# A. Conspectus of problems

#### DOI

10.34663/9783945561638-06

#### Citation

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

For the following list of calculations and problems in al-Hawārī’s book, we have adopted our transcription of the Arabic notation. This is explained in our commentary, especially at 219.1. We had introduced the algebraic notation in our commentary at 211.2, and notation for fractions is explained in our commentary in the chapter on fractions, beginning at 135.1. Here is a quick guide to the notation:

The notation is a transcription of how it appears, or in some cases would have appeared, in the manuscripts. With the exception of the figures for double false position, we put the notation in red, as it is in many Arabic manuscripts. It should be read right to left.

The reversed letter “ ” stands for “less”, and indicates that the number to its left is removed from the number to its right. So, an apotome that we would write as is shown as . See our commentary at 86.1. Binomials have no sign for “+”, so the modern is shown as .

Instead of the letter jīm above a number to indicate square root, we use the modern “ ” which functions similarly.

We write “=” for the “equals” in algebraic equations, and for no other purpose. This sign functions like the elongated lām, its counterpart in Arabic manuscripts.

Because the transcribed notation for algebra is entirely different from our notation, we include the versions of the calculations in modern notation as well, even if it sometimes makes little sense (as in the calculations from through 222.1).

An “L” is placed after the reference for examples taken from Ibn al-Bannāʾ’s Lifting the Veil.

