A. Conspectus of problems

Mahdi Abdeljaouad; Jeffrey Oaks

A. Conspectus of problems

Chapter structure

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TY  - CHAP
T1  - A. Conspectus of problems
T2  - Al-Hawārī’s Essential Commentary: Arabic Arithmetic in the Fourteenth Century
A1  - Mahdi Abdeljaouad
A1  - Jeffrey Oaks
AB  - Al-Hawārī completed his Essential Commentary in Marrakesh in 1305. This modest book was written with the goal of supplying numerical examples to the rules of calculation explained in the famous Condensed Book on the Operations of Arithmetic of his teacher Ibn al-Bannāʾ. It is partly because of its modesty that al-Hawārī’s book is a good vehicle for presenting the various facets of practical Arabic arithmetic, which in al-Hawārī’s time included not just computational techniques but also algebra and other problem-solving methods. In this book we give an edition, English translation, and running commentary of al-Hawārī’s commentary, together with a general introduction to Arabic arithmetic to situate it in context. We cover not just the rules for operating on known and unknown numbers, but also their conceptual and historical background. Numbers in Arabic arithmetic were conceived as amounts or measures of something, and thus could be any positive quantity, including fractions and irrational roots. Thus they were neither founded in the Greek notion of multitudes of an indivisible unit, nor were they defined axiomatically as they are today. This way of understanding numbers informed both their ways of expressing numbers and algebraic expressions and how they performed their calculations. On the historical front, we witness the ways that different local arithmetical traditions interacted, and how Greek arithmetic, even if fundamentally incompatible with its Arabic counterparts, was partially absorbed into practical books like al-Hawārī’s.
PB  - Max-Planck-Gesellschaft zur Förderung der Wissenschaften
PY  - 2021
SN  - 978-3-945561-63-8
L1  - https://edition-open-sources.org/media/sources/14/6/sources14_chap06.pdf
LK  - https://edition-open-sources.org/sources/14/6/index.html
C1  - by-sa
DO  - 10.34663/9783945561638-06
LA  - english
ER  -

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 “ $\scalebox {-1}[1]{\ell }$ ” 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 $\sqrt{5}-\sqrt{3}$ is shown as $\leavevmode {\color {red}\sqrt{3}~\scalebox {-1}[1]{\ell }~\sqrt{5}}$ . See our commentary at 86.1. Binomials have no sign for “+”, so the modern $5+\sqrt{21}$ is shown as $\leavevmode {\color {red}\sqrt{21}~5}$ .

Instead of the letter jīm above a number to indicate square root, we use the modern “ $\leavevmode {\color {red}\sqrt{~~}\,}$ ” 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: $\leavevmode {\color {red}1\over 2}$ , $\leavevmode {\color {red}1~\,3\over 2~\,8}$ , $\leavevmode {\color {red}{1\over 4}~{1\over 9}}$ , $\leavevmode {\color {red}7~\bullet ~6\over 8~\bullet ~7}$ , $\leavevmode {\color {red}{1\over 9}~\scalebox {-1}[1]{\ell }~{5\over 6}}$
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 $\leavevmode {\color {red}2\over 3}$ : 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	$\leavevmode {\color {red}2~\scalebox {-1}[1]{\ell }~5~\scalebox {-1}[1]{\ell }~7~\scalebox {-1}[1]{\ell }~8~\scalebox {-1}[1]{\ell }~10}$ 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 $\leavevmode {\color {red}{1\over 3}}$ by $\leavevmode {\color {red}{1\over 4}14}$ to get $\leavevmode {\color {red}{3\over 4}4}$ Cast out sevens to check: multiply $\leavevmode {\color {red}1\over 3}$ by $\leavevmode {\color {red}1\over 4}$ to get $\leavevmode {\color {red}1~0\over 3~4}$
