Difference between revisions of "מלאכת המספר"

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|The author said: When I saw the lengthiness of the discussions of the ancient scholars, the composers of the books of number, and that the necessary methods and teachings in astronomy, geometry, and the ratios of music are explained in those books merely in great difficulty; likewise the seven types of [operations] with integers as well as with fractions, the roots and the proportions of numbers; and since the teaching that is [based] on brief and comprehensive methods is better chosen for the student as well as for the teacher, for every lengthiness of words ''and loquacity are exhaustion of the body'' [Ecclesiastes 12, 12].

|The author said: When I saw the lengthiness of the discussions of the ancient scholars, the composers of the books of number, and that the necessary methods and teachings in astronomy, geometry, and the ratios of music are explained in those books merely in great difficulty; likewise the seven types of [operations] with integers as well as with fractions, the roots and the proportions of numbers; and since the teaching that is [based] on brief and comprehensive methods is better chosen for the student as well as for the teacher, for every lengthiness of words ''and loquacity are exhaustion of the body'' [Ecclesiastes 12, 12].

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|Therefore, I Yiẓḥaq b. R. Moshe ʽEli ha-Sefaradi [= the Spanish] from the city of Oriola of the kingdom of Aragon, at the request of my friends, who studied astronomy and geometry, because they used those methods with a great difficulty and bother, I shook out my lap and wrote this short treatise that encompasses all that is necessary for this science, arithmetic, that is called arishmetika, by the grace of God to me [Psalms 57, 2], and ''according to the good hand of my God upon me'' [Nehemiah 2, 8].

|Therefore, I Yiẓḥaq b. R. Moshe ʽEli ha-Sefaradi [= the Spanish] from the city of Oriola of the kingdom of Aragon, at the request of my friends, who studied astronomy and geometry, because they used those methods with a great difficulty and bother, I shook out my lap and wrote this short treatise that encompasses all that is necessary for this science, arithmetic, that is called arishmetika, by the grace of God to me [Psalms 57, 2], and ''according to the good hand of my God upon me'' [Nehemiah 2, 8].

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|I have written in it comprehensive and short ways in all seven types of [operations] of number, its properties and the existence of the ratios in all that is possible in this science.

|I have written in it comprehensive and short ways in all seven types of [operations] of number, its properties and the existence of the ratios in all that is possible in this science.

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|This is by demonstrative ways that are loved by the intellectuals, in a manner that anyone who endeavors in studying this short book will comprehend all that is necessary for this science, will be saved from the non-useful lengthiness and from the loss of time, and will attain in it whatever his heart desires; since it is for mathematics ''as the light of dawn shines ever brighter until the perfect'' [Proverbs 4, 18].

|This is by demonstrative ways that are loved by the intellectuals, in a manner that anyone who endeavors in studying this short book will comprehend all that is necessary for this science, will be saved from the non-useful lengthiness and from the loss of time, and will attain in it whatever his heart desires; since it is for mathematics ''as the light of dawn shines ever brighter until the perfect'' [Proverbs 4, 18].

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|With the help of God, Blessed is He, I will begin and say:

|With the help of God, Blessed is He, I will begin and say:

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::*Chapter seven: knowing the ratio of the six proportional numbers [= the proportional hexad].

::*Chapter seven: knowing the ratio of the six proportional numbers [= the proportional hexad].

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|style="text-align:right;"|<big>מלאכת</big> המספר היא מלאכה תורה למנות הרבה אחדים והבדליהם וסגולותם ובנקלה יקויימו בזכירה{{#annotend:weza}}

|style="text-align:right;"|<big>מלאכת</big> המספר היא מלאכה תורה למנות הרבה אחדים והבדליהם וסגולותם ובנקלה יקויימו בזכירה{{#annotend:weza}}

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|style="text-align:right;"|<big>אחדות</big> הוא יסוד וחלק ראשון מהמספר וכל מספר יורכב ממנו אבל הוא חוץ לכל מספר{{#annotend:dYUi}}

|style="text-align:right;"|<big>אחדות</big> הוא יסוד וחלק ראשון מהמספר וכל מספר יורכב ממנו אבל הוא חוץ לכל מספר{{#annotend:dYUi}}

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|style="text-align:right;"|ולכן יגדר המספר בשהוא קבוץ אחדים{{#annotend:I7x1}}

|style="text-align:right;"|ולכן יגדר המספר בשהוא קבוץ אחדים{{#annotend:I7x1}}

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!<span style=color:Green>The written ranks [= decimal places] and their writing order</span>

!<span style=color:Green>The written ranks [= decimal places] and their writing order</span>

|style="text-align:right;"|<big>הפרק</big> הראשון במין הראשון מהמספר והוא הקבוץ

|style="text-align:right;"|<big>הפרק</big> הראשון במין הראשון מהמספר והוא הקבוץ

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|style="text-align:right;"|<big>קבוץ</big> הוא חבור שני מספרים או יותר במספר אחד כולל לכולם{{#annotend:qlUB}}

|style="text-align:right;"|<big>קבוץ</big> הוא חבור שני מספרים או יותר במספר אחד כולל לכולם{{#annotend:qlUB}}

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*Then all the numerals are summed as units.

*Then all the numerals are summed as units.

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|This sum necessarily belongs to one of three categories, as you know: either it is units, or tens, or consists of both.

|This sum necessarily belongs to one of three categories, as you know: either it is units, or tens, or consists of both.

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*If it is tens, we write a zero and shift the 10 to the first rank next to it, which is the rank of the tens.

*If it is tens, we write a zero and shift the 10 to the first rank next to it, which is the rank of the tens.

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:<math>\scriptstyle5243+8962</math>

:<math>\scriptstyle5243+8962</math>

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|The proof of this is that we cast out the nines from the sum and keep the remainder. We do the same with the addend. If the remainders from both are equal, then the addition we made is correct, otherwise it is not.

|The proof of this is that we cast out the nines from the sum and keep the remainder. We do the same with the addend. If the remainders from both are equal, then the addition we made is correct, otherwise it is not.

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|This is enough for the first type.

|This is enough for the first type.

|style="width:45%; text-align:right;"|<big>הפרק</big> הב' במין הב' שהוא חסור

|style="width:45%; text-align:right;"|<big>הפרק</big> הב' במין הב' שהוא חסור

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!<span style=color:Green>Description of the procedure:</span>

!<span style=color:Green>Description of the procedure:</span>

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*If they are equal, we write zero beneath them, as a sign that nothing remains.

*If they are equal, we write zero beneath them, as a sign that nothing remains.

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:<math>\scriptstyle4282-2432</math>

:<math>\scriptstyle4282-2432</math>

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:*Next, we add 1 to the 2 that is in the fourth rank; it is 3. We subtract it from the 4, which is the upper digit; 1 remains beneath.

