Commentary on al-Hawārī’s Commentary

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10.34663/9783945561638-05

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Abdeljaouad, Mahdi and Oaks, Jeffrey (2021). Commentary on al-Hawārī’s Commentary. In: Al-Hawārī’s Essential Commentary: Arabic Arithmetic in the Fourteenth Century. Berlin: Max-Planck-Gesellschaft zur Förderung der Wissenschaften.

59.1 [Introduction]

A copyist wrote the introductory words “In the name of God…said:”. Al-Hawārī first praises God and Muḥammad, and then Abū Yaʿqūb, Marinid ruler of Morocco from 1286 to 1307. Last, praise is given to Abū Muḥammad ʿAbdallah Ibn Abū Madyan, an intellectual and an important government minister from 1302 to 1307.

59.15 Al-Hawārī asked permission from Ibn al-Bannāʾ to write this commentary, which Ibn al-Bannāʾ granted. The latter had already written a commentary on his own book, titled Lifting the Veil from the Face of the Operations of Arithmetic. What the Condensed Book and Lifting the Veil lacked, according to al-Hawārī, were ample numerical examples of the rules.

61.1 Four of the manuscripts we consulted show the phrase “may God forgive him”, which agrees with manuscripts of the Condensed Book. The Medina manuscript has instead “may God preserve his splendor, his reputation, and keep his memory whole”. This version implies that Ibn al-Bannāʾ is already deceased.

61.9 Part I. Known numbers.

63.1 Chapter I.1. Whole numbers.

65.1 Section I.1.1. The divisions of numbers and their ranks.

65.2 Euclid defined “number” in the beginning of Book VII of the Elements as “a multitude composed of units”.1 Because these units are indivisible, Euclidean numbers are restricted to positive integers. This is in contrast with the numbers of practical Arabic arithmetic, which include fractions and irrational roots. Al-Fārisī, a Persian contemporary of Ibn al-Bannāʾ, gives the definition of number of “the arithmeticians” as “a quantity you obtain from one by repetition or partition or both, and it is clear by this meaning that the type is divided into whole numbers and their fractions”.2 Ibn a-Bannāʾ condenses and combines the Greek and Arabic definitions: “A number is a collection of units, and it is divided according to how it is produced into two kinds: whole and fractional”. Later, he made a philosophical apology for fractional numbers in Lifting the Veil in which he claimed that Euclid’s definition is not really a definition at all, but merely an expression of “what is in the soul”.3 Later commentators entered into the discussion, including al-Mawāḥidī (ca. 1382), Ibn Qunfudh (1370) and Ibn Ghāzī (1483).4

Al-Hawārī will explain the various ways of representing fractions in Chapter 2, beginning at 131.1. The examples he gives here are, in the notation described later in the book, 1\over 2, 3~\,1\over 8~\,2 (equivalent to our 7\over 16), {1\over 4}~{1\over 9} (our 13\over 36), 6~\bullet ~7\over 7~\bullet ~8 (our 3\over 4), and {1\over 9}~\scalebox {-1}[1]{\ell }~{5\over 6} (our 13\over 18).5

65.6 Reading right to left, the first place is the units place. Zero was not considered to be a digit, but was instead a sign indicating an empty place in the representation of a number.

65.10 Evenly-even numbers are the powers of two starting with 2, and evenly-odd numbers are the double of an odd number \ge 3. Evenly-evenly-odd numbers are numbers in between: they are the product of some power of two \ge 4 by an odd number \ge 3. The classification of even numbers into evenly-even, evenly-odd, and evenly-evenly-odd comes from Greek number theory. These definitions are taken from Nicomachus’s Arithmetical Introduction, translated into Arabic by Thābit ibn Qurra in the late ninth century.6

Euclid’s definitions of these terms in Elements Book VII are different, and do not correspond to the standard Greek definitions. He defines a number to be “even-times even” if it is the product of two even numbers, and “even-times odd” if it is the product of an even number by an odd number. This way a number can be both, like 28=14\cdot 2=7\cdot 4. Also, Euclid has nothing corresponding to the “evenly-evenly-odd”.

Note the wording “each of the sixteens”. When we take half of 32, the result is a single number, 16. For al-Hawārī, halving 32 means to partition it into a pair of 16s. Numbers in medieval Arabic arithmetic admit multiplicity. See our comments at 163.2 below.

66.7 The word we translate as “prime” is awwal. This word also means “first, foremost”, etc., and so is close to our “prime” or “primal”. Ibn al-Bannāʾ and al-Hawārī also use the word aṣamm (“deaf”) to mean “prime”, though other arithmeticians, like al-Uqlīdisī and al-Baghdādī, use that word to mean “irrational”. The use of the word “deaf” for “prime” has to do with Arabic ways of expressing fractions, which we explain below at 134.2. We describe the words for “irrational” at 163.2. A prime number can also be called basīṭ (“simple”), in contrast to composite numbers, which are formed from more than one prime.

An “oddly-odd” number is the product of two odd numbers, both \ge 3. This time Euclid, not Nicomachus, is probably the source. Definition VII.10 in Elements reads “An odd-times odd number is that which is measured by an odd number according to an odd number”, i.e., it is the product of two odd numbers. Nicomachus has no such definition. He calls composite odd numbers simply “composite” (sýnthetós, translated as murakkab by Thābit).7

The word for “composite” in Ibn al-Bannāʾ and al-Hawārī is similarly murakkab, or some related form. (Sometimes we translate these words as “composed” or “composition”.) The associated verb rakiba serves to multiply the factors together to produce the number composed of those factors. See below at 196.16 and 211.13 for related forms of this word in the context of proportions and algebra.

66.13 Al-Hawārī presents another classification of numbers. All numbers are either prime or composite, and composite numbers come in three types: (a) perfect squares, (b) products of two or more different numbers, and (c) perfect cubes. He does not mention that some numbers fall into more than one category. The number 36, for example, is a square ( 6^2) and a product of two different numbers ( 4\cdot 9). The classification of composite numbers is explained for even numbers starting at 66.17, then again for odd numbers at 67.17.

We have translated the adjective majdhūr as “has a root”. A number “has a root” if its square root is rational, like 9 or 16\over 25. We could have translated it as a single word like “rootable”, but that seemed too awkward. Later, first at 163.14, we will encounter the same concept for ranks. A rank is a majdhūra if a number of that rank can have a root. This is true for the units, the hundreds, ten thousands, and every other rank after that.

66.17 The words ḍilʿ (“side”), musaṭṭaḥ (“surface” or “plane”), and murabbaʿ (“square”) are geometric terms used in arithmetic in a metaphorical sense. They derive from the corresponding Greek words in the number theory books VII to IX in Euclid’s Elements, and are first encountered in Definition 16 at the beginning of Book VII. They do not imply any underlying geometric conception of number.

67.12 The word mukaʿʿab can mean a geometrical cube, or, as here, it can be an arithmetical term meaning “[perfect] cube”, like 27 or 8\over 125. The related word kaʿb in the present passage means “cube root”, but it is also the name given to the third power of the unknown in algebra (first encountered at 221.1), which we translate as “cube”. The meanings are clear by the context.

68.11 Perhaps al-Hawārī is thinking of the fact that “root” and “cube root” are particular to dimensions 2 and 3 respectively, while “side” is the term for the n-th root for any particular n.

To “decompose” (ḥalla) a number means to express it as the product of two or more numbers. The opposite operation is to “compose” (rakiba) two or more numbers into their product.

Ibn al-Bannāʾ and al-Hawārī explain a rule for extracting square roots numerically beginning at 166.1. Similar rules for extracting cube roots were well known in their time, but al-Hawārī remarks here that going through the work is “of little benefit”. Instead, he briefly explains how to find the cube root of a perfect cube by factoring. For example, to find the cube root of 216, one can break it down as 3\times 72, then to 3\times 9\times 8, and then to 2\times 2\times 2\times 3\times 3\times 3. Then one can piece it together as 6\times 6\times 6, so 6 is the cube root.

68.18 Ibn al-Bannāʾ uses a few different words to explain the base ten system for writing numbers with the Indian figures:

Rank (martaba). This word indicates the position of a digit in a number. The Arabic word suggests a ranking of the digits, as al-Hawārī explains at 68.18. Ranks are sometimes designated by ordinal numbers, like first, second, third, etc., or by the names “units”, “tens”, “hundreds”, etc. For example, the “7” in 17,285 is the fourth rank, or the rank of thousands.

Place, position (manzila). Like martaba, this word indicates the position of a digit in a number. But here the Arabic word evokes the image of a place or a home, “because the numbers reside in them”, as remarked at 68.18. Again, the “7” in 17,285 is in the fourth place, or the thousands place. Sometimes the word mawdhiʿ, also meaning “place”, is used with the same meaning (at 108.12, 109.4).

Index (uss). This is the number indicating the position of the digit. The index of units is 1, of tens is 2, of hundreds is 3, of thousands is 4, etc. The Arabic word uss is used today to indicate the exponent in mathematics. We could have translated it as “power”, but the numbers would be off by one. For us 10 is the first power of ten, while the corresponding index is 2. Later, in the chapter on algebra, the word uss is used to mean the power of the unknown, and there it matches our exponents. The uss of the second degree unknown is 2, for instance. Thus we translate it as “power” there.

Name (ism). The name of a digit is “units”, “tens”, “hundreds”, “thousands”, “ten thousands”, etc., depending on its place. For instance, the name of the 7 in 17,285 is “thousands”.

Species (nawʿ or jins). There are three species of number: units, tens, and hundreds. These are repeated for the thousands, for the thousand thousands (i.e., millions), etc. So the species of the “7” in 17,285 is units, and the species of the “1” is tens. The same two words nawʿ and jins are used for the different “species”, or what we would call the powers of the unknown, in algebra.

At 68.18 Ibn al-Bannāʾ limits “rank” and “place” to units, tens, and hundreds, these being repeated for the thousands, thousand thousands, etc. But at 70.2, 70.23, 72.4, and 105.1 al-Hawārī regards these terms as progressing indefinitely instead of repeating.

One word absent in the book is ḥarf, meaning “digit”. Other medieval books, such as Principles of Indian Reckoning of Kūshyār ibn Labbān, use ḥarf to mean “digit”. The word is also used by our authors, but with the meanings of “letter”, “particle”, or “conjunction”. In many instances, beginning in the section on addition, we translate martaba as “digit”. This is not technically correct since the rank is a location and the digit is a number that is placed there, but it makes the reading easier. We also often translate manzila as “digit”, too, first at 74.9; and in the passages at 111.7 and 112.5, we render ʿadad (“number”) as “digit” where appropriate.

There were several Arabic words that played the roles of our words “type”, “kind”, “species”, “division”, or “variety”. The ones appearing in our book are nawʿ, jins, ḍarb, qism, and ṣanf. These words were more or less interchangeable in Arabic mathematics, whether for types of number, of fraction, of addition (and other operations), of proportion, of equation, etc. It is partly from the ways numbers were classified, and from the descriptions of the two types each of multiplication and division (95.3, 117.9), that we can recognize numbers in Arabic mathematics as being numbers of something. See our commentary at 95.3 below.

69.2 By the thirteenth century CE, and perhaps much earlier, two distinct styles of writing the Indian numerals had developed in the Islamic world.8 The Western forms, written in the Maghreb and al-Andalus, ultimately led to the European forms 123456789 and 0, while the Eastern forms led to the current Arabic forms of the numerals, ١٢٣٤٥٦٧٨٩ and ٠. There are of course variations within each style.

Naturally, al-Hawārī wrote the Western forms, and it is these that are described by the poet. The poem makes use of the similarities between the shapes of the numerals and the shapes of letters of the Arabic alphabet to teach the student how to write the numerals. The copyist of the Medina manuscript was not familiar with the Western way of writing the numerals, so he followed the instructions in the poem to the letter (pun intended). Figure 1 below shows the numerals 987654321, with the Eastern forms written underneath.

Fig. 1: Digits in the Medina MS.

Fig. 1: Digits in the Medina MS.

Starting from the right, the “1” looks like the letter alif, which is just a vertical line. The “2” is written as the letter ḥā, and the “3” is the ligature of the letters ḥā and jīm, pronounced ḥajja as if it were a word. The “4” should be the ligature of ʿayn and wāw, pronounced ʿuw, but the copyist forgot to add the waw. See below in Figure 2 for the “4” in the Istanbul manuscript. The “5” is shown as an ʿayn alone, so it looks like the “4” in this figure. The “6” is shaped like the letter (the copyist failed to close the loop), the “7” like an anchor, and the “8” is said to be a couple zeros (small circles), one above the other, connected by a vertical line (the alif). The line is not evident in the figure, and is generally not written at all. The Tunis manuscript writes the “8” the normal way, as , but just below it follows the instructions with the vertical line: (and afterward the copyist wrote the Eastern forms of the digits). Finally, the “9” is written as the letter wāw.

The copyist of the Istanbul manuscript was already familiar with the Western forms. These are shown in Figure 2.

Fig. 2: Digits in the Istanbul MS.

Fig. 2: Digits in the Istanbul MS.

For comparison, Figure 3 shows the number 9367184225 from the same manuscript, with the Eastern forms above the Western forms. The Eastern “5”, on the right in black, is mistakenly written as a “4”.

Fig. 3: More digits, from the Istanbul MS.

Fig. 3: More digits, from the Istanbul MS.

70.2 There were no words for “million”, “billion”, etc. in Arabic arithmetic. They wrote instead “thousand thousand”, “thousand thousand thousand”, etc.

70.8 The places/ranks of a number were spoken in Arabic in a different order than they are in English. We preserve the Arabic order in our translation from here to the end of the chapter (through page 72 of our edition) to give the reader a feel for how the numbers were expressed.

Arabic numbers less than one hundred were spoken with the units first, like “four and sixty” instead of “sixty-four”. So “four and sixty thousands” is 64,000, and “four and sixty thousands and three hundred thousands” is 364,000, though this was often spoken as “three hundred thousands and four and sixty thousands”. In Arabic, the hundreds (like “three hundred” and “five hundred”) are compound words, so we translate “three hundred” instead of “three hundreds”. The word for "Thousands" is stated separately from its number, and is made plural when there is more than one of them. They wrote “a thousand” for one, and “two thousands”, “three thousands”, etc. for more than one.

The plural becomes more complicated when we get to the millions. When there are more than ten of something, the plural form of an Arabic noun is written the same as the singular form. So we read “four trees”, but fourteen of them reads like “fourteen tree”. Thus, what appears to be “ten thousands thousand” should be understood as “ten thousands thousands”. If there are 10,000 of the second word “thousand”, it must be plural.

Also, where we say “one hundred”, “one thousand”, etc., in Arabic the “one” is not written. They expressed these numbers with the implied indefinite article, as “[a] hundred”, “[a] thousand”, etc.

71.12 Subsection on knowing the index of the repeated number.

72.1 A 1, 2, or 3 is needed for the rule, so if the number is divisible by three, then three is regarded as the remainder.

73.1 Section I.1.2. Addition.

73.2 The act of adding two numbers, according to al-Hawārī’s definition, should result in a single expression. Sometimes, though, the sum can only be expressed with more than one expression. For example, adding 978 to 456 yields one expression, 1434 (at 75.9), as does adding “a root of two to a root of eight” ( \sqrt{2}+\sqrt{8}), which gives “a root of 18” ( \sqrt{18}, at 179.16). The example at 180.10 shows an addition that results in two expressions: adding a root of three ( \sqrt{3}) to a root of fifteen ( \sqrt{15}) can only be expressed as “a root of three and a root of fifteen”. We write this as \sqrt{3}+\sqrt{15}, but as we explain below at 219.1, the operation of addition is not inherent in this composite expression. The current section covers addition of whole numbers, which always results in one expression.

Four different verbs are used in the book to mean “to add”: jamaʿa, ḥamala, zāda, and ḍāfa. Lane’s definitions of jamaʿa begin “to collect; bring, or gather together”.9 He starts off his definitions of zāda with “to increase, or augment, or grow”, while the various meanings of ḥamala begin with “to bear it, carry it, take it up and carry it, convey it, or carry it off or away”. Lane gives no definition of ḍāfa that relates to addition, but Wehr has “to be added, be annexed, be subjoined, be attached”.10 There are similar variations for words meaning “exceed”/“surpass” (zāda, faḍala, ʿalā) and “sum” (majmū and related forms, jumla).

Ibn al-Bannāʾ covers five types of addition in the Condensed Book:

(1)Adding numbers with no known relation. He covers the basic process of adding numbers in Indian notation beginning at 74.9.

(2)Adding sequences of numbers with a known disparity, at 76.7.

2a. In one kind of disparity, the ratio between consecutive terms is constant.

2b. In the second kind, the difference between consecutive terms is constant.

(3)Adding consecutive numbers, their squares, and their cubes, at 79.13.

(4)Adding consecutive odd numbers, their squares, and their cubes, at 80.5.

(5)Adding consecutive even numbers, their squares, and their cubes, at 80.20.

The rules for summing finite series, extending from 76.7 to 82.4, are (mostly) originally Greek in origin, but were probably borrowed from some intermediate Arabic source.11

73.7, 73.17 Ibn al-Bannāʾ’s distinction between a disparity in quantity (kamm), in which the difference between consecutive terms is constant, and a disparity in quality (kayf), in which the ratios of consecutive terms is constant, comes from Nicomachus.12 The terms “quantity” and “quality” appear again in the passage at 92.17, but with different meanings.

The Arabic for “geometric progression” is nisba handasiyya, literally “geometric relation”. Ibn al-Bannāʾ writes “known disparity” instead of “known relation” because the word nisba (“relation”) might imply the geometric progression. See below at 193.1 for more on the word nisba.

74.17 Al-Hawārī gives two examples of the first type of addition, starting with the addition of 4043 to 2685. First, the two numbers are written on two lines, like this:

Then the units 3 and 5 are added, and the result is put above (we put changes from the previous figure in red):

The tens are next. Because 4+8=12 has two digits, 1 is added to the 6 in the hundreds place of the lower number, and a 2 is put above the tens place:

There is nothing to add to the 7, so 7 is put above the hundreds place:

Finally, the thousands place is 6, from the sum of 2 and 4. The answer is 6,728:

Because this method requires erasing and replacing, it was intended to be worked out on a dust-board or wax tablet, and not with ink.

Operating on zero

The rules for operating on numbers expressed in Indian notation call for the addition, subtraction and multiplication of digits, and sometimes one or both of these digits is a zero. The zero signifies a place where there is no number at all, so we should ask what it meant to operate on it. For this we need to understand that the operations themselves were thought of in a more material sense than our binary operations on abstract sets. Even the notion of a set is a modern one – there was no word for “set” in Greek, Latin, Arabic, Sanskrit, or medieval Italian. Premodern mathematicians, Europeans and Indians included, had no concept of a set as an object.

