Difference between revisions of "האריתמטיקה של ניקומכוס"

{{#annotpage: author="Nicomachus of Gerasa", translator="Qalonymos ben Qalonymos", place="South of France", time_from="1286", time_until="1329"}}

!style="text-align:right;"|<big>המאמר הראשון מספר הארתמאיטי</big>

=== [Prologue by al-Kindī's Student] ===

|May God fulfill your wishes, carry out your fantasies and increase His grace and His pleasantness upon you.

|<span style=color:red>[The history of the successive translations of the text:]</span>

|You think it has a precious benefit and a wonderful guidance.

|I inform you, may God grant you honor, that many times I perhaps wished to describe something by exaggeration and lengthiness, in order to reach the intention of the author of the book.

|Then, I came across to a concise statement I heard from our master, '''Abū Yusūf''', who already succeeded to explain what you had requested to explain and interpret what you had intended to interpret, in spite of his brevity, as a long speech would not have achieved and an exaggerate description would not have surpassed.

|More than once I have heard our master saying that the philosophy of these two men, i.e. '''Ptolemy''' and '''Nicomachus''', is best clarified in the introduction to their works - for Ptolemy in his introduction to the '''Almagest''' and for Nicomachus in the introduction to the arithmetic book - for the introductions of these two works stimulate elevate places of philosophy and they are of valuable virtue in knowledge.

|But, I append to each chapter in this description a discussion attached to that chapter that states the true meaning of the thesis.

|May God, the exalted, lead you to what will rescue your soul.

|'''Nicomachus''' '''said''': The first among the ancients who scrutinize the science and deepened in it - the first of them was Pythagoras - defined philosophy by saying that philosophy is the the love of science, as its name indicates this, for its translation is "the love of science".

|By saying "as its name indicates this", he meant because the term philosophy is composed of two words "filo" and "sofia" - "filo" is loving and "sofia" is science - so the term is composed from "loving science".

|'''He''' [= Nicomachus] '''said''': Before Pythagoras, whoever was skilled in any craft or medicine was called "wise", in a general term, without investigating the principle by which one is worthy to be named by the term "science"

|'''Pythagoras''', reserving this name to its meaning and to the root from which it is derived, has called "science" specifically - excluding the sciences that derived from it - the true knowledge of permanent things, i.e. the species and the types.

|style="text-align:right;"|אמנם פיתאגוריש למה שייחד זה השם לעיניינו והשרש אשר ממנו נגזר קרא אמתת הידיעה בדברים המתמידים ר"ל המינים והסוגים החכמה בפרט מבלתי מה שיצא מזה מן החכמות

|'''He''' [= Nicomachus] '''said''': the appropriate is to call the science "sofia" and the love of it "philosophy", i.e. the love of science.

:<span style=color:red>[A list of definitions of philosophy given by the ancients:]</span>

*Derivation of its name, as loving science, since the filosof is composed of filo, which is loving, and sofia, which is the science.

*It has been further defined from its effect: it has been said that it is the attention paid to death. The death is twofold: natural death, which is when the soul leaves the usage of the body; and a voluntary death, which is the abolition of desires by the sword of the virtuous choices - this is the death aimed for here, for the abolition of desires is the path leading to the excellence. Because of this many of the most eminent philosophers have said that pleasure is necessarily bad. For, when the soul engages in two activities - one of them is sensual and the other of the intellectual - and what is called "pleasure" is what happens when it indulges in the sensual pleasures - the soul then ceasing to engage in the intellect.

:As an example, it is said that since things are either bodies or non-bodies - and the bodies are substances, while the non-bodies are either substances or accidents - but, man is endowed with a soul, a body and an accident - and the soul is a substance, non-body - thus, when the man knows himself, he knows the body with its accidents, the first accident, and the substance that is a non-body; and when he knows all of this together, he knows it all. This is the cause for which the wise men have called the man a microcosm.

|[Al-Kindī] defines philosophy by that philosophy is knowledge of the eternal universal things - their essences, their beings, their causes, and their whys, according to the human perception capacity.

|He has also defined knowledge by saying that knowledge is the apprehension of the purpose of things that are ascribed to knowledge, i.e. those that are permanent, whose existence does not change, whose essence does not change, and who never depart from their quality. These are the primary natural species and genera, which - when the individuals are associated and defined by them - deserve the name of existing, since their species give them names and definitions.

|Likewise, he has made the faculty, which is called the nature, as the cause of the movement of the movings that are resting after the movement.

|The movings are the bodies, and because of this the bodies are called nature, since those that receive their movement and their rest by nature are therefore always deficient and changing.

|Although, the knowledge of the individuals is called science metaphorically, since it is the path towards the knowledge of those permanent, eternal and primary things, which are not subject to change and nor removal of their nature and which are said to exist truly and distinctly.

|Because of this, '''Plato''' said in the '''Timaeus''': What is the thing that always exists, without ever becoming? And what is the thing that always becomes without ever being?  

|style="text-align:right;"|ולזה אמר אפלטון בספר טימאוש מה הדבר אשר הוא נמצא לעולם ואין לו התהוות כלל ומה הדבר אשר יתהווה לעולם ואין לו מציאות כלל

|'''He''' [= Nicomachus] '''said''': It is appropriate for us and necessary to us, if we desire the perfection of man - as far as it is possible - and the permanent survival that it will be achieved by philosophy alone not by anything else.

|Among the existing things, some are said to truly exist, and some are called such by homonymity.  

|We say that among the perceptible existences, there are those whose parts are unified, their members are integrated - this is as the rank of the animal, whose parts are unified in it, or the tree, and their like - those are called magnitudes in essence.

|For, the plurality, starting from a limited origin and multiplied, does not have, by nature, an end, at which it would stop, and it does not multiplied a single time, but it always continues to grow.

:The numerous is numerous compared to what is less than it, and the few is few compared to what is more than it.

:The few, compared to what is more than it, is numerous compared to what is less than it.

::For example, the ten is numerous compared to what less than it, and few compared to what is more than it.

|style="text-align:right;"|ונאמר עתה שהכמה המספרי יחלק לשני חלקים

::For, the even is not divisible into two parts of equal units, due to its relation to another thing.

:<span style=color:red>[1. Arithmetic – investigation of the absolute quantity]</span>

|style="text-align:right;"|הארתמאיתיקא והיא תחקור מהכמה הניפרד

:<span style=color:red>[2. Music – investigation of the relative quantity]</span>

|<span style=color:red>[The science of quality is investigated by two arts:]</span>

:<span style=color:red>[1. Astronomy – investigation of what is moving]</span>

:<span style=color:red>[2. Geometry – investigation of what is resting]</span>

|'''He''' [= Nicomachus] '''said''': There is no other way to arrive to the knowledge of the species that have been said to truly exist, except through these four arts which are: arithmetic, geometry, astronomy and music.

|This has already been said by '''Androcydes the Pythagorean'''.

|'''Archytas of Taranto''', too, at the beginning of his Art of the Composition of Melodies, also said: The ancients did well to see, from these sciences, that it is by no means impossible that they will have for everything a unique and true opinion.

|Similarly, '''Plato''', at the end of the thirteenth treatise of the book he wrote on laws, which is the book that people call the "Philosophy", examining and defining what properer for a true philosopher to be, and the knowledge of these four, summarized it, after having explained it at long, saying that any shape, sum of numbers, harmonic music, and what is indicated by the movements of the planets, its knowledge should necessary be practice individually by the measure of their harmony.

|style="text-align:right;"|עם שאפלטון בסוף האופן השלשה עשר מהספר אשר הניח בנימוסים והוא הספר אשר יקראוהו האנשים הפילוסופיא בחקרו וגדרו איך ראוי שיהיה אשר הוא פילוסוף באמת כן בידיעת אלו הארבעה אחר כן שב ואמר בקצרה אחר שהפליג בזה והקדים שכל תמונה וקיבוץ מספר והסכמת חבור ומה שיורו בו ממרוצות הכוכבים אמנם ראוי שיורגל ידיעתו כפי שעור חבורם אחד אחד

|Their acquisition should not be neglectful, for who has united these sciences in the described mode, is the one who is called a true scholar, and is described as indulging in his science.

|It is not hidden that these sciences are like bridges through which our thoughts pass from those perceptible apprehended things, to the existing intelligible things, they transfer our thoughts from those corporeal things, with which we have grown and which are familiar to us, to things which are foreign to us, which are not familiar to our senses, and which, by their subtlety, are related to our souls.

|The best suited of what was chosen in the introduction to the investigation of the sciences and their acquisition is what '''Plato''' mentioned in '''the Book of Laws''' that Socrates told him, and he tries to be clever by putting forward authentic and useful reasons for these sciences that are necessary for the sciences that are useful for the improvement of human lives, for he said:

|style="text-align:right;"|והיותר ראוי ממה שבחרנוהו בהקדמת העיון בחכמות וקנינם מה שזכרו אפלאטון בספר הנימוסים שסקראט אמר לו והוא יתחכם כשיביא עלות אמיתיות מועילות בענין אלו החכמות מתחייבות מצד החוכמות מועילות בתיקון חיי בני אדם אחר שיאמר

|On this '''Plato''' replied, blaming him: do you think, seeing that I am indulgent towards you, that I will not dispute you?

|But, I will add support: there is no way to say that the eye of the soul, i.e. which is blind and closed by other corporeal activities, spreading a veil over it, is opened and awakened due to these sciences.

|The truth of the eye of the soul is more excellent than ten thousand bodily eyes, for it is only through it that the truth of things can be seen.

:So for any thing that is anterior, by natural precedence, to another thing, when the anterior disappears, the other disappears with its disappearance, but it does not disappear with the disappearance of what is posterior to it.

:The same, when reversing the statement, the same meaning is found: when a thing is necessary, its necessity implies the necessity of what is anterior to it, by natural precedence; but, when the anterior is necessary, its necessity does not imply the necessity of what is below it in precedence.

::Indeed, when, a triangle, a quadrilateral, an octahedron, or a pentadecagon, are necessary in geometry, the art of arithmetic is already used, since the four, the three, the eight and the fifteen come from arithmetic.

::On the other hand, reversing the statement, one finds that the three, the four, and the other numbers already exist in nature and in thought, even if the quantities, whose names are derived from them, are not known or do not exist.

:Since when the arithmetic disappear, the geometry also disappears, but the arithmetic does not disappear with the disappearance.

:*As the money is anterior to the abundance of the money and the substance is anterior to the size of the substance.

:These are the all the combinations of music - they could not be combined without the existence of arithmetic.

|May you succeed by all that the Creature has lavished upon you, and may He, by His grace, lead you to His will. Amen.

|All things, whose composition is improved by nature, are found to be subject to a numerical order

|Such as the elements, the times, the movements of the spheres, and the rest of the natural amendments.

|By this, the number is necessary in a rank of allegory and metaphor from which the thing is understood.

|But, since every compound is compounded out of different things undoubtedly and out of things that are also exist.

|As, the non-existent - their composition is not imaginable.

|Even for those that exist, but are not similar, it is impossible to compound them.

|It follows that the compound things are compounded out of existing different suitable things

|For the exalted Creator has compounded them, with their diversity, in a complicated composition, which has no separation in it, as it will be clarified in what follows.  

|style="text-align:right;"|{{#annot:||By9R}}המספר בהעברה מן המאמר הוא קבוץ האחדים או כלל הכמות המורכב מהאחדים {{#annotend:By9R}}

:'''He''' [= Nicomachus] '''said''': Indeed, the '''odd number''' is that which does not admit this, but is forever divisible into two unequal parts.

::Except for the two, of the even numbers, which, when divided, are dissolved to units, but their parts do not admit the excess and deficiency.

===== The Discussion on the Units and the Rulership of the Unity  =====

|'''Every number''' is half its two sides when they are summed.

*i.e. that which exceeds this number by one, and that which decreases from this number by one when they are summed;

*and also half of what follows, when what decreases from one of the two sides is as the measure of what exceeds in the other side.  

*and so on always, until the deficiency of the smaller side reaches to the unity, beyond which it cannot pass.

|Indeed, the '''unit''' is followed by the smallest number alone, which is two.  

|style="text-align:right;"|ואמנם האחדות הנה לא ילוה לו רק הפחות במספרים אשר הוא שנים

:As it is not a number that has two sides, therefore it is half its one side, i.e. half of the two.

*The first two types, which are the even-times-even and the even-times-odd, are dissimilar due to the difference between the even and the odd by definition.

:each of its half [is divisible] into two halves,

|style="text-align:right;"|והיו להם שני אמצעיים והם ח' וי"ו

::You find that nothing will be prevented from what was described:

|style="text-align:right;"|ויהיה שמיניתו ח' אשר הוא האמצעי

*It is also necessary for the terms of the even-times-even number, that when they are placed in the order that was given as an example, and the number of terms was even, they will have two means.

:and [to the product of the terms] that follow what follow [the extremes one by the other],

:Examine this rule in the terms that were counted, whose number is even, whose beginning is 1 and end is 128, and you will find them as been said.

*It is clarified from this that the terms of even-times-even numbers, when they are summed from one, their sum is odd number, as they are one less than the greater extreme, which is always an even number.

|The number that is an even-times-odd number, is divided into two parts of similar units, by the nature of its type, as it is an even number, but it stops there and none of its halves cannot be divided once more into halves.

:::Since the natural odd numbers that form these even numbers by their doubling, are added two by two,

*It is necessary for the even-times-odd number that the type of the value of each of its parts differs from the type of the names of their derivation.

::for its third, which is derived from 3 that is odd, is 2 <math>\scriptstyle{\color{blue}{\frac{1}{3}\sdot6=2}}</math>

:Thus, it is clear that the even-times-even number differs from the even-times-odd number, since the parts of the even-times-even number are named after the type of their value.

:It is thought that because of this, this type is called even-times-odd since the type of the names of their parts differs from the type of their value.

::Passing three terms further from 18 - the fourth term has a third: <math>\scriptstyle{\color{blue}{\frac{1}{3}\sdot30=10}}</math>

:*One find their thirds by their successive order are the terms of the even-times-odd number from their beginning by their order, none of them is missing.

*It is comprehended concerning the the even-times-odd numbers, that when its terms are in an even number, and they have two means, the sum of the two means is equal to the sum of the two extremes, and to the sum of the terms that follow the extremes, and to the sum of the terms that follow the terms that follow [the extremes].

:As explained previously concerning the terms of the even-times-even numbers, except that the property there was by multiplication, and here it is by addition.

:Double the one mean of terms of the even-times-even numbers is equal to the sum of the two extremes, and to the sum of the terms that follow the extremes, and to the sum of the terms that follow the terms that follow [the extremes].

|Hence, the difference of the even-times-even numbers and the even-times-odd numbers is already clear with regard to diffrence between the property of the product and the property of the sum of the two extremes, the single mean, or the two means.

|Furthermore, the even-times-even number and the even-times-odd number differ from each other also in that the even-times-odd number alone is divisible [= its other divisors are indivisible], while the only indivisible part of the even-times-even number is the smallest part, i.e. 1 [= its other divisors are divisible]

|The even-times-even-times-odd number is the third type of even numbers

*The similarity it adopts from the even-times-even number: it is divisible into two equal parts by the nature granted from its type.

:Hence, as it can be divided into two equal parts more than once it is similar to the even-times-even number and differ from the even-times-odd number.

:As its halving does not end with the unit it is similar to the even-times-odd number and differ from an even-times-even number.

|By every property, by which it is similar to one of the two types, it differs from the other type.

:Only the greatest part of the even-times-odd number is divisible [= its other divisors are indivisible].

:The only indivisible part of the even-times-even number is the smallest part [= i.e. 1 its other divisors are divisible]<br>

*[A property that the even-times-even-times-odd number] shares with the other:

:It has parts that are denominated by a number of the same type as the type of their value - by this it is similar to the even-times-even number.

:It has parts whose derivation differs from their value by the type - by this it is similar to the even-times-odd number.

|How this type is produced and grows:

:*The same is done with the third and fourth terms or whichever of the given terms and arranging all as is seen in this diagram:

:*A lengthwise line was added breadthwise [column 1], next to the line of the natural odds [column 2], in which the even-times-odd numbers were placed by the order of their terms, starting from 6, in order to demonstrate all the properties mentioned of the three types of even numbers, since this diagram includes them together.

===== The Mentioning of the Odd Number and The Explanation of its Categories =====

|The odd number, as said before, is that which is divided by any division into two unequal parts greater and less, and as its division is closer as possible to equality, one of them exceeds the other by one  

|It is divided naturally into two types and a third type occasionally occurs

*one of the naturally two types is called incomposite prime

====== The Mentioning of the Prime Incomposite Number ======

|The first type of odd numbers is incomposite, that has no divisor but the one.

::Such as: 3; 5; 7; 11; 13; 17

|These numbers and others like them has no divisor and no fractional part but that of which the numerator is one and the denominator is derived from the whole number itself.

|style="text-align:right;"|אלו המספרים ומה שימצא כמותם לא ימצא להם מספר ימנם ואין להם חלק כלל ימנם בלתי האחדים שנגזר להם חלק מכלל המספר

|It is incomposite, since it cannot be divided into equal parts, as it is produced from the sequence of units.

|style="text-align:right;"|ואמנם יהיה למספר הנפרד חלק ימנהו כאשר היה נכפל מזה המין מהנפרד ר"ל הנפרד הראשון הבלתי מורכב

:An even number necessarily has a divisor, even if its divisors are greatly diminished, since two is its divisor and the number of times by which two divides it is also its divisor.

*If it is multiplied an odd number of times, the product is an odd number <math>\scriptstyle\left(2m-1\right)\sdot\left(2n-1\right)</math>, so that the multiplied odd number divides it by the number of times by which it is multiplied, and the number of times, by which the prime incomposite odd number divides the multiplied odd number, divides also the product.

|style="text-align:right;"|ואם נכפל פעמים מספרם נפרד יחודש ממנו נפרד ימנהו אותו המספר הנפרד הנכפל בשעור אחדי הפעמים אשר נכפל בם וימנהו ג"כ כלל אחדי הפעמים אשר נכפל בם אותו המספר הנפרד הראשון הבלתי מורכב רצוני לומר בשעור אחדי אותו הנפרד הנכפל

====== The Mentioning of the Secondary Composite Odd Number ======

|style="text-align:right;"|<big>בזכר המספר הנפרד השני המורכב</big>

|The composite odd number has a divisor beside one, and that number is a fractional part of it.

|style="text-align:right;"|המספר הנפרד המורכב הוא אשר לו עם האחד מספר ימנהו וזה המספר הוא חלק לו

|style="text-align:right;"|וכבר אמרנו שהמספר הנפרד הראשון אמנם התילד מהמשך האחדים לבד

|Hence, this type of odd number consists of the prime odd number.

|style="text-align:right;"|הנה זה הנפרד אם כן מורכב מהנפרד הראשון

::*9 is composed from the duplication of 3 three times <math>\scriptstyle{\color{blue}{9=3\times3}}</math>.

::*15 is composed from the duplication of 5 three times, or from the duplication of 3 five times <math>\scriptstyle{\color{blue}{15=5\times3=3\times5}}</math>.

::*21 is composed from the duplication of 7 three times, or from the duplication of 3 seven times <math>\scriptstyle{\color{blue}{21=7\times3=3\times7}}</math>.

|The first type that has a fractional part that is derived from its name is not a number, it is the unity, for this is the property of the type that is the number, i.e. that it consists of the units. [?]

|style="text-align:right;"|והמין הראשון שיש לו חלק נגזר השם משמו אינו מספר והוא האחדות כי זה סגולת סוג הסוגים אשר הוא המספר ר"ל שהוא מורכב מהאחדים

|The secondary composite is unique by having a fractional part that is not derived from its name.

|style="text-align:right;"|ואמנם מה שנתיחד בו השני המורכב הנה הוא שיש לו חלק אינו נגזר משמו ימנהו

====== The Quality of the Third Type of Odd Number - The Relatively Prime Number ======

|It was already said in the categorization of the odd number that this type is created as odd accidentally, as it is by a comparison between two composite numbers that have no common number that counts them [= common divisor], but each of them, when it returns to its nature, has a number that counts it, which is a part [= divisor] of it.

::Such as 9 that is composed, as said, from the duplication of 3 three times, when it is compared with 25, which is composed from 5 five times, each of these two numbers is prime and incomposite compared to its companion, as they have no common number that counts them [=common divisor].  

|How these types, which are the types of the odd number, are recognized and how each type is sorted from the natural odd numbers, so that the prime incomposite is known from the secondary composite - this is done by technique that '''Eratosthenes''' called the "sieve":

|style="text-align:right;"|אמנם איך יוכרו אלו המינים אשר הם מיני הנפרד ואיך יוברר מין מין מהם מהנפרד הטבעי עד שיודע הראשון הבלתי מורכב מהשני המורכב הנה זה בתחבולה יקראה ארסטשאש המכבר

:By taking the natural odds, beginning from 3, by their natural succession, as in the table above.

::*Starting and leaving two intermediary terms after the first term, so that the fourth term is a secondary composite - the first term is 3, the fourth is 9, which is a secondary composite, as it is composed of 3 three times.

::*Leaving again two intermediary terms, so that the fourth term from it is also a secondary composite, which is 15 that is composed of 5 three times, or from 3 five times.

::Proceeding always like this, leaving two intermediary terms and the fourth is a secondary composite odd number, until reaching the end of the given natural odd numbers.

::*Then, starting from five, which is the second term of the natural odds, leaving four intermediary terms after it, so that the sixth term from it is a secondary composite, which is 15.

::*Again, leaving four intermediary terms after 25 also, and the sixth term from it is a secondary composite odd number.

::Proceeding like this - as long as there is a term after the first, continue leaving two intermediary terms for each term.

|When one does that and writes the terms of the secondary composite odds that are recognized by this technique, nothing is missing, and what remains of those that have no sign, after completing this and extracting them according to the described description, are the prime incomposite odd numbers.

|style="text-align:right;"|כי כשתעשה זה ורשמת על מדרגת הנפרד השני המורכב אשר תעמד עליהם בזאת התחבולה לא יבצר ממך דבר והיה מה שישאר אחר כלותך מזה ולקיחתו כפי זה הרושם אשר תארתי לך מאשר לא יפל בם סימן הם נפרדים ראשונים בלתי מורכבים

:Know that when you start from the first term, leaving two intermediary terms between each secondary composite and secondary composite:

::The first composite term is composed of the term, from which it began to be multiplied, i.e. by the measure of its units.

:And when you start from the second term, leaving four intermediary terms between each composite and composite, the resulting composite numbers will be composed of the second term, from which you started, multiplied by each of the others.

|Since it is already clear how to separate the secondary composite numbers from the prime incomposite odds by this "sieve", until the two natural types of the types of the odd number are recognized, each of them by itself, we shall continue to the recognizing of the third accidental type and discover how it is known by comparing two numbers, each of which is secondary composite by itself, whether they are prime each in comparison to its companion, i.e. that they have no common number that counts them [=common divisor], or if they do have a common number that counts them, how it is found.

|style="text-align:right;"|ואחר שכבר התבאר לך איך תברור בזה המכבר המספרים השניים המורכבים מהנפרדים הראשונים הבלתי מורכבים עד שיוכרו שני המינים הטבעיים ממיני הנפרד כל אחד מהם לבדו נמשיך לזה הכרת המין השלישי המקרי ונגלה איך יודע כשנקיש בין שני מספרים כל אחד מהם בעצמו שני מורכב אם הם עם ההקשה כל אחד מהם אצל חברו ראשון ר"ל שאין להם מספר משותף ימנם ואם היה להם מספר משותף ימנם איך נמצאהו

:And when what remains from the greater is less than the smaller, we subtract from the smaller the multiples of the remainder from the greater that are contained in it.

:Then, we subtract from the remainder of the greater the multiples of the remainder from the smaller that are contained in it.

:We do not stop subtracting from the greater the multiples of the less that are contained in it, until we reach either to the unity, or to two numbers such that the smaller of them counts the greater.

*If they reach to the unity, each of them is prime incomposite compared to the other.

*If we reach to two numbers such that the smaller of them counts the greater, then the smaller is the common number that counts the two numbers that are compared to each other.

::Example for numbers that each of them is a secondary composite by itself, but in comparison with the other it is prime incomposite: 9 and 25.

::*Subtracting the sevenths from 9, 2 remains.

::Hence, 9 and 25, each is prime incomposite compared with its companion.

::Example for numbers that each of them is a secondary composite by itself, and in comparison with each other they have a common number [= common divisor] that counts them, so they are secondary composite by it: 21 and 35.

::7 counts 14, therefore 21 and 35 have a common number that counts them [common divisor], which is 7, and they are in comparison secondary composite by it.

|The even-times-odd number is the divisible into two halves once that rests on an odd number, its example:

|The even-times-even-times-odd number is the divisible more than once that does not end in one but in an odd number, its example:

|The prime incomposite odd number is that which is not counted by any number at all and has no part but the one, its example:

|The secondary composite number is that which has a number that counts it, which is its part, and the number that counts it is solely an odd number, its example:

|The accidental number is by comparing two composite odd numbers that have no relatively common number at all, its example:

|This third type is as a mean between the first two types, which are as separate extremes, since they are in the greatest degree of distinction for their surpassing over equality by addition and subtraction.

:::Such as 12, 24, and their similar:

|As it is impossible to grasp the end of the departure from equality by addition in nature, as is said in the introduction of the book, so the subtraction in the quantity of the having magnitude is endless in comparison, except by a general expression.

|The equality is one mean between two things whose end is impossible to be grasped

|Yet, the perfect is rarely found, not as much as the first two types are multiplied.  

|The perfect even number is obtained by a technique that will be mentioned now.

====== The Discussion on the Quality of the Perfect Number and its Extraction ======

|This number resembles an animal, whose organs are balanced, mediator in its form.

|How these perfect numbers are produced, how we recognize their existing number and how many of them there are in every given [rank] - it is as described:

::The same is done from the pollowing terms of the even-times-even - setting beneath each term the sum that is summed from it and from what precedes it.  

::From these odd numbers all the prime incomposite numbers are recognized, and marked, and the others are left aside.

|style="text-align:right;"|הנה יוכרו מאלו המספרים הנפרדים כל מספר ראשון הבלתי מורכב ונרשום עליו רושם ונעזוב מה שזולתו

|style="text-align:right;"|וכאשר תחפש בזאת התחבולה כל מה שהנחת ממדרגות מספרי זוג הזוג הנה כבר הוצאת כל המספרים השלמים אשר הם נכללים תוך מספר המדרגה האחרונה ממה שהוצאת מהם ואותם המספרים הם אשר העמדתם בטור השלישי תחת הנפרדים הראשונים הבלתי מורכבים והם נלוים על משך סדרם בטבע לא ידולג ביניהם מספר שלם כלל

|The units that are left over from the tens of the perfect numbers are necessarily 6 and 8 alternately - these numbers are always found with this quality.

|Examine all that was mentioned concerning their quality and their extraction method in this diagram.

=== The Discussion on the Quality of the Relative Quantity and its Division into the Equal and the Unequal ===

|style="text-align:right;"|<big>הדבור בתאר הכמה הצירופיי והחלקו אל השוה ובלתי שוה</big>

|After elaborating the discussion on the absolute quantity, completing its discription and the explanation of its interprations, now the relative quantity will be discussed, and described by a proper description in accordance with what was described beforehand

|style="text-align:right;"|ואחר אשר הרחבנו המאמר על הכמה הנפרד והשלמנו תאריו ובאור פירושיו אנו נקח עתה לדבר בכמה המצטרף ונתארהו תאר נאות ומסכים למה שתארנו בו מה שקדם לפניו

|style="text-align:right;"|והכמה המצטרף יחלק חלוקה ראשונה לשני חלקים אחד מהם השווי והאחר לא שווי

|There is no supplement nor deficiency in equality, but one of the two relatives is equal to the other.

:As the hundred, when compared with the hundred, the ten with the ten,

|The species of the given numbers that differ in comparison with one another, in inequality, is divided in a second division, into two parts - one of them is greater, and the other is smaller, and they are different by name and antithetical by matter.

|The smaller is divided, like the greater related to it, into five species, each species is parallel to a species of the greater, ascribed and named according to its corresponding, i.e. the species of the smaller are called: the submultiple, the subsuperpaticular, the subsuperpartient, the submultiple-superparticular, and the submultiple-superpartient.

:He did not mean by superparticular ratio to the multiple superparticular ratio, but he meant the ratio between the two numbers, of which the greater is as the smaller once and as a part of the smaller.

:This is because he mentioned the numbers that are ascribed to the nine spheres, and related between them, relating to each supreme sphere, conceived by the intellect, their natures, of which the numbers ascribed, are of the numbers that are given to the spheres of the four elements.

::Since he explained why the sphere of these two stars is one and the number ascribed to them is one, by saying that these two stars are included in one matter, which is that their distance from the sun is only in the circles of their encirclements, and the sum of the movments of their centers and the sun is one, therefore one number is ascribed to them.

:*the sphere of the Saturn - 72

:His theory about the attributions to these spheres, that ascribed these numbers, was not by accident, without a nesessary reason, but it was resulted from a necessary discussion that approves that this is the greater first relation described of these numbers to these spheres in this ratio.

:When he related between these numbers, he mentioned that <math>\scriptstyle{\color{blue}{108:24}}</math> is not a simple ratio.

:'''He''' [= Nicomachus] '''said''': because of this we say that the fixed stars are far from the corporeal activities, that have essence.

|This is the explanation of that he did not mean by superparticular ratio to the multiple superparticular ratio.

|I did not elaborate this discussion, thinking that the similar of the author's discussions were [not] unknown to you, in your casuistry and diligence when studying his talk, and your love to this craft, all the more so that you are of those who have his books

|But, I wanted to remind you of it.

|I have no doubt that our book will fall into the hand of the one who is ignorant concerning the teacher's opinion, of which you yourself know.

|And when the discussion has no capacity of bringing to the essence of the thesis, the thoughts become confused, the conception falls, the truths disappear, and the axioms are non-existent.  

===== The Discussion on the Multiple Ratio - its Nature and it Production =====

|Now, we shall preface the discussion on the multiple ratio, since it is anterior by nature to the other four remaining ratios.

|style="text-align:right;"|ואמנם עתה נקדים המאמר על יחס בעל הכפלים אחר שהוא היותר קודם בטבע משאר היחסים הארבעה הנשארים

*When relating the beginning of the numbers, which is 2, to 1, which is the cause of the number, its double is found, and this is called the double ratio. <math>\scriptstyle{\color{blue}{2:1}}</math>

|style="text-align:right;"|וזה שאנו כאשר הקשנו תחלת המספרים אשר הוא ב' אל האחד אשר הוא עלת המספר מצאנוה כפל לו וזה הוא הנקרא הכפל השניי

*If relating 1 also to the second number, which is 3, the triple ratio is created. <math>\scriptstyle{\color{blue}{3:1}}</math>

|style="text-align:right;"|ואם הקשנו האחד גם כן אל המספר השני אשר הוא ג' יתחדש הכפל השלישי

*If it is related to the third number, which is 4, the quadruple ratio is created. <math>\scriptstyle{\color{blue}{4:1}}</math>

|style="text-align:right;"|ואם הקשנוהו אל המספר השלישי והוא ד' יתחדש הכפל הרביעי

|style="text-align:right;"|עוד כן כאשר יוקש אל מדרגות המספרים הטבעיים יתחדשו כפליים יקרא בשם המספר אשר אליו יוקש האחד

|It is clarified that the first ratio that subsists in number is the double ratio, when the beginning of the numbers, which is 2, is in comparison with the one, whose duplication generates the number.

|It is clarified also that the double ratio is the first of the multiple ratios by nature, as it is by relating the one to the beginning of the numbers.

|The species of the multiple ratios are endless in potentia, as they are growing with the growth of the natural numbers.

|We will start first with the description of the production the multiples, without the rest of the ratios

|style="text-align:right;"|ונתחיל ראשונה בספור התילד בעל הכפלים בלתי שאר היחסים

*The first of them by nature will prefaced:

|style="text-align:right;"|ויהיו מדרגותיו כמספר המדרגות אשר סדרת אותם בטור סדר המספרים הטבעיים

|style="text-align:right;"|עוד תקיש הראשון ממדרגות הזוג הטבעי בראשון ממדרגות המספר הטבעי והשני בשני והשלישי בשלישי וכן במה שאחר זה שאנו נמצאם יחד ביחס הכפל השני

::The two terms that follow this term are skipped and the third is taken and placed second in the other line.

::Likewise, the two terms that follow each taken term are skipped and the third is taken and arranged in the terms of the other line.

|style="text-align:right;"|ואחר נקיש בין המדרגות מהטור המוצא ובין המדרגה הראשונה מטור המספר הטבעיי עוד השנית בשנית והשלישית בשלישית הנה אנו נמצאם ביחס הכפל השלישי

:Thus it is clear that the triple ratio is generated by multiplying each of the terms of the natural numbers by three and placing these products in the terms of another line.

|style="text-align:right;"|הנה אם כן כבר התבאר שאמנם יתחדש יחס הכפל השלישיי מהכאת כל מדרגה ממדרגות המספרים הטבעיים בשלשה והעמדת אלו ההגעות במדרגות טור אחר

:It is necessary for the line of the greater numbers in this ratio, and any ratio that is derived from the duplication of an odd number to a number, that it will be an odd number, then an even number, then another odd number, as the observation shows.

|style="text-align:right;"|אמנם הכפל הרביעיי אמנם יתחדש מהכאת כל מדרגה ממדרגות המספר הטבעי בד' עוד יוקיש מה שיצא מן ההכאה אל אותה המדרגה

===== The Discussion on the Second Simple Ration which is the Superparticular Ratio =====

|style="text-align:right;"|<big>הדבור ביחס השני הפשוט והוא יחס המוסיף חלק</big>

|Since the discussion on the multiple ratio has already been brought, because it is simplier, and anterior by nature, as been explained, its successive by natural succession, which is the superparticular ratio is discussed:

|style="text-align:right;"|ואחר שכבר הבאנו הדבור על יחס הכפלים אחר שהוא יותר פשוט ויותר קודם בטבע כמו שבארנו נאמ' על אשר ימשך לו המשכות טבעי והוא המוסיף חלק

|When comparing the beginning of the numbers, which is 2, with its successive, which is 3, the three is to two as its similar and its half.[<math>\scriptstyle{\color{blue}{3=2+\left(\frac{1}{2}\sdot2\right)}}</math>]

|Hence, the superparticular ratio is formed by comparing the beginning of the numbers with its successive.

|style="text-align:right;"|הנה יתחדש המוסיף חלק מההקשה בין תחלת המספרים ואשר ימשך לו

|style="text-align:right;"|והתבאר שהחלק הראשון בטבע הוא המוסיף חצי

|style="text-align:right;"|עוד המוסיף שליש בהקשת המספר השלישי אשר הוא ד' אל המספר השני אשר הוא ג' כי ד' כמו ג' וכמו שלישיתו

|style="text-align:right;"|עוד המוסיף רביע