:*Next, we add 1 to the 2 that is in the fourth rank; it is 3. We subtract it from the 4, which is the upper digit; 1 remains beneath.

|style="text-align:right;"|ואחר כך בב' שהיא במדרגה הד' {{#annot:term|178,1206|0QE7}}נוסיף{{#annotend:0QE7}} א' ויהיו ג' ונחסרם מהד' שהיא האות העליון וישאר למטה א&#x202B;'

|style="text-align:right;"|ואחר כך בב' שהיא במדרגה הד' {{#annot:term|178,1206|0QE7}}נוסיף{{#annotend:0QE7}} א' ויהיו ג' ונחסרם מהד' שהיא האות העליון וישאר למטה א&#x202B;'

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|428<span style="color:red>2</span>||rowspan="3"|<math>\scriptstyle\xrightarrow{{\color{red}{2-2}}={\color{blue}{0}}}</math>||42<span style="color:red>8</span>2||rowspan="3"|<math>\scriptstyle\xrightarrow{{\color{red}{8-3}}={\color{blue}{5}}}</math>||4<span style="color:red>2</span>82||rowspan="3"|<math>\scriptstyle\xrightarrow{{\color{red}{2+\left(10-4\right)=2+6}}={\color{blue}{8}}}</math>||<span style="color:red>4</span>282||rowspan="3"|<math>\scriptstyle\xrightarrow{{\color{red}{4-\left(2+1\right)=4-3}}={\color{blue}{1}}}</math>||4282

|428<span style="color:red>2</span>||rowspan="3"|<math>\scriptstyle\xrightarrow{{\color{red}{2-2}}={\color{blue}{0}}}</math>||42<span style="color:red>8</span>2||rowspan="3"|<math>\scriptstyle\xrightarrow{{\color{red}{8-3}}={\color{blue}{5}}}</math>||4<span style="color:red>2</span>82||rowspan="3"|<math>\scriptstyle\xrightarrow{{\color{red}{2+\left(10-4\right)=2+6}}={\color{blue}{8}}}</math>||<span style="color:red>4</span>282||rowspan="3"|<math>\scriptstyle\xrightarrow{{\color{red}{4-\left(2+1\right)=4-3}}={\color{blue}{1}}}</math>||4282

| ||&#8199;&#8199;&#8199;<span style="color:#0000FF>0</span>||&#8199;&#8199;<span style="color:#0000FF>5</span>0||&#8199;<span style="color:#0000FF>8</span>50||<span style="color:#0000FF>1</span>850

| ||&#8199;&#8199;&#8199;<span style="color:#0000FF>0</span>||&#8199;&#8199;<span style="color:#0000FF>5</span>0||&#8199;<span style="color:#0000FF>8</span>50||<span style="color:#0000FF>1</span>850

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|By this order it should be done, even if the ranks are numerous.

|By this order it should be done, even if the ranks are numerous.

|style="width:45%; text-align:right;"|<big>הפרק</big> הג' במין הג' והוא הכפול ודרכים כוללים למצאות המספרים השלמים

|style="width:45%; text-align:right;"|<big>הפרק</big> הג' במין הג' והוא הכפול ודרכים כוללים למצאות המספרים השלמים

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|style="text-align:right;"|<big>כפול</big> הוא קבוץ איזה ב' מספרים שיהיו שוים{{#annotend:jEGw}}

|style="text-align:right;"|<big>כפול</big> הוא קבוץ איזה ב' מספרים שיהיו שוים{{#annotend:jEGw}}

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::<math>\scriptstyle2\times5372</math>

::<math>\scriptstyle2\times5372</math>

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|style="text-align:right;"|ובדרך ה{{#annot:term|159,1348|XTTg}}כפול{{#annotend:XTTg}} הזה ימצאו {{#annot:term|75,1268|cqAo}}המספרים השלמים{{#annotend:cqAo}}

|style="text-align:right;"|ובדרך ה{{#annot:term|159,1348|XTTg}}כפול{{#annotend:XTTg}} הזה ימצאו {{#annot:term|75,1268|cqAo}}המספרים השלמים{{#annotend:cqAo}}

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|style="text-align:right;"|וגדר המספר השלם הוא כל מספר שיבנה מקבוץ כל חלקיו שבשילקח כל אחד מחלקיו ויקובצו יבנו אותו לא פחות ולא יתר{{#annotend:1ldm}}

|style="text-align:right;"|וגדר המספר השלם הוא כל מספר שיבנה מקבוץ כל חלקיו שבשילקח כל אחד מחלקיו ויקובצו יבנו אותו לא פחות ולא יתר{{#annotend:1ldm}}

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|style="text-align:right;"|ואם יהיה מספר ראשון אז {{#annot:term|185,1255|I01K}}נכה{{#annotend:I01K}} אותו הכפל שלקחנו עם כפלו פחות א' והעולה מהכאה זו הוא {{#annot:term|75,1268|K1qA}}מספר שלם{{#annotend:K1qA}}

|style="text-align:right;"|ואם יהיה מספר ראשון אז {{#annot:term|185,1255|I01K}}נכה{{#annotend:I01K}} אותו הכפל שלקחנו עם כפלו פחות א' והעולה מהכאה זו הוא {{#annot:term|75,1268|K1qA}}מספר שלם{{#annotend:K1qA}}

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|style="width:45%; text-align:right;"|<big>הפרק הרביעי</big> במין הרביעי שהוא חלוק באמצע

|style="width:45%; text-align:right;"|<big>הפרק הרביעי</big> במין הרביעי שהוא חלוק באמצע

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|style="text-align:right;"|&#x202B;[חלוק באמצע]&#x202B;<ref>P1095 om.</ref> והוא חלוק איזה מספר שיהיה בשני חלקים שוים{{#annotend:MyS6}}

|style="text-align:right;"|&#x202B;[חלוק באמצע]&#x202B;<ref>P1095 om.</ref> והוא חלוק איזה מספר שיהיה בשני חלקים שוים{{#annotend:MyS6}}

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:<math>\scriptstyle262144\div2</math>

:<math>\scriptstyle262144\div2</math>

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:<math>\scriptstyle1048876\div2</math>

:<math>\scriptstyle1048876\div2</math>

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|The proof of this is doubling: if after it is doubled it becomes equal to the halved, then it is correct, otherwise it is incorrect.

|The proof of this is doubling: if after it is doubled it becomes equal to the halved, then it is correct, otherwise it is incorrect.

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|This is enough for the type of halving.

|This is enough for the type of halving.

|style="width:45%; text-align:right;"|<big>פרק</big> ה' במין הה' והוא הרבוע וקצת מסגולותיו

|style="width:45%; text-align:right;"|<big>פרק</big> ה' במין הה' והוא הרבוע וקצת מסגולותיו

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|You should know that whoever wants to be proficient in this type [of operation], must memorize this table that is called the multiplication table.

|You should know that whoever wants to be proficient in this type [of operation], must memorize this table that is called the multiplication table.

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|10||9||8||7||6||5||4||3||2||1

|10||9||8||7||6||5||4||3||2||1

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|י||ט||ח||ז||ו||ה||ד||ג||ב||א

|י||ט||ח||ז||ו||ה||ד||ג||ב||א

|ק||צ||פ||ע||ס||נ||מ||ל||כ||י

|ק||צ||פ||ע||ס||נ||מ||ל||כ||י

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|What is received from one of these two tables, is received from the other, there is no difference between them, except for the extensiveness versus brevity.

|What is received from one of these two tables, is received from the other, there is no difference between them, except for the extensiveness versus brevity.

