Difference between revisions of "עיר סיחון"

Introduction

ומחשבתי הטובה יצרף אותה למעשה
למען רחמיו וחסדיו אשר נפלאו ועצמו מספר כל מעמדיו
יהי שמו משובח לעד ומפואר לנצח

Numbers 21, 27: בֹּאוּ חֶשְׁבּוֹן; תִּבָּנֶה וְתִכּוֹנֵן, עִיר סִיחוֹן = "Come to Ḥeshbon, may it be built and established as the city of Siḥon"

Table of Contents

Chapter One – The Nature of the One and Foundation of Numbers

Numeration

• Introduction – the nature of the one and the foundation of all numbers

• One is not affected by change or multiplicity, but is the cause of any change and multiplicity

• One is eternal and the origin of all numbers, by which all numbers are generated

A learned person may draw hints from the numerical one regarding the secret of the unity of God

• Two

• Three

• Four

• Five

• Six

• Seven

• Nine

• Ten

ten in the rank of tens is like one in the rank of units

• Twenty

• Thirty

• Forty

• Fifty

• Sixty

• Seventy

• Eighty

• Ninety

all the names of the tens are borrowed from the units

Hundred as an inclusive name for ten tens

Thousand as an inclusive name for ten hundreds

• the nine names of the units

• ten

• hundred

• thousand

Chapter Two – the Decimal Ranks and the Ten Digits

The Positional Decimal System

• twenty: 20

• hundred: 100

Chapter Three – Addition

• We write each number of the addition request one beneath the other, each in a row of its own, rank by rank, then we draw a line.

• We start from the rank of units and sum all the units.

• We write the sum under [the] line, if it is less than ten.

• If [the sum] is tens, we write a digit in the column of the second rank, as the number of the tens of the sum and we write a zero alone under the line.

• If the sum is [units] and tens, we should write [the units] under the line and in the second rank we write the number of the tens as is stated.

• We proceed this way and do with the rest of the ranks as we did with the rank of units, writing the sums of the ranks successively by the order beneath the line.

• The [total] sum is what comes out from the sum of the ranks arranged under the line.

• If there is one or two ranks in the lines of the addend numbers, of which the whole column are zeros, zeros should be written correspondingly beneath the line as the number of these ranks.

• We wish to know how much is the sum of nine thousand, two hundred and eight, with three thousand, eight hundred and one.

We write these two numbers in two lines, according to this diagram:

the two lines of the numbers

9 2 0 8  3 8 0 1

the ranks of the sum

1 3 0 0 9

• first rank: 9

The sum of the first rank is smaller than ten, hence we write the result beneath the line, so it is 9.

• second rank: 0

Since there is no digit but zeros in the second rank, we write one zero beneath the line, after the 9 that we wrote for the digits of the first rank.

• third rank: 0

Since the sum of the third rank is ten, we write 1 beneath the lines of the numbers in the following fourth rank and another zero in the third [rank] beneath the line, next to the zero that we wrote after the 9.

• fourth-fifth ranks: 13

The sum of the fourth rank is thirteen, therefore we write 1 for the ten in the fifth rank; we place the rest, which is three, beneath the line next to the last zero and after the three we write the 1, from which we have formed the fifth rank.

Chapter Four – Subtraction of a Small Number from a Greater Number

• We consider all the digits, in whichever rank they are, as if they are units.

• We start from the digit in the last [= highest] rank of the number that is in the bottom line and subtract it from the corresponding digit of the number that is in the upper line.

• We write the remainder above it, if something remains.

• If there is no [remainder], we subtract it by writing a zero above it.

• Then, we proceed with all the other ranks as the procedure in this rank, moving back from rank to rank until they are complete.

• If there is a digit in one of the ranks of the number in the bottom line [= subtrahend] that is greater than the corresponding digit in the upper line [= minuend], there is necessarily one digit in the number of the upper line that follows the [digit] that is in the rank in which we stand.

We do it in this way: we subtract 1 from the digit that follows this rank and write above it what remains from the digit from which we took the 1. If nothing remains from it, as there was only 1 there, we erase it.

It is known that the 1 that we have is worth ten in relation to the preceding rank, hence, we consider the 1 as ten and add to this ten the number of the digit in the rank of the upper number, on which we stand. Now we have enough reserve to subtract the bottom digit from all this and we write the remainder above the corresponding upper digit.

• We proceed like this until we reach the first rank and what remains above the line of the upper number is the remainder of the subtraction.

• Whenever we cannot subtract the bottom digit from the upper [digit], we give it supplement from the next rank by the aforesaid way, even if there are zeros separating between it and the following digit, or if we have to subtract the bottom digit from one zero that corresponds it in the upper line.

We shift back by this way: we subtract 1 from the closest digit and write the remainder above it, if something remains from it, then we place this 1 above the preceding zero, so it becomes ten, we subtract 1 from it and 9 remains there above the zero. We place the 1 above the further preceding zero, so it is ten, we subtract 1 from it and nine remains above the zero. So on, we proceed until reaching the digit or the zero of the rank on which we stand, then we do as the aforementioned rule.

• Example: we wish to subtract ninety-nine from five thousand eighty-three.

We write the two lines of the numbers according to this diagram: the greater number in the upper line and the smaller number in the bottom line correspondingly.

4 9 9 1  5 0 8 3 9 2

• Since the digit in the [highest] rank of the bottom line is greater that the corresponding digit in the upper line, we do as the aforesaid way: we subtract 1 from the last digit of the upper line, which is 5 that is next to the digit in the upper rank, on which we stand, after the zero, and 4 remains instead of the 5.

• The digit becomes ten when we place it above the zero; we subtract one from the ten and nine remains.

• We place this one above the rank, on which we stand, which is 8, the total is eighteen. We subtract from it the last digit of the bottom line, which is 9; nine remains. We write it in its position above the 8.

• Then, we subtract the 2 of the bottom line from the corresponding 3 in the upper line and 1 remains. We write it above it.

Chapter Five – Multiplication of Units by Themselves or by Other Units and Multiplication of any Number by Itself or by Other

• For example: three times four

• When we say three times three

• We say also: ten times twenty

• Or ten times ten

Multiplication of Units by Units

• Two times one is two

• Three times [one] is three

• Likewise, all the other numbers that are multiplied by one do not change or duplicated.

• Two by two is four

• Two by three is six

• Two by four is eight

• Two by five is ten

• Two by six is twelve

• Two by seven is fourteen

• Two by eight is sixteen

• Two by nine is eighteen

• Three by three is nine

• Three by four is twelve

• Three by five is fifteen

• Three by six is eighteen

• Three by seven is twenty-one

• Three by eight is twenty-four

• Three by nine is twenty-seven

• Four by four is sixteen

• Four by five is twenty

• Four by six is twenty-four

• Four by eight is twenty-eight

• Four by eight is thirty-two

• Four by nine is thirty-six

• Five by five is twenty-five

• Five by six is thirty

• Five by seven is thirty-five

• Five by eight is forty

• Five by nine is forty-five

• Six by six is thirty-six

• Six by seven is forty-two

• Six by eight is forty-eight

• Six by nine is fifty-four

• Seven by seven is forty-nine

• Seven by eight is fifty-six

• Seven by nine is sixty-three

• Eight by eight is sixty-four

• Eight by nine is seventy-two

• Nine by nine is eighty-one

Written Multiplication

• We write the two number that we wish to multiply by each other, line beneath line, rank beneath rank and draw a line under the two lines of the numbers.

• We start from the first digit of the upper line and multiply all the upper digits by all the bottom digits successively.

• We write all the products beneath the line in the appropriate ranks, digit by digit, as required.

• We always count how many ranks there are from the digit we multiply on the upper line to the multiplied digit of the bottom line, including these two digits:

• If the product of a digit by another is equal to tens, we write a digit [that mark] the number of the tens according to the counted number of the ranks.

• If it does not reach ten, we write the product on the rank that is one less than the [counted] number [of the ranks].

• If it is equal to units and tens, we write the number of the tens according to the number of the ranks between the multiplied digits and the units one rank before that.

Summing the interim multiples

the lines of the numbers

9 0 2 2 4 6

the interim products

1 3 5 4 1 2   8 6 4 8

the total product

2 2 1 8 9 2

• first rank: 2

• second rank: 9

• third rank: 8

• fourth rank: 1

• fifth rank: 2

• sixth rank: 2

The result: 221892

Chapter Six – Division of a Great Number by a Smaller Number

• The one who wants to do this, writes the line of the greater number first, then he leaves a space of a whole line and write the smaller number in another line, corresponding to the first column of the greater number, each rank corresponds to its similar.

• Thereafter, he looks and sees how many times he can subtract the number of the digit on the last rank in the line of the smaller number [= the divisor] from the digit on the last rank of the greater number [= the dividend] and he subtracts them from it.

However, he should be wise and see if there is a number left, from which he can subtract all the other ranks in the line of the smaller number that precede the last one as the number of times he subtracts this last rank from the last digit in the line of the greater number. If he sees that there will be no number left that will be enough for it, he does not subtract it so many times, but as he sees that is enough to do what is said.

• As the number of times of subtraction he writes a digit corresponding them in the middle between the two lines of the smaller and greater numbers.

• He counts from the last rank in the line of the greater number backwards according to the number of the ranks in the line of the smaller number and beneath the rank where the count ends he writes the number of times of subtraction.

• If there is anything left of the digit from the subtraction of the smaller number after he subtracted from it, he writes the remainder above it.

If nothing is left of it, he writes a zero above its.

• Then, he subtracts all the other digits in the line of the smaller number that precede the digit of the last rank, one by one, from the digits that he finds written in the line of the greater number, as much as the times of subtraction from the last digit.

He should be careful throughout to write what remains after the subtraction in its place

If nothing is left of it, he writes a zero above its.

• If he happens to have a digit in one rank of the line of the [smaller] number that he can not subtract properly from the digit written in the line of the greater number, but there are still digits after the rank, on which he stands, he assists it from digits of those ranks, since it is known that each rank is ten times greater than the preceding rank, as is known in the previous chapters. Hence, he takes from their number as needed and writes what remains in its place and zeros in places where nothing remains.

• After he completes the subtraction of all the digits in the line of the smaller number from the line of the greater number the same number of times, if that which is not divided yet is greater than the number of the line of the small number, he looks again, sees and measures how many times he can subtract the digit in the last rank of the line of the smaller number from the last digit of what is left in the line of the greater number and as the number of times he subtracts it from it so are the times he has to subtract the rest of the ranks in the line of the smaller number from what remains then in the line of the greater number.

Therefore, he should be careful first when looking for the value and evaluation to maintain his practice according to the rule and justice.

• As the number of times he subtract the digits that are in the line of the smaller number from the digits of the greater number, he writes a digit corresponding them in the middle between the two lines of the smaller and greater numbers.

• He starts to count from the line of the greater number, from the digit at which he began to look in the second time, according to the number of the ranks in the line of the smaller number and beneath the rank where the count ends he writes the number of times of subtraction.

He should be careful throughout not to forget to write what remains from the line of the greater number.

Or to write a zero above the empty place, where nothing is left.

• Afterwards, he examine if what is left in the line of the greater number is still greater than the number of the line of the smaller number. If it is so, he returns again and sees as we have done twice.

• He does it as many times until the number of what is left in the line of the greater number is smaller than the number in the line of the smaller number.

• If it happens in the beginning of the examination that the number of the last digit of the ranks in the line of the smaller number is greater than the last digit in the line of the greater number, so that we cannot subtract it from it even once, he should shift this whole digit one rank backwards and it becomes a number of tens in this rank, whereupon it will be enough to subtract the digit of the smaller number as many times from it.

He thinks, looks and measures how many times he can subtract it from there, so that the remainder will be enough to subtract from it the other digits in the line of the smaller number the same number of times.

• He always writes the number of times of subtraction in the middle between the two lines of the smaller and greater numbers as the number of the ranks of the smaller number.

• He starts to count their number from the digit that is in the rank at which he began to look for the subtraction in the line of the greater number.

• We wish to divide one hundred and twenty-five by eleven.

We write the two lines of the numbers according to this diagram:

We write first the greater number, then we leave a space and write the smaller number in another line correspondingly.

0   0 1 4 1 2 5   1 1   1 1

• When we look and take the 1 that is in the last rank of the line of the smaller number and the 1 that is in the last rank of the line of the greater number, we know clearly that we can subtract one from the other only once.

Since the ranks of the line of the smaller number are two, we write 1 for this one time beneath the 2 that is in the line of the greater number, because it is second rank to the digit 1 of this same line, from which we start looking.

As nothing remains from this 1, we write a zero above it.

• Then, we take the 1 that in the first rank of the line of the smaller number that precedes the other, with which we have started, in writing and we subtract it once also from the 2 that is in the line of the greater number; 1 remains above it.

By that we have completed the subtraction of the digits in the line of the smaller number from the line of the greater number once of each equally.

When we look at the remainder in the line of the greater number, we see that it is fourteen and this number is more than the number of the line of the digits of the smaller number, which is only eleven.

• Because of this, we go back and take the last 1 from the ranks in the line of the smaller number and subtract it once from the 1 that remains above the 2 in the line of the greater number. Then we write zero above it.

For this subtraction one time, we write 1 beneath the 5, which is the first rank in the line of the greater number, since it is second to the digit, from which we started looking at the subtraction.

• Then, we subtract the first 1 that is in the line of the smaller number once also from the 5, which is the first rank in the line of the greater number; 4 remains above the 5.

Now it is visible and clear to the eye that what remains in the line of the greater number is much smaller than the number in the line of the smaller number.

The number of the middle line that we have formed from the number of the times of subtraction is eleven.

What remains in the line of the greater number is four.

Therefore, we can say that we have found that the smaller number is eleven times in the greater number plus a remainder of four.

• We wish to divide one hundred by nine.

We write them according to this diagram:

0   0 1 1 1 0 0   1 1     9

[Illustration of the procedure:]

				0
		01		011
100		100		100
		1		11
9		9		9

• When we take the 9, which is the smaller number, we see that we cannot subtract it from the 1 that is in the third rank of the greater number.

Therefore, we shift the 1 to the zero that precedes it in writing, so it equals ten and from this ten we subtract 9; 1 remains above this zero.

We write 1 beneath it, for the one time that we have subtracted the 9 from the ten, as it is the first rank of the beginning of looking at the subtraction and the smaller number has only one rank.

We write a zero above the 1 that we have shifted from its position.

• We look again to subtract the 9, which is the smaller number, from the 1 that is left above the zero in the line of the greater number. We should shift it to the other similar zero, so it equals ten.

When we subtract from it 9 once, 1 remains above the first zero in the line if the greater number.

For this one time that we have subtracted the 9 from the ten, we write 1 beneath the first zero, as it is first to the looking at the subtraction.

By this the division is complete and we have found that there are eleven times nine in one hundred and one is added to them.

• We wish to divide one hundred by twelve.

We write them according to this diagram:

0   0 2 4 1 0 0     8   1 2

[Illustration of the procedure:]

				0
		02		024
100		100		100
		8		8
12		12		12

• We wish to divide nine hundred and one by thirty-two.

We write them according to this diagram:

0 0   2 2   3 6 5 9 0 1   2 8   3 2

• We wish to divide eight hundred and ninety-one by forty.

We write them according to this diagram:

0 1   8 9 1   2 2   4 0

[Illustration of the procedure:]

		0		01
891		891		891
		2		22
40		40		40

• We wish to divide three hundred and twenty-one by nine.

We write them according to this diagram:

0   0 5 6 3 2 1   3 5     9

[Illustration of the procedure:]

				0
		05		056
321		321		321
		3		35
9		9		9

• We wish to divide five hundred and eighty-three thousand, six hundred and ninety-six by seven hundred and sixty-four.

