Epistles of the Brethren of Purity

Praise be to God, and peace be upon His servants whom He has chosen.

Know, O virtuous and compassionate brother, that since it is the doctrine of our noble brethren, may God support them, to examine all the sciences of the beings that exist in the world, including substances and accidents, simple things and abstract things, single things and composite things; and to investigate their principles, the number of their genera and species, their properties, their arrangement and order as they presently are, and the manner of their coming into being and arising from one cause, one principle, and one Originator, exalted is His majesty; and since they use numerical examples and geometrical demonstrations to clarify these matters, as the Pythagorean sages used to do, we needed to place this epistle before all our other epistles.

In it, we shall mention something of the science of number and its properties, which is called arithmetic, as a kind of entrance and preliminary introduction, so that the path may be made easier for learners who seek wisdom, which is called philosophy, and so that it may become more accessible to beginners who are studying the mathematical sciences.

So we say:

On Philosophy and the Order of the Sciences

Philosophy begins with the love of the sciences. Its middle is the knowledge of the realities of existing things, according to human capacity. Its end is speech and action in accordance with knowledge.

The philosophical sciences are of four kinds. The first is mathematics, the second is logic, the third is the natural sciences, and the fourth is the divine sciences.

Mathematics is of four kinds. The first is arithmetic, the second is geometry, the third is astronomy, and the fourth is music.

Music is the knowledge of the composition of sounds, through which the principles of melodies are derived.

Astronomy is the science of the stars, by means of the demonstrations mentioned in the book of the Almagest.

Geometry is the science of measurement, by means of the demonstrations mentioned in the book of Euclid.

Arithmetic is the knowledge of the properties of number and of the meanings among existing things that correspond to them, as mentioned by Pythagoras and Nicomachus.

The first of the philosophical sciences to begin with is mathematics. The first part of mathematics is the knowledge of the properties of number, because it is the easiest of the sciences to grasp. Then comes geometry, then composition, then astronomy, then logic, then the natural sciences, and then the divine sciences.

This, then, is the first thing we shall say concerning the science of number, as an entrance and preliminary introduction.

On the One and the Many

Words indicate meanings, and meanings are the things named. Words are names. The most general of all words and names is our saying: “thing.”

A thing is either one or more than one.

“One” is said in two ways: either truly or metaphorically.

The truly one is the thing that has no part whatsoever and is not divisible. Everything that is indivisible is one from the aspect by which it is indivisible. You may also say: the one is that in which there is no other, insofar as it is one.

As for the metaphorically one, it is any totality that is called one, as when one says: one ten, one hundred, or one thousand.

The one is one by unity, just as the black is black by blackness. Unity is an attribute of the one, just as blackness is an attribute of the black.

Multiplicity, however, is a collection of ones. The first multiplicity is two, then three, then four, then five, and whatever exceeds that, however far it may go.

On Number, Numbered Things, and Calculation

Multiplicity is of two kinds: either number or numbered things. The difference between them is that number is the quantity of the forms of things in the soul of the counter, whereas numbered things are the things themselves. Calculation is the gathering together and separating of number.

Number is of two kinds: whole numbers and fractions.

The one that comes before two is the root and principle of number. From it all number arises, both whole numbers and fractions, and to it number returns when dissolved.

As for the arising of whole number, it occurs by increase. As for fractions, they arise by division.

The Arising of Whole Number

An example of the arising of whole number is as follows. When another one is added to the one, it is then said that they are two. When another one is added to them, that whole is called three. When another one is added to it, it is called four. When another one is added to it, it is called five. In this same manner, whole number arises by increase, one by one, however far it may go.

The Return of Number to the One

As for the reduction of number back to the one, it is according to this example. If one is taken from ten, nine remain. If one is removed from nine, eight remain. If one is subtracted from eight, seven remain. In this way, one after another is removed until only one remains.

From the one, however, nothing can be removed, because it has no part whatsoever.

Thus it has become clear how whole number arises from the one and how it dissolves back into it.

The Arising of Fractions

As for the arising of fractional number from the one, it is according to this example. When the whole numbers are arranged in their natural order, which is: one, two, three, four, five, six, seven, eight, nine, ten, and then the one is indicated within each totality, it becomes clear how fractional number arises from the one.

When the one is indicated within the two, the one is then called a half.

When it is indicated within the totality of three, it is called a third.

When it is indicated within the totality of four, it is called a fourth.

When it is indicated within the totality of five, it is called a fifth.

Likewise, there are the sixth, the seventh, the eighth, the ninth, and the tenth.

Again, when the one is indicated within the totality of eleven, it is called one part of eleven. Within twelve, it is called half of a sixth. Within thirteen, it is called one part of thirteen. Within fourteen, it is called half of a seventh. Within fifteen, it is called a third of a fifth.

According to this example, the rest of the fractions are to be understood.

Thus it has become clear how number arises from the one, both whole number and fractions together, and how the one is the root of both of them.

The Four Ranks of Whole Number

Know, my brother, that whole number is arranged in four ranks: units, tens, hundreds, and thousands. The units are from one to nine, the tens from ten to ninety, the hundreds from one hundred to nine hundred, and the thousands from one thousand to nine thousand.

All of these are contained within twelve simple words. From one to ten there are ten words, along with the word “hundred” and the word “thousand,” making twelve simple words altogether.

As for the remaining expressions, they are derived from these, or compounded from them, or repeated. The repeated expressions are like twenty from ten, thirty from three, forty from four, and the like.

The compounded expressions are like two hundred, three hundred, four hundred, and five hundred, for they are composed of the word “hundred” together with the other units. Likewise, two thousand, three thousand, and four thousand are composed of the word “thousand” together with the other expressions from the units, tens, and hundreds, as one says five thousand, seven thousand, twenty thousand, one hundred thousand, and the rest.

Its form is this:

The Abjad Ranks of Number

As for the units, they are أ ب ج د ه و ز ح ط ي. As for the tens, they are ك ل م ن س ع ف ص. As for the hundreds, they are ق ر ش ت ث خ ذ ض ظ. As for the thousands, they are غ بغ جغ دغ هغ وغ زغ حغ طغ يغ.

The Fourfold Arrangement of Number

Know that the fact that number is arranged in four ranks, namely units, tens, hundreds, and thousands, is not a necessary matter inherent in the nature of number, as are its being even and odd, whole and fractional, and its divisions one beneath another. Rather, it is a conventional arrangement established by the sages through their own choice.

They did this so that numerical matters would correspond to the ranks of natural things. For most natural things have been made by the Creator, exalted be His praise, in arrangements of four.

Examples of this are the four natures: heat, coldness, moisture, and dryness; the four elements: fire, air, water, and earth; the four humours: blood, phlegm, and the two biles, yellow bile and black bile; the four seasons: spring, summer, autumn, and winter; the four directions; the four winds: east, west, south, and north; the four astrological pivots: the ascendant, the descendant, the midheaven, and the lower heaven; and the four generated beings: minerals, plants, animals, and human beings.

In this way, most natural things are found to be arranged in fours.

The Four Spiritual Ranks Above Nature

Know that these natural things have mostly become arranged in fours through the care of the Creator, exalted be His praise, and according to the requirement of His wisdom, so that the ranks of natural things may correspond to the spiritual things that are above natural things. These spiritual things are not bodies.

The things above nature are in four ranks. The first is the Creator, exalted is His majesty. Beneath Him is the Universal Active Intellect. Beneath that is the Universal Soul. Beneath that is Prime Matter. None of these are bodies.

Know, my brother, may God support you and us with a spirit from Him, that the relation of the Creator, exalted be His praise, to existing things is like the relation of the one to number.

The relation of the Intellect to existing things is like the relation of two to number. The relation of the Soul to existing things is like the relation of three to number. The relation of Prime Matter is like the relation of four.

The Root of All Number in One Through Four

Know, my brother, may God support you and us with a spirit from Him, that all number, including its units, tens, hundreds, thousands, and whatever exceeds them however far it may go, has its root in one through four. These are:

This is because all other numbers are composed from these, arise from these, and have these as their root.

The explanation of this is that if one is added to four, it becomes five. If two are added to four, it becomes six. If three are added to four, it becomes seven. If one and three are added to four, it becomes eight. If two and three are added to four, it becomes nine. If one, two, and three are added to four, it becomes ten.

The same rule applies to all other numbers among the tens, hundreds, thousands, and whatever exceeds them however far it may go.

Likewise, the roots of script are four, and all other letters are composed from them. Speech is composed from letters, as we shall explain later. Consider this, and you will find that what we have said is true and correct.

Whoever wishes to know how the Creator, exalted be His praise, originated things in the Intellect, how He caused them to exist in the Soul, and how He formed them in Matter, should consider what we have mentioned in this chapter.

The Origin of Intellect, Soul, Matter, and Creation

Know, my brother, that the first thing the Creator, exalted be His praise, originated and brought forth from the light of His unity was a simple substance called the Active Intellect, just as He produced two from one by repetition.

Then He produced the Universal Celestial Soul from the light of the Intellect, just as He produced three by adding one to two.

Then He produced Prime Matter from the movement of the Soul, just as He produced four by adding one to three.

Then He produced all other created beings from Matter and arranged them through the mediation of the Intellect and the Soul, just as He produced all the rest of number from four by adding to it what came before it, as we have shown previously.

The One and Divine Unity

Know, my brother, may God support you with a spirit from Him, that if you reflect on what we have mentioned concerning the composition of number from the one that comes before two, and its arising from it, you will find it among the clearest proofs of the unity of the Creator, exalted be His praise, and of the manner in which He originated things and brought them forth.

For although the existence and composition of number can be conceived from the one that comes before two, as we have explained earlier, the one itself has not changed from what it was, nor has it been divided.

Likewise God, mighty and exalted, although He is the One who originated things from the light of His unity, brought them forth, and caused them to arise, and although through Him they have their subsistence, permanence, completion, and perfection, He has not changed from the unity in which He was before originating and bringing them forth, as we have explained in the Epistle on Intellectual Principles.

Thus we have informed you, through what we have mentioned, that the relation of the Creator, exalted be His praise, to existing things is like the relation of the one to number. Just as the one is the root of number, its source, its beginning, and its end, so too God, mighty and exalted, is the cause of things, their Creator, their beginning, and their end.

Just as the one has no part and no equal in number, so too God, exalted be His praise, has no equal and no likeness among His creation.

And just as the one encompasses all number and numbers it, so too God, exalted is His majesty, knows all things and their essences. Exalted is God far above what the wrongdoers say.

The Sixteen Ranks of Number According to the Pythagoreans

Know, my brother, that among most nations the ranks of number are four, as we have already mentioned. As for the Pythagoreans, they are arranged in sixteen ranks. Their form is this: