Arithmetic: The sciences concerned with numbers

This is the knowledge of the properties of numbers combined in arithmetic or geometric progressions.

Arithmetic progression is where:

• each number is always higher by one than the preceding number
• the sum of the first and last numbers of the progression is equal to the sum of any two numbers in the progression that are equally far removed from the first and the last number, respectively, of the progression.
• Or, the sum of the first and last numbers of a progression is twice the middle number of the progression, if the total number of numbers (in the progression) is an odd number. It can be a progression of even and odd numbers, or of even numbers, or of odd numbers.
• Or, if the numbers of a geometrical progression are such that the first is half of the second and the second onehalf of the third, and so on,
• or if the first is 1/3 of the second and the second one-third of the third, and so on, the result of multiplying the first number by the last number of the progression is equal to the result of multiplying any two numbers of the progression that are equally far removed from the first and the last number, respectively, of the progression.
• Or, (the result of multiplying the first number by the last number of a geometrical progression,) if the number of numbers (in the progression) is odd, is equal to the square of the middle number of the progression. For instance, the progression may consist of the powers of two= two, four, eight, sixteen.
• Or, there are the properties of numbers that originate in the formation of numerical muthallathah (triangle), murabba’ah (square), mukhammasah (pentagon), and musaddasah (hexagon) progressions, 607 where the numbers are arranged progressively in their rows by 608 adding them up from one to the last number.

Thus, a muthallath(ah) is formed.

Other muthallathahs are placed successively in rows under the “sides.” Then, each muthallathah is increased by the “side” in front of it. Thus, a murabba’ah is formed.

Then, each murabba’(ah) is increased (by the “side”) in front of it. Thus, a mukhammasah is formed, and so on.

The various progressions of “sides” form figures. Thus, a table is formed with vertical and horizontal rows.

The horizontal rows are constituted by the progression of the numbers (one, two, etc.), followed by the muthallathah, murabba’ah, mukhammasah progressions, and so on.

The vertical rows contain all the numbers and certain numerical combinations. The totals and (the results of) dividing some of the numbers by others, both vertically and horizontally, (reveal) remarkable numerical properties.

They have been evolved by the inductive method. The problems connected with them have been laid down in the systematic treatments of (arithmeticians).

The same applies to special properties originating in connection with even numbers, odd numbers, the powers of two, odd numbers multiplied by two,609 and odd numbers multiplied by multiples of two.

They are dealt with in this discipline, and in no other discipline.This discipline is the first and most evident part of mathematics.

It is used in the proofs of the mathematicians. 611 Both early and later philosophers have written works on it.

Most of them include it under mathematics in general and, therefore, do not write monographs on it.

This was done by Avicenna in the Kitab ash-Shifa’ and the Kitab an-Najah, and by other early scholars.

The subject is avoided by later scholars, since it is not commonly used in practice), being useful in (theoretical mathematical) proofs rather than in practical calculation.

They handled the subject the way) it was done, for instance, by Ibn al-Banna, in the Kitab Raf al-hijab.

They extracted the essence of the subject (as far as it was useful) for the theory of (practical) calculation and then avoided it. And God knows better.

The Craft of Calculation

A subdivision of arithmetic is the craft of calculation.

It is a scientific craft concerned with the counting operations of “combining,” and “separating.”

• Addition is the “combining” of the units.
• Multiplication is the increase of a number as many times as there are units in another number.
• Subtraction is the “separating” of one number from another and seeing what remains
• Division is the separating a number into equal parts of a given number.

These operations may concern either whole numbers or fractions. A fraction is the relationship of one number to another number. Such relationship is called fraction.

Or they may concern “roots.” “Roots” are numbers that, when multiplied by themselves, lead to square numbers. 614

Numbers that are clearly expressed are called “rational,” and so are their squares.

They do not require (special) operations in calculation. Numbers that are not clearly expressed are called “surds.”

Their squares may be rational, as, for instance, the root of three whose square is three.

Or, they may be surds, such as the root of the root of three, which is a surd. They require (special) operations in calculation. Such roots are also included in the operations of “combining” and “separating.”

This craft is something newly created. It is needed for business calculations.

Scholars have written many works on it. They are used in the cities for the instruction of children.

The best method of instruction is to begin with (calculation), because it is concerned with lucid knowledge and systematic proofs.

As a rule, it produces an enlightened intellect that is trained along correct lines.

Whoever applies himself to the study of calculation early in his life will as a rule be truthful, because calculation has a sound basis and requires self-discipline.

Soundness and self-discipline will, thus, become character qualities of such a person. He will get accustomed to truthfulness and adhere to it methodically.

In the contemporary Maghrib, one of the best simple works on the subject is the small work by al-Hassar.

Ibn al-Banna’ al-Marrakushi deals with the (subject) in an accurate and useful brief description (talkhis) of the rules of calculation.

Ibn al-Banna’ later wrote a commentary on it in a book which he entitled Raf al-hijab.

The Raf al-hijab is too difficult for beginners, because it possesses a solid groundwork of (theoretical) proofs. It is an important book.

We have heard our teachers praise it. It deserves that. In the work, the author competed with the Kitab Fiqh alhisab by Ibn Mun’im, and the Hamid by al-Ahdab.

He gave a resume of the proofs dealt with in these two works, but he changed them in as much as, instead of ciphers, he used clear theoretical reasons in the proofs.

They bring out the real meaning and essence of (what in the work itself isexpressed by calculations with ciphers). All of them are difficult.

The difficulty here lies in the attempt to bring proof. This is usually the case in the mathematical sciences.

All the problems and operations are clear, but if one wants to comment on them - that is, if one wants to find the reasons for the operations - it causes greater difficulties to the understanding than (does) practical treatment of the problems.