0 0       0 3 2       4 6 0 0     5 8 2 1   0 9 1 8 5 0 5 8 3 6 9 6       7 6 4       7 6 4

• We wish to divide five hundred and eighty-three thousand, six hundred and ninety-six by one thousand and eighty.

We write them according to this diagram:

0         0 4 0 4     5 8 3 6 9 6       5 4 0     1 0 8 0

והנשאר על אותיות הטור העליון עולה ארבע מאות ותשעים ושש

Methods of Checking - Multiplication, Division, Addition, Subtraction

• Checking the result of division: multiplication

• Checking the result of multiplication: division

• Checking the result of addition: subtraction

• Checking the result of subtraction: addition

Chapter Seven – Roots

One versus the integers

• For all integers - the product of an integer by itself is greater than the integer itself

Except for one - one is not affected or changing when multiplied by itself

Therefore it is a square as well as the root of itself

• Another advantage of one over all other numbers

For all other numbers - the sum of the preceding number and the succeeding number is equal to twice the middle number

One does not have a preceding extreme, only subsequent extreme - its single extreme is equal to its double

square numbers

• The product of a square by a square is a square

• Square ranks and non-square ranks

• The first number of the units - the number one - is a square number

• The first number of the tens - the number ten - is not a square number

• The first number of the hundreds - the number one hundred - is a square number

• The first number of the thousands - the number one thousand - is not a square number

And so on

Extracting roots - written procedure

0     1 0 0 2 2 5   2 5

[Illustration of the procedure:]

				0		0
		1		100		100
225		225		225		225
		12		25		15

0     9 2 5   6 0

[Illustration of the procedure:]

		0		0		0
925		925		925		925
		36		60		30

0       2 0     6 1 0 7 0 5 6   1 6 4     8

[Illustration of the procedure:]

				0		0
				20		20
		06		0610		0610
7056		7056		7056		7056
		86		164		164
		1				8

0 3       3 8 6     7 6 5 4 3   4         1 4

[Illustration of the procedure:]

				03
		3		386
76543		76543		76543
		24		47
				14

3 6   0 4 0 7 3 6 4 3   5 4 6

[Illustration of the procedure:]

	36		36
	0407		0407
	3643		3643
	546		276

0           3 6       0 9 9 0     5 8 3 6 9 6     4         1 1 2

[Illustration of the procedure:]

				0
				36
		09		0990
583696		583696		583696
		74		746
		1		112

0       2   1 0 6 0 9 6 1 5 2 4

[Illustration of the procedure:]

	0 0		36
	2 10		2 10
	6096		6096
	1524		764

0 0       0 1 6 0 0 0 8 2 4 4 6 4   9 1 8 0 8

0     1 0 0   1 0

[Illustration of the procedure:]

		0
100		100
		1

0     4 0 0   2 0

[Illustration of the procedure:]

		0
400		400
		2

Chapter Eight – Proportions

1) Rule of Three

2) Rule of Four

3) Arithmetic proportion

• 1; 2; 3; 4

• 2; 4; 6

This kind is clear and explained - thus, there is no need to elaborate on that

4) Harmonic proportion

Chapter Nine – Fractions

Introduction

Operations with fractions

Multiplication of fractions

Introduction

the common denominator of the fractions is

Multiplication of fractions by fractions

• We wish to multiply four fifths by four fifths

• It is known that the fifth is derived from the five

• We wish to multiply two quarters by three quarters

• We wish to multiply three quarters by four fifths.

Check:

Multiplication of integers by fractions

Multiplication of integers and fractions by fractions of one type

נוסיף עליהם הארבע חמשיות ויהיו תשעה עשר חמשיות
נכפול אותם על השלש חמשיות יהיו חמשים ושבע חמשיות חמשית
והחמשים מהם הם שני שלמים והשבע חמשיות חמשית הנשארות הן הם חמשית אחת ושתי חמשיות חמשיות

Multiplication of integers and fractions by fractions of different types

Multiplication of integers and fractions by integers and fractions of one type

Multiplication of integers and fractions by integers and fractions of different types

Division of fractions

Division of fractions by fractions

Division of integers and fractions by integers and fractions

Addition of fractions

Subtraction of fractions

Chapter Ten – Approximating the Roots of Non-Square Numbers

Introduction: square numbers and non-square numbers

Sexagesimal fractions

• The astronomers divided the zodiac into 12 zodiacal signs

The reason for dividing by 12: twelve is the smallest number that has numerous fractions [= divisors, in modern terminology]:

• Each zodiacal sign was divided into 30 degrees

The reason for dividing by 30: thirty is the smallest number that has as many fractions [= divisors, in modern terminology] as it has:

• Each degree was divided into 60 minutes

The fractions [= divisors] of 60:

• Each minute was divided into 60 seconds

• Each second was divided into 60 thirds

• Each third was divided into 60 fourths

Multiplication of Sexagesimal Fractions

Written procedure

Extracting Roots of Deaf Numbers

1) First approximation:

2) Second approximation:

3) Third approximation - repeating the procedure of the second approximation - the most accurate approximation

1) First approximation:

Further approximations are as in the first case

• We wish to [know] the measure of each side of the square that is five.

• We wish to know the measure of the sides of the square that is seven.

• We wish to know the measure of the square that is six.

Chapter Eleven – Word Problems

• Multiplication of a number by itself

• Multiplication of units and tens by units and tens
  

Word Problems

• Question: we summed all the successive numbers from one to twenty and it is the sum. How much is the sum?

• We want to know how much are the numbers summed up up to eleven.

There are other ways, but the above is the easiest and the right way

• Reverse question: the sum of the successive numbers starting from one is 210. What is the last number of the summed [numbers]?

איזהו המחובר האחרון מהמחוברים

If there is no number such the - the calculation is mistaken

• Question: a cane 5 cubits tall, is standing next to a wall of the same height.
  

If we shift its [top] two cubits down from the top of the wall, so that it will stand on a slope, how far will be the bottom end of the cane from the foot of the wall?

אם נשפיל אותה מראש הכותל אמתים כדי התעמד בשיפוע כמה הרחיק ראש הקנה התחתון מיסוד הכותל

If the distance is a non-square number - using an approximation

• Question: I gave a messenger 30 dinar and one pašuṭ and ordered him to hire workers as much as his money allows, so that the payment of the one equals the payment of his friend, there will be no worker whose payment is one pašuṭ, and there will be no fractions in one's payment.
  

We want to know how many workers he could hire

נרצה לדעת כמה פועלי' יוכל לשכור

pešuṭim

19 workers for at 19 pešuṭim each

• Question: a wall collapsed. We rebuild it with an extension, so that it will be higher than what it was by half the size it had at the beginning, and its sixth, and ninth. With the whole extension, its height was 50 cubits. How much was its original height?
  

False Position:

Rule of Four:

cubits

נחלקם על האמצעי הנודע שהוא מאה ותשעים ושנים נמצאנו שם שמנה ועשרים פעמים וישארו מהם עשרים וארבעה חלקים שלא נתחלקו והמה חלקים ממאה ותשעים ושנים בשלם אשר חלקנו עליו
ועל כן נוכל להשיב כי מדת גבהות החומה בראשונה היו שמנה ועשרים אמות ועשרים וארבעה חלקים ממאה ותשעים ושנים במאה

Check:

• Question: the town wall was 100 cubits high. Its third and its quarter had collapsed.
  

How high is what remains

כמה גובה הנשאר

False Position:

Rule of Four:

cubits

נחלקם על האמצעי הנודע נמצאנו שם ארבעים ואחד פעמים ונשארו מהם שמנה שלה נתחלקו שהם חלקים משנים עשר באמה שהם שתי שלישיותיה

Check:

• Question: the landlord collected his grain and gave a heave offering by law from what he collected at first.
  

Afterwards he gave the first tithe from what remained and from what remained then, he gave a second tithe, and he was left with 40 measures of grain.

How much was the grain at first?

כמה היה הכרי מתחלה

offering by law:

first tithe:

False Position -

second tithe:

after the offering by law:

after the first tithe:

after the second tithe:

Rule of Four:

the amount of grain at first:

נחלקם על החשבון הראשון הנודע נמצאנו שם ששים ושתים פעמי' ונשארו שם שלא נתחלקו שלשת אלפים ותשע מאות ועשרים ושנים והם חלקים משלשת אלפים ותשע מאות וששים ותשעה אשר חלקנו עליו וכזה היה סכום המדות אשר היו בכדי כשהתחיל בעל הבית לתרום

• Question: Reuven demands from Shimon a hundred measures, which he says he owes him according to an oral [agreement].
  

Shimon says: I do not owe you a hundred measures, but as much as I owe you, with the same amount, and one-half of it, and a quarter of it, plus one will make a hundred.

How much did he admit he owes him?

ויאמר אליו איני חייב לך מאה מנה אבל כאותם שאני חייב לך ואחדים כמותם ומחציתם ורביעיתם ועם אחד יהיו מאה
נרצה לדעת כמה הודה לו מתביעתו

False Position:

Rule of Four:

the debt:

חלקנום על האמצעי הנודע שהוא עשרים ושנים ויצא בחילוק ששה ושלשים וככה הוא מספר המינים שהודה שחייב לו

• Question: a silversmith has sold to a moneychanger a silver chain that is worth three dinar of one coin or five dinar of another coin or seven dinar of yet another [coin].
  

The silversmith asked the moneychanger to pay him its price with these three coins an equal amount of each.

How much is this amount?

ושאל הצורף לשולחני שיתן לו בדמיו משלש המטבעות האלה מכל אחד בחלק שוה
ונבקש לדעת מספר החלק ההוא

Common Denominator:

dinar

Check:

parts of the coin of 5:

parts of the coin of 7:

dinar

נחלקם על שבעים ואחד שהם חלקי הדינר השלם ונמצאם שם שלשה פעמ' והנם שלשה והם דינרי' שלמים

• Question: a third, a fifth, and a seventh are summed together, how much is [their sum] in relation to the whole?
  

Common Denominator:

• Question: a buyer [bought] four fifths of a liṭra for one pašuṭ, and sold his possession at one pašuṭ for five ninths of a liṭra.
  

He earned 11 pešuṭim

How much was [his] money?

כמה היה הממון

Common Denominator:

the amount of money:

• If he earned 22 pešuṭim
  

• If he earned 33 pešuṭim
  

• Question: a seller bought 20 liṭra of peanut for 20 dinar.
  

Then he went and sold 10 liṭra of them at one dinar for five quarters of a liṭra. It turned out that he lost in doing so.

Afterwards many buyers came and overcharged and he sold the remaining 10 liṭra at one dinar for three quarters of a liṭra.

Did he gain or lose?

והלך ומכר מהן עשרה ליטרי' לערך חמש רביעיות ליטר' בדינר ונמצא שהוא מפסיד בזה
ואחר כן באו הרבה קונים והפקיעו השערים ומכר העשרה ליטרי הנשארות לערך שלש רביעיות ליטר' בדינר
ועתה בא אלינו לשאול אם הרויח או הפסיד או אם יצא הפסדו בשכרו

number of dinar for 10 liṭra at ⅘ for dinar: dinar

number of dinar for 10 liṭra at ¾ for dinar: dinar

total number of dinar: = 21 dinar and 4 pešuṭim

the profit: = 16 pešuṭim

• Question: three invested 46 dinar - one contributed 12 dinar, the second contributed 15 dinar, the third contributed 19 dinar, and they earned together 20 dinar. How much should each one take [from the profit]?
  

Rule of Four:

• Question: one hired Reuven and Shimon for 10 days to do a work for him any one of them in turns so that the work will not cease.
  

He agreed with Reuven that if he will do the work alone the whole 10 days he would pay him 2 dinar and to Shimon he said that if he will do the work alone the whole days he would pay him 5 dinar.

What they did? They did the work together so that when one was tired his friend replaced him and did the work while the other was resting and if the second was tired, the first returned to his work and the second was resting.

Each one wrote how many days, or parts of days, he worked.

They did so the whole ten days, and then when they came to the employer he paid both of them and gave each of them money equally.

How much money did they receive and how many days did each of them work?

והתנה עם ראובן שאם יעסוק הוא במלאכה לבדו כל העשרה ימים שיתן לו ב' דינרים
ולשמעון אמר שאם יעסוק הוא לבדו במלאכה כל מספר הימים ההם שיתן לו ה' דינרים
מה עשו החזיקו שניהם במלאכה והוא שובת היה עיף השני והראשון חוזר למלאכתו והשני שובת וכל אחד ואחד כתב כמה ימים או חלקי ימים עבד
ועשו זה כל העשרה ימי'
וכשבאו אל השוכר פרע את שניהם ונתן להם מעות לכל אחד בשוה ונרצ' לדעת כמה מעות נטלו וכמה ימי עבודת כל אחד ואחד

the total number of days that both work for one dinar:

dinar for each

Reuven must work 5 days for one dinar

Rule of Four:

days for dinar

Reuven's working days: days

Shimon must work 2 days for one dinar

Rule of Four:

days for dinar

Shimon's working days:

Check: days

• Question: I hired a messenger for 13 dinar and we agreed that he will walk for me from now on for 20 days 11 parsa [1 parsa = ca. 4 kilometers] a day, but the messenger embezzled or got injured and walked only 5 parsa a day for 7 days.
  

How much should his payment be according to the terms?

והשליח מעל או שנאנס ולא הלך כי אם חמש פרסאות בכל יום עד שבעה ימים
נרצה לידע כמה משפט דמי שכירותו לפי התנאי

If he would have walked 5 parsa the whole 20 days:

Rule of Four:

dinar

Since he walked 5 parsa for 7 days only:

Rule of Four:

גם כן נכפול אותו על הי' חלקים יהיו שבעים חלקים
וכאשר אנו צריכים לחלק העולה מהכפלה הזאת על האמצעי הנודע נשיב הכל ממתכונת אחת ונעשה מהל"ה חלקים מי"א ויהיו שפ"ה
נחבר אליהם הע' חלקים שגם הם המה חלקים מי"א ויהיו תנ"ה
ואחרי כן נשיב גם כן האמצעי הנודע שהוא עשרים חלקים מי"א ויהיו ר"כ
נחלק תנ"ה עליהם נמצאם שם פעמים וישארו ט"ו שלא נתחלקו והם שלש רביעיות חלק אחד מאחד עשר חלקים מר"ך שהוא השלם אשר חלקנו עליו

dinar

• Question: Jacob's four wives married him on the same day. On that day, he prepared for each of them a ketuba [= Jewish marriage contract] according the Jewish law.
  

The name of the one is Leah and her ketuba amount is 4000 zehuvim [= golden coins];

the name of the second is Zilpah and he ketuba amount is 3000 zehuvim;

the name of the third is Rachel and her ketuba amount is 2000 zehuvim;

and the name of the fourth is Bilhah and her ketuba amount is 1000 zehuvim.

Later Jacob died and nothing was left of his property but 4000 zehuvim.

The widows came to the court in order to divide the money that remained between them

שם האחת לאה וכתובתה ארבעת אלפים זהובים
ושם השנית זלפה וכתובתה שלשת אלפים זהובים
ושם השלישית רחל וכתובתה אלפים זהובים
ושם הרביעית בלהה וכתובתה אלף זהובים
לימים מת יעקב ולא נשאר מנכסיו כי אם ארבעת אלפים זהובים באו האלמנות לבית דין לחלוק להן הממון הנשאר

• The division according the sages of Israel:

Bilhah:

Rachel:

Zilpah:

Leah:

Check: