Difference between revisions of "ספר המספר / אברהם אבן עזרא"

Prologue

Numeration

והנה הַסְּפָר^[20] תשעה^[21] מספרים

• The resemblance of the ranks to the rank of units

• units – as foundations of all numbers

For nine is the end of every number and these are called units that are the first rank.

• tens and their resemblance to the units

Because ten resembles to one and twenty resembles to two.

ועשרים דומים^[29] לשנים^[30] שהם שני^[31] עשרות

The exceptional pronunciation of the number twenty

It would be appropriate to call it "‛esrayim", as "matayim" [= two hundred] is called from "meah" [= one hundred] and "alpayim" [= two thousand] from "eleph" [one thousand], but for the following numbers that are šelošim [thirty] to tiš‛im [ninty] it has been treated in their usage.

רק בעבור^[38] חבריו הבאים אחריו^[39] שהם^[40] שלשים^[41] עד תשעים נהגו^[42] כמנהגם‫^[43]

So šelošim is derived from šaloš [= three] and so all of them.

• hundreds and their resemblance to the units

One hundred resembles to one as well as to ten; two hundred resembles to two as well as to twenty.

ומאתים^[51] דומה^[52] לשנים גם לעשרים‫^[53]

• thousand and upwards

likewise the thousand and the ten thousand, because they are the beginning of decades of the numbers that follow them, which are 1, 10, 100; 2, 20, 200.

• The sign that the nine units are building all numbers – the presentation of the multiplication of units by nine through the arrangement of the nine units in a circle

The sign for this: when you draw a circle and write nine digits around it.

You multiply nine by itself; the reason is its being a square whose length is as its width; see it and so it is:

• The square is eighty-one, so the one is on the left of the nine, which is the beginning of the units; and the 8, which corresponds to eighty in the decade, is on the right of the nine.

• If you multiply 9 by 8, the product is 72; 2 is on its left and 7, which corresponds to 70, is on its right.

• If you multiply 9 by 7, the product is 63; 3 is on its left and 6, which corresponds to 60, is on its right.

• If you multiply 9 by 6, the product is 54, 4 is on its left and 5, which corresponds to 50, is on its right.

Since the number five is mean between the 9 digits, therefore it is called round number, for it circles around itself as its square contains five.

• When you multiply 9 by 5, the matter rotates in a circle, for the units are to the right of 9 and the decades to its left; because the product is 45; 5 is on the right side of 9 and the decades are on its left, which is 4 for the 40.

• When you multiply 9 by 4, the product is 36; 3 corresponds to thirty.

• When you multiply 9 by 3, the product is 27; 2 corresponds to twenty.

• When you multiply 9 by 2, the product is 18; 1 corresponds to ten.

The Positional Decimal System

• The written numerals

Therefore, the wise men of India have denoted all their numbers by nine and they make forms for the 9 digits. I wrote instead of them: א ב ג ד ה ו ז ח ט [the first nine Hebrew letters]

ואני כתבתי במקומם^[147] א' ב' ג' ד' ה' ו' ז' ח' ט‫'^[148]

• The written ranks and their writing order

• tens

• units and tens

If you have a number of units and the beginning of the decades, which are tens, always write first the number of the units, then the number of the decades.

• tens

If there is no number in the units, but it has a number in the second rank, which is the tens, one puts a shape of a wheel first to indicate that there is no number in the first rank and then write the number that he has in the tens afterwards.

• hundreds

• tens and hundreds

If his decade consists of the hundreds and the tens, one writes a wheel in the first [rank], then the number of the tens in the second [rank] and the number of hundreds in third [rank].

• thousands and upwards

If one has a number of thousands [he writes it] in the fourth [rank], the number of tens of thousands in the fifth and the number of the hundreds of thousands in the sixth.

• Threefold cycle of the decimal ranks – units, tens, hundreds

For 1, 10, 100 repeat as thousands in the fourth, as thousands of thousands in the seventh and as thousands of thousands of thousands in the tenth and so on endlessly.

• units and hundreds

If there is a number of units and hundreds, but no tens, one writes the number of the units in the first [rank], a wheel in the second [rank] and the number of hundreds in third [rank].

• zero – a placeholder digit

In this way one puts at first or in the middle two wheels or as much as needed endlessly.

Table of Contents

• Chapter One: multiplication of a number by itself or by other; multiplication of one number by two numbers or more; multiplication of multiple numbers by multiple [numbers]

• Chapter Two: division of decades by units; two decades by units; large decades by small decades; decades and units by units; including a discussion on the checking of multiplication and division

• Chapter Three: addition of a number to a number; units to decades or decades to decades

• Chapter Four: subtraction of a number from a number; units from decades or decades from decades; including a discussion on the checking of addition and subtraction

Chapter One – Multiplication

Shortcuts

I have already mentioned what the ranks of the number are. When you have two numbers to multiply a decade by a decade, be it by itself, or by another, look for their analogous in the first rank, see how much is the product of the multiplication of one by the other and keep it. Then, see how many ranks the two numbers are, whether they are in the same rank or two [different] ranks, and know what the sum of the numbers of the two ranks is. Always subtract one for the foundation and put the reserved number in the remaining number of the ranks.

בין שיהיה^{[I 8]} על עצמו‫^{[I 9]}
או^{[I 10]} על אחר‫^{[I 11]}
בקש^{[I 12]} דמיונו במעלה הראשונה ^{[I 13]}וראה כמה^{[I 14]} המחובר מכפל^{[I 15]} זה על זה ושמור^{[I 16]} אותו‫^{[I 17]}
ואחר כך^{[I 18]} ראה כמה מעלות שני^{[I 19]} החשבונות^{[I 20]} בין שיהיה^{[I 21]} במעלה אחת או^{[I 22]} בשתים^{[I 23]} מעלות ודע כמה המחובר ממספר^{[I 24]} שתים^{[I 25]} מעלות^{[I 26]} וגרע לעולם אחד^{[I 27]} למוסד
ובקש במספר^{[I 28]} המעלה^{[I 29]} הנשאר^{[I 30]} במספר^{[I 31]} השמור‫^{[I 32]}

• Example: we wish to multiply thirty by two hundred.

The number that is analogous to thirty is three and the number that is analogous to two hundred is two. We multiply 2 by 3, the result is six and this is the reserved number. We seek the ranks: thirty is of the second rank, which are the tens; we take two for it. Since the two hundred is of the third rank, for they are the hundreds, we take three for it and add the two to them; they are five. We subtract one to the foundation; four remain. As we already know that the fourth rank is for the thousand and the reserved number is six, the result is six thousand.

כפלנו ב'^{[I 43]} על ג' והנה^{[I 44]} עלו^{[I 45]} ששה וזהו^{[I 46]} החשבון השמור‫^{[I 47]}
ונשוב לבקש המעלות והנה שלשים מהמעלה^{[I 48]} השנית שהם עשרות^{[I 49]} והנה נקח לו^{[I 50]} שנים‫^{[I 51]}
ובעבור כי המאתים^{[I 52]} מהמעלה^{[I 53]} השלישית כי הם^{[I 54]} מאות נקח לו שלשה^{[I 55]} ונחבר עליו^{[I 56]} השנים^{[I 57]} ויהיה^{[I 58]} חמשה
נחסר^{[I 59]} אחד^{[I 60]} למוסד ישארו^{[I 61]} ארבעה וכבר ידענו^{[I 62]} כי המעלה הרביעית^{[I 63]} היא לאלף‫^{[I 64]}
והמספר השמור^{[I 65]} היה^{[I 66]} ששה^{[I 67]} והנה^{[I 68]} העולה^{[I 69]} ששת אלפים‫^{[I 70]}

• Another example: we wish to multiply two hundred by seven hundred.

We multiply two by seven; it is 14 and this is the reserved. two hundred is of the third rank and 7 hundred is also of to the third rank; we take for them 6 and subtract one, it is 5. The beginning of the fifth rank is ten thousand and the reserved number is 14, so we take ten thousand of this number, the result is one hundred thousand and forty thousand.

והנה מאתים^{[I 77]} מהמעלה השלישית וז' מאות גם כן^{[I 78]} מהמעלה השלישית^{[I 79]} נקח^{[I 80]} להם^{[I 81]} ששה ונחסר^{[I 82]} אחד^{[I 83]} הנה^{[I 84]} חמשה
וראש^{[I 85]} המעלה^{[I 86]} החמישית^{[I 87]} עשרת^{[I 88]} אלפים והשמור היה^{[I 89]} י"ד
הנה נקח במספר^{[I 90]} הזה^{[I 91]} עשרת^{[I 92]} אלפים^{[I 93]} ויהיה^{[I 94]} העולה^{[I 95]} מאה^{[I 96]} אלף^{[I 97]} וארבעים אלף‫^{[I 98][I 99]}

You can proceed endlessly according to this procedure.

If there are two numbers whose distance from a decade is as much as the distance of two other numbers, only that the one is deficient and the other exceeding, know how much the square of the decade, always subtract from it the square of the excess and the deficit and the remainder is the sought number.

דע כמה^{[I 111]} מרובע^{[I 112]} מספר^{[I 113]} הכלל^{[I 114]} וגרע ממנו לעולם מרובע החשבון היתר והחסר^{[I 115]} והנשאר הוא^{[I 116]} החשבון^{[I 117]} המבוקש

• Example: we wish to multiply 29 by 31.

The decade is thirty, its square is 9 hundred, because three by three are nine; the excess and the deficit is one, its square one; we subtract it from the square of the decade, the remainder is the sought after, which is 899.

• Another example: we wish to multiply 66 by 54.

The decade is 60, the deficit and the excess is 6, the square of the decade is 3 thousand and 600. We subtract from it 36, which is the square of the deficit and the excess, the remainder is the sought after.

• Another example: one number is 250 and the other number 350.

The decade is 300, its square is 90 thousand. We subtract from it the square of 50, which is the deficit and the excess, and it is 2500. The remainder is the sought after.

According to this procedure we can proceed with other numbers that are similar to these, such that the deficit is the same as the excess.

Another important method that I have found is with thirds: We take one-third of the number and we know how much its square is. We take its closest higher decade and subtract from it the square of the third; the remainder is the sought after.

• Example: we wish to know how much is the square number of 3.

We take its third, which is one; its square is one; ten is its closest decade. We subtract from it the square of one, which is the third; 9 remain and this is the sought after.

• Another example: we wish to know the square of 15.

Its one-third is 5, the square of which is 25, the analogous number in the closest decade is 250. Subtract from it the square of 5, which is the third; 225 remain.

• Another example: we wish to know how much is the square of 24.

Its one-third is 8, the square of which is 64 and its analogous number in the closest higher rank is 640. We subtract from it the square of the third, which is 64; 576 remain and this is the sought after.

If the number does not have a whole third and there is an excess of one, subtract the one from the number and calculate the sought number in the procedure that I have shown you. Add to the result the number that has a [whole] third and the [original] number itself, so the sum is the sought after.

חסר^{[I 207]} האחד^{[I 208]} מהמספר^{[I 209]} והוצא^{[I 210]} המספר^{[I 211]} המבוקש כמשפט^{[I 212]} שהראיתיך ומה שיעלה^{[I 213]} הוסף^{[I 214]} עליו המספר שיש לו^{[I 215]} שלישית^{[I 216]} והמספר בעצמו והמחובר הוא המבוקש

• Example: we wish to know the square of 7.