Addition for al-Hawārī was not an operation on R+ satisfying the commutative and associative axioms. It was simply the appending of a number with another number, or the gathering of numbers together, which were all regarded as amounts of something (dirhams, men, hours, etc.). To add five to three was like combining the five silver dirhams in one purse with the three silver dirhams in another purse, or like extending a length of five adhruʿ by three more adhruʿ, or like adding three mathāqīl of grain to five mathāqīl. Not even Euclid found it necessary to provide a definition for addition conceived like this, though some Arabic arithmetic books, al-Hawārī’s included (at 73.2), characterize the operation.

For the operation of addition, Ibn al-Bannāʾ provides a special instruction when there is a zero present (74.9): “Then you add each digit of one of the addends to its counterpart in the other. If there is no counterpart, then the answer is the addend, as if it had a counterpart”. Al-Hawārī follows this rule in the present calculation: “Nothing corresponds to the seven in the upper line, so it is considered to be the sum of that rank and that of its counterpart as if it had something”. Adding nothing to 7 to get 7 does not mean that 0 assumes the role of an operable quantity. Instead, no addition takes place at all. Think of it like combining the money in two purses: one with 7 dirhams and the other empty. There is no act of combining to perform. Subtraction works similarly. In the passage at 83.19 al-Hawārī is faced with the subtraction of 0 from 9: “So we subtract this nothing of the minuend from the nine of the subtrahend, leaving nine”. No subtraction takes place when taking nothing away from an amount, so it leaves the amount unchanged.

Multiplication by zero is explained in the passage at 114.4: “multiplying the number by the zero or the zero by the number is identical. It comes from voiding the number or duplicating zero. Neither of these gives a number, so its sign is always a zero”. The word behind “voiding” (taṣfīr) is related to the word for “zero” (ṣifr). The former could have been translated as “emptying” or “zeroing”, and “zero” could be replaced with “nothing”. This duplication conforms to the standard definition of multiplication, given by Ibn al-Bannāʾ at 95.2: “Multiplication consists of the duplication of one of two numbers by however many units are in the other”. Duplicating nothing a number of times surely gives nothing, so the multiplication makes sense even if zero, being nothing, is not a number.

See the passages in the translation at 83.19,84.13, 90.7, 90.18, 123.3, and 215.14 for other operations with zero and/or nothing.

75.9 Al-Hawārī’s second example shows addition starting from the highest power term. He adds 978 to 456, first writing one above the other as before:

Working from the hundreds place, 9+4=13, so 13 is placed above:

Next, 7 + 5 = 12, so a 2 is put above the 7, and 1 is added to the 3 next to it:

Finally, 8 + 6 = 14, so the 4 is put above the 8, and 1 is added to the 2 next to it to get the answer, 1,434:

76.7 For the second type of addition, Ibn al-Bannāʾ works with the famous chessboard problem. In some books, a grain of wheat is placed in the first square, two grains in the second, four in the third, etc. Ibn al-Bannāʾ simply places numbers in the squares, as did Abū Kāmil when he wrote about it in the late ninth century CE at the end of his Book on Algebra.13 A 1 is placed in the first square, a 2 in the second, a 4 in the third, and continuing so that each square has double the previous square, like this:

Ibn al-Bannāʾ gives the rule for finding the sum of the numbers from the first square up to the 2^nth square. Al-Hawārī gives the example for the 2^4th = 16th square. The following iteration is performed:

Take the 1 in the first square. Add 1 to get 2.

Square it to get 4. This is 1 more than what is in the first two squares ( 1 + 2), and it is also what is in the third square (4).

Then square the 4 to get 16. This is 1 more than what is in the first four squares ( 1+2+4+8), and it is also what is in the fifth square (16).

Square the 16 to get 256. This is one more than what is in the first 8 squares ( 1+2+4+8+16+32+64+128), and it is also what is in the 9th square (256).

Square the 256 to get 65,536. This is 1 more than what is in the first 16 squares, and it is also what is in the 17th square.

So, the sum of the numbers in the first 16 squares is 65,535.

The figure shown in the translation is the one found in the Medina, Tehran, and Tunis manuscripts. The Istanbul and Oxford manuscripts show this figure instead (only Istanbul has the 65536 written on the left):

77.9 Ibn al-Bannāʾ then gives a variation in which the first square has a number other than 1, and the rule for filling out the remaining squares in the chessboard is the same: each square is double the one before it. For example, if the first square has a 3, then the succeeding squares are 6, 12, 24, etc. The rule to find the sum of the first 2^n squares is to follow the procedure as if a 1 were in the first square, then one multiplies the result by the number that is in the first square. Al-Hawārī gives the example of adding 4+8+16+32+64+128+256+512. The sum of the first eight squares starting with 1 is 1+2+4+8+16+32+64+128=255. Multiplying this by 4 gives 1020, which is the required sum.

78.1 Another variation is when the ratio of consecutive terms is some number other than 1\over 2. The example given by al-Hawārī starts with 16, and each square is 2/3 of the succeeding square. He gives the first five numbers: 16, 24, 36, 54, 81. Putting Ibn al-Bannāʾ’s rhetorical rule into modern form, the sum will be {16\cdot (81-16)\over 24-16}+81=211. We leave the general rule as an exercise for the reader.

79.1 Ibn al-Bannāʾ then gives a rule for summing sequences of numbers in which the difference, rather than the ratio, of consecutive terms is constant. Al-Hawārī’s example is to add the 6 numbers starting with 10, and with a difference of 3. If we were to write it all out the sum would be 10+13+16+19+22+25, but we are working only with the known numbers 6, 10, and 3. The rule begins by finding the last number, which in this case is 3\cdot (6-1)+10=25. Then (25+10)\cdot ({1\over 2} of 6)=105 is the sum. Ibn al-Bannāʾ’s rhetorical rule can be expressed in modern notation this way: if there are n numbers starting with a and with a difference of d, then the last number b is d(n-1)+a, and the sum is (b+a)\cdot {1\over 2}n.

79.13 The third type of addition covers consecutive numbers, their squares, and their cubes. It is true that the first of these is a special case of the second type of addition, but here the first and last numbers are both given, and the rule is then used to find the sums of the squares and the cubes.

To add the numbers from 1 to 10, multiply half of the 10 by one more than the 10: 5\times 11=55. In modern notation, \displaystyle {\sum _{k=1}^nk=\tfrac{1}{2}k\cdot (k+1)}.

To add the squares of these numbers, or 1+4+9+16+\cdots +100, the rule is ({2\over 3}\cdot 10+{1\over 3})\cdot 55=385. In general, \displaystyle {\sum _{k=1}^nk^2=\left(\tfrac{2}{3}n+\tfrac{1}{3}\right)\sum _{k=1}^nk}.

To add consecutive cubes, square the sum of the numbers. The example is 1+8+27+\cdots +1000=55^2=3025. In general, \displaystyle {\sum _{k=1}^nk^3=\left(\sum _{k=1}^nk\right)^2}.

Medieval Arabic mathematicians wrote “[a] square” of a number, with the implied indefinite article, rather than “the square” because their numbers admit multiplicity. See below at 163.2 for a more detailed explanation for the case of roots.

80.5 The fourth type of addition is to add the consecutive odd numbers, their squares, and their cubes. For the first of these, square half of one more than the last number. Al-Hawārī’s example is to find 1+3+5+7+9. The answer is ({1\over 2}(9+1))^2=25. In general, if the last number is n, then the sum 1+3+5+\cdots +n= ({1\over 2}(n+1))^2.

The sum of the squares of consecutive odd numbers up to n is {1\over 6}n(n+1)(n+2). In the example, al-Hawārī finds that 1^2+3^2+\cdots +9^2={1\over 6}9\cdot 10\cdot 11=165.

If we let a be the sum of the odd numbers up to n, then the sum of the cubes of the odd numbers to n^3 is a(2a-1). Al-Hawārī calculates that 1^3+3^3+5^3+7^3+9^3=25\cdot 49=1225.

80.20 The fifth and last type of addition deals with consecutive even numbers, their squares, and their cubes. Al-Hawārī adds the even numbers from 2 to 10 by calculating {1\over 2}(2+10)\cdot {1\over 2}10=30.

For the squares 2^2+4^2+\cdots +10^2 one adds 2\over 3 of 10 to 2\over 3 of 1, and the result is multiplied by the sum, 30: 6{2\over 3}+{2\over 3}=7{1\over 3}, and 7{1\over 3}\cdot 30=220.

Al-Hawārī gives an alternative rule, using the example of adding the squares 2^2+4^2+\cdots +12^2. Here one multiplies a sixth of the last number (12) by the product of the next two numbers (13 and 14). Taking 1\over 6 of 12 gives 2, and 2 by 182 is 364, which is the required sum. This happens to be the same rule he gave for adding the odd squares.

The sum of the even cubes 2^3+4^3+\cdots +10^3 “is given by multiplying the sum by its double”. We already know that the sum 2+4+\cdots +10 is 30, so we multiply 30 by 60 to get 1800, which is the answer.

83.1 Section I.1.3. Subtraction.

Just as with addition, several verbs are used for subtraction. The most common, and the one which appears in chapter titles and instructions, is ṭaraḥa. We translate it as “to subtract”. The verb ṭaraḥa is also used for what we call “casting out”, as in casting out nines to check the answer to a calculation. Two other common verbs for subtraction are saqaṭa, “to drop”, and naqaṣa, which we also translate as “to subtract”. Rarer are the verbs nazala, “to remove”, and dhahaba, “to take away”.

To announce the result of a subtraction the verb baqiya (“to remain, leave”) is used. The word for “remainder” is bāqī. Often we translate a phrase whose literal meaning is “[there] remains” or “what remains” as “the remainder”. The word for the “residue” after casting out nines or eights or sevens, covered starting at 87.15, is the related word baqiya, and we translate bāqiya as “residual”. These two words appear one time each to mean “remaining” and “remainder”, respectively, at 110.16 and 118.20.

83.2 Ibn al-Bannāʾ writes of two kinds of subtraction. The first is the subtraction of one number from another with Indian numerals, and the second is “casting out” to check the answer of a calculation. Between these two kinds al-Hawārī inserts a description of repeated subtractions, starting at 86.1, that he took from Lifting the Veil.

Two examples are given for the first kind. Al-Hawārī begins with the example 5035-4968 and proceeds from the highest rank to the lowest. First, the greater number is written above the smaller:

The 4 is subtracted from the 5, and the result is written above:

For the hundreds place, there is nothing (i.e., a 0) in the minuend, so one takes nothing away from the 9 in the subtrahend, leaving 9. This is subtracted from the 1 above the 5, which is really 10 since we are now working in the hundreds place. 9 from 10 leaves 1, so the first 1 is replaced with a 0, and this new 1 is placed above the hundreds place:

In the tens place the 3 is smaller than the 6, so we subtract 3 from 6 to get 3, and this is subtracted from the 10 above, leaving 7. The figure then becomes:

The situation is similar for the units place. Since 5 is less than 8, we subtract their difference, which is 3, from the 70 above. This leaves 67 as the answer:

84.13 Al-Hawārī gives a second example that starts with the units place. He begins with:

The 3 is less than 9, so we add 10 to the 3 and then subtract 9, leaving 4. This is placed above the 3. To compensate for the added 10, a 1 is added to the 6 next to the 9:

For the tens place we have a similar situation: 4 is less than 7. So add 10 to the 4 and subtract, leaving 7. Then add one to the 4 in the bottom row to get:

Next, taking 5 from 5 leaves nothing, so a 0 is placed above them:

Taking 3 from 6 leaves 3, so the figure becomes:

The answer is the 3,074 on top.

85.16 Rūmī signs were used in a system of calculation practiced in Western North Africa and al-Andalus. See 1.3 in the Introduction.

86.1 The part on repeated subtractions is taken from Ibn al-Bannāʾ’s commentary, and is not mentioned in the Condensed Book. It is not one of the two categories of subtraction he mentions at 83.2 at the start of the chapter. In modern notation the expression “ten less eight less seven less five less two” is 10-(8-(7-(5-2))). This is explained in words from the inside out: “subtract two from five, and the remainder from seven,…” and in an English version of the Arabic notation it would be 10~\ell ~8~\ell ~7~\ell ~5~\ell ~2, where the \ell stands for “less”. We are keeping the direction of the Arabic figures, so we write it in the translation as 2~\scalebox {-1}[1]{\ell }~5~\scalebox {-1}[1]{\ell }~7~\scalebox {-1}[1]{\ell }~8~\scalebox {-1}[1]{\ell }~10. See the discussion at 219.4 below for an explanation of the word “less” (illā).

In two of the five manuscripts we consulted, Istanbul and Oxford, the numbers are separated by the word illā. Here it is in the Istanbul manuscript: . In the Medina manuscript only the last part of the word is drawn, so it looks like an upside-down “ \ell ”: . The two other manuscripts, Tunis and Tehran, put three dots in place of the word illā for this figure. Here is the figure from the Tunis manuscript: .14 Both manuscripts, curiously, write the illā in other instances.

86.15 One could simply perform the operations as stated: subtract 2 from 5, then subtract the result from 7, etc. But Ibn al-Bannāʾ gives three other rules to work it out. For his first rule he distinguishes between the minuend, in this case 10, and the subtrahends, which here are 8, 7, 5, and 2. Add the even subtrahends (the 7 and 2) to the 10, and from this subtract the odd subtrahends, giving 19-13=6.

86.18 Another way is to collapse the string of numbers in groups of three. We “subtract the middle from the sum of the extremes, leaving the remainder as one number”. Taking the 10, 8, and 7, calculate 10+7-8=9, and replace all three with the 9 to get 2~\scalebox {-1}[1]{\ell }~5~\scalebox {-1}[1]{\ell }~9. Repeating the process gives 9+2-5=6. One does not need to start with the first three numbers. This works starting from any three consecutive numbers.

87.9 A third way is to perform alternating subtractions and additions, beginning with the 10. Subtract 8 from 10, then add 7 to the remainder, then subtract 5, and finally add 2.

87.11 See our extended discussions below, at 219.1 and 219.4, for an explanation of “appended” (zāʾid) and “deleted” (nāqiṣ). They do not mean “positive” and “negative”. Because numbers were numbers of something counted or measured, negative numbers would have been meaningless to medieval arithmeticians.

87.15 The second kind of subtraction is what we call in English “casting out”. In “casting out nines”, which is still taught today, the remainder from division by 9 can be found by adding the digits and removing multiples of 9. Al-Hawārī gives the example of 6435. He adds the digits one by one, casting out nines as he goes. So 6+4=10, and removing a 9 leaves 1. Then 1+3=4, and 4+5=9. “This is cast out entirely”, meaning nothing remains. Nothing is literally no number at all, and not our modern number 0.

Al-Hawārī also gives examples of the rules for casting out eights and casting out sevens. Because multiples of 200 are divisible by 8, one only needs to deal with the first three places in casting out eights. Al-Hawārī’s example is 5393. The 5000 is cast out entirely, as is the 200 from the 300. The remainder from the 100 is 4, then 2 is multiplied by the 9, and to these are added the 3, giving 25. Casting eights from this leaves 1. In general, the remainder of a number of the form 1ab (i.e., 100+10a+b) is the same as the remainder of 4+2a+b.

88.10 Casting out sevens is more complicated. No multiple of 10 is divisible by 7, so all digits must be taken into account. The remainders of each power of 10 are different for the first six powers, after which they repeat. These remainders must be memorized, and al-Hawārī illustrates Ibn al-Bannāʾ’s rule of expressing this sequence of remainders in abjad form. The letters appearing there are:

Arabic Transliteration Value
alif A 1
B 2
jīm J 3
dāl D 4
H 5
wāw W 6

The remainders of 1, 10, 100, 1000, 10000, and 100000 are 1, 3, 2, 6, 4, and 5, respectively. At 88.14 he gives a short poem designed to help the student memorize these letters.

88.17 Indian numerals were sometimes called al-ghubār, or “dust” numerals, after the dust-board on which they were commonly written.

89.4 The example al-Hawārī gives is to cast out sevens from 23,786,435. He writes the digits above their corresponding letters:

For us, it is easier to use the Indian numerals:

Multiplying the digits in the corresponding places, and casting out sevens if necessary, gives:

For example, in the thousands place we multiply 6 by 6 to get 36, and casting out sevens leaves 1. The digits in the top row add up to 22, and casting out sevens again leaves 1.

90.3 Ibn al-Bannāʾ gives two variations for casting out sevens that do not require the memorization of the sequence of abjad numerals. The first is an iteration: first multiply the highest power digit by three (the residue of 10), cast out sevens if necessary, and then add the result to the previous digit. Al-Hawārī’s example is 58,064. Starting with the highest power term, multiply 5 by 3 to get 15. Cast out sevens to get 1. Then add this 1 to the 8 to get 9. Now repeat: multiply 9 by 3 to get 27, leaving 6 after casting out. Add 6 to nothing (the 0) to get 6. Multiply by 3 to get 18, and cast out sevens, leaving 4. Add 4 to 6 to get 10. Thrice 10 is 30; cast out to get 2, and add it to the 4 to get 6. This is the answer.

90.15 The other variation takes into account two digits at a time. Casting out sevens from the 58 in 58,064 leaves 2. Although al-Hawārī does not write it, the residue of 58,064 is the same as the residue of 2,064. Next, the residue of 20 is 6, so we now consider 664. The residue of 66 is 3, so now the number is 34. Its residue is 6, which is the answer.

91.1 Subsection on the way to test [calculations] by casting-out.

Next, Ibn al-Bannāʾ turns to applications of these techniques of casting out to check the results of arithmetical operations. These include addition, subtraction, multiplication, and division/denomination. For the latter two it works even for fractions. Here are a couple of al-Hawārī’s examples.

91.4 For addition, he works with the example 43+64=107. He could cast out nines, eights, or sevens, and for this and subsequent examples he chooses sevens. The residue of 43 is 1, and the residue of 64 is also 1. Add them to get 2, which should be (and is) the residue of 107. If the residues added to 7 or more, one would cast out a 7 to make it less than 7.

92.3 In subtraction there is a problem if the residue of the minuend is smaller than the residue of the subtrahend. Take for example 29-13=16. The residue of the 29 is 1, and the residue of the 13 is 6. We cannot subtract 6 from 1, so we add 7 to the 1 and then subtract: 8-6=2, and this 2 is the residue of the remainder 16. For casting out nines, add nine to the residue of the minuend, and for casting out eights, add eight.

