Difference between revisions of "מלאכת המספר"

Introduction

Table of content

In the first book we discuss the seven types of numerical [operations].

In the second book we discuss the methods, ratios, problems and answers needed in this science.

In the third book we discuss some techniques and premises common to arithmetic and geometry.

• The first book

It is divided into four sections:

Section one: the definition of arithmetic, the definition of number, the units, a few premises needed, and the number of the types of numerical [operations].

Section two: we discuss the six types of numerical [operations] with integers.

Section three: we discuss the seven types of numerical [operations] with fractions and the multiplication techniques of fractions in astrology [= sexagesimal fractions].

Section four: we talk about guiding ways for finding roots of square and cube numbers.

• Section one of the first book: the definition of arithmetic, the definition of number, the units, a few premises needed, and the number of the types of numerical [operations].

• Section two.

It is divided into six chapters:

• Chapter one: we talk in it about the first type of numerical [operations], which is addition.

• Chapter two: the second type, which is subtraction.

• Chapter three: the third type, which is doubling, and general ways of knowing integers.

• Chapter four: the fourth type, which is halving.

• Chapter five: the fifth type, which is multiplication and some of its properties.

• Chapter six: the sixth type, which is division.

• Section three of book one.

It is divided into eight chapters:

• Chapter one: guiding methods of the writing fractions, their definition, and their arrangement.

• Chapter two: addition of fractions.

• Chapter three: subtraction of fractions.

• Chapter four: doubling fractions.

• Chapter five: halving fractions.

• Chapter six: multiplication of fractions.

• Chapter seven: methods of multiplication of fractions in astrology [= sexagesimal fractions].

• Chapter eight: division of fraction.

• Section four of the first book.

It is divided into four chapters:

• Chapter one: giving guiding ways for finding roots of square numbers or approximate roots of non-square numbers.

• Chapter two: finding the roots of fractions alone or fractions and integers together.

• Chapter three: giving one inclusive method for finding the roots of numbers by adding zeros.

• Chapter four: guiding ways for finding roots of cube numbers or approximate roots of non-cube numbers.

• Book Two

It is divided into two sections:

The section one: we talk about general methods and proportions of this science.

The section two: we discuss some theoretical and practical problems and guiding answers of this science.

• Section one of the second book.

It is divided into seven sections:

• Chapter one: proportions of integers.

• Chapter two: guiding ways for finding proportions of fractions.

• Chapter three: guiding ways for finding the numerator in the proportions of fractions.

• Chapter four: giving a general example for all the teaching methods of the denominator and the numerator in the context of knowing the ratios of fractions.

• Chapter five: knowing the ratio of the four proportional numbers [= the rule of three] in the two sciences - arithmetic and geometry.

• Chapter six: knowing the ratio of every three proportional numbers [= proportional triad].

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

• The second section of the second book.

It is divided into seven chapters:

• The chapter one: knowing the exchange of measurements, weights, liquid measures and currencies according to the change of places.

• The chapter two: knowing the relation of two numbers that have the property that if we subtract one from the greater number and add it to the smaller number, they become equal; and if if vice versa, the greater becomes double the smaller, or more if we wish.

• Chapter three: the relation of two numbers, such that if the greater gives one to the smaller, the smaller becomes double the greater; and if vice versa, the greater becomes three times [the smaller].

• Chapter four: knowing the whole, whichever it may be, from the sum of its two different parts.

• Chapter five: giving general ways for finding any unknown number by the method of double false position.

• Chapter six: the teaching of partnership.

• Chapter seven: some questions and answers.

• Book three: we talk in it about some principles of geometry.

It is divided into three sections:

Section one: knowing the measure of the line.

Section two: knowing the measure of the surface.

Section three: knowing the measure of the solid.

• Section one of book three.

It is divided into three chapters:

• Chapter one: some principles of geometry, definition of a line, and knowing the measure of the line as a height.

• Chapter two: knowing the measure of the line in the surface [= length].

• Chapter three: knowing the measure of the line as a depth.

• Section two of book three: knowing the measure of the surface.

It is divided into five chapters:

• Chapter one: knowing the area of the equilateral triangle.

• Chapter two: knowing the area of the isosceles triangle.

• Chapter three: knowing the area of the scalene triangle.

• Chapter four: knowing the area of the quadrilateral and the area of the square.

• Chapter five: knowing the measure of the circle according to the opinion of the sages.

• Section three of book three: in it one chapter on knowing the volume of any solid.

Book One: Numbers

Section One of the First Book: the definition of arithmetic, the definition of number, the units, a few premises needed and the number of the types of numerical [operations].

• Definition of arithmetic: arithmetic is a science that teaches to count many units, their differences and properties that are easily applied by memory.

• Definition of unit: unit is a foundation and the first part of the number, every number consists of it, but it is apart from every number.

For, 2 or 3 cannot be conceived without 1.

As 2 is nothing but the double of 1.

And 3 is nothing but three times 1.

But, 1 can be conceived without the 2 or 3 being conceived.

• Definition of number: the number is defined as a sum of units.

The Positional Decimal System

The first numeral, or digit, as you wish to call it, indicates one; the second indicates two; the third - three; and so on until 9.

The tenth is called sifra [= zero] and it is not worth anything in itself, but indicates a rank that designate a greater numerical value to the next digit.

• The first digit in the line is called the image of the units, because it indicates the units, whether it is the first of the nine [digits] or any of them.

• The second digit in the line is called the image of the tens.

• The third digit is called the image of the hundreds.

• The fourth is of the thousands.

• The fifth is of the tens of thousands.

• The sixth is of the hundreds of thousands.

• The seventh is of the thousands of thousands

• The eighth is of the tens thousands of thousands.

• The ninth is of the hundreds thousands of thousands.

• The tenth digit in the line is called the image of the thousands thousands of thousands, because it indicates them.

And so on from ten to ten.

• In a way that 1 in the place of the units equals one; in the place of the tens: ten; in the place of the hundreds: one hundred, in the place of the thousands: one thousand; and so on by the order from rank to rank.

• 2 in the [place of the] units equals two; in the place of the tens: twenty; in the place of the hundreds: two hundreds, in the place of the thousands: two thousands; and so on by the order from rank to rank.

• The units is any number that is less than 10.

• The tens is ant number equal to ten or tens.

• The consisting of both is any number that has units and tens together.

Section Two of Book One: [Integers]

Chapter One: The First Type of Numerical [Operations], which is Addition

• The units should be written corresponding to the units, the tens corresponding to the tens, the hundreds corresponding to the hundreds and so on by the order, every rank corresponding to its similar.

• Then all the numerals are summed as units.

• If it is units, we write it beneath the units.

• 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 units and tens together, we write the units beneath the units as stated, and the tens in the rank of the tens.

• As can be seen in this diagram:

5 2 4 3    8 9 6 2

1 4 2 0 5

Chapter Two: The Second Type [of Numerical Operations], which is Subtraction

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

As subtracting 6 from 6; nothing remains.

• If the upper digit is greater, we subtract what is beneath from the rank that is above and write the remainder.

As subtracting 5 from 6; 1 remains. We write 1 beneath.

• If the upper digit is less than the one that is beneath, we examine the bottom digit: how much is its complement to ten and we add it to the corresponding upper digit. The sum of these two is called the "remainder". We write it beneath the upper digit.

When we want to subtract the digit that follows it from its corresponding upper [digit], we should add 1 to the bottom digit that follows.

This procedure is necessary, since the preceding upper digit was less than the bottom [digit] and the addition of the 1 that we said is as if we subtract one from the digit that is next to the upper [digit] that is less than its corresponding bottom [digit].

• As can be seen in this diagram:

4 2 8 2 2 4 3 2

1 8 5 0

• You should know, as we said, that since the [first] upper digit is equal to the bottom [digit], when one is subtracted from the other, nothing remains, so we write zero.

• In the second rank, since the upper digit is greater than the bottom [digit], we see how much is the complement of the bottom to the upper. We know that it is 5, so we write 5 beneath.

• Then, in the third rank, since the bottom digit is greater than the upper [digit], we should know how much is its complement, i.e. of the bottom [digit], to ten. We know that it is 6. We add to it the upper digit, which is 2; the sum of both is 8 and this is the remainder. We write it beneath the 4.

• 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.

Chapter Three: The Third Type [of Numerical Operations], which is Doubling, and General Methods for Finding the Perfect Numbers

• If double any digit is less than ten, we write it.

• If its double is ten integer, we write 0 beneath and we are left with 1 to add it to double the digit that follows.

• If it is more than ten, we write the excess [over ten] and we shift the ten to the rank that follows, exactly as we did in the addition operation.

• As can be seen in this diagram:

5 3 7 2

1 0 7 4 4

As 7; or 31; or 3.

• Example: we take the first double, which is 2. Since its double minus 1 is 3, which is a prime number, we multiply the first double of this type, which is 2, by its double minus 1, which is 3; the result is six and it is a perfect number. This is the perfect number that is in the rank of units.

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

Chapter Four: The Fourth Type [of Numerical Operations], which is Halving

As writing 4 beneath 8.

• As this diagram:

2 6 2 1 4 4

1 3 1 0 7 2

• Another diagram:

1 0 4 8 8 7 6

5 2 4 4 3 8

Chapter Five: The Fifth Type [of Numerical Operations], which is Multiplication and Some of its Properties

• If the multiplier consists of one digit, we write one line [of the product] and we start from the units.

• If the multiplier consists of two digits, we write two lines, the first line starts with the units and the second starts with the tens.

• If the multiplier consists of three digits, we write three lines, the first starts with the units, the second with the tens, and the third starts with the hundreds.

• First, we multiply the digit of the units of the multiplier by the digit of the units of the multiplied. This multiplication yields either ten, or more, or less.

• If it is ten, we write 0 and keep the ten for the rank that follows, there its numerical value becomes one.

• If it is more than ten, we write the excess over 10, and we shift the ten to the close rank that follows, there its numerical value becomes one.

• If it is less than ten, we write it in its place.

The same should be done with each of the digits of the multiplier by the digits of the multiplied.

So, if the digits of the multiplier are one, there is one line [of the product]; if two, there are two lines; and if more, there are more.

• Then, we sum up all the lines that we wrote and this is the product we wanted of the two numbers.

1 2 3 4       1 2

2 4 6 8 1 2 3 4

1 4 8 0 8

1 2 3 4       5

6 1 7 0

• You already know that the product of the units by units is written in the [rank of] units.

• The product, from which the tens are generated, is this:

• The product of the units of the multiplier by the tens of the multiplied.

• Also, the tens of the multiplier by the units of the multiplied.

These two products alone generate the tens.

• The products, from which the hundreds are generated, are three:

• The first is the units of the multiplier by the hundreds of the multiplied.

• The second is the tens of the multiplier by the tens of the multiplied.

• The third product is the hundreds of the multiplier by the units of the multiplied.

The sum of these three [products] should be written in the rank of hundreds.

• The products, from which the thousands are generated, are four:

• The first is the product of the units of the multiplier by the thousands of the multiplied.

• The second is the tens of the multiplier by the hundreds of the multiplied.

• The this is the product of the hundreds of the multiplier by the tens of the multiplied.

• The fourth is the thousands of the multiplier by the units of the multiplied.

The sum of these four products should be written in the rank of thousands.

• The position of the thousands is the fourth position, so we do 4 products.

• Likewise for the tens of thousands, which is the fifth position, so 5 products are needed.

Sums

• The first way is as we stated, and the way to know it is that we examine the last number, if it is even or odd.

• If it is even number, we take its half, multiply it by the last number plus 1, and we get the sum of all of them.

Example: we wish to know the sum from one 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.

• If the last number is odd, we take its half plus one half, multiply it by the last number, and we get the sum of all of them.

Example: we wish to know the sum from one to 13.

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.

This way even if the numbers are very many.

• The second way is that if the numbers are all even, starting from 2, then 4, then 6, and so on, even if they are very many in this succession.

To know the sum of all of them, we take half the last even number, multiply it by its other half plus one and the result is the sum of all the even numbers.

Example of this: if the last even number is 12.

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.

• If the numbers are all odd, starting from 1, then 3, then 5, then 7, and so on, even if they are very many in this succession.

To know the sum of all of them, we take half the last number plus one half, multiply it by itself and the result is the sum of all of them.

Example of this: we wish to know the sum of all the odd numbers from one to 15.

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.

Do this way even if the numbers are very many.

Shortcuts

• The first is when we wish to know the product of any two numbers that are less than ten, we do as follows: we look at the greater number, by how much is it less than ten, and as much as the greater is less than 10, that many times we subtract the smaller number from its product by ten.

• Example: we wish to know 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.

This is the way for others.

• As we arranged our procedure according to the greater number, so we can arrange it also according to the smaller number.

• 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.

All is the same procedure, but it is easier when we arrange our procedure according to the greater number.

• If it is in the rank of tens, when we wish to multiply any number by ten, we add zero to it.

• Example: if we want to know how much is 10 times 12, we add 0 to 12; it is 120, and in the positional system they are written as 120.

If the product is 20 by any number, we multiply the multiplied number by 2, then add 0.

• Example: 20 times 5. We double the 5; it is 10. We add zero; it is 100.

The same for the others.

If we want to multiply thirty by any number, we triple it, then add 0 to it.

• Example: 30 times 20. We triple 20; it is 60. We add zero; it is 600 and this is its shape by the positional system: 600.

This way for all the tens.

• If we wish to multiply any number by a hundred, we add two zeros to it, so it is multiplied by a hundred.

• If we wish to multiply any number by a thousand, we add three zeros to it.

And so on, we always add one zero for each rank.

Chapter Six: The Sixth Type which is Division

• Example: we wish to divide 144 by 8.

Do as this diagram:

0   0 6 0 1 4 4

1 8

8

As you see in this diagram:

The 1 is a ten and the 4 is four; their sum is 14. We say: how many times is 8 in 14? We see that it is found once and 6 remains.

We write the one as the quotient, beneath the 4; and we write the remaining 6 above the 4 of the dividend.

Since only 6 remains from the 14, we write 0 above the 1 and the remaining 6 are tens for what follows.

So, there is still 64 remains to divide. We divide it by 8; it is 8 in the quotient.

We write it beneath the first 4 of the dividend.

We say: 8 times 8 is 64. When we subtract it from the dividend, which is 64, nothing remains.

Hence, we write 0 above the 6 and 0 above the 4.

The example: if the divisor is 8 and what remains to divide is 3, we take the ratio of 3 to 8; the ratio is 3-eighths.

Or, we convert every one of the remaining three into 8 parts; they are 24 parts. We divide them by the divisor that is 8; each part receives 3-eighths.

• As can be seen in this diagram:

0 0 0   1 2 3 0 9 8 7 6

8 2 3

1 2

Suppose we wish to divide 9876 into 12 parts.

• 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.

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.

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.

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.

• 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.

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.

Therefore, we write 2 as the quotient.

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, 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.

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

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 check also if the 2, which is the right [digit] of the divisor, is found so many times in the right [digit] of the dividend. We find that it is found [that mach].

Therefore, we write 3 as the quotient.

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

We multiply also the 3 of the [quotient] by the right [digit] of the divisor, which is 2; the product is 6. We subtract it from the 6, which is the right [digit] of the dividend; 0 remains. We write 0 above the 6.

00		000
123		1230
9876		9876
823		823
12		12

So, all is divided and the quotient is 823.

Section Three of Book One: [Fractions]

Chapter One: Guiding Methods of the Writing Fractions, their Definition, and their Arrangement

As a half, or a third, or a quarter, etc.

Example: when we say 2-thirds, or 3-quarters.

The 2 and 3 are discontinuous quantity, because they state the number and the number is a discontinuous quantity.

The thirds and quarters are continuous quantity, because they state the part, or parts of whichever whole, so they indicate a continuous quantity.

Example: if we wish to write 2-thirds.

We write 2; beneath the 2 a line; and beneath the line 3. As is shown in this diagram:

If you wish to write a half, write it as follows:

The thirds - write like this:

The quarters - like this:

Three-quarters - we write as follows:

Four-fifths- we write as follows:

Chapter Two: Addition of Fractions

• As the one who wants to add or to know the sum of a half, a third, and a quarter.

The number in which all the fractions written in the above diagram are found is 24. It is generated this way: we multiply 2 by 3; the result is 6, and all this, i.e. 6, by 4; it is 24.

In this numbers there are a half, a third, a quarter and it is as their whole.

Because, as one takes from the whole whichever part he wants, so we take whichever part we want from this number 24 that is as a whole.

So, we take the half, the third, and the quarter this way:

We start taking the half this way: we multiply the digit that is above the line of the half, which is 1, by the digit of 3 that is beneath the 1 indicating the third; it is 3. We multiply the 3 by 4 that is beneath the digit indicating the quarter; it is 12, which is a half of the number that we said.

Then, we take the third this way: we multiply the digit of the third that is above the line by the digit of the half that is beneath the line, which is 2; the product is 2. We multiply the 2 by 4; the result is 8, which is a third of this number.

Then, we multiply the 1 that is above the 4 by the 2 that is beneath the line indicating the half; it is 2. We multiply this 2 by 3 that is beneath the line indicating the third; the result is 6, which is a quarter of this number, i.e. the 24.

So, we have three numbers that are a half, a third, and a quarter, which are 12, 8, and 6; their total sum is 26.

The whole, to which the 26 is related, is 24. Therefore, we divide 26 by 24; the result is one integer and 2 remains that cannot be divided. We relate it to 24 and we find that it is one part of 12 of the whole. Hence, sum of the three aforementioned fractions is 1 integer and one part of 12 parts of the whole.

• Suppose we wish to sum up 2-thirds, 3-quarters, 4-fifths, and 5-sixths.

Their diagram is as follows:

The first thing that is needed is that we find the number that is the whole, i.e. in which all these fractions are found. We do that by multiplying all the digits that are beneath the lines like this:

First, we multiply 3 by 4; it is 12. We multiply the 12 by 5; the result is 60. We multiply 60 by 6; the result is 360. This is the number, in which all these fractions are found, and it is the whole that is also called the common denominator.

We should extract all the fractions from it this way:

First, we extract the first fraction in the said diagram, which is two-thirds, this way: we take the 2 that is above the 3 and multiply it by 4; it is 8. We multiply the 8 by 5 that is beneath 4; it is 40. We multiply 40 by 6 that is beneath the 5; the result is 240 and this number is called two-thirds of the common denominator, i.e. the same common denominator that we said is the whole.

Then, we extract the 3-quarters, this way: we take the 3 that is above the 4 and multiply it by all the bottom digits except for the digit that is beneath it, which is 4. As follow: 3 by 3, it is 9; 9 by 5, it is 45; 45 by 6, the result is 270 and this number indicates 3-quarters.

Then, we extract the 4-fifths, this way: we take the 4 that is above the 5 and multiply it by all the bottom digits except for the 5 that is beneath the 4. As follow: we multiply 4 by 3, it is 12; 12 by 4, it is 48; 48 by 6, the result is 288 and this number indicates 4-fifths.

Then, we extract the five-sixths by taking the 5 that is above the 6 and multiply it by all the bottom [digits] except for the 6 that is beneath it, as follows: we multiply 5 by 3, it is 15; 15 by 4, it is 60; 60 by 5, the result is 300 and this number indicates 5-sixths.

We extracted all the said fractions. Now we should sum up all these said fractions and we do this as follows: we take the numbers 240, 270, 288, and 300 and we sum up all; their sum is 1098 and this is the sum of all the said fractions.

We divide all this sum by 360 that is the common denominator, which is the whole as we said; the result is 3 integers and one part of 20 parts of a whole and this is the sum of all the said fractions. Here is its diagram:

• Example: you have already seen in the above diagram that 18 remains that cannot be divided as it is less than the divisor, which is 360. Therefore, we look for the ratio between 18 and 360 and we find that it is the ratio of 1 to 20, so we said above that is is 1 of 20 parts of the whole.

The way to find this ratio is easy: we take these two numbers - 18 above and 360 beneath, and we examine into how many parts the 18 can be divided. We divide the 360 into this number of parts.

• Example: as the 18 is divided into 3 parts and the quotient is 6, so we divide the 360 also into 3 parts and the result is 120.

We divide the 6 again into 3 parts; the result is 2. We divide the 120 into 3 also; the result is 40.

We divide the 2 again into half; the result is 1. We divide the 40 into half also; the result is 20. So, it is one [part] of 20 parts of the whole.

Chapter Three: Subtraction of Fractions

• As seen in this diagram: suppose we want to subtract from three-quarters of the whole two-thirds of the whole and we wish to know how much remains.

We arrange them as follows:

The example of this diagram: the 4 from one side by the 3 from the other side; the result is 12 and this is the number that we call the common denominator. It is called "common denominator", because the two numbers that are beneath the lines take part in it as said.

Now, we should extract 3-quarters of the common denominator this way: we multiply the 3 that is above the 4 by the 3 that is beneath the 2; the result is 9 and these are the 3-quarters of the common denominator, which is 12.

Then, we extract the 2-thirds this way: we multiply the 2 that is above the 3 by the 4 that is beneath the 3; the result is 8 and these are the two-thirds of the common denominator.

First we have found the common denominator, which is the whole.

Second, we have found its 3-quarters, which is 9.

Third, we have found its two-thirds, which is 8.

Since our intention is to subtract two-thirds from 3-quarters, as we have them, we subtract the 8, which is 2-thirds, from the 9, which is 3-quarters; one remains.

We should know what is the one that remains. In order to know this, we divide it by the divisor, or the common denominator, which is 12, then we know what it is. The division method is:

So, it is 1 [part] of 12 parts of the whole.

As seen in this diagram:

The first thing that should be extracted is the common denominator, which is the product of 3 by 12; it is 36.

From this we extract 2-thirds this way: we multiply the 2 that is above the 3 by the 12 that is beneath the 1; it is 24 and these are 2-thirds of the common denominator.

We extract also 1 [part] of 12 parts of the whole this way: we multiply the 1 that is above the 12 by the 3 that is beneath the 2; it is 3 and this is 1 [part] of 12 parts of the whole.

Now, we sum up the 3 with the 24; it is 27.

We divide the 27 by the common denominator, exactly as the way in the first type; it is 27 parts of 36 of the whole.

These parts should be 3-quarters of the whole. To know this, you should examine this diagram:

We do as follows: we multiply the numbers each by its opposite, i.e. 4 by 27 and 3 by 36. If the products are equal it is a proof that this number is 3-quarters of the whole.

Chapter Four: Doubling Fractions

For example: we wish to double 2-thirds, or 5-sixths, or 4-fifths, or any of the fractions we want.

• As the one who wants to double this number, which is 3-quarters.

We double the 3 that is above the line; the result is six. We divide it by 4 that is beneath the line; the result of division is 1 integer and a half, which is double the 3-quarters.

Chapter Five: Halving Fractions

• As the one who wants to halve one-quarter, whose form is this:

We double the 4 that is beneath; it is:

Chapter Six: Multiplication of Fractions

1) Integer or integers alone by fraction alone.

2) Integer or integers and fraction together by fraction.

3) Integer or integers and fraction together by integer or integers alone.

4) Integer or integers and fraction together by integer and fraction together.

5) Fraction by fraction alone

• Example: we wish to multiply 4 integers by 2-thirds.

We do as follows: we write two numbers on both sides this category.

denominator: First, we extract the denominator like this: we take the number that is beneath the line and it is the denominator. It is 3 in this diagram.

numerator: Then, we multiply the 4 integers by 2 that is above the line; it is 8.

We divide this 8 by the denominator that we said it is 3; the result is 2 integers and 2-thirds and this is the product of the said 4 integers by two-thirds.

• As the one who wants to multiply 4 and a half by two-thirds.

denominator: First, we extract the denominator as follows: we multiply the 2 that is beneath the 1 by the 3 that is beneath the 2; the result is 6 and this 6 is the denominator.

numerator: Then, we multiply the 4 integers by 2 that is next to them, beneath the 1; the result is 8. We add to it the 1 that is above the 2; it is 9 and it is the numerator of this side. We multiply the 9 by the 2 of the other side; the result is 18.

We divide this 18 by the denominator, which is 6; the quotient is 3 and this 3 is the product of 4 integers and a half by 2-thirds.

• As the one who wants to multiply 4 and a third by 6.

denominator: Since there is only one type of fractions here, the 3 that is beneath the 1 is the denominator.

numerator: After we write the method of the denominator, we multiply the 4 integers mentioned by 3 that is beneath the 1; it is 12. We add to it the 1 that is above the 3; it is 13. We multiply the 13 by the 6 integers mentioned; the result is 78.

We divide the 78 by 3, which is the said denominator; the result of division is 26 and this is the product of 4 integers and one third by 6 integers.

• As the one who wants to multiply this number:

Example: [we take] the 4 that is beneath the 3 and multiply it by the 2 that is beneath the 1; it is 8 and this is the denominator. We keep it by itself.

numerator: Then, we multiply the 4 integers by the 4 that is beneath the 3 next to them; the result is 16. We add to it the 3 that is above the 4 next to it; the result is 19. We keep it.

We turn to the other side and multiply the 2 integers by the 2 that is beneath the 1 next to them; the result is 4. We add to it the 1 that is above the 2 next to it; the result is 5. We keep it.

So, we have three things: the first is the denominator; the second is the integers of one side that we have converted to the type of the fractions that are next to them and they are 19; the third is the integers of the other side that we have converted to the type of the fractions that are next to them and they are 5.

Now, we multiply the 5 by the 19; the result is 95.

We divide it by the said denominator, which is 8; the result of division is 11 integers and 7-eighths and this is the product that we wanted.

• As the one who wants to multiply 2-thirds by 4-fifths.

denominator: We extract the denominator, which is the product of the digits that are beneath the lines that are 3 and 5; the result is 15. We keep it.

numerator: Then, we multiply the digits that are above the lines that are 2 and 4; the result is 8.

We divide the 8 by the denominator, which is 15. Since the divisor is greater than the dividend, we relate the dividend to the divisor; we find 8 parts of 15 of one integer and this is the product of 2-thirds and 4 fifths.

Chapter Seven: Multiplication of Fractions in Astrology [= Sexagesimal Fractions]

Example: if we multiply seconds by thirds, we say: the decimal value of the seconds is 2 and the decimal value of the thirds is 3; their sum is 5, so the result of their multiplication are fifths.

• Here you have an example of some of them: suppose we want to multiply 2 degrees, 24 minutes, and 43 seconds by 3 degrees, 3 minutes. and 8 seconds.

As seen in this diagram:

Example: if the result of multiplication are 124 seconds, we write 4 seconds in the column of the seconds, and we write the 120 seconds that are 2 times sixty that are 2 minutes in the column of the minutes.

Chapter Eight: Division of Fraction

1) Fraction by fraction.

2) Fraction by integer.

3) Fraction by integer and fraction.

4) Integer and fraction by integer and fraction.

5) Integer by fraction.

6) Integer and fraction by fraction.

7) Integer and fraction by integer.

8) Integer by integer and fraction.

• As the one who wants to divide 3-quarters by 2-thirds.

We write each one next to the other as this:

denominator: First of all, we extract the denominator, by that we multiply 2, which is above the 3, by 4, which is beneath the 3; the result is 8 and this is the denominator.

numerator: Then, we should extract the [numerator], by that we multiply the 3 that is above the 4 by the 3 that is beneath the 2; the result is 9 and this is the numerator.

Next, we divide the numerator by the denominator; the result is 1 and an eighth of a whole and this is the result of division of 3-quarters by 2-thirds that we wanted.

• Suppose we wish to divide 3-eighths by 2 integers.

As this diagram:

denominator: First, we extract the denominator as follows: we multiply the integer, which is 2, by 8, which is beneath the 3; the result is 16 and this is the denominator.

numerator: Then, we extract the numerator, by taking the 3 that is above the 8 and this is the numerator.

We divide this numerator, which is 3, by 16, which is the denominator; the result are 3 parts of 16 parts of one whole and this is what we wanted.

• As the one who wants to divide 2-thirds by 4 integers and a half.

We write each one as in this diagram:

denominator: We extract the denominator as follows: we multiply 4 by 2, which is beneath the 1; the result is 8. We add to it the 1 that is above the 2; it is 9. We multiply the 9 by 3 that is beneath the 2, next to the 1; the result is 27 and this is the denominator.

numerator: We extract the numerator, by that we multiply the 2 that is above the 3 by the 2 that is beneath the 1 of the other side; the result is 4 and this is the numerator.

We divide it by the denominator, which is 27; the result of division is 4 parts of 27 parts of the whole and this is what we wanted.

We write each one as said above.

• As the one who wants to divide 9 and a half by 3 and 4-fifths. Like this:

denominator: We extract the denominator as follows: we multiply 3 by 5, which is beneath the 4; the result is 15. We add to it the 4 that is above the 5; it is 19. We multiply it by 2 that is beneath the 1 in the other side; the result is 38 and this is the denominator. We keep it.

numerator: We extract the numerator by that we multiply 9 by 2 that is beneath the 1 next to it; the result is 18. We add also the 1 that is above the 2 next to it; the result is 19. We multiply this by 5 that is beneath the 4 of the other side; the result is 95 and this is the numerator.

Then, we divide the numerator by the said denominator; the result of division is 2 integers and a half and this is what we wanted.

We write them as said.

• As the one who wants to divide 12 by 4-ninths, like this:

denominator: In this category the denominator is always the digit that is above the line, so the denominator now is the 4 that is above the 9.

numerator: The numerator is the product of the 12 integers by the 9 that is beneath the 4; so the numerator is 108.

We divide it by 4, which is the denominator; it is 27 in the division we wanted.

We write each one as said above.

• Example: we wish to divide 4 and one-third by 5-sixths, like this:

denominator: We extract the denominator, by that we multiply 5, which is above the 6, by 3, which is beneath the 1 next to it; the result is 15 and this is the denominator.

numerator: Then, we extract the numerator by that we multiply 4 by 3 that is beneath the 1 next to it; the result is 12. We add the 1 that is above the 3 next to it; it is 13. We multiply this by 6 that is beneath the 5 of the other side; the result is 78 and this is the numerator.

We divide it by the denominator, which is 15; the result division is 5 integers and one-fifth and this is what we wanted.

We write them as said above.

• We wish to divide 3 and one-third by 4, like this:

denominator: First, we extract the denominator, by that we multiply 4 by 3, which is beneath the 1 in the other side; the result is 12 and this is the denominator in this category.

numerator: We extract the numerator by that we multiply the 3 integers by the 3 that is beneath the 1 next to it; it is 9. We add to it the 1 that is above the 3 next to it; the result is ten and this is the numerator.

We divide it by the denominator, which is 12; the result is 5-sixths and this is what we wanted.

We write each one as said above.

• Example: we wish to divide 50 by two and a half, like this:

denominator: First, we extract the denominator as follows: we multiply 2 by 2, which is beneath the 1 next to it; the result is 4. We add to it the 1 that is above the 2 next to it; the result is 5 and this is the denominator. We keep it.

numerator: Then, we extract the numerator by that we multiply the 50 by the 2 that is beneath the 1 of the other side; the result is 100 and this is the numerator.

We divide it by the denominator, which is 5; the result division is 20 for each one of the two integers of the denominator.

Example: in this category, when we divide 50 by 2 and a half; the result of division is 20 to the whole.

So, we say: if each of the integers receives 20, the half should receive ten, and according to this we divide the fifty.

• Example: you already know that the quotient of the last [example] of this type was 20 and the divisor was two and a half. So, if you multiply two and a half by 20, the result is fifty.

Section Four of the First Book: [Roots]

Chapter One: Giving Guiding Ways for Finding Roots of Square Numbers or Approximate Roots of Non-Square Numbers

The first is the root of a square.

The second is called the root of a cube.

So, 2 is the root of 4, because when it is multiplied by itself it generates four.

Such as 2 times 2; or 5 times 5; or 10 times 10; and so on for the others.

• Example: 4 that has a root is multiplied by 9 that has a root also yields 36, whose root is 6.

• Example: 18 is divided by 8; the result is [2 and a quarter]; its root is one and a half.

If we multiply 18 by 8, the result is 144 and the root of this number is 12, even though 18 and 8 do not have roots.

With the exception of the units, which is the first rank, that has a root that is not generated from other [ranks], but from its product by itself, because one time 1 is 1.

The second rank, which is the tens, does not have a root, because there is no number, whose product by itself generates it.

The third rank, which is the hundreds, has a root, because the product of ten by itself generates 100.

The fourth rank, which is the [thousands], does not have a root, because there is no number, whose product by itself generates it.

The fifth rank, which is the tens of thousands, has a root, because the hundred multiplied by itself generates it, for 100 times 100 yields ten thousand.

The minutes, since they are in the first rank of fractions, do not have a root.

The seconds that are in the second rank of fractions have a root, because they are generated from the multiplication of the minutes by themselves.

The thirds that are in the third rank do not have a root, because they are not generated from the multiplication of others by themselves.

The fourths that are in the fourth rank have a root, because they are generated from the multiplication of the seconds by themselves.

As halves, thirds, fifths, sixths and their like.

As the product of a half by itself that generates a quarter, which is called a half of a half.

The product of a third by itself that generates a ninth, which is called a third of a third.

As the product of a fifth by itself that generates one [part] of 25, which is called a fifth of a fifth.

The tenth, the eleventh, the twelfth, the thirteenth, the fourteenth, and their like do not have a root, for they are fractions of first degree and they are not formed from others.

The tenth of a tenth, the eleventh of an eleventh, the twelfth of a twelfth, the thirteenth of a thirteenth, the fourteenth of a fourteenth, the fifteenth of a fifteenth, and their like have a root.

Such as a half of a half of a half, or a thirteenth of a thirteenth of a thirteenth, and all their like.

Written calculations

If it is odd, write beneath the last rank a number whose product by itself is a number that is equal to the number that is above it, or as close as possible, if an equal number cannot be found.

If [the number of] the ranks is even, one writes beneath the rank that precedes the last rank a number, such that when multiplied by itself yields a number that is equal to the number that is above it, or as close as possible, if a number equal to it cannot be found.

If nothing remains, we write zero above them.

If there is a remainder, what remains is written above the preceding digit, which is the right of the two [digits].

If there are no ranks left, then the first and the third [bottom] digits are the root of the upper number.

• Example: suppose we want to look for the root of a number, which is 5625.

0       2     0 7 0 0 5 6 2 5

7 4 5

Since the [number of] ranks is even, we should begin from the digit that precedes the last, which 6.

• We should write a number, such that when it is multiplied by itself, it yields the same number, or the smallest closest number, when no number is found that is equal to the one that is above, which is 56.

The number that is closest to 56 is 7, because when it is multiplied by itself, it yields 49.

When 49 is subtracted from the number that is above, which is 56, 7 remains.

We write the remaining 7 above the 6, which is one of the digits of 56; and above the 5 we write zero.

We double the 7 that we wrote below as a root; it is 14. We write the ten beneath the 7, erase the 7, and write the 4 in the following rank, beneath the 2.

• Then, beneath the next digit, we write a number, which is 5, such that when it is multiplied by the doubled and by itself, it erases, or consumes the one that is above, as follows:

When multiplying 5 by the one, which is ten of the doubled, the result is 5. When it is subtracted from what is above, which is 7, 2 remains above the 7.

Then, we multiply the mentioned 5 by 4; it is 20. When it is subtracted from what is above, which is 22, 2 remains.

We write 0 above the upper 2 that is tens for the 2 that follows.

Next, we multiply the 5 by itself; we get 25. When it is subtracted from what is above, which is 25, nothing remains.

We receive that the root of the number is 75, as seen in this diagram.

Since nothing remains, we write 0'0' above the 2 and the 5 and this is what we wanted.

Chapter Two: Finding the Roots of Fractions Alone or Fractions and Integers Together

• Example: if the fractions, whose root you are looking for, are quarters, their root is halves.

If they are ninths, their root is thirds.

• As the one who multiplies thirds by themselves that yield ninths, which is a number that is smaller than the thirds that is its root.

• So, in astrology, 3 minutes are not the root of 9 minutes, since the product of minutes by themselves is seconds that are of a different type.

• Example: if they are thirds that do not have a root, we convert them to ninths that have a root.

• Likewise in astrology, if they are minutes and we wish to extract their root, we convert them to seconds and then extract their root.

Chapter Three: Giving one inclusive Method for Finding the Roots of Numbers by Adding Zeros

• Example: if you wrote six zeros and you find that there are more than three digits in the root, which is half the six that is [the number of] the zeros, take the [exceeding ranks] as integers as said and multiply the remaining that are three ranks by sixty.

So, if the preceding product was minutes, the next [product] is seconds and the next product is thirds and so on, until only three zeros remain.

• Example: we wish to know the approximate root of two

We write 2 and add six zeros before it.

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

1 2 4 8 1 2 4     2 2 8

• First of all, we extract the root in the way that we said in the chapter on roots: we find that its root according to this way is 1414 and a bit more.

• Since there are four digits here, we take the fourth digit as integer, which is one, and we multiply the rest of the digits that are as the number of half the zeros by sixty. The result of multiplication is 24840.

• We take the two last digits, which are 2 and 4, that are more than [the number of] half the zeros and they are minutes. We multiply the remaining 840 by sixty; the result of multiplication is 50400.

• We take what is more than [the number of] half the zeros; it is 50 and this 50 are seconds. We multiply the remaining 400 by sixty; the result of multiplication is 24000.

• We take the 24 that is more than [the number of] half the zeros and they are thirds. Three zeros remain that are half the original zeros that we added.

So, in this procedure we have found that the approximate root of the required number, which is 2, is 1 integer, 24 minutes, 50 seconds, and 24 thirds.

Chapter Four: Giving Guiding Ways for Finding Roots of Cube Numbers or Approximate Roots of Non-Cube Numbers

• As 2, which is a cube root of 8. For, the product of 2 by its square, which is 4, generates 8, which is the cube number of 2.

• Likewise 3 is a cube root of 27, because the product of 3 by its square, which is 9, generates 27, which is the cube number of 3.

• Also 1000 is a cube root of 10, as it is generated from the product of 10 by its square, which is 100.

• 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.

• The second rank, which is the rank of tens, do not have a cube root, because no number can be found that when multiplied cubically generates ten.

• The third rank, which is the rank of hundreds, do not have a cube root, because no number can be found that generates it when multiplied cubically.

• The fourth rank, which is the rank of thousands, have a cube roots, because it is generated by cubing the ten.

• The fifth rank, which is the tens of thousands, do not have a cube root, for the reason mentioned.

• The sixth rank, which is the hundreds of thousands, do not have a cube root also, for the reason mentioned.

• The seventh rank, which is the thousands of thousands, have a cube roots, because it is generated by cubing the hundred.

Written calculations

Example: if it is 2, we turn it to 6.