + 9 + ... AND OTHER SYSTEMS OF WEIGHTSÄ Ä ÁÁà Ã7.L.3.ÁÁ1 + 3 + 9 + ... AND OTHER SYSTEMS OF WEIGHTSÄ Ä ÁÁà Ã7.AO.ÁÁDIVINATION OF A PERMUTATIONÄ Ä ÁÁOn. p. 717, Tropfke says TabarÀ…À lived in the 2nd half of the 11C in À\Àmul, south©east of the Caspian Sea. He cites two works. ÁÁ1. À°ÀumÀ]Àr©nÀ]Àmeh. Ed. TaqÀ…À BÀ…ÀneÀ±À. Teheran, 1966. [Tropfke translates the title as Arithmetic (Rechenbuch) and says it is in Persian.] ÁÁ2. MiftÀ]Àh al©muÀÀÀ]ÀmalÀ]Àt. Ed. Mohammed AmÀ…Àn RiyÀ]ÀhÀ…À. Teheran, 1970. [Tropfke translates the title as Key of Transactions and says it is in Persian.] ÁÁTropfke cites material by the item number and page, sometimes adding problem numbers, e.g. TabarÀ…À [1; 5 & 7] is the first book, pp. 5 & 7. ÁÁP. 17. TabarÀ…À [1; 5 &7] is cited as an early user of zero, probably as al©sifr. ÁÁP. 515. This is in Section 4.1.1 Buying and Selling (Profit and Loss Calculation). In the last paragraph on p. 515, Tropfke says: "In the mathematical writings of the Arabs which have been edited so far ÀMÀ which admittedly form a vanishing fraction of those named in the Fihrist ÀMÀ there is proportionately little on mercantile calculation. Some is found in al©UqlÀ…ÀdisÀ…À [2; 237©241], al©KaraÀwÀÀ…À [1; 83, No. 11]; an abundance of the problems belonging to this section only occurs in TabarÀ…À [2; 105©109, 119©121, 123 f. = IV, No. 8, 11©16, 32©36, 40, 45]. ÁÁPp. 528©529. Section 4.1.3. Work and Service Calculations. The first examples are simple, but then we get: ÁÁThe following very remarkable problem is given by TabarÀ…À [2; 96 = III, No. 17]: "Calculation of ditchwork. The payment for digging a 15 ell deep well is 30 dirhems. What is the cost of a 10 ell deep well?" The solution given by TabarÀ…À is clarified in the following way in another place [2; 227 = VI, 61]: "Apportionment of wells and cisterns. In order to dig a well, the earth must be lifted out. For the first ell, the earth comes up one ell; for the second ell, the earth comes up two ells; for the third, three ells; etc. until the end. For this calculation, one must use the series of of natural numbers. So we take 1 as first term, to which 2 as second term gives 3, to which 3 as third term gives 6 as sum, etc. In this way, we do each time, and that is the 'bast' (literally 'extension'). Example: the well depth is 10 ells; what is the 'bast'? 1 + 2 = 3, 3 + 3 = 6, ..., 45 + 10 = 55. That is the 'bast' of a well 10 ells deep." So, in the given problem, the 'bast' for wells of depth 10 and 15 ells is determined, giving 55 and 120, which yields a ratio of 55/120 = 1/3 [+] 1/8. "We multiply that by 30; this yields 13 [+] 1/2 [+] 1/4. That is the payment for a well of 10 ells, when the other well costs 30 dirhems." The problem is reminiscent of a Babylonian one, where the work payment in digging a canal decreases corresponding to the depth of the layers. ÁÁI had missed these examples of TabarÀ…À's which are nearly two centuries earlier than my previous earliest example. I have amended section 7.H.7 of my à ÃSourcesÄ Ä to include this. ÁÁP. 538. This is part of 4.1.6 Interest Calculations. The following is a fairly straightforward problem. ÁÁThe following example is given by the Persian TabarÀ…À [2; 132f. = IV, No. 51]: On the determination of the initial capital and the raising of triple profit from it. If one asks us: a man had money and gained profit; and from that money, he made a triple profit; all these three profits, excepting the initial capital, made 19 dirhems, and the first profit gained is 4 dirhem. Now say, how much was the initial capital of the man and how much is all that he has made as profit. ÁÁThis is not entirely clear and one needs to look at the calculation to understand what is intended. From the equations, one sees that he is assuming the interest rate is the same each time, i.e. if k is the initial capital and zÃÃ1ÄÄ, zÃÃ2ÄÄ, zÃÃ3ÄÄ are the three profits or interests received, then zÃÃ1ÄÄ / k = zÃÃ2ÄÄ / (k + zÃÃ1ÄÄ) = zÃÃ2ÄÄ / (k + zÃÃ1ÄÄ + zÃÃ2ÄÄ). However, the calculation assumes that the relative profits are the same, that is: (zÃÃ2ÄÄ - zÃÃ1ÄÄ) / zÃÃ1ÄÄ  =  (zÃÃ3ÄÄ - zÃÃ2ÄÄ) / zÃÃ2ÄÄ, which gives us zÃÃ1ÄÄ, zÃÃ2ÄÄ, zÃÃ3ÄÄ  = 4, 6, 9 and the interest rate turns out to be 50%. ÁÁP. 558. This is in 4.1.9 Partnership Calculation (Proportional Division). These are straightforward division problems. Tropfke simply cites TabarÀ…À [2: 106, 130f. = IV, No. 10, 48] as an example. ÁÁP. 564. This is in 4.1.12 Money Changing and Conversion of Measures. Again these are pretty straightforward and TabarÀ…À [2: 86©90 = III 9, 10, 11] is simply cited as an example. ÁÁP. 573. This begins a new main section: 4.2 Problems of Recreational Mathematics which begins with section 4.2.1 Linear Problems with One Unknown which is considerably subdivided. ÁÁPp. 573©574 is section 4.2.1.1 Hau Calculation. 'Hau' is the Egyptian term for 'heap' or 'pile' and was used as the assumed value in problems solved by false position, which must reduce to the form ax = b. ÁÁP. 574 cites TabarÀ…À [2; 116f., No. 27] as an example. ÁÁPp. 574©575 is section 4.2.1.1.1 God Greet You Problems. (The name comes from the traditional opening line of these problems, which continue with something like "If there were as many of us again and half as many more and half of that half more and one more, then we would be 100." ÁÁP. 575 cites TabarÀ…À [ 2; 129f. No. 47] as an example where a number of doves is wanted. ÁÁPp. 576©577 is section 4.2.1.1.3 The Pole in the Water. Here a typical problem is: A column is 1/8 in the ground, 1/3 in the water, 1/4 in the bog and 7 hastas are to be seen in the air. What is the length of the column? ÁÁThe Persian TabarÀ…À computes in his exercises, as well as the length of a tree [2: 109f., 119, 122, No. 18, 31, 38], where, in No. 31, remarkably, the algebraic root of the length l of the tree, that is, ÀÀl, towers into the air, but also the weight of a fish [2: 121f., No. 37]. ÁÁP. 578 is the beginning of 4.2.1.1 Cistern Problems (Work Problems) which begins with 4.2.1.2.1 Proper Cistern Problems and cites TabarÀ…À [2 101, No. 1] as an example. ÁÁPp. 582©586 is 4.2.1.3 Box Problems The Porters in the Garden (or Orchard). On p. 582, he gives the formulae for the three basic cases. Case 1 has a merchant on a trip making a rate of profit of aÃÃ1ÄÄ on his money, then spending bÃÃ1ÄÄ, etc. winding up with s. Case 3 is the typical 'apple garden' problem: someone is coming out of a garden with a bag of apples and the three porters each demand a fraction and some more. A typical example is that each porter demands half of the apples plus half an apple more. ÁÁP. 585 cites TabarÀ…À [2: 177f. 128, No. 28, 45] for the first case as a merchant's trip and for the third case as porters in an orange garden. ÁÁThis is section 7.E in my à ÃSourcesÄ Ä which has been revised in light of this material. ÁÁPp. 588©598 is 4.2.1.4 Motion Problems which is a big topic and is subdivided by area and the problems are classified by type. His type I has motion in a straight line. Subtype A c has one person with alternating forward and backward movement. Subtype B a has tow persons meeting at constant velocities. Subtype C a has one person overtaking another with constant velocities. Subtype C b is the same with variable velocities (usually in arithmetic progressions). ÁÁP. 593. 4.2.1.4.3 Arabs. ÁÁAmong the Arabs, motion problems are found in al©KaraÀwÀÀ…À and the Persian TabarÀ…À in his work ÃÃKey of TransactionsÄÄ. Here are the most important cases of motion problems, each represented by an example [2; 103f.]. ÐФ˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐ I C a (No. 4). 2 couriers are travelling, one after the other; the first travels 6 parasangs per day, the other 9. When do the tow meet, if the second courier sets out 4 days later than the first? I C a (No. 5). A courier goes 30 parasangs per day. The second goes 1 parasang on the first day, 2 parasangs on the 2nd day, 3 parasangs on the 3rd day, etc., in arithmetic progression. When do the two couriers met, if they set out at the same time? I B a (No. 6). One boat travels from an eastern city to a western city in 5 days. Another boat travels from this same western city to the eastern in 7 days. When do they meet, if they start out at the same time? I A c (No. 7). A boat comes forward 18 parasangs per day and goes backward 12 parasangs per day. It comes and goes for 40 days. How many days does it come and how many days does it go? Ðа¤˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐ ÁÁThe last is not a clear example. I