92.17 An example for the multiplication of fractions is {1\over 3}\times 14{1\over 4}=4{3\over 4}. The residues of the multipliers are 1\over 3 and 1\over 4. Ibn al-Bannāʾ does not say so, but these are the residues of the numerators. If, for instance, the multiplier were 3{1\over 4}, he would have found the residue to be 6\over 4, since 3{1\over 4} as a single fraction is 13\over 4, and the residue of 13 is 6.

The product of the 1\over 3 by the 1\over 4 is 1\over 12, or as Ibn al-Bannāʾ puts it, “a third of a fourth”. The numerator of this fraction is 1. The product 4{3\over 4} is 19\over 4, and Ibn al-Bannāʾ takes the residue of the numerator, which is 5. But these are fourths, not twelfths, so he multiplies by 3 to get 15 (for 15\over 12), whose residue is 1. This agrees with the residue of the multipliers.

The numerator 1 of “a third of a fourth” is equal to the numerator of the answer, so they are equal in quantity. The kinds of fractions they are, thirds of fourths, are the same, making them equal in quality. This terminology comes from Aristotle’s Categories, probably via Ibn Sīnā.15 See above at 73.7 for another use of the words “quantity” and “quality”.

93.6 As Ibn al-Bannāʾ will explain in the section on division at 118.14, the term “division” is used for the division of a greater number by a smaller number, and “denomination” for the division of a smaller number by a greater number. For the division a\div b= c, where a>b, the a is the dividend and b the divisor. If a<b, then a is the denominated [number] and b is the denominating [number] or what we call the denominator. For both kinds, c is the quotient or result. So “the result and the divisor or denominating number” are c and b, and the “dividend or denominated number” is a.

The Arabic words for the “denominator” of a fraction (imām, and less frequently maqām) are unrelated to the verb “denominate” (sammā) and related nouns such as “denominating [number]” (musammā minhu).

93.10 For both division and denomination, al-Hawārī gives an example with whole numbers and an example with fractions.

The Arabic word for the “numerator” of a fraction is basṭ, or occasionally the related word mabsūṭ. These words are unrelated to the word for “number”, which is ʿadad. We translate the related verb basaṭa as “to numerate”. Its meaning is to find the numerator of a fraction that is “a combination of two or more names” (135.1). For example, to numerate the fraction “five sixths and three fourths” (at 93.15) means to express it as “thirty-eight fourths of a sixth” ( 38\over 4\cdot 6). Finding the numerator is described at length in the first section on fractions, beginning at 135.8.

93.15 The notation for distinct fractions shows one next to the other, as explained later at 136.8 and 139.1. This problem is to divide {3\over 4}~{5\over 6} by {1\over 2}. We would write {3\over 4}~{5\over 6} as {5\over 6}+{3\over 4}, and the result of the division as 3{1\over 6}. The residue of 3{1\over 6}={19\over 6} is 5, or 5\over 6. The residue of 1\over 2 is 1, or 1\over 2. Multiplying them, one gets 5\over 12, or, as Ibn al-Bannāʾ says, “five halves of a sixth”. The residue of the dividend, {3\over 4}~{5\over 6}={38\over 24}, is 3, with a denominator of 24. So the 5\over 12 must be converted to 24ths, making it 10\over 24, which is 3\over 24 after casting out again. This matches the residue of the 38\over 24.

94.1 Al-Hawārī gives an example of checking the result of the denomination of whole numbers even if it might be superfluous in practice. The example is to denominate 11 with 15, which for us gives the fraction 11\over 15. The residue of the denominated number 11 is of course the same as the residue of the numerator 11 of the result, so there is nothing to check. The only aspect that makes this appear to be a problem is that al-Hawārī follows common Arabic practice by expressing the result not as 11\over 15, but as “three fifths and two thirds of a fifth”. We might write this as {3\over 5}+{2\over 3}{1\over 5}, but for al-Hawārī it would be shown as 2~\,3\over 3~\,5 (the notation for this fraction is explained at 123.18). The numerator and denominator of this fraction must be calculated in order to find its residue, but this brings us right back to the 11 and 15 we started with.

94.5 Checking the result of the denomination of fractions requires some work. Writing Ibn al-Bannāʾ’s example in notation, it is to denominate 2~\,2\over 3~\,6 with 1~\,5\over 3~\,8, which results in 2\over 3. After finding the numerators, the problem remains to denominate 8\over 6\cdot 3 with 16\over 8\cdot 3. The product of the residues of 2\over 3 and 16\over 8\cdot 3 should equal the residue of 8\over 6\cdot 3, but we need to be sure the denominators are the same to make it work. The residue of the numerator of 2\over 3 is 2. The residue of the numerator of 16\over 8\cdot 3 is also 2, and multiplying the 2 (as 2\over 3) by the 2 (as 2\over 8\cdot 3) gives 4\over 3\cdot 3\cdot 8. Because the 6 in the denominator of the 8\over 6\cdot 3 is lacking in the 4\over 3\cdot 3\cdot 8, we multiply the numerator and denominator of the latter by 6 to get 24\over 3\cdot 3\cdot 6\cdot 8, and its residue is 3, which is the “answer”. We now turn our attention to 8\over 6\cdot 3. The residue of the 8 is 1, and this must be multiplied by 3 and then by 8 to make the denominators match. This gives 24\over 3\cdot 3\cdot 6\cdot 8, and its residue is 3, which agrees with the answer.

95.1 Section I.1.4. Multiplication.

95.2 Arabic arithmetic books often define multiplication as “the duplication of one of two numbers by however many units are in the other”, as Ibn al-Bannāʾ phrased it here in the Condensed Book.16 This definition by “duplication” may come from Book VII of Euclid’s Elements: “A number is said to multiply a number when that which is multiplied is added to itself as many times as there are units in the other, and thus some number is produced”.17 In fact, Abū l-Wafāʾ (tenth century) writes in his arithmetic book: “Euclid stated the meaning of multiplication in Book VII of his work The Elements, as did Nicomachus of Gerasa in the Arithmetic. They said that multiplication is the duplication of one of two numbers by the amount of what is in the other in units”.18

It would not matter so much that Nicomachus, in fact, gives no definition of multiplication in his Arithmetical Introduction if it were not for what Abū l-Wafāʾ says about division: “Division according to the example (qiyās) of Euclid and Nicomachus is the partitioning (tafrīq) of one of two numbers by the amount of units in the other”.19 Neither Euclid nor Nicomachus defines division in their books, and if this statement is not intended to make such a claim, it at least implies that this definition, too, originates with the Greeks.

Contrary to what Abū l-Wafāʾ’s testimony might imply, there is evidence that the “duplication” definition of multiplication was already circulating among Arabic arithmeticians before they began reading Euclid. The definition appears in al-Khwārazmī’s algebra book, written sometime between 813 and 833 CE. This book presents the practical algebra connected with finger-reckoning, and the only hint of Greek influence is the presence of letters to label points in some diagrams.20 Later, al-Khwārazmī copied the definition into his book on calculating with Indian numerals. This way of characterizing multiplication is rather intuitive, and it is entirely possible that arithmeticians developed or learned it independent of Euclid.

It may have been his desire to tie practical Arabic arithmetic with the Greek tradition that led Abū l-Wafāʾ to imply a false origin for both definitions. We have found another instance of this kind of “false history” in the tenth century philosopher al-Fārābī. In the third part of his Enumeration of the Sciences he classifies the various branches of mathematics. He describes algebra as being “concerned with the ways of figuring out how to find numbers that apply Euclid’s principles on the rational and irrational in the tenth book of the Elements…”21 Yet not one of the extant books on algebra written before him, and to our knowledge none written after him, show any sign of applying principles from Book X to algebra.22 Al-Fārābī’s wishful connection of algebra to Euclid’s Elements seems to have been a way to legitimize the prevalence of irrational numbers in algebra and to suggest a Greek inspiration for the art.

Whatever its source, the “duplication” definition of multiplication was devised with whole numbers in mind. Abū l-Wafāʾ criticizes it on this point, and shows through an example of the price of cloth that it also works for fractions. He justifies this extension by appealing to propositions about lines, reinterpreted numerically, from Book II of the Elements.23 Ibn al-Bannāʾ gives a different definition for the multiplication of fractions, below at 149.2. Neither he nor al-Hawārī defines the multiplication of irrational numbers, even if these multiplications are performed in the book (183.2). Definitions in practical books like Ibn al-Bannāʾ’s should be read more as intuitive characterizations that introduce students to the concepts rather than precise definitions of the kind that Aristotle or David Hilbert would have approved.

95.3 Ibn al-Bannāʾ distinguishes between two types of multiplication. In the first type, one puts a copy of the multiplicand in place of each unit of the multiplier. The example is “three men: each of them has five dirhams. You multiply five by three, which gives fifteen dirhams”. Each of the three men is substituted with five dirhams, to get a total of fifteen dirhams. There is a change in meaning (maʿnan) because the three (men) became fifteen (dirhams). Also, the units shift from (three) men to (fifteen) dirhams, which is a change in terms (lafẓ).

The example for the second type is “five dirhams, how many thirds does it contain?”. This type is called “conversion” because the units for the amount “five dirhams” is converted from dirhams to thirds of a dirham. There are 5 of the former and 15 of the latter. The total amount “five dirhams” remains the same, so there is no change in meaning. But the shift in units, from dirhams to thirds of a dirham, is a change in terms. “All of what is in the multiplier in units”, or three (thirds of a dirham), “is equal to the one [dirham] of the multiplicand”. Thus the “[number of] units in the multiplier”, or three, “is the number of what is in one of the multiplicand in parts”, or, more loosely translated, “is the number in each part [i.e., each dirham] of the multiplicand”. Conversion of fractions is covered later, at 157.9, and the two corresponding types of division are explained beginning at 117.9.

Ibn al-Bannāʾ’s description of the first type, at 95.3, is problematic. It reads “in putting down the multiplier, each one of them is equal to the one of the multiplicand”. Reading the example for this type at 95.10, it could be translated as: “in putting down the multiplier, each one of them stands for (mithl) the single entity (wāḥid) of the multiplicand”. One should substitute for each unit in the multiplier the entire multiplicand, not just its unit (“the one”). But this translation is at odds with the description of the second type of multiplication, which begins: “all of what is in the multiplier in units is equal to the one of the multiplicand”. The same phrase “the one of the multiplicand” in this case must mean its unit, and not the multiplicand as a single entity. Perhaps the description of the first type is misstated.

In Lifting the Veil, Ibn al-Bannāʾ completes his descriptions and examples by writing “The first type is combining (tarkīb) and the second is dissecting (tafṣīl)”, though al-Hawārī does not copy it. In the example of the first type the three copies of five dirhams are combined, and in the example of the second type each of the five dirhams is dissected into three pieces.

Modern mathematicians work with only one kind of multiplication because our numbers all belong to the same abstract set. Instead, in Arabic arithmetic a number is a number of something, whether it be men, dirhams, mithqals, or abstract units. This is why Ibn al-Bannāʾ distinguishes between a change in terms and a change in meaning. (See also our comments on division, below at 117.9.) This idea of different kinds of number is also behind the names of the powers of ten (hundreds, thousands, etc.), the names of fractions (thirds, fourths, etc.), and the names of the powers of the unknown in algebra (numbers, things, māls, cubes, etc.). So “three men” was a particular kind of 3, like “three boats”, “three thousands”, “three fifths”, and “three things”.

Arabic authors often designated numbers by their kind, and several examples are found in al-Hawārī’s book. Just above, at 95.7, we read “the units in the multiplier”; at 114.12, we find “the ranks of the result”; at 208.1; “if we make the starlings twenty-four”; and at 215.15, “we square half of the things”. These nouns mean, respectively, “the number of units”, “the number of ranks”, “the number of starlings”, and “the number of things”. If one has 24 starlings, for example, then that collection of birds is an instance of the number 24, and one can indicate their number by saying merely “the starlings”.

This idea that a number has two aspects, its meaning (value) and its term (the kind of number), breaks down with irrational roots, though our authors do not discuss it. This is because it makes no sense to have an irrational number of anything, like \sqrt{5} bricks, for example. This becomes problematic in algebra, where the “coefficient” (the “number”, or meaning) of a term had to remain rational even when multiplying by an irrational number. See the discussion below in the last paragraph of our commentary to 219.1.

95.15 Rules for multiplying numbers, Ibn al-Bannāʾ tells us, fall into one of three categories: those with shifting, those with half-shifting, and those without any shifting. Methods involving shifting are designed for the dust-board or wax tablet, where it is easy to erase and rewrite the digits. Those without shifting were intended for the lawḥa with ink pen, and will also work on paper. He also describes some techniques of mental arithmetic at the end of the chapter, beginning at 108.9, that he took mainly from Ibn al-Yāsamīn’s Grafting of Opinions.

95.17 The first multiplication rule with shifting is called “sleeper multiplication”, perhaps because the numbers are placed horizontally as if they are sleeping on a bed. There is another method of “sleeper multiplication”, without shifting, that features numbers written horizontally. It is described at 103.14.

96.3 Al-Hawārī gives the example of multiplying 43 by 54. They are arranged on the dust-board like this:

The overall scheme will be to multiply the 4 of the 43 by the 54, then the 54 will be shifted to the right, and then the 3 from the 43 will be multiplied by the 54. First the 4 from the 43 is multiplied by the 5, and the 20 is placed above:

The 4 from the 43 is then multiplied by the 4 below, giving 16. The 1 from the 16 is added to the 0 we just wrote, and the 6 of the 16 replaces the 4:

The 54 is then shifted to the right one unit, giving this figure:

Now we multiply the 3 by the 5 below. The resulting 15 is added to the 16 above to get 31:

Now the 3 by 4 gives 12. The 1 is added to the 1 on top, and the 2 replaces the 3:

The result of the multiplication of 43 by 54 is 2,322.

It is important to know when to add a digit and when to replace a digit. One adds to digits that have been calculated, while one replaces digits from the original problem. This way, the digits of the multiplicand, 43, are replaced with the digits of the evolving calculated number.

97.1 The other method of multiplication with shifting is called vertical multiplication. It works the same way as the sleeper multiplication just described. Al-Hawārī’s example is to multiply 42 by 37. The digits are arranged vertically, with units on top:

Like before, we first multiply the 4 of the 42 by the 3 of the 37 to get 12. This is placed next to the 3, with the 2 above the 1:

The 4 is then multiplied by the 7, giving 28. This will be placed to the left, too. As before, we replace the digit 4 of the multiplicand with the 8, and we add the 2 to the calculated 2 that lies below it:

Now the multiplier is shifted up one unit:

The 2 on the left will now be multiplied by the 37. First, 2\times 3=6, so we add the 6 to the calculated 8, giving 14. Because of the extra digit, the 1 is added to the 4 below:

Finally, 2\times 7=14. The 4 replaces the 2, and the 1 is added to the 4 below it:

The result of multiplying 42 by 37 is 1,554.

98.4 The method by half-shifting is a technique for squaring numbers. Al-Hawārī’s example is to multiply 463 by itself. First, the number is written with dots between the digits:

The 4 is squared, and the resulting 16 is written above:

Next, double the 4 to get 8, and put this in place of the first dot:

This 8 is regarded as being shifted, and it is multiplied by the 6 to its right to get 48. This result is added above the 8, treating it like 160+48:

Next the 6 is squared, giving 36. This is added to the 2080 above:

Now the 6 is doubled to get 12, and this is shifted one place to the right so that it replaces the “6 •”. One must also shift the 8, treated as 80 since it is one place to the left, that was doubled before. This is added to the 12, making 92:

Now the last 3 must be multiplied by this shifted 92. The 3 by the 9 gives 27, which is added to the 16 above the 89, making 43:

Next 3 by 2 is 6, and this is added above the 2:

Last, we square the 3 to get 9, and add this above the 3:

The result of multiplying 463 by itself is 214,369.

99.14 There are several different ways to perform the third kind of multiplication, without any shifting. Ibn al-Bannāʾ first explains “table multiplication”, which we call lattice multiplication. Al-Hawārī’s example is to multiply 435 by 287. These are drawn around a grid with diagonals:

Ibn al-Bannāʾ mentions that the 287 can be placed on the left or the right. Most manuscripts, and our translation, show it on the right, but putting it on the left makes the final addition easier. Next, in each of the nine squares we put the product of the column digit by the row digit. For example, for the upper left square 7\times 4=28, so a 2 goes under the diagonal and an 8 above the diagonal. The spaces are then filled out like this:

We then add the numbers between the diagonal lines. The upper right shows a 5, which is the units. Between the first and second diagonal lines are a 1, 3, and 0. These add to 4, which is the tens place of the answer. Adding the numbers between the next two diagonals gives 8+2+4+4=18, so the hundreds place is an 8, and the 1 is carried to the next sum. Continue like this to get the answer 124,845.

101.16 For “vertical multiplication” al-Hawārī multiplies 183 by 347. These are arranged vertically, separated by some space:

The units place of the answer will be in the top row between the numbers, the tens place in the row below that, etc. First, we multiply the 3 from the 183 one by one by the digits of 347. The 21 from 3\times 7 is put on the right like this:

Next, 3\times 4=12. The 2 is added to the 2 in the tens place to become 4, and the 1 is placed below:

So far we have 141. As in the Oxford manuscript, we will cross out discarded digits. Next, the 3 is multiplied by the 3 to get 9, which is added to the 1 on the bottom to make 10:

Each digit on the left is then multiplied by the 8, bringing the diagram to this state:

Last, the 1 is multiplied by the 7, 4, and 3:

Thus 183\times 347=63{,}501.

103.14 The horizontal version of “vertical multiplication” is called “sleeper multiplication”, like the method described above at 95.17. Al-Hawārī multiplies 253 by 987, first writing them on two lines like this:

First, the 3 is multiplied by the digits in 987, beginning with the 7. The 21 is written above:

Next is 3\times 8=24. The 4 is added to the 2 of the 21, making 6, which is placed above, and the 2 is put to the left:

So far the result is 261. Again, we cross out defunct digits as we go, but most manuscripts just leave them alone. Next, 3\times 9=27, and this is added the same way:

Now the 5 is multiplied by the 7, 8, and 9, giving this figure:

Finally, after multiplying the digits by 2, the final figure is:

The result of multiplying 253 by 987 is 249,711.