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|In order that the two numbers we want to multiply will be well distinguished, we call the bottom number "multiplier" and the upper number "multiplied". We write in a line any multiplied number we want and beneath it we write the multiplier.

|In order that the two numbers we want to multiply will be well distinguished, we call the bottom number "multiplier" and the upper number "multiplied". We write in a line any multiplied number we want and beneath it we write the multiplier.

|style="text-align:right;"|ואם יהיו פחות מי' נכתוב אותם במקומם

|style="text-align:right;"|ואם יהיו פחות מי' נכתוב אותם במקומם

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|style="text-align:right;"|וכן צריך לעשות מכל אחד מאותיות הפועל עם אותיות הפעול

|style="text-align:right;"|וכן צריך לעשות מכל אחד מאותיות הפועל עם אותיות הפעול

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!<span style=color:Green>Multiplication with recollection</span>

!<span style=color:Green>Multiplication with recollection</span>

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|style="text-align:right;"|<big>ויש</big> דרך אחר לרבע

|style="text-align:right;"|<big>ויש</big> דרך אחר לרבע

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|style="text-align:right;"|דע כי כמו שה{{#annot:term|186,1656|uhV0}}פועל{{#annotend:uhV0}} בהכאתו ב{{#annot:term|608,1655|vkxf}}פעול{{#annotend:vkxf}} עושה כל כך טורים כאותיות שיש בו כמו שאמרנו למעלה ואחר כך ב{{#annot:term|154,1211|ThwY}}קבוץ{{#annotend:ThwY}} טוריו מורה הרבוע כן גם כן אם נרצה להקל מלעשות הרבה טורים באופן שמה שנשיג בהרבה טורים נוכל להשיג בטור א' כבר יש לנו דרך לזה והיא זאת

|style="text-align:right;"|דע כי כמו שה{{#annot:term|186,1656|uhV0}}פועל{{#annotend:uhV0}} בהכאתו ב{{#annot:term|608,1655|vkxf}}פעול{{#annotend:vkxf}} עושה כל כך טורים כאותיות שיש בו כמו שאמרנו למעלה ואחר כך ב{{#annot:term|154,1211|ThwY}}קבוץ{{#annotend:ThwY}} טוריו מורה הרבוע כן גם כן אם נרצה להקל מלעשות הרבה טורים באופן שמה שנשיג בהרבה טורים נוכל להשיג בטור א' כבר יש לנו דרך לזה והיא זאת

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|style="text-align:right;"|<big>דע</big> שהתחלת פעולת הפועל היא בהכאת האחדות שלו באחדות הפעול ומקום ההכאה הזאת היא מקום האחדות

|style="text-align:right;"|<big>דע</big> שהתחלת פעולת הפועל היא בהכאת האחדות שלו באחדות הפעול ומקום ההכאה הזאת היא מקום האחדות

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|style="text-align:right;"|וה{{#annot:term|156,1256|MFpo}}הכאה{{#annotend:MFpo}} שממנה יולדו [העשרות צריך שתתחיל במקום העשרות

|style="text-align:right;"|וה{{#annot:term|156,1256|MFpo}}הכאה{{#annotend:MFpo}} שממנה יולדו [העשרות צריך שתתחיל במקום העשרות

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|style="text-align:right;"|וההכאה שממנה יולדו]&#x202B;<ref>P1095 om.</ref> המאות צריך שתתחיל במדרגת המאות

|style="text-align:right;"|וההכאה שממנה יולדו]&#x202B;<ref>P1095 om.</ref> המאות צריך שתתחיל במדרגת המאות

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|style="text-align:right;"|וכן מהאחרות&#x202B;<ref>P1095 מהאחדות</ref> כסדר

|style="text-align:right;"|וכן מהאחרות&#x202B;<ref>P1095 מהאחדות</ref> כסדר

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|style="text-align:right;"|ולדעת ההכאה שממנה יולדו העשרו' וההכאה שממנה יולדו המאות וכן מה{{#annot:term|203,1344|THnl}}מדרגות{{#annotend:THnl}} האחרות זו היא הדרך

|style="text-align:right;"|ולדעת ההכאה שממנה יולדו העשרו' וההכאה שממנה יולדו המאות וכן מה{{#annot:term|203,1344|THnl}}מדרגות{{#annotend:THnl}} האחרות זו היא הדרך

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|style="text-align:right;"|<big>כבר</big> ידעת שהכאת האחדות באחדו' כתיבתה היא באחדות

|style="text-align:right;"|<big>כבר</big> ידעת שהכאת האחדות באחדו' כתיבתה היא באחדות

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|style="text-align:right;"|והעולה מאלו הד' הכאות צריך להכתב במדרגת אות האלפים

|style="text-align:right;"|והעולה מאלו הד' הכאות צריך להכתב במדרגת אות האלפים

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|style="text-align:right;"|וכסדר הזה בכל אותיות הטור שיהיו כי לפי המקום יהיו ההכאות

|style="text-align:right;"|וכסדר הזה בכל אותיות הטור שיהיו כי לפי המקום יהיו ההכאות

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|style="text-align:right;"|ובעבור שמקום האלף הוא מקום רביעי לכן עשינו ד' הכאות

|style="text-align:right;"|ובעבור שמקום האלף הוא מקום רביעי לכן עשינו ד' הכאות

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|style="text-align:right;"|וכמו כן בעשרות אלפי' שהוא מקום חמישי צריך ה' הכאות

|style="text-align:right;"|וכמו כן בעשרות אלפי' שהוא מקום חמישי צריך ה' הכאות

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|style="text-align:right;"|<big>ובעבור</big> שבזה המין ימצאו קצת סגולות מיוחדות נאמר אותם הנה והם אלו

|style="text-align:right;"|<big>ובעבור</big> שבזה המין ימצאו קצת סגולות מיוחדות נאמר אותם הנה והם אלו

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::<math>\scriptstyle\sum_{i=1}^{12} i=1+\ldots+12</math>

::<math>\scriptstyle\sum_{i=1}^{12} i=1+\ldots+12</math>

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::We take a half of 12, which 6, multiply it by 13, which is the last number plus 1; it is 78 and this is the sum from 1 to 12.

::We take a half of 12, which 6, multiply it by 13, which is the last number plus 1; it is 78 and this is the sum from 1 to 12.

::<math>\scriptstyle{\color{blue}{\sum_{i=1}^{12} i=1+\ldots+12=\left(\frac{1}{2}\sdot12\right)\sdot\left(12+1\right)=6\sdot13=78}}</math>

::<math>\scriptstyle{\color{blue}{\sum_{i=1}^{12} i=1+\ldots+12=\left(\frac{1}{2}\sdot12\right)\sdot\left(12+1\right)=6\sdot13=78}}</math>

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::<math>\scriptstyle\sum_{i=1}^{13} i=1+\ldots+13</math>

::<math>\scriptstyle\sum_{i=1}^{13} i=1+\ldots+13</math>

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::We take 6 and a half plus one half, which is 7, and multiply it by 13; the result is 78 and this is the sum of all of them.

::We take 6 and a half plus one half, which is 7, and multiply it by 13; the result is 78 and this is the sum of all of them.

|style="text-align:right;"|נקח ו' וחצי וחצי יותר שהם ז' ונכה אותם על י"ג ויעלו צ"א וכך הוא המקובץ מכולם

|style="text-align:right;"|נקח ו' וחצי וחצי יותר שהם ז' ונכה אותם על י"ג ויעלו צ"א וכך הוא המקובץ מכולם

::<math>\scriptstyle{\color{blue}{\sum_{i=1}^{13} i=1+\ldots+13=\left[\left(\frac{1}{2}\sdot13\right)+\frac{1}{2}\right]\sdot13=\left[\left(6+\frac{1}{2}\right)+\frac{1}{2}\right]\sdot13=7\sdot13=91}}</math>

::<math>\scriptstyle{\color{blue}{\sum_{i=1}^{13} i=1+\ldots+13=\left[\left(\frac{1}{2}\sdot13\right)+\frac{1}{2}\right]\sdot13=\left[\left(6+\frac{1}{2}\right)+\frac{1}{2}\right]\sdot13=7\sdot13=91}}</math>

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::<math>\scriptstyle\sum_{i=1}^6 2i=2+\ldots+12</math>

::<math>\scriptstyle\sum_{i=1}^6 2i=2+\ldots+12</math>

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::We take its half, which is 6, and multiply it by 7, which is half the number plus 1; the result is 42 and this is the sum of all of them.

