Difference between revisions of "לקוטים מספר פראלוקא"

Introduction

Section One - Theoretical Arithmetic

impossible

תמצא לי ב' מספרים אחרים אשר מרובעהם מקובצים יעלו עשרים

גם חלק כ"ב על ה' ויצא ד' וב' חמישיות וזהו המספר השני

Section Two - Practical Arithmetic

Check

Subtraction

Multiplication

Progressions

49) A way to know how much is the sum of the numbers beginning from one, by knowing half the sum.

• Example: we wish to know the sum of all the numbers from 1 to 12.

50) A way to sum from 1 by the succession of the numbers

It is the required, whether for even or for odd.

• Example: we wish to sum from 1 to 6.

• If you want to sum from 1 to 7.

51)

52)

53)

54)

57)

58)

59)

וכן אם תרצה לחבר כל המרובעים שהם מא' עד י"ב

60)

61)

62)

Motion Problems

69) Question: One man walks 32 miles every day; another man starts following him after 6 days and reaches him in 25 days.

I ask: how many miles did the latter walk each day?

אשאל כמה מילין הלך האחרון בכל יום

70) Question: a man walks 30 miles every day; at the end of 5 days another [man] starts following him and walks 35 every day.

I ask: in how many days will he reach him?

אשאל בכמה ימים ישיגנו

71) Question: a man walks, we do not know how many [miles] every day; at the end of 4 days [another] man starts following him and walks 35 every day. He reaches him at the end of twenty days.

I ask: how many [miles] does the former walk every day?

אשאל כמה הולך הראשון בכל יום

מתי יפגעו זה את זה

74) Question: A man has to walk ten miles. During the day he walks 2 [miles] and at night he goes back one [mile].

I ask: how many days will he walk?

אשאל בכמה ימים ילך

75) Question: A dog is sixty cubits away from a rabbit and 5 steps of the dog are equal to 7 of the rabbit.

I ask: when will the dog reach the rabbit?

‫^[37]אשאל מתי ישיג הכלב הארנבת

77) If you wish to sum up all the cubes from 1 to 12; i.e. 1, 8, 27, 64, 125.

78) If you wish to sum up all the squares from 1 to ten; i.e. 1, 4, 16, 25.

Question: to extract the root of fractions and integers or fractions alone.

• Example: we wish to extract the root of 9 parts of 16.

81) Question: find me 9 different squares such that their sum is a square number.

Cubic Roots

82) Question:

Fractions

83)

84) If you wish to multiply 30 by 5 sixths.

Multiply 30 by 5 then divide by 6 and the result is 25.

85) If you wish to subtract 3 fifths from 7 eighths.

• Common denominator: take a number that consists of 5 and 8, which is 40.

• Take its 3 fifths, they are 25.

• Take its 7 eighths, they are 35.

• Subtract 24 from 35, the remainder is 11 and they are 11 parts of the denominator, which is 40.

• As if you say: sum 2 thirds with 3 quarters

• Common denominator: take a number that consists of 3 and 4, which is 12 and it is the denominator.

• Take its 2 thirds, they are 8.

• And its 3 quarters, they are 9.

Sum 9 and 8, they are 17.

• Divide them by 12, which is the denominator and the result is 1 and 5 parts of 12.

86) If you wish to divide one half by one third.

• Common denominator: take the denominator, which is 6.

• Take its third, it is 2.

• And its half, it is 3.

• Divide 3 by 2, it is 1 and a half and it is the required.

• If you wish to divide 4 by 2 thirds.

87)

88) Which number will you sum with 23 so that the result will be 39?

• Subtract 23 from 39, the remainder is 16.

89) Which number will you sum with 6 and a half so that the result will be 13 and 2 fifths?

• Subtract 6 and a half from 13 and 2 fifths, the remainder is 6 and [9] parts of [10].

90) Which number will you divide by 5 so that the result will be 17?

• Multiply 5 by 17, the result is 85.

91) Which number will you divide by 4 and a seventh so that the result will be 2 and one eighth?

• Multiply 4 and a seventh by 2 and one eighth, the result is 8 and 45 parts of 56.

92) By which number will you divide 36 so that the result will be 9?

• Do as follows: divide 36 by 9, the result is 4 and this is the required.

93) By which number will you divide 12 and a quarter so that the result will be 3 and a half?

• Divide 12 and a quarter by 3 and a half, the result is 4 and a half.

94)

95) Question: how many parts of 18 there are in 5 ninths?

96)

97) Divide 7 and a half by a certain number so that the result will by 5 sixths of 17 and a half.

98)

99) By how many fifths should we multiply 3 and 2 thirds so that the result will be 7 and 3 quarters?

100) Divide 7 and a half by so and so ninths so that the result will be 2 and a [quarter].

101)

102)

103)

104)

105)

106)

107) Question:

108)

109) Question:

110) Question:

111) Question:

Ratios

112) Now we will discuss the ratios

The reason is that the double ratio is generated from the ratio of 1 to 2, for 3 is as 2 plus its half.

The triple ratio is generated from the ratio of 1 to 3, for it is greater than 2.

• Quadruple ratio:

• Quintuple ratio:

113) Question: find me three proportional numbers, such that their product by each other is the same as their sum.

Do as follows: 1, 2, 3 are related, because the excess of the second over the first is 1 and the excess of the third over the second is 1.

Multiply 1 by 2; it is 2. Multiply it by 3; it is 6. Also the sum of 1, 2, 3 is 6.

114)

If the ratio is generated from the ratio of 1 to 2, it is called double ratio.

If it is generated from the ratio of 1 to 3, it is called triple ratio.

If it is generated from the ratio of 1 to 4, it is called quadruple ratio.

The sesquialter ratio is called the ratio of 3 to 2, for 3 is as 2 plus its half.

The sesquitertian ratio is [generated] from the ratio of 4 to 3, for 4 is as 3 plus its third.

The sesquiquartan ratio is [generated] from the ratio of 5 to 4, for 5 exceeds over 4 by its quarter.

The sesquiquintan ratio is [generated] from the ratio of 6 to 5, for 6 is as 5 plus its fifth added to it.

If the ratio is of 5 to 3 it is called superbipartient ratio, for 5 is as 3 plus 2.

If the number adds 3 over the other, it is called supertripartient, such as 8 to 5, for 8 is as 5 plus 3.

If it adds 4 [over the other], such as the ratio of 9 to 5, it is called superquadripartient.

If it added 5, such as , it is called superquintipartient ratio.

115)

116) Question:

117)

118)

119)

כן יעשה מהמספרים אחרים שיהיו דופלא כגון ו' י"ב כ"ד אם תחברם יהיו מ"ב חלקם על ו' יצא ז' כמו בראשנה חלקם על י"ב יצא ג' וחצי חלקם על כ"ד יצא א' וג' רביעיות כבראשנה

120)

121)

122)

123)

124)

125)

126)

127)

128)

129)

130) If you have four proportional numbers, then the product of all of them by each other is the same as the [square of the product of the] first by the fourth.

• Example: 2, 4, 8, 16.

Now, multiply 2 by 4; it is 8; 8 by 8 is 64; and 16 times 64 is one thousand and 24.

If you multiply 2, which is the first, by 16, which is the fourth; it is 32 and its square is one thousand and 24.

131)

132)

133)

134)

135)

136)

137)

138)

139) Question:

140) Question:

141)

142)

143) Divide 19 into three proportional numbers, such that if you multiply the first by the second, then [the product] by the third, the result is 216.

144) Divide ten into three proportional numbers, such that the product of the first by the third is the same as 3 times the product of the first by the second.

Do as follows: since 3 is mentioned, take 1 for the first, 3 for the second, and for the third take it 3 times, which 9.

Sum them; it is 13.

Divide 10 by 13; the result is 10 parts of 13 and this is the first.

Multiply it 3 times; the result is 2 and 4 parts of 13.

Multiply it by 3; the result is 6 and 12 parts of 13 and this is the third.

145) Question: divide ten into four proportional parts, such that the product of the first part by the fourth is three times the product of the first part by the third.

Do as follows: we suppose that the first is 1, the second is 3, the third is 9 and the fourth is 27.

The total is 40.

Therefore, say: if 40 gives me 1, what will ten give you and how much will it add? The method is known.

We find that the first is a quarter, the second is 3-quarters, the third is 9-quarters, the fourth is 27-quarters.