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 $\leavevmode {\color {red}{3\over 4}~{5\over 6}}$ by $\leavevmode {\color {red}1\over 2}$ to get $\leavevmode {\color {red}{1\over 6}3}$ . Cast out sevens to check: multiply $\leavevmode {\color {red}5\over 6}$ by $\leavevmode {\color {red}1\over 2}$ , then convert to 4ths of 6ths to get $\leavevmode {\color {red}3~0\over 4~6}$
94.1	Denominate 11 with 15 to get $\leavevmode {\color {red}2~\,3\over 3~\,5}$ . Cast out sevens to check: multiply 4 by 1 to get 4
94.5 L	Denominate $\leavevmode {\color {red}{2~\,2\over 3~\,6}}$ with $\leavevmode {\color {red}1~\,5\over 3~\,8}$ to get $\leavevmode {\color {red}2\over 3}$ . 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 $\leavevmode {\color {red}{1~\,2\over 2~\,6}\,20}$
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 $\leavevmode {\color {red}{1\over 3}2}$
121.4	Apportionment. Wealth of donors: 4, 5, 6 dinars, amount of loan: 10 dinars
122.5	Apportionment. Wealth of donors: $\leavevmode {\color {red}{1\over 3}4}$ , $\leavevmode {\color {red}{1\over 4}5}$ , $\leavevmode {\color {red}{1\over 6}6}$ dinars, amount of loan: 12 dinars
123.22	Common denomination: denominate 11 with 15 to get $\leavevmode {\color {red}{2~\,3\over 3~\,5}}$
124.12,14,17	Other denominations: denominate 4 with 12 to get $\leavevmode {\color {red}1\over 3}$ ; denominate 9 with 15 to get $\leavevmode {\color {red}3\over 5}$ ; denominate 10 with 16 to get $\leavevmode {\color {red}5\over 8}$
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 $\leavevmode {\color {red}3\over 8}$

Part 1, Chapter 2. On fractions

Passage	Example
134.2 L	Language of parts: $\leavevmode {\color {red}1\over 11}$ , $\leavevmode {\color {red}1\over 17}$
134.8	Numerator denominator: $\leavevmode {\color {red}1\over 4}$ , $\leavevmode {\color {red}2\over 4}$ , $\leavevmode {\color {red}3\over 4}$ ; and $\leavevmode {\color {red}1\over 7}$ through $\leavevmode {\color {red}6\over 7}$
135.1	Fractions with two or more names: $\leavevmode {\color {red}1~\,2\over 7~\,8}$ ; $\leavevmode {\color {red}1~\,2~\,4~\,8\over 3~\,6~\,7~\,9}$
135.10 L	A related fraction: $\leavevmode {\color {red}2~\,4~\,5\over 3~\,5~\,6}$
136.8 L	A distinct fraction: $\leavevmode {\color {red}{4\over 5}\,{5\over 6}}$
137.1 L	A portioned fraction: $\leavevmode {\color {red}5~\bullet ~3\over 6 ~\bullet ~4}$
137.13	The numerator of $\leavevmode {\color {red}1\over 7}$ is 1
138.5	The numerator of $\leavevmode {\color {red}2~\,3~\,4~\,5\over 3~\,5~\,7~\,8}$ is 596
139.2	The numerator of $\leavevmode {\color {red}{4\over 6}~{1~\,5\over 2~\,7}}$ is 122
139.11	The numerator of $\leavevmode {\color {red}3~\bullet ~5~\bullet ~7\over 10\,\:\bullet \,\:6\,\:\bullet \,\:9\,}$ is 105
140.8	The numerator of $\leavevmode {\color {red}{1\over 9}~\scalebox {-1}[1]{\ell }~{6\over 8}}$ is 46
141.1	The numerator of $\leavevmode {\color {red}{1\over 3}~\scalebox {-1}[1]{\ell }~{1~\,6\over 2~\,7}}$ is 26
141.11	The numerator of the connected fraction $\leavevmode {\color {red}{1\over 5}~\scalebox {-1}[1]{\ell }~{1\over 7}~\scalebox {-1}[1]{\ell }~{1\over 4}~\scalebox {-1}[1]{\ell }~{1\over 3}\,5}$ , also written $\leavevmode {\color {red}{1\over 5}\,{1\over 7}\,{1\over 4}\,\scalebox {-1}[1]{\ell }\,{1\over 3}\,5}$ , is 912
142.6	The numerator of the distinct fractions $\leavevmode {\color {red}{1\over 5}~\scalebox {-1}[1]{\ell }~{1\over 7}~\scalebox {-1}[1]{\ell }~{1\over 4}~\scalebox {-1}[1]{\ell }~{1\over 3}\,5}$ is 1991
143.5	The numerator of $\leavevmode {\color {red}{3~5\over 4~6}\,5}$ is 143
143.13	The numerator of $\leavevmode {\color {red}10\,{6\over 8}\,{4\over 7}}$ is 740
144.10	The numerator of $\leavevmode {\color {red}{3\over 6}\,5\,{4\over 9}}$ is 147
145.4	The numerator of $\leavevmode {\color {red}{4\over 7}\,7\,{2\over 3}}$ is 106
147.4	Add $\leavevmode {\color {red}{6\over 8}\,{4\over 5}\,3}$ to $\leavevmode {\color {red}{1~3~\,\,4\,\over 2~8~10}}$ to get $\leavevmode {\color {red}{1~7~\,\,9\,\over 2~8~10}4}$
147.15	Subtract $\leavevmode {\color {red}{1\over 3}~\scalebox {-1}[1]{\ell }~2\,{7\over 10}}$ from $\leavevmode {\color {red}{5\,\bullet \, 3\over 6\,\bullet \, 4}\,4}$ to get $\leavevmode {\color {red}{1~3~\,\,5\,\over 2~6~10}3}$
149.8	Multiply $\leavevmode {\color {red}{1\over 3}\,{3\over 4}}$ by $\leavevmode {\color {red}{1~\,4~\,3\over 5~\,6~\,9}}$ to get $\leavevmode {\color {red}{1~\,0~\,0~\,4\over 4~\,5~\,6~\,9}}$
150.2	Multiply $\leavevmode {\color {red}{1\over 8}\,4\,{1\over 3}}$ by $\leavevmode {\color {red}10\,{2\,\bullet \, 1\over 3\,\bullet \, 5}}$ to get $\leavevmode {\color {red}{5\over 6}1}$
151.4	Divide $\leavevmode {\color {red}{1\over 3}\,6}$ by $\leavevmode {\color {red}3\,{7\,\bullet \, 4\over 8\,\bullet \, 5}}$ to get $\leavevmode {\color {red}{1~\,0\over 7~\,9}\,3}$
151.14	Denominate $\leavevmode {\color {red}{2\over 9}~\scalebox {-1}[1]{\ell }~{1\over 4}\,3}$ with $\leavevmode {\color {red}{3\over 5}\,{2\over 8}\,6}$ to get $\leavevmode {\color {red}{5~\,\,50\,\over 9~137}}$
152.11	Divide $\leavevmode {\color {red}{2~\,\,9\,\over 3~\,10}\,8}$ by $\leavevmode {\color {red}{1~\,\,5\,\over 3~\,10}}$ to get $\leavevmode {\color {red}{1~\,6\over 2~\,8}\,16}$
153.3	Denominate $\leavevmode {\color {red}{1\over 3}\,2}$ with $\leavevmode {\color {red}{2\over 3}\,6}$ to get $\leavevmode {\color {red}{\,1~~3~\over 2~\,10}}$
153.12 L	Divide 5 by $\leavevmode {\color {red}5\over 6}$ to get 6; denominate $\leavevmode {\color {red}5\over 6}$ with 5 to get $\leavevmode {\color {red}1\over 6}$
154.6ff	Restore $\leavevmode {\color {red}1\over 2}$ to $\leavevmode {\color {red}9\over 10}$ ; $\leavevmode {\color {red}1~\,2\over 2~\,7}$ to $\leavevmode {\color {red}{1\over 2}5}$ ; $\leavevmode {\color {red}5~\bullet ~2\over 7~\bullet ~3}$ to 10; 5 to $\leavevmode {\color {red}{4\over 6}10}$ ; $\leavevmode {\color {red}{1~\,3\,\over 2~10}4}$ to 8; $\leavevmode {\color {red}{1\over 3}~\scalebox {-1}[1]{\ell }~{3\over 5}3}$ to $\leavevmode {\color {red}{3\over 5}12}$
155.11ff	Reduce $\leavevmode {\color {red}7\over 10}$ to $\leavevmode {\color {red}1\over 3}$ ; 8 to $\leavevmode {\color {red}{1\over 2}2}$ ; 10 to $\leavevmode {\color {red}3\over 4}$ ; $\leavevmode {\color {red}{1\over 4}7}$ to $\leavevmode {\color {red}{4\over 6}3}$ ; $\leavevmode {\color {red}{4~\,9\,\over 7~10}11}$ to 5; $\leavevmode {\color {red}{1\over 3}2}$ to $\leavevmode {\color {red}1\over 9}$
157.2 L	Convert $\leavevmode {\color {red}{3\over 4}~{5\over 6}}$ to tenths. Answer: $\leavevmode {\color {red}{5~\,5\,\over 6~10}1}$
157.12 L	How many tenths are in $\leavevmode {\color {red}{3\over 4}~{5\over 6}}$ ? Again, it is $\leavevmode {\color {red}{5~\,5\,\over 6~10}1}$
158.1 L	How many tenths are in 5? Answer: 50
158.11	How many ninths are in $\leavevmode {\color {red}{4\over 10}~{6\over 8}}$ ? Answer: $\leavevmode {\color {red}{4~\,\,\,3\,~\,1\over 8~\,10~\,9}\,1}$

Part 1, Chapter 3. On roots

Passage	Example
163.4 L	Examples: $\leavevmode {\color {red}\sqrt{10}}$ , $\leavevmode {\color {red}\sqrt{1\over 2}}$ , $\leavevmode {\color {red}\sqrt{{1\over 2}10}}$ , $\leavevmode {\color {red}\sqrt{\sqrt{10}}}$
166.13	$\leavevmode {\color {red}\sqrt{625}}$ is 25
167.8	$\leavevmode {\color {red}\sqrt{20}}$ is approximately $\leavevmode {\color {red}{1\over 2}4}$
167.14	$\leavevmode {\color {red}\sqrt{54}}$ is approximately $\leavevmode {\color {red}{1~\,2\over 2~\,7}7}$
168.1	$\leavevmode {\color {red}\sqrt{92}}$ is approximately $\leavevmode {\color {red}{3\over 5}9}$
168.16	$\leavevmode {\color {red}\sqrt{92}}$ is approximately $\leavevmode {\color {red}{1~\,5~\,\,\,5\,\over 2~\,6~\,10}\,9}$