::We take its half, which is 6, and multiply it by 7, which is half the number plus 1; the result is 42 and this is the sum of all of them.

|style="text-align:right;"|נקח חציו שהוא ו' ונכהו על ז' שהוא חצי המספר בתוספת א' ויעלה למ"ב וכך הוא הקבוץ של כולם

|style="text-align:right;"|נקח חציו שהוא ו' ונכהו על ז' שהוא חצי המספר בתוספת א' ויעלה למ"ב וכך הוא הקבוץ של כולם

::<math>\scriptstyle{\color{blue}{\sum_{i=1}^6 2i=2+\ldots+12=\left(\frac{1}{2}\sdot12\right)\sdot\left[\left(\frac{1}{2}\sdot12\right)+1\right]=6\sdot\left(6+1\right)=6\sdot7=42}}</math>

::<math>\scriptstyle{\color{blue}{\sum_{i=1}^6 2i=2+\ldots+12=\left(\frac{1}{2}\sdot12\right)\sdot\left[\left(\frac{1}{2}\sdot12\right)+1\right]=6\sdot\left(6+1\right)=6\sdot7=42}}</math>

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::<math>\scriptstyle\sum_{i=1}^8 \left(2i-1\right)=1+\ldots+15</math>

::<math>\scriptstyle\sum_{i=1}^8 \left(2i-1\right)=1+\ldots+15</math>

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::We take half the last number, which is 7 and a half, plus one half; it is 8. We multiply it by itself; the result is 64 and this is the sum of all of them.

::We take half the last number, which is 7 and a half, plus one half; it is 8. We multiply it by itself; the result is 64 and this is the sum of all of them.

|style="text-align:right;"|נקח חצי האחרון שהוא ז' וחצי וחצי יותר ויהיו ח' ונכהו בעצמו ויעלו ס"ד וכך הוא המקובץ מכולם

|style="text-align:right;"|נקח חצי האחרון שהוא ז' וחצי וחצי יותר ויהיו ח' ונכהו בעצמו ויעלו ס"ד וכך הוא המקובץ מכולם

::<math>\scriptstyle{\color{blue}{\sum_{i=1}^8 \left(2i-1\right)=1+\ldots+15=\left[\left(\frac{1}{2}\sdot15\right)+\frac{1}{2}\right]^2=\left[\left(7+\frac{1}{2}\right)+\frac{1}{2}\right]^2=8^2=64}}</math>

::<math>\scriptstyle{\color{blue}{\sum_{i=1}^8 \left(2i-1\right)=1+\ldots+15=\left[\left(\frac{1}{2}\sdot15\right)+\frac{1}{2}\right]^2=\left[\left(7+\frac{1}{2}\right)+\frac{1}{2}\right]^2=8^2=64}}</math>

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::<math>\scriptstyle8\times9</math>

::<math>\scriptstyle8\times9</math>

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::We find that the greater number, which is 9, is less than 10 by one. So, we subtract 8 once from 80, which is its product by ten; 72 remains and this is the product of 8 by 9.

::We find that the greater number, which is 9, is less than 10 by one. So, we subtract 8 once from 80, which is its product by ten; 72 remains and this is the product of 8 by 9.

|style="text-align:right;"|ומצאנו שהמספר היותר גדול שהוא ט' הוא פחות מי' אחד ולכן נחסר ח' פעם אחת מפ' שהוא עשיריתו וישארו ע"ב וזאת היא ההכאה מח' בט&#x202B;'

|style="text-align:right;"|ומצאנו שהמספר היותר גדול שהוא ט' הוא פחות מי' אחד ולכן נחסר ח' פעם אחת מפ' שהוא עשיריתו וישארו ע"ב וזאת היא ההכאה מח' בט&#x202B;'

::<math>\scriptstyle{\color{blue}{8<9<10\longrightarrow8\times9=\left(10\sdot8\right)-\left[\left(10-9\right)\sdot8\right]=80-\left(1\sdot8\right)=80-8=72}}</math>

::<math>\scriptstyle{\color{blue}{8<9<10\longrightarrow8\times9=\left(10\sdot8\right)-\left[\left(10-9\right)\sdot8\right]=80-\left(1\sdot8\right)=80-8=72}}</math>

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:*Example: as we said regarding the product of 8 by 9: by how much is the 9 less than 10, so we do when we examine by how much is 8 less than 10, which is by 2. So we subtract 2 times 9 from 90; 72 remains.

:*Example: as we said regarding the product of 8 by 9: by how much is the 9 less than 10, so we do when we examine by how much is 8 less than 10, which is by 2. So we subtract 2 times 9 from 90; 72 remains.

::<math>\scriptstyle{\color{blue}{8<9<10\longrightarrow8\times9=\left(10\sdot9\right)-\left[\left(10-8\right)\sdot9\right]=90-\left(2\sdot9\right)=72}}</math>

::<math>\scriptstyle{\color{blue}{8<9<10\longrightarrow8\times9=\left(10\sdot9\right)-\left[\left(10-8\right)\sdot9\right]=90-\left(2\sdot9\right)=72}}</math>

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::<math>\scriptstyle10\times12=120</math>

::<math>\scriptstyle10\times12=120</math>

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::<math>\scriptstyle20\times5=\left(2\sdot5\right)0=100</math>

::<math>\scriptstyle20\times5=\left(2\sdot5\right)0=100</math>

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::<math>\scriptstyle30\times20=\left(3\sdot20\right)0=600</math>

::<math>\scriptstyle30\times20=\left(3\sdot20\right)0=600</math>

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|style="width: 45%; text-align:right;"|<big>פרק שישי</big> במין השישי שהוא ה{{#annot:term|157,1223|W8wP}}חלוק{{#annotend:W8wP}}

|style="width: 45%; text-align:right;"|<big>פרק שישי</big> במין השישי שהוא ה{{#annot:term|157,1223|W8wP}}חלוק{{#annotend:W8wP}}

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|In this type [of operation] one should start from the highest rank.

|In this type [of operation] one should start from the highest rank.

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:<math>\scriptstyle144\div8</math>

:<math>\scriptstyle144\div8</math>

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|style="text-align:right;"|<big>וצריך</big> שתדע כי כשהאות המחולק תהיה פחות מהמחלק כמו בזה המשל אז נקח ב' אותיות מהמחולק ונקח היותר גדולה בשם עשרה והאחרת בשם אחדות

|style="text-align:right;"|<big>וצריך</big> שתדע כי כשהאות המחולק תהיה פחות מהמחלק כמו בזה המשל אז נקח ב' אותיות מהמחולק ונקח היותר גדולה בשם עשרה והאחרת בשם אחדות

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|style="text-align:right;"|כמו שאתה רואה בזאת הצורה

|style="text-align:right;"|כמו שאתה רואה בזאת הצורה

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::<span style=color:Green>[Illustration of the procedure:]</span>

::<span style=color:Green>[Illustration of the procedure:]</span>

::{|

::{|

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::<math>\scriptstyle\longrightarrow{\color{Violet}{18\ the\ result}}</math>

::<math>\scriptstyle\longrightarrow{\color{Violet}{18\ the\ result}}</math>

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|In this way you proceed whenever anything remains for you to divide.

|In this way you proceed whenever anything remains for you to divide.

|style="text-align:right;"|ובדרך הזה תעשה בכל עת שישאר לך דבר לחלק

|style="text-align:right;"|ובדרך הזה תעשה בכל עת שישאר לך דבר לחלק

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|style="text-align:right;"|כנראה בצורה הזאת

|style="text-align:right;"|כנראה בצורה הזאת

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:*Since the divisor is of two numerals, we take the two last digits of the dividend, which are 9 and 8 - the 8 are as units and the 9 are as tens, which are 98 above the 12.

:*Since the divisor is of two numerals, we take the two last digits of the dividend, which are 9 and 8 - the 8 are as units and the 9 are as tens, which are 98 above the 12.

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::We examine how many times the leftmost digit of the divisor, which is 1, can be subtracted from the leftmost digit of the dividend, which is 9. We see that it is 9 times in it.

::We examine how many times the leftmost digit of the divisor, which is 1, can be subtracted from the leftmost digit of the dividend, which is 9. We see that it is 9 times in it.

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::Indeed, since we say that as many times as the left [digit] of divisor is found in the left [digit] of the dividend, so many times the right [digit] of the divisor should be subtracted from the right [digit] of the dividend, and that is not enough, thus we write 8 as the quotient and subtract it from the 9; 1 remains above the 9.

::Indeed, since we say that as many times as the left [digit] of divisor is found in the left [digit] of the dividend, so many times the right [digit] of the divisor should be subtracted from the right [digit] of the dividend, and that is not enough, thus we write 8 as the quotient and subtract it from the 9; 1 remains above the 9.

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::We say: 8 times 2, which are 16, when we subtract them from 18, 2 remain above the 8 and zero above the 1 of the dividend.

::We say: 8 times 2, which are 16, when we subtract them from 18, 2 remain above the 8 and zero above the 1 of the dividend.

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:*Now we go back and take the 2 and the 7 - the 2 on the left and the 7 on the right as said - and we arrange them as we arranged the two first digits, saying: how many times the left [digit] of the divisor, which is 1, can be subtracted from the left [digit] of the dividend, which is 2. We find that it can be subtracted twice.

:*Now we go back and take the 2 and the 7 - the 2 on the left and the 7 on the right as said - and we arrange them as we arranged the two first digits, saying: how many times the left [digit] of the divisor, which is 1, can be subtracted from the left [digit] of the dividend, which is 2. We find that it can be subtracted twice.

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::We find also that the right [digit] of the divisor, which is 2, can be subtracted twice from the right [digit] of the dividend, which is 7.  

::We find also that the right [digit] of the divisor, which is 2, can be subtracted twice from the right [digit] of the dividend, which is 7.  

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::We multiply the 2 of quotient by the 1, which is the left digit of the divisor; the product is 2. We subtract it from the 2, which is the left [digit] of the dividend; 0 remains.

::We multiply the 2 of quotient by the 1, which is the left digit of the divisor; the product is 2. We subtract it from the 2, which is the left [digit] of the dividend; 0 remains.

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::We multiply the 2, which is the quotient by the 2, which is the right [digit] of the divisor; the product is 4. We subtract it from the 7; 3 remain above the 7.

::We multiply the 2, which is the quotient by the 2, which is the right [digit] of the divisor; the product is 4. We subtract it from the 7; 3 remain above the 7.

::<span style=color:Green>[Illustration of the procedure:]</span>

::<span style=color:Green>[Illustration of the procedure:]</span>

::{|

::{|

|<span style="color:red>12</span>&#8199;&#8199;||12&#8199;&#8199;||&#8199;<span style="color:red>12</span>&#8199;||&#8199;12&#8199;

|<span style="color:red>12</span>&#8199;&#8199;||12&#8199;&#8199;||&#8199;<span style="color:red>12</span>&#8199;||&#8199;12&#8199;

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:*Let 6 be the right [digit] of the dividend and 3 be the left [digit] of the dividend.

:*Let 6 be the right [digit] of the dividend and 3 be the left [digit] of the dividend.

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::We examine how many times the 1, which is the left [digit] of the divisor, can be subtracted from the 3, which is the left [digit] of the dividend. We find that it is 3 times.

::We examine how many times the 1, which is the left [digit] of the divisor, can be subtracted from the 3, which is the left [digit] of the dividend. We find that it is 3 times.

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::We multiply 3 by 1; the product is 3. We subtract it from the 3 of the dividend; 0 remains.

::We multiply 3 by 1; the product is 3. We subtract it from the 3 of the dividend; 0 remains.

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|&#8199;&#8199;<span style="color:red>12</span>||&#8199;&#8199;12

|&#8199;&#8199;<span style="color:red>12</span>||&#8199;&#8199;12

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|style="text-align:right;"|וכן אם ירבו האותיות הרבה מאד נעשה כסדר הזה

|style="text-align:right;"|וכן אם ירבו האותיות הרבה מאד נעשה כסדר הזה

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|style="text-align:right;"|וזה יספיק בו' מיני השלמים

|style="text-align:right;"|וזה יספיק בו' מיני השלמים

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|style="text-align:right;"|<big>וקודם</big> שנדבר [מהם]&#x202B;<ref>P1095 om.</ref> צריך שנאמר מהו שבר ואיך יסודר בכתיבתו ובאיזה חשבון ישבר

|style="text-align:right;"|<big>וקודם</big> שנדבר [מהם]&#x202B;<ref>P1095 om.</ref> צריך שנאמר מהו שבר ואיך יסודר בכתיבתו ובאיזה חשבון ישבר

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:<math>\scriptstyle\frac{3}{4}-\frac{2}{3}</math>

:<math>\scriptstyle\frac{3}{4}-\frac{2}{3}</math>

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:<math>\scriptstyle2\times\frac{3}{4}</math>

:<math>\scriptstyle2\times\frac{3}{4}</math>

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|In this operation you only need to double the digit that is beneath the line and the fraction is the halved.

|In this operation you only need to double the digit that is beneath the line and the fraction is the halved.

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:<math>\scriptstyle\frac{1}{4}\div2</math>

:<math>\scriptstyle\frac{1}{4}\div2</math>

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:<math>\scriptstyle4\times\frac{2}{3}</math>

:<math>\scriptstyle4\times\frac{2}{3}</math>

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:<math>\scriptstyle\left(4+\frac{1}{2}\right)\times\frac{2}{3}</math>

:<math>\scriptstyle\left(4+\frac{1}{2}\right)\times\frac{2}{3}</math>

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:<math>\scriptstyle\left(4+\frac{1}{3}\right)\times6</math>

:<math>\scriptstyle\left(4+\frac{1}{3}\right)\times6</math>

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:<math>\scriptstyle\left(2+\frac{1}{2}\right)\times\left(4+\frac{3}{4}\right)</math>

:<math>\scriptstyle\left(2+\frac{1}{2}\right)\times\left(4+\frac{3}{4}\right)</math>

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|<span style=color:Green>denominator:</span> First of all, we extract the denominator, which is the product of the digits that are beneath the fraction lines as follows:

|<span style=color:Green>denominator:</span> First of all, we extract the denominator, which is the product of the digits that are beneath the fraction lines as follows:

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|If we divide the seconds by 60, they become minutes.

|If we divide the seconds by 60, they become minutes.

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|The same for the others, when we divide them by 60 they are converted to the preceding rank.

|The same for the others, when we divide them by 60 they are converted to the preceding rank.

