Difference between revisions of "השאלות החרשות/האלמות"

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:<math>\scriptstyle\begin{cases}\scriptstyle x^2+x=n^2\\\scriptstyle x^2+2x=m^2\end{cases}</math>

:<math>\scriptstyle\begin{cases}\scriptstyle x^2+x=n^2\\\scriptstyle x^2+2x=m^2\end{cases}</math>

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:<math>\scriptstyle\begin{cases}\scriptstyle x^2+x=n^2\\\scriptstyle x^2+3x=m^2\end{cases}</math>

:<math>\scriptstyle\begin{cases}\scriptstyle x^2+x=n^2\\\scriptstyle x^2+3x=m^2\end{cases}</math>

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|style="text-align:right;"|תניח המרובע שלך מרובע

|style="text-align:right;"|תניח המרובע שלך מרובע

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:<math>\scriptstyle\begin{cases}\scriptstyle49+a=n^2\\\scriptstyle49+2a=m^2\end{cases}</math>

:<math>\scriptstyle\begin{cases}\scriptstyle49+a=n^2\\\scriptstyle49+2a=m^2\end{cases}</math>

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:<math>\scriptstyle\begin{cases}\scriptstyle x^2-2x=n^2\\\scriptstyle x^2-3x=m^2\end{cases}</math>

:<math>\scriptstyle\begin{cases}\scriptstyle x^2-2x=n^2\\\scriptstyle x^2-3x=m^2\end{cases}</math>

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:<math>\scriptstyle\begin{cases}\scriptstyle a^2-b=n^2\\\scriptstyle a^2-\left(1+\frac{1}{2}\right)b=m^2\end{cases}</math>

:<math>\scriptstyle\begin{cases}\scriptstyle a^2-b=n^2\\\scriptstyle a^2-\left(1+\frac{1}{2}\right)b=m^2\end{cases}</math>

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|style="text-align:right;"|תניח המרובע שלך מרובע

|style="text-align:right;"|תניח המרובע שלך מרובע

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:<math>\scriptstyle x-x^2=n^2</math>

:<math>\scriptstyle x-x^2=n^2</math>

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|style="text-align:right;"|תניח שרשו איזה שתרצה מהדברים אם תרצה יהיה דבר או אם תרצה יהיה יותר או אם תרצה יהיה פחות

|style="text-align:right;"|תניח שרשו איזה שתרצה מהדברים אם תרצה יהיה דבר או אם תרצה יהיה יותר או אם תרצה יהיה פחות

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|style="text-align:right;"|ותנחיהו דבר

|style="text-align:right;"|ותנחיהו דבר

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:<math>\scriptstyle{\color{blue}{x^2=x-x^2}}</math>

:<math>\scriptstyle{\color{blue}{x^2=x-x^2}}</math>

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:The square is a quarter of a dirham.

:The square is a quarter of a dirham.

:<math>\scriptstyle{\color{blue}{x^2=\frac{1}{4}}}</math>

:<math>\scriptstyle{\color{blue}{x^2=\frac{1}{4}}}</math>

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|style="text-align:right;"|ואם נניח שרשו שני דברים

|style="text-align:right;"|ואם נניח שרשו שני דברים

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|style="text-align:right;"|והמרובע חמישית מחמישית דרהם

|style="text-align:right;"|והמרובע חמישית מחמישית דרהם

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:<math>\scriptstyle\begin{cases}\scriptstyle x+x^2=n^2\\\scriptstyle x-x^2=m^2\end{cases}</math>

:<math>\scriptstyle\begin{cases}\scriptstyle x+x^2=n^2\\\scriptstyle x-x^2=m^2\end{cases}</math>

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|style="text-align:right;"|ותעשה אותה כמעשה

|style="text-align:right;"|ותעשה אותה כמעשה

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|style="text-align:right;"|שתבקש מספר שיחזיק שרש ואם תקבץ עמו מספר אחר יחזיק שרש ואם תגרע המספר הראשון המחזיק שרש מהמספר האחר הנוסף יחזיק הנשאר שרש

|style="text-align:right;"|שתבקש מספר שיחזיק שרש ואם תקבץ עמו מספר אחר יחזיק שרש ואם תגרע המספר הראשון המחזיק שרש מהמספר האחר הנוסף יחזיק הנשאר שרש

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|style="text-align:right;"|וכאשר תמצא זה המספר תחלקהו על המספר הנוסף והעולה לחלק יהיה שרש המרובע

|style="text-align:right;"|וכאשר תמצא זה המספר תחלקהו על המספר הנוסף והעולה לחלק יהיה שרש המרובע

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|style="text-align:right;"|וכאשר תוסיף על ארבעה חמשה יהיה תשעה ומחזיק שרש ושרשו שלשה

|style="text-align:right;"|וכאשר תוסיף על ארבעה חמשה יהיה תשעה ומחזיק שרש ושרשו שלשה

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|style="text-align:right;"|וכאשר תגרע הארבעה מהחמשה ישאר אחד ומחזיק שרש והוא אחד

|style="text-align:right;"|וכאשר תגרע הארבעה מהחמשה ישאר אחד ומחזיק שרש והוא אחד

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|style="text-align:right;"|וכאשר תרצה לדעת שעור המרובע תחלק ארבעה על חמשה ויעלה לחלק ארבעה חומשים והוא שרש המרובע

|style="text-align:right;"|וכאשר תרצה לדעת שעור המרובע תחלק ארבעה על חמשה ויעלה לחלק ארבעה חומשים והוא שרש המרובע

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|style="text-align:right;"|והמרובע יהיה שלשה חומשים וחמישית מחמישית

|style="text-align:right;"|והמרובע יהיה שלשה חומשים וחמישית מחמישית

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|style="text-align:right;"|ובארתי לך הנה המלאכה להוציא אלו המספרים

|style="text-align:right;"|ובארתי לך הנה המלאכה להוציא אלו המספרים

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|style="text-align:right;"|ושניתי האדרהם בדבר בעבור שיעלה בהכאה דרהם אחד וכאשר תכוין עמו ישאר מרובע ישוה דבר

|style="text-align:right;"|ושניתי האדרהם בדבר בעבור שיעלה בהכאה דרהם אחד וכאשר תכוין עמו ישאר מרובע ישוה דבר

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:Confront them; the square equals two things.

:Confront them; the square equals two things.

:<math>\scriptstyle{\color{blue}{x^2=2x}}</math>

:<math>\scriptstyle{\color{blue}{x^2=2x}}</math>

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|style="text-align:right;"|ואם נניח שרש המרובע חצי דבר ואדרהם אחד

|style="text-align:right;"|ואם נניח שרש המרובע חצי דבר ואדרהם אחד

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|style="text-align:right;"|והמספר האחר חמשה ושני דברים ויהיה שרש המרובע ששה עשר חלקים מס"ה חלקים באחד

|style="text-align:right;"|והמספר האחר חמשה ושני דברים ויהיה שרש המרובע ששה עשר חלקים מס"ה חלקים באחד

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|style="text-align:right;"|וכל מה שיפול בידך מזה האופן תעשהו במעשה הזה ותקח הדרך אשר תבחר

|style="text-align:right;"|וכל מה שיפול בידך מזה האופן תעשהו במעשה הזה ותקח הדרך אשר תבחר

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|style="text-align:right;"|ועתה אבאר לך זאת השאלה בהבנת דרך אחרת

|style="text-align:right;"|ועתה אבאר לך זאת השאלה בהבנת דרך אחרת

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|style="text-align:right;"|<big>דע</big> כי כל מספר שיחלק לשני חלקים באופן שיהיה לכל חלק שרש ולשני חלקים אחרים שלא יהיה להם אטימו

|style="text-align:right;"|<big>דע</big> כי כל מספר שיחלק לשני חלקים באופן שיהיה לכל חלק שרש ולשני חלקים אחרים שלא יהיה להם אטימו

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|style="text-align:right;"|ועתה אבאר לך זה למען תבינהו

|style="text-align:right;"|ועתה אבאר לך זה למען תבינהו

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:<math>\scriptstyle a^2+b^2=5</math>

:<math>\scriptstyle a^2+b^2=5</math>

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

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

|style="text-align:right;"|תקח שרש החלק האחד ותוסיף עליו דבר ותכהו בעצמו והיוצא יהיה החלק האחד

|style="text-align:right;"|תקח שרש החלק האחד ותוסיף עליו דבר ותכהו בעצמו והיוצא יהיה החלק האחד

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:Subtract it from five and the other part remains.

:Subtract it from five and the other part remains.

|style="text-align:right;"|ותגרעהו מחמשה וישאר החלק האחר

|style="text-align:right;"|ותגרעהו מחמשה וישאר החלק האחר

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: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.

: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.

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: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.

: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.

:<math>\scriptstyle{\color{blue}{a_2=x+1=2+\frac{1}{5}}}</math>

:<math>\scriptstyle{\color{blue}{a_2=x+1=2+\frac{1}{5}}}</math>

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: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.

: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.

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|style="text-align:right;"|ויהיה החלק האחד ארבעה דרהמי וי"ב חלקים מי"ג חלקים באחד ותשעה חלקים מקס"ט

|style="text-align:right;"|ויהיה החלק האחד ארבעה דרהמי וי"ב חלקים מי"ג חלקים באחד ותשעה חלקים מקס"ט

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|style="text-align:right;"|וכן תעשה לכל אשר יבוא לך מזה המין תעשה עמו כמעשה הזה ויחזיק מהחלקים כמה שתרצה

|style="text-align:right;"|וכן תעשה לכל אשר יבוא לך מזה המין תעשה עמו כמעשה הזה ויחזיק מהחלקים כמה שתרצה

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|style="text-align:right;"|והוא שלעולם תקח שרש האחד מהשני חלקים ותוסיף עליו דבר ותכה זה בעצמו והיוצא הוא האחד מהשני חלקים אשר בקשת

|style="text-align:right;"|והוא שלעולם תקח שרש האחד מהשני חלקים ותוסיף עליו דבר ותכה זה בעצמו והיוצא הוא האחד מהשני חלקים אשר בקשת

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|style="text-align:right;"|ותכוין עמו החלק שהוא מספר המחזיק שרש נשנה מהאלגו ודבר ויעלו לך מן החלקים עד אין להם תכלית

|style="text-align:right;"|ותכוין עמו החלק שהוא מספר המחזיק שרש נשנה מהאלגו ודבר ויעלו לך מן החלקים עד אין להם תכלית

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

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

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

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

:<math>\scriptstyle\begin{cases}\scriptstyle a+b=10\\\scriptstyle20+a=n^2\\\scriptstyle50-b=m^2\end{cases}</math>

:<math>\scriptstyle\begin{cases}\scriptstyle a+b=10\\\scriptstyle20+a=n^2\\\scriptstyle50-b=m^2\end{cases}</math>

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

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

:<math>\scriptstyle{\color{blue}{50-b=50-\left(30-x^2\right)=x^2+20}}</math>

:<math>\scriptstyle{\color{blue}{50-b=50-\left(30-x^2\right)=x^2+20}}</math>

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:When you want to find it, suppose the square is greater than twenty and less than thirty [<math>\scriptstyle{\color{blue}{20<x^2<30}}</math>], for this number has many [solutions].

:When you want to find it, suppose the square is greater than twenty and less than thirty [<math>\scriptstyle{\color{blue}{20<x^2<30}}</math>], for this number has many [solutions].

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:<math>\scriptstyle\begin{cases}\scriptstyle a+b=10\\\scriptstyle20+a=n^2\\\scriptstyle50+b=m^2\end{cases}</math>

:<math>\scriptstyle\begin{cases}\scriptstyle a+b=10\\\scriptstyle20+a=n^2\\\scriptstyle50+b=m^2\end{cases}</math>

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

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

:<math>\scriptstyle{\color{blue}{a+20=x^2}}</math>

:<math>\scriptstyle{\color{blue}{a+20=x^2}}</math>

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|The procedure of extracting any solution you want is as I told you before.

|The procedure of extracting any solution you want is as I told you before.

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

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

:<math>\scriptstyle{\color{blue}{A^2=26+\frac{30}{289}}}</math>

:<math>\scriptstyle{\color{blue}{A^2=26+\frac{30}{289}}}</math>

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: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.

: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.

:<math>\scriptstyle{\color{blue}{b+50=B^2=53+\frac{{\color{red}{59}}}{289}=\left(7+\frac{5}{17}\right)^2}}</math>

:<math>\scriptstyle{\color{blue}{b+50=B^2=53+\frac{{\color{red}{59}}}{289}=\left(7+\frac{5}{17}\right)^2}}</math>

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:<math>\scriptstyle\begin{cases}\scriptstyle3-x^2=n^2\\\scriptstyle2+x^2=m^2\end{cases}</math>

:<math>\scriptstyle\begin{cases}\scriptstyle3-x^2=n^2\\\scriptstyle2+x^2=m^2\end{cases}</math>

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:<math>\scriptstyle\begin{cases}\scriptstyle10-x^2=n^2\\\scriptstyle20-x^2=m^2\end{cases}</math>

:<math>\scriptstyle\begin{cases}\scriptstyle10-x^2=n^2\\\scriptstyle20-x^2=m^2\end{cases}</math>

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:When you subtract it from ten, a square remains.

:When you subtract it from ten, a square remains.

:<math>\scriptstyle{\color{blue}{10-n^2=x^2}}</math>

:<math>\scriptstyle{\color{blue}{10-n^2=x^2}}</math>

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:<math>\scriptstyle\begin{cases}\scriptstyle20+x^2=n^2\\\scriptstyle30+x^2=m^2\end{cases}</math>

:<math>\scriptstyle\begin{cases}\scriptstyle20+x^2=n^2\\\scriptstyle30+x^2=m^2\end{cases}</math>

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

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

:<math>\scriptstyle{\color{blue}{a^2+20=x^2}}</math>

:<math>\scriptstyle{\color{blue}{a^2+20=x^2}}</math>

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:<math>\scriptstyle\begin{cases}\scriptstyle10+x^2=n^2\\\scriptstyle10-x^2=m^2\end{cases}</math>

:<math>\scriptstyle\begin{cases}\scriptstyle10+x^2=n^2\\\scriptstyle10-x^2=m^2\end{cases}</math>

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|If the two numbers in this problem are the same, and we suppose one of them can be divided into two parts, each of which has a root, then [the problem] has a countless number of solutions.

|If the two numbers in this problem are the same, and we suppose one of them can be divided into two parts, each of which has a root, then [the problem] has a countless number of solutions.

: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.

: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.

:<math>\scriptstyle{\color{blue}{x^2=\left(\sqrt{1}+\sqrt{9}\right)^2-10=4^2-10=16-10=6}}</math>

:<math>\scriptstyle{\color{blue}{x^2=\left(\sqrt{1}+\sqrt{9}\right)^2-10=4^2-10=16-10=6}}</math>

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:<math>\scriptstyle{\color{blue}{m^2=10-x^2=10-6=4}}</math>

:<math>\scriptstyle{\color{blue}{m^2=10-x^2=10-6=4}}</math>

:<math>\scriptstyle{\color{blue}{\left(\sqrt{a^2}-\sqrt{b^2}\right)^2=\left(\sqrt{9}-\sqrt{1}\right)^2=4=m^2}}</math>

:<math>\scriptstyle{\color{blue}{\left(\sqrt{a^2}-\sqrt{b^2}\right)^2=\left(\sqrt{9}-\sqrt{1}\right)^2=4=m^2}}</math>

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:<math>\scriptstyle8x+109-x^2=n^2</math>

:<math>\scriptstyle8x+109-x^2=n^2</math>

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|When you have such a question, multiply half [the number of] the roots by itself and add [the product] to the dirham.

|When you have such a question, multiply half [the number of] the roots by itself and add [the product] to the dirham.

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|If their sum can be divided into two parts, each of which has a root, the problem is solvable and has a countless number of solutions.

|If their sum can be divided into two parts, each of which has a root, the problem is solvable and has a countless number of solutions.

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|If their sum cannot be divided into two parts, each of which has a root, the problem has no solution and no solving procedure.

|If their sum cannot be divided into two parts, each of which has a root, the problem has no solution and no solving procedure.

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:Multiply it by itself; it is sixteen.

:Multiply it by itself; it is sixteen.

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

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

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

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

:<math>\scriptstyle{\color{blue}{x^2+100=8x+109}}</math>

:<math>\scriptstyle{\color{blue}{x^2+100=8x+109}}</math>

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:<math>\scriptstyle\begin{cases}\scriptstyle8x+x^2=n^2\\\scriptstyle2x-x^2=m^2\end{cases}</math>

:<math>\scriptstyle\begin{cases}\scriptstyle8x+x^2=n^2\\\scriptstyle2x-x^2=m^2\end{cases}</math>

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

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

:<math>\scriptstyle{\color{blue}{x^2=3+\frac{1}{5}+\left(\frac{1}{5}\sdot\frac{1}{5}\right)}}</math>

:<math>\scriptstyle{\color{blue}{x^2=3+\frac{1}{5}+\left(\frac{1}{5}\sdot\frac{1}{5}\right)}}</math>

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: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 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.

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:<math>\scriptstyle2x+49-x^2=n^2</math>

:<math>\scriptstyle2x+49-x^2=n^2</math>

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:If you want, multiply the two roots by themselves and the product of the square [<math>\scriptstyle{\color{blue}{x^2}}</math>].

:If you want, multiply the two roots by themselves and the product of the square [<math>\scriptstyle{\color{blue}{x^2}}</math>].

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

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

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|If the number have no root, return to the first procedure I explained to you before this question.

|If the number have no root, return to the first procedure I explained to you before this question.

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:<math>\scriptstyle\begin{cases}\scriptstyle x^2+x=n^2\\\scriptstyle x^2-x=m^2\end{cases}</math>

:<math>\scriptstyle\begin{cases}\scriptstyle x^2+x=n^2\\\scriptstyle x^2-x=m^2\end{cases}</math>

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|When you find this number, divide it by the added number and the quotient is the root of the square.

|When you find this number, divide it by the added number and the quotient is the root of the square.

:<math>\scriptstyle{\color{OliveGreen}{x=\frac{a^2}{b}}}</math>

:<math>\scriptstyle{\color{OliveGreen}{x=\frac{a^2}{b}}}</math>

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

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

:<math>\scriptstyle{\color{blue}{25+24=49=7^2}}</math>

:<math>\scriptstyle{\color{blue}{25+24=49=7^2}}</math>

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: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 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.

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|colspan=2|

|colspan=2|

|-

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|I will further teach you the extraction of these numbers, so that your heart will be wiser about them:

|I will further teach you the extraction of these numbers, so that your heart will be wiser about them:

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|I have already shown you that for every square that has a root, when you add its two roots and one dirham to it, it has a root.

|I have already shown you that for every square that has a root, when you add its two roots and one dirham to it, it has a root.

:We suppose it is a thing minus two dirham.

:We suppose it is a thing minus two dirham.

:<math>\scriptstyle{\color{blue}{\sqrt{x^2-\left(2x+1\right)}=x-2}}</math>

:<math>\scriptstyle{\color{blue}{\sqrt{x^2-\left(2x+1\right)}=x-2}}</math>

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

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

:<math>\scriptstyle{\color{blue}{a^2=25\quad b=24}}</math>

:<math>\scriptstyle{\color{blue}{a^2=25\quad b=24}}</math>

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:<math>\scriptstyle\begin{cases}\scriptstyle x^2+2x=n^2\\\scriptstyle x^2-3x=m^2\end{cases}</math>

:<math>\scriptstyle\begin{cases}\scriptstyle x^2+2x=n^2\\\scriptstyle x^2-3x=m^2\end{cases}</math>

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|When you find that number, divide it by half the additional number, or by a third of the subtracted number, and the quotient is the root of the square.

|When you find that number, divide it by half the additional number, or by a third of the subtracted number, and the quotient is the root of the square.