169.4	$\leavevmode {\color {red}\sqrt{12}}$ is approximately $\leavevmode {\color {red}{1~\,3\over 4~\,7}3}$
169.10,17 L	$\leavevmode {\color {red}\sqrt{625}}$ is 25; $\leavevmode {\color {red}\sqrt{729}}$ is 27
170.6	$\leavevmode {\color {red}\sqrt{100}}$ is 10
170.16	$\leavevmode {\color {red}\sqrt{1~\,4\over 6~\,6}}$ is $\leavevmode {\color {red}{5\over 6}}$
171.1	$\leavevmode {\color {red}\sqrt{{1\over 4}12}}$ is $\leavevmode {\color {red}{1\over 2}3}$
171.6	$\leavevmode {\color {red}\sqrt{3~\,4\over 6~\,9}}$ is $\leavevmode {\color {red}{1~\,5~\,3~\,13\over 2~\,6~\,9~\,19}}$
171.15	$\leavevmode {\color {red}\sqrt{{1~\,7\over 2~\,8}10}}$ is approximately $\leavevmode {\color {red}{1~\,2~\,16\over 4~\,8~\,53}\,3}$
172.6	$\leavevmode {\color {red}\sqrt{{1~\,7\over 2~\,8}10}}$ is approximately $\leavevmode {\color {red}{4\over 13}3}$
172.12	$\leavevmode {\color {red}\sqrt{1~\,4\over 2~\,7}}$ is approximately $\leavevmode {\color {red}{3~\,5~\,\,\,8\,\over 4\,~7~\,11}}$
173.13 L	Binomials: $\leavevmode {\color {red}\sqrt{3}~5}$ ; $\leavevmode {\color {red}\sqrt{3}~\sqrt{5}}$
173.16 L	Apotomes: $\leavevmode {\color {red}\sqrt{3}~\scalebox {-1}[1]{\ell }~5}$ ; $\leavevmode {\color {red}\sqrt{3}~\scalebox {-1}[1]{\ell }~\sqrt{5}}$
174.5	1st & 4th binomials: $\leavevmode {\color {red}\sqrt{21}~5}$ , $\leavevmode {\color {red}\sqrt{2}~2}$
174.8	2nd & 5th binomials: $\leavevmode {\color {red}\sqrt{45}~5}$ , $\leavevmode {\color {red}\sqrt{72}~5}$
174.11	3rd & 6th binomials: $\leavevmode {\color {red}\sqrt{18}~\sqrt{10}}$ , $\leavevmode {\color {red}\sqrt{8}~\sqrt{7}}$
175.11	$\leavevmode {\color {red}\sqrt{\sqrt{60}~8}}$ is $\leavevmode {\color {red}\sqrt{3}~\sqrt{5}}$
175.19	$\leavevmode {\color {red}\sqrt{\sqrt{60}~\scalebox {-1}[1]{\ell }~8}}$ is $\leavevmode {\color {red}\sqrt{3}~\scalebox {-1}[1]{\ell }~\sqrt{5}}$
176.6	$\leavevmode {\color {red}\sqrt{\sqrt{55}~8}}$ is $\leavevmode {\color {red}\sqrt{{1\over 2}2}~\sqrt{{1\over 2}5}}$ ; $\leavevmode {\color {red}\sqrt{\sqrt{55}~\scalebox {-1}[1]{\ell }~8}}$ is $\leavevmode {\color {red}\sqrt{{1\over 2}2}~\scalebox {-1}[1]{\ell }~\sqrt{{1\over 2}5}}$
176.10	$\leavevmode {\color {red}\sqrt{\sqrt{112}~7}}$ is $\leavevmode {\color {red}\sqrt{\sqrt{{3\over 4}1}}~\sqrt{\sqrt{{3\over 4}85}}}$ ; $\leavevmode {\color {red}\sqrt{\sqrt{112}~\scalebox {-1}[1]{\ell }~7}}$ is $\leavevmode {\color {red}\sqrt{\sqrt{{3\over 4}1}}~\scalebox {-1}[1]{\ell }~\sqrt{\sqrt{{3\over 4}85}}}$
176.15	$\leavevmode {\color {red}\sqrt{\sqrt{14}~\sqrt{32}}}$ is $\leavevmode {\color {red}\sqrt{\sqrt{1\over 2}}~\sqrt{\sqrt{{1\over 2}24}}}$ ; $\leavevmode {\color {red}\sqrt{\sqrt{14}~\scalebox {-1}[1]{\ell }~\sqrt{32}}}$ is $\leavevmode {\color {red}\sqrt{\sqrt{1\over 2}}~\scalebox {-1}[1]{\ell }~\sqrt{\sqrt{{1\over 2}24}}}$
176.20	$\leavevmode {\color {red}\sqrt{\sqrt{30}~7}}$ is $\leavevmode {\color {red}\sqrt{\sqrt{{3\over 4}4}~\scalebox {-1}[1]{\ell }~{1\over 2}3}~\sqrt{\sqrt{{3\over 4}4}~{1\over 2}3}}$ ; $\leavevmode {\color {red}\sqrt{\sqrt{30}~\scalebox {-1}[1]{\ell }~7}}$ is $\leavevmode {\color {red}\sqrt{\sqrt{{3\over 4}4}~\scalebox {-1}[1]{\ell }~{1\over 2}3}~\scalebox {-1}[1]{\ell }~\sqrt{\sqrt{{3\over 4}4}~{1\over 2}3}}$
177.5	$\leavevmode {\color {red}\sqrt{\sqrt{20}~3}}$ is $\leavevmode {\color {red}\sqrt{\sqrt{{1\over 4}~{1\over 2}2}~\scalebox {-1}[1]{\ell }~\sqrt{5}}~\sqrt{\sqrt{{1\over 4}~{1\over 2}2}~\sqrt{5}}}$ ; $\leavevmode {\color {red}\sqrt{3~\scalebox {-1}[1]{\ell }~\sqrt{20}}}$ is $\leavevmode {\color {red}\sqrt{\sqrt{{1\over 4}~{1\over 2}2}~\scalebox {-1}[1]{\ell }~\sqrt{5}}~\scalebox {-1}[1]{\ell }~\sqrt{\sqrt{{1\over 4}~{1\over 2}2}~\sqrt{5}}}$
177.11	$\leavevmode {\color {red}\sqrt{\sqrt{11}~\sqrt{10}}}$ is $\leavevmode {\color {red}\sqrt{{1\over 2}~\scalebox {-1}[1]{\ell }~\sqrt{{3\over 4}2}}~\sqrt{\sqrt{{3\over 4}2}~{1\over 2}}}$ ; $\leavevmode {\color {red}\sqrt{\sqrt{10}~\scalebox {-1}[1]{\ell }~\sqrt{11}}}$ is $\leavevmode {\color {red}\sqrt{{1\over 2}~\scalebox {-1}[1]{\ell }~\sqrt{{3\over 4}2}}~\scalebox {-1}[1]{\ell }~\sqrt{\sqrt{{3\over 4}2}~{1\over 2}}}$
179.7,11	Add $\leavevmode {\color {red}\sqrt{3}}$ to $\leavevmode {\color {red}\sqrt{27}}$ to get $\leavevmode {\color {red}\sqrt{48}}$
179.16	Add $\leavevmode {\color {red}\sqrt{2}}$ to $\leavevmode {\color {red}\sqrt{8}}$ to get $\leavevmode {\color {red}\sqrt{18}}$
179.20	Add half of $\leavevmode {\color {red}\sqrt{20}}$ to two $\leavevmode {\color {red}\sqrt{5}}$ s to get $\leavevmode {\color {red}\sqrt{45}}$
180.10	Add $\leavevmode {\color {red}\sqrt{3}}$ to $\leavevmode {\color {red}\sqrt{15}}$ to get $\leavevmode {\color {red}\sqrt{15}~\sqrt{3}}$
180.15	Add half of $\leavevmode {\color {red}\sqrt{\sqrt{80}}}$ to $\leavevmode {\color {red}{1~\bullet ~1\over 4~\bullet ~3}}$ of $\leavevmode {\color {red}\sqrt{684}}$ to get $\leavevmode {\color {red}\sqrt{\sqrt{{1~4\over 2~\,8}\,22}}~\sqrt{\sqrt{5}}}$
181.6	Subtract $\leavevmode {\color {red}\sqrt{8}}$ from $\leavevmode {\color {red}\sqrt{32}}$ to get $\leavevmode {\color {red}\sqrt{8}}$
181.12,16	Subtract $\leavevmode {\color {red}\sqrt{12}}$ from $\leavevmode {\color {red}\sqrt{27}}$ to get $\leavevmode {\color {red}\sqrt{3}}$
182.4	Subtract $\leavevmode {\color {red}\sqrt{8}}$ from $\leavevmode {\color {red}\sqrt{10}}$ to get $\leavevmode {\color {red}\sqrt{8}~\scalebox {-1}[1]{\ell }~\sqrt{10}}$
183.4	Multiply $\leavevmode {\color {red}\sqrt{8}}$ by $\leavevmode {\color {red}\sqrt{9}}$ to get $\leavevmode {\color {red}\sqrt{72}}$
183.7	Multiply $\leavevmode {\color {red}\sqrt{\sqrt{5}}}$ by $\leavevmode {\color {red}\sqrt{\sqrt{7}}}$ to get $\leavevmode {\color {red}\sqrt{\sqrt{35}}}$
183.11	Multiply $\leavevmode {\color {red}\sqrt{\sqrt{\sqrt{3}}}}$ by $\leavevmode {\color {red}\sqrt{\sqrt{\sqrt{8}}}}$ to get $\leavevmode {\color {red}\sqrt{\sqrt{\sqrt{24}}}}$
183.15	Multiply 3 by $\leavevmode {\color {red}2~\scalebox {-1}[1]{\ell }~\sqrt{7}}$ to get $\leavevmode {\color {red}6~\scalebox {-1}[1]{\ell }~\sqrt{63}}$
184.1	Multiply 3 by $\leavevmode {\color {red}\sqrt{7}}$ to get $\leavevmode {\color {red}\sqrt{63}}$
184.4	Multiply 2 by $\leavevmode {\color {red}\sqrt{\sqrt{3}}}$ to get $\leavevmode {\color {red}\sqrt{\sqrt{48}}}$
184.11	Multiply 2 by two $\leavevmode {\color {red}\sqrt{7}}$ s to get $\leavevmode {\color {red}\sqrt{112}}$
184.18	Multiply 5 by three $\leavevmode {\color {red}\sqrt{\sqrt{2}}}$ s to get $\leavevmode {\color {red}\sqrt{\sqrt{101250}}}$
185.6	Multiply $\leavevmode {\color {red}2\over 3}$ by half of $\leavevmode {\color {red}\sqrt{20}}$ to get $\leavevmode {\color {red}\sqrt{{2\over 9}2}}$
185.12	Multiply $\leavevmode {\color {red}\sqrt{5}}$ by half of $\leavevmode {\color {red}\sqrt{\sqrt{40}}}$ to get $\leavevmode {\color {red}\sqrt{{1\over 2}62}}$
186.1	Duplicate $\leavevmode {\color {red}\sqrt{3}}$ twice to get $\leavevmode {\color {red}\sqrt{12}}$
186.4	Duplicate $\leavevmode {\color {red}\sqrt{7}}$ five times to get $\leavevmode {\color {red}\sqrt{175}}$
186.8	Half of $\leavevmode {\color {red}\sqrt{10}}$ is $\leavevmode {\color {red}\sqrt{{1\over 2}2}}$
186.11	$\leavevmode {\color {red}4~\bullet ~1\over 8~\bullet ~3}$ of $\leavevmode {\color {red}\sqrt{\sqrt{60}}}$ is $\leavevmode {\color {red}\sqrt{\sqrt{1~2~0\over 2~6~9}}}$
187.4	Divide $\leavevmode {\color {red}\sqrt{20}}$ by $\leavevmode {\color {red}\sqrt{3}}$ to get $\leavevmode {\color {red}\sqrt{{2\over 3}6}}$
187.7	Denominate $\leavevmode {\color {red}\sqrt{3}}$ with $\leavevmode {\color {red}\sqrt{8}}$ to get $\leavevmode {\color {red}\sqrt{3\over 8}}$
187.10	Divide $\leavevmode {\color {red}\sqrt{\sqrt{6}}}$ by $\leavevmode {\color {red}\sqrt{\sqrt{2}}}$ to get $\leavevmode {\color {red}\sqrt{\sqrt{3}}}$
187.14	Denominate $\leavevmode {\color {red}\sqrt{\sqrt{18}}}$ with $\leavevmode {\color {red}\sqrt{\sqrt{32}}}$ to get $\leavevmode {\color {red}\sqrt{\sqrt{1~4\over 2~8}}}$
188.6	Divide $\leavevmode {\color {red}\sqrt{\sqrt{14}}}$ by $\leavevmode {\color {red}\sqrt{2}}$ to get $\leavevmode {\color {red}\sqrt{\sqrt{{1\over 2}3}}}$
188.11	Divide two $\leavevmode {\color {red}\sqrt{15}}$ s by 2 to get $\leavevmode {\color {red}\sqrt{15}}$
188.15	Divide half of $\leavevmode {\color {red}\sqrt{24}}$ by $\leavevmode {\color {red}\sqrt{2}}$ to get $\leavevmode {\color {red}\sqrt{3}}$
189.1	Divide 12 by $\leavevmode {\color {red}\sqrt{3}~5}$ to get $\leavevmode {\color {red}\sqrt{9~~9\over 11~\,11}~\scalebox {-1}[1]{\ell }~{8\over 11}2}$
189.11	Divide 10 by $\leavevmode {\color {red}\sqrt{7}~\scalebox {-1}[1]{\ell }~ 3}$ to get $\leavevmode {\color {red}\sqrt{175}~15}$

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: $\leavevmode {\color {red}{t\atop 7}={m\atop 3}}$ ; $\leavevmode {\color {red}\hbox{20}={m\atop 5}}$ ; $\leavevmode {\color {red}\hbox{12}={t\atop 3}}$ ( ; ; )
212.6	Composite equations: $\leavevmode {\color {red}\hbox{24}={t\atop 10}~{m\atop 1}}$ ; $\leavevmode {\color {red}{t\atop 5}=\hbox{4}~{m\atop 1}}$ ; $\leavevmode {\color {red}\hbox{5}~{t\atop 4}={m\atop 1}}$ ( ; ; )
213.7	Solve $\leavevmode {\color {red}{t\atop 15}={m\atop 3}}$ to get $\leavevmode {\color {red}t\atop 1}$ is 5 and $\leavevmode {\color {red}m\atop 1}$ is 25 ( $3x^2=15x\Rightarrow x=5$ , )
213.13	Solve $\leavevmode {\color {red}\hbox{18}={m\atop 2}}$ to get $\leavevmode {\color {red}m\atop 1}$ is 9 and $\leavevmode {\color {red}t\atop 1}$ is 3 ( $2x^2=18\Rightarrow x^2=9$ , )
214.1	Solve $\leavevmode {\color {red}\hbox{20}={t\atop 5}}$ to get $\leavevmode {\color {red}t\atop 1}$ is 4 and $\leavevmode {\color {red}m\atop 1}$ is 16 ( $5x=20\Rightarrow x=4$ , )
214.9	Solve $\leavevmode {\color {red}\hbox{15}={t\atop 2}~{m\atop 1}}$ to get $\leavevmode {\color {red}t\atop 1}$ is 3 and $\leavevmode {\color {red}m\atop 1}$ is 9 ( $x^2+2x=15\Rightarrow x=3$ , )
215.6	Solve $\leavevmode {\color {red}{t\atop 6}=\hbox{8}~{m\atop 1}}$ to get $\leavevmode {\color {red}t\atop 1}$ is 4 and $\leavevmode {\color {red}m\atop 1}$ is 16, or $\leavevmode {\color {red}t\atop 1}$ is 2 and $\leavevmode {\color {red}m\atop 1}$ is 4 ( $x^2+8=6x\Rightarrow x=4$ , or , )
215.14	Solve $\leavevmode {\color {red}{t\atop 6}=\hbox{9}~{m\atop 1}}$ to get $\leavevmode {\color {red}m\atop 1}$ is 9 and $\leavevmode {\color {red}t\atop 1}$ is 3( $x^2+9=6x\Rightarrow x^2=9$ , )
216.13	Solve $\leavevmode {\color {red}\hbox{3}~{t\atop 2}={m\atop 1}}$ to get $\leavevmode {\color {red}t\atop 1}$ is 3 and $\leavevmode {\color {red}m\atop 1}$ is 9 ( $x^2=2x+3\Rightarrow x=3$ , )
217.10	$\leavevmode {\color {red}\hbox{36}={t\atop 6}~{m\atop 2}}$ simplifies to $\leavevmode {\color {red}\hbox{18}={t\atop 3}~{m\atop 1}}$ ( $2x^2+6x=36\Rightarrow x^2+3x=18$ )
218.1	$\leavevmode {\color {red}\hbox{6}={t\atop 2}~{m\atop {1\over 2}}}$ simplifies to $\leavevmode {\color {red}\hbox{12}={t\atop 4}~{m\atop 1}}$ ( ${1\over 2}x^2+2x=6\Rightarrow x^2+4x=12$ )
219.2	Add $\leavevmode {\color {red}m\atop 1}$ , $\leavevmode {\color {red}t\atop 6}$ , and $\leavevmode {\color {red}\hbox{10}}$ to get $\leavevmode {\color {red}\hbox{10}~{t\atop 6}~{m\atop 1}}$ ( $6x+x^2+10\Rightarrow x^2+6x+10$ )