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|style="text-align:right;"|כנראה בצורה זו

|style="text-align:right;"|כנראה בצורה זו

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!44||26||42||21||7

!44||26||42||21||7

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!רביעיים||שלישיים||שניים||ראשונים||מעלות

!רביעיים||שלישיים||שניים||ראשונים||מעלות

|In order to add a further explanation, I write this second diagram which involves a longer procedure and study.  

|In order to add a further explanation, I write this second diagram which involves a longer procedure and study.  

|style="text-align:right;"|&#x202B;[וכדי להוסיף לך ביאור כתבתי זאת הצורה השנית שיש בה יותר מלאכה ועיון&#x202B;]&#x202B;<ref>P1095 om.:</ref>

|style="text-align:right;"|&#x202B;[וכדי להוסיף לך ביאור כתבתי זאת הצורה השנית שיש בה יותר מלאכה ועיון&#x202B;]&#x202B;<ref>P1095 om.:</ref>

{|class="wikitable" style="margin-left: auto; margin-right: auto; border: none; color: blue; text-align:center;"

{|class="wikitable" style="margin-left: auto; margin-right: auto; border: none; color: blue; text-align:center;"

!36||32||32||49||57||35||1

!36||32||32||49||57||35||1

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{|class="wikitable" style="margin-left: auto; margin-right: auto; border: none; text-align:center;"

{|class="wikitable" style="margin-left: auto; margin-right: auto; border: none; text-align:center;"

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:<math>\scriptstyle\frac{3}{4}\div\frac{2}{3}</math>

:<math>\scriptstyle\frac{3}{4}\div\frac{2}{3}</math>

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:<math>\scriptstyle\frac{3}{8}\div2</math>

:<math>\scriptstyle\frac{3}{8}\div2</math>

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:<math>\scriptstyle\frac{2}{3}\div\left(4+\frac{1}{2}\right)</math>

:<math>\scriptstyle\frac{2}{3}\div\left(4+\frac{1}{2}\right)</math>

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:<math>\scriptstyle\left(9+\frac{1}{2}\right)\div\left(3+\frac{4}{5}\right)</math>

:<math>\scriptstyle\left(9+\frac{1}{2}\right)\div\left(3+\frac{4}{5}\right)</math>

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:<math>\scriptstyle12\div\frac{4}{9}</math>

:<math>\scriptstyle12\div\frac{4}{9}</math>

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:<math>\scriptstyle\left(4+\frac{1}{3}\right)\div\frac{5}{6}</math>

:<math>\scriptstyle\left(4+\frac{1}{3}\right)\div\frac{5}{6}</math>

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:<math>\scriptstyle\left(3+\frac{1}{3}\right)\div4</math>

:<math>\scriptstyle\left(3+\frac{1}{3}\right)\div4</math>

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:<math>\scriptstyle50\div\left(2+\frac{1}{2}\right)</math>

:<math>\scriptstyle50\div\left(2+\frac{1}{2}\right)</math>

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|style="text-align:right;"|וראשונה נדבר מהשרש המרובע

|style="text-align:right;"|וראשונה נדבר מהשרש המרובע

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|style="text-align:right;"|ולכן ב' הם שרש ד' בעבור כשיוכה בעצמו יוליד ארבעה

|style="text-align:right;"|ולכן ב' הם שרש ד' בעבור כשיוכה בעצמו יוליד ארבעה

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|Hence, every number that is multiplied by itself generates a number that has a root and this number is called a square.

|Hence, every number that is multiplied by itself generates a number that has a root and this number is called a square.

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|style="text-align:right;"|ובעבור שלא נשאר כלום נכתוב 0'0' על ב' ועל ה' וזה מה שרצינו

|style="text-align:right;"|ובעבור שלא נשאר כלום נכתוב 0'0' על ב' ועל ה' וזה מה שרצינו

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::<span style=color:Green>[Illustration of the procedure:]</span>

::<span style=color:Green>[Illustration of the procedure:]</span>

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::{|

| ||&#8199;<span style="color:#0000FF>1</span>&#8199;&#8199;||&#8199;1&#8199;&#8199;

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*For, the first numerical rank, which is the rank of units, has a cube root. Because, the one that is cubically multiplied generates one, which is a cube number.

*For, the first numerical rank, which is the rank of units, has a cube root. Because, the one that is cubically multiplied generates one, which is a cube number.

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:<math>\scriptstyle\sqrt[3]{12167}</math>

:<math>\scriptstyle\sqrt[3]{12167}</math>

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:Therefore, we know that this number is a [perfect] cube and its root is 23.

:Therefore, we know that this number is a [perfect] cube and its root is 23.

|style="text-align:right;"|וא"כ ידענו שהמספר הזה הוא מעוקב ושורשו כ"ג

|style="text-align:right;"|וא"כ ידענו שהמספר הזה הוא מעוקב ושורשו כ"ג

::<span style=color:Green>[Illustration of the procedure:]</span>

::<span style=color:Green>[Illustration of the procedure:]</span>

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::{|

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::<math>\scriptstyle\longrightarrow{\color{Violet}{the\ result\ \sqrt[3]{12167}=23}}</math>

::<math>\scriptstyle\longrightarrow{\color{Violet}{the\ result\ \sqrt[3]{12167}=23}}</math>

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:<math>\scriptstyle\sqrt[3]{571787}</math>

:<math>\scriptstyle\sqrt[3]{571787}</math>

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:Therefore, nothing remains after the extraction of this root, so it is deduced that this number is a perfect cube and its root is 83.

:Therefore, nothing remains after the extraction of this root, so it is deduced that this number is a perfect cube and its root is 83.

|style="text-align:right;"|וא"כ לא נשאר דבר אחר הוצאת השרש הזה יראה המספר הזה הוא מעוקב שלם ושורשו הוא פ"ג

|style="text-align:right;"|וא"כ לא נשאר דבר אחר הוצאת השרש הזה יראה המספר הזה הוא מעוקב שלם ושורשו הוא פ"ג

::<span style=color:Green>[Illustration of the procedure:]</span>

::<span style=color:Green>[Illustration of the procedure:]</span>

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| ||rowspan="6"|<math>\scriptstyle\xrightarrow{\begin{align}&\scriptstyle{\color{red}{571-{\color{blue}{8}}^3=571-512=}}{\color{green}{59}}\\&\scriptstyle{\color{red}{3\times8=}}{\color{blue}{24}}\\\end{align}}</math>||&#x200B;||rowspan="6"|<math>\scriptstyle\xrightarrow{\begin{align}&\scriptstyle{\color{red}{59-\left(8\times2\times{\color{blue}{3}}\right)=59-\left(8\sdot6\right)=59-48=}}{\color{green}{11}}\\&\scriptstyle{\color{red}{117-\left[\left(8\times4\times{\color{blue}{3}}\right)+\left(3\times2\times{\color{blue}{3}}\right)\right]=}}\\&\scriptstyle{\color{red}{117-\left[\left(8\sdot12\right)+\left(3\sdot6\right)\right]=}}\\&\scriptstyle{\color{red}{117-\left(96+18\right)=117-114=}}{\color{green}{3}}\\&\scriptstyle{\color{red}{38-\left(3\times4\times{\color{blue}{3}}\right)=38-\left(3\sdot12\right)=38-36=}}{\color{green}{2}}\\&\scriptstyle{\color{red}{27-{\color{blue}{3}}^3=27-27=}}{\color{green}{0}}\\\end{align}}</math>||&#8199;<span style="color:LimeGreen>00</span>&#8199;&#8199;&#8199;

| ||rowspan="6"|<math>\scriptstyle\xrightarrow{\begin{align}&\scriptstyle{\color{red}{571-{\color{blue}{8}}^3=571-512=}}{\color{green}{59}}\\&\scriptstyle{\color{red}{3\times8=}}{\color{blue}{24}}\\\end{align}}</math>||&#x200B;||rowspan="6"|<math>\scriptstyle\xrightarrow{\begin{align}&\scriptstyle{\color{red}{59-\left(8\times2\times{\color{blue}{3}}\right)=59-\left(8\sdot6\right)=59-48=}}{\color{green}{11}}\\&\scriptstyle{\color{red}{117-\left[\left(8\times4\times{\color{blue}{3}}\right)+\left(3\times2\times{\color{blue}{3}}\right)\right]=}}\\&\scriptstyle{\color{red}{117-\left[\left(8\sdot12\right)+\left(3\sdot6\right)\right]=}}\\&\scriptstyle{\color{red}{117-\left(96+18\right)=117-114=}}{\color{green}{3}}\\&\scriptstyle{\color{red}{38-\left(3\times4\times{\color{blue}{3}}\right)=38-\left(3\sdot12\right)=38-36=}}{\color{green}{2}}\\&\scriptstyle{\color{red}{27-{\color{blue}{3}}^3=27-27=}}{\color{green}{0}}\\\end{align}}</math>||&#8199;<span style="color:LimeGreen>00</span>&#8199;&#8199;&#8199;

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::<math>\scriptstyle\longrightarrow{\color{Violet}{the\ result\ \sqrt[3]{571787}=83}}</math>

::<math>\scriptstyle\longrightarrow{\color{Violet}{the\ result\ \sqrt[3]{571787}=83}}</math>

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:<math>\scriptstyle\sqrt[3]{12812904}</math>

:<math>\scriptstyle\sqrt[3]{12812904}</math>

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:*So, we multiply the 2 cubically; it yields 8.

:*So, we multiply the 2 cubically; it yields 8.

::<math>\scriptstyle{\color{blue}{2^3=8}}</math>

::<math>\scriptstyle{\color{blue}{2^3=8}}</math>

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::We shift the 2 also, and write it in the next rank, so it is placed beneath the upper 4.

::We shift the 2 also, and write it in the next rank, so it is placed beneath the upper 4.

|style="text-align:right;"|והב' ג"כ נסיעה למקום הסמוך אליה ויעמוד תחת הד' העליונה

|style="text-align:right;"|והב' ג"כ נסיעה למקום הסמוך אליה ויעמוד תחת הד' העליונה

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| ||rowspan="6"|<math>\scriptstyle\xrightarrow{\begin{align}&\scriptstyle{\color{red}{12-{\color{blue}{2}}^3=12-8=}}{\color{green}{4}}\\&\scriptstyle{\color{red}{3\times2=}}{\color{blue}{6}}\\\end{align}}</math>||&#x200B;||rowspan="6"|<math>\scriptstyle\xrightarrow{\begin{align}&\scriptstyle{\color{red}{48-\left(2\times6\times{\color{blue}{3}}\right)=48-\left(2\sdot18\right)=48-36=}}{\color{green}{12}}\\&\scriptstyle{\color{red}{121-\left(3\times6\times{\color{blue}{3}}\right)=121-\left(3\sdot18\right)=121-54=}}{\color{green}{67}}\\&\scriptstyle{\color{red}{72-{\color{blue}{3}}^3=72-27=}}{\color{green}{45}}\\&\scriptstyle{\color{red}{3\times3=}}{\color{blue}{9}}\\\end{align}}</math>||&#8199;<span style="color:LimeGreen>0</span>&#8199;&#8199;&#8199;&#8199;&#8199;&#8199;

| ||rowspan="6"|<math>\scriptstyle\xrightarrow{\begin{align}&\scriptstyle{\color{red}{12-{\color{blue}{2}}^3=12-8=}}{\color{green}{4}}\\&\scriptstyle{\color{red}{3\times2=}}{\color{blue}{6}}\\\end{align}}</math>||&#x200B;||rowspan="6"|<math>\scriptstyle\xrightarrow{\begin{align}&\scriptstyle{\color{red}{48-\left(2\times6\times{\color{blue}{3}}\right)=48-\left(2\sdot18\right)=48-36=}}{\color{green}{12}}\\&\scriptstyle{\color{red}{121-\left(3\times6\times{\color{blue}{3}}\right)=121-\left(3\sdot18\right)=121-54=}}{\color{green}{67}}\\&\scriptstyle{\color{red}{72-{\color{blue}{3}}^3=72-27=}}{\color{green}{45}}\\&\scriptstyle{\color{red}{3\times3=}}{\color{blue}{9}}\\\end{align}}</math>||&#8199;<span style="color:LimeGreen>0</span>&#8199;&#8199;&#8199;&#8199;&#8199;&#8199;

|style="text-align:right;"|ולכן נכתוב ספרא על הד' העליונה ועל הו' העליונה

|style="text-align:right;"|ולכן נכתוב ספרא על הד' העליונה ועל הו' העליונה

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::<span style=color:Green>[Illustration of the procedure:]</span>

::<span style=color:Green>[Illustration of the procedure:]</span>

::{|

::{|

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::<math>\scriptstyle\longrightarrow{\color{Violet}{the\ result\ \sqrt[3]{12812904}=234}}</math>

::<math>\scriptstyle\longrightarrow{\color{Violet}{the\ result\ \sqrt[3]{12812904}=234}}</math>

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:<math>\scriptstyle2:3=5:X</math>

:<math>\scriptstyle2:3=5:X</math>

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:<math>\scriptstyle5:\left(7+\frac{1}{2}\right)=2:X</math>

:<math>\scriptstyle5:\left(7+\frac{1}{2}\right)=2:X</math>

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:<math>\scriptstyle\frac{2}{3}:\left(7+\frac{4}{9}\right)=\frac{4}{13}:X</math>

:<math>\scriptstyle\frac{2}{3}:\left(7+\frac{4}{9}\right)=\frac{4}{13}:X</math>

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:<math>\scriptstyle\frac{2}{3}:\left(4+\frac{1}{2}\right)=6:X</math>

:<math>\scriptstyle\frac{2}{3}:\left(4+\frac{1}{2}\right)=6:X</math>

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:<math>\scriptstyle\frac{2}{3}:8=9:X</math>

:<math>\scriptstyle\frac{2}{3}:8=9:X</math>

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:<math>\scriptstyle\left(5+\frac{2}{3}\right):\left(6+\frac{3}{4}\right)=\left(8+\frac{5}{12}\right):X</math>

:<math>\scriptstyle\left(5+\frac{2}{3}\right):\left(6+\frac{3}{4}\right)=\left(8+\frac{5}{12}\right):X</math>

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:<span style=color:Green>denominator</span>: To find the denominator, we multiply 5 by 3 that is beneath the 2 next to it ; the result is 15. We add the 2 that is above the 3; the result is 17. We multiply it by 4 that is beneath the 3 of the second number; the result is 68. We multiply it by 12 that is beneath the 5 of the third number; the result is 816 and this is the denominator.

:<span style=color:Green>denominator</span>: To find the denominator, we multiply 5 by 3 that is beneath the 2 next to it ; the result is 15. We add the 2 that is above the 3; the result is 17. We multiply it by 4 that is beneath the 3 of the second number; the result is 68. We multiply it by 12 that is beneath the 5 of the third number; the result is 816 and this is the denominator.

:<math>\scriptstyle{\color{blue}{\left[\left[\left(5\sdot3\right)+2\right]\sdot4\right]\sdot12=\left[\left(15+2\right)\sdot4\right]\sdot12=\left(17\sdot4\right)\sdot12=68\sdot12=816}}</math>

:<math>\scriptstyle{\color{blue}{\left[\left[\left(5\sdot3\right)+2\right]\sdot4\right]\sdot12=\left[\left(15+2\right)\sdot4\right]\sdot12=\left(17\sdot4\right)\sdot12=68\sdot12=816}}</math>

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:<math>\scriptstyle\left(5+\frac{2}{7}\right):4=20:X</math>

:<math>\scriptstyle\left(5+\frac{2}{7}\right):4=20:X</math>

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:<math>\scriptstyle20:\left(15+\frac{5}{37}\right)=\left(5+\frac{2}{7}\right):X</math>

:<math>\scriptstyle20:\left(15+\frac{5}{37}\right)=\left(5+\frac{2}{7}\right):X</math>

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:<math>\scriptstyle\frac{2}{3}:\frac{4}{9}=\frac{4}{13}:X</math>

:<math>\scriptstyle\frac{2}{3}:\frac{4}{9}=\frac{4}{13}:X</math>

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::<math>\scriptstyle\frac{2}{3}:8=9:X</math>

::<math>\scriptstyle\frac{2}{3}:8=9:X</math>

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::<math>\scriptstyle\left(5+\frac{2}{3}\right):\left(6+\frac{3}{4}\right)=\left(8+\frac{5}{12}\right):X</math>

::<math>\scriptstyle\left(5+\frac{2}{3}\right):\left(6+\frac{3}{4}\right)=\left(8+\frac{5}{12}\right):X</math>

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::<math>\scriptstyle1:\frac{4}{5}=\left(2+\frac{1}{2}\right):X</math>

::<math>\scriptstyle1:\frac{4}{5}=\left(2+\frac{1}{2}\right):X</math>

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::<math>\scriptstyle\frac{2}{3}:\left(4+\frac{1}{2}\right)=6:X</math>

::<math>\scriptstyle\frac{2}{3}:\left(4+\frac{1}{2}\right)=6:X</math>

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:<span style=color:Green>numerator</span>: In the example of this category, we extract the numerator by that we multiply 4 by the 2 that is beneath the line next to it; the result is 8. We add to it the 1 that is above the 2; the result is 9. We multiply it by the 6, which is the integer of the third number; the result is 54 and this were the numerator, if there were no fraction in the first number. Since there is a fraction in it, we multiply the 54 that we have by the digit that is beneath the line of that fraction, which is 3; the result is 162 and this is the numerator.

:<span style=color:Green>numerator</span>: In the example of this category, we extract the numerator by that we multiply 4 by the 2 that is beneath the line next to it; the result is 8. We add to it the 1 that is above the 2; the result is 9. We multiply it by the 6, which is the integer of the third number; the result is 54 and this were the numerator, if there were no fraction in the first number. Since there is a fraction in it, we multiply the 54 that we have by the digit that is beneath the line of that fraction, which is 3; the result is 162 and this is the numerator.

:<math>\scriptstyle{\color{blue}{\left[\left[\left(4\sdot2\right)+1\right]\sdot6\right]\sdot3=\left[\left(8+1\right)\sdot6\right]\sdot3=\left(9\sdot6\right)\sdot3=54\sdot3=162}}</math>

:<math>\scriptstyle{\color{blue}{\left[\left[\left(4\sdot2\right)+1\right]\sdot6\right]\sdot3=\left[\left(8+1\right)\sdot6\right]\sdot3=\left(9\sdot6\right)\sdot3=54\sdot3=162}}</math>

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!The sixth category: one of the last numbers is an integer alone and the other is a fraction alone.

!The sixth category: one of the last numbers is an integer alone and the other is a fraction alone.

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::<math>\scriptstyle9:\frac{2}{3}=8:X</math>

::<math>\scriptstyle9:\frac{2}{3}=8:X</math>

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:<math>\scriptstyle\frac{2}{3}:\left(4+\frac{1}{2}\right)=6:X</math>

:<math>\scriptstyle\frac{2}{3}:\left(4+\frac{1}{2}\right)=6:X</math>

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:<span style=color:Green>numerator:</span> Then, we extract the numerator by that we multiply the 4 of the second number by the 2, which is beneath the 1 next to it; the result is 8. We add the 1 that is above the 2 of the second number; the result is 9. We multiply it by the 6 that is the third number; the result is 54. We multiply it also by 3 that is beneath the 2 of the first number; the result is 162 and this is the numerator.

:<span style=color:Green>numerator:</span> Then, we extract the numerator by that we multiply the 4 of the second number by the 2, which is beneath the 1 next to it; the result is 8. We add the 1 that is above the 2 of the second number; the result is 9. We multiply it by the 6 that is the third number; the result is 54. We multiply it also by 3 that is beneath the 2 of the first number; the result is 162 and this is the numerator.

:<math>\scriptstyle{\color{blue}{\left[\left[\left(4\sdot2\right)+1\right]\sdot6\right]\sdot3=\left[\left(8+1\right)\sdot6\right]\sdot3=\left(9\sdot6\right)\sdot3=54\sdot3=162}}</math>

:<math>\scriptstyle{\color{blue}{\left[\left[\left(4\sdot2\right)+1\right]\sdot6\right]\sdot3=\left[\left(8+1\right)\sdot6\right]\sdot3=\left(9\sdot6\right)\sdot3=54\sdot3=162}}</math>

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:<math>\scriptstyle6:\left(40+\frac{1}{2}\right)=\frac{2}{3}:X</math>

:<math>\scriptstyle6:\left(40+\frac{1}{2}\right)=\frac{2}{3}:X</math>

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:<math>\scriptstyle9:\frac{2}{3}=8:X</math>

:<math>\scriptstyle9:\frac{2}{3}=8:X</math>

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==== Chapter Seven: Knowing the Ratio of the Six Proportional Numbers [= the Proportional Hexad] ====

==== Chapter Seven: Knowing the Ratio of the Six Proportional Numbers [= the Proportional Hexad] ====

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|There is another way of ratios, which is that if there are six proportional numbers, such that the ratio of the first to the second is as the ratio of the third to the fourth and the ratio of the fifth to the second is as the ratio of the sixth to the fourth, then the ratio of the sum of the first and the fifth to the second is as the ratio of the sum of the third and the sixth to the fourth.

|There is another way of ratios, which is that if there are six proportional numbers, such that the ratio of the first to the second is as the ratio of the third to the fourth and the ratio of the fifth to the second is as the ratio of the sixth to the fourth, then the ratio of the sum of the first and the fifth to the second is as the ratio of the sum of the third and the sixth to the fourth.

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*Example of six proportional numbers that are:

*Example of six proportional numbers that are:

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|style="text-align:right;"|בעבור שבהרבה מקומות יש חלוף באלו הדברי' הנזכרי' צריך שנדבר מאי זה דרך כללי שבו נוכל לדעת כל חלוף מאלה שנרצה

|style="text-align:right;"|בעבור שבהרבה מקומות יש חלוף באלו הדברי' הנזכרי' צריך שנדבר מאי זה דרך כללי שבו נוכל לדעת כל חלוף מאלה שנרצה

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|style="text-align:right;"|ולדעת זה נסדר כל אלו המספרי' ככה

|style="text-align:right;"|ולדעת זה נסדר כל אלו המספרי' ככה

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|6||3||9||6||4

|6||3||9||6||4

|İskenderun||İskenderun||Bursa ||Bursa ||Constantinople

|İskenderun||İskenderun||Bursa ||Bursa ||Constantinople

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|ו||ג||ט||ו||ד

|ו||ג||ט||ו||ד

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|Since our question is how much six from Alexandria are worth in Constantinople, we should know first how much the three from Alexandria are worth in Constantinople.

|Since our question is how much six from Alexandria are worth in Constantinople, we should know first how much the three from Alexandria are worth in Constantinople.