wonder if 'days' in the last sentence should be 'parasangs'. This material is in sections 10.A and 10.H of my à ÃSourcesÄ Ä, which have been amended. ÁÁP. 602 is 4.2.1.5.3 The Broken Service Contract. A typical example, from al-KaraÀwÀÀ…À, is that the payment to a servant for one month shall be 35 dirhems and a ring. He works 3 days and takes just the ring. What is the value of the ring? This leads directly to x = 3/30 (x + 35) and I consider this too simple to include in my à ÃSourcesÄ Ä. However TabarÀ…À [2; 112, No. 22] is the second oldest example cited by Tropfke, and features money and a ring, as does the earlier example of al-KaraÀwÀÀ…À. ÁÁP. 603 is 4.2.1.5.4 The Lazy Worker. A worker receives an amount for each day he works and has to forfeit an amount for each day he doesn't work. At the end of some period, his total wages are given. How many days did he work? ÁÁTabarÀ…À [2; 113, No. 23] is cited, again as the second oldest example. ÁÁThis is section 7.AK in my à ÃSourcesÄ Ä, now revised. ÁÁP. 604 is 4.2.1.5.5 What Time Is It? A typical response is that if you take 3/5 of the time and multiply by 4, that is the time remaining in the day. I.e. x + 4 (3/5) x = 12. ÁÁTabarÀ…À [2; 118f., No. 29, 30] is cited, again as the second oldest examples. ÁÁI haven't included this in my à ÃSourcesÄ Ä. ÁÁP. 604 starts 4.2.2 Linear Problems with Several Unknowns. Some of these are indeterminate. ÁÁP. 606 starts 4.2.2.1 The Found Purse. See section 7.R.1 of my à ÃSourcesÄ Ä for an explanation. ÁÁOn p. 607, TabarÀ…À [2; 129, No. 46] is cited as the second oldest example. ÁÁP. 608 starts 4.2.2.3 One Alone Cannot Buy (Buying a Horse). See section 7.R.2 of my à ÃSourcesÄ Ä for an explanation. ÁÁOn p. 609, TabarÀ…À [2; 133f., 150f., No. 52, 13] is cited. ÁÁP. 609 starts 4.2.2.4 Giving and Taking. This is usually called the Ass and Mule problem. See section 7.R of my à ÃSourcesÄ Ä for an explanation. ÁÁOn p. 611, TabarÀ…À [2; 145f., No. 7] is cited. ÁÁP. 613 starts 4.2.2.6 The Problem of the Hundred Fowls and the Banquet Problems. If roosters cost 5, hens 3 and chicks 1/3, how can you buy 100 fowls for 100? This has been an extremely popular problem. See section 7.P.1 of my à ÃSourcesÄ Ä. ÁÁOn p. 614, TabarÀ…À [2; 110f. = IV, No. 20]; cows, sheep, hens is cited. ÁÁP. 616 starts 4.2.3 Problems of Computational Geometry whose first subsection starts of P. 617 as 4.2.3.1 The Pythagorean Theorem in Geometrical Problems. P. 4.2.3.1.1 The Leaning Ladder and 4.2.3.1.3 The Leaning Reed are essentially the same problem, one being the other turned upside down and slightly shifted. Hence I treat them both in section 6.BF.2 in my à ÃSourcesÄ Ä. ÁÁOn p. 621, TabarÀ…À [2; 124, No. 42] is cited for the leaning reed. ÁÁP. 624 starts a new main section 4.2.4 Problems with Sequences and Series with first subsection 4.2.4.1 Arithmetic Series. These frequently were used in motion problems and the citation of TabarÀ…À on p. 627 is to problem no. 5 of those already discussed on p. 593 (my section 10.A). ÁÁP. 628 starts subsection 4.2.4.2 Geometric Series. ÁÁP. 629 starts 4.2.4.2.3 The Weights Problem. I've broken this down into binary and ternary forms in my sections 7.L.2.c and 7.L.3. ÁÁOn p. 634, we have the following. ÁÁThis problem type first appears in the Persian TabarÀ…À [2; 125ff., No. 43]. Here all weights from 1 to 10,000 pounds are weighed out with the help of stones of weight 1, 3, 9, ..., 3ÃÃ9ÄÄ. ÁÁP. 635 simply repeats the citation. ÁÁP. 642 starts 4.2.6 The Divination of Numbers. The first subsection is 4.2.6.1 Backwards Calculation. These are very simple and I haven't included them in my à ÃSourcesÄ Ä, except for some that use more mathematics. The first chapter of Rouse Ball starts with a number of these and I once sent some material to Coxeter about them ÀMÀ but it's not very detailed. ÁÁP. 642 cites TabarÀ…À [2; 115f. = IV 25, 26] as the earliest of this type. ÁÁP. 646 starts 4.2.6.5 Where is the Ring? P. 647 cites TabarÀ…À [2; 113ff. = IV, No. 24] as the second earliest example of this type. ÁÁP. 648 starts 4.2.6.7 Distribution of Three Objects. This is my section 7.AO. ÁÁOne of the same type of problem, with m = 18 = x + y + z and n = 16 is given already by TabarÀ…À [2: 109 = IV, No. 17]. ÁÁThis is the earliest example he cites.