It has no third, so we subtract from it one, which is the excess, one-third of the remainder is two, the square of which is four and the closest analogous decade is 40. We subtract from it 4, which is the square of the third; 36 remain; which is the square of 6. We add to it the 6, which has a third, and the 7 that was our number originally, both together are 13, the sum is 49, and this is the square of 7.

• Another example: we wish to know how much is the square of 22.

We subtract one; 21 remain, its third is 7, the square of which is 49 and its closest decade is 490. We subtract from it 49, which is the square of the third; 441 remain, which is the square of 21. We add 21 as well as 22 together, which are 43; the sum is 484 and this is the square of 22.

If there are 2 between our number and the number that has a third, we do the opposite by adding one to our number and know how much is the square of the number that has a third. We subtract from it the number that has a third and the number that we had, and the remainder is the sought after.

נעשה להפך שנוסיף^{[I 266]} על המספר שלנו^{[I 267]} אחד ונדע כמה מרובע^{[I 268]} מספר^{[I 269]} שיש לו שלישית^{[I 270]} ונחסר^{[I 271]} ממנו כמספר^{[I 272]} שיש לו שלישית^{[I 273]} וכמספר^{[I 274]} שהיה לנו והנשאר הוא המבוקש

• Example: we wish to know how much is the square of 23.

Since it does not have a whole third, we add one; it is 24, its third is 8, the square of which is 64 and the analogous number is 640. We subtract from it 64, which is the square of the third; 576 remain, which is the square of 24. We subtract from this number 47, which is the sum of 24 with 23; 529 remain and this is the sought after.

• units by units - one step    

• units by two-digit number - two steps

• units by three-digit number - three steps

• two-digit number by two-digit number - four steps

• Example: we wish to multiply 13 by 28.

• two-digit number by two-digit number with the same digit of tens - three steps

• Example: we wish to multiply 13 by 16.

• Example: we wish to multiply 24 by 26.

• three-digit number by three-digit number - nine steps

• even number by even number - the product is even

the digit of the units indicates if the number is even

every product of ten is even

• even number by odd number - the product is even

• odd number by odd number - the product is odd

Written calculations

• The product of two digits is equal to units and tens

• Example: we wish to multiply 127 by 355.

עוד כפלנו^{[I 576]} ז' על ה' השני התחתון^{[I 577]} עלו^{[I 578]} ל"ה^{[I 579]} כתבנו^{[I 580]} ה'^{[I 581]} במעלה השנית^{[I 582]} תחת ג'^{[I 583]} וג' בשלישית‫^{[I 584]}
עוד כפלנו^{[I 585]} ז' הראשון^{[I 586]} על ג' התחתון^{[I 587]} עלו^{[I 588]} כ"א כתבנו^{[I 589]} א'^{[I 590]} בשלישית תחת ג'^{[I 591]} וב' ברביעית^{[I 592]}

	127		127		127
	355		355		355
	2335		2335		2335
	15		115		115
	1		1		61

עוד כפלנו^{[I 603]} ב'^{[I 604]} העליון^{[I 605]} על ה'^{[I 606]} השנית^{[I 607]} התחתון^{[I 608]} היו^{[I 609]} גם כן^{[I 610]} י' כתבנו^{[I 611]} א'^{[I 612]} תחת ב'^{[I 613]} ברביעית‫^{[I 614]}
עוד כפלנו^{[I 615]} ב'^{[I 616]} העליון^{[I 617]} על ג'^{[I 618]} התחתון^{[I 619]} והיו^{[I 620]} ו' כתבנו אותו^{[I 621]} תחת א'^{[I 622]} ברביעית‫^{[I 623]}

	127		127		127
	355		355		355
	2335		2335		2335
	15		115		115
	61		61		61
	5		55		355

עוד כפלנו^{[I 632]} א'^{[I 633]} על ה'^{[I 634]} השני^{[I 635]} התחתון^{[I 636]} עלו^{[I 637]} ה'^{[I 638]} כתבנוהו^{[I 639]} ברביעית‫^{[I 640]}
עוד כפלנו^{[I 641]} א'^{[I 642]} העליון על ג' התחתון^{[I 643]} היו^{[I 644]} ג' כתבנוהו^{[I 645]} בחמישית^{[I 646]} אחר ב'‫^{[I 647]}

וכפול ז' על ה' עלו ל"ה וג' עשרו' וג' עשרו' היו בידך בין הכל עלו ל"ח כתו' ח' במדרגה השנית ושמור ג' מאות
עוד כפול ז' על ג' שהם עלו כ"א וג' שהיו בידך היו כ"ד מאו' כתו' ד' במדרגת המאות והכ' שהם ב' אלפי' כתו' במדרגת האלפי' כי עתה אין עוד אות בטור השפל לכפול האות הראשון עמו

וכפול ב' על ה' השני עלו י' וא' שהיה שמור הנה י"א כתו' א' במדרגת המאות ובידך שמור א' שהוא אלף
עו' כפול ב' על ג' הנה הם ו' וא' הנה ז' כתו' ו' במדרגת האלפי'

וא' על ה' השני וכתו' במדרגת האלפי' ה'
וכפול א' על ג' וכתו' במדרגת העשרת אלפי' ג'
וראה ועשה

Chapter Two – Division

Introduction - Preliminary definitions

• Every number is half of the sum of the numbers on either side - one is half the sum of its one side

reason: 12 lunar months, & 12 is the smallest number that has a half, third, quarter, sixth and half sixth

reason: 30 has a half, third, fifth, sixth and tenth

reason: 360 is close to the number of days in one year & it has a half, third, quarter, fifth, sixth, eighth, ninth, and tenth

• multiplication of degrees by degrees = degrees

• division of degrees by degrees = degrees

Written calculations

• Example: we wish to divide 9 thousands by 70.

נשיבם אחורנית והם עשרים נחלק^{[II 116]} על ז' והנה נתן^{[II 117]} לו^{[II 118]} ב' ונכתוב אותו אחורנית לפני הנכתב בראשונה^{[note 33]} ונשארו לנו ו‫'
נשיבהו^{[II 119]} אחורנית יהיו^{[II 120]} ששים נחלקנו^{[II 121]} על ז' נתן לו^{[II 122]} ח' תכתבהו^{[II 123]} אחורנית^{[note 34]}
ונשארו לנו ד'^{[II 124]} והם במעלה השנית והם מ'^{[II 125]} והמספר המחולק עליו^{[II 126]} גדול ממנו^{[II 127]}

• Example: we wish to divide 20 thousands by 90.

נשיבם^{[II 147]} אחורנית במעלה השלישית^{[II 148]} והם^{[II 149]} כ' נחלקם^{[II 150]} על ט'^{[II 151]} והנה^{[II 152]} ב' ונשארו^{[II 153]} ב‫'^{[II 154]}
נשיבם אחורנית במעלה השנית^{[II 155]} והם עשרים נחלק על ט'^{[II 156]} והנם^{[II 157]} ב'^{[II 158]} ונשארו^{[II 159]} ב' שהם עשרים כי הם^{[II 160]} במעלה השנית בטור העליון^{[II 161]} וזה המספר^{[II 162]} פחות ממספרנו^{[II 163]} על^{[II 164]} כן נכתוב גלגל אחורנית כי לא עלו אחדים^{[II 165]} כי לא יצא לחוץ

• Example: we wish to divide 4 thousands and 32 by 30.

נשיבהו^{[II 182]} אחורנית במעלת^{[II 183]} המאות^{[note 36]} והיו^{[II 184]} י'^{[II 185]} נחלקנו^{[II 186]} על ג' ועלה בחלוק^{[II 187]} ג' נשאר^{[II 188]} לנו א‫'^{[II 189]}
השבנוהו^{[II 190]} אחורנית במעלת העשרות ויהיו^{[II 191]} י' חברנו אותו עם הכתוב במעלה^{[II 192]} השנית^{[II 193]} היו^{[II 194]} י"ג^{[II 195]} חלקנו אותו^{[II 196]} על ג' ונתנו^{[II 197]} לו^{[II 198]} ד'^{[II 199]} נשאר^{[II 200]} לנו^{[II 201]} א‫'^{[II 202]}
השיבונוהו^{[II 203]} אחורנית^{[II 204]} במעלה הראשונה^{[II 205]} היו^{[II 206]} י"ב^{[II 207]} שלא^{[II 208]} יתחלקו כי הנשאר^{[II 209]} פחות מאותו^{[II 210]} המחולק^{[II 211]} עליו וכבר^{[II 212]} יצא לחוץ

• Example: we wish to divide 8213 by 353.

והנה^{[II 254]} כאשר^{[II 255]} חלקנו ח' שהוא^{[II 256]} אחרון בטור העליון על^{[II 257]} ג' שהוא אחרון^{[II 258]} בטור השפל והנה נתנו^{[II 259]} לו שנים ובעבור שהיה^{[II 260]} הג'^{[II 261]} שבטור השפל שלישי^{[II 262]} החזרנו^{[II 263]} לשלישי^{[II 264]} ממנו אחורנית והגיע למעלת^{[II 265]} העשרות ונשאר לנו במספר הח'^{[II 266]} שנים
והנה הנחנו שם אחד^{[II 267]} כי אחד יספיק לנו והחזרנו האחד^{[II 268]} אצל השנים והיו^{[II 269]} י"ב וחסרנו^{[II 270]} ממנו י' שהוא כפל חמשה האמצעי שבטור השפל והנה נשארו ב' על הב‫'^{[II 271]}
נניח שם אחד ונחזיר אחד אחורנית על אחד שהוא שלישי ויהיו^{[II 272]} י"א נסיר ממנו ו'^{[II 273]} לג' שהוא ראשון שבטור^{[II 274]} השפל^{[II 275]} נשארו ה‫'^{[II 276]}
והנה^{[II 277]} לכל הג' השפלים מה שראוי להם

נשיבם על הה'^{[II 290]} אחורנית^{[II 291]} יהיו כ"ה נתן לחמשה האמצעי^{[II 292]} שבטור השפל^{[II 293]} ט"ו ונשארו^{[II 294]} עשרה
נשיב אחד אחורנית על הג'^{[II 295][note 38]} וישארו ט' והא' עם הג'^{[II 296]} י"ג^{[II 297]} ונקח^{[II 298]} מהם^{[II 299]} ט'^{[II 300]} וישארו^{[II 301]} ד‫'^{[II 302]}
וט' על הה'^{[II 303]} כי לא יכולנו^{[II 304]} לקחת בראשונה^{[II 305]} הג' מהי'^{[II 306]} אע"פ שיספיק לו^{[II 307]} כי אינו מעלתו עכשיו^{[II 308]} אע"פ שלקח ממנו בראשונה^{[II 309]} כי בראשונה^{[II 310]} היה^{[II 311]} שלישי^{[II 312]} לו כי האחרון שבטור השפל לקח מן הח' אבל עתה לקח מן^{[II 313]} הב'^{[II 314]} א'^{[II 315]} ואחר שהוא שלישי צריך הוא^{[II 316]} שיקח^{[II 317]} ממעלתו^{[II 318]} השלישית^{[II 319][II 320]} והנה^{[II 321]} הנשאר^{[II 322]} ד' ט‫'^{[II 323]}

• Another example.