:<math>\scriptstyle{\color{OliveGreen}{x=\frac{a^2}{\frac{1}{2}b}=\frac{a^2}{\frac{1}{3}\left(b+\frac{1}{2}b\right)}}}</math>

:<math>\scriptstyle{\color{OliveGreen}{x=\frac{a^2}{\frac{1}{2}b}=\frac{a^2}{\frac{1}{3}\left(b+\frac{1}{2}b\right)}}}</math>

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: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.

: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.

:<math>\scriptstyle{\color{blue}{x=\frac{121}{\frac{1}{2}\sdot48}=\frac{121}{\frac{1}{3}\sdot72}=\frac{121}{24}=5+\left(\frac{1}{3}\sdot\frac{1}{8}\right)}}</math>

:<math>\scriptstyle{\color{blue}{x=\frac{121}{\frac{1}{2}\sdot48}=\frac{121}{\frac{1}{3}\sdot72}=\frac{121}{24}=5+\left(\frac{1}{3}\sdot\frac{1}{8}\right)}}</math>

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

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

:<math>\scriptstyle{\color{blue}{a^2=49\quad b={\color{red}{32}}\quad b+\frac{1}{2}b=48}}</math>

:<math>\scriptstyle{\color{blue}{a^2=49\quad b={\color{red}{32}}\quad b+\frac{1}{2}b=48}}</math>

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

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

:<math>\scriptstyle{\color{blue}{x=\frac{49}{16}=3+\left(\frac{1}{2}\sdot\frac{1}{8}\right)}}</math>

:<math>\scriptstyle{\color{blue}{x=\frac{49}{16}=3+\left(\frac{1}{2}\sdot\frac{1}{8}\right)}}</math>

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

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

:<math>\scriptstyle{\color{blue}{x^2=9+\frac{3}{8}+\frac{1}{256}}}</math>

:<math>\scriptstyle{\color{blue}{x^2=9+\frac{3}{8}+\frac{1}{256}}}</math>

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: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 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].

|-

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|colspan=2|

|colspan=2|

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: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.

: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.

|-

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|colspan=2|

|colspan=2|

|

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: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.

: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.

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|colspan=2|

|colspan=2|

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

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

:<math>\scriptstyle{\color{blue}{\sqrt{x^2-\left[3x+\left(1+\frac{1}{2}\right)\right]}={\color{red}{x}}-2}}</math>

:<math>\scriptstyle{\color{blue}{\sqrt{x^2-\left[3x+\left(1+\frac{1}{2}\right)\right]}={\color{red}{x}}-2}}</math>

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

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

:<math>\scriptstyle{\color{blue}{2x+1=12}}</math>

:<math>\scriptstyle{\color{blue}{2x+1=12}}</math>

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:<math>\scriptstyle10x-8-x^2=n^2</math>

:<math>\scriptstyle10x-8-x^2=n^2</math>

|-

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|When you have such a question, multiply half [the number of] the roots by itself and subtract the dirham from the product.

|When you have such a question, multiply half [the number of] the roots by itself and subtract the dirham from the product.

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|-

|If the remainder can be divided into two parts, each of which has a root, the problem has as many solutions as you wish.

|If the remainder can be divided into two parts, each of which has a root, the problem has as many solutions as you wish.

|-

|-

|If the remainder cannot be divided into two parts, each of which has a root, the problem has no solution and no solving procedure.

|If the remainder cannot be divided into two parts, each of which has a root, the problem has no solution and no solving procedure.

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:<math>\scriptstyle x^2+1=10x-8</math>

:<math>\scriptstyle x^2+1=10x-8</math>

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:<math>\scriptstyle260-6x-x^2=n^2</math>

:<math>\scriptstyle260-6x-x^2=n^2</math>

|-

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|When you have such a question, multiply half [the number of] the roots by itself and add it to the dirham.

|When you have such a question, multiply half [the number of] the roots by itself and add it to the dirham.

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:<math>\scriptstyle\begin{cases}\scriptstyle x^2+2x=n^2\\\scriptstyle x^2+2x+3\sqrt{x^2+2x}=m^2\end{cases}</math>

:<math>\scriptstyle\begin{cases}\scriptstyle x^2+2x=n^2\\\scriptstyle x^2+2x+3\sqrt{x^2+2x}=m^2\end{cases}</math>

|style="text-align:right;"|כז <big>ואם</big> יאמרו לך אלגו המחזיק שרש אם תוסיף עליו שני שרשיו יחזיק שרש<br>

|style="text-align:right;"|כז <big>ואם</big> יאמרו לך אלגו המחזיק שרש אם תוסיף עליו שני שרשיו יחזיק שרש<br>

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:Its procedure is that you assume your square is a square [<math>\scriptstyle{\color{blue}{x^2}}</math>].

:Its procedure is that you assume your square is a square [<math>\scriptstyle{\color{blue}{x^2}}</math>].

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: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.

: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.

:<math>\scriptstyle{\color{blue}{\sqrt{x^2+2x}=\left(1+\frac{1}{2}\right)x}}</math>

:<math>\scriptstyle{\color{blue}{\sqrt{x^2+2x}=\left(1+\frac{1}{2}\right)x}}</math>

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: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.

: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.

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|colspan=2|

|colspan=2|

|-

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|And so on for any addition you want in this way, when you assume an addition that relates to the question, as if we say that the ratio of the question is that its six roots are the same as two roots and their half, more and more are being added endlessly.

|And so on for any addition you want in this way, when you assume an addition that relates to the question, as if we say that the ratio of the question is that its six roots are the same as two roots and their half, more and more are being added endlessly.

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:<math>\scriptstyle\begin{cases}\scriptstyle x^2+3x=n^2\\\scriptstyle x^2+3x+6\sqrt{x^2+3x}=m^2\end{cases}</math>

:<math>\scriptstyle\begin{cases}\scriptstyle x^2+3x=n^2\\\scriptstyle x^2+3x+6\sqrt{x^2+3x}=m^2\end{cases}</math>

|style="text-align:right;"|כח <big>וכן</big> אם יאמרו לך אלגו המחזיק שרש אם תוסיף עליו שלשת שרשיו יחזיק שרש<br>

|style="text-align:right;"|כח <big>וכן</big> אם יאמרו לך אלגו המחזיק שרש אם תוסיף עליו שלשת שרשיו יחזיק שרש<br>

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

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

:<math>\scriptstyle{\color{blue}{\sqrt{x^2+3x}=2x}}</math>

:<math>\scriptstyle{\color{blue}{\sqrt{x^2+3x}=2x}}</math>

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

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

:<math>\scriptstyle{\color{blue}{3\sqrt{x^2+3x}=6x}}</math>

:<math>\scriptstyle{\color{blue}{3\sqrt{x^2+3x}=6x}}</math>

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*If you add 48 roots, it has a root.