146) Question: weigh for me with four stones up to 15.

Take 1, 2, 4 and 8; they are 15 and with these four you can weigh up to 15.

If a fifth is added to them one can weigh up to 31.

If you want another ratio, as if you say: weigh with four stones from 1 up to 40.

Take 1, 3, 9, 27 and they are 40.

If you add another stone, you can weigh up to 121 and so on.

If you want to weigh with four stones up to 35.

False Positions

147) Question: a man spent a third of his money and was left with 18.

How much was his money?

כמה ממונו

False Position:

Rule of Three:

148) Question: a man bought something but he did not know how much it is.

He knew that he earns 3⅓ soldi for every florin he spends.

The fund and the profit were 50 florins.

How much did he spend?

אמנם הוא יודע כי הוא מרויח ג' סולדי ושליש בעד כל פיורינו שפזר
ועולה הקרן והריוח עשרים פרחי‫'
כמה פזר

1 florin = 100 soldi

False Position:

florin

the profit of 30 florin: soldi = 1 florin

Rule of Three:

149) Question:

Rule of Three:

150) Question:

Rule of Three:

151) Question:

152) Question:

Rule of Three:

153) Question:

154) Question:

155) Question:

Euclidean Propositions

156) Euclid, Elements, Book II, proposition 1: If you have two numbers and you divide one of the two numbers, then you multiply each part by the second number, the total sum is the same as the product of the first number by the second.

• Example: the two numbers are 10 and 12 and we divide 12 to 2, 3, 5, 2, so their sum is 12.

As written in the first proposition of the second section of Euclid.

157) Euclid, Elements, Book II, propositions 2: If you have a number and you divide iy into parts as you wish, if you multiply each part by the divided number, then sum all the [products] they are equal to the divided number multiplied by itself.

• Example: we wish to divide 12 into three parts 5, 3, 4.

158) Euclid, Elements, Book II, proposition 3: If you divide any number into two parts, then multiply any of the two parts by the divided number and keep the product, so will be the result if you multiply the same part by itself, then add to it the product of the one part by the other.

• Example: we have 12 and you divide it to 8 and 4.

159) Euclid, Elements, Book II, proposition 4: If you divide any number into two and multiply each part by itself, you multiply also one [part] by the other and multiply [the product] by 2, then you sum all, the result is equal to the divided number multiplied by itself.

• Example: we divide 12 to 4 and 8.

160) Euclid, Elements, Book II, proposition 5: If you divide any number into two equal parts and into two unequal parts, the product of the equal parts one by the other is as the product of the unequal parts one by the other, when you add to it the [square] of the difference between the [equal] part and the [unequal part].

• Example: we divide 12 into two equal parts 6 and 6, and into two unequal parts 4 and 8.

161) Euclid, Elements, Book II, proposition 6: If you divide any number into two equal parts, then add an additional [number] to the divided number, multiply [the sum] by the additional [number] and add [the product] to the square of one of the parts, I say that it is equal to the square of half [the number] with the additional [number].

• Example: we have 12, you divide it to 6 and 6, then add to 6 whichever number you wish. Suppose that we wish to add 4 to 6.

162) Euclid, Elements, Book II, proposition 7: If you have a number and you divide it into two unequal parts, you multiply the greater part by itself and multiply also the divided number by itself, then sum both products, the result is equal to the product of the greater number by the divided number multiplied by two, then you add to it the square of smaller part.

• Example: we wish to divide 12 to 4 and 8.

163) Euclid, Elements, Book II, proposition 8: If you have a number and you divide it into unequal two parts, if you add one of the parts to the whole number, i.e. the divided number, [its square] is equal to the same part that you added multiplied by the whole number, then multiplied by four, when you add to it the square of the second part.

• Example: we wish to divide 12 to 4 and 8.

164) Euclid, Elements, Book II, proposition 9: If you divide any line into two equal parts and into two unequal parts, then you multiply each of the unequal parts [by itself] and sum them, they are twice [the sum of] the product of the equal parts plus the [square of] excess of the greater [part] over [half the number].

• Example: we wish to divide 12 into two equal parts 6 and 6 and into two unequal parts 8 and 4.

As explained in proposition 36 of the tenth [section] of Euclid.

165) Euclid, Elements, Book II, proposition 10: If you divide any number into two equal parts, then add another number to the divided number, square it and add to it the square of the [number] you added, it is twice [the sum of] the square of the additional [number] plus half [the number] with the square of half [the number].

• Example: we have 12, you divide it to 6 and 6, then add to 12 another number. Suppose that we add 8 to it.

166)

167)

168)

169) Euclid, Elements, Book II, proposition 4: If you have a number and you divide it into two unequal parts, I say that if you sum the [parts] that are multiplied by themselves, then you multiply the smaller [part] by the greater, multiply the product by 2 and sum it with the reserved, the result is equal to the [original] number multiplied by itself.

Divide a Number Problems

170) If you have a number and you divide it into two unequal numbers, multiply each one by itself and sum [the products], then multiply the smaller by the greater, multiply [the product] by 2 and subtract the product from the [sum of the] two products. The remainder is equal to the square of the greater number after you subtract the smaller from it.

Let the number be ten. Divide it into 8 and 2.

The sum of the products of these two numbers, i.e. 2 by 2, which is 4, and 8 by 8, which is 64, is 68. Keep it.

Then, multiply 2 by 8; it is 16.

Multiply it by 2; it is 32.

Subtract it from 68; the remainder is 36.

Likewise, [the square of] the excess of the greater [number] over the smaller, which is 6 by 6, is 36.

171) Divide a number into two unequal parts, divide the greater by the smaller, then multiply the quotient by the greater; the result is the same as the product of the greater by itself divided by the smaller.

The number ten. Divide it into 8 and 2.

Divide 8 by 2; the result is 4.

Multiply 4 by 8; it is 32. Keep it.

Then, multiply the greater by itself and divide the product of 8 by 8, which is 64, by the smaller number, which is 2; the result is 32 and it is equal to the reserved 32.

172)

173)

174)

Find a Number Problems

175) Question: find me two numbers, such that when you sum the squares of each by itself they are 58 and the product of the one by the other is 21.

Do as follows: multiply 21 by 2; it is 42.

Add it to 58; it is 100.

Now, take its root; it is 10.

Divide ten into two parts, such that the product of one by the other is 21.

One is 3, the other is 7; together they are 10.

Multiply [3] by itself; it is 9.

Multiply 7 by itself; it is 49.

Their sum is 58.

176) Question:

177) Question: find me two numbers, such that when they are multiplied by each other they are 16 and the excess of the greater multiplied by itself over the other multiplied by itself is 60. How much is the greater and which is the smaller?

17[8]) Question: find me [two] number[s], such that the one is 8 and 8 ninths more than the other and the product of one by the other is but one.

Do as follows: take a half of 8 and 8-ninths; it is 4 and 4-ninths.

Multiply it by itself; it is 19 and 61 parts of 81.

Add one to it, i.e. what you want the product of one by the other to be; the result is twenty and 61 parts of 81.

Take its root; it is 4 and 5-ninths.

Add it to 4 and 4 ninths; it is nine and this is the greater number.

The smaller is 4 and 5-ninths.

Subtract 4 and 4-ninths from it; the remainder is one-ninth.

You see that the greater exceeds the smaller by 8 and 8-ninths.

The product of one by the other is exactly one.

Divide a Number Problems

179) If you divide one number into two unequal parts:

As 10: divide it into 2 and 8. The sum of the squares of each by itself is 68. The difference between 2 and 8 is 6 and its square is 36. Subtract it from 68; the remainder is 32 and this is double the product of the greater number multiplied by the smaller [number]; because their product by each other is 16. Double it; it is 32.

הנה ההפרש שיש מב' לח' הוא ו' ומרובעו ל"ו
תוציאם מס"ח הנשאר ל"ב והוא כפל מכפל המספר הגדול כפול על הקטן
כי כפל זה על זה הוא י"ו כפלם ויהיו ל"ב

180) If you divide one number into two unequal parts:

Find a Number Problems

181) Find me two numbers, such that the sum of the two is 10 and the sum of their squares exceeds by 28 over their product by each other.

Do as follows: divide 10 into half; it is 5.

Multiply it by itself; it is 25.

Subtract it from 28; 3 remains.

Divide it by 3; the result is one and its root is one.

If you subtract 1 from 5, 4 remains and this is one of the two numbers.

If you add one to 5, it is 6 and this is the second number.

You see that the sum of the products of these numbers by themselves is 52.

Subtract from it the product of the greater by the smaller, which is 24; 28 remains as we said.

182) Question: divide ten into two parts, such that the square of the greater exceeds the square of the smaller by 60.

Do as follows: square ten; it is 100.

Subtract 60 from it; the remainder is 40.

Divide 40 by double 10, which is twenty; the result is 2 and this is the smaller part.

The complement to ten is 8 and this is the greater part.

183) Question: find me two numbers, such that when multiplied by themselves, then you sum these products and you a to them also the product of the smaller by the greater, the result is 84.

Square ten; it is 100.

Subtract 84 from it; the remainder is 16 and this is the product of the smaller by the greater.

Then, say: divide ten into two parts, such that their products by each other is 16.

You find that one part is 2 and the other is 8.

Divide a Number Problems

184)

185) Question: Divide ten into two unequal parts, [such that when] you divide the greater [part] by the smaller [part] and the smaller by the greater, then multiply [the sum] by twenty, the result is 85.

Do as follows: divide 10 into 2 parts, 2 and 8.

Divide 2 by 8; the result is a quarter.

Divide 8 by 2; the result is 4.

Together they are 4 and a quarter.

Then, divide 85 by twenty; the result is also 4 and a quarter.

Multiply 4 and a quarter by twenty; the result is 85.

186) Question:

187) Question:

Find a Number Problems

188) Question:

189) Question:

190) Question:

191) Question:

192) Question:

193) Question:

194) Question:

195) Question:

196) Question:

197) Question:

198) Question:

199) Question:

200) Question: produce from 29 two parts such that if you subtract the root of the smaller part from the root of the larger 3 will remain

201) Question: produce from 10 two parts such that if you subtract the root of the smaller part from the root of the larger 2 will remain

202) Question: produce from 29 two parts such that if you subtract the root of the smaller part from the root of the larger 3 will remain.

203) Question: produce from twenty two parts such that if you add to the smaller part twice its root, the square of the result is the same as the product of the two numbers multiplied by each other.

204) Question: produce from 13 two parts such that the sum of their roots is five.

205) Question:

206) Question:

207) Question:

208) Question:

209) Question:

210)

211) If you divide ten into two parts

• for example 6 and 4

גם חלק י' על ו' ויצא א' וב' שלישיות
ומחוברים הם ד' ושתות ושמרם

חלק ד' על ו' ויצא ב' שלישיות
ומחוברים הם ב' ושתות אחד

212) If you divide 12 by 2 and by 6

ושניהם מחוברים הם ק"ח
חלקם על כפל ג' על ו' שהם י"ח ויצא ו' והם שוים אל הו' השמורים

213) If you multiply a square by a square the result is a square

If you multiply a cube by a cube the result is a cube

214) Question:

215) Question: We have 3 proportional numbers, such that [the root of] the product of the first by the [third] is the mean.

How much is the first and the third?

כמה הוא הא' והג‫'

• Example: we have 2, 8, 32.

The product of 2 by 32 is 64; its root is 8 and this is the mean.

216) Question: A man has an amount of money, such that at the end of 3 years it is nine.

How much was it in the beginning?

217) Sometimes it happens that if you multiply a root of a root of a number [by another], even though these numbers have no real root, that the result of the multiplication has a real root of a root.

• As saying: multiply the root of the root of 96 by the root of the root of 216.

The root of 96 has no discrete root.

Whereas 216 has a discrete root of a root, which is 4.

Therefore, multiply 96 by 216; the result is twenty thousand and 736. Its root is 144; the root of 144 is 12 and this is the required.

218) If you multiply the root of a number by the root of a number, multiply one of the numbers by the other number, then take the root of the product and the result is the required.

• Example: we wish to multiply the root of four by the root of nine.

Multiply 4 by 9; it is 36. Take its root; it is 6 and this is the required.

Also if no one has a root or one has a root

• As if you multiply the root of 12 by the root of 10, because neither of them has a root.

Multiply 12 by 10; it is 120 and the root of 120 is the required.

• Also if you multiply the root of 9 that has a root by the root of 10 that has no root.

Multiply 10 by 9; the result is 90 and its root is the required.

Also if you wish to multiply the root of the root of a number by the root of the root of another number, multiply these two numbers by each other, then take the root of the root of the product and this is the required.

As saying: take the root of the root of 3 and multiply it by the root of the root of 27.

Multiply 27 by 3; it is 81. Take the root of [the root of] 81 and this is the required.

219) Question: find me 2 non-square numbers, such that the product of the root of the root of one of them multiplied by the root of the root of the other has no root.

As if you say: of 10. Multiply it [by itself]; it is 100.

Then, take any number you want, such as 5.

Take 100 and divide it by 5; the result is twenty.

You see that 5 times [twenty] is 100, as the square of ten.

Therefore, if you multiply the root of 5 by the root of twenty, the result is the root of 100, which is ten.

We already said that if you multiply the root of the root of 10 by the root, which is coza, of the root of twenty, it is the root of the root of 100, that is ten.

220) Question: divide 16 into 3 proportional numbers that relate in a ratio of 1 to 5 called quintuple ratio.

Do as follows: take 1 for the first, 5 for the second, 25 for the third. Sum them; it is [3]1. Divide 16 by it; the result is 16 parts of 31 and this is the first.

Do the same for the second and the third.

You can do the same for all the ratios, whether it is a quintuple ratio, or a quadruple ratio, or any other number.

221)

Quadratic Equations

222) Question: find me a number such that if you multiply it by 4 it will be the same as it multiplied by itself.

Take one root [] and multiply it by 4; it is 4 roots [] .

Then, square one root; it is one square [].

This is equal to 4 roots.

Divide the number of roots by the number of squares; you receive 4 and this is the required.

223) Question: find me a number such that when multiplied by itself and the result is multiplied by 5 it is 45.

We suppose it is one root [].

Multiply it by itself; the result is one square [].

Then, multiply it by 5; it is 5 squares equal 45.

Divide 45 by the number of the squares; the result is 9. Extract the root of 9; it is 3 and this is the required.

224) Question: find me a number such that its third multiplied by 5 will be twenty.

We suppose it is one root [].

Take its third; it is a third of one root [].

Multiply it by 5; it is one root and 2-thirds of a root equal twenty.

Divide twenty by 1 and 2-thirds; the result is 12 and it is the required.

225) Question: find me a number such that when you add it to its square the result is 22.

We suppose it is one root [].

Square it; it is one square [].

Add one root to it; it is one square plus one root equal 12.

[Halve] the root; it is half a root.

Multiply it by itself; the result is a quarter.

Add the required number, which is 12; it is 12 and a quarter.

Hence, the root of 12 and a quarter minus a half equal the square; i.e. 3 and this is the required.

226) Question: find me a number such that when you add to it 12 the result is equal to its square.

Do as follows:

We suppose it is one root [].

Add 12 to it; it is one root plus 12 equal one square.

[Halve] one root; it is half [a root].

Multiply it by itself; the result is a quarter.

Add 12 to it; the result is 12 and a quarter.

The root of 12 and a quarter plus a half is the required, i.e. 4.

227) Question: find me a number such that when multiplied by 5 it is the same as its square if you add to it 4.

Do as follows:

We suppose it is one root [].

Square it; it is one square [].

Add 4 to it; it is one square plus 4 equal 5 roots.

[Halve the number of] the roots, which is 5; it is 2 and a half.

Multiply it by itself; it is 6 and a quarter.

Subtract 4 from it; 2 and a quarter remain.

The root of 2 and a quarter is 1 and a [half].

We add 2 and a half to it; it is the required number, i.e. four.

228) Question: find me a number such that if you add to it a quarter of its square it is three.

We suppose it is one root [].

Square it; it is one square [].

Its quarter is a quarter of a square [].

Add one root to it; the result is one root plus one quarter of a square and this is equal to three.

Normalization: You see that you have less than one square, because it is only a quarter of a square. So, consider it as one square, i.e. divide all by one quarter.

You receive one [square] plus 4 roots and this is equal to 12.

This way you find that the root is equal to a root of 16 minus two.

As say: 3 squares and a half [] or one square and a quarter []: it should be divide by 3 and a half or by 1 and a quarter.

229) Question: find me a number such that if you multiply it by itself, then multiply the result by itself it is equal to the square of the number multiplied by 4

We suppose [it is] one root [].

Multiply it by itself; it is one square [].

Square the result again; it is a square of a square [].

This is equal to 4 times the square, i.e. 4 squares.

Divide 4 by 1; the result is 4. Extract its root; it is 2 and this is the required.

230) Question: find me a number such that if you multiply it by itself, then multiply the result once again by itself it will be the same as it is multiplied by 8.

We suppose the required is one root [].

Multiply it by itself; it is one square [].

Multiply it also by itself; the result is a square of a square [].

This is equal to one root multiplied by [8].

Divide it by 1; the result is 8. So, the cube root of 8 is equal to the root and this is the required.

231) Question: find me a number such that the square of its square is 81.

Do as follows:

We suppose it is one root [].

Square it; it is one square [].

Square it again; it is a square of a square [].

This is equal to 81.

So, extract the root of the root of 81; it is 3 and this is the required.

232) Question: find me a number such that if you add its square to the square of its square the result is 20.

We suppose [it is] one root [].

Take its square; it is one square [].

Take the square of its square; it is a square of a square [].

Plus one square they are equal to twenty.

Take half the square; the result is half a square.

Square it by itself; the result is a quarter of a square.

Add twenty to it; the result is twenty and a quarter.

Subtract half a square from it; the result is twenty and a quarter minus a half; the root of the remainder is the value of the thing.

The root is 4 and a half. Subtract a half from it; the remainder is 4 and this is the required.

I.e. if you extract the root of 4, which is 2, it is the required.

233) Question: find me a number such that the square of its square is the same as its square plus 12

Do as follows:

We suppose the required is one root [].

Square it; it is one [square ].

Square it [again]; it is a square of a square [].

This is equal to one square plus 12.

[Halve] the square; the result is half a square.

Square it; it is a quarter of a square.

Add it to 12; it is 12 and a quarter.

Add a half to the [root] of 12 and a quarter; the result is a root of 12 and a quarter plus a half.

[The root] of 12 and a quarter plus a half is 4, since the root of 12 and a quarter is 3 and a half.

Extract its [root]; it is 2 and this is the required.

234) Question: find me a number such that if you add 4 to the square of its square the result is the same as its square multiplied by 5

Do as follows:

We suppose the number is one root [].

Square it; it is one square [].

Square it [again]; it is a square of a square [].

Add four to it; it is a square of a square plus 4 and this is equal to [5] squares.

[Halve the number of] the squares, which is five; the result is 2 and a half.

Multiply it by itself; it is 6 and a quarter.

Subtract four from it; 2 and a quarter remain.

Add 2 and a half to this, which is half [the number of] the squares; the result is 2 and a half plus a root of 2 and a quarter; all this is equal to the square, i.e. a root of 2 and a half plus a root of a quarter.

I.e. 1 and a quarter, which is the root of 2 and a quarter, add to it 2 and a half; the result is 4.

Extract its root and it is the required.

Partnership Problems

235) Question: two form a partnership.

The first contributed a certain amount.

The second contributed twice as much as him and five more.

They earned 120.

The first deserves 30 minyanim of the profit.

How much should the second receive and how much did each contribute?

והנה הראשון שם סכום מה
והשני שם ב' פעמים כמוהו יותר חמשה ולשני
והרויחו ק"כ
ולראשון מגיע לו מריוח ל' מנינים
כמה יגיע לשני וכמה שם כל אחד ואחד

the first contributed:

the second contributed:

they contributed together:

Rule of Four:

the first contributed:

the second contributed:

236) Question:

237) Question:

238) Question:

239) Question: two form a partnership.

One contributes 3000 and he should receive ⁵/₉ from the profit.

The other contributes 600 and he should receive ⁴/₉ from the profit.

It happened that the first added to the partnership so that he should receive ⁷/₉ from the profit.

האחד שם ג' אלפים ויש לו לקחת מריוח ה' תשיעיות
והאחר שם ו' מאות ויש לו לקבל מריוח ד' תשיעיות
נזדמן כי הראשון מוסיף ממון בשותפות באופן כי יש לו לקחת מהריוח ז' תשיעיות
ראה כמה ישים השני אם לא היה מתעסק בשותפות באופן כי יקבל ד' תשיעיות