## Part 1. On known numbers

#### Chapter 1. On whole numbers

Passage Example
65.2 Examples of whole numbers: 15, 18, etc.
65.2 Examples of fractions: , , , ,
65.6,10,17, 66.1 Even: 10, 50; evenly-even: 32; evenly-odd: 14; evenly-evenly-odd: 28
66.7 Odd prime: 11, 29; Oddly-odd: 15, composed of 3 by 5
66.17 Even square: 36 is 6 by 6
67.3 Even, composed of two unequal numbers (a surface): 18 is 3 by 6, and also 2 by 9
67.3 Even, composed of three unequal numbers (a surface): 24 is 3 by 4 by 2
67.12 Even cube: 64 is 4 by 4 by 4
67.19 Odd square: 25 is 5 by 5
68.1 Odd, composed of two unequal numbers (a surface): 35 is 5 by 7; 105 is 3 by 5 by 7
68.8 Odd cube: 27 is composed of 3 by 3 by 3
70.8, 70.23 Sample figures for numbers: 9367184225 and 84725
71.6 143 has three places
71.16 The rank of 10000000 is 8
72.4 The name of 1000000000 is thousands thousands of thousands
74.17 Adding from the units place: add 4043 to 2685 to get 6728
75.9 Adding from the highest place: add 978 to 456 to get 1434
75.20 The most one can get by adding is one extra place: add 9 to 9 to get 18
76.15 Chessboard: add the first sixteen squares to get 65535
77.11 Chessboard: add the first eight squares with 4 as the first square to get 1020
78.4 Adding five numbers with a ratio of : add 16, 24, 36, 54, 81 to get 211
79.4 Adding six numbers starting at 10, with a difference of three: add 10, 13, 16, 19, 22, 25 to get 105
79.13 Add 1, 2, 3, …, 10 to get 55
79.18 Add the squares of 1, 2, 3, …, 10 to get 385
80.1 Add the cubes of 1, 2, 3, …, 10 to get 3025
80.5 Add 1, 3, 5, 7, 9 to get 25
80.10 Add the squares of 1, 3, 5, 7, 9 to get 165
80.15 Add the cubes of 1, 3, 5, 7, 9 to get 1225
80.20 Add 2, 4, 6, 8, 10 to get 30
81.4 Add the squares of 2, 4, 6, 8, 10 to get 220
81.9 Add the squares of 2, 4, 6, 8, 10, 12 to get 364
81.15 Add the cubes of 2, 4, 6, 8, 10 to get 1800
83.16 Subtracting from the highest place: subtract 4968 from 5035 to get 67
84.13 Subtracting from the units place: subtract 3469 from 6543 to get 3074
85.8 The most one can get by subtracting is one fewer place: subtract 1 from 10 to get 9
86.9 L is 6
87.19 Casting out nines from 6435 gives nothing
88.5 Casting out eights from 5393 gives 1
89.4 Casting out sevens from 23786435 gives 1
90.7 Casting out sevens from 58064 gives 6
91.4 Add 43 to 64 to get 107 Cast out sevens to check: add 1 to 1 to get 2
91.16 Subtract 74 from 96 to get 22 Cast out sevens to check: subtract 4 from 5 to get 1
92.10 Multiply 12 by 16 to get 192 Cast out sevens to check: multiply 5 by 2 to get 3
92.17 L Multiply by to get Cast out sevens to check: multiply by to get
93.10 Divide 1488 by 12 to get 124 Cast out sevens to check: multiply 5 by 5 to get 4
93.15 L Divide by to get . Cast out sevens to check: multiply by , then convert to 4ths of 6ths to get
94.1 Denominate 11 with 15 to get . Cast out sevens to check: multiply 4 by 1 to get 4
94.5 L Denominate with to get . Cast out sevens to check: multiply 2 by 2, adjust for the denominators to get 3
95.10 L Meanings of multiplication: 3 men, each has 5 dirhams; 5 dirhams, how many thirds?
96.3 Multiplication by shifting: multiply 43 by 54 to get 2322
97.4 Vertical multiplication: multiply 42 by 37 to get 1554
98.15 Multiplication by half-shifting: multiply 463 by itself to get 214369
100.5 Lattice multiplication: multiply 435 by 287 to get 124845
102.1 Vertical multiplication (no shifting): multiply 183 by 347 to get 63501
104.10 Sleeper multiplication (no shifting): multiply 253 by 987 to get 249711
107.6 Multiply 444 by 333 to get 147852
108.13 Multiplication by excess: multiply 12 by 15 to get 180
109.1 Multiply 13 by 17 to get 221
109.10 Multiplication by denomination: multiply 6 by 12 to get 72
110.6 Another method of multiplication by denomination: multiply 24 by 8 to get 192
110.12 Multiply 12 by 15 to get 180
111.1 Multiply 3 by 15 to get 45
111.15 Multiplication by nines: multiply 444 by 999 to get 443556
112.10 Another method of multiplication by nines: multiply 999 by 9354 to get 9344646
113.1 Multiplication by squaring: multiply 17 by 19 to get 323
113.9 Another squaring method: multiply 25 by 15 to get 375
113.19 Another squaring method: multiply 36 by 14 to get 504
114.8 Multiplication with zeros: multiply 30 by 140 to get 4200
117.16 L, 118.1 L Meanings of division: Divide 15 dirhams among 3 men; Divide a piece of wood of 15 spans by a piece of wood of 3 spans
119.1 Divide 245 by 12 to get
120.5 Divide 44 among 11 men to get 4
120.10 Divide 96 by 12 to get 8
120.16 Divide 35 by 15 to get
121.4 Apportionment. Wealth of donors: 4, 5, 6 dinars, amount of loan: 10 dinars
122.5 Apportionment. Wealth of donors: , , dinars, amount of loan: 12 dinars
123.22 Common denomination: denominate 11 with 15 to get
124.12,14,17 Other denominations: denominate 4 with 12 to get ; denominate 9 with 15 to get ; denominate 10 with 16 to get
124.20ff Finding divisors: 50, 36, 66, 42, 64, 68, 14, 26, 81, 39, 123, 77, 221
129.6 Restore 8 to 19; reduce 50 to 6
129.8 Restore 3 to 6: divide 6 by 3 to get 2
129.12 Reduce 8 to 3. Denominate 3 with 8 to get

#### Part 1, Chapter 2. On fractions

Passage Example
134.2 L Language of parts: ,
134.8 Numerator denominator: , , ; and through
135.1 Fractions with two or more names: ;
135.10 L A related fraction:
136.8 L A distinct fraction:
137.1 L A portioned fraction:
137.13 The numerator of is 1
138.5 The numerator of is 596
139.2 The numerator of is 122
139.11 The numerator of is 105
140.8 The numerator of is 46
141.1 The numerator of is 26
141.11 The numerator of the connected fraction , also written , is 912
142.6 The numerator of the distinct fractions is 1991
143.5 The numerator of is 143
143.13 The numerator of is 740
144.10 The numerator of is 147
145.4 The numerator of is 106
147.15 Subtract from to get
149.8 Multiply by to get
150.2 Multiply by to get
151.4 Divide by to get
151.14 Denominate with to get
152.11 Divide by to get
153.3 Denominate with to get
153.12 L Divide 5 by to get 6; denominate with 5 to get
154.6ff Restore to ; to ; to 10; 5 to ; to 8; to
155.11ff Reduce to ; 8 to ; 10 to ; to ; to 5; to
157.2 L Convert to tenths. Answer:
157.12 L How many tenths are in ? Again, it is
158.1 L How many tenths are in 5? Answer: 50
158.11 How many ninths are in ? Answer:

#### Part 1, Chapter 3. On roots

Passage Example
163.4 L Examples: , , ,
166.13 is 25
167.8 is approximately
167.14 is approximately
168.1 is approximately
168.16 is approximately
169.4 is approximately
169.10,17 L is 25; is 27
170.6 is 10
170.16 is
171.1 is
171.6 is
171.15 is approximately
172.6 is approximately
172.12 is approximately
173.13 L Binomials: ;
173.16 L Apotomes: ;
174.5 1st & 4th binomials: ,
174.8 2nd & 5th binomials: ,
174.11 3rd & 6th binomials: ,
175.11 is
175.19 is
176.6 is ; is
176.10 is ; is
176.15 is ; is
176.20 is ; is
177.5 is ; is
177.11 is ; is
179.20 Add half of to two s to get
180.15 Add half of to of to get
181.6 Subtract from to get
181.12,16 Subtract from to get
182.4 Subtract from to get
183.4 Multiply by to get
183.7 Multiply by to get
183.11 Multiply by to get
183.15 Multiply 3 by to get
184.1 Multiply 3 by to get
184.4 Multiply 2 by to get
184.11 Multiply 2 by two s to get
184.18 Multiply 5 by three s to get
185.6 Multiply by half of to get
185.12 Multiply by half of to get
186.1 Duplicate twice to get
186.4 Duplicate five times to get
186.8 Half of is
186.11 of is
187.4 Divide by to get
187.7 Denominate with to get
187.10 Divide by to get
187.14 Denominate with to get
188.6 Divide by to get
188.11 Divide two s by 2 to get
188.15 Divide half of by to get
189.1 Divide 12 by to get
189.11 Divide 10 by to get

## Part 2. Finding unknown numbers

#### Chapter 1. Solving problems by proportion

Passage Example
195.16 Example of four proportional numbers:
198.7 L
199.1
200.1
201.1
202.3
203.1
203.11 L
205.5 L
207.6 L

#### Part 2, Chapter 2. Solving problems by algebra

Passage Example
211.15 Simple equations: ; ; ( ; ; )
212.6 Composite equations: ; ; ( ; ; )
213.7 Solve to get is 5 and is 25 ( , )
213.13 Solve to get is 9 and is 3 ( , )
214.1 Solve to get is 4 and is 16 ( , )
214.9 Solve to get is 3 and is 9 ( , )
215.6 Solve to get is 4 and is 16, or is 2 and is 4 ( , or , )
215.14 Solve to get is 9 and is 3( , )
216.13 Solve to get is 3 and is 9 ( , )
217.10 simplifies to ( )
218.1 simplifies to ( )
219.2 Add , , and to get ( )
219.5 L Add to to get ( )
219.7 L Add to to get ( )
219.10 L Add to to get ( )
219.12 L Add to to get ( )
220.1 Subtract from to get ( )
220.3 Subtract from to get ( )
220.9 Subtract from to get ( )
221.1 Subtract from to get ( )
221.13 Subtract from to get ( )
222.1 Subtract from to get ( )
223.2 simplifies to ( )
223.7 simplifies to ( )
223.14 simplifies to ( )
223.17 simplifies to ( )
224.1 simplifies to ( )
224.4 simplifies to ( )
225.8 The power of the māl māl is 4; of the māl cube is 5; of the māl māl māl is 6; of the māl cube māl cube is 10
225.13 The power of the māl cube māl māl is 9; of the cube māl cube cube māl māl is 15
226.1 A term for 4 is a māl māl; for 7 is a cube māl māl; for 6 is a māl māl māl or a cube cube
226.3 A term for 8 is a māl māl māl māl or a cube māl cube, or a cube cube māl, or a māl cube cube, etc.
226.7 A term for 9 is a cube cube cube or a cube māl māl māl, etc.
226.12 Multiply by to get ( )
226.17 Multiply by to get ( )
227.4 Multiply by to get ( )
227.7 Multiply 6 by to get ( )
227.10 Multiply 7 by to get ( )
227.17 simplifies to ( )
228.1 simplifies to ( )
228.4 simplifies to ( )
228.11 Multiply by to get ( )
228.15 Multiply by to get ( )
229.4 Divide by to get ( )
229.8 Divide by to get ( )
229.13 Divide by to get 4 ( )
229.17 Divide by 4 to get ( )
230.4 Divide by to get ( )
230.8 Divide by 2 to get ( )
230.13 Divide by to get ( )
230.17 Divide by to get ( )