For the following list of calculations and problems in alHawārī’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 alBannāʾ’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; evenlyeven: 32; evenlyodd: 14; 
66.7  Odd prime: 11, 29; Oddlyodd: 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): 
67.3  Even, composed of three unequal numbers (a surface): 
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): 
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: 
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
: 
79.4  Adding six numbers starting at 10, with a difference of three: 
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: 
84.13  Subtracting from the units place: 
85.8  The most one can get by subtracting is one fewer place: 
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 
91.16  Subtract 74 from 96 to get 22 
92.10  Multiply 12 by 16 to get 192 
92.17 L  Multiply
by
to get

93.10  Divide 1488 by 12 to get 124 
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 halfshifting: 
100.5  Lattice multiplication: multiply 435 by 287 to get 124845 
102.1  Vertical multiplication (no shifting): 
104.10  Sleeper multiplication (no shifting): 
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: 
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: 
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; 
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, 
122.5  Apportionment. Wealth of donors:
,
,
dinars, 
123.22  Common denomination: denominate 11 with 15 to get 
124.12,14,17  Other denominations: denominate 4 with 12 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
, 
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 
147.4  Add to to get 
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
; 
176.15  is
; 
176.20  is
; 
177.5  is
; 
177.11  is
; 
179.7,11  Add to to get 
179.16  Add to to get 
179.20  Add half of to two s to get 
180.10  Add to 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: 
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 
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; 
225.13  The power of the māl cube māl māl is 9; 
226.1  A term for 4 is a māl māl; for 7 is a cube māl māl; 
226.3  A term for 8 is a māl māl māl māl or a cube māl cube, 
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 ( ) 