104.1 For rūmī calculation see 85.16 above and 1.3 in the Introduction.

106.11 This next method, called “repetition”, is just a curiosity of calculation. It works when the digits in each number are all the same and they both have the same number of digits, like 7777 by 9999, or, in al-Hawārī’s example, 444 by 333. These are written on two lines like this:

Under this we put a 1, 2, 3, etc., until we get to the end of one number and the beginning of the next. Then the numbers descend back to 1. These numbers will serves as multipliers:

Multiply the 3 of 333 by the 4 of 444 to get 12. Multiplying the 12 by the 1 on the left gives 12, which is put above like this:

Next, the 12 is multiplied by the 2 to get 24, and this is added above, one place to the right. The 2 is added to the 2 of the last 12 to make 4, so we replace it as we did in the last scheme:

The 12 is then multiplied by the 3, giving 36, and this is added above the same way:

Continuing, we multiply the 12 by the 2, and then finally the 12 by the 1. The final figure is:

So the product of 333 by 444 is 147,852.

108.9 The techniques of multiplication presented so far are meant to be worked out in Indian notation on a board, and can be applied to any two (positive) integers. The remaining techniques are either shortcuts originating in finger-reckoning, or they are board techniques that work for specific kinds of numbers. Many of these rules were copied by Ibn al-Bannāʾ from Ibn al-Yāsamīn’s Grafting of Opinions, a book that is a kind of hybrid between Indian calculation and finger reckoning. Even though that book is expressly devoted to calculation with Indian numerals, it is organized like a book on finger reckoning in that it covers multiplication and division before passing to addition and subtraction; also, it still retains the rules of mental calculation that Ibn al-Bannāʾ copied into his book. In the following breakdown of Ibn al-Bannāʾ’s remaining rules, we indicate which come from finger-reckoning (FR) and which were taken from Ibn al-Yāsamīn (Y):

In addition, at 109.16 al-Hawārī gives a variation on a finger-reckoning rule that he attributes to al-Yāsamīn.

Multiplication “by excess” is a trick for doing the calculation mentally when at least one of the numbers is between 10 and 19. To multiply 12 by 15, think of the 12 as 2 more than 10. Divide the 2 by the 10, which is 1\over 5, and multiply this by the 15 to get 3. Then add this 3 back to the 15 to get 18, and multiply by 10 to get the answer, 180. Al-Hawārī then does it again, switching the roles of the 12 and 15. In general, to multiply a number of the form 10+a ( a is a digit) by another number b, one calculates ({a\over 10}b+b)\cdot 10. Al-Hawārī gives a second example, 13\times 17, that gives fractions. Try it in your head! 3\over 10 of 17 is 51\over 10, or 5{1\over 10}. Add this to the 17 to get 22{1\over 10}. Finally, multiply by 10 to get the product, which is 221.

109.7 Another trick for mental calculation is called “denomination”. Al-Hawārī’s example is to multiply 6 by 12. First denominate (we would say “divide”) one of the numbers with their sum. Dividing 6 by 18 gives 1\over 3. Then multiply this by the other number, 12, to get 4. Last, multiply this 4 by the sum, 18, giving 72.

Written algebraically, to multiply a by b one performs these operations: {a\over a+b}\cdot b\cdot (a+b). This method can only be useful if the two numbers have a common divisor. Otherwise, the multiplication of a\over a+b by b will require one to find a\cdot b, which is the original problem.

109.16 A variation on this rule can be written in modern notation as a\cdot b=(a-{a\over a+b}a)(a+b). Al-Hawārī gives Ibn al-Yāsamīn as his source, though this rule is not in that author’s Grafting of Opinions. We do not know its origin.

109.19 Another denomination method is to divide one of the two numbers by a power of ten, and then multiply the result by the other number. This is then multiplied by the power of ten to get the product. Al-Hawārī first works out the example 24\times 8. He divides the 8 by 10 to get 4\over 5. Then {4\over 5}\times 24=19{1\over 5}. Multiplying this by 10 gives the product 192. The advantage here is that the numbers are kept small until the end when the power of ten is multiplied back.

110.5 The Arabic word we translate as “power of ten” is ʿaqd. The use of this word in arithmetic originated in finger-reckoning. Historian A. S. Saidan wrote: “In Arabic ʿuqda is the name of the finger-joint, and ʿaqd in this sense should mean ‘to bend the finger-joint’”.24 He later described the word in more detail:

This placement [of the fingers] is called ʿaqd, plural ʿuqūd. Thus the finger-reckoner understood numbers as formed of places, namely units, tens, hundreds, etc., each place having one or the other of the nine ʿuqūd: one, two,… nine. With this understanding the word ʿuqūd came to mean what we may now call digits. But in usage ʿaqd and place were not always clearly distinguished.25

The first occurrence of the word ʿaqd in al-Hawārī’s book is in a rule of Ibn al-Bannāʾ given above at 87.17. There it takes the meaning of “power of ten” or “place”. The other seven instances of the word are in the present chapter covering finger reckoning rules, from 109.19 to 111.3. When reading “power of ten” here one should keep in mind its association with the positioning of the fingers.

The definition at 110.5 is not accurate. The only non-zero rank should be a ten or a hundred or the like, not just the first (awwal). The way the condition is stated, a number like 310 would be a “simple power of ten”.

110.12 Al-Hawārī then works out 12\times 15, which he solves by working through 15\div 10=1{1\over 2}, 1{1\over 2}\times 12=18, 18\times 10=180. He then works it out again, this time subtracting 5 from the 15 so that the first operation gives a whole number. He takes 15-5=10, then 10\div 10=1, 1\times 12=12, and 12\times 10=120. To compensate for the subtracted 5, he calculates 5\times 12=60, and this is added to the 120 to get 180, which is the product.

111.1 Next al-Hawārī finds 3\times 15 similarly, but by adding rather than subtracting. Adding 2 to the 3 gives 5, and dividing that by 10 gives 1\over 2. This 1\over 2 may not be a whole number, but it is easier to work with than 3\over 10. Then {1\over 2}\times 15= 7{1\over 2}, and “we raise each [digit] by ten” to get 75. From this we must subtract 2\times 15, or 30, to get 45, which is the answer. With practice, methods like these prove to be quite useful.

111.7 Recall the “repetition” method at 106.11, which was covered just before the techniques of mental multiplication. That method requires that the number of digits in the multipliers be the same, and that all the digits in each multiplier be equal, as in al-Hawārī’s example 333\times 444. This next method, called “nines”, works when one of the two multipliers consists of all 9s. Al-Hawārī works through the example 444\times 999. We put them on two lines, and above them we put a row of dots equal to the sum of the number of places of the two numbers, in this case six:

First, 4\times 9=36. The 6 replaces the right-hand dot, and the 3 replaces the middle dot of the remaining dots:

Next, the difference 9-4=5 replaces the dots between the 3 and 6:

Last, the remaining dots are replaced with 4s:

Then 444\times 999=443{,}556.

To see why this works, note that multiplying a number of the form aaa (i.e., 100a+10a+a) by 999 is the same as aaa by 1000-1, which is of the form aaa000-aaa. The digits of the answer must then be a, a, a-1, 9-a, 9-a, and 10-a.

112.5 Next is another method of multiplying by a number expressed with all 9s. This one has no restriction on the other number. Al-Hawārī’s example is 999\times 9{,}354. Add to the 9,354 as many 0s as there are 9s in 999 to get 9,354,000. Then subtract the 9,354 to get 9,344,646, which is the desired product.

112.16 The method called “squaring” derives from the fact that [{1\over 2}(b+a)]^2-[{1\over 2}(b-a)]^2=ab. The method is much simpler to apply mentally than the modern formula suggests. To multiply 17 by 19, al-Hawārī squares half their sum, 18, to get 324. From this he subtracts a square of half the difference between them, which is 1, to get 323. This is the desired product.

Incidentally, in his Lifting the Veil Ibn al-Bannāʾ appropriates this rule as the foundation for arithmetical proofs for the rules for solving the three composite algebraic equations. The same rules are stated and illustrated in the present work starting at 214.7, but without proofs.26

113.6 Another “squaring” method entails squaring one of the two numbers and multiplying or dividing the result by their ratio. To multiply 25\times 15, al-Hawārī squares the 25 to get 625. Because 25 is the greater of the two numbers, its square is multiplied by the ratio of 15 to 25, or 3\over 5, to get 375, which is the answer. He then works it out by squaring the 15 to get 225. Because 15 is the smaller number, its square is divided by the ratio 3\over 5, again giving 375. Algebraically, a\cdot b=a^2\cdot {b\over a}.

113.16 In this next method, Ibn al-Bannāʾ makes use of rules that can be written in modern notation as ab=b^2-(b-a)b and ab=a^2+(b-a)a, for a<b. Al-Hawārī multiplies 36 by 14 using both rules. For the first rule, the difference 22 is multiplied by 36 to get 792. This is taken from 1296, a square of 36, leaving the answer 504. For the second rule, the difference 22 is multiplied by 14 to get 308. This is added to 196, a square of 14, to get the same answer, 504.

Al-Hawārī did not pick a good example to illustrate the utility of this trick. In one step in the first rule he has to multiply 22 by 36, which itself is no easier than finding 36 by 14 directly. The rules have an advantage if the difference between a and b is a nice number, like in the example of 16 by 26. The difference is 10, which multiplied by 26 easily gives 260. Subtract this from 676, a square of 26, to get 416. Or multiply the 10 by the 16 to get 160, and add this to 256, a square of 16, to again get 416.

114.4 The last trick deals with multiplying multiples of powers of 10. To multiply 30 by 140 al-Hawārī first multiplies 3 by 14 to get 42, and to this he adds back the zeros to get the answer, which is 4,200.

115.1 Students should memorize the multiplications of the whole numbers from 1 to 10.

117.1 Section I.1.5. Division.

117.2 Ibn al-Bannāʾ may not have taken into account non-integers in his definition of multiplication at 95.2 above, but he did so for division. He copied two definitions from Ibn al-Yāsamīn, one at 117.2, which calls for the decomposition of the dividend into equal parts, and the other at 117.5, based instead on ratio. Al-Hawārī copies Ibn al-Bannāʾ’s explanations from Lifting the Veil that the first “applies to discrete quantities”, that is, to whole numbers, while the second “concerns continuous quantities”, which in his case are the numbers of the arithmeticians that include fractions and irrationals. This second definition is echoed in Ibn al-Bannāʾ’s definition of a fraction, below at 133.1, but there the numbers in the ratio are assumed to be whole numbers. Neither definition is known from Greek sources (but see above at 95.2 for a discussion of Abū l-Wafāʾ’s attempt to link the first definition of division to Euclid and Nicomachus). At 117.7 Ibn al-Bannāʾ then gives the definition of division that is observed by “most people”, which instead counts how many divisors are in the dividend.

117.9 The two meanings of division that he then offers stem from the two definitions at 117.2 and 117.7. The first is “the division of a type by another type, like dirhams by men”, and the second is “the division of a type by the same type”. This distinction corresponds to the notion of a change in terms for the two kinds of multiplication, above at 95.3.

117.16 The examples of the two meanings, here and at 118.1, mirror the examples for multiplication at 95.10 and 95.12. For the first meaning, Ibn al-Bannāʾ divides 15 dirhams equally among five men, and for the second, he divides a piece of wood of fifteen spans into pieces of wood of three spans. The explanations given here correspond to those for multiplication at 95.3 and 95.6: decomposing (the 15 dirhams) and uniting (the 15 spans into groups of three) are the opposites of the combining (5 dirhams of 3 men) and dissecting (5 dirhams into thirds of a dirham) that we saw for multiplication.

118.14 A distinction was often made in medieval Arabic arithmetic between dividing a greater number by a smaller number and dividing a smaller number by a greater number. Books written in the finger-reckoning tradition typically “divide” (qasama) the greater by the smaller, and “relate” (nasaba) the smaller to the greater.27 Nasaba is the verb associated with nisba, the word for “ratio”. Ibn al-Bannāʾ and al-Hawārī use qasama similarly, but they follow al-Ḥaṣṣār and Ibn al-Yāsamīn by “denominating” (from sammā) the smaller with the greater. To denominate 3 with 7 means to give 3 the denomination, or name, “sevenths”. Saying the result “three sevenths” is like saying “three cents”, but with a different denomination. The prepositions are different between division and denomination, too. One divides a number by (ʿalā) a smaller number, while one denominates a number with (min) a greater number.

118.17 Al-Hawārī illustrates the method of dividing a greater number by a smaller number with the example 245\div 12. First, the numbers are put on two lines like this, with the highest power terms lined up:

The 12 goes into the 24 of the 245 two times, so we write a 2 under the 12. Because the 2 and 4 of 245 are exhausted, they are replaced with zeros:

Next, shift the 12 one place to the right:

The 12 does not go into the 5 at all, so a zero is put below and the remainder is 5:

The remaining 5/12 is “two sixths and half a sixth”, which is written as 1~\,2\over 2~\,6. This kind of fraction is explained at 138.1. The final answer is {1~\,2\over 2~\,6}\,20, read right-to-left.

119.18 The rules Ibn al-Bannāʾ gives from here up to and including 124.8 are from finger-reckoning. The first technique is to partition the dividend and then add the respective quotients. Al-Hawārī’s example is to divide 44 by 11. He breaks up the 44 into 22 and 22, then he divides 22 by 11 to get 2 for each of the 22s. The quotient is then 2+2=4. This technique might seem more useful with a problem like dividing 399 by 19. Thinking of 399 as 380+19, we divide each number by 19 to get 20+1=21, which is the answer. In general, (a+b)\div c=(a\div c)+(b\div c).

The second technique is to factor the divisor and divide by each of them in turn. To divide 96 by 12, al-Hawārī decomposes the 12 into 2\cdot 6. He finds that 96\div 2=48, and this he divides by the 6 to get 8, which is the answer. In general, a\div (b\cdot c)=(a\div b)\div c.

The third technique is to “reconcile” (wafiqa) the dividend and the divisor, which means to cancel common divisors. To divide 35 by 15, al-Hawārī takes a fifth of the dividend and the divisor to change the problem into 7\div 3, which gives “two and a third”. In general, ab\div ac= b\div c.

120.20 In “apportionment” (muḥāṣṣa), a certain quantity is deducted proportionally from two or more amounts. In al-Hawārī’s example three people want to give a total of ten dinars to a bankrupt friend, and they do it in such a way that each of them gives an amount proportional to how much each of them has, and they have 4, 5, and 6 dinars, respectively. In modern notation, we want to find three numbers a, b, and c such that a+b+c=10 and {a\over 4}={b\over 5}={c\over 6}.

Al-Hawārī begins by adding up the wealth of the three people: 4+5+6=15. These “surpass them”; in other words, they surpass the three “apportioned parts” that will be given, and which will add up to ten dinars. For the first friend’s gift al-Hawārī multiplies the 10 by the 4 dinars to get 40 dinars, and he divides the result by 15 to get his share, or “apportioned part”, which is 2{2\over 3} dinars. The same procedure gives the other shares. For the second friend this is 5\cdot 10\div 15=3{1\over 3} dinars, and for the third it is 6\cdot 10\div 15=4 dinars. This works because the first must give 4/15 of the total amount of 10 dinars, the second 5/15, and the third 6/15, and the proportions add to 1.

121.14 Al-Hawārī mentions four variations on this method, each one being a reordering of the operations. For example, where the share of the first friend was calculated above by 10\times 4, then \div 15, the first variation is to calculate 4\div 15, then \times 10. The second is to calculate 10\div 15, then \times 4. In the third, we find 15\div 4, then we divide 10 by the result. In the fourth we find 15\div 10, then we divide 4 by the result. These variations were probably devised for cases in which one or another would be easier. For example, if instead of 4, 15, and 10, we had 8, 26, and 22, it would be easier to apply the first variation, by dividing 8 by 16 to get a half, then multiplying this by the 22 to get 11. The original way has us multiplying 8 by 22 and then dividing by 16, which is more difficult.

122.2 In case there are fractions, multiply the numbers corresponding to the wealth of the friends by the least common multiple of their fractional parts. And if the fractions have common divisors, divide by those factors. The amount of money given to the bankrupt friend remains the same.

By “they are all different (mutabāyna)” in the passage at 122.10 Ibn al-Bannāʾ means they are relatively prime. Also, we translate ishtirāk, a word whose ordinary meaning is “common”, as “common divisor”.

Al-Hawārī considers the situation in which a total of 12 dinars is given by three friends who have 4{1\over 3}, 5{1\over 4}, and 6{1\over 6} dinars, respectively. He first calculates the least common multiple of the denominators by collecting factors: a 3 from the third, two 2s from the 4, and nothing from the 6 because we already have a 2 and a 3. So the least common multiple is 12. The 4{1\over 3}, 5{1\over 4}, and 6{1\over 6} are then multiplied by 12 to get 52, 63, and 74. Next he checks to see if there is a common factor among these numbers that can be cancelled out. In this case, the numbers are relatively prime. He then finishes the problem, following the rule in the first example.

123.18 Next, Ibn al-Bannāʾ gives the rule for expressing the result of a denomination as a related fraction (see below at 135.10 and 138.1 for this type). We saw one example of this notation already at 119.7, where “two sixths and half a sixth” was written as 1~\,2\over 2~\,6.

123.22 The example is to denominate 11 with 15. Decompose the 15 into 5 and 3 and put them under a line like this:

Then divide the 11 by the 3 and put the remainder, 2, above it:

Then divide the quotient 3 from 11\div 3 by the 5. Since 3 is less than 5, 3 is the remainder, and it too is written above:

This is spoken as “three fifths and two thirds of a fifth”, and we might write it in our notation like this: {3\over 5}+{2\over 3\cdot 5}.

This process was followed in situations where the denominating number is composite and greater than ten. It was convention that the denominators descend. One said “three fifths and two thirds of a fifth”, 2~\,3\over 3~\,5, rather than “two thirds and a fifth of a third” 1~\,2\over 5~\,3. Also, it was preferred to start with the largest possible denominator, so we read “half a sixth” more often than “a third of a fourth”.

124.8 There are three “lesser known” ways to denominate. Al-Hawārī’s example of the first is to denominate 4 with 12. He switches the numbers and divides 12 by 4, then denominates 1 with the result. Here 12\div 4=3, and the denomination gives 1\over 3. This works best in simple cases where there is a common factor. It would not work so well with a problem like 11\div 15, since 15\div 11=1{4\over 11}, and it would be too difficult to denominate 1 by this number. This method uses the property that {a\over b}=1/{b\over a}.

His example for the second way is to denominate 9 with 15. Denominating 1 with the 15 gives “a third of a fifth”, or 1~\,0\over 3~\,5. Multiplying this by 9 gives “three fifths”, or 3\over 5. Here, to find a\div b, one first finds 1\div b, then the result is multiplied by a.

In the third way, al-Hawārī denominates 10 with 16. The idea is to turn denomination into division by multiplying the 10 by some number to make it greater than 16. He chooses 8, to get 10\times 8=80. Dividing this by 16 gives 5, and to compensate for the multiplied 8 he denominates the result with 8 to get 5\over 8. This works best when the new numerator (here 80) is a multiple of the denominator, and the method is convenient only when there is a common factor (in this case 2). In modern notation, the motive becomes somewhat lost: if a<b then {a\over b}={ac\over b}/c.

