Part 18

Arithmetic: The sciences concerned with numbers Icon

January 19, 2022

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. 603
    • 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. 604
    • 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). 605
    • 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. 606
  • 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.610 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, 612 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 613

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 615 works on the subject is the small work by al-Hassar. 616 Ibn al-Banna’ al-Marrakushi deals with the (subject) in an accurate and useful brief description (talkhis) of the rules of calculation. 617 Ibn al-Banna’ later wrote a commentary on it in a book which he entitled Raf al-hijab. 618 The (Raf al-hijab) 619 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 620 the (work), the author competed with the Kitab Fiqh alhisab by Ibn Mun’im,621 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). 622 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.

Algebra of al-Khuwarizmi

Another subdivision of arithmetic is algebra. This discovers the unknown from the known data, if there exists a relationship between them requiring it.

Special technical terms have been invented in algebra for the various multiples (powers) of the unknown.

  • The first of them is called “number,” 624 because by means of it one determines the unknown one is looking for, discovering its value from the relationship of the unknown to it.
  • The second is (called) “thing,” 625 because every unknown as such refers to some “thing.” It also is called “root,” because (the same element) requires multiplication in second degree (equations). The third is (called) “property.” 626 It is the square of the unknown. Everything beyond that depends on the exponents of the two (elements) that are multiplied. 627

Then, there is the operation that is conditioned by the problem. One proceeds to create an equation between two or more different (units) of the (three) elements (mentioned).

The various elements are “confronted,” and “broken” portions (in the equation) are “set” 628 and thus become “healthy.” The degrees of equations are reduced to the fewest possible basic forms. Thus, they come to be three. Algebra revolves around these three basic forms:

  • number
  • thing
  • property

When an equation consists of one (element) on each side, the value of the unknown is fixed. The value of “property” or “root” becomes known and fixed, when equated with “number.” 630 A “property” equated with “roots” is fixed by the multiples of those “roots.” 631

When an equation consists of one (element) on one side and two on the there is a geometrical solution for it by multiplication in part on the other, unknown side of the equation with the two (elements).

Such multiplication in part fixes the (value) of (the equation). Equations with two (elements) on one side and two on the other are not possible. 633

The largest number of equations recognized by algebraists is six. The simple and composite equations of “numbers,” “roots,” and “properties” come to six. 6 34

The first to write on this discipline was Abu ‘Abdallah al-Khuwarizmi.635 Afterwads, there was Abu Kamil Shuja’ b. Aslam.636

His book on the six problems of algebra is one of the best books written on the subject.

  • Many Spanish scholars wrote good commentaries on it, the best being the book of al-Qurashi.637

Great eastern mathematicians have extended the algebraic operations beyond the six types and brought them up to more than 20. For all of them, they discovered solutions based on solid geometrical proofs. 639

Business (arithmetic)

Another subdivision of arithmetic is business (arithmetic). This is the application of arithmetic to business dealings in cities. These business dealings may concern the sale (of merchandise), the measuring (of land), the charity taxes, as well as other business dealings that have something to do with numbers.

In this connection, one uses both arithmetical techniques, 640 (and one has to deal) with the unknown and the known, and with fractions, whole (numbers), roots, and other things.

In this connection, very many problems have been posed. Their purpose is to give (the student) exercise and experience through repeated practice, until he has the firm habit of the craft of arithmetic.

Spanish mathematicians have written numerous works on the subject. The best known of these works are the business arithmetics of az-Zahrawi, 641 Ibn as-Samh, 64 2 Abu Muslim b. Khaldun, 643 who were pupils of Maslamah al-Majriti, and others.

Inheritance laws 644

Another subdivision of arithmetic is inheritance laws. It is a craft concerned with calculation, that deals with determining the correct shares of an estate for the legal heirs.

It may happen that there is a large number of heirs, and one of the heirs dies and his portions have to be (re-)distributed among his heirs. Or, the individual portions, when they are counted together and added up, may exceed the whole estate. 645

Or, there may be a problem when one heir acknowledges, but the others do not acknowledge, (another heir, and vice versa). All this requires solution, in order to determine the correct amount of the shares in an estate and the correct share that goes to each relative, so that the heirs get the amounts of the estate to which they are entitled in view of the total amount of the shares of the estate.

A good deal of calculation comes in here. It is concerned with whole (numbers), fractions, roots, knowns and unknowns; it is arranged according to the chapters and problems of inheritance law.

This craft, therefore, has something to do with jurisprudence, namely, with inheritance law, as far as it is concerned with the laws concerning the legal shares of inheritance, the reduction of the individual shares (‘awl), the acknowledgement or non-acknowledgement (of heirs), wills, manumission by will, and other problems.

It has also a good deal 646 to do with arithmetic, in as much as it is concerned with determining the correct amount of the shares in accordance with the law evolved by the jurists.

It is a very important discipline. The scholars who cultivate it have produced traditions attesting to its excellence, such as, for instance= “The fara’id (inheritance laws) constitute one-third of (religious) scholarship, and they are the first science to be abolished,” 647 and other such traditions. However, as was mentioned before, I am of the opinion that according to their plain meaning, all those traditions refer to individual “obligations” (fara’id), and not to the inheritance laws (fara’id).

The latter are too few in number to constitute one-third of (religious) scholarship, whereas individual obligations are numerous.Scholars, in early and late times, have written extensive works on the subject. Among the best works on the subject according to the school of Malik are the book of Ibn Thabit, the Mukhtasar of Judge Abul-Qasim al-Hawfi, and the books of Ibn al-Munammar, al-Ja’di, 648 az-Zawdi, 649 and others.

But al-Hawfi is pre-eminent. His book is preferable to all the others. A clear and comprehensive commentary on it was written by one of our teachers, Abu ‘Abdallah Muhammad b. Sulayman as-Satti, 650 the leading shaykh of Fez.

The Imam alHaramayn wrote works on the subject according to the school of ash-Shafi’i. They attest to his great scholarly capability and his firm grounding in scholarship. The Hanafites and the Hanbalites also (wrote works on the subject). The positions of scholars in scholarship vary. 651