219.5 L	Add $\leavevmode {\color {red}{t\atop 1}~\scalebox {-1}[1]{\ell }~{m\atop 1}}$ to $\leavevmode {\color {red}\hbox{10}}$ to get $\leavevmode {\color {red}{t\atop 1}~\scalebox {-1}[1]{\ell }~\hbox{10}~{m\atop 1}}$ ( $(x^2-x)+10\Rightarrow x^2+10-x$ )
219.7 L	Add $\leavevmode {\color {red}{m\atop 1}~\scalebox {-1}[1]{\ell }~{m\atop 2}}$ to $\leavevmode {\color {red}\hbox{10}}$ to get $\leavevmode {\color {red}\hbox{10}~{m\atop 1}}$ ( $(2x^2-x^2)+10\Rightarrow x^2+10$ )
219.10 L	Add $\leavevmode {\color {red}{t\atop 2}~\scalebox {-1}[1]{\ell }~{m\atop 1}}$ to $\leavevmode {\color {red}t\atop 10}$ to get $\leavevmode {\color {red}{t\atop 8}~{m\atop 1}}$ ( $(x^2-2x)+10x\Rightarrow x^2+8x$ )
219.12 L	Add $\leavevmode {\color {red}{t\atop 5}~\scalebox {-1}[1]{\ell }~{m\atop 1}}$ to $\leavevmode {\color {red}t\atop 10}$ to get $\leavevmode {\color {red}{t\atop 5}~{m\atop 1}}$ ( $(x^2-5x)+10x\Rightarrow x^2+5x$ )
220.1	Subtract $\leavevmode {\color {red}t\atop 1}$ from $\leavevmode {\color {red}m\atop 1}$ to get $\leavevmode {\color {red}{t\atop 1}~\scalebox {-1}[1]{\ell }~{m\atop 1}}$ ( $x^2-x\Rightarrow x^2-x$ )
220.3	Subtract $\leavevmode {\color {red}\hbox{10}}$ from $\leavevmode {\color {red}{t\atop 3}~{m\atop 2}}$ to get $\leavevmode {\color {red}\hbox{10}~\scalebox {-1}[1]{\ell }~{t\atop 3}~{m\atop 2}}$ ( $(2x^2+3x)-10\Rightarrow 2x^2+3x-10$ )
220.9	Subtract $\leavevmode {\color {red}{t\atop 1}~\hbox{2}}$ from $\leavevmode {\color {red}{t\atop 3}~\scalebox {-1}[1]{\ell }~{m\atop 1}}$ to get $\leavevmode {\color {red}\hbox{2}~\scalebox {-1}[1]{\ell }~{t\atop 4}~\scalebox {-1}[1]{\ell }~{m\atop 1}}$ ( $(x^2-3x)-(2+x)\Rightarrow x^2-4x-2$ )
221.1	Subtract $\leavevmode {\color {red}{t\atop 5}~\scalebox {-1}[1]{\ell }~\hbox{52}}$ from $\leavevmode {\color {red}\hbox{30}~{c\atop 2}}$ to get $\leavevmode {\color {red}{t\atop 22}~\scalebox {-1}[1]{\ell }~{t\atop 5}~{c\atop 2}}$ ( $(2x^3+30)-(52-5x)\Rightarrow 2x^3+5x-22$ )
221.13	Subtract $\leavevmode {\color {red}{t\atop 4}~\scalebox {-1}[1]{\ell }~\hbox{12}}$ from $\leavevmode {\color {red}{t\atop 2}~\scalebox {-1}[1]{\ell }~{m\atop 3}}$ to get $\leavevmode {\color {red}\hbox{12}~\scalebox {-1}[1]{\ell }~{t\atop 2}~{m\atop 3}}$ ( $(3x^2-2x)-(12-4x)\Rightarrow 3x^2+2x-12$ )
222.1	Subtract $\leavevmode {\color {red}{m\atop 2}~\scalebox {-1}[1]{\ell }~{c\atop 1}}$ from $\leavevmode {\color {red}{t\atop 4}~\scalebox {-1}[1]{\ell }~\hbox{30}}$ to get $\leavevmode {\color {red}{t\atop 4}~\scalebox {-1}[1]{\ell }~{c\atop 1}~\scalebox {-1}[1]{\ell }~{m\atop 2}~\hbox{30}}$ ( $(30-4x)-(x^3-2x^2)\Rightarrow 30+2x^2-x^3-4x$ )
223.2	$\leavevmode {\color {red}{t\atop 1}~\hbox{2}={t\atop 3}~\scalebox {-1}[1]{\ell }~{m\atop 1}}$ simplifies to $\leavevmode {\color {red}{t\atop 4}~\hbox{2}={m\atop 1}}$ ( $x^2-3x=2+x\Rightarrow x^2=2+4x$ )
223.7	$\leavevmode {\color {red}{t\atop 5}~\scalebox {-1}[1]{\ell }~\hbox{24}={t\atop 3}~\scalebox {-1}[1]{\ell }~{m\atop 1}}$ simplifies to $\leavevmode {\color {red}\hbox{24}={t\atop 2}~{m\atop 1}}$ ( $x^2-3x=24-5x\Rightarrow x^2+2x=24$ )
223.14	$\leavevmode {\color {red}{t\atop {1\over 2}2}~\scalebox {-1}[1]{\ell }~{m\atop 1}=\hbox{10}~\scalebox {-1}[1]{\ell }~{m\atop 1}}$ simplifies to $\leavevmode {\color {red}{t\atop {1\over 2}2}=\hbox{10}}$ ( $x^2-10=x^2-2{1\over 2}x\Rightarrow 10=2{1\over 2}x$ )
223.17	$\leavevmode {\color {red}{t\atop 4}~\scalebox {-1}[1]{\ell }~\hbox{51}=\hbox{10}~{m\atop 1}}$ simplifies to $\leavevmode {\color {red}\hbox{41}={t\atop 4}~{m\atop 1}}$ ( $x^2+10=51-4x\Rightarrow x^2+4x=41$ )
224.1	$\leavevmode {\color {red}{t\atop 1}~\scalebox {-1}[1]{\ell }~{m\atop 2}~\hbox{10}={t\atop 5}~{m\atop 1}}$ simplifies to $\leavevmode {\color {red}\hbox{10}~{m\atop 1}={t\atop 6}}$ ( $x^2+5x=10+2x^2-x\Rightarrow 6x=x^2+10$ )