124.20 Ibn al-Bannāʾ now turns to the decomposition of numbers into their prime factors. This will be useful later in the manipulations of the denominators of fractions. He gives several rules, some of which warrant explanation because of the way they are worded.

The first case is a number that “does not begin with units”. Recall that the units are 1, 2,…, 9. The 0, signifying nothing, stands for an empty place. So a number that “does not begin with units” has a 0 in the units place, and is divisible by 10, 5, and 2. This is what he means when he says that it “has a tenth and a fifth and a half”.

125.5 One way of finding factors is to consider the residue after casting out nines or eights or sevens. Ibn al-Bannāʾ first covers various cases for a number whose units digit is even. The result of casting out nines is also the remainder after dividing by 9, so from this we can tell if the number is divisible by 3 or 9. If it is cast out entirely by nines – in other words, if nothing is left – then the number “has a ninth and a sixth and a third”, or, in our language, it is divisible by 9, 6, and 3.

Al-Hawārī gives the example of 36. He does not mention that it is also divisible by 18. There is no need to say it is divisible by 2 since the number is already known to be even. If the residue is 3 or 6 then the number is divisible by 3, and because we already knew it was even, it is also divisible by 6. Al-Hawārī gives the examples 66 and 42.

If the residue is some other number (1, 2, 4, 5, 7, or 8), then cast out eights. If it is cast out entirely then the number is divisible by 8 and 4, and if the residue is 4, then it is divisible by 4. If the residue is some other number, then cast out sevens. Because 7 is prime, the only rule is that if it is cast out entirely then the number is divisible by 7.

Similar rules are then given for odd numbers.

126.14 Subsection on finding deaf parts.

If all these rules have been applied and the original number is still not decomposed completely, “then look for deaf parts by dividing by them”. (We explain the term “deaf part”, meaning “prime”, below at 134.2.)

127.9 To find deaf parts, Ibn al-Bannāʾ explains how to draw a table of odd numbers to make the sieve. This is the famous “sieve of Eratosthenes”, after the third century BC Greek mathematician. Ibn al-Bannāʾ’s description ultimately derives from Nicomachus’s Arithmetical Introduction.28 Al-Hawārī draws the table to find the prime numbers less than 145. The first number in the table is 3, which is prime. Putting a bar over every third number after that gives:

The next number in the table without a bar is the 5, so it is prime. Put a bar likewise above every fifth number after it, starting with 15, 25, etc. Then do the same for the next number without a bar, which is 7, then for 11. There is no need to do this for 13 or any greater prime, since their squares are greater than 145. The table should now look like this:

Numbers with a bar are composite, and the remaining numbers are prime. Al-Hawārī writes that “these deaf parts are counted only by one”, or as we would say, prime numbers are divisible only by 1. One can say, for example, that 5 counts 40 because one can count to 40 by fives. But one cannot count to a prime number by anything but ones.

128.1 Al-Hawārī writes that he is filling the table to 145, but below that he writes that we know, presumably from the table, that 151 is prime. The Oxford and Medina manuscripts show the table to 145, while the Tunis and Tehran manuscripts show it to 199. The table in the Istanbul manuscript is drawn with 15 columns and 10 rows. The copyist must have been confused. He began by filling in the odd numbers, from 3 to 37, and then continued by writing in only prime numbers from 83 to 157. The rest of the squares are blank, and no marks are drawn above any number.

129.1 Section I.1.6. Restoration and reduction.

In problems involving proportion it is often necessary to restore or reduce a number to another number. “Restoration” (al-jabr) is what we do when we want to increase a number to a greater number, and “reduction” (al-ḥaṭṭ) is for reducing a number to a smaller number. These operations are applied, for instance, to set the “coefficient” of the highest power in an algebraic equation to 1 (at 217.1). Al-jabr is also the word used in the simplification of subtractions and equations in algebra (in the text at 211.2, 220.5, 223.1, and in our commentary at 223.1). Restoration and reduction of fractions is covered at 154.1.

131.1 Chapter I.2. Fractions.

Fractions in Arabic arithmetic are numbers that result from partitioning the unit. Ibn al-Bannāʾ, for example, writes in the passage at 137.1 the fraction “fifteen…parts of twenty-four parts of the unit” ( 15\over 24). The unit is partitioned into 24 parts, and the fraction is 15 of those parts. But to define fractions, Ibn al-Bannāʾ feels obliged to respect the Greek notion that the unit is indivisible. Because of this he identifies fractions with ratios of whole numbers, so the fraction just mentioned is considered to be the ratio of 15 to 24. A ratio according to Euclid is “a sort of relation in respect of size between two magnitudes of the same kind”.29 Ratios are not mathematical objects in themselves, but are only relations between such objects. This way, the unit maintains its integrity. Ibn al-Bannāʾ’s definition only serves to provide Arabic fractions with a semblance of a Greek foundation. It had no impact on actual calculations, and in Arabic arithmetic books, al-Hawārī’s included, fractions were understood to be fractional portions of the unit.30

Euclid defines “part” and “parts” at the beginning of Elements, Book VII, the first of his three books on number theory: “A number is a part of a number, the less of the greater, when it measures the greater…” For example, four is “a part” of twelve because twelve can be broken into three equal parts, each one of them four. He continues, “…but parts when it does not measure it”.31 Eight, for example, is “parts” of fourteen because, if we consider the number two as one part, then eight is four of the seven parts making up fourteen, or, as we might say, eight is four-sevenths of fourteen. Similarly, five is “parts” of twelve where one part is the unit.

The Greek term meaning “part” is translated into Arabic as juzʾ. This is the same word meaning “part” used in the statements of simple fractions in Arabic with (usually prime) denominators greater than ten, like Ibn al-Bannāʾ’s 15\over 24 quoted above. The big difference is that in Greek “a part” is always a positive integer that is part of a greater integer, while in Arabic fractions it is the unit itself that is partitioned.

We describe the different ways Arabic arithmeticians expressed fractions below at 134.2. It is there that we explain Ibn al-Bannāʾ’s phrase “the part and its name”.

133.3 Al-Hawārī goes further than his teacher in addressing the ontology of fractions. Had he worked with the common notion of fractions as parts of a divisible unit, he could have written that a fraction like “three sixths” is named in terms of sixths, much like “three cats” is named in terms of cats. In both cases the “three” modifies the name. But al-Hawārī follows Ibn al-Bannāʾ in formally identifying fractions with ratios. He argues that the ratio of three to six is not named in terms of either number, nor in terms of the two together.32

Al-Hawārī’s distinction between “sensible” and “intelligible” ultimately comes from Aristotle. For Aristotle, mathematical objects are sensible objects. We experience lines, squares, spheres, and numbers through our senses as attributes of the physical things we see and touch. But a ratio, being a relation between two objects, cannot be apprehended through the senses. It is an intelligible object that can only be imagined in the mind.33 Al-Hawārī’s statement that ratios are intelligible objects is the only philosophical observation he makes in the book. We suspected that he may have copied it from somewhere, so we checked the obvious places: Aristotle, Ibn Sīnā, and Ibn al-Bannāʾ. Aristotle makes no comment on the ontology of ratios in his extant works, nor did we find anything in Ibn al-Bannāʾ’s writings. But we did find this remark of Ibn Sīnā on relations in general: “As for predicating [the quiddity of the relative] with respect to another, this occurs only in the mind”.34

We set this aside for a moment and continue with the rest of the passage. Al-Hawārī next paraphrases Ibn al-Bannāʾ’s Lifting the Veil35 when he explains the word “fraction” by comparing a fractional number with fractured land. Perhaps he was thinking of the incremental increase of fractions with the same denominator, like 1\over 17, 2\over 17, 3\over 17, etc., that model the incremental strata of some rocky landscape. He also compares fractions with geometric magnitudes, probably because lines, surfaces, and bodies can be partitioned into arbitrarily many parts. Al-Hawārī contrasts these continuous magnitudes with “discrete quantities”, echoing the notion in Aristotle and Euclid that numbers do not admit fractions due to the indivisibility of the unit.

We contend that Ibn al-Bannāʾ related nearly the entire passage at 133.3, from “For example…” to “…similar abstractions”, verbally to al-Hawārī. Ibn al-Bannāʾ was familiar with Ibn Sīnā’s work, and al-Hawārī paraphrased rather than quoted Ibn al-Bannāʾ’s comparison of fractions with terrain from Lifting the Veil.

134.1 Section I.2.1. The names of fractions and numerating them.

134.2 Fractions in most Arabic books on Indian arithmetic, al-Hawārī’s included, are borrowed from finger-reckoning. This particular system has its origin in the ancient Egyptian practice of expressing fractions as sums of unit fractions, like writing 2\over 5 as the sum of 1\over 3 and 1\over 15. It was later modified, probably by Greek calculators, with approximation techniques based in sexagesimal arithmetic.36

At some point before the ninth century CE the restriction to unit fractions was dropped. One could now say fractions like “five sevenths” and “two ninths”. But in Arabic fractions could only be formed from nine names, or “heads” (ruʾūs): “a half”, “a third”, “a fourth”, up to “a tenth”. There were no words in Arabic for “eleventh”, “twelfth”, “thirteenth”, etc., so fractions with denominators greater than ten were expressed when possible by combinations of the heads. For instance, al-Khwārazmī wrote “a fifth and four fifths of a fifth” for 9\over 25 and “a ninth and a tenth” for 19\over 90.37 The various ways of combining the heads already present in ninth century texts remained common throughout the medieval period, and are explained by al-Hawārī in the present section.

Numbers in medieval Arabic that could be expressed in words were called “expressible” (munṭaq).38 These included integers and fractions reducible to combinations of the heads. Numbers inexpressible in words, for which only approximations could be found, were often called “deaf” (aṣamm). These included irrational roots, and also fractions irreducible to combinations of the heads, that is, fractions whose denominators contain prime factors greater than ten. The problem of deaf fractions was overcome some time before the early ninth century with the introduction of “parts”. For example, al-Khwārazmī wrote 4\over 13 as “four parts of thirteen parts of a dirham”.39 With the dirham divided into thirteen equal parts, the fraction is four of those parts. So now, in addition to being able to say “four fifths” and “four sevenths”, one could say “four parts”. Thus, “a part” became the tenth name for the naming of fractions.

This accounts for the origin of the term “deaf parts” for prime numbers, above at 126.14. To express the result of a denomination as a related fraction, explained at 123.18 above, one needs to find the prime factorization of the denominated number. If this number contains a factor not divisible by 2, 3, 5, or 7, then one needs to look for “deaf parts”, that is, to find the greater prime denominators required for the fraction. The example at 151.14 results in “fifty parts of one hundred thirty-seven parts and five ninths of a part of one hundred thirty-seven parts”. The deaf parts here are the 137 parts. Prime numbers do not quite coincide with the denominators of deaf fractions, since the former include 2, 3, 5, and 7, but the association was close enough.

In arithmetic, the naming of fractions with parts typically terminates with “of a unit/one” or “of a dirham” (units were often counted in dirhams, a silver coin, in many calculations). In one problem al-Karajī writes “thirty-five parts of eighty-three parts of a unit” and then, just after, “thirty-four parts of eighty-three parts of a dirham”.40 Often, as in al-Hawārī, the designation is left off altogether. So his “a part of eleven” is short for “a part of eleven parts of one/a unit/a dirham”. Also, it is not just abstract units that are partitioned in Arabic fractions. In the inheritance problems solved by al-Khwārazmī it is sometimes a share of the estate, as in “twenty-three parts of fifty-nine parts of a share”,41 and in the context of algebra Abū Kāmil writes “fifteen parts of thirty-nine parts of a thing”, where the “thing” is the name of the first degree unknown.42

Grammatically, the ten names function like other Arabic nouns. Saying “three fourths” or “three parts” is like saying “three apples”. The “fourths”, “parts”, and “apples” are the names, or kinds of object counted, while the “three” tells how many there are (the same applies also to “shares” and “things”). Our own words “numerator” and “denominator” reflect this idea. The word “denominator” derives from the Latin dēnōmināre, “to call, to name”, and our word “name” comes from the related Latin word nōmen. Our “numerator” is the “number” of this name.

There was no rule that the language of parts could only be used for deaf fractions. Al-Khwārazmī and Abū Kāmil routinely use it to name fractions like 25\over 36, 14\over 15, 5\over 12, and 4\over 25. And when convenient for the calculations the fraction can even be improper, like al-Khwārazmī’s “twenty-eight parts of thirteen of a dirham”.43 Al-Hawārī, though, works with combinations of the heads whenever possible, and his numerator is always less than his denominator.

Although schoolbooks today often still explain fractions in terms of parts, mathematicians define them in terms of division. The value of al-Khwārazmī’s “four parts of thirteen parts of a dirham” may be equal to the result of dividing 4 by 13, but the two ways of regarding fractions are not equivalent. With quotients in mind, we allow numerators and denominators to be irrational, like 1\over \sqrt{2} and \sqrt{2}\over 2. But these numbers cannot be fractions in medieval Arabic. The first makes no sense because one cannot partition the unit (or anything else, for that matter) into an irrational number of parts. And the second does not work because the number of parts making up a fraction cannot be irrational. To have \sqrt{2} parts is just as meaningless as having \sqrt{2} loaves of bread. One can perform the corresponding divisions, however, to get the acceptable “root of a half” ( \sqrt{1\over 2}). We explain this below at 188.1.

135.1 When the denominator is a composite number greater than ten it was common to express it with “two or more names”. Al-Hawārī’s first example is “two eighths and a seventh of an eighth”, which in notation looks like this: 1~\,2\over 7~\,8. We can write this as a modern fraction by working it out to get 15\over 56. Although it may sound overly complicated to us, saying “two eighths and a seventh of an eighth” gives a good idea of the magnitude of the fraction. We know that “two eighths” is a fourth, and we are adding to that the small amount “a seventh of an eighth”. The fraction is expressly presented as being just a little over 2\over 8, while with our 15\over 56 this is not immediately clear. One of us even found himself thinking in terms of these fractions while measuring wood with a ruler scaled in inches. The fractional part was clearly three fourths and a fourth of a fourth, and it would have been much less transparent as well as superfluous to convert 1~\,3\over 4~\,4 to 13\over 16. We can all grasp the magnitudes of the fractions 1\over 2, {1\over 3},\ldots ,{1\over 10}, and combinations of these are often easier to understand than fractions with large composite denominators.

An example in which the related fraction is not easier to grasp is Ibn al-Bannāʾ’s artificial example 2~\,4~\,5\over 3~\,5~\,6 (at 135.10 below), which is equal to 89\over 90. And just as our fractions do not admit of a unique representation ( {1\over 2}={2\over 4}={3\over 6} etc.), related fractions can be written in many ways. One example is al-Hawārī’s 3~\,4\over 6~\,9 in the example at 171.6, which is equal to 1\over 2.

Al-Hawārī uses the language of parts when the denominator is not prime in only two instances, at 155.2 and 156.7. There the denominators are 87 and 93.44

135.8 To expand on Ibn al-Bannāʾs remark, al-Hawārī copies Lifting the Veil from 135.10 to 137.10 to explain how to find the numerators of three kinds of fractions: related, distinct, and portioned. These types are covered again immediately after by al-Hawārī himself, along with other types. Finding the numerator is necessary to perform every kind of operation: addition, subtraction (including casting out, above at 93.9), multiplication, division/denomination, conversion, and finding square roots.

135.10 Ibn al-Bannāʾ works with the example 2~\,4~\,5\over 3~\,5~\,6 to show how to find the numerator and denominator of a related fraction. These fractions are expressed with more than one name, or denominator, so that each part is related to, or is a fraction of, what precedes it. In this example, the 5 in the top is multiplied by the second denominator, also a 5, to get 25, and this is added to the 4 to get 29. So the first two terms of the fraction are equivalent to 29\over 5\cdot 6, or 29\over 30. Then this 29 is multiplied by the 3 to get 87, which is added to the 2 above, resulting in the numerator 89. The fraction is the same as our 89\over 90.

136.8 Sometimes two fractions are gathered with the conjunction “and” (wa), like the example “five sixths and four fifths”. In notation, one is written next to the other like this: {4\over 5}\,{5\over 6}. This is called a distinct fraction, and its value is the sum of the individual fractions. To find the numerator, Ibn al-Bannāʾ multiplies the 5\over 6 by the denominator 5 of the other fraction to get 25\over 30 (though he does not name this fraction). He then multiplies the 4\over 5 by the 6 to get 24\over 30. Together they are 49\over 30, or forty-nine “parts of thirty” (or “fifths of sixths” or “sixths of fifths”), so the numerator is 49.

137.1 A portioned fraction is one like “three fourths of five sixths”, written as 5~\bullet ~3\over 6 ~\bullet ~4. This may be equivalent to the product of the fractions, but it was conceived and stated as a fraction of a fraction. In modern notation the example becomes 15\over 24.

137.11 Al-Hawārī now returns to commenting on the Condensed Book. Ibn al-Bannāʾ names five different kinds of fraction, and al-Hawārī shows how to find the numerator of each.

137.13 Simple fractions. The numerator of a simple fraction, like “a seventh”, written 1\over 7, is the number above the line. From the example at 139.2, we know that for Ibn al-Bannāʾ simple fractions are not restricted to those with a numerator of 1. So a fraction like “four sixths” ( 4\over 6) is also simple, and its numerator is the 4 above the line.

Combined fractions. This may not have been considered as a separate type since neither al-Hawārī nor Ibn al-Bannāʾ describe it. A combined fraction is a fraction whose representation has all 0s in the numerator except on the extreme left, like Ibn al-Yāsamīn’s 21\,\,0~\,0~\,0\over \,2~\,6~13~7. In modern notation, the fraction is 21\over 2\cdot 6\cdot 13\cdot 7, or 21\over 1092, and the numerator is 21. (Ibn al-Yāsamīn preferred to write his fractions in reverse order, so he shows it like this: \,0~\,\,0~\,0~21\over 7~13~6~\,2.45)

138.1 Related fractions. A related fraction is the common type with “two or more names”, which we have seen several times already. Al-Hawārī’s example is 2~\,3~\,4~\,5\over 3~\,5~\,7~\,8. Ibn al-Bannāʾ gave two rules, the first being the same as the one from Lifting the Veil cited above at 135.10. The fraction can be written in modern form as {596\over 3\cdot 5\cdot 7\cdot 8}={596\over 840}. The second rule is explained as clearly as the first.

139.1 Distinct fractions. Distinct fractions are those that are added, or gathered together. They have already been described above at 136.8. Al-Hawārī finds the numerator for “five sevenths and half a seventh and four sixths”. In notation this is {4\over 6}~{1~\,5\over 2~\,7}, which we would write as {11\over 14}+{4\over 6}. Combining them is equivalent to our method of cross multiplication, so the numerator is 11\cdot 6+4\cdot 14= 122.

139.10 Portioned fractions. The numerator of fractions of fractions is the product of the individual numerators. These were described above at 137.1.

140.1 Excluded fractions of disconnected type. These are called “excluded” because they are expressed as a fraction removed or excluded from a greater fraction. For an explanation of the terms “less”, “diminished”, and “excluded”, see our remarks at 219.4 below. They are “disconnected” because the second fraction is not taken of the first fraction, but is independent of it. The example here is “six eighths less a ninth”, written as {1\over 9}~\scalebox {-1}[1]{\ell }~{6\over 8}. This fraction is drawn in the Medina manuscript as . (See our commentary above at 86.1 for an explanation of the sign for “less”.) In modern notation we write it as {6\over 8}-{1\over 9}. By “is not taken from what precedes it”, al-Hawārī means that this is not {6\over 8}-{1\over 9}\left({6\over 8}\right). The procedure is like that for distinct fractions, but involves taking the difference rather than the sum of the products. Here 6\times 9=54, and 1\times 8=8. The numerator is 54-8=46.

140.14 Excluded fractions of connected type. Al-Hawārī converts the example “a half less its third” to the disconnected version “a half less a sixth”. He then finds the numerator of “six sevenths and half a seventh less its third”. By “its third” he means that you take away a third of the six sevenths and half a seventh. In notation it is written just like an excluded fraction of disconnected type, {1\over 3}~\scalebox {-1}[1]{\ell }~{1~\,6\over 2~\,7}, though the meaning is different. In modern notation we would write 1~\,6\over 2~\,7 as 13\over 14, and the whole fraction is equivalent to our {13\over 14}-{1\over 3}\left({13\over 14}\right). Take the numerator of the greater part, which is 1~\,6\over 2~\,7, to get 13, and multiply it by the denominator of the smaller part, which is 3, to get 39. Then multiply the two numerators: 1\times 13=13. The numerator is then 39-13=26, and the fraction is equal to 26\over 42.

The word “connected” for excluded fractions takes the same meaning as the word “related” for related fractions, in that the fraction that follows is taken as a portion of the preceding fraction. So it is natural that in cases where the diminished amount is itself a related fraction the notation was sometimes written with a single bar. For this example, if the “less” were instead an “and”, it would have been “six sevenths and half a seventh and a third of a half of a seventh”, or 1~\,1~\,6\over 3~\,2~\,7. When the last fraction is taken away instead of added, as in “six sevenths and half a seventh less its third”, it might be written as 1~\scalebox {-1}[1]{\ell }~1~\,6\over 3~~~\,2~\,7 instead of the ambiguous {1\over 3}~\scalebox {-1}[1]{\ell }~{1~\,6\over 2~\,7}. Four of the five manuscripts we consulted show the latter, but the Tehran manuscript shows the fraction with the bar extending all the way across: (this manuscript shows the Eastern forms of the numerals).

It was not a problem to have one notation with two possible meanings. While working through a problem, one knows whether the fraction is connected or disconnected. Notation was used by the individual to perform calculations, not to communicate the work to others or to preserve a record for future consultation. A rhetorical version served those purposes, and there the ambiguity is erased by the language.

141.7 Al-Hawārī then gives some “additional remarks” of Ibn al-Bannāʾ’s. Recall from 86.1 above that an expression like “ten less eight less seven less five less two” is 10-(8-(7-(5-2))). Here Ibn al-Bannāʾ explains that when the word “and” (wa) appears before each “less”, then the numbers are all taken away from the first term. Al-Hawārī’s example is “five and a third less its fourth and less its seventh and less its fifth”, and is written in notation as {1\over 5}~\scalebox {-1}[1]{\ell }~{1\over 7}~\scalebox {-1}[1]{\ell }~{1\over 4}~\scalebox {-1}[1]{\ell }~{1\over 3}\,5. We might write it as 5\,{1\over 3}-{1\over 4}\left(5\,{1\over 3}\right)-{1\over 7}\left(5\,{1\over 3}\right)-{1\over 5}\left(5\,{1\over 3}\right). The figure can be reduced to {1\over 5}\,{1\over 7}\,{1\over 4}\,\scalebox {-1}[1]{\ell }\,{1\over 3}\,5, with a single “less”.

The “particle of exclusion” is most often the word “less” (illā), as it is here. But sometimes in Arabic mathematics other words, meaning “except” or “other than” (ghayr or siwa), are used. The Arabic word for “detached” (munfaṣil) is related to the word we later translate as “apotome” (munfaṣila) in the chapter on roots. See our remarks below at 173.10.

142.6 Here al-Hawārī gives a disconnected example. If we said “five and a third less a fourth of one and less a seventh of one and less a fifth of one” it would be equivalent to the modern 5{1\over 3}-{1\over 4}-{1\over 7}-{1\over 5}, or 5{1\over 3}-{83\over 140}={1991\over 420}. He does not show the notation for this.

142.10 Again quoting Ibn al-Bannāʾ, al-Hawārī explains the meaning of the repeated “less” and cites the same rule given at 86.9 for whole numbers. He does this for the connected type (like “…less its fifth”) and the disconnected type (like “…less a fifth of one” or simply “…less a fifth”). He does not mention that the work starts from the last term.

143.1 The “three of its fourths” must become disconnected from the “five sixths”. So the fraction is rephrased as an excluded fraction of disconnected type.

143.5 Al-Hawārī finds the numerator for the mixed fraction {3~5\over 4~6}\,5, or 5\,{23\over 24} in modern notation. He first multiplies the whole number 5 by the two denominators to get 120. He then adds this to the numerator of the fraction, which is 23, to get 143. We can write the number as the fraction 143\over 24.

143.13 If the whole number comes after the fraction (that is, it is placed on the left), then the number is the fraction of the whole number. Al-Hawārī’s example is 10\,{6\over 8}\,{4\over 7}, which is “four sevenths and six eighths of ten” which we might write as \left({4\over 7}+{6\over 8}\right)\cdot 10. The numerator of the fractions is 74, since together they are 74\over 56. Multiply this 74 by the 10 to get the numerator 740.

144.4 If the whole number is written between two fractions, then it might be attached to the first fraction or to the second. If it is attached to the first, or “to what precedes it”, then one proceeds as in al-Hawārī’s example {3\over 6}\,5\,{4\over 9}, which is “four ninths of five, and three sixths”. That is, take 4\over 9 of 5, then add 3\over 6. Al-Hawārī multiplies the 4 by the 5 to get 20, the numerator of the first part. We can now look at it as the addition of 3\over 6 to 20\over 9. He effectively cross multiplies, adding 20\times 6 to 3\times 9 to get the numerator 147.

145.1 The whole number might be attached to the second fraction, or “what is after it”. Al-Hawārī’s example is “two thirds of seven and four sevenths”, or {4\over 7}\,7\,{2\over 3}. In modern notation this is equivalent to {2\over 3}\cdot \left(7\,{4\over 7}\right). First we find the numerator of the mixed fraction 7\,{4\over 7} to get 53. The problem is now {2\over 3}\cdot {53\over 7}. Then the 53 is multiplied by the 2 to get 106, which is the numerator.

As we explained above at 140.14, the ambiguity of the notation is not problematic for the person working through the calculations.

146.3 Al-Hawārī often does not cancel common factors in his fractions. The most glaring examples could have been simplified easily, like “four sixths” at 139.2, “six eighths” at 140.8, and “three sixths” at 144.10. But here Ibn al-Bannāʾ states that one must decompose the numerator and denominator into their prime factors and cancel the common factors, and al-Hawārī explains it for the case of portioned fractions.

147.1 Section I.2.2. Adding and subtracting fractions.

147.2 Now we turn to operating on fractions, beginning with addition. To add two fractions Ibn al-Bannāʾ gives the rule that one multiplies the numerator of each fraction by the denominator of the other, and then the sum of the results is divided by the product of the denominators. Al-Hawārī’s example is to add {6\over 8}\,{4\over 5}\,3 to 1~3~\,\,4\,\over 2~8~10. He gives instructions to put the addend over the augend. The operations are clearer if we write the first fraction as 182\over 40 and the second as 71\over 160. He multiplies 182 by 160 to get 29,120. Next he multiplies 71 by 40 to get 2,840. Adding 29,120 to 2,840 gives 31,960, which is the numerator of the sum. The denominator is the product of the denominators of the two numbers: 40 \times 160=6{,}400. So the sum is 31,960\over 6,400, which he prefers to express as “four and nine tenths and seven eighths of a tenth and half an eighth of a tenth”, and to write as {1~7~\,\,9\,\over 2~8~10}4.

Al-Hawārī then remarks “And its answer is given by (bi-) five”. This might be read as an approximation, since the answer, as we would write it, is 4{159\over 160}. But phrases like this follow the sample calculations for the other operations on fractions (subtraction, multiplication, division, denomination), and the “answer” in these cases is often not close to the number given. At 149.8, he writes “and its answer is given by one” for the number that as a decimal is approximately .3246. At 150.2, he writes “And its answer is given by two” for the number 1{5\over 6}, and at 151.4 he has “And its answer is four” for 3{1\over 63}. All we can say is that the stated “answer” is the smallest whole number greater than the calculated value. Then in two cases, at 147.15 and 151.14, he writes “And its answer is given by removal/subtraction (ṭarḥ)”. In these two questions one of the numbers is diminished. We have not been able to decipher the meanings of these phrases, which in any case are given after the exact answer to the question is found.

147.15 Al-Hawārī’s next example is to subtract {1\over 3}~\scalebox {-1}[1]{\ell }~2\,{7\over 10} from {5\,\bullet \, 3\over 6\,\bullet \, 4}\,4. In our notation this is {111\over 24}-{32\over 30}. Al-Hawārī rhetorically works through (111\cdot 30)-(32\cdot 24)=2,562. This is the numerator of the difference. The denominators of the two fractions are already given as 6, 4, 3, and 10. Al-Hawārī silently cancels a 6, and instead of working with “thirds of fourths of tenths” he switches to the more customary “halves of sixths of tenths”. The result is stated as “three and five tenths and three sixths of a tenth and half a sixth of a tenth”, or {1~3~\,\,5\,\over 2~6~10}3.

149.1 Section I.2.3. Multiplying fractions.

149.2 Ibn al-Bannāʾ characterizes, or defines, the multiplication of fractions as “the portioning of one of the two fractions by the amount of the other”. Said a little differently, one takes the portion of one number according to the fraction of the other, which is of course also the meaning of portioned fractions described above at 137.1. So to multiply three fourths by two thirds, for example, one takes two thirds of the three fourths to get one half. As noted in the text, this definition is different from the definition of multiplication for whole numbers given at 95.2.

Both definitions work for the multiplication of a whole number by a fraction. The first is to find “the whole number by the amount of the fraction”. To multiply 10 by 3\over 5, for example, take 3\over 5 of 10 to get 6. By the second definition, one duplicates the fraction as many times as there are units in the whole number. Duplicating 3\over 5 ten times again gives 6.

149.6 The rule for multiplying fractions is just what we know it should be: multiply the numerators and divide the result by the product of the denominators. Al-Hawārī sets up his examples the same way he did for adding and subtracting fractions, with one number placed above the other. To multiply {1\over 3}\,{3\over 4} by {1~\,4~\,3\over 5~\,6~\,9} he multiplies the numerators 13 and 111 to get 1443, which he divides by the denominators 3, 4, 5, 6, and 9. This gives “four ninths and a fourth of a fifth of a sixth of a ninth”, or 1~\,0~\,0~\,4\over 4~\,5~\,6~\,9. He would have arrived at this form by canceling the common 3 to get 481\over 4\cdot 5\cdot 6\cdot 9, and then by following the rule for denomination given above at 123.18.

150.2 In the next example the numbers are set up similarly. The notational versions are written one above the other, the numerators are found, and the multiplication follows. Here the numerators of {1\over 8}\,4\,{1\over 3} and 10\,{2\,\bullet \, 1\over 3\,\bullet \, 5} are 33 and 20, respectively, so their product is 660. The denominators are given as 8, 3, 3, and 5. Everything but a leftover 2 and a 3 cancel, so the answer is “one and five sixths”.

151.1 Section I.2.4. Division and denomination.

151.2 This rule is also what we would expect. To divide or denominate a\over b by or with c\over d, divide/denominate ad by/with bc. Al-Hawārī’s example for division is to “divide six and a third by four fifths of seven eighths of three”:

The numerator of the top fraction is 19, and this is multiplied by the denominator of the bottom number, which is 40, to get 760. The other numerator and denominator are 84 and 3, and their product is 252. The answer is what we get from dividing 760 by 252, which is “three and a seventh of a ninth”, or {1~\,0\over 7~\,9}\,3. Here the denominators 7 and 9 do not come from the denominators in the given numbers. To get them, note that 252 goes into 760 three times with a remainder of 4. The fraction 4/252 is the same as 1/63, and 63 is 7\times 9. Thus the remainder is “a seventh of a ninth”.

151.14 The example for denomination is to “denominate three and a fourth less two ninths of it with six and two eighths and three fifths”. The calculations are easier to follow if we rewrite the problem as {91\over 36}\div {274\over 40}. After cancelling eights, the denominator of the quotient is 1233, which is 9\cdot 137.

152.9 There is no need to do all this work if the denominators of the two fractions are the same. In this case, just divide the numerator of the top number by the numerator of the bottom number. Al-Hawārī again gives examples for both division and denomination, and his calculations are clear.

153.7 If the numerators are equal, then for division one divides the denominator of the divisor by the denominator of the dividend, and similarly for the case of denomination. The example for division is to “divide five by five sixths”. Al-Hawārī notes that the denominator of a whole number is 1, so he divides 6 by 1 to get 6. Similarly, denominating 5\over 6 with 5 gives “a sixth”.

154.1 Section I.2.5. Restoration and reduction.

This chapter is the version for fractions of Section I.1.6 (at 129.1 above). Ibn al-Bannāʾ breaks up his explanations of restoration into six “problems”: (1) restoring a fraction to a fraction, (2) restoring a fraction to a whole number and a fraction, (3) restoring a fraction to a whole number, (4) restoring a whole number to a whole number and a fraction, (5) restoring a whole number and a fraction to a whole number, and (6) restoring a whole number and a fraction to a whole number and a fraction. He omits the two cases of (a) a whole number to a fraction, and (b) a whole number and a fraction to a fraction, since in both cases the first number is necessarily greater than the second, so the problem belongs to reduction. There are also six types of reduction, this time omitting the two that necessarily belong to restoration.

The rule is the same as that given above in Section I.1.6. Reviewing one example each for restoration and reduction should suffice. For the third type of restoration, al-Hawārī restores “two thirds of five sevenths so that it gives ten”. The answer is found by dividing 10 by 5~\bullet ~2\over 7~\bullet ~3, which gives 21. This means that to restore 5~\bullet ~2\over 7~\bullet ~3 to 10, one multiplies it by 21. The example for the first type of reduction is to “reduce seven tenths so that it becomes a third”. Denominate 1\over 3 with 7\over 10 to get 10\over 21, which is “three sevenths and a third of a seventh”, or 1~\,3\over 3~\,7. So, to reduce 7\over 10 to 1\over 3 one multiplies the 7\over 10 by 1~\,3\over 3~\,7.

157.1 Section I.2.5. Converting.

Sometimes one wants to change a fraction from one name to another, like from thirds to fifths. Two thirds, for example, is the same as three and a third fifths. We might awkwardly write this as 3{1\over 3}\over 5, but it would have been expressed by al-Hawārī as “three fifths and a third of a fifth”, which in notation is 1~\,3\over 3~\,5. Al-Hawārī begins the section on converting (taṣrīf) by quoting from Lifting the Veil.

157.2 Ibn al-Bannāʾ speaks about two kinds of conversion. The first type “concerns only the name”. Ibn al-Bannāʾ’s example is to convert “five sixths and three fourths” ( {3\over 4}~{5\over 6}) to tenths. “Naming the fraction in tenths” results in “one and five tenths and five sixths of a tenth”, or {5~\,\,\,5\,\over 6~\,10}\,1. Al-Hawārī will illustrate Ibn al-Bannāʾ’s rule below at 158.11.

157.9 The second type “concerns how many of that name, taken as units, are in the whole [fraction]”. The example asks how many tenths are in {3\over 4}~{5\over 6}. Ibn al-Bannāʾ relates this to the problem of converting whole numbers in the section on multiplication above at 95.6. The answer is found by multiplying the fraction by 10, so it “does not require division by the denominator of the converted number” (158.1).

158.7 Al-Hawārī then returns to give the rule in the Condensed Book. His example is to convert six eighths and four tenths to ninths. The numerator and denominator of the fraction “to be converted” are 92 and 80. The procedure is to multiply the 92 by the 9, giving 828, and then this is divided by the 80. We would write the quotient as 10{28\over 80}, but in Arabic it is {4~\,\,\,3\,\over 8~\,10}\,10. Al-Hawārī writes “four eighths” instead of “a half” because he wants to keep the same denominators as the original fractions. Dividing this by 9 is easy: 10\div 9 is {1\over 9}1, and to divide 4~\,\,\,3\,\over 8~\,10 by 9 one just puts a 0\over 9 on the right: 4~\,\,\,3\,~\,0\over 8~\,10~\,9. Gathering them together gives the answer: {4~\,\,\,3\,~\,1\over 8~\,10~\,9}\,1. For this problem, al-Hawārī did not bother to rearrange the denominators in descending order.

161.1 Chapter I.3. Roots.

163.1 Section I.3.1. Taking a root of a whole number and a root of a fraction.

163.2 The term for “square root” in Arabic was simply “root” (jadhr or jidhr). A root of a number might be rational, like \sqrt{25}=5 and \sqrt{4\over 9}={2\over 3}, or it might be irrational, like \sqrt{3} and \sqrt{{17\over 25}\,1}. In Ibn al-Bannāʾ and al-Hawārī, as in many Arabic authors, the word for “rational” is munṭaq, which literally means “expressible”. One can say \sqrt{4\over 9} as “two thirds”, while one cannot say the result of taking a square root of 3 other than as “a root of three”. Our authors write ghayr munṭaq (“inexpressible”) to mean “irrational”. Since irrational numbers in medieval mathematics are all surds, we translate this term by “surd” (recognition of transcendental numbers came only in the nineteenth century). Some other Arabic texts write aṣamm (“deaf”) for “irrational”, but our authors use this word to mean “prime” (see our comments at 66.7 and 134.2 above). The corresponding words in other languages have the same association with speech. In Greek the word for “rational” was rhêtos, meaning “something that may be said or spoken”, while “irrational” was either alogos (“deprived of speech”) or arrhêtos (“unsayable”). In medieval Latin the word surdus, meaning “deaf”, was used for “irrational”, and Italian texts show the related word sordo. The Latin surdus is the origin of our word “surd”.

Roots are expressed in medieval Arabic in a slightly different way than they are in English today. Where we speak of “the root of ten”, they said “a root of ten”. In modern arithmetic numbers are unique. There is only one 3, one \sqrt{10}, etc. But just as in Greek arithmetic,46 in medieval arithmetic numbers admitted multiplicity. One can have twelve roots of ten, for example. So when an Arabic arithmetician takes the square root of a number, the result is expressed with the implied indefinite article. Read “a root of ten” to mean a single \sqrt{10}. This may seem to be just a linguistic curiosity, but it becomes relevant in the duplication of roots and for understanding their preference for single roots as opposed to multiple roots, discussed below at 179.1. Al-Hawārī will, however, sometimes write “the root” when he is pointing to a specific quantity in a calculation. This is done with the same intention as when he writes, for example, “we add the four in the hundreds rank…” (at 75.11). Other words, like “difference”, “square”, and “ratio”, will lack the definite article, too. Sometimes we insert a “the” where there is none in the Arabic to make the reading easier. We did this in the passage at 174.1, for example, where neither “the” in our translation “the difference between the squares” is in the original (with the exception of “the result”, the other instances of “the” in that passage are present in the Arabic).

163.4 Ibn al-Bannāʾ differentiates between irrational roots that can be expressed with the word “root” once, and those that require the word more than once. A number in the first category, like \sqrt{3{9\over 10}}, is said to be “rational in square” because its square is rational, while “a root of a root of ten” is an example of a “medial” root. In our notation this number is \sqrt{\sqrt{10}}, which we usually write as \root 4  \of {10}. Rational numbers are also rational in square, so when we want to speak of an irrational number that is rational in square we say it is “rational in square only”.

The phrase “rational in square” is more literally translated as “expressible in power (fī l-quwwa)”. The phrase fī l-quwwa is a translation of the Greek term dynámei, which ordinarily means “in power” or “in value”, but was applied in mathematics to mean the size of the square on a line or of a number.

The word “medial” (muwassaṭ) is translated from the Greek mésos. Reinterpreting Euclid’s geometric definition of mésos in Elements Proposition X.21 in arithmetical terms, medial numbers are those that take the form \sqrt{\sqrt{p}}, where p is a non-square rational. These “roots of roots” are called “medial” because they are the mean proportion between 1 and a number rational in square only. For example, the medial \sqrt{\sqrt{10}} satisfies 1:\sqrt{\sqrt{10}}::\sqrt{\sqrt{10}}:\sqrt{10}. Ibn al-Bannāʾ’s characterization, if taken to the letter, diverges from the Greek. Other numbers, like \sqrt{\sqrt{\sqrt{3}}} and \sqrt{2+\sqrt{8}} are also expressed with the word “root” more than once, though it is unlikely that he considered them to be medial, too.

163.11 Vowels are generally not indicated when writing Arabic, so the word for “root” shows only the three consonants j-dh-r. Some pronounce this as jidhr and others jadhr. Ibn al-Bannāʾ preferred the latter.

163.14 Ranks, as well as numbers, are said to either have a root or not have a root. Every other rank, starting with the units, has a root (the units, hundreds, ten thousands, etc.), while the remaining ranks have no root (the tens, thousands, hundred thousands, etc.). As Ibn al-Bannāʾ explains, a rank has a root “if there is a number in it that has a root”. For example, the number 400 is in the hundreds rank, and it has a root, but none of the numbers 1,000, 2,000, up to 9,000 have a root.

164.3 – 165.16 Here al-Hawārī copies Ibn al-Bannāʾ’s rules for determining when a whole number might have a (rational) square root. For example, if the units digit (the first digit) is a 5, and the tens digit is not a 2, then the number does not have a root.

Squaring numbers of the form 9n+r for r=1, 2,\ …, 8 and then casting out nines shows that the remainder, when it is not cast out entirely, must be a 1, 4, or 7. Determining the remainders of the squares after casting out eights and sevens works similarly.

166.1 The examples that al-Hawārī and Ibn al-Bannāʾ give for extracting roots are too small to show fully how the rule works, and no figures are shown in the book. Fortunately, the same method is explained by Ibn al-Yāsamīn with greater numbers and with some figures. We give one of his examples here.47

To find a root of 876,096 we begin by writing the number on a line, and we mark the first place with an “r” for “root”, the second place with an “n” for “no root”, alternating “r”s and “n”s to the last place of the number:

We start with the last “r” on the left. The number below it is 87, and the greatest square that can be subtracted from it is 81. So we put its root, 9, under it, and we subtract 81 from 87 and replace the 87 with the remainder:

The 9 below is then doubled and shifted one place to the right:

Now we look for the greatest digit n such that when we multiply it by the number of the form “18 n” (i.e., 180+n) it will cancel as much of the 660 above it as possible. This digit is 3. We put the 3 below the 0, and then start the multiplication. Instead of multiplying the 3 by the 183 at once, Ibn al-Yāsamīn, Ibn al-Bannāʾ, and al-Hawārī perform the multiplication one digit at a time. First we multiply the 3 by the 1 to get 3, and this is subtracted from the 6 above:

Next, we multiply the 3 by the 8 to get 24, and this is subtracted from the 36 above:

Then the 3 is multiplied by itself, and the 9 is subtracted from the 120 above:

Now the 3 is doubled, and the 186 is shifted one place to the right:

Next, we want the greatest digit n such that when it is multiplied by “186 n” it cancels as much of the 11196 above as possible. This digit is a 6. We put it under the 6, and we multiply it by 1866, again one digit at a time. Multiplying it by 1 gives 6, and we take 6 away from the 11 above it:

Then we multiply the 6 by the 8, and we take 48 away from the 51 above it:

The 6 is then multiplied by the 6 next to it to get 36, and this is subtracted from the 39 above:

And finally, the 6 is multiplied by itself to get 36, and this cancels the 36 above. The answer is found by adding the 6 to half of the 1860 that follows it, giving 936. So \sqrt{\hbox{876,096}}=936.

166.9 To find a fractional approximation to a root of a non-square number, Ibn al-Bannāʾ gives different rules depending on whether or not the remainder is greater than the integer part of the root. The rule for remainders less than or equal to the root is the well-known approximation n+{r\over 2n}, where n is the integer part of the root and r is the remainder. The rule for greater remainders is the same, but based at the next root up. In modern notation, the value obtained by this rule is n+{r+1\over 2n+2}, and we can calculate that it is equal to (n+1)+{r^{\prime }\over 2(n+1)} where the remainder r^{\prime } is now negative, counting back from (n+1)^2 instead of forward from n^2.

Ibn al-Bannāʾ gives proofs/derivations for this method in Lifting the Veil. These are not copied by al-Hawārī, so we translate part of one proof here to show the reasoning behind it. This part covers the case where the approximation to the root is smaller than the true root. Ibn al-Bannāʾ’s arguments are arithmetical, and to make the reading easier we give modern algebraic equivalents in brackets. In it, he approximates the fraction f that is added to the known root n of a square n^2 so that together they equal the unknown root m of a greater square m^2. The “surplus” m^2-n^2 is our remainder r.

And its cause is that a root of the number is divided into two parts, a root of the smaller and a fraction [ m=n+f]. Multiplying that by itself is like multiplying each one by itself and one of them by double the other [ (n+f)^2= n^2+f^2+f\cdot 2n]. So the surplus between48 the number and the square of the smaller is equal to a square of the fraction and a product of the fraction by double a root of the smaller [ m^2-n^2=f^2+f\cdot 2n]. You are allowed to drop a square of the fraction, and you make the surplus equal to a product of the fraction by double a root of the smaller [ f^2 will be small, so we can suppose that m^2-n^2=f\cdot 2n]. So divide the surplus by double a root of the smaller, resulting in the fraction by approximation [ f={m^2-n^2\over 2n}].49

He continues: “And it is clear that the resulting fraction is greater than the true fraction, so the approximation is always in excess of the required root of the number”.50 This is true for both rules, whether the remainder exceeds the root or not. Reinterpreting the rules in terms of modern functions, the approximations for the first rule lie on the line tangent to f(x)=\sqrt{x} at x=n^2, where n is the integer part of the root. The approximations from the second rule lie on the line tangent to f(x)=\sqrt{x} at x=(n+1)^2. We know that the approximations will always be greater than the actual values because the graph of f(x)=\sqrt{x} is concave down. Ibn al-Bannāʾ’s reasoning is simpler.

Examples are calculated at 167.8, 167,14, and 168.1. Because n can be any approximation to the root, not just a whole number, Ibn al-Bannāʾ understood that the rule can be iterated. This is done beginning at 168.8.

166.13 Al-Hawārī’s first example of calculating a square root is to find \sqrt{625}. He sets it up and begins the process according to Ibn al-Bannāʾ’s instructions:

The next digit to be found is the units digit. Because al-Hawārī presumes that 625 is a perfect square, he knows that it must be a 5: no other unit would produce the 5 in 625.

167.3 He then multiplies the 5 by 4 to get 20, which leaves a 2 when confronted with the 22 above it. Then the 5 is squared, which cancels the 25 left above:

By “the five and the doubled two after it” he means the 5 in the units place on the lower line, and half of the 4 (really 40) to its left. The root is then 25.

167.8 Al-Hawārī’s next example is not a perfect square. He finds an approximation to a root of 20 using the rule from 166.9:

Since the number above is not exhausted and is not greater than the root, he adds to the root, which is the 4 below, the remainder above divided by twice the root, or 4\div (2\cdot 4), to get 4{1\over 2}. A square of 4{1\over 2} is 20{1\over 4}, which is close to 20.

167.14 The same rule is applied to approximate a root of 54, since the remainder, 5, is less than the root, 7.

168.1 Al-Hawārī’s next example is to find a root of 92. On the dust-board, it would have progressed like this:

This time, the remainder 11 is greater than the root 9: so he adds 1 to the 11 to get 12, and he adds 2 to double the root to get 20. Then the fractional part is 12/20, or 3/5. The approximation is then {3\over 5}\,9. As al-Hawārī notes, its square is {4~\,0\over 5~\,5}\,92. We would write it as 92{4\over 25} or 92.16.

168.8 In the approximation method just described, one finds the integer part of the root first, and the approximation is calculated as a fraction to be added on. This approximation is greater than the desired root, so Ibn al-Bannāʾ’s rule for obtaining an even closer approximation is calculated by subtracting off another fraction.

168.10 In the example of approximating \sqrt{92}, the “close smaller square” is a square of the integer part of the first approximation 9{3\over 5}, or 9, which is 81, and which is less than 92. The “close greater square” is a square of 9{3\over 5}, or 92{4\over 25}, which is greater than 92.

168.16 To obtain a better approximation with this greater square, he denominates the 4\over 25 with double the root 9{3\over 5} to get 1\over 120, or “half a sixth of a tenth”. He subtracts this from the root 9{3\over 5} to get {1~\,5~\,\,\,5\,\over 2~\,6~\,10}\,9, or, in our terms, 9{71\over 120}. This approximation is more accurate: its square as a decimal is 92.0000694…

169.1 Another way to get a better approximation is to multiply the original number by some large square number, take its root, and then divide by a root of that large square. The example is to approximate a root of 12 by first multiplying the 12 by 16, taking its root, and then dividing the result by 4. The approximation reached this way is {1~\,3\over 4~\,7}\,3, whose square as a decimal begins 12.00127… If we were to approximate \sqrt{12} by the rule at 166.9 we would get {1\over 2}3, whose square is 12.25.

169.8 Ibn al-Bannāʾ extends the technique for finding roots of whole numbers by allowing one to put down numbers with fractional parts, and he gives two examples in Lifting the Veil that are copied by al-Hawārī at 169.10 and 169.17. The statement that “the remainder will be smaller than the remainder with whole numbers” is best illustrated in the example at 169.17. The remainder after the first step is 3\over 4, which is smaller than the 3 that would have been the remainder had he worked with whole numbers. When working with fractions, there is no smallest possible (positive) remainder because one can always find a closer fraction to the root.

169.10 In Lifting the Veil, Ibn al-Bannāʾ shows how to use the standard rule from 166.1 with fractions to find \sqrt{729} and \sqrt{625}. Al-Hawārī copies them in reverse order, doing \sqrt{625} first. Instead of choosing the greatest whole number whose square does not exceed the number above, now a fractional part can be added to that number. The work proceeds like this:

Ibn al-Bannāʾ finds the answer by taking half of the 50 in the last figure.

169.17 The next example is to find a root of 729. The work goes like this:

From the second to the third figure he calculates that \frac{3}{4} of 100 is 75, so he adds this to the 29 to get 104. In the last step 2\cdot 50+2^2=104, so the number on top is exhausted. The answer is then half of the 50 together with the 2, which is 27.

170.6 The example of calculating \sqrt{100} by adding half the zeros back to \sqrt{1} is clear.

170.10 Al-Hawārī gives examples of taking roots of fractions. There are two methods to do this. The first is that “you multiply the numerator by the denominator and you divide a root of the result by the denominator”. This method is preferred when the denominator is not a perfect square, because you avoid dividing by an ugly fractional approximation of a root. The second way is to “divide a root of the numerator by a root of the denominator”. This way is easier if the denominator is a perfect square. In modern notation, in the first method the square root of the fraction a\over b is found by calculating \sqrt{ab}\div b, and in the second by \sqrt{a}\div \sqrt{b}.

170.14 Ibn al-Bannāʾ classifies four kinds of fraction, depending on whether the numerator and/or the denominator is a square.

As a single fraction, the example at 170.16 is 25\over 36, and at 171.1 it is 49\over 4. The fraction at 171.6 is 27\over 54, which is the same as 1\over 2. The approximation for its root is equal to 1451\over 2052, which is correct to four decimal places. Applying the standard method to 1\over 2 gives an approximation of 3\over 4, which is much less accurate.

The fraction at 171.15 is 175\over 16. For the first method, al-Hawārī multiplies the denominator by the numerator to get 2800, and he takes a root of the product. The result, {1~\,48\over 2~\,53}\,52, is then divided by 16. For the second method, he calculates a root of 175 as {3\over 13}13, and he divides this by 4 to get {4\over 13}3. He is wrong when he writes “This method is closer than the first [method]”. The first method is correct to five places, while the second is correct only to three.

The fraction at 172.12 is 9\over 14.

173.4 Binomials and apotomes.

The only mention of binomials and apotomes in Ibn al-Bannāʾ’s Condensed Book is his rule for finding their roots at 173.4. He does not explain what binomials and apotomes are, and he omits their classification into the six types and how to find examples of them. All that should precede the calculations. He may have inserted this rule as an afterthought, since the heading for the current section, given above at 163.1, does not cover roots of irrational numbers: “on taking a root of a whole number and a root of a fraction”.

173.10 Ibn al-Bannāʾ provided the missing material in his Lifting the Veil, and al-Hawārī copied it from there into his book. This includes the definitions of binomials and apotomes (173.12-18), the classification of the six types (174.1-13), and the finding of the six binomials and six apotomes (174.14-175.10). Al-Hawārī then finds a root of a sample binomial and its apotome by Ibn al-Bannāʾ’s rule (175.11-21).

Then, at 176.1, al-Hawarī gives a variation on Ibn al-Bannāʾ’s rule and applies it to find roots of examples of each of the six types of binomial and apotome (176.6-177.15). After calculating each root, al-Hawārī adds a brief description of it. These descriptions are translated from Euclid’s Elements, and they are found with this same particular wording in a number of Arabic geometry and arithmetic books written before al-Hawārī’s time. In the translation, we put the descriptions in quotation marks to indicate the borrowing. We do not know al-Hawārī’s immediate source for these descriptions, but they may originate from al-Ḥajjāj’s translation of the Elements.51 We single them out in part because the meanings of the terms “medial” and “bimedial” become altered in the arithmetical setting of al-Hawārī’s book. In Euclid’s geometrical setting medial/bimedial areas are intended, meaning that it is their sides (square roots) that are medial. Al-Hawārī’s \sqrt{20} (at 177.5), for example, corresponds to a medial area because its “side”, \sqrt{\sqrt{20}}, is medial. Reading the Arabic literally, it seems that the number \sqrt{20} itself is being called medial.

We should give two related definitions before proceeding. Two (positive) numbers are said to be commensurable if their ratio is rational, like \sqrt{12} and \sqrt{27}, or any two rational numbers. Two numbers are commensurable in square if their squares are commensurable. Examples include \sqrt{\sqrt{12}} and \sqrt{\sqrt{27}}, 5 and \sqrt{13}, and any two commensurable numbers. Our authors write “numerical ratio” (nisba ʿadadiya, at 174.17) or “in ratio” (min nisba, at 180.8) to mean “commensurable”, and a word whose ordinary meaning is “different” (mutabāyna, at 180.11 and 182.5) for “incommensurable”. There may not have been standard terms for these meanings.

The nomenclature and theory of binomials and apotomes that was part of Arabic arithmetic derive from a numerical reading of Book X of Euclid’s Elements.52 Our word “binomial”, like the original Greek (ek) duo onomatōn, means “two names”. In Arabic, the term for “binomial” is dhū l-ismīn, meaning “two unified names”. A binomial is a number that can only be expressed in the form “ x and y”, where the incommensurable numbers x and y are either rational or rational in square.53 Examples are “eight and a root of sixty” and “a root of five and a root of three”, which we would write as 8+\sqrt{60} and \sqrt{5}+\sqrt{3}, respectively. Our 8+\sqrt{60} contains the arithmetical operation of addition, but as we explain at 219.1 below, the premodern “eight and a root of sixty” consists instead of two numbers gathered together.

Ibn al-Bannāʾ’s verb for forming a binomial is waṣala, “to join” two numbers together (pp. 174-5). Sometimes the related noun muttaṣil, “joining”, takes the place of the usual term for “binomial”. At 188.19, Ibn al-Bannāʾ spells it out as “[a] joining of two names”, and in three instances just after that we find it as merely “joining”. On that page we translate these variants as “binomial” to make the reading clearer, while at 177.9 and 177.15 the word is better rendered as “joining”.

