19.
The first of them is arithmetic. Arithmetic is the knowledge of the properties of numbers combined in arithmetic or geometric progressions.
For instance, in an arithmetic progression, in which 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 one-half of the second and the second onehalf of
the third, and so on, or if the first is one-third 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
The same applies to special properties originating in
connection with even numbers, odd numbers, the powers of two, odd
numbers multiplied by two,
This discipline is the first and most evident part of
mathematics. It is used in the proofs of the mathematicians.
A subdivision of arithmetic is the craft of calculation. It is a scientific craft concerned with the counting operations of "combining," and "separating." The "combining" may take place by (adding the) units. This is addition. Or it may take place by increasing a number as many times as there are units in another number. This is multiplication. The "separating" may take place by taking away one number from another and seeing what remains. This is subtraction. Or it may take place by separating a number into equal parts of a given number. This is division.
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. 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. It has been said that 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
God guides with His light whomever He wants (to guide).
Another subdivision of arithmetic is algebra. This is a
craft that makes it possible to discover 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,"
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"
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."
When an equation consists of one (element) on one side
and two on the other,
The largest number of equations recognized by algebraists
is six. 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.
We have heard that great eastern mathematicians have
extended the algebraic operations beyond the six types and brought them
up to more than twenty. For all of them, they discovered solutions based
on solid geometrical proofs.
God "gives in addition to the creatures whatever He
wishes to give to them."
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, 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,
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.
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. And it has also a good deal
It is a very important discipline. The scholars who
cultivate it have produced traditions attesting to its excellence, such
as, for instance: "The
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
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,
"God guides whomever He wants to guide." |
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