*If you add 48 roots, it has a root.

:<math>\scriptstyle{\color{OliveGreen}{X+48\sqrt{X}=n^2}}</math>

:<math>\scriptstyle{\color{OliveGreen}{X+48\sqrt{X}=n^2}}</math>

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|Whichever you have of this species, when the secondary roots are greater than the primary roots, is solved by this procedure.

|Whichever you have of this species, when the secondary roots are greater than the primary roots, is solved by this procedure.

|-

|-

|However, if they are less than the primary roots they are not solved by this procedure, but they have other methods by which they are solved as follows:

|However, if they are less than the primary roots they are not solved by this procedure, but they have other methods by which they are solved as follows:

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:<math>\scriptstyle\begin{cases}\scriptstyle x^2+2x=n^2\\\scriptstyle x^2+2x+\sqrt{x^2+2x}=m^2\end{cases}</math>

:<math>\scriptstyle\begin{cases}\scriptstyle x^2+2x=n^2\\\scriptstyle x^2+2x+\sqrt{x^2+2x}=m^2\end{cases}</math>

|style="text-align:right;"|כט <big>כאשר</big> יאמרו לך אלגו המחזיק שרש אם תוסיף עליו שני שרשים יחזיק שרש<br>

|style="text-align:right;"|כט <big>כאשר</big> יאמרו לך אלגו המחזיק שרש אם תוסיף עליו שני שרשים יחזיק שרש<br>

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: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.

: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.

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|colspan=2|

|colspan=2|

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:<math>\scriptstyle\begin{cases}\scriptstyle x^2+4x=n^2\\\scriptstyle x^2+4x+2\sqrt{x^2+4x}=m^2\end{cases}</math>

:<math>\scriptstyle\begin{cases}\scriptstyle x^2+4x=n^2\\\scriptstyle x^2+4x+2\sqrt{x^2+4x}=m^2\end{cases}</math>

|style="text-align:right;"|ל <big>וכן</big> אם יאמרו לך אלגו המחזיק שרש<br>

|style="text-align:right;"|ל <big>וכן</big> אם יאמרו לך אלגו המחזיק שרש<br>

אם תוסיף עליו ארבעת שרשיו יחזיק שרש<br>

אם תוסיף עליו ארבעת שרשיו יחזיק שרש<br>

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:Add to it its four roots; it is a square and four things [<math>\scriptstyle{\color{blue}{x^2+4x}}</math>].

:Add to it its four roots; it is a square and four things [<math>\scriptstyle{\color{blue}{x^2+4x}}</math>].

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: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 [<math>\scriptstyle{\color{blue}{x^2+4x+2\sqrt{x^2+4x}=x^2+4x+\sqrt{4x^2+16x}}}</math>].

: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 [<math>\scriptstyle{\color{blue}{x^2+4x+2\sqrt{x^2+4x}=x^2+4x+\sqrt{4x^2+16x}}}</math>].

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: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 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.

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|colspan=2|

|colspan=2|

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

:<math>\scriptstyle{\color{blue}{\left(1-2x\right)^2=1+4x^2-4x=4x^2+16x}}</math>

:<math>\scriptstyle{\color{blue}{\left(1-2x\right)^2=1+4x^2-4x=4x^2+16x}}</math>

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|-

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:<math>\scriptstyle\begin{cases}\scriptstyle x^2-4x=n^2\\\scriptstyle x^2-4x-2\sqrt{x^2-4x}=m^2\end{cases}</math>

:<math>\scriptstyle\begin{cases}\scriptstyle x^2-4x=n^2\\\scriptstyle x^2-4x-2\sqrt{x^2-4x}=m^2\end{cases}</math>

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

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

:<math>\scriptstyle{\color{blue}{2x=\frac{1}{2}\sdot4x}}</math>

:<math>\scriptstyle{\color{blue}{2x=\frac{1}{2}\sdot4x}}</math>

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

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

:<math>\scriptstyle{\color{blue}{\left(\frac{1}{2}x\right)^2=\frac{1}{4}x^2=x^2-4{\color{red}{x}}}}</math>

:<math>\scriptstyle{\color{blue}{\left(\frac{1}{2}x\right)^2=\frac{1}{4}x^2=x^2-4{\color{red}{x}}}}</math>

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|-

|-

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:<math>\scriptstyle\begin{cases}\scriptstyle x^2+x=n^2\\\scriptstyle x^2+1=m^2\end{cases}</math>

:<math>\scriptstyle\begin{cases}\scriptstyle x^2+x=n^2\\\scriptstyle x^2+1=m^2\end{cases}</math>

|style="text-align:right;"|לב <big>ואם</big> יאמרו לך אלגו המחזיק שרש אם תוסיף עליו שרשו יחזיק שרשו<br>

|style="text-align:right;"|לב <big>ואם</big> יאמרו לך אלגו המחזיק שרש אם תוסיף עליו שרשו יחזיק שרשו<br>

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:Add its root to it; it is a square plus a thing [<math>\scriptstyle{\color{blue}{x^2+x}}</math>], and it has a root.

:Add its root to it; it is a square plus a thing [<math>\scriptstyle{\color{blue}{x^2+x}}</math>], and it has a root.

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|

:Add one dirham to the square; it is a square plus one dirham [<math>\scriptstyle{\color{blue}{x^2+1}}</math>], and it has a root.

:Add one dirham to the square; it is a square plus one dirham [<math>\scriptstyle{\color{blue}{x^2+1}}</math>], and it has a root.

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: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.

: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.

|-

|-

|colspan=2|

|colspan=2|

|-

|-

|

|

:<math>\scriptstyle\begin{cases}\scriptstyle x^2-5=n^2\\\scriptstyle x^2-5+\sqrt{x^2-5}=m^2\end{cases}</math>

:<math>\scriptstyle\begin{cases}\scriptstyle x^2-5=n^2\\\scriptstyle x^2-5+\sqrt{x^2-5}=m^2\end{cases}</math>

|style="text-align:right;"|לג <big>ואם</big> יאמרו לך אלגו המחזיק שרש אם תגרע ממנו חמשה דרהמי יחזיק הנשאר שרש<br>

|style="text-align:right;"|לג <big>ואם</big> יאמרו לך אלגו המחזיק שרש אם תגרע ממנו חמשה דרהמי יחזיק הנשאר שרש<br>

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:Knowing this is that you assume your square is a square [<math>\scriptstyle{\color{blue}{x^2}}</math>].

:Knowing this is that you assume your square is a square [<math>\scriptstyle{\color{blue}{x^2}}</math>].

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: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 [<math>\scriptstyle{\color{blue}{x^2-5+\sqrt{x^2-5}}}</math>].

: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 [<math>\scriptstyle{\color{blue}{x^2-5+\sqrt{x^2-5}}}</math>].

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

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

:<math>\scriptstyle{\color{blue}{x^2-5=4+\frac{103}{168}+\frac{121}{28224}}}</math>

:<math>\scriptstyle{\color{blue}{x^2-5=4+\frac{103}{168}+\frac{121}{28224}}}</math>

|

|

: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.

: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.

|-

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|colspan=2|

|colspan=2|

|-

|-

|

|

:<math>\scriptstyle\begin{cases}\scriptstyle x^2+4x=n^2\\\scriptstyle x^2-\left(2x+1\right)=m^2\end{cases}</math>

:<math>\scriptstyle\begin{cases}\scriptstyle x^2+4x=n^2\\\scriptstyle x^2-\left(2x+1\right)=m^2\end{cases}</math>

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

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

:<math>\scriptstyle{\color{blue}{6\sdot\sqrt{x^2+4x}=\sqrt{36x^2+144x}}}</math>

:<math>\scriptstyle{\color{blue}{6\sdot\sqrt{x^2+4x}=\sqrt{36x^2+144x}}}</math>

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|-

|-

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:<math>\scriptstyle\begin{cases}\scriptstyle x^2-2x=n^2\\\scriptstyle x^2-2x+\sqrt{x^2-2x}=m^2\end{cases}</math>

:<math>\scriptstyle\begin{cases}\scriptstyle x^2-2x=n^2\\\scriptstyle x^2-2x+\sqrt{x^2-2x}=m^2\end{cases}</math>

|style="text-align:right;"|לה <big>ואם</big> יאמרו לך אלגו המחזיק שרש אם תגרע ממנו שני שרשיו יחזיק שרש<br>

|style="text-align:right;"|לה <big>ואם</big> יאמרו לך אלגו המחזיק שרש אם תגרע ממנו שני שרשיו יחזיק שרש<br>

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:Add its root to it; it is a square minus two roots and a root of a square minus two roots [<math>\scriptstyle{\color{blue}{x^2-2x+\sqrt{x^2-2x}}}</math>] that has a root.

:Add its root to it; it is a square minus two roots and a root of a square minus two roots [<math>\scriptstyle{\color{blue}{x^2-2x+\sqrt{x^2-2x}}}</math>] that has a root.

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

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

:<math>\scriptstyle{\color{blue}{x=2+\left(\frac{1}{4}\sdot\frac{1}{10}\right)}}</math>

:<math>\scriptstyle{\color{blue}{x=2+\left(\frac{1}{4}\sdot\frac{1}{10}\right)}}</math>

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: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.

: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.

|style="text-align:right;"|ותגרע ממנו שני שרשיו והם ארבעה וחצי עשירית ישאר חצי עשירית וחלק אחד מאלף ת"ר

|style="text-align:right;"|ותגרע ממנו שני שרשיו והם ארבעה וחצי עשירית ישאר חצי עשירית וחלק אחד מאלף ת"ר

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:<math>\scriptstyle\begin{cases}\scriptstyle x^2+3x+1=n^2\\\scriptstyle x^2-\left(3x-2\right)=m^2\end{cases}</math>

:<math>\scriptstyle\begin{cases}\scriptstyle x^2+3x+1=n^2\\\scriptstyle x^2-\left(3x-2\right)=m^2\end{cases}</math>

|style="text-align:right;"|לו <big>ואם</big> יאמרו לך אלגו שיחזיק שרש אם תוסיף עליו שלשת שרשיו ודרהם אחד יחזיק שרש<br>

|style="text-align:right;"|לו <big>ואם</big> יאמרו לך אלגו שיחזיק שרש אם תוסיף עליו שלשת שרשיו ודרהם אחד יחזיק שרש<br>

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:Subtract from the square its three roots minus two dirham. A square and two dirham minus three roots remain [<math>\scriptstyle{\color{blue}{x^2-\left(3x-2\right)=x^2+2-3x}}</math>] and it has a root.

:Subtract from the square its three roots minus two dirham. A square and two dirham minus three roots remain [<math>\scriptstyle{\color{blue}{x^2-\left(3x-2\right)=x^2+2-3x}}</math>] and it has a root.

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:<math>\scriptstyle{\color{blue}{x^2+3x+1=13+\frac{4}{9}=\left(3+\frac{2}{3}\right)^2}}</math>

:<math>\scriptstyle{\color{blue}{x^2+3x+1=13+\frac{4}{9}=\left(3+\frac{2}{3}\right)^2}}</math>

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:<math>\scriptstyle\begin{cases}\scriptstyle{\color{red}{x^2-\left(x-1\right)=n^2}}\\\scriptstyle x^2-\left(1-x\right)=m^2\end{cases}</math>

:<math>\scriptstyle\begin{cases}\scriptstyle{\color{red}{x^2-\left(x-1\right)=n^2}}\\\scriptstyle x^2-\left(1-x\right)=m^2\end{cases}</math>

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: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.

: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.

:<math>\scriptstyle{\color{blue}{\left(x-\frac{1}{2}\right)^2=x^2+\frac{1}{4}-x=x^2+x-1}}</math>

:<math>\scriptstyle{\color{blue}{\left(x-\frac{1}{2}\right)^2=x^2+\frac{1}{4}-x=x^2+x-1}}</math>

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

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

:<math>\scriptstyle{\color{blue}{x=\sqrt{x^2}=\frac{5}{8}}}</math>

:<math>\scriptstyle{\color{blue}{x=\sqrt{x^2}=\frac{5}{8}}}</math>

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:<math>\scriptstyle\begin{cases}\scriptstyle x^2+\left(2-x\right)=n^2\\\scriptstyle x^2-\left(3-x\right)=m^2\end{cases}</math>

:<math>\scriptstyle\begin{cases}\scriptstyle x^2+\left(2-x\right)=n^2\\\scriptstyle x^2-\left(3-x\right)=m^2\end{cases}</math>

|style="text-align:right;"|לח <big>ואם</big> יאמרו לך אלגו המחזיק שרש אם תוסיף עליו שני דרהמי פחות שרש האלגו יחזיק שרש<br>

|style="text-align:right;"|לח <big>ואם</big> יאמרו לך אלגו המחזיק שרש אם תוסיף עליו שני דרהמי פחות שרש האלגו יחזיק שרש<br>

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: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.

: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.

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

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

:<math>\scriptstyle{\color{blue}{x^2=1+\frac{4}{5}+\left(\frac{4}{5}\sdot\frac{1}{5}\right)}}</math>

:<math>\scriptstyle{\color{blue}{x^2=1+\frac{4}{5}+\left(\frac{4}{5}\sdot\frac{1}{5}\right)}}</math>

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:<math>\scriptstyle\begin{cases}\scriptstyle x^2+x+1=n^2\\\scriptstyle x^2+2x+2=m^2\end{cases}</math>

:<math>\scriptstyle\begin{cases}\scriptstyle x^2+x+1=n^2\\\scriptstyle x^2+2x+2=m^2\end{cases}</math>

|style="text-align:right;"|לט <big>ואם</big> יאמרו לך אלגו המחזיק שרש אם תוסיף עליו שרשו ודרהם אחד יחזיק שרש<br>

|style="text-align:right;"|לט <big>ואם</big> יאמרו לך אלגו המחזיק שרש אם תוסיף עליו שרשו ודרהם אחד יחזיק שרש<br>

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

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

:<math>\scriptstyle{\color{blue}{x=\sqrt{x^2}=\frac{7}{8}}}</math>

:<math>\scriptstyle{\color{blue}{x=\sqrt{x^2}=\frac{7}{8}}}</math>

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== <span style="color:Green>Word Problems</span> ==

== <span style="color:Green>Word Problems</span> ==

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

: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?

: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<br>

ואמר הראשון לשלשה אם תתנו לי החצי מאשר תחזיקו יהיה לי ערך הבהמה<br>

ושאל השני לשלשה השלישית<br>

ושאל השני לשלשה השלישית<br>

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:Knowing this is that you assume the first has a thing [<math>\scriptstyle{\color{blue}{x}}</math>], the second a dinar [<math>\scriptstyle{\color{blue}{y}}</math>], the third a dirham [<math>\scriptstyle{\color{blue}{u}}</math>], and the fourth a fals [<math>\scriptstyle{\color{blue}{v}}</math>]

:Knowing this is that you assume the first has a thing [<math>\scriptstyle{\color{blue}{x}}</math>], the second a dinar [<math>\scriptstyle{\color{blue}{y}}</math>], the third a dirham [<math>\scriptstyle{\color{blue}{u}}</math>], and the fourth a fals [<math>\scriptstyle{\color{blue}{v}}</math>]

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: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.

: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.

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: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.  

: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.  

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: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.

: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.

:<math>\scriptstyle{\color{blue}{y+\frac{1}{3}x+\frac{1}{3}u+\frac{1}{3}v}}</math>

:<math>\scriptstyle{\color{blue}{y+\frac{1}{3}x+\frac{1}{3}u+\frac{1}{3}v}}</math>

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:<math>\scriptstyle{\color{blue}{x-\frac{1}{3}x=\frac{2}{3}x}}</math>

:<math>\scriptstyle{\color{blue}{x-\frac{1}{3}x=\frac{2}{3}x}}</math>

|style="text-align:right;"|ותכוין בהם ויהיה זה שתגרע שלישית דבר מדבר שלם וישאר שני שלישי דבר

|style="text-align:right;"|ותכוין בהם ויהיה זה שתגרע שלישית דבר מדבר שלם וישאר שני שלישי דבר

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

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

:<math>\scriptstyle{\color{blue}{\frac{1}{2}v-\frac{1}{3}v=\frac{1}{6}v}}</math>

:<math>\scriptstyle{\color{blue}{\frac{1}{2}v-\frac{1}{3}v=\frac{1}{6}v}}</math>

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: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.

: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.

:<math>\scriptstyle{\color{blue}{y-\frac{1}{2}y=\frac{1}{2}y=\frac{2}{3}x+\frac{1}{6}u+\frac{1}{6}v}}</math>

:<math>\scriptstyle{\color{blue}{y-\frac{1}{2}y=\frac{1}{2}y=\frac{2}{3}x+\frac{1}{6}u+\frac{1}{6}v}}</math>

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

:<math>\scriptstyle{\color{blue}{y=1\frac{1}{3}x+\frac{1}{3}u+\frac{1}{3}v}}</math>

:<math>\scriptstyle{\color{blue}{y=1\frac{1}{3}x+\frac{1}{3}u+\frac{1}{3}v}}</math>

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

:<math>\scriptstyle{\color{blue}{\frac{1}{4}y=\frac{1}{3}x+\left(\frac{1}{2}\sdot\frac{1}{6}\right)u+\left(\frac{1}{2}\sdot\frac{1}{6}\right)v}}</math>

:<math>\scriptstyle{\color{blue}{\frac{1}{4}y=\frac{1}{3}x+\left(\frac{1}{2}\sdot\frac{1}{6}\right)u+\left(\frac{1}{2}\sdot\frac{1}{6}\right)v}}</math>

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: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.

: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.

:<math>\scriptstyle{\color{blue}{\frac{1}{5}y=\left[\frac{1}{5}+\left(\frac{1}{5}\sdot\frac{1}{3}\right)\right]x+\left(\frac{1}{5}\sdot\frac{1}{3}\right)u+\left(\frac{1}{5}\sdot\frac{1}{3}\right)v}}</math>

:<math>\scriptstyle{\color{blue}{\frac{1}{5}y=\left[\frac{1}{5}+\left(\frac{1}{5}\sdot\frac{1}{3}\right)\right]x+\left(\frac{1}{5}\sdot\frac{1}{3}\right)u+\left(\frac{1}{5}\sdot\frac{1}{3}\right)v}}</math>

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: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.

: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

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

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: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.

: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.

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: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.

: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.

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: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.

: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

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

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:<math>\scriptstyle{\color{blue}{\left[\left(1+\frac{2}{3}\right)-\left(\frac{1}{3}+\frac{1}{4}\right)\right]x+\left(\frac{2}{3}-\frac{1}{3}\right)v=\left[1+\left(\frac{1}{2}\sdot\frac{1}{6}\right)\right]x+\frac{1}{3}v}}</math>

:<math>\scriptstyle{\color{blue}{\left[\left(1+\frac{2}{3}\right)-\left(\frac{1}{3}+\frac{1}{4}\right)\right]x+\left(\frac{2}{3}-\frac{1}{3}\right)v=\left[1+\left(\frac{1}{2}\sdot\frac{1}{6}\right)\right]x+\frac{1}{3}v}}</math>

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: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.

: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.

:<math>\scriptstyle{\color{blue}{\left[\left[1+\left(\frac{1}{2}\sdot\frac{1}{6}\right)\right]-\frac{2}{3}\right]u=\left(\frac{1}{4}+\frac{1}{6}\right)u=\left[1+\left(\frac{1}{2}\sdot\frac{1}{6}\right)\right]x+\frac{1}{3}v}}</math>

:<math>\scriptstyle{\color{blue}{\left[\left[1+\left(\frac{1}{2}\sdot\frac{1}{6}\right)\right]-\frac{2}{3}\right]u=\left(\frac{1}{4}+\frac{1}{6}\right)u=\left[1+\left(\frac{1}{2}\sdot\frac{1}{6}\right)\right]x+\frac{1}{3}v}}</math>

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

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

:<math>\scriptstyle{\color{blue}{u=\left(2+\frac{3}{5}\right)x+\frac{4}{5}v}}</math>

:<math>\scriptstyle{\color{blue}{u=\left(2+\frac{3}{5}\right)x+\frac{4}{5}v}}</math>

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: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.

: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.

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

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

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: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.

: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.

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: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.

: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.

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:<math>\scriptstyle{\color{blue}{\left[1+\frac{1}{5}+\left(\frac{2}{5}\sdot\frac{1}{5}\right)\right]v-1\frac{1}{5}v=\frac{2}{5}\sdot\frac{1}{5}v}}</math>

:<math>\scriptstyle{\color{blue}{\left[1+\frac{1}{5}+\left(\frac{2}{5}\sdot\frac{1}{5}\right)\right]v-1\frac{1}{5}v=\frac{2}{5}\sdot\frac{1}{5}v}}</math>

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: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.

: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.

:<math>\scriptstyle{\color{blue}{3\frac{2}{5}x-\left[1+\left(\frac{4}{5}\sdot\frac{1}{5}\right)\right]x=\left[2+\frac{1}{5}x+\left(\frac{1}{5}\sdot\frac{1}{5}\right)\right]x=\frac{2}{5}\sdot\frac{1}{5}v}}</math>

:<math>\scriptstyle{\color{blue}{3\frac{2}{5}x-\left[1+\left(\frac{4}{5}\sdot\frac{1}{5}\right)\right]x=\left[2+\frac{1}{5}x+\left(\frac{1}{5}\sdot\frac{1}{5}\right)\right]x=\frac{2}{5}\sdot\frac{1}{5}v}}</math>

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: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 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.

:<math>\scriptstyle{\color{blue}{u=2\frac{3}{5}x+\frac{4}{5}v=\left[\left(2+\frac{3}{5}\right)\sdot1\right]+\left(\frac{4}{5}\sdot28\right)=25}}</math>

:<math>\scriptstyle{\color{blue}{u=2\frac{3}{5}x+\frac{4}{5}v=\left[\left(2+\frac{3}{5}\right)\sdot1\right]+\left(\frac{4}{5}\sdot28\right)=25}}</math>

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: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.

: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.

:<math>\scriptstyle{\color{blue}{3\frac{{\color{red}{2}}}{5}x+1\frac{1}{5}v=\left[\left(3+\frac{2}{5}\right)\sdot1\right]+\left(1+\frac{1}{5}\sdot28\right)=37}}</math>

:<math>\scriptstyle{\color{blue}{3\frac{{\color{red}{2}}}{5}x+1\frac{1}{5}v=\left[\left(3+\frac{2}{5}\right)\sdot1\right]+\left(1+\frac{1}{5}\sdot28\right)=37}}</math>

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|This procedure is the one that is known to the arithmeticians and it is a complete understanding, except that it is very subtle and the number is extracted with great difficulty.

|This procedure is the one that is known to the arithmeticians and it is a complete understanding, except that it is very subtle and the number is extracted with great difficulty.

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|I solve this with a direct procedure and full understanding:

|I solve this with a direct procedure and full understanding:

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|style="text-align:right;"|וזה שנניח כל אשר יחזיקו דרהם אחד

|style="text-align:right;"|וזה שנניח כל אשר יחזיקו דרהם אחד

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|style="text-align:right;"|ונניח שיהיה לאחד מזה הדרהם דבר

|style="text-align:right;"|ונניח שיהיה לאחד מזה הדרהם דבר

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: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.

: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.

:<math>\scriptstyle{\color{blue}{\frac{2}{3}u=\frac{1}{2}x+\frac{1}{2}y-\frac{1}{3}y=\frac{1}{2}x+\frac{1}{6}y}}</math>

:<math>\scriptstyle{\color{blue}{\frac{2}{3}u=\frac{1}{2}x+\frac{1}{2}y-\frac{1}{3}y=\frac{1}{2}x+\frac{1}{6}y}}</math>

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

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

:<math>\scriptstyle{\color{blue}{u=\frac{3}{4}x+\frac{1}{4}y}}</math>

:<math>\scriptstyle{\color{blue}{u=\frac{3}{4}x+\frac{1}{4}y}}</math>

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: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.

: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.

:<math>\scriptstyle{\color{blue}{u+\left[\frac{1}{4}\sdot\left(y-u\right)\right]=\frac{3}{4}u+\frac{1}{4}y=\frac{1}{2}x+\frac{1}{2}y}}</math>

:<math>\scriptstyle{\color{blue}{u+\left[\frac{1}{4}\sdot\left(y-u\right)\right]=\frac{3}{4}u+\frac{1}{4}y=\frac{1}{2}x+\frac{1}{2}y}}</math>

וישוה ערך הבהמה מהחלק האחד שהוא חצי דבר וחצי דרהם

וישוה ערך הבהמה מהחלק האחד שהוא חצי דבר וחצי דרהם

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: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.

: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.

:<math>\scriptstyle{\color{blue}{\frac{3}{4}u=\frac{1}{2}x+\frac{1}{2}y-\frac{1}{4}y=\frac{1}{2}x+\frac{1}{4}y}}</math>

:<math>\scriptstyle{\color{blue}{\frac{3}{4}u=\frac{1}{2}x+\frac{1}{2}y-\frac{1}{4}y=\frac{1}{2}x+\frac{1}{4}y}}</math>

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: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.

: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.

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: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.

: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.

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: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.

: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

וישוה לערך מהחלק האחד שהוא חצי דבר וחצי דרהם

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:<math>\scriptstyle{\color{blue}{\left(\frac{1}{3}\sdot\frac{1}{10}\right)y=\left(2+\frac{1}{3}+\frac{1}{10}\right)x}}</math>

:<math>\scriptstyle{\color{blue}{\left(\frac{1}{3}\sdot\frac{1}{10}\right)y=\left(2+\frac{1}{3}+\frac{1}{10}\right)x}}</math>

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:<math>\scriptstyle{\color{blue}{y=73x}}</math>

:<math>\scriptstyle{\color{blue}{y=73x}}</math>

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