Al-Hawārī does not show notation for roots, but other authors do. In his 1370 commentary on Ibn al-Bannāʾ’s Condensed Book, Ibn Qunfudh writes these two binomials in notation as and .54 Reversing the order of the terms, we can transcribe them as “ 8~\sqrt{60}” and “ \sqrt{5}~\sqrt{3}”, respectively. The letter jīm () without the dot, for jadhr (“root”), is placed above the 60, 5, and 3 to indicate square root. We have already seen this way of writing gathered numbers – one after the other – several times for distinct fractions, first at 136.8.

Our word “apotome” comes from the Greek word apotomē, meaning “something cut off”, and the Arabic translation is munfaṣil, meaning an amount from which something has been “detached”. Examples are “eight less a root of sixty” and “a root of five less a root of three”. We write these in modern notation as 8-\sqrt{60} and \sqrt{5}-\sqrt{3}, but again, our operation of subtraction distorts the premodern idea of a diminished quantity. An apotome was regarded as a single quantity from which something has been removed. The “eight less a root of sixty”, for example, should be thought of as a deficient eight, and not as the subtraction of something from eight. We explain this more thoroughly below at 219.4. Ibn Qunfudh shows these two examples in notation with the word illā (“less”) between the numbers: and .55 We transcribe them as 8~\ell ~\sqrt{60} and \sqrt{5}~\ell ~\sqrt{3}. This notation is the same as we have seen for excluded fractions of disconnected type, first at 140.8, and with an image from the Medina manuscript in our commentary at 140.1. Further, Ibn al-Bannāʾ and al-Hawārī use the term munfaṣilat with the meaning of “detached” for the excluded parts of these fractions in the passages at 141.7, 142.6, and 142.10.

Binomials and apotomes are not restricted to the arithmetic of fractions and roots. In algebra, too, the same concepts and notation are behind polynomials. At 183.15, al-Hawārī makes a direct comparison between the quadratic irrationals of this section and algebraic expressions: “The principle behind multiplying appended and deleted terms will be covered in [the chapter on] algebra”.

Euclid’s theory of binomials and apotomes is expressed in a geometric context, so he necessarily takes into account the dimensions of his magnitudes. For example, instead of the multiplication of two numbers, he writes of the formation of a rectangle from two sides. He defines a binomial as a line, and lines by their nature cannot have “square roots”. So, Euclid defines its square root as the side of a square equal in area to a rectangle having the binomial as one of its sides and a rational line (his version of a unit) for the other side. This theory was transferred to the setting of numbers in Arabic arithmetic. Numbers are homogeneous and thus dimensionless, so some of Euclid’s distinctions become superfluous.

Because we will be interspersing the notation for binomials and apotomes with calculations in modern notation, we will express the binomials and apotomes with “ +” and “ -” from now until the end of the section. To transcribe them in notation that better reflects the notation in the texts, omit the “ +” and replace the “ -” with an “ \ell ”. We write them this way, right-to-left, in the conspectus in Appendix A.

174.1 In Lifting the Veil, Ibn al-Bannāʾ explains the classification of the six types of binomial and apotome, and al-Hawārī gives examples of each. There are three basic types. Writing the greater term first, in order they are p\pm \sqrt{q}, \sqrt{p}\pm q, and \sqrt{p}\pm \sqrt{q} where both p and q are rational, and where one of these values appears under a root it is not a perfect square. Each of the basic types is divided into two subtypes stemming from a characteristic of its root. Writing the general binomial in modern notation as \sqrt{P}+\sqrt{Q}, in which P and Q are both rational, P>Q, and at most one is a square, al-Hawārī’s rule for calculating \sqrt{\sqrt{P}+\sqrt{Q}} at 176.1 can be written as

A root of the associated apotome, \sqrt{\sqrt{P}-\sqrt{Q}}, is

In one of the two subtypes \sqrt{P} is commensurable with \sqrt{P-Q}, so the two terms can be combined into a single root. This holds for the first three of the six types of binomial/apotome. In the other subtype the two are incommensurable, and this holds for the last three of the six types.

Ibn al-Bannāʾ’s classification follows that of Euclid, who gives constructions for each of the six types of binomial in Propositions X.48-53, and for the six types of apotome in Propositions X.85-90.

Just as the digits in a number come in ranks, so do numbers expressed with roots. Two numbers are of the same rank if they are the same “distance” from being rational. Thus \sqrt{5} and \sqrt{21} are of the same rank, as are \sqrt{\sqrt{5}} and \sqrt{\sqrt{21}} and the pair \sqrt{\sqrt{\sqrt{5}}} and \sqrt{\sqrt{\sqrt{21}}}.

174.14 Euclid then gives constructions for roots of the six binomials in Propositions X.54-59, and for roots of the six apotomes in Propositions X.91-96. Ibn al-Bannāʾ gives numerical rules in place of Euclid’s constructions. We rewrite them below in modern notation. Al-Hawārī does not give examples here, so we provide our own. In these descriptions it is assumed that a, b, and c are positive rational numbers, and numbers under roots are positive non-squares.

A number is a first binomial if it can be written in the form a+\sqrt{a^2-b^2}, and a first apotome if it can be written as a-\sqrt{a^2-b^2}. If we let a=5 and b=2, then the binomial is 5+\sqrt{21} and the corresponding apotome is 5-\sqrt{21}.

A number is a fourth binomial if it can be written in the form a+\sqrt{a^2-b}, where b is not a perfect square. If a=5 and b=8, then the fourth binomial is 5+\sqrt{17} and the associated apotome is 5-\sqrt{17}.

A number is a second binomial if it can be written in the form \sqrt{a^2(a^2-b^2)}+(a^2-b^2). A second apotome can be written in the form \sqrt{a^2(a^2-b^2)}-(a^2-b^2). For example, if we let a=3 and b=2, the binomial is \sqrt{45}+5 and the associated apotome is \sqrt{45}-5. Medieval mathematicians would say “a root of forty-five” instead of “three roots of five” because the latter is a collection of three numbers. They preferred the single number \sqrt{45}. See below at 179.1 for more on this. Ibn al-Bannāʾ made an error in defining this type by writing “a root of their difference” instead of simply “their difference”.

Ibn al-Bannāʾ also made an error in defining the fifth binomial. In our notation his description translates into \sqrt{a^2+b^2}+a. In fact, it should be of the form \sqrt{a^2+b}+a, where \sqrt{a^2+b} is incommensurable with \sqrt{b}. If we make a=5 and b=3, the fifth binomial is \sqrt{28}+5 and the apotome is \sqrt{28}-5. Al-Hawārī’s example of the fifth binomial at 177.5 below is correct.

A number is a third binomial if it can be written in the form \sqrt{ca^2}+\sqrt{c(a^2-b^2)}. If we let a=5, b=2, and c=3, then the third binomial is \sqrt{75}+\sqrt{63} and the associated apotome is \sqrt{75}-\sqrt{63}.

A number is a sixth binomial if it can be written in the form \sqrt{a^2+b}+\sqrt{b}, and if the two roots are incommensurable. If a=6 and b=7, then the sixth binomial is \sqrt{43}+\sqrt{7} and the associated apotome is \sqrt{43}-\sqrt{7}.

175.11 Al-Hawārī finds roots of the first binomial 8+\sqrt{60} and its associated apotome 8-\sqrt{60} (from a=8 and b=2) by the method given by Ibn al-Bannāʾ at 173.4. We can rewrite this as the modern formula:

For the binomial, he subtracts 1\over 4 of a square of the smaller ( 1\over 4 of \sqrt{60}^2 is 15) from 1\over 4 of a square of the greater ( 1\over 4 of 8^2 is 16), leaving 1. Its root is 1. This 1 is added to half of the greater term ( 1\over 2 of 8 is 4) to get 5, and it is also subtracted from the 4 to get 3. Then \sqrt{5}+\sqrt{3} is a root of 8+\sqrt{60}. He then finds a root of the associated apotome 8-\sqrt{60} by the same procedure to get \sqrt{5}-\sqrt{3}.

176.1 – 177.15 Al-Hawārī gives the variation on the rule for finding roots of binomials and apotomes that we described above at 174.1, and he applies it to his examples for each of the six types.

176.1 To find a root of the first binomial 8+\sqrt{55} (with a=8 and b=3), take the difference of their squares 8^2-(\sqrt{55})^2 to get 9, and take its root to get 3. Now figure {1\over 2}(8+3) and {1\over 2}(8-3), and take their roots. A root of the binomial is then \sqrt{5{1\over 2}}+\sqrt{2{1\over 2}}, and a root of the apotome 8-\sqrt{55} is \sqrt{5{1\over 2}}-\sqrt{2{1\over 2}}. Euclid (Propositions X.54, 91) and the Arabic translation call the root of the binomial “a binomial” and the root of the apotome “an apotome”, which is indeed what they are. Al-Hawārī expands these phrases to “one of the binomials” and “one of the six apotomes”.

176.10 Next, al-Hawārī finds a root of \sqrt{112}+7, which is an example of the second binomial ( a=4, b=3). Following the same procedure, (\sqrt{112})^2-7^2=63. Then {1\over 2}(\sqrt{112}+\sqrt{63}) and {1\over 2}(\sqrt{112}-\sqrt{63}) are \sqrt{85{\frac{3}{4}}} and \sqrt{1{\frac{3}{4}}}. (For these steps, al-Hawārī calculated \sqrt{112}+\sqrt{63}=\sqrt{343} and \sqrt{112}-\sqrt{63}=\sqrt{7}. Instructions on how to add and subtract roots are given below, starting at 179.1.) So a root of the binomial \sqrt{112}+7 is \sqrt{\sqrt{85{\frac{3}{4}}}}+\sqrt{\sqrt{1{\frac{3}{4}}}}.

Euclid (Proposition X.55) and the Arabic translation call a root of the second binomial “the first bimedial”. Euclid defines this term in Proposition X.37: “If two medial straight lines commensurable in square only and containing a rational rectangle be added together, the whole is irrational; and let it be called a first bimedial straight line”. Reinterpreting this in our arithmetical context, the root is called a bimedial because it is composed of the two medial numbers \sqrt{\sqrt{85{\frac{3}{4}}}} and \sqrt{\sqrt{1{\frac{3}{4}}}}. They are commensurable in square only because the ratio of their squares, \sqrt{85{\frac{3}{4}}}/\sqrt{1{\frac{3}{4}}}, is 7, but their ratio \sqrt{\sqrt{85{\frac{3}{4}}}}/\sqrt{\sqrt{1{\frac{3}{4}}}} is \sqrt{7}, which is irrational. The two medial numbers contain a rational rectangle because their product, \sqrt{\sqrt{85{\frac{3}{4}}}}\cdot \sqrt{\sqrt{1{\frac{3}{4}}}}=3{1\over 2}, is rational.

A root of the second apotome \sqrt{112}-7 is \sqrt{\sqrt{85{\frac{3}{4}}}}-\sqrt{\sqrt{1{\frac{3}{4}}}}. Euclid (Proposition X.92) and the Arabic translation call this “a first apotome of a medial [straight line]”.56 Because the root was considered to be a diminished \sqrt{\sqrt{85{\frac{3}{4}}}}, it does not consist of “two names”. Thus it is medial and not bimedial.

176.15

Al-Hawārī’s example of the third binomial is \sqrt{32}+\sqrt{14} ( a=4, b=3, and c=2). Its root is \sqrt{\sqrt{24{1\over 2}}}+\sqrt{\sqrt{1\over 2}}. This root is called a “second bimedial” in Euclid (Proposition X.56) and “the binomial of the second bimedial” in the Arabic translation. The second bimedial is defined in Elements, Proposition X.38: “If two medial straight lines commensurable in square only and containing a medial rectangle be added together, the whole is irrational; and let it be called a second bimedial straight line”.57 The bimedial numbers \sqrt{\sqrt{24{1\over 2}}} and \sqrt{\sqrt{1\over 2}} are commensurable in square only because the ratio of their squares \sqrt{24{1\over 2}}/\sqrt{1\over 2} is 7, while their ratio is the irrational number \sqrt{7}. Their product \sqrt{\sqrt{24{1\over 2}}}\cdot \sqrt{\sqrt{1\over 2}} is \sqrt{3{1\over 2}}, which is irrational. A “medial area” (i.e., a “medial rectangle”) “is the area which is equal to the square on a medial straight line”.58 In arithmetical terms, the \sqrt{3{1\over 2}} is a medial area because its square root \sqrt{\sqrt{3{1\over 2}}} is medial. In arithmetic there is no use for such a designation, because, unlike geometric magnitudes, numbers do not possess dimension. A “medial area” translated to arithmetic becomes simply a number rational in square only, or, in other terms, a number of the form \sqrt{p} where p is a non-square rational.

A root of the corresponding apotome is \sqrt{\sqrt{24{1\over 2}}}-\sqrt{\sqrt{1\over 2}}. Euclid (Proposition X.93) and the Arabic translation call this “a second apotome of a medial [straight line]”.

176.20 Next, al-Hawārī finds a root of 7+\sqrt{30}, which is an example of the fourth binomial ( a=7, b=19). Following the same procedure he gets \sqrt{3{1\over 2}+\sqrt{4{\frac{3}{4}}}}+\sqrt{3{1\over 2}-\sqrt{4{\frac{3}{4}}}}. Again, following Euclid (Propositions X.57, X.94) the Arabic translation calls this “the major” and the corresponding apotome “the minor”. A major is defined in Elements Proposition X.39: “If two straight lines incommensurable in square which make the sum of the squares on them rational, but the rectangle contained by them medial, be added together, the whole straight line is irrational: and let it be called major”.59 In al-Hawārī’s example, \sqrt{3{1\over 2}+\sqrt{4{\frac{3}{4}}}} is incommensurable in square with \sqrt{3{1\over 2}-\sqrt{4{\frac{3}{4}}}} because the ratio of their squares, (3{1\over 2}+\sqrt{4{\frac{3}{4}}})/(3{1\over 2}-\sqrt{4{\frac{3}{4}}}), is 2{4\over 15}+\sqrt{4{31\over 225}}, which is irrational. The sum of their squares is 7, which is rational. The rectangle contained by them, or their product, is \sqrt{7{1\over 2}}, which is a medial area when interpreted in terms of geometry. The minor is defined similarly for the apotome in Proposition X.76.

177.5 The example for the fifth binomial is to find a root of 3+\sqrt{20} ( a=3, b=11). Its root by the same procedure is \sqrt{\sqrt{5}+\sqrt{2{\frac{3}{4}}}}+\sqrt{\sqrt{5}-\sqrt{2{\frac{3}{4}}}}. Euclid (Proposition X.58) called this “[the line whose] power is a rational and a medial [area]”.60 Here the rational area is the 3, and the medial area is the \sqrt{20}. The Arabic translation is quite literal: “[the number whose] power is a rational and a medial”, with the result that term “medial” would be misinterpreted in the arithmetical setting. The number \sqrt{20} is rational in square, not medial like its square root.

Euclid (Proposition X.95) calls the corresponding apotome \sqrt{20}-3 “[a straight line] which produces with a rational [area] a medial whole”.61 In our numbers, the root \sqrt{\sqrt{5}+\sqrt{2{\frac{3}{4}}}}-\sqrt{\sqrt{5}-\sqrt{2{\frac{3}{4}}}} produces by squaring it the quantity \sqrt{20}-3 which, if one adds back the rational area 3, gives the medial area \sqrt{20}. The Arabic translation describes it as “the joining with a rational to become a whole medial”.

The Greek and Arabic words behind “power” in the descriptions “[the line/number whose] power is a rational and a medial” warrant some explanation. The Greek term is dynaménē, deriving from dynamis, a word meaning “power” or “value”, and in a mathematical setting, “square”. In geometry, a dynamis is a characteristic of a line. One did not make reference to a dynamis directly by naming its opposite vertexes, but rather to a side in respect of dynamis. As Jens Høyrup proposed, a dynamis is “a square identified with its side” or “a line seen under the aspect of a square” (Høyrup 1990, 210). The word often occurs in the dative form, dynámei (“in power”, or “in square”), as we described above at 163.4 for the phrase “rational in square”. Thus, our added words “[the line/number whose]” in the description of the fifth binomial are implied in Euclid’s description.

The forms of the word dynamis in Euclid were translated into Arabic by corresponding forms of quwwa, a word which also means “power”. This word, too, is a characteristic of a side or a number. Gustav Junge and William Thomson, in an appendix to their edition and translation of the medieval Arabic translation of Pappus of Alexandria’s Commentary on Book X of Euclid’s Elements, wrote this of quwwa:

The Dictionary of Technical Terms (Calcutta, A. Sprenger, Vol. II, p. 1230, top.) defines it as “Murabbaʿu-l-Khaṭṭi”, i.e., “the square of the line”, “the square which can be constructed upon the line”, and goes on to say that the mathematicians treat the square of a line as a power of the line, as if it were potential in that line as a special attribute. Al-Ṭūsī (Book X, Introd., p. 225, l. 9.) says: — “The line is a length actually (reading “bi-l-fiʿli” for “bi-l-ʿalqi”)62 and a square (murabbaʿun) potentially (bi-l-quwwati) i.e., it is possible for a square to be described upon it. (Pappus 1930, 181)

So here, too, an Arabic reader familiar with Euclid would have understood the description as meaning “[the number whose] power is a rational and a medial”. But for a young arithmetic student studying al-Hawārī’s book, we guess that the meaning of this description, and of the others, too, would have been obscure, and would have been read as being mere labels.

As a last remark on this passage, al-Hawārī writes the fraction “three fourths” in his calculations of the fourth and sixth binomials and apotomes, but here he writes it in the colloquial way reminiscent of the unit fractions from finger-reckoning as “a half and a fourth”.

177.11 The example for the sixth binomial is \sqrt{10}+\sqrt{11} ( a=1, b=10), and its root is \sqrt{{1\over 2}+\sqrt{2{\frac{3}{4}}}}+\sqrt{\sqrt{2{\frac{3}{4}}}-{1\over 2}}. It is called by Euclid (Proposition X.59) “[the line whose] power [consists of] two medial [areas]”,63 where in this case the medial areas are the \sqrt{11} and the \sqrt{10}. The Arabic translation is again literal: “[the number whose] power is a bimedial”. And as before, the word “bimedial” takes a different meaning in an arithmetical context.

Euclid (Proposition X.96) calls the root of the corresponding apotome, here \sqrt{{1\over 2}+\sqrt{2{\frac{3}{4}}}}-\sqrt{\sqrt{2{\frac{3}{4}}}-{1\over 2}}, “[a straight line] which produces with a medial [area] a medial whole”.64 The root produces by squaring it the apotome \sqrt{11}-\sqrt{10}. This is a deficient, or incomplete, \sqrt{11}, so to make it whole we add to it the medial area \sqrt{10} that it lacks to produce the whole medial area \sqrt{11}. The Arabic translation has “the joining with a medial to become a whole medial”.