224.4	$\leavevmode {\color {red}{t\atop 7}~\scalebox {-1}[1]{\ell }~\hbox{7}~{m\atop 3}={t\atop 2}~\scalebox {-1}[1]{\ell }~\hbox{10}~{m\atop 2}}$ simplifies to $\leavevmode {\color {red}{t\atop 5}~\hbox{3}={m\atop 1}}$ ( $2x^2+10-2x=3x^2+7-7x\Rightarrow x^2=3+5x$ )
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 $\leavevmode {\color {red}t\atop 5}$ by $\leavevmode {\color {red}t\atop 7}$ to get $\leavevmode {\color {red}m\atop 35}$ ( $5x\times 7x\Rightarrow 35x$ )
226.17	Multiply $\leavevmode {\color {red}t\atop 10}$ by $\leavevmode {\color {red}m\atop 6}$ to get $\leavevmode {\color {red}c\atop 60}$ ( $10x\times 6x^2\Rightarrow 60x^3$ )
227.4	Multiply $\leavevmode {\color {red}t\atop 1}$ by $\leavevmode {\color {red}c\atop 1}$ to get $\leavevmode {\color {red}mm\atop 1}$ ( $x\times x^3\Rightarrow x^4$ )
227.7	Multiply 6 by $\leavevmode {\color {red}m\atop 4}$ to get $\leavevmode {\color {red}m\atop 24}$ ( $6\times 4x^2\Rightarrow 24x^2$ )
227.10	Multiply 7 by $\leavevmode {\color {red}mc\atop 3}$ to get $\leavevmode {\color {red}mc\atop 21}$ ( $7\times 3x^5\Rightarrow 21x^5$ )
227.17	$\leavevmode {\color {red}{m\atop 10}~{c\atop 4}={mm\atop 3}}$ simplifies to $\leavevmode {\color {red}\hbox{10}~{t\atop 4}={m\atop 3}}$ ( $3x^4=4x^3+10x^2\Rightarrow 3x^2=4x+10$ )
228.1	$\leavevmode {\color {red}{t\atop 20}~{m\atop 10}={c\atop 3}}$ simplifies to $\leavevmode {\color {red}\hbox{20}~{t\atop 10}={m\atop 3}}$ ( $3x^3=10x^2+20x\Rightarrow 3x^2=10x+20$ )
228.4	$\leavevmode {\color {red}{t\atop 39}={m\atop 10}~{c\atop 1}}$ simplifies to $\leavevmode {\color {red}\hbox{39}={t\atop 10}~{m\atop 1}}$ ( $x^3+10x^2=39x\Rightarrow x^2+10x=39$ )
228.11	Multiply $\leavevmode {\color {red}t\atop 5}$ by $\leavevmode {\color {red}{t\atop 4}~\scalebox {-1}[1]{\ell }~\hbox{13}}$ to get $\leavevmode {\color {red}{m\atop 20}~\scalebox {-1}[1]{\ell }~{t\atop 65}}$ ( $5x\times (13-4x)\Rightarrow 65x-20x^2$ )
228.15	Multiply $\leavevmode {\color {red}{t\atop 2}~\scalebox {-1}[1]{\ell }~\hbox{8}}$ by $\leavevmode {\color {red}{m\atop 4}~\scalebox {-1}[1]{\ell }~\hbox{7}}$ to get $\leavevmode {\color {red}{m\atop 32}~\scalebox {-1}[1]{\ell }~{t\atop 14}~\scalebox {-1}[1]{\ell }~\hbox{56}~{c\atop 8}}$ ( $(8-2x)\times (7-4x^2)\Rightarrow 8x^3+56-14x-32x^2$ )
229.4	Divide $\leavevmode {\color {red}m\atop 10}$ by $\leavevmode {\color {red}t\atop 2}$ to get $\leavevmode {\color {red}t\atop 5}$ ( $10x^2\div 2x\Rightarrow 5x$ )
229.8	Divide $\leavevmode {\color {red}c\atop 15}$ by $\leavevmode {\color {red}t\atop 3}$ to get $\leavevmode {\color {red}m\atop 5}$ ( $15x^3\div 3x\Rightarrow 5x^2$ )
229.13	Divide $\leavevmode {\color {red}m\atop 12}$ by $\leavevmode {\color {red}m\atop 3}$ to get 4 ( $12x^2\div 3x^2\Rightarrow 4$ )
229.17	Divide $\leavevmode {\color {red}t\atop 12}$ by 4 to get $\leavevmode {\color {red}t\atop 3}$ ( $12x\div 4\Rightarrow 3x$ )
230.4	Divide $\leavevmode {\color {red}{m\atop 3}~\scalebox {-1}[1]{\ell }~{c\atop 12}}$ by $\leavevmode {\color {red}t\atop 2}$ to get $\leavevmode {\color {red}{t\atop {1\over 2}1}~\scalebox {-1}[1]{\ell }~{m\atop 6}}$ ( $(12x^3-3x^2)\div 2x\Rightarrow 6x^2-1{1\over 2}x$ )
230.8	Divide $\leavevmode {\color {red}{t\atop 3}~\scalebox {-1}[1]{\ell }~{m\atop 10}}$ by 2 to get $\leavevmode {\color {red}{t\atop {1\over 2}1}~\scalebox {-1}[1]{\ell }~ {m\atop 5}}$ ( $(10x^2-3x)\div 2\Rightarrow 5x^2-1{1\over 2}x$ )
230.13	Divide $\leavevmode {\color {red}m\atop 6}$ by $\leavevmode {\color {red}c\atop 3}$ to get $\leavevmode {\color {red}2\over {t\atop 1}}$ ( $6x^2\div 3x^3\Rightarrow {2\over x}$ )
230.17	Divide $\leavevmode {\color {red}m\atop 10}$ by $\leavevmode {\color {red}{t\atop 1}~\scalebox {-1}[1]{\ell }~\hbox{3}}$ to get $\leavevmode {\color {red}{m\atop 10}\over {{t\atop 1}~\scalebox {-1}[1]{\ell }~\hbox{3}}}$ ( $10x^2\div (3-x)\Rightarrow {10x^2\over 3-x}$ )