Difference between revisions of "השאלות החרשות/האלמות"

Prologue

• The lines that are cutting the pentagon and decagon, each of which is circumscribed by a known circle, or both are circumscribed by a known circle.

• Extracting the diameter of the circle circumscribing the known pentagon and decagon, or inscribed by known pentagon and decagon.

• Extracting the measure of one of the parts of the circumference of the known circle [= arc].

• Extracting the measure of the sides of the regular pentagon and decagon, when their area is known; and the sides of the triangles whose area is known, when they are in a regular pentagon or a regular decagon.

וצלעות המשולשים ידועי השעור כאשר היו במחומש או במעושר שוה הצלעו' והזיות

Indeterminate Problems

1) When you are told: a square that has a root, if you add five to it, it has a root. How much is the square?

This is an indeterminate problem

It has many solving methods - we will state two of them and how they are executed, so you will understand from them how to apply whichever of them you wish.

The first method is that you assume that your square is a square that has a root [= perfect square], whose root is a thing [].

Add five to it; it is a square plus five and it should have a root.

You already know that its root is greater than a thing, since the root of the square is a thing.

Suppose its root is a thing plus a number.

When this number is multiplied by itself, its square is less than the additional five.

We suppose it is a thing plus one dirham.

Multiply it by itself; it is a square and one dirham plus two things equal a square and five dirham.

Subtract the square and one dirham from the square and five dirham; two things remain equal to four dirham.

The thing is two and it is the root of the square.

The square is four dirham.

When you add five to it, it becomes nine dirham and its root is three.

If you assume its root is a thing plus two dirham [].

Multiply it by itself; it is a square and four dirham plus four things equal a square and five dirham.

Subtract the square and four dirham from the square and five dirham; four things remain equal to one dirham.

The thing is equal to a quarter of a dirham and it is the root of the square.

The square is half an eighth of a dirham.

When you add five dirham to it, it becomes five plus half an eighth and its root is two dirham and a quarter.

If you wish, suppose the root of the square is a thing plus half a dirham [].

Or a thing plus one dirham and two thirds [].

Or any other number you wish, provided that when the number you add to the thing is multiplied by itself, it is less than 5

Whatever you ask of this kind is solved by the same procedure.

2) When you are told: a square that has a root, if you subtract ten dirham from it, what remains has a root.

This is an indeterminate problem

You already know that the root of the square is a thing [].

The root of the square minus ten dirham is less than a thing.

Suppose it is a thing minus one dirham [].

Or minus two dirham [].

Or minus a square []

Add either a number and fractions, or fractions.

We suppose it is a thing minus one dirham.

Multiply it by itself; it is a square and one dirham minus two things.

So, it is a square and 11 dirham minus two things equal a square.

• Restoration:

Restore the two things and add them to the square, then subtract a square from a square; 11 dirham remain equal to two things.

The thing is five and a half and it is the root of the square.

The square is thirty and a quarter.

When you subtract ten dirham from it, tewnty and a quarter remains.

Its root is four and a half.

If you suppose the root of your square minus ten dirham is a thing minus two dirham [].

• Confrontation:

Multiply it by itself and confront it with the square minus ten dirham.

The root of the square is three and a half.

The square is 12 and a quarter.

If you subtract ten from it, two and a quarter remains.

Its root is one and a half.

If you suppose the number that is subtracted from the thing is such that when multiplied by itself, its product is greater than the dirham that are subtracted from one square [].

Know that the root of the square minus this number is then the same as the number minus the thing.

Since the root of the square minus ten is the same as the number minus the thing.

Also the product of the number minus the thing by itself is the same as the product of the thing minus the number by itself.

3) If you are told: a square that has a root, if you add three times its root to it, it has a root. How much is the square?

This is an indeterminate problem.

Suppose the root of the square plus three things is a thing plus one dirham [].

Or, a thing plus half the dirham [].

Or, a thing plus one dirham [] [?]

We suppose it is a thing plus one dirham [].

Multiply it by itself; it is a square and one dirham plus two things equal a square and three things.

Subtract a square and two things from a square and three things; one thing remains equal to one dirham and this is the root of the square.

The square is one dirham.

If you add three times its root to it, it is four dirham.

Its root is two.

4) If you are told: a square that has a root, when we subtract six times its root from it, it has a root.

This is an indeterminate problem.

Knowing this is that we assume the square is a square [].

We subtract its root from it six times; a square minus its six roots remain.

Its root is less than a thing.

Suppose it is a thing minus four dirham [].

Or, minus five [dirham] [].

Or, minus three and a half [dirham] [].

Or, minus any number you want provided that it is greater than half the six roots that are subtracted from the square.

We suppose it is a thing minus four dirham [].

We multiply them by themselves; it is a square and 16 dirham minus eight things equal a square minus six things.

• Confrontation:

Confront them; the thing is eight.

The square is sixty-four.

Subtract from it six times its root, which is 48; 16 remains.

Its root is four.

5) If you are told: a square that has a root, if you add to it ten times its root plus ten dirham, it has a root.

This is an indeterminate problem.

Knowing this is that you assume your square is a square [].

Add ten times its root to it plus twenty dirham.

The root of it is a thing plus a number [].

Such that when it is multiplied by itself, the sum of the products by the things and the dirham is more than the twenty dirham, but no less and no more than the ten [things] and the twenty [dirham].

ולא יעבור שיהיה פחות משניהם רצוני מהעשרה ומהעשרים ולא יותר שמשניהם

Because then, when you confront it with the product of the square and ten things plus twenty dirham, a thing and one dirham remain equal to the thing.

Suppose it is a thing plus four and a half dirham [].

Multiply it by itself; it is a square and nine things plus twenty dirham and a quarter equal a square and ten things plus twenty dirham.

• Confrontation:

Confront them; the thing is a quarter of a dirham and it is the root of the square.

The square is half an eighth.

If you add to it ten times its root and twenty dirham, it is twenty-two, a half and half an eighth.

Its root is four dirham, a half and a quarter.

6) If you are told: a square that has a root, if you subtract from it eight times its root and thirty dirham, the remainder has a root.

This is an indeterminate problem.

Knowing this is that you assume your square is a square [].

Subtract from it eight times its root and thirty dirham; the remainder is a square minus eight things and minus thirty dirham.

It root is less than a thing [].

We suppose it is a thing minus five dirham [].

Or, minus six [].

Or, minus any number you want provided that it is greater than half the eight things that are subtracted from the square.

If you wish, suppose its root is a number minus a thing provided that the number is greater than half the eight things that are subtracted from the square.

We suppose it is a thing minus five dirham [].

Multiply it by itself; it is a square and 25 dirham minus ten things equal a square minus eight things and thirty dirham.

• Confrontation:

Confront them; the thing is equal to 2[7] and a half; and this is the root of the square.

The square is 756 dirham and a quarter.

Subtract from it eight times its root and thirty dirham; 506 and a quarter remain.

Its root is 2[2] and a half.

7) If you are told: a square that has a root, if you add its root to it, it has a root; and if you add its two roots to it, it has a root.

This is an indeterminate problem.

Solve it according to the following procedure:

The number that has a root is one, and the added numbers are 24 and 48.

You already know that when you add 48 to one, it is 49 and its root is seven.

If you add 24 to one, it is 25 and its root is five.

If you wish to know how much is the square, divide one by 24; the result is a third of an eighth and this is the root of the square.

The square is one part of 576.

If you add its root to it, it is a third of an eighth and one part of 576; it has a root and its root is five parts of 24.

If you add to the square its two roots, it is half a sixth and one part of 576; it has a root and its root is seven parts of 24.

The square is a [third] of an eighth [?].

If you wish, suppose the sought-after number that has a root is 49 and the added parts are 120 and 240.

You know that when you add 120 to 49, it is 169 and its root is 13.

If you add 240 to it, it is 289 and its root is 17.

When you wish to know the value of the square, divide 49 by 120; the quotient is a fifth, an eighth and half a sixth, and this is the root of the square.

The square is two thousand and 401 parts of 14 thousand and 400.

If you add its root to it, which is five thousand and 880 [parts] of 14 thousand and 400, it is eight thousand and 281 [parts] of 14 thousand and 400, which has a root and its root is 91 parts of 120.

If you add to the square its two roots, which is 11 thousand and 760 [parts] of 14 thousand and 400, it is [14 thousand and 161 parts of] 14 thousand and 400, which has a root and its root is 1[1]9 parts of 120.

8) If you are told: a square, if you add its root to it, it has a root; and if you add its three roots to it, it has a root.

[?]

The required number that has a root is one and the numbers added to it are eight and 24.

Divide one by eight; the result is an eighth and this is the root of the square.

The square is an eighth of an eighth.

If you add to it its root, which is an eighth, it is an eighth and an eighth of an eighth, which has a root and its root is three-eighths.

If you add to it its three roots, which is three-eighths, it is three-eighths and an eighth of an eighth, which has a root and its root is five-eighths.

If you wish, suppose the number that has a root is 169 and the numbers added to it are 120 and 360.

Because, when you add 120 to 169, it is 289, which has a root and its root is seventeen.

If you add 360 to it, it is 529, which has a root and its root is twenty-three.

If you wish to know the value of the square, divide 169 by 120; the result is one and 49 parts of 120 and this is the root of the square.

The square is one integer and 14 thousand and 161 parts of 14 thousand and 400.

If you add its root to it, the result is three integers plus 676 thousand and 920 parts of a thousand of a thousand and 728 thousand, which has a root and its root is one and 101 parts of 120.

If you add its three roots to it, which is four dirham and twenty-seven parts of 120, it is six dirham plus three thousand [and one] parts of 14 thousand and 400, which has a root and its root is [two] and [5]9 [parts] of 120.

Suppose your square is a square [].

Sum it with two of its roots and one dirham; the result has a root and its root is a thing and one dirham.

• [Defining: ]

• Confrontation:

Confront them; the thing is equal to a quarter of a dirham; and this is the root of the square.

The square is half an eighth.

It is as if we assume the number added to it is two things and one dirham; so it is one and a half dirham, because the thing is a quarter of a dirham.

The dirham is sixteen.

Therefore, the one dirham and a half is twenty-four parts and this is the number that when you add it to one, it has a root.

If you add it twice, which is 48, it also has a root.

Suppose the root of the square plus four things and two dirham is a thing, plus one dirham and two-thirds of a dirham.

Multiply this root by itself; it is a square, two dirham and seven-ninths of a dirham, plus three things and a third of a thing equal a square plus four things and two dirham.

• Confrontation:

Confront them; the thing is one dirham and a sixth; and this is the root of the square.

The square is 49 parts of 36 of one dirham; and this is the required part that has a root.

• If you wish to know the number that when you add it to 49, it has a root; and if you add it to it twice, it has a root.

You know that the first added number is assumed to be two roots and one dirham and we have already explained that the root is one dirham and a sixth, and the dirham is 36 parts. So, the two roots and one dirham are three and a third.

Multiply three and a third by 36; it is 120, and this is the number that when you add it to 49, it has a root.

If you add 120 twice to 49, it is 240, which has a root.

Proceed as I explained.

9) If you are told: a square that has a root, if you subtract its two roots from it, the remainder has a root; and if you subtract its three roots from the square, the remainder has a root.

This is an indeterminate problem.

Solve it according to the following procedure:

Look for a number that has a root [], such that when you subtract from it another number, the remainder has a root []; and when you subtract from it that number and its half, the remainder has a root []

Because the three roots are as twice the number and its half.

The minuend number that has a root is 25; the number subtracted from it is 16, which is generated from two roots, each root is eight; the other number, which is three roots, is 24.

When you subtract 16 from 25, nine remains, which has a root and it is three.

When you subtract 24 from it, one remains, and its root is one.

When you wish to know the value of the square, divide 25 by eight, the quotient is three and an eighth and this is the root of the square.

The square is nine, six-eighths, and an eighth of an eighth.

Subtract its two roots from it, which is six and a quarter; the remainder is three and an eighth of an eighth, which has a root and it is one and seven-eighths.

Subtract from the square its three roots, which is nine and three-eighths; the remainder is three-eighths and an eighth of an eighth, which has a root and it is five-eighths.

• When you wish to find a number that has a root, such that when you subtract from it another number, the remainder has a root; and if you subtract from it again the other number and its half, the remainder has a root.

Suppose your square is a square [].

Subtract its two roots minus one dirham from it; the remainder is a square and one dirham minus two roots and its root is a thing minus one dirham.

Subtract from this square its two roots minus one dirham and their half; the remainder is a square and one dirham and a half minus three roots.

[]

• Confrontation:

Confront them as I explained to you; the thing is two dirham and a half; and this is the root of the square.

The square is six and a quarter and this is 25 parts of four [parts] of one dirham.

We suppose the number subtracted from it is two roots minus one dirham, which is four dirham.

Each dirham is four parts, so they are sixteen

The required number that has a root is 25; the number subtracted from it is 16; the other subtracted number is 24.

Proceed in the rest of the problem as I have shown you.

As I explained to you, you can solve this problem and similar ones with other methods as you wish.

10) If you are told: a square that has a root, if you subtract it from its root, the remainder has a root.

This is an indeterminate problem.

We suppose the square is a square [].

We subtract it from its root, which is a thing; a thing minus a square remains [] and it should have a root.

Suppose its root is as many things as you want - if you wish, it can be a thing, or more, or less.

Suppose it is a thing [].

Multiply it by itself; it is a square equals a thing minus a square.

• Confrontation:

Confront them as I instructed; the thing is equal to half a dirham; and this is the root of the square.

The square is a quarter of a dirham.

Suppose its root is two things [].

The thing is equal to a fifth dirham.

The square is a fifth of a fifth of a dirham.

Subtract it from its root, which is a fifth of a dirham; the remainder is four-fifths of a fifth of a dirham, which has a root and it is two-fifths of a fifth of a dirham.

11) If you are told: a square that has a root, if you add its root to it, it has a root; and if you subtract the square from its root, the remainder has a root.

This is an indeterminate problem.

Solve it according to the following procedure:

Look for a number that has a root [], such that when you add another number to it, it has a root []; and when you subtract the former number that has a root from the other number added, the remainder has a root []

The number that has a root is four and the added number is five.

When you add five to four, it is nine, which has a root and its root is three.

When you subtract four from five, one remains, which has a root and it is one.

When you want to know the value of the square, divide four by five; the quotient is four-fifths and it is the root of the square.

The square is three-fifths and a fifth of a fifth.

Then, we suppose its root is a thing minus one dirham.

I subtract the dirham from the thing, because the product has one dirham and when you confront them, what remains is squares equal roots.

Then, multiply a thing minus one dirham by itself; it is a square and one dirham minus two things equal two things and one dirham minus a square.

• Confrontation:

Confront them; the square equals two things.

The thing is two.

The square is four; and this is the number that has a root.

The other number is four and one dirham, which is five.

If we suppose the root of the square is half a thing plus one dirham.

• Confrontation:

Multiply it by itself, then confront [the product] with the product of [the root of] two things and one dirham minus a square [by itself].

The result is the number that has a root, which is sixteen [parts of 25].

[?]

12) If you are told: divide five dirham into two parts, so that each part has a root.

You already know that one of the parts is one and the other is four.

When you want to divide it into two other parts, so that each has a root:

Take the root of the first part, add a thing to it, and multiply it by itself; the result is the first part.

Subtract it from five and the other part remains.

We take the first part, which is one.

Its root is one.

We add a thing to it; it is one and a thing.

We multiply it by itself; it is a square, one dirham and two things; and this is the first part of the two required parts.

Subtract it from five; the remainder is four minus a square and minus two things; this is the other part and it should have a root.

We already know that its root should be less than two dirham.

We suppose it is two things minus [one and a third of a dirham].

We multiply them by themselves; it is four dirham, a square and seven-ninths of a square minus five things and a third of a thing equal four dirham minus a square and minus two things.

• Confrontation:

Confront them as you know; the thing is one dirham and one-fifth.

We assumed the root of the first part of the two is a thing and one dirham, so the root of the first part is two dirham and a fifth.

Multiply it by itself; it is four dirham, four-fifths, and a fifth of a fifth; this is the first part you are looking for.

The other is what remains from five, which is four-fifths of a fifth.

If you want to divide it into two other parts:

Suppose the smaller part, which is a root of four dirham minus a square and minus two things, is two dirham minus [one thing and a half].

• Confrontation:

Multiply them by themselves, then confront [the product] with the product of [the root of] four dirham minus a square minus two things [by itself].

Do as I told you.

The root of one part is two dirham and three parts of thirteen.

The root of the other part is two parts of 13.

Multiplying each of the two by itself:

One part is four dirham, 12 parts of 13, and nine parts of 169.

13) If you are told: divide ten into two parts.

Add the one part to twenty and the sum has a root.

Subtract the other part from fifty and the remainder has a root.

This is an indeterminate problem.

Its procedure is that we assume the first part is a square minus twenty [].

So that when you add it to twenty, it has a root; it is a square.

Its root is a thing.

The other part remaining from ten is thirty minus a square.

Subtract it from fifty; a square and twenty dirham remain and this should have a root.

When you want to find it, suppose the square is greater than twenty and less than thirty [], for this number has many [solutions].

One solution is that you suppose the root of the square is five and a sixth.

The square is 26, two-thirds, and a quarter of a ninth.

Subtract twenty dirham from it; the remainder is six, two-thirds, and a quarter of a ninth, and this is one of the two parts.

When you add it to twenty, it has a root and its root is five and a sixth.

The other part is what remains from ten, which is three dirham, a quarter and half a ninth.

When you subtract it from fifty, the remainder is 46, two-thirds and a quarter of a ninth; this has a root and its root is six dirham, a half and a third.

When we want to know how we assume the root of the square is five and a sixth:

Suppose the root of a square and twenty dirham is one dirham and two-thirds plus one thing.

Multiply them by themselves; it is two dirham and seven-ninths, plus a square and three things and a third of a thing equal a square and twenty dirham.

• Confrontation:

Confront them; the thing is five and a sixth.

14) If you are told: divide ten into two parts, add the first part to twenty and it has a root; add the other part to fifty, and it has a root.

This is an indeterminate problem.

Its procedure is that we assume the first part is a square minus twenty [].

So that when you add it to twenty, it has a root.

The other part that remains from ten is thirty minus a square.

When you add it to fifty, it is eighty dirham minus a square, which should have a root.

As if it is said: divide eighty into two parts, each of which has a root.

Divide eighty into two parts, each of which has a root.

Suppose one of the parts is greater than twenty and less than thirty; and the other part is greater than fifty and less than sixty.

One of the solutions is that you assume the first part is 26 dirham and 30 parts of 289.

The other part remaining from eighty is 53 and 259 parts of 289.

When you want to know the measure of each part of ten:

Subtract twenty from the first part of eighty; the remainder is six dirham and [two hundred] and thirty parts of 289, and this is the first part of ten.

The other part is what remains from ten, which is three dirham and 259 parts of 289.

When we add the greater of the two parts of ten to twenty, it is twenty-six and [two hundred and] thirty parts of 289 and it has a root, which is five dirham and three parts of seventeen.

When we add the smaller part of ten to fifty, it is fifty-three dirham and [59] parts of 289 and it has a root, which is seven dirham and five parts of seventeen.

15) If you are told: a square, if you subtract it from three dirham, the remainder has a root; and if you add it to two dirham, the sum has a root.

This is an indeterminate problem.

Its procedure is that you sum two with three; they are five.

Divide them into two parts, each of which has a root.

Suppose the first part is greater than two and less than three.

Subtract two from the first part; what remains is a square.

If you want, suppose the square is twenty-three parts of 169.

If you want, suppose it is 146 [parts] of 169.

If you want, suppose it is 431 parts of 625.

16) If you are told: a square, if you subtract it from ten dirham, the remainder has a root; and if you subtract it from twenty, the remainder has a root.

This is an indeterminate problem.

Its procedure is that you assume your square is ten dirham minus a square.

Or, if you want, twenty dirham minus a square.

Suppose it is so, in order that when you subtract it from one of them, a square remains and it has a root, which is a thing.

We suppose it is ten dirham minus a square.

When you subtract it from ten, a square remains.

If you subtract it from twenty, a square and ten dirham remain, which should have a root.

You already know that its root is greater than a thing.

We suppose its root is a thing plus two dirham.

Or, a thing plus three

Or, whichever numbers you wish, as I explained you before.

So that when we multiply it by itself and confront it with a square and ten dirham, the square is less than ten dirham.

Suppose it is a thing plus two dirham.

Multiply it by itself; it is a square and four dirham plus four things equal a square and ten dirham.

• Confrontation:

Confront them; the root of the square is one dirham and a half.

The square is two dirham and a quarter.

We assumed the square is ten dirham minus a square.

So, the required square is seven dirham, a half and a quarter.

If we subtract it from ten, two and a quarter remains and its root is one and a half.

If we assume the root of the square and ten dirham is a thing and three dirham:

The square is nine dirham, a half, a third, and five-sixths of a sixth.

17) If you are told: a square, if you add it to twenty, it has a root; and if you add it to thirty, it has a root.

This is an indeterminate problem.

Its procedure is that you assume the square is a square minus twenty dirham.

Or, if you want, a square minus thirty.

We suppose it is a square minus twenty dirham.

When it is added to twenty, the result is a square.

Its root is a thing.

We add it to [thirty]; it is a square and ten dirham and this should have a root.

The square is greater than twenty.

Suppose its root is a thing plus one dirham:

• Confrontation:

Multiply them by themselves, then confront [the product] with the product of [the root of] a square and ten dirham [by itself].

The square is twenty dirham and a quarter.

Subtract twenty dirham from it; one quarter remains and it is the required square.

If you add it to twenty, it has a root.

has a root

If you want, say: the square is 75 dirham and half an eighth.

If you add it to twenty, it has a root, and its root is nine dirham, a half and a quarter.

If you add it to thirty, it has a root, which is ten dirham and a quarter.

18) If you are told: a square, if you add it to ten, it has a root; and if you subtract it from ten, it has a root.

Divide ten into two parts, each of which has a root:

It is one and nine.

Take the roots of one and nine; they are four.

Multiply it by itself; it is sixteen.

Subtract the ten from it; six remains and this is the square, which when you add it to ten, it has a root; and if you subtract it from ten, it has a root.

The reason in this question is that when you subtract the square, which is six, from ten, four remains and four is the product of the excess of [the root of] one of the two parts over the root of the other part, so there is no mistake.

19) If you are told: a square, if you subtract it from its eight roots plus 109 dirham, it has a root.

Multiply half [the number of] the roots by itself; it is sixteen.

Add it to 109; it is 125.

Divide 125 into two parts, each of which has a root:

If you want, suppose the first part is 25 and the other is a hundred.

If you want, suppose the first is four and the other is 121.

Take 25 and add it to the square; it is a square and 25.

We say: a square and 25 dirham equal eight things and 109 dirham.

Subtract 25 from 109; the square remains equal to eight things and 84 dirham.

Half [the number of] the things is four.

Multiply it by itself; it is sixteen.

Add it to 84; it is one hundred and it is one part.

Extract the root of one hundred; it is ten.

Add half [the number of] the root to it, which is four; it is fourteen and this is the root of the square.

The square is 196.

Subtract it from its eight roots and 109 dirham, which is 221; 25 remains and its root is five.

If we take the first part, which is one hundred dirham, and add it to the square:

• Confrontation:

We confront the sum with its eight roots and 109 dirham.

The square is 81.

If we take the other part, which is four, and add it to the square:

• Confrontation:

We confront it with its eight roots and 109.

The square is 225.

If you want it is 36, or if you want four.

20) If you are told: a square, if you add its eight roots to it, it has a root; and if you subtract it from its two roots, the remainder has a root.

This is an indeterminate problem.

Its knowledge is that we assume the roots added to the square are sixteen things and 64 dirham.

So that the sum has a root and its root is a thing and eight dirham.

When we assume the eight roots are 16 things and 64 dirham, [], then you know that each root is two things and eight dirham [] and the two roots are four things and 16 dirham []

It is as if it is said: a square, if you subtract it from its four roots plus sixteen, the remainder has a root.

Proceed as I instructed:

The square is thirty-six.

Again, say: we assume the root of the square, which is two things and eight dirham, is twenty.

Divide 36 by twenty; the result is one dirham and four-fifths and this is the root of the required square.

The square is three dirham, a fifth, and a fifth of a fifth.

If you add its eight roots to it, which is fourteen dirham and two-fifths, it is seventeen, three-fifths, and a fifth of a fifth; and its root is four and a fifth.

If you subtract the square, which is three dirham, a fifth, and a fifth of a fifth, from its two roots, which is three dirham and three-fifths, the remainder is a fifth and four-fifths of a fifth; and its root is three-fifths.

Do the same with the other part; the square will be one dirham.

21) If you are told: a square, if you subtract it from its two roots plus 49 dirham, the remainder has a root.

If you want, do as I instructed.

If you want, multiply the two roots by themselves and the product of the square [].

The reason why we did so is because 49 has a root.

22) If you are told: a square, if you add its root to it, the sum has a root; and if you subtract its root from it, the remainder has a root.

This is an indeterminate problem.

Solve it according to the following procedure:

Look for a number that has a root [], such that when you add to it another number, it has a root; and when you subtract that number from it, [the remainder] has a root.

The number that has a root is 25, the number added to it is 24, and the subtracted number is 24.

You already know that when you add 24 to 25, it is 49 and its root is seven.

And when you subtract 24 from 25, one remains and its root is one.

If you want to know the value of the square, divide 25 by 24; it is one and a third of an eighth and this is the root of the square.

The square is one, two-thirds [of an eighth], and one part of 576.

If you add its root to it, which is one and [a third of an eighth], it is two dirham, an eighth, and one part of 576 and it has a root, which is one, a third and an eighth.

If you subtract its root from it, which is one and a third [of an eighth], 25 part of 576 remain and it has a root, which is an eighth and half a sixth.

If you want, suppose the required number that has a root is 169, the added number is 120, and the subtracted number is 120.

והמספר הנוסף ק"כ והמספר הנגרע ממנו ק"כ

You know that when you add 120 to 169, the sum has a root, which is seventeen.

And when you subtract 120 from it, the remainder has a root, which is seven.

When you want to know the value of the square, divide 169 by 120; the quotient is one, a fifth, an eighth and half a sixth and this is the root of the square.

The square is one and 14161 parts of 14400.

If you add its root to the square, which is one, a fifth, an eighth, and half a sixth, it is three dirham and 5641 parts of 14400 and it has a root, which is one and [101] parts of 120.

If you subtract its root from it, which is one, a fifth, an eighth, and half a sixth, 8281 parts of 14400 remain and it has a root, which is 91 parts of 120.

We suppose it is a thing minus two dirham.

Multiply it by itself; it is a square and four dirham minus four things equal a square minus two things and one dirham.

• Confrontation:

Confront them; the thing is equal to two dirham and a half and this is the root of the square.

The square is six and a quarter.

We suppose the number added to it is two roots and one dirham, which is six, since the root is two and a half.

The required number is six and a quarter, the added number is six, and the subtracted number is six.

Convert them into quarters; the required number is 25, the added number is 24, and the subtracted number is 24.

We suppose the root of the square minus two things and one dirham is a thing minus dirham and a half.

Multiply it by itself; it is a square and two dirham and a quarter minus three things equal a square minus a thing and one dirham.

• Confrontation:

Confront them; the thing is equal to three dirham and a quarter.

[Convert them into quarters], the thing is thirteen-[quarters] and this is the root of the required square.

[The square] is 169, every sixteen of which are one unit.

The number added to it is two roots and one dirham, which is [seven] and a half, because the root is three dirham and a quarter. Every dirham is sixteen parts, so they are 120 parts.

23) If you are told: a square, if you add its two roots to it, it has a root; and if you subtract its three roots from it, it has a root.

You should look for a number that has a root [], such that when you add to it a certain number, it has a root; and when you subtract that number and its half from it, the remainder has a root.

Since the three roots are two roots plus their half.

The number that has a root is 112, the number added to it is 48, and the subtracted number is 72.

When you want to know the [unknown] number, divide 121 by a half of 48 if you want, or by a third of 72 if you want, which is 24; the quotient is five and a third of an eighth and it is the root of the square.

The square is 25, a quarter, a sixth, and one part of 576.

If you add to it its two roots, which are ten dirham and half a sixth, it is 35 dirham, [a half] and one part of 576 and it has a root, which is five dirham, a half, a third, [and] an eighth.

If you subtract from the square its three roots, which are 15 and one eighth, the remainder is ten plus a sixth, an eighth, and one part of 576 and it has a root, which is three plus an eighth and half a sixth.

If you wish, assume the required number that has a root is 49 and the subtracted number is 48.

Divide 49 by sixteen; the quotient is three and half an eighth and this is the root of the square.

The square is nine plus three-eighth and one part of 256.

If you wish, assume the required number that has a root is 625, the added number is 336, and the subtracted number is 504.

Divide 625 by a half of 336 or by a third of 504, which is 168; the quotient is three, two-thirds, a quarter of a seventh, and an eighth of a seventh, and this is the root of the square.

The square is thirteen dirham, 141 parts of 1[68] and 25 parts of 28224.

If you add to it its two roots, which are seven dirham and [7]4 parts of 1[68], it is 21 dirham, 47 parts of 168, and 25 parts of 28224 and it has a root, which is four dirham and 103 parts of 16[8].

If you subtract from it its three roots, the remainder is two dirham, 114 parts of 168, and 25 parts of 28224 and it has a root, which is one dirham and 10[7] parts of 168.

I have already shown you that every square [plus its two roots and one dirham] has a root, which is one thing and one dirham.

Suppose the number added to the square is two things and one dirham, and the number subtracted from it is its three roots and one and a half dirham.

Subtract from the square its three roots and one and a half dirham; the remainder is a square minus three roots and one dirham.

Suppose its root is [a thing] minus two dirham.

Multiply it by itself; it is a square and four dirham minus four things equal a square minus three things plus one dirham and a half.

• Confrontation:

Confront them; the thing is five and a half.

The square is thirty and a quarter.

We suppose the number added to it is two things and one dirham, which is twelve.

We suppose the number subtracted from it is three things plus one dirham and a half, which is eighteen.

Convert them into quarters; The required number that has a root is 121, the additional number is 48, and the subtracted number is 72.

24) If you are told: a square that has a root, if you subtract it from its ten roots minus eight dirham, the remainder has a root.

We multiply half [the number of] the roots in this question, which is five, by itself; it is twenty-five.

Subtract the eight dirham from it; seventeen remains.

Divide it into two parts, each of which has a root; they are sixteen and one.

If you want, say: a square and sixteen dirham equal ten roots minus eight dirham.

If you want, say: a square and one dirham equal ten roots minus eight.

It is as if you say: a square and sixteen dirham equal ten roots minus eight dirham.

The square is 36 if you wish, or [six]teen if you wish.

25) If you are told: a square and one dirham equal ten roots minus eight dirham.

The square is one if you wish, or eighty-one if you wish.

26) If you are told: a square that has a root, if you subtract it from 260 minus six roots, the remainder has a root.

The product of half [the number of] the roots by itself in this question is nine, because their half is three.

Add it to 260; it is 269.

Divide it into two parts, each of which has a root; they are 169 and one hundred.

If you want, say: a square and 169 equal 260 dirham minus six things.

If you want, say: a square and one hundred dirham equal 260 dirham minus six things.

The square is 49 dirham.

And if you said: a square and one hundred dirham equal 260 dirham minus six things, the square is one hundred dirham.

27) If you are told: a square that has a root, if you add its two roots to it, it has a root; and if you add to the sum its three roots, it has a root.

ואם תוסיף על המקובץ שלשת שרשיו יחזיק שרש

Its procedure is that you assume your square is a square [].

Add its two roots to it; it is a square and two roots, which has a root and its root is a thing and a half.

The reason I assumed its root is a thing and a half is that the three roots are the same as the two roots plus their half.

Multiply a thing and a half by it self; it is two squares and a quarter equal a square and two roots.

• Confrontation:

Confront them; the root of the square is one dirham and three-fifths.

The square is two dirham, two-fifths and [four-fifths] of a fifth.

If you add to it its two roots, which is three and a fifth; it is five dirham, three-fifths and four-fifths of a fifth.

It is known that its root is the same as the root of the primary square plus its half, which is two dirham and two-fifths.

The two roots are the same as the square plus its quarter.

Three roots of five dirham, three-fifths and [four-fifths] of a fifth are necessarily the same as the square plus its quarter, which is seven and a fifth.

When you add them to five dirham, three-fifths and four-fifths of a fifth, it is twelve, four-fifths and four-fifths of a fifth.

It is known that its root is the same as one and a half the root of five, three-fifths and four-fifths of a fifth, which is three dirham and three-fifths.

Four and a half roots of 12 dirham, four-fifths and four-fifths of a fifth are necessarily the same as the square plus its quarter, which is 16 and a fifth.

When you add them to 12 dirham, four-fifths and four-fifths of a fifth, it is 29 and four-fifths of a fifth.

It is known that its root is the same as one and a half the root of 12, four-fifths and four-fifths of a fifth, which is five dirham and two-fifths.

Because, when we add to 29 and four-fifths of a fifth its six roots and three-quarters of its root, the sum has a root and it is the same as one and a half of the root of 29 and four-fifths of a fifth, which is eight [and a tenth].

And when you add to the sum its ten roots and an eighth of its root, the sum has a root and it is the same as eight [and a tenth] plus its half, which is [12...]

28) If you are told: a square that has a root, if you add its three roots to it, it has a root; and if you add to the sum its six roots, it has a root.

ואם תקבץ עם העולה ששת שרשיו יחזיק שרש

Its procedure is that you assume your square is a square [].

Add its three roots to it; it is a square and three roots and its root is two things.

The reason we assumed its root is two things is that six [things] are the same as three roots.

Multiply two things by two things; it is four squares equal a square and three things.

• Confrontation:

Confront them; the thing is equal to one dirham and this is the root of the square.

The square is one dirham.

• We say that when you add to the sum [its] twelve roots, it has a root.

• If you add to the sum its root, it has a root.

• If you add to this sum its twenty-four roots, the sum has a root.

• If you add 48 roots, it has a root.

• If you add 96 roots, it has a root.

29) If you are told: a square that has a root, if you add two roots to it, it has a root; and if you add to the sum its root, it has a root.

ואם תוסיף על המקובץ שרשו יחזיק שרש

To know this, assume your square is a square [].

Add its two roots to it; it is a square and two roots [].

Add to the sum its root, which is a root of a square and two roots; it is a square and two roots plus a root of a square and two roots [], which should have a root.

We suppose its root is a thing and half a dirham [].

Multiply it by itself; it is a square and a thing plus a quarter of a dirham equal a square and two things plus a root of a square and two roots.

Subtract a square and a root from a square and two roots; a root and a root of a square and two roots remain equal to a quarter of a dirham.

Subtract a root from a quarter of a dirham; a quarter of a dirham minus a root remain equal to a root of a square and two roots.

Multiply a quarter of a dirham minus a root by itself; it is one part of 16 plus a square minus half a thing equal a square and two roots.

• Confrontation:

Confront them; the root is a quarter of a tenth.

The square is one part of 1600.

We assumed that the root is a thing and half a dirham [], so that the square will be consumed and the things remain equal to numbers.

30) If you are told: a square that has a root, if you add its four roots to it, it has a root; and if you add to the sum its two roots, it has a root.

אם תוסיף עליו ארבעת שרשיו יחזיק שרש
ואם תוסיף על המתקבץ שני שרשיו יחזיק שרש

Knowing this is that you assume your square is a square [].

Add to it its four roots; it is a square and four things [].

Add to the square and four things its two roots, which is a root of four squares and 16 things; it is a square and four things plus a root of four squares and 16 things [].

Subtract a square and two things from a square and four things; one dirham remains equal to two things and a root of four squares and 16 things.

Subtract the two things from the dirham; one dirham minus two things remain equal to a root of four squares and 16 things.

Multiply one dirham minus two things by themselves; they are one dirham plus four squares minus four roots equal four squares and 16 things.

• Confrontation:

Confront them; the thing is half a tenth of a dirham.

The square is one part of four hundred.

31) If you are told: a square, if you subtract its four roots from it, it has a root; and if you subtract from the remainder its two roots, it has a root.

ואם תגרע ואם תגרע מהנשאר שני שרשיו יחזיק שרש

Knowing this is that you assume your square is a square [].

Subtract its four roots from it; a square minus four things remain and it has a root, which is half a thing [].

The reason I have assumed it is half a thing is that two roots are half of four.

Multiply half a square by itself; it is a quarter of a square equals a square minus four [things].

• Confrontation:

Confront them; the root of the square is five and a third.

The square is twenty-eight and four-ninths.

32) If you are told: a square that has a root, if you add its root to it, it has a root; and if you add to one dirham to it, it has a root.

ואם תוסיף עליו דרהם אחד יחזיק שרש

Knowing this is that you assume your square is a square [].

Add its root to it; it is a square plus a thing [], and it has a root.

Add one dirham to the square; it is a square plus one dirham [], and it has a root.

Suppose a root of the square plus a thing [] and a root of the square and one dirham minus half a dirham [].

Multiply them by themselves; they are a square plus one dirham and a quarter minus a root of a square and one dirham equal a square and a thing.

Subtract a square from a square; one dirham and a quarter minus a root of a square and one dirham remain equal to a root.

• Restoration:

Restore the dirham and a quarter with the root of a square and one dirham by adding it to the root [of the square]; it is a root and a root of a square and one dirham equal one dirham and a quarter.

Subtract the root from one dirham and a quarter; one dirham and a quarter minus a thing remain equal to a root of a square and one dirham.

Multiply a dirham and a quarter minus a thing by themselves; they are a dirham, a half and a half of an eighth [plus a square] minus two roots and a half equal a square and one dirham.

• Confrontation:

Confront them; the thing equals nine parts of forty and this is the root of the square.

The square is 81 parts of 1600.

We have assumed its root is a square and one dirham minus half a dirham [] so that after the confrontation the roots will be equal to [dirham].

33) If you are told: a square that has a root, if you subtract five dirham from it, the remainder has a root; and if you add to the remainder its root, the sum has a root.

ואם תוסיף על הנשאר שרשו יחזיק המקובץ שרש

Knowing this is that you assume your square is a square [].

Subtract five dirham from it; a square minus five dirham [] remain.

Add to the square minus five dirham its root, which is a root of a square minus five dirham; it is a square minus five dirham plus a root of a square minus five dirham [].

Suppose its root is a thing [minus half dirham]. []

Multiply then by themselves; they are a square and a quarter of a dirham minus a root equal a square minus five dirham [plus] a root of a square minus five dirham.

• Restoration:

Restore the square and the root with five; add the five to the square and the quarter of a dirham [minus a thing]; it is five dirham and a quarter [minus a thing] equal a root of a square minus five dirham.

Multiply five dirham and a quarter [minus a thing] by themselves; they are twenty-seven dirham, a half and a half of an eighth, plus a square minus ten roots and half a root equal a square minus five dirham.

• Confrontation:

Confront them; the thing is equal to three dirham and seventeen parts of 1[68] and this is the root of the square.

The square is nine dirham, 103 parts of 168 , and 121 parts of 28224.

• Check:

When you subtract five from it, four dirham, 103 parts of 168 and 103 parts of 28224 remain.

Add to it its root, which is two dirham and [25 parts of 168]; it has a root, which is 2 dirham and 10[1] parts of 168.

34) If you are told: a square that has a root, if you add its four roots to it, it has a root; and if you subtract its two roots plus one dirham from it, the remainder has a root.

ואם תגרע ממנו שני שרשיו ודרהם אחד יחזיק הנשאר שרש

Knowing this is that you assume your square is a square [].

Add its four roots to it; it is a square and four roots [] that has a root.

Subtract from the square its two roots and one dirham; a square minus two roots and one dirham remain [] and this has a root.

Suppose its root is a root of a square and four things minus three dirham [], as I explained to you.

Multiply it by itself; it is a square plus nine dirham and four roots minus six roots of a square and four things equal a square minus two roots and one dirham.

• Confrontation:

Confront them; it is a square and six roots plus ten dirham equal a square and six roots of a square and four things.

Six roots of a square and four things is a root of 36 squares and 144 things.

Subtract a square from a square; six roots and ten dirham remain equal to a root of 36 squares and 144 things.

• Confrontation:

Confront them; the thing is equal to four dirham plus a sixth and this is the root of the square.

The square is seventeen dirham and a quarter of a ninth.

35) If you are told: a square that has a root, if you subtract its two roots from it, it has a root; and if you add to the remainder its root, it has a root.

ואם תוסיף על הנשאר שרשו יחזיק שרש

Knowing this is that you assume your square is a square [].

Subtract its two roots from it; a square minus two roots [] remain.

Add its root to it; it is a square minus two roots and a root of a square minus two roots [] that has a root.

Suppose its root is a thing minus one and a half dirham [].

Multiply them by themselves; they are a square plus two dirham and a quarter minus three roots equal a square minus two roots and a root of a square minus two roots.

• Confrontation:

Confront them; after the confrontation they are a thing and a root of a square minus two roots equal two dirham and a quarter.

Subtract a thing from two dirham and a quarter; two dirham and a quarter minus a thing remain equal to a root of a square minus two roots.

Multiply two dirham and a quarter minus a thing by themselves; they are five dirham and a half of an eighth plus a square minus four things and a half equal a square minus two roots.

• Confrontation:

Confront them; the thing is two dirham and a quarter of a tenth and this is the root of the square.

The square is four, a tenth, and [one] part of one thousand and 600.

• Check:

Subtract from it its two roots that are four and half a tenth; half a tenth and one part of one thousand and 600 remain.

Add to it its root, which is a fifth and a quarter of a tenth; it is a quarter, a quarter of a tenth and one part of a thousand and 600, which has a root and it is a half and a quarter of a tenth.

ומחזיק שרש והוא חצי ורביע העשירית

36) If you are told: a square that has a root, if you add to it its three roots and one dirham, it has a root; and if you subtract its three roots minus two dirham from it, the remainder has a root.

ואם תגרע ממנו שלשת שרשיו פחות שני דרהמי יחזיק מה שישאר שרש

Knowing this is that you assume your square is a square [].

Add to it its three roots and one dirham. It is a square and three roots plus one dirham [] that has a root.

Subtract from the square its three roots minus two dirham. A square and two dirham minus three roots remain [] and it has a root.

Suppose its root is a root of a square plus three roots and one dirham minus three dirham [].

Multiply them by themselves; the result is a square and three roots plus ten dirham minus a root of 36 squares and 36 dirham plus 108 things equal a square and two dirham minus three roots.

• Confrontation:

Confront them; after the confrontation it is six things and eight dirham equal a root of 36 squares, 36 dirham and [1]08 things.

Multiply six things and eight dirham by themselves; they are 36 squares and 96 things plus 64 dirham equal 36 squares and 36 dirham plus 108 things.

• Confrontation:

Confront them; the thing is two dirham and a third and this is the root of the square.

The square is five dirahm and four-ninths.

• Check:

If you add to it its three roots and one dirham, it is thirteen dirham and four-ninths and its root is three dirham and two-thirds.

If you subtract from it its three roots minus two dirham, four-ninths remain and its root is two-thirds.

37) If you are told: a square that has a root, if you subtract one dirham minus the root of the square from it, it has a root; [and if you subtract its root minus one dirham from it, it has a root].

Knowing this is that you assume your square is a square [].

Add to it one dirham minus a root of the square. It is a square and one dirham minus a root of the square [] that has a root.

If you subtract one dirham minus a root of the square from the square, a square and a root minus one dirham [] remains and it has a root.

Suppose the root of the square and one dirham minus a thing is one dirham minus a root of the square and a root minus one dirham. []

Multiply them by themselves; they are a square and a root minus two roots of a square and a root minus one dirham equal a square and one dirham minus a root.

• Restoration:

Restore them with the root by adding it to the square and the root; restore the one with two roots of a square and a root minus one dirham by adding it to the square and one dirham; subtract a square from a square; one dirham and [two roots] of a square and a root minus one dirham remain equal two things.

Subtract one dirham from two things; two things minus one dirham remain equal two roots of a square and a root minus one dirham.

It half, which is a thing minus half a dirham, is equal to a root of a square and a root minus one dirham.

Multiply a thing minus half a dirham by themselves; they are a square and a quarter of a dirham minus a thing equal a square and a root minus one dirham.

• Confrontation:

Confront them; the thing is five-eighths of a dirham and this is the root of the square.

The square is three-eighths and one-eighth of an eighth.

38) If you are told: a square that has a root, if you add two dirham minus the root of the square to it, it has a root; and if you subtract three minus the root of the square from it, it has a root.

ואם תגרע ממנו שלשה פחות שרש האלגו יחזיק שרש

To know this, assume your square is a square [].

Add two dirham minus a root to it; it is a square and two dirham minus a root [] and it has a root.

Subtract three minus a root from it; the remainder is a square and a root minus three dirham [] and it has a root.

Suppose the root of the square and two dirham minus a root is one dirham plus a root of a square and a root minus three dirham.

Multiply them by themselves; they are a square and a root minus two dirham plus two roots of a square [and a root] minus three dirham equal a square and two dirham minus a root.

• Restoration:

Restore the square and two dirham with the root by adding it to the square and the root minus two dirham and subtract a square from a square; two roots and [two roots] of a square and a root minus three dirham remain equal four dirham.

Subtract the two roots from the four dirham; four dirham minus two roots remain equal two roots of a square and a root minus three dirham.

Take its half; it is two dirham minus a root equal a root of a square and a root minus three dirham.

Multiply two dirham minus a thing by themselves; they are four dirham and a square minus four things equal a square and a root minus three dirham.

• Confrontation:

Confront them; the thing is equal to one dirham and two-fifths.

The square is one dirham, four-fifths, and four-fifths of a fifth.

39) If you are told: a square that has a root, if you add its root plus one dirham to it, it has a root; and if you add to the square its two roots plus two dirham, it has a root.

ואם תוסיף על האלגו שני שרשיו ושני דרהמי יחזיק שרש

Knowing this is that you assume your square is a square [].

Add its root plus one dirham to it; it is a square and a root plus one dirham [] and it has a root.

Add to the square its two roots plus two dirham []; it has a root.

Suppose its root is half a dirham plus a root of a square plus a root and one dirham that is equal to a square and two roots plus two dirham.

Subtract a square plus a root and one dirham and a quarter from a square plus two roots and two dirham; a thing plus three-quarters of a dirham remain.

וישאר דבר ושלשה רביעי דרהם

Multiply them by themselves; they are a square, plus a thing and a half, plus half a dirham and half an eighth equal a square and a root plus [one] dirham.

• Confrontation:

Confront them; the thing is seven-eighths and this is the root of the square.

The square is [six] eighths and one-eighth of an eighth.

Word Problems

If You Give Me Problems

40) If you are told: four people gathered to buy a beast.

The first said to the three [others]: if you give me half of what you have I will have the price of the beast.

The second asked the three [others] for a third.

The third asked the three [others] for a quarter.

The fourth asked the three [others] for a fifth of what they have

How much is the price of the beast and how much money does each of them have?

ואמר הראשון לשלשה אם תתנו לי החצי מאשר תחזיקו יהיה לי ערך הבהמה
ושאל השני לשלשה השלישית
ושאל השלישי לשלשה הרביעית
ושאל האיש הרביעי לשלשה החומש מאשר להם
כמה יהיה ערך הבהמה וכמה ממון כל אחד מהם

Knowing this is that you assume the first has a thing [], the second a dinar [], the third a dirham [], and the fourth a fals []

The first, who has a thing, says to the three: give me half of what you have, I will add it to what I have and I will have the price of the beast.

Half of what they have is half a dinar, half a dirham, and half a fals. When they are added to the thing that he has, it is a thing, half a dinar, half a dirham, and half a fals; this is the price of the beast.

The second, who has one dinar, says to the three: give me a third of what you have, I will add it to what I have and I will have the price of the beast.

A third of what they have is: a third of a thing, a third of a dirham, and a third of a fals. When they are added to the dinar that he has, he has in his hand one dinar, a third of a thing, a third of a dirham, and a third of a fals; this is the price of the beast.

It is equal to the thing, half a dinar, half a dirham, and half a fals.

• Confrontation:

Confront them by subtracting a third of a thing from a whole thing; two-thirds of a thing remain.

Subtract a third of a dirham from half a dirham; one-sixth of a dirham remains.

Subtract a third of a fals from half a fals; one-sixth of a fals remains.

Subtract half of [a dinar] from one dinar; half dinar remains, which is equal to two-thirds of a thing, a sixth of a dirham, and a sixth of a fals.

The dinar equals a thing and one-third of a thing, one-third of a dirham, and one-third of a fals.

A quarter of a dinar equals one-third of a thing, half a sixth of a dirham, and half a sixth of a fals.

A fifth of a dinar equals one-fifth of a thing and a fifth of a third of a thing, a fifth of a third of a dirham, and fifth of a third of a fals.

Suppose the price of the beast is a thing, half a dinar, half a dirham, and half a fals.

Remove half the dinar from the price of the beast and put half the worth of the dinar instead, which is two-thirds of a thing, one-sixth of a dirham, and one-sixth of a fals. The price of the beast is a thing and two-thirds of a thing, two-thirds of a dirham and two-thirds of a fals.

ויהיה ערך הבהמה דבר ושני שלישים מדבר ושני שלישי דרהם ושני שלישי אלפלס

We have already removed the dinar and placed a thing and a third of a thing, a third of a dirham and a third of a fals instead.

The third, who has one dirham, says: give me a quarter of what you have, I will add it to what I have and I will have the price of the beast.

A quarter of what they have is: a quarter of a thing, a quarter of a dinar, and a quarter of a fals. When you add them to the dirham that he has, then remove the amount of the dinar and put instead the value of a quarter of a dinar as explained above, what is in his hand is a third and a quarter of a thing, one dirham and half a sixth of a dirham, and a third of a fals; this is the price of the beast. It is equal to a thing and two-thirds of a thing, two-thirds of a dirham, and two-thirds of a fals.

ויהיה שוה לדבר ושני שלישי דבר ושני שלישי דרהם ושני שלישי אלפלס

• Confrontation:

Confront them by subtracting a third and a quarter of a thing from a thing and two-thirds of a thing; and a third of a fals from two-thirds of a fals; a thing and half a sixth plus one-third of a fals remain.

Subtract two-thirds of a dirham from a dirham and half a sixth; a quarter and a sixth of a dirham remain that are equal to a thing and half a sixth of a thing plus one-third of a fals.

The dirham equals two things and three-fifths of a thing, plus four-fifths of a fals.

Remove two-thirds of a dirham from the price of the beast and put two-thirds the value of the dirham instead, which is a thing, a third and two-fifths of a thing, plus one-third and one-fifth of a fals. The price of the beast is three things and two-fifths of a thing, plus one fals and a fifth of a fals.

ויהיה ערך הבהמה שלשה דברים ושלשה ^ושני חומשי דבר ופלס אחד וחומש

We have already removed the dirham and placed two things and three-fifths of a thing, plus four-fifths of a fals instead.

Remove a third of the dirham from the value of the dinar and put a third of the worth of the dirham instead, which is two-thirds and a fifth of a thing, plus a fifth and a third of a fifth of a fals. The value of the dinar is two things and a fifth, plus three-fifths of a fals.

ויהיה ערך הבהמה הדינר שני דברים וחומש ושלשה חומשי אלפלס

It turns out that the first has a thing []

The second that has a dinar, has two things and a fifth, plus three-fifths of a fals.

The third that has a dirham, has two things and three-fifths of a thing, plus four-fifths of a fals.

The price of the beast is three things and two-fifths, plus one fals and a fifth.

The fourth man, who has one fals, says to the three: give me a fifth of what you have, I will add it to what I have and I will have the price of the beast.

A fifth of what they have is: a thing and four-fifths of a fifth of a thing, plus a fifth of a fals and two-fifths of a fifth of a fals. When we add them to the fals that he has, he gets one fals, a fifth and two-fifths of a fifth of a fals, plus a thing and four-fifths of a fifth of a thing. All this is equal to the price of the beast, which is three things and two-fifths of a thing, plus one fals and a fifth of a fals.

וישוה כל זה ‫^[42]ערך הבהמה שהוא שלשה דברים ושני חומשי דבר ופלס וחומש

• Confrontation:

Subtract a fals and a fifth of a fals from a fals, a fifth, and two-fifth of a fifth of a fals; two-fifths of a fifth of a fals remain.

Subtract a thing and four-fifths of a fifth of a thing from three things and two-fifths of a thing; two things, a fifth and a fifth of a fifth of a thing remain that are equal to two-fifths of a fifth of a fals.

The fals equals twenty-eight things.

Suppose the thing is one and this is what the first has.

The second [has] nineteen, since it is three-fifths of what the fourth has plus twice of what the first has and [a fifth] of it.

The third [has] twenty-five, since it is four-fifths of what the fourth has plus twice of what the first has and three-fifths of it.

The price of the beast is thirty-seven, since it is as what the fourth has and a fifth of it plus three times of what the first has and [two-fifths] of it.

We suppose all they have is one dirham [].

We suppose the first has a thing [] of this dirham.

Then, the three are left with a dirham minus a thing [].

When we add its half to what the first has, which is a thing, the price of the beast is half a thing plus half a dirham.

Suppose the second has one dinar [].

When we add a third of a dirham to the dinar, the price of the beast is two-thirds of a dinar plus a third of a dirham. This is equal to the price of the beast in the first stage, which is half a thing plus half a dirham.

וישוה ערך הבהמה מהחלק האחד שהיה חצי דבר וחצי דרהם

Subtract a third of a dirham from half a dirham; two-thirds of a dinar remain equal to half a thing plus a sixth of a dirham.

The dinar is three-quarters of a thing plus a quarter of a dirham.

When the dinar is removed, three-quarters plus a quarter of a dirham remain instead.

Suppose the third has one dinar [].

Then, the three are left with a dirham minus a dinar [].

When we add its quarter to what the third has, which is a dinar, the price of the beast is three-quarters of a dinar plus a quarter of a dirham. This is equal to the price of the beast in the first stage, which is half a thing plus half a dirham.

וישוה ערך הבהמה מהחלק האחד שהוא חצי דבר וחצי דרהם

Subtract a quarter of a dirham from half a dirham; half a thing plus a quarter of a dirham remain equal to three-quarters of a dinar.

The dinar is equal to two-thirds of a thing plus a third of a dirham.

The three have a total of two things, a quarter and a sixth of a thing, plus a third and a quarter of a dirham.

We assumed the four have a dirham, hence the fourth is left with a quarter and a sixth of a dirham minus two things, a quarter and a sixth of a thing.

The fourth man says to the three that have two things, a quarter and a sixth of a thing, plus a third and a quarter of a dirham: give me a fifth of what you have, I will add it to what I have and I will have the price of the beast.

A fifth of what they have is twenty-nine parts of sixty parts of a thing and 7 parts of 60 of a dirham.

When you add this to what he has, which is a quarter and a sixth of a dirham minus two things, a quarter and a sixth of a thing, the fourth has 32 parts of sixty of a dirham minus a thing and fifty-six parts of sixty parts of the thing, which is the price of the beast. This is equal to the price in the first stage, which is half a thing plus half a dirham.

וישוה לערך מהחלק האחד שהוא חצי דבר וחצי דרהם

• Confrontation:

Confront them and it becomes after the confrontation a third of a tenth of a dirham equals two things, a third and a tenth of a thing.

The dirham is equal to seventy-three things.

Suppose the value of the dirham is 73 and this is the total that the four have.

The value of the thing is one and this is what the first has.

The second has nineteen, since what he has is equal to a quarter of a dirham and three-quarters of a thing.

The third has twenty-five, since what he has is equal to a third of a dirham and two-thirds of a thing.

The fourth has twenty-eight, since what he has is equal to a quarter and a sixth of a dirham minus two things, a quarter and a sixth of a thing.

If you wish, sum up what the first, the second, and the third have; it is forty-five. Subtract it from 73; what remains is what the fourth man has, which is twenty-eight.

The price of the beast is thirty-seven, since it is assumed to be half of a dirham plus half of what the first has.

41) If you are told: five people gathered to buy a beast.

The first said to the second and the third: give me half of what you have and I will have enough to buy it.

The second said to the third and the fourth: give me a third of what you have and I will have the price of the beast.

The third said to the fourth and the fifth: give me a quarter of what you have and I will have the price of the beast.

The fourth said to the fifth and the first: give me a fifth of what you have and I will have the price of the beast

The fifth said to the first and the second: give me a sixth of what you have and I will have the price of the beast

How much is the price of the beast and how much does each of the five have?

ואמר הראשון לשני ולשלישי תנו לי החצי מאשר לכם ויהיה לי ‫^[46]כדי לקנותה
ואמר השני לשלישי ולרביעי תנו לי שלישית אשר לכם ויהיה לי ערך הבהמה
ואמר השלישי לרביעי ולחמישי תנו לי רביעית אשר לכם ויהיה לי ערך האלפאגה
ואמר הרביעי לחמישי ולראשון תנו לי חמישית אשר לכם ויהיה לי ערך האלפאגה
ואמר החמישי לראשון ולשני תנו לי הששית מאשר לכם ויהיה לי ערך הבהמה
כמה יהיה ערך הבהמה וכמה יהיה לכל אחד מהחמשה

To know this we assume the first has one thing [], the second a dinar [], and the third a dirham [].

We do not need more than these three species in this problem and its similar, in order to extract the number.

The first, who has a thing, says to the second and the third, who have a dinar and a dirham: give me half of what you have, I will add it to what I have and I will have the price of the beast.

Half of what they have is half a dinar and half a dirham. So, he has one thing, half a dinar, and half a dirham; this is the price of the beast.

The second, who has one dinar, says to the third and the fourth: give me a third of what you have, and with what I have, which is one dinar, I will have the price of the beast, which is one thing, half a dinar, and half a dirham.

Subtract one dinar from the price of the beast; one thing and half a dirham remain minus half a dinar and this is a third of what the third and the fourth have.

Multiply it by three; it is three things, a dirham and a half minus a dinar and a half and this is all that the third and the fourth have.

Subtract what the third has, which is one dirham; what the fourth has remains, which is three things and half a dirham minus one dinar and a half.

The third, who has one dirham, says to the fourth and the fifth: give me a quarter of what you have, and with what I have, I will have the price of the beast, which is one thing, half a dinar, and half a dirham.

Subtract what the third has, which is one dinar, from the price of the beast; one thing and half a dinar remain minus half a dirham and this is a quarter of what the fourth and the fifth have.

Multiply it by four; it is four things and two dinar minus two dirham and this is all that the fourth and the fifth have.

Subtract from it what the fourth has, which is three things and half a dirham minus one dinar and a half; what the fifth has remains, which is one thing and three dinar and a half minus two dirham and a half.

The fourth, who has three things and half a dirham minus one dinar and a half, says to the fifth and the first, who have two things and three dinar and a half minus two dirham and a half: give me a fifth of what you have, and with what I have, I will have the price of the beast.

A fifth of what they have is two-fifths of a thing and seven-tenths of a dinar minus half a dirham.

When you add this to what he has, it is three things and two-fifths minus four-fifths of a dinar.

This is equal to the price of the beast in the previous stage, which is a thing and half a dinar plus half a dirham.

• Confrontation:

Confront them and the dirham becomes equal to four things and four-fifths of a thing minus two dinar and three-fifths of a dinar.

Subtract half a dirham from the price of the beast and replace it with half the value of the dirham, which is two things and two-fifths of a thing minus one dinar and three-tenths of a dinar; the price of the beast is three things and two-fifths of a thing minus four-fifths of a dinar.

The first has a thing [].

The second has one dinar [].

The third has four things and four-fifths of a thing minus two dinar and three-fifths of a dinar [].

The fourth has five things and two-fifths of a thing minus two dinar and four-fifths of a dinar [].

The reason we assumed this for the fourth is because the fourth has three things and a half minus one dinar and a half, so subtract half a dirham of what he has and replace it with half of what the dirham is worth.

As we have explained, the fifth should have ten dinar minus eleven things.

He says to the first and the second, who have one thing and one dinar: give me a sixth of what you have, I will add it to what I have, and I will have the price of the beast.

A sixth of what they have is a sixth of a thing plus a sixth of a dinar.

So, the fifth has ten dinar and a sixth minus ten things and five-sixths of a thing and this is the price of the beast.

This is equal to the price of the beast in the previous stage, which is three things and two-fifths of a thing minus four-fifths of a dinar.

• Confrontation:

Confront them and the thing becomes forty-seven after the confrontation, which is what the first has.

The dinar is sixty-one and this is what the second has.

The third has sixty-seven, because it is four times and four-fifths of what the first has minus two times and three-fifths of what the second has.

The fourth has eighty-three, because he has five times and two-fifths of what the first has minus two times and four-fifths of what the second has.

The fifth has ninety-three, because it is ten times of what the second has minus eleven times of what the first has.

The price of the beast is 111, because it is three times and two-fifths of what the first has minus four-fifths of what the second has.

If you wish to extract what the fourth and the fifth have after this procedure:

You already know that the first has 47, the second has 61, and the third has 67.

Subtract what the second has from the price of the beast; fifty remains. Multiply it by three; it is 150. Subtract from it what the third has, which is 67; the remainder is what the fourth has, which is 83.

If you want to know what the fifth has: subtract what the third has from the price of the beast; 44 remains. Multiply it by four; it is 176. Subtract from it what the fourth has, which is 83; 93 remains and this is what the fifth has.

If you wish, suppose the first has one thing []; the second [has] one dinar []; the third [has] one dirham []; and the fourth [has] one fals [].

The first says to the second and the third: give me a half of what you have, I will add it to what I have, and I will have the price of the beast.

So, the price of the beast is a thing, half a dinar and half a dirham.

The second says to the third and the fourth: give me a third of what you have; with what I have I will have the price of the beast.

Its price in this stage is a dinar, a third of a dirham and a third and a fals.

It is equal to the price of the beast in the previous stage, which is a thing, half a dinar and half a dirham.

• Confrontation:

Confront them and the fals becomes equal to three things and half a dirham minus one dinar and a half, which is what the fourth has.

Suppose the fifth has one fals [].

The third, who has one dirham, says to the fourth and the fifth, who have one fals, three things and half a dirham minus two dinar and a half: give me a quarter of what you have, and with what I have, I will have the price of the beast.

A quarter of what they have is three-quarters of a thing, a quarter of a fals and an eighth of a dinar minus 3-eighths of a dinar.

When it is added to the dirham he has, [the sum] is [the same as] the price of the beast. So, the price of the beast in this stage is one dirham, an eighth of a dirham, a quarter of a fals, and three-quarters of a thing minus three-eighths of a dinar.

This is equal to the price of the beast in the first stage, which is a thing, half a dinar and half a dirham.

• Confrontation:

Confront them and the fals becomes equal to a thing plus three dinar and a half minus two dirham and a half, which is what the fifth has.

Do as I showed you with the rest of the question.

42) If you are told: five people gathered.

The first said to the second: if you give me half of what you have I will have the price of the beast.

The second said to the third: if you give me a third of what you have I will have the price of the beast.

The third said to the fourth: if you give me a quarter of what you have I will have the price of the beast.

The fourth said to the fifth: if you give me a fifth of what you have I will have the price of the beast

The fifth said to the first: if you give me a sixth of what you have I will have the price of the beast

How much is the price of the beast and how much does each of them have?

ואמר השני לשלישי אם תתן לי השלישית מאשר בידך יהיה לי ערך הבהמה
ואמר השלישי לרביעי אם תתן לי הרביעית מאשר בידך יהיה לי ערך הבהמה
ואמר הרביעי לחמישי אם תתן לי החמישית מאשר בידך יהיה לי ערך הבהמה
ואמר החמישי לראשון אם תתן לי ששית אשר בידך יהיה לי ערך הבהמה
כמה יהיה ערך הבהמה וכמה יהיה אלגו מכל אחד מהחמשה

To know this, assume the first has a thing []; the second [has] one dinar ; and the third [has] one dirham [].

The first, who has one thing, says to the second, who has one dinar: if you give me a half of what you have, I will have the price of the beast.

So, the price of the beast is a thing and half a dinar.

The second, who has one dinar, says to the third, who has one dirham: if you give me a third of what you have, I will have the price of the beast.

Then, the price of the beast in this stage is one dinar and a third of a dirham.

This is equal to the price of the beast in the first stage, which is a thing and half a dinar.

• Confrontation:

Confront them and the dirham becomes equal to three things minus one dinar and a half, which is what the third has.

Suppose the fourth has one dirham [].

The third, who has three things minus a dinar and a half, says to the fourth, who has one dirham: if you give me a quarter of what you have, I will have the price of the beast.

So, the price of the beast in this stage is three things and a quarter of a dirham minus a dinar and a half.

This is equal to the price of the beast in the first stage, which is a thing and half a dinar.

• Confrontation:

Confront them and the dirham becomes equal to eight dinar minus eight things, which is what the fourth has.

Suppose the fifth has one dirham [].

The fourth, who has eight dinar minus eight things, says to the fifth, who has one dirham: if you give me a fifth of what you have, I will have the price of the beast.

Then, the price of the beast in this stage is eight dinar and a fifth of a dirham minus eight things.

This is equal to the price of the beast in the first stage, which is a thing and half a dinar.

• Confrontation:

Confront them and the dirham becomes equal to 45 things minus 37 dinar and a half, which is what the fifth has.

He says to the first, who has a thing: if you give me a sixth of what you have, I will have the price of the beast.

The sixth of what the first has is a sixth of a thing.

We add it to what he has; the price of the beast in this stage is 45 things and a sixth of a thing minus 37 dinar and a half.

This is equal to the price of the beast in the first stage, which is a thing and half a dinar.

• Confrontation:

Confront them and it becomes after the confrontation 44 things and a sixth equals 38 dinar.

Convert all into sixths:

The thing is 228 and this is what the first has.

The dinar is 265 and this is what the second has.

The third has three things minus one dinar and a half, which is 286 and a half.

Convert all into halves:

The thing, which is what the first has, is 456.

The dinar, which is what the second has, is 530.

The third has 573.

The fourth has eight dinar minus eight things, which is 592.

The fifth has 45 things minus 37 dinar and a half, which is 645.

The price of the beast is [7]21.

43) If you are told: four people gathered to buy a beast.

The first said to the second: if you give me one dirham I will have twice as much of what remains with you.

The second said to the third: if you give me two dirham I will have three times as much of what remains with you.

The third said to the fourth: if you give me three dirham I will have four times as much of what remains with you.

The fourth said to the first: if you give me four dirham I will have five times as much of what remains with you.

ואמר הראשון לשני אם תתן לי דרהם אחד יהיה לי הכפל ממה שנשאר לך
ואמר השני לשלישי אם תתן לי שני דרהמי יהיה לי שלשה כפלים ממה שנשאר לך
ואמר השלישי לרביעי אם תתן לי שלשה דרהמי יהיה לי ארבעה כפלים ממה שנשאר לך
ואמר הרביעי לראשון אם תתן לי ארבעה דרהמי יהיה לי חמשה כפלים ממה שנשאר לך

To know this, assume the first has a thing [] and the second [has] one dinar [].

When he first takes one dirham from the second, the first has a thing and one dirham; this should be the same as double what is left for the second, which is two dinar minus two dirham.

• Confrontation:

Confront them and the dinar becomes equal to half a thing plus a dinar and a half, which is what the second has.

Suppose the third has one dinar [].

When the second takes two dirham from the third, the second has [half] a thing plus three dirham and a half; this should be equal to three times what is left for the third, which is three dinar minus six dirham.

• Confrontation:

Confront them and the dinar becomes equal to a sixth of a thing plus three dirham and a sixth, which is what the third has.

We suppose the fourth has one dinar [].

When the third takes two dirham from him, he has a sixth of a thing plus six dirham and a sixth; this should be equal to four times what is left for the fourth, which is four dinar minus twelve dirham.

• Confrontation:

Confront them and the dinar becomes equal to a third of an eighth of a thing plus four dirham, a quarter, a sixth, and an eighth of a dirham, which is what the fourth has.