נשיבם כלם^{[II 337]} אחורנית לשני לו שהוא ג' ויהיו^{[II 338]} שם^{[II 339]} ל"ג נתן לט'^{[II 340]} ג' יהיו^{[II 341]} כ"ז נשארו^{[II 342]} ו' על הג‫'^{[II 343]}
נקח^{[II 344]} מהם א'^{[II 345]} ונניח ה' נשיבהו^{[II 346]} על הח' שהוא שלישי^{[II 347]} לאחרון^{[II 348]} שבטור העליון ועם הח' יהיו^{[II 349]} י"ח נתן^{[II 350]} אותם^{[II 351]} כלם לו' שהוא שלישי שבטור השפל נכתוב^{[II 352]} על הח' גלגל לפי שלא נשאר^{[II 353]} על הח'^{[II 354]} מאומה

נקח מן הג'^{[II 362]} שעל הג' אחד וישארו שנים נתן האחד על הגלגל והם י' נתן לט' שבטור^{[II 363]} השפל ט' נשאר על הגלגל א‫'
נשיבהו אחורנית על הא' שהוא רביעי וראשון^{[II 364]} בטור העליון והם י"א נתן לו ו' נשארו^{[II 365]} ה' על הא‫'

• Another example.

נקח מהם אחד^{[II 396]} ישארו^{[II 397]} ב' על הה'^{[II 398]} נשיבהו^{[II 399]} אחורנית על הד' יהיו י"ד^{[II 400]} יקח^{[II 401]} ט'^{[II 402]} ישארו ה‫'
יש^{[II 403]} לד'^{[II 404]} שבטור^{[II 405]} השפל לקחת מן השלישי שבטור העליון למען כי שלישי הוא^{[II 406]} ולא נוכל כי השלישי^{[II 407]} שבטור העליון גלגל הוא^{[II 408]} נשיב^{[II 409]} מן הה' שהנחנו על הד' אחד על הגלגל^{[II 410]} יהיו י' יקח^{[II 411]} ד' ישארו^{[II 412]} ו' על הגלגל
יקח הה'^{[II 413]} שהוא ראש בטור השפל והוא רביעי^{[II 414]} יקח^{[II 415]} מהרביעי שבטור^{[II 416]} העליון שהוא ט' ישארו^{[II 417]} על ט'^{[II 418]} ד‫'

נשיב^{[II 429]} מהם ז' על הו'^{[II 430]} אחורנית^{[II 431]} וא' נשאר על הד' ויהיו^{[II 432]} ע"ו נקח^{[II 433]} לט'^{[II 434]} ע"ב נשארו^{[II 435]} ד' במקום הו‫'
אם^{[II 436]} נשיב מהם ג'^{[II 437]} ואמת^{[II 438]} כי הד' שבטור^{[II 439]} השפל יוכלו לקחת^{[II 440]} מהג'^{[II 441]} עם הד' שאחריהם^{[II 442]} שהם ל"ד אך לא ישאר כי אם שנים וכשנשיב^{[II 443]} אותו על הג'^{[II 444]} יהיו כ"ג לבד^{[II 445]} ויש לה'^{[II 446]} שיקחו^{[II 447]} מ‫'^{[II 448]}
לפיכך נשיב כל הד' אחורנית על הד' ונכתוב גלגל במקום הו' כנגד הגלגל^{[II 449]} שבטור העליון והם מ"ד יקחו^{[II 450]} הד' ל"ב^{[II 451]} נשארו^{[II 452]} י"ב
נקח מהם ד'^{[II 453]} כי לא יהיה די^{[II 454]} לנו^{[II 455]} בפחות וישארו ח' על הד' שהם על הט' ועם^{[II 456]} הג'^{[II 457]} הם^{[II 458]} מ"ג^{[II 459]} יקחו^{[II 460]} הה' מ‫'^{[II 461]}

• Another example: we wish to divide 68 thousands, 9 hundred and 21 by 7 thousands and 53.

נשיב אותו^{[II 482]} כלו^{[II 483]} אחורנית^{[II 484]} על הח' ויהיו^{[II 485]} ס"ח הנה^{[II 486]} חלקנו מן הח'^{[II 487]} שהוא רביעי לטור העליון^{[II 488]} נתן לז'^{[II 489]} ט'^{[II 490]} נוציאנו^{[II 491]} לחוץ^{[II 492]} על הג'^{[II 493]} שהוא רביעי בטור^{[II 494]} השפל נשארו^{[II 495]} ה' על הח‫'^{[II 496]}
הנה^{[II 497]} השני לז' לא יקח מאומה^{[II 498]} כי הוא גלגל
יש לה'^{[II 499]} שיקח^{[II 500]} מן ב'^{[II 501]} הרביעי^{[II 502]} בטור העליון שהוא שלישי לחלוק^{[II 503]} לא^{[II 504]} יוכל^{[II 505]} לקחת
נקח^{[II 506]} מן הט' שאחריו ה' נשיבם^{[II 507]} על הב'^{[II 508]} יהיו^{[II 509]} נ"ב^{[II 510]} וד' נשאר^{[II 511]} על הט' והה' מנ"ב^{[II 512]} יקחו מ"ה^{[II 513]} ישארו^{[II 514]} ז' על הב‫'^{[II 515]}
מן^{[II 516]} הז' נקח ג' ונשיבם^{[II 517]} אחורנית על א'^{[II 518]} והם^{[II 519]} ל"א יקחו ג' כ"ז^{[II 520]} נשאר^{[II 521]} ד' על א‫'^{[II 522]}

• Another example: we wish to divide 680402 by 2009.

• Example: we wish to divide 9 sevens by 4 nines.

• casting out by 9

• inverse operation - multiplication

• Example: we wish to divide 83521 by 903.

• Another example: we wish to divide 11350 by 110.

Chapter Three – Addition

Sums

• Example: we wish to know how much are the numbers that are summed from 1 to 11?

We know that the half of 11 is 5 and a half, we add to it one half, it is 6, then we multiply 11 by 6, they are 66 and this is the sum.

• Another example for an even number [of terms]: how much is the sum up to 18?

We take its half, which is 9, multiply 18 by it and the result is 162, then we have to multiply 18 again by one half, the result is 9, add it to 162, the result is 171 and it is the sum.

• Example for an odd number [of terms]: how much is the sum up to 11?

We add 1 and they are 12, half 11 is 5 and a half, we multiply 12 by 5 and a half, the result is 66 and so is the sum.

• For an even number [of terms]: to 18.

Its half is 9, we add one to 18, they are 19, we multiply 19 by 9, the result is 171.

• Example: we know that the square of 11 is 121, then we add to it 11, which is the root and it is the last number, the result is 132 and its half is 66.

• Example: one asks: I sum all the numbers until they reach a known number and the sum is 465. How much is the last number?

Hence, we double 465 and the result is 930. It is known that the preceding square is 900 and its root is 30, so it is the last summed number, for between the square and the double there is none but 30, thus it is the last number.

• Example: we wish to know how much is the sum up to 12?

Half the number is 6, the square is 36, double it, they are 72, add to it 6, which is the root, the result is 78 and it is the required.

• Example: we wish to know how much are the squares up to seven?

We already know that the sum is 28, thus we turn to know how much are two thirds of a seven plus one third - they are five, for two thirds of a six are four and we have two thirds more plus one third, so we multiply the sum by 5 and the result is 140 which is the required.

• Another example: how much is the sum of the squares up to 12?

It is known that its sum is 78 and its two thirds are 8, we multiply it by 78, the result is 624, we add to it a third of the sum, which is 26, the result is 650 and it is the required.

• Example: we wish to know how much is the sum of the cubes up to 5?

The sum up to 5 is 15, take its square, it is 225 and it is the [sum] of the cubes up to 5.

• Example: the sum of 3, 6, 12

They are 24 minus 3, thus 21 is the sum.

• Another example: 1, 2, 4

They are 8 minus 1, thus 7 is the sum.

• If it is said: how much is the sum of the squares up to 8?

• Question: how much is the sum from 4 to 9?

Written calculations

Addition of Sexagesimal Fractions

Chapter Four – Subtraction

Written calculations

• Example: