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.

1. “Number”

This determines the unknown that one is looking for, discovering its value from the relationship of the unknown to it.

2. “Thing” [Variable]

Every unknown refers to some “thing.”

It also is called “root” because the same element requires multiplication in second degree equations.

3 “Property”

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 those 3 basic forms.

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.”

A “property” equated with “roots” is fixed by the multiples of those “roots.”

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 2 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.

The largest number of equations recognized by algebraists is 6.

The simple and composite equations of “numbers,” “roots,” and “properties” come to six.

The first to write on this discipline was Abu ‘Abdallah al-Khuwarizmi.

Afterwards, there was Abu Kamil Shuja’ b. Aslam.636

His book on the 6 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.

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.

Business Arithmetic

Another subdivision of arithmetic is business arithmetic.

This applies arithmetic to business dealings in cities.

These dealings may concern the sale of merchandise, the measuring of land, the charity taxes, and other dealings that involve numbers.

One uses arithmetic techniques, deals with the unknown and the known, and with fractions, whole numbers, roots, and other things.

In this connection, very many problems have been posed to give the student exercise and experience through repeated practice.

Spanish mathematicians have written numerous works on the subject.

The best known of these works are the business arithmetics of az-Zahrawi, Ibn as-Samh, Abu Muslim b. Khaldun, who were pupils of Maslamah al-Majriti, and others.

Inheritance laws

Another subdivision of arithmetic is inheritance laws.

It is a craft concerned with calculation that determines the correct shares of an estate for the legal heirs.

There might be many 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.

Or, there may be a problem when one heir acknowledges, but the others do not acknowledge, (another heir, and vice versa).

All this requires a solution.

A lot 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, inheritance law and its problems.

It has also a good deal 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