Difference between revisions of "האנציקלופדיה של אבו אלצלת"

Book One

Introduction - Basic Definitions

• Arithmetic is a theoretical science, its subject is the absolute number and its types, its purpose is the understanding of its concepts, of what follows from it, of its properties and its general essential accidents.

• The essence of the number means the number is a sum of units and its definition is that it is a divisible quantity that exists from units.

• Some understand unities as units, since it indicates the nature of the number, as the sum of the units is not the number itself, whose intention here is its attribute and its definition and in general the announcement of its essence, but the counted that are the subjects of the number, as a sum of people, or horses, or others.

Whereas the sum of unities is the number itself that is summed from the duplication of the unity, by which it is said "one" about each thing of the counted.

• The absolute number is divided first into two types: even number and odd number.

• The even number is any number that is divisible into two equal parts, such as 2 and 4.

• The odd number is any number that is indivisible into two equal parts.

Properties of the Absolute Number

• The first of them and the most famous: every number is half [the sum of] its two sides, which are the numbers that are next to it at the same distance from the side of the subtraction and the addition:

For example:

• Its double is equal to [the sum of] its two sides

• Its half [is equal to] a quarter of [the sum of] its two sides

• For every number, its square is equal to the product of its two close sides, one of them by the other, plus one

Such as:

• 

• Also, for every number, its square exceeds the product of its two sides that are at the same distance, whichever they are, one of them by the other, by the square of the number of the terms between them

• For every number, its distance from its double in terms:

If one takes it as the first of the terms - it is the same as the number plus one:

If one takes the number that follows as the beginning of the terms, it is as the number of units the are in it:

The example for this: what is between four and eight -

sometime 4, 5, 6, 7, 8 - which are five [terms]

and sometimes 5, 6, 7, 8, which is as the units that are in [four].

• For every number, its distance from its thrice is the same as the measure of its units multiplied by two:

either plus one:

or without the addition of the one, as is known:

Such as the two, whose distance from 6, which is its thrice, is as its multiplication by two plus one:

or without the addition:

• For every number, its distance from its fourfold is the same as its multiplication by three plus [one]:

or without the addition [of the one]

Example:

Example:

Example:

Example:

Example:

Example:

• ?

Example:

Example:

Example:

Example:

Example:

Example:

Example:

Example:

Example:

Example:

Example:

Example:

Example:

Example:

Example:

Example:

Arithmetic Progression

• If their number is an odd number they have undoubtedly a mean.

This mean is always half the two summed sides

The meaning of the two sides is two numbers, or one number, whose interval from the mean, by the order, is an equal interval, one of them on the deficit side and the other on the surplus side.

Such as:

• 9 and 1 that are the two sides of 5 and 5 is half their sum

• and it is also half [the sum of] 8 and 2 that are also its two sides

• and half [the sum of] 7 and 3

• and of 6 and 4

and the most remote extremes are 9 and 1.

Every mean number is their half [= half its sides]

• If the number of the terms is even, there are two means together, instead of one mean.

The sum of the two means is as the sum of the two sides whichever they are

For example: 4 and 5 that are between 1 and 8, when they are summed they are equal to 1 and 8, to 2 and 7, and to 3 and 6

Sums

• Among the properties relating to the sum of progressions: when one is added to the value of the last term, starting from one, then multiplied by half the last term, the result is equal to the total sum.

Example: 4 is the last term

[The sum] from 1 to five:

Also, the sum of every two extremes of succesive terms, either [starting] from the 1 or from other, when multiplied by half [the number of] the terms, or its half is multiplied by the whole [number of] the terms, the product is as the total sum of these terms.

If the first term is 2 and the last term is 6:

• Among the properties relating to the sum of successive numbers, such that do not exceed by one, but by two, or three, or other than this, as long as they follow the same rule, whichever is their beginning, the product of the number of the terms minus one by the number of the excess, such as two, or three, or any other excess, plus the first term is equal to the last term.

And if [the excess] is added once more and multiplied by the number of the terms as it is, it is as double the sum.

Example: five succesive numbers , starting from 4 , between every two numbers there are three, so that the excess is four , which is the last of them and how much is their sum?

so the terms are:

multiplying its half by [the number of] the terms, or the whole by half [the number of] the terms, so that it yields the answer of the question.

• Among the properties relating to the sum: all successive numbers, beginning from one, when they are summed from the first to the last, then backwards from the last to the first, such as 1, 2, 3, 4, 3, 2, 1, their sum is equal to the square of the last number.

For the sum of the given example is 16:

This is because the sum of double the terms that precede the last successive term plus the last term is equal the square of the last term.

• Among the properties relating to the sum: when summing the successive numbers from one, the first sum is one and a half times the last term, the second sum is twice the last term, the third sum is two and a half times the last term, the fourth sum is three times the last term, the fifth sum is three and a half times the last term and so on endlessly.

Example:

• 

Also, all the successive numbers, when they are summed together, the first sum is as the consecutive term, the second sum is one and a half times the consecutive term, the third sum is twice the consecutive term and so on endlessly.

Example:

• 

• Among the properties relating to the sum: when summing the successive odds, beginning from one, then summing the successive evens, [beginning] from two, by their numbers, the first sum of the evens is one and a half times the first sum of the odds, the second sum is one and a third times, the third sum is one and a quarter times - each sum exceeds by a part that is denominated by the number of terms.

Example:

• 

Types of Numbers

• Either by the excess of the natural succession of the types of number, which is one by one.

• Or by the excess of the natural succession of the odds and the evens, which is two by two.

• for every odd number, when one is added to it, it becomes even.

[]

• When one is further added to it, it becomes odd.

[]

• Then, when one is added to it, it becomes even.

• Between the odd and the following odd there are two.

• Between the even and the following even there are two.

• It follows necessarily that every mean of the odd terms in the natural succession and the even terms in this succession is half the sum of the two sides, whichever they are.

Since they are the sides of this mean itself in the natural succession of the numbers.

• The sum of every two means is as the sum of every two sides.

Since these two means have two sides of the number that falls between them in the succession of the natural numbers, it follows necessarily that their sum is equal to the sum of these two other sides, as explained above.

This property does not applied between the successive odds and the successive evens alone, but between all the numbers that are exceeding by equality.

Hence, this property is found also in the succession of the terms of the types of the odd numbers

Odd Number

[odd ≠ even × even]

[odd ≠ even × odd]

[odd − odd ≠ odd]

[odd − even ≠ even]

• Among its properties: its sum by the way of succession, [starting] from the one, is always a square.

Such as:

• 

• Among its properties: the side of each of these squares is the number of the terms

Such as:

• the four, which is the sum of two terms and its root is two

• the nine, which is the sum of three terms and its root is three

• Among its properties: when wishing to know the value of a number whose position is known, starting from the one - multiplying the number of the term by two, then subtracting one.

For example:

The property of the mean and the two means, with the two sides, is as known.

• Among its properties: each of the units returns every sixth term.

For example:

• 

• Among its properties: for every prime odd , when extracting from it the interval, whose value is that number, its end is a composite number that consists of deficient primes.

Such as:

• three, for the third from it is nine , which is composite.

• five, for the fifth from it, which is 15 , is composite.

• Another property: the first of the incomposite numbers, which is three, when extracting from it the first interval of its value, it yields a number that has a root , but afterwards not

And the second [incomposite number], which is five, yields, by the second interval of that value, a number that has a root , but afterwards not

And so on.

• Another property: the fourth after the first that has a root, which is one , has a root, which is nine , then the eighth after the first that has a root , then the twelfth after the second that has a root , then the sixteenth after the third that has a root , and so on by adding four

For every term and rank, whose value has a root, the product of the root [by itself] is equal to double the number of the rank plus one.

Because, the first square number is nine, which is in the fourth term of the odd numbers:

And 25 is in the twelfth term from 3:

• Starting with a square [table] of five by five:

9	7	5	3	1
19	17	15	13	11
29	27	25	23	21
39	37	35	33	31
49	47	45	43	41

• For every two symmetric diagonals, whether they are the principal diagonals of the diagram or not, the sums of both diagonals are equal.

For the principal diagonals: the sum of each diagonal in this diagram is 125.

[]

If they are not the principal diagonals, as these two symmetric diagonals:

• The sum of the two extremes of each diagonal is equal to the sum of the two extremes of the corresponding symmetric diagonal.

• the sum of all the cells in every square equals the 4th power of the number of cells on the side of the square.

• For, when one builds a square, whose side is two cells, and these are its numbers:

3	1
7	5

The sum is []

• And when its sides are of three cells, such as:

5	3	1
11	9	7
17	15	13

Their sum is: []

The sum [of the numbers] on the principal diagonal is equal to the cube of the number [of cells on the side of the square]:

• For example, in the greater table, whose [side is] five cells, its principal diagonal is: []

• On the second [table], the principal diagonal is: []

• On the third [table], the principal diagonal is: []

And so on.

• When one builds a triangular table, as this diagram:

                  1
               5     3
            11    9      7
         19    17    15    13
      29    27    25    23    21
   41    39    37    35    33    31
55    53    51    49    47    45    43

                  א
               ג      ה
            ז      ט    יא
         יג    טו    יז    יט
      כא    כג    כה    כז    כט
   לא    לג    לה    לז    לט    מא
מג    מה    מז    מט    נא    נה    נט

• the numbers on the altitude of the triangle from top to bottom are the successive square numbers

• the sum of the numbers in each row breadthwise is a cubic number

Such as: ; and:

Even Number

• Among them: the sum of its terms is equal to the square of the number of the terms plus its side.

Such as:

And as:

• Every even number that exceeds a prime odd number by one, is equal to the sum of the divisors of the square of this prime number.

Such as 4:

And also 6:

• For an even number, such that when 3 is subtracted from it, the remainder is a prime odd number, this even number consists of the divisors of double this odd number.

Such as: 8

Types of Even and Odd Numbers

Types of Even Numbers

Even-Times Even

• It has no part that is denominated by [= it is not divisible by] an odd number, or an even number that is not an even-times-even-number

• There is no smaller even-times-even number that does not divide it.

• For every even-times-even number, its square is and even-times-even number.

• When the first even number, which is two, is subtracted from it, the result is an even-times-odd number.

→ even-times-odd number

Such as 8:

• Every even-times-even is deficient, and its deficiency is one [= the sum of its parts is the number minus 1.

]

• Among the properties of the even-times-even number: its terms follow by the geometric proportion.

Since they follow by the doubling way.

And their excesses are unequal.

But, each excess is equal to that over which it exceeds.

The difference between the excesses is the excess itself.

• It follows necessarily from its terms being in one ratio, that they are proportional when they are divided and when they are multiplied equally.

• It follows necessarily that the product of whichever mean by itself is as the product of one of the two sides by the other.

As the ratio of the smaller side to the mean is as the ratio of the mean to the greater side.

• It follows necessarily that the product of one of the two means by the other is as the product of one of the two sides by the other.

As the ratio of the smaller side to the smaller mean is as the ratio of the greater mean to the greater side.

Such as the terms: 4; 8; 16; 32; 64

• Since the even-times-even numbers are arranged by a continuous ratio, it follows necessarily that the squares and the cubes among them have sequences, as the third from a square is a square, and the fourth from a cube is a cube.

• Among its properties: the perfect numbers are generated from it.

• The amicable numbers are the numbers that are compounded each from the parts of its amicable number, as the amicable number is compounded from its own parts.

Such as: 220; 284

142

71

4

2

1

220

None of them has parts other than those that were mentioned.

Their production technique:

• Summing the even-times-even numbers together with 1, so that the sum is a prime number, and when the last term is added to it or subtracted from it, the result of the addition [] and the result of the subtraction [] are prime numbers.

• The product of the result of addition by the result of subtraction, multiplied by the last added term, is a number that has an amicable number:

• Its amicable number is the number that resulted from adding the product of the sum of the mentioned result of addition and the result of subtraction, multiplied by the last of the summed terms, to the number that was found first, which is the amicable number, and they are amicable numbers.

Even-Times Odd

• It is divided by an even number only with an odd number, and by an odd number only with an even number.

[even-times-odd ÷ even = odd; even-times-odd ÷ odd = even]

• Its even part is denominated by a name of an odd number.

Such as:

• Its odd part is denominated by a name of an even number.

Such as:

• When the first even number, which is 2, is added to it, it yields an even-times-even[-times-odd] number.

• Know that its production is from the products of the successive odd numbers by 2.

• The difference between each term and the successive term is double the difference between the natural odd numbers.

• The excess of its terms is four.

• There is none that has a square root among them, nor a cube.

Since, for every one that has a square root and every cube: if it is an odd number, it is divided by an odd number with an odd number; if it is an even number, it is divided by an even number with an even number.

This is already known, as the excess is four.

The beginning either from 2, or from 6, as the property will be explained.

Hence, returning to 2, recurring cyclically.

• It follows necessarily that there is one cycle that repeats in the others and their units are similar to the units of the first cycle [i.e. 2, 6, 0, 4, 8]

• When starting from 6, which has a third [= divisible by 3] that is denominated by three, the third successive term after it, which is 18, has a third [= divisible by 3], and the third successive term after 18, which is 30, has a third, and so on endlessly.

[→ are even-times-odd numbers divisible by 3]

• The successive term after 6 is 10 and its part is denominated by the successive odd after 3, which is 5, for 10 has a fifth [= divisible by 5], hence, when starting from 10, the fifth successive term after it has a fifth [= divisible by 5], and so on as much as one wishes.

[→ are even-times-odd numbers divisible by 5]

• The successive term after 10 is 14 and its part is denominated by the successive odd after 5, for it has a seventh [= divisible by 7], hence, the seventh successive term after it is found [divisible by 7], and so on.

[→ are even-times-odd numbers divisible by 7]

• Among the properties of these terms: the sum of two, which is the first even-times-odd number, with every term that is denominated by a square number, yields a square number.

Such as:

• 

Also, the successive term after two, which is six, and it is the second even-times-odd number, when it is summed with every term, beginning from one, that is denominated by a square number, the sum is a square number.

Such as:

• 

• The product of the name of every term by four, when the first number is subtracted from it, is the value of that term.

For instance:

It can be reversed by saying that every term of them, when two is added to it, then divided by four, the result is the number of its term from the first.

• Double the product of the number of the terms by itself is equal to the sum of the terms.

For four terms:

• The sum of the first and the second is a cube, 8, [], and its term is already known, also the cube of the cube and so on [??]

• Constructing a square [table] of six by six from the even-times-odd numbers

22	18	14	10	6	2
46	42	38	34	30	26
70	66	62	58	54	50
94	90	86	82	78	74
118	114	110	106	102	98
142	138	134	130	126	122

• Among the properties of this square table: the units of the beginning of each row breadthwise are the same as the units of its end.

If there is a zero in one of [the extremes], there is a zero in the other [extreme] as well.

• the sum of the two extremes on the principal diagonal is equal to the sum of the two extremes on the other principal diagonal.

Such as:

• The sum of the extremes on the principal diagonal always has a root.

• The sum of two numbers at the same distance from the two extremes on the principal diagonal is equal to the sum of the two extremes on the principal diagonal, and therefore it also has a root.

• The excess of each [consecutive] line over the beginning of the [preceding] line is the same.

For the excess of [46 over 22] is as the excess of [70 over 46]:

Even-Times-Even-Times Odd

• It is said to be similar to the even-times-odd, since it does not admit the repeated halving until reaching 1, except by a fraction [= not divisible by 2 repeatedly].

• It is similar to the even-times-even, since its first halving does not yield two odds, and its halving does not cease by another division [= divisible by 2 more than once]

• Its production is by multiplying the even-times-even numbers, beginning from four, by the successive odd numbers.

The greater this even number is, the more it admits halving.

• [Among the even-times-even-times-odd numbers] there are superabundant, deficient, and perfect numbers:

• 68 is deficient number, and it is [an even-times-even-times-odd number]

• The perfect number, such as: 28.

• The superabundant number, such as: 12

• [Among the even-times-even-times-odd numbers] there are also squares:

• The production of these squares that are [even-times-even-times-odd numbers] :

• Multiplying the first even number by the first odd number, until they are six , which is the root of the first square.

• Multiplying it by the second odd number, until it is ten , which is the root of the second square.

And so on.

• When an [even-times-even-times-odd] term is subtracted from its successive term, the result is an even-times-even number.

Such as:

This is for what is produced from the products of four by odd numbers.

And such as:

This is for what is produced from the products of eight by odd numbers.

The Two - is it an even-times-even number or an even-times-odd number?

• It is considered as an even-times-odd number from the aspect that its halving does not end in an even number.

• Some considered it as an even-times-even number as well as an even-times-odd number,and as the beginning of both [types].

• a true even is the number that is divided into the odd number by halving.

• The true even-times-odd number is the divided into the odd number by halving.

• The even-times-even number is that whose half is an even number, and each half is halved by an even number, except the one.

• The even number is necessarily halved.

• The even-times-odd number is that whose half is an odd number, and cannot be halved.

• The odd is either a number, or one, from the aspect of its being indivisible into two parts

• If one wants to define the two as matching the two names together:

• The definition of the even-times-even number should be stated as that which cannot be halved into an odd number.

But, the two are so, hence the categorization is not by adequacy.

• If one wants to extract the two from both names together:

• The definition of the even-times-even number should be stated as that which is halved into an even number.

• The definition of the even-times-odd number should be stated as that which is halved into an odd number.

But, the two do not match one of the two names according to the adequacy of the categorization.

Types of Odd Numbers

Composite Numbers

• After three, every third number is a composite number

• After five, every fifth number is a composite number

• And so on endlessly.

• Example for the first [case]: 9; 15; 21

• Example for the second [case]: 15; 25; 3[5]

Likewise when starting from seven and nine according to this way.

• One finds that three counts:

• The first composite of their terms by the first number of the odds, as nine

• The second by the consecutive odd number, which is five

• The third by the third odd number, which is seven

• Five also counts:

• The its consecutive by the first number of the odds, which is three, as fifteen

• The second by itself, as twenty five

• The third by its consecutive [odd], which is seven, as thirty five

Relatively Prime Numbers

As for the composite in relation to itself that is prime in relation to the other:

It is every square of a prime number in relation to another square of a prime number among these successive odd numbers.

[= if p and q are prime numbers, then and are relatively prime]

Superabundant / Deficient / Perfect Numbers

• There is one [perfect number] in the units: 6

• One in the tens: 28

• One in the hundreds: 496

• One in the thousands: 8128

• And so on, in each rank there is one [perfect number]

• The units [of a perfect number] are necessarily 6 or 8, even if it does not determined by the experience that they are recurring repeatedly.

• Among the properties of the perfect number: when it is multiplied by eight and one is added to [the product], [the result] has a root.

[If a is a perfect number → 8a+1 is a square number]

When its root is divided by four and a quarter is added to the quotient, it is an even-times-even number, such that when multiplied by its double minus one the product is the perfect number.

[If a is a perfect number → is an even-times-even number; and ]

Example:

is an even-times-even number

• Some people applied an examination for the extraction of the perfect, superabundant and the deficient numbers:

For every even-times-even number, when it is multiplied by a prime number, whichever it may be:

• If the even-times-even number is greater than half this prime number by one half, the product is always a perfect number.

[If → is a perfect number]

Such as:

• 

• If it is greater than its half by more than one half, the product is always a superabundant number.

[If → is a superabundant number]

• If less than its half, whichever the deficiency is, [the product is always] a deficient number.

[If → is a deficient number]

• The example for the first [case]: [superabundant]

• The example for the second [case]: [deficient]

• For every number of the perfect numbers that is multiplied by a prime number, such that this prime number is not a divisor of the perfect number:

Such as:

The total sum:

It exceeds the 43 by 12, which is double the 6

• Every number divisible by 2 is always deficient.

• All the prime numbers are undoubtedly deficient.

• All the even-times-even numbers are deficient by 1.

• Every number divisible by 2 and 3 is always superabundant, except for the number 6.

• every number divisible by two and two other numbers [whose sum is divisible by three] is superabundant [the wording is very vague]

Example: is superabundant

• every even-times-odd number divisible by 2, 5, and 7 is superabundant

• every even-times-odd number that consists of a composite odd number is superabundant

Example: 18 and 30 are superabundant

• an even-times-odd number that consists of a prime number is deficient

• among the even-times-even-times-odd numbers there are deficient, superabundant and perfect numbers

superabundant number: 36

deficient number: 44

perfect number: 28

• there is no perfect number among the odd numbers

most of the odd numbers are deficient

• an odd number is superabundant only if it is a product of four consecutive odd numbers

Example: is superabundant

Book Two

Ratios

• The Simple is either double or multiple; either superparticular or superpartient

• The Compounded is the double superparticular, or the double superpartient, or the multiple superparticular, or the multiple superpartient.

[The multiple] and the superpartient meaning what is more than one duplication, or more than one part, even if it is [only] double, or two parts.

Simple Ratios

Multiple Ratio

• The first multiple is the double, in which the addition is one time the other.

• Its beginning is from the numbers one and two

The smaller are added by the sequence of the successive numbers

The greater, which are the doubles, by the sequence of the successive even numbers, exceeding two by two

• Then the triple, in which the addition is two times.

• Its beginning is from three and one

The smaller are added by the sequence of the successive numbers

The greater [are added] three by three

Such as:

Superparticular Ratio

• The first of the superparticular is that which adds to the other its half [= sesquialter].

• Its beginning is from three and two

The less are added by the sequence of the successive even numbers, since they have a half [= divisible by 2].

The greater [are added] three by three.

Such as:

• After the sesquialter is the sesquitertian

• Its beginning is from four and three

The less are added three by three.

Such as:

The greater [are added] four by four.

Table of Multiple and Superparticular Ratios

Let there be a table of ten by ten lines:

• []: The second line is in the double ratio to the first line.

• []: The third [line] is in the triple ratio.

• []: The third line is in the sesquialter ratio to the second line.

• []: The fourth [line] is in the sesquitertian ratio to the third [line].

• []: The fifth [line] is in the sesquiquartan ratio to the fourth [line].

• []: The excesses of the second line over the first line differ by number, but not differ by ratio.

• The excess of the first rubric of [the second line] over the first rubric of the first line is one.

• The excess of the second rubric of [the second line] over the second rubric of the first [line] is two.

According to this way the property of each rubric to its preceding []

• []: When the third is related to the first in every sequence they are found by the order of the even numbers.

• The first [rubric] of each third [line] exceeds by two over the first [rubric] of each first [line].

• The second by four.

• The third by six.

And so on according to this way.

• The excess of the first rubric of the fourth [line] over the first rubric of each first [line] is three by three.

• The excess of the second [rubric] of the fourth [line] over the second [rubric] of the first [line] is six by six.

• According to this way the excess of each rubric over the [other] rubric on both directions is three by three []

• The excess of the fourth over the second, between them one line, as the excess of the second over the first in ratio [?]

[?]

• The excess of the sixth over the third, between them two lines, as the excess of the third over the second in ratio [?]

[?]

• Every number on the principal diagonal is a square.

Such as:

• The sum of every two consecutive square numbers plus the numbers on their inverse diagonal is a square number.

Such as:

• The sum of every two consecutive square numbers exceeds the sum of the numbers on their inverse diagonal by one.

• Double the sum of every two consecutive square numbers minus one is necessarily a square number.

• The product of every number in a line by a number in another line is equal to the product of the corresponding by the corresponding.

Such as: 2, which is the second in the first [line], by 20, which is the last of the second [line], that is the same as 4, which is the second in the second [line], by 10, which is the last of the first [line]

• The product of every number on the principal diagonal by its inverse number on the other side of the principal diagonal is equal to the product of the corresponding numbers by each other, meaning on the inverse diagonal.

Such as:

Also:

And so on by this way.

Superpartient Ratio

Such as:

• The superbisextan which is reduced to the sesquitertian.

• The superbiquartan which is reduced to the sesquialter.

• Every superbipartient that is denominated by an even number [is reducible].

[]

• The supertrisextan is reduced to the sesquialter.

[]

For example:

• If the addition is of two thirds [superbitertian], the three are set, then two are added to them, which are five, so their beginning is three and five.

[]

• If the addition is of three quarters [supertriquartan], the four are set, then three are added to them, and they are four and seven that are their beginning.

[]

• Finding the smaller numbers in the superbipartient ratios :

• Superbitertian ratio

• the less are added three by three

• the greater [are added] five by five

• Superbiquartan ratio , which is not absolute

• the less are added four by four

• the greater [are added] six by six

According to this way the less are by themselves and the greater are by themselves.

• By this order, the superbiquintan

i.e. comparing the superbitertian to the superbiquartan, then to the superbiquintan

• the less are added one by one.

• the greater are also added one by one.

• When examine the absolute [superbipartient] ratios, they are by the sequence of the successive odd numbers.

[]

Such as:

• five to three, which is the superbitertian

• seven to five, which is the superbiquintan

• nine to seven, which is the superbiseptian

Such as:

• the supertriquartan

Their less and their greater are added by the mentioned rule.

• By this way the addition of three fifthes [supertriquintan]

The relation between the [supertripartient ratios] is as said for the former [= the superbipartient ratios]

[]

Such as:

Compounded Ratios

Multiple Superparticular Ratio

Such as:

• The double sesquialter, for its production is by multiplying the two by two and adding one to it, so they are two and five.

[]

• The double sesquitertian, whose production is by multiplying the three by two and adding one to it, so they are three and seven.

[]

• The double sesquiquartan - they are four and nine

[]

• Finding the numbers of the first [= the double sesquialter ratio ]:

• The less are added two by two, by the sequence of the successive even numbers

• The greater are added five by five.

So they are in the double sesquialter:

• Finding the numbers of the second, which is the double sesquitertian ratio :

• The less are added three by three.

• The greater [are added] seven by seven.

Such as:

• Finding the numbers of the third, which is the double sesquiquartan ratio :

• The less are four by four

• The greater [are] nine by nine

So they are according to the succession of:

[]

• The less are added one by one.

• The greater are added two by two, according to the multiple [n+2]

So they are: and so on in this way.

Multiple Superpartient Ratio

• The double [superbipartient] ratios - their production should be done as was already done [in the multiple superparticular ratios], except adding two instead of one.

• The double superbitertian ratio - starting from three and eight.

• The double superbiquartan ratio, which is not absolute - from four and ten.

• The double superbiquintan ratio - from five and twelve.

The greater are also added two by two, and the less [are added] one by one.

The sequence and the order [of the additions] are found in one category as the sequence of the numbers given in the double [superparticular] ratio - the less and the greater are added by their number, yet the numbers of the less are as in the [double superbitertian] ratio, double superbiquartan ratio, the double superbiquintan ratio, and the rest of them.

• The double supertripartient

The first of them is the three quarters [the double supertriquartan].

The production is by the same way, except that for the three parts [the supertripartient] three are added, and for the four parts [the superquadripartient] four [are added].

• The beginning of the double supertripartient is the double supertriquartan - starting from four and eleven.

• The double supertriquintan - starting from five and thirteen.

So on in this way.

The excesses of the terms are as before.

When referring to one category, the less and the greater are added as themselves also.

The number of the less is as it is and the number of the greater is [another?] number.

• The triple superparticular ratios, the triple superbitertian ratios, or the triple superpartient ratios

The production of these is as already done, except for not doubling once, but multiplying by the number of the multiples, then the part or the parts are treated as before.

• The triple superparticular ratios

• The beginning of the triple sesquitertian is three and ten.

• The beginning of the triple sesquiquartan is four and thirteen.

The less are added one by one and the greater three by three.

When taken breadthwise:

• The triple sesquialter:

All are also added by their number.

The less follow the addition of the successive even numbers.

• The triple sesquitertian:

The rule is kept.

• The triple superbipartient

• Their beginning is the triple superbitertian - starting from three and eleven.

• The beginning of the triple superbiquartan from four and fourteen.

• The beginning of the triple superbiquintan from five and seventeen.

The excesses of the less are by the sequence of the natural numbers and the greater [are added] three by three.

When taken breadthwise:

• The triple superbitertian:

The order is kept.

• The triple supertripartient

• The beginning is the triple supertriquartan - starting from four and fifteen.

• The triple supertriquintan - starting from five and eighteen.

The property is in this way.

When examined breadthwise:

• The triple supertriquartan:

The order is kept.

Tables of Ratios

• Setting a table of two columns

• In one of them the successive odds, starting from five and ending at twenty-one.

• In the second the successive numbers, starting from three and ending at eleven.

From what is between them, ratios will be explained:

3	5
4	7
5	9
6	11
7	13
8	15
9	17
10	19
11	21

• Observing what is in each rubric of the first column related to its corresponding in the other column:

• It is in the superbitertian ratio

• Then the supertriquartan ratio

• Then the superquadriquintan ratio

And so on.

• Observing the sequence of what is in the first column: they are in the absolute superbipartient ratio.

• Observing the sequence of what is in the second column: they are in the superparticular ratio.

• When placing instead of the second [column], starting from three, another [column], starting from two and following the sequence of the natural numbers, the ratio of the first rubric in the first column to its corresponding in the second [column] is the double superparticular ratio.

• []:The superparticular ratio is resulted, as already known, from the third and second columns.

• The superbipartient ratios:

• []: The superbitertian ratio from the fifth and the third columns.

• []: The superbiquartan ratio from the sixth and the fourth columns.

• []: The superbiquintan ratio from the seventh and the fifth columns.

And so on.

• [The supertripartient ratios:]

• []: From the seventh and the fourth columns = the supertriquartan ratio.

• []: From the eighth and the fifth columns - the supertriquintan ratio.

And so on.

• [The superquadripartient ratios:]

• []: From the ninth and the fifth columns - the superquadriquintan ratio.

• []: From the tenth and the sixth columns - the superquadrisextan ratio.

• According to this way, the double superparticular ratios are also extracted from this table:

• []: The first of them, the double sesquialter ratio, from the fifth and the second columns.

• []: The second, the double sesquitertian ratio, from the seventh and the third columns.

• []: The third, the double sesquiquartan ratio, from the ninth and the fourth columns.

• The double superbipartient ratios:

• []: The double superbitertian [ratio], from the eighth and the third [columns].

• []: The double superbiquartan [ratio], from the tenth and the fourth [columns].

• The [double] supertripartient ratios and the rest of the ratios are extracted when conducting in the way that was established.

Producing the Ratios from Equality

A table with three numbers, then three other number, three by three.

They are multiplied for examination and figurative illustration for the experience.

It is joined breadthwise by another table divided similarly.

4	2	1		1	1	1
8	4	2		2	2	2
9	6	3		3	3	3
16	8	4		4	4	4

• Taking the first and placing it in the first rubric of each line breadthwise []

• Summing the first and the second and placing the [sum] in the second rubric of the second table, so there is 2 in the line of the units []

• Summing the first, the third and double the second and placing the [sum] in the third rubric, so there is 4 in the line of the units []

• Placing the second rubric as a basis []

• First the multiple ratios:

• []: what is on the second row is in the double ratio.

• []: what is on the third row is in the triple ratio.

• []: what is on the fourth row is in the quadruple ratio.

It goes one endlessly.

• []: the numbers that are on the third row are in the sesquialter ratio to what is on the second row.

• []: what is on the fourth row to what is on the third row are in the sesquitertian ratio.

And so on.

• []: what is in the second rubric of the second row to what is in the first rubric is in the quadruple ratio.

• []: what is in the third rubric to what is in the second rubric is in the double sesquiquartan ratio.

• []: what is in the fourth rubric to what is in the third rubric is in the superseptinona ratio.

• Taking the first [after switching C_1 with C_3] and placing it first on the third [table] []

• Summing the first and the second and placing the [sum] in the third [table] []

• Summing the first, which is 4, the third, which is 1, and double the second, which is 4, and placing the [sum] in the third rubric []

• The numbers of the line follow the sesquialter ratio, they are produced from the double ratio, and the are both named after the two.

• When this procedure is applied on the line of the triple ratio lengthwise the result are three numbers in the sesquitertian ratio, for both are named after the three.

• The same for the fourth line, for it will produce the sesquiquartan ratio.

• When the setting of the numbers in sesquialter ratio is reversed, the double sesquialter ratio is produced.

• And the double sesquitertian ratio [is produced] from the sesquitertian ratio.

Reducing the Ratios to Equality

• Assuming three numbers in a successive ratio.

• Keeping the smaller as it is [].

• Subtracting it from the mean and setting the remainder as a mean term [].

• Subtracting the smaller and double the second from the greater and setting the remainder as a third term [].

For example: the numbers are first in the double superbitertian ratio, such as

The result are succesive numbers [] in the superbitertian ratio.

Proceeding with [these numbers ] by this procedure, and the result are , which are successive numbers in the sesquialter ratio.

Proceeding with these numbers [] by this procedure, and the result are , which are in the double ratio.

Proceeding with this procedure, and the result are , restored to equality.

Composing a Ratio from Two Ratios

For example: let

• AB : AD = sesquitertian ratio

• AD : AG = sesquialter ratio

• AB : AG = double ratio, which undoubtedly consists of both these ratio.

• Proposition: for every sesquialter ratio, when a sesquitertian ratio is added to it, the sum is the same result [= double ratio]; and every double ratio bears the division and separation into these to ratios.

For, if not:

Let:

• HC : HZ = sesquialter ratio

• HW : HC = sesquitertian ratio

→ [supposition:] HW : HZ = double ratio

[proof:]

• BD : WC = AD : HC

by distinction: AG : HZ = GD : ZC

by equality: BD : DG = WC : CZ

by composition: BG : DG = WZ : CZ

DG : GA = CZ : ZH

by equality: BG : GA = WZ : ZH

by expansion: AB : AG = HW : ZH

• Setting whichever two numbers

• HZ : HW = double ratio

→ [supposition:] The sesquialter ratio is between Z an C

[proof:]

• If not, it is external, such as Z : T

When attaching to them the other ratio, such as T : W, the primary compound ratio is restored.

→ HW : HZ = HW : HC

Hence, what is greater than HW is as HW [→ contradiction]

→ It is interior, such as C

• Proposition: the triple ratio [AD : AB] is composed from the double ratio and the sesquialter ratio.

The proof:

• AG = 2AB

BG = AB = ½AG

• GD = ½AG

AB = BG = GD

• → AD = 3AB

• If GD = ⅓AG

Then AD = [2⅔]AB

[The proof:] Dividing AG into three segments on H and Z.

• AH = GD = ⅓AG = ⅓(2AB)

→ ½AH = ⅓AB

→ AD = AG + GD = 2AB + ⅓(2AB)

• If AG : AB = sesquitertian ratio

Then AD : AB = double ratio

[The proof:] Dividing AB into two segments on H.

• AH = BG

AH = HB = BG

• GD = one of the three segments of AG

the four segments are equal [AH = HB = BG = DG]

→ BD = AB

→ the excess of AD over AB is the same [i.e. AD = 2AB]

• If AG : AB = sesquitertian ratio

And AD : AG = sesquioctavian ratio

Then AD : AB = sesquialter ratio

[The proof:] Dividing AB into thirds on Z and H.

AZ = ZH = HB = BG

half of each is ⅛AG = GD

BD = 3GD

AB = 6GD

→ [AB : BD] is sesquialter ratio

BD : DG = AB : BG

by conversion: BD : AB = DG : BG

DG = ½BG

→ BD = ½AB

→ AD : AB = sesquialter ratio

• If AG : AB = sesquiquartan ratio

And AD : AG = sesquiquintan ratio

Then AD : AB = sesquialter ratio

[The proof:] AB is divided into quarters, each segment is the same as BG, so that the five segments are equal and BD = ½AB.

• If AG : AB = sesquiquintan ratio

And AD : AG = sesquisextan ratio

Then AD : AB = superbiquintan ratio

This is clear when AB is divided into five segments and what was done is applied.

• The composed from the sesquisextan ratio [] and the sesquiseptan ratio [] is the sesquitertian ratio []

• The composed from the sesquiseptan ratio [] and the sesquioctavian ratio [] is the superbiseptan ratio []

• The composed from the sesquioctavian ratio [] and the sesquinona ratio [] is the sesquiquartan ratio []

• The composed from the sesquinona ratio [] and the sesquidecima ratio [] is the superbinona ratio []

• The composed from the sesquidecima ratio [] and the sesquiundecima ratio [] is the sesquiquintan ratio []

• The composed from the sesquiundecima ratio [] and the sesquiduodecima ratio [] is the superbiundecima ratio []

• The composed from the sesquiduodecima ratio [] and the sesquitercdecima ratio [] is the sesquisextan ratio []

• The composed from the sesquitertdecima ratio [] and the sesquiquartdecima ratio [] is the superbitertdecima ratio []

• The composed from the sesquiquartdecima ratio [] and the sesquiquintdecima ratio [] is the sesquiseptian ratio []

• When AG : AB = sesquidecimoquinta ratio

And AD : AG = sesquiquartan ratio

Then AD : AB = sesquitertian ratio

[The proof:] when AB is divided into 15 segments.

AD = 16 segments.

GD = ¼AD = 4 segments.

BD = 5 segments.

AB = [15] segments.

AD = 20 segments.

→ BD = ⅓AB

The same practice when AG : AB = sesquialter ratio

Numbers as Geometric Shapes

• Linear Numbers

• Plane Numbers

• Solid Numbers

Linear Numbers

Plane Numbers

Triangular Numbers

• The first of them is three, and its shape is:

• Then six

Two is joined to one, and the first triangle is produced, which is three, so the shape is:

For example, when joining to this a numerical line of four units, the shape will be another triangle, like this:

• The first of the triangular numbers is 3 and its side is 2.

• The second triangular number is 6 and its side is 3.

• The third triangular number is 10 and its side is 4.

• The fourth triangular number is 15 and its side is 5.

• Every triangle exceeds its preceding by its own side.

• Their sides differ according to the sequence of the numbers, from one, including one, and whichever number is summed from them is a triangle [].

• For every triangle, its side exceeds its rank by one.

When you are told: what is the side of the tenth triangle from the beginning of the triangular numbers?

Hence, the number of the side and the number of the rank will be one and the same [when one is included].

• If you are told to say that it is a square, or a cube in actu, it is not a triangle, nor a pentagon, and none of those, not in potential and not in actu, only by similarity of the name - do not pay attention to what they say.

• Every triangular number is half the product of its position by the exceeding over it by one []

So, if you are told: what is the value of the fifth potential triangle?

is the fifth triangle [in potential].

• The side of a triangle is the smaller among two successive numbers, such that the product of one of them by the other is double its triangle [].

So, if you are told: what is the side of the triangle 15?

→ its side is five.

Square Numbers

Such as:

• 

As these shapes:

• Their production is from the sum of the successive odd numbers with the one.

Such as:

• which is the first of the square numbers.

• which is the second square number.

• which is the third square number.

• Among the properties of the squares: when they are summed from the first square, their sum is greater than the last square by the sum of the preceding squares .

• The sum of the square of one with the square of two exceeds the square of two by the square of one .

• The sum of the squares of one, two and three exceeds the square of three by the sum of the two squares of one and two .

• When the way to produce the squares was applied, it was called run and return [Ezekiel 1, 14]

• That is by starting from one and summing the ranks as much as one wishes, then summing in descending [order], and the total sum is a square .

• which is the first square number.

• which is the second square number.

• which is the third square of the square numbers.

• The completion of this way is that the sum of all the successive numbers plus the sum that is lacking the last rank is a square .

• Also, double the sum of all successive numbers minus the last number is a square .

• [The sum of] every two successive triangles that are summed together, [beginning] from one, three, and six, is a square, and this is also the production of the squares.

• Every square is [generated] from a triangle of its rank plus a triangle of its rank minus one.

• For every two squares, the side of one of which is multiplied by [the side of] the other, then [the product] is doubled and added to the two squares, the total sum is a square .

such as:

• For every square, when its root is added to it twice plus one , or when it is added to its same and a quarter of its same , or thrice its same , or when its three quarters are subtracted from it - the results are squares.

• There is no square whose half or double are squares.

• The sum of the successive squares, beginning from one, is not a square at all.

• The units of the number that has a root are necessarily either 1, or 4, or 5, or 6, or 9:

• If it is 1: the units of its side are 1 or 9.

• If they are 4: [the units of its side] are 2 or 8.

• If they are 5: [the units of its side] are 5.

• If they are 6: [the units of its side] are 4 or 6.

• If they are 9: [the units of its side] are 3 or 7.

• The check [= modulo] of the squares, according to the way of the people of India, is necessarily either one, or four, or seven, or nine.

• One: either one, or eight.

• Four: either two, or seven.

• Seven: either four, or five.

• Nine: either three, or six, or nine.

Polygonal Numbers

• []: The first of them is the five, which is composed as this:

It is the first pentagonal and its side is two.

• []: The second pentagonal is that whose side is the second number, which is three, so the pentagonal formed from it is twelve, like this:

• []: The third number, which is four, the pentagonal that is formed from it is twenty two, like this:

• Their production is by summing the numbers exceeding three by three, beginning from one.

Such as the numbers: with the one

• which is the first pentagonal number.

• which is the second pentagonal number.

• which is the third pentagonal number.

• They are produced by summing the squares with the triangles, meaning summing the square that is in its rank with the triangle that is in the preceding rank.

• As finding that from the sum of the second square, which is nine, with the first triangle, which is three, the second pentagonal, twelve, is produced .

• The hexagonal is produced by summing the pentagonal that is in its rank with the triangle that is in the preceding rank:

• The second hexagonal number = = the second pentagonal number + the first triangular number.

• The second heptagonal number = = the second hexagonal number + the first triangular number.

And so on.

• The pentagonal numbers are produced by taking the squares that are in their rank, and adding to the squares half their sides as many times as the number of the ranks:

• The first square number + half its side = [] = the first pentagonal number.

• The second square number + twice the half of its side = [] = the second pentagonal number.

• The third square number + three times the half of its side = [] = the third pentagonal number.

• For the hexagonal one takes the side, instead of half the side.

• For the heptagonal the side and its half instead of this.

• For the octagonal - two sides

For example:

[hexagonal number = ]

• The first square number + its side = [] = the first hexagonal number.

• The second square number + twice its side = [] = the second hexagonal number.

[heptagonal number = ]

• The first square number + one and a half times its side = [] = the first heptagonal number.

• The second square number + [three] times its side = [] = the second heptagonal number.

• The square is composed from two triangles.

• The pentagon is composed from two triangles.

• The hexagon - from four.

• The heptagon - from five.

For example:

• The second pentagonal, which is 12, consists of twice the first triangle and once the second triangle.

• The third pentagonal [consists of] twice the second triangle and once the third triangle.

• The second hexagonal, which is 15, consists of the second pentagonal that consists of three triangles, plus the first triangle that precedes it.

Hence, it is possible to say that one is also in the rank of a triangle.

For the first pentagonal consists of four and one []

Also, every measured square is divided into two triangles, therefore, it is proper that one is instead of one triangle.

• Every hexagon is a triangle, but not vice versa.

• = the first hexagonal number = the second triangular number.

• = the second hexagonal number = the fourth triangular number.

• = the third hexagonal number = the sixth triangular number.

Not every triangle is a hexagonal number.

Such as , which are triangles, but not hexagonal numbers.

• Every triangular number whose position is an even number is not a hexagonal number.

• Finding the position of the triangle according to the hexagon: subtracting one from double the position of the hexagon [starting from one] and vice versa, adding one to the position of the triangle, then taking its half.

• Every pentagonal number is half the product of the sum of the product of its position minus one, multiplied by the difference between the numbers from which the [pentagonal numbers] are produced, which is three, plus two, multiplied by the position of the pentagonal number.

[pentagonal number = ]

For example:

• 

which is the fourth pentagonal number.

• Every pentagonal number is the same as the product of its position, beginning from one, by itself, plus the [product of] half its side by its position among the pentagonal numbers [without one].

[pentagonal number = ]

The same example [= the fourth pentagonal number]:

• 

They are composed from the numbers that are exceeding four by four.

Such as:

As said for the pentagonal numbers.

They are composed from the numbers that are exceeding five by five.

• For every surface that succeeds the triangle, when added to the triangle, the surface that succeeds this [surface] by the number of sides, is generated.

Such as:

• The first triangular number, which is three, when summed with the second square, will be a pentagonal number.

• When it [= the first triangular number] is summed with the second pentagonal, which is 12, it will be a hexagonal number, which is 15.

So on in this way.

• The excess of every surface over the preceding [is a triangle ?]

• For every perfect number, there is a triangle or hexagon [after] it.

By this, there is also a way for reaching the extraction of the ranks of the perfect numbers.

When you are told: the first perfect number, which triangle or hexagon is it?

Examine the arrangement known in particular:

• The first even-times-even number, for which the known arrangement is noticed, is four:

a prime number, appropriate for the procedure.

• the second hexagonal number and the third triangular number.

• [The even-times-even number] that follows four is eight:

prime numbe, appropriate for the requested [procedure]

• the fourth hexagonal number and the seventh triangular number.

• [The even-times-even number] that follows eight is sixteen:

composite number, not appropriate for the procedure

• [The even-times-even number] that follows sixteen is thirty two:

prime number, appropriate for the procedure

• the 16th hexagonal number and the 31th triangular number.

So on according to this rule.