cannot be kasar (fraction) with respect to more than four LECTURE parties. Durr-ul-Mukhtár, page 877.

V.

XCIX. The tamásul (or similarity) of two Principle. numbers is the equality of one to the other (a).*. Vide Sirájiyyah, page 25.

(a.) As three to three: such two numbers are called Example. mutamásil (equal).—Sharífiyyah, page 59.

C. The tadákhul (or concordance) between two Principle. different numbers is when the smaller of the two numbers exactly measures the larger or exhausts it (b). Or we call it tadákhul between two numbers, when the larger of the two numbers is divided exactly by the smaller (c). Or we define it thus when one (number) or more equal to the smaller being increased upon it or added to it, the same (i. e., the smaller number) becomes equal to the larger (d). Or it is tadákhul when the smaller (number) is an aliquot part of the larger (e),—as three of nine (ƒ)† Vide Ibid.

tion.

(b.) That is, when you measure the larger by the Explanasmaller number twice or more times, nothing remains (in hand) of the larger number, as six and three; for, when you twice measure six by three, the former is totally exhausted in like manner, when you measure nine by three, the number nine is exhausted by those measures, consequently these (i. e., such) two numbers are technically called mutadákhil (one contained in the other).—Sharífiyyah, page 59.

(c.) That is divided in such a manner as to leave no Explanafraction;-as the number six is divided by three, as well as tion. by two without leaving a fraction.-Ibid.

(d.) For instance, when an equal number is once added Example. to three, it becomes six, when twice, it becomes nine.-Ibid.

* Numbers are said to be "mutamásil," or equal, where they exactly agree.-Macn. M. L., Chap. I, Sect. v, Princ. 70. See ante, p. 201.

†They are said to be mutadákhil or concordant, where one number being multiplied exactly measures the other.-Ibid, Princ. 71, See ante, p. 201,

V.

LECTURE (e.) These different definitions (of tadákhul) are merely (different) wordings; for, if the smaller number measures (or exhausts) the larger, the former is technically called an aliquot part of it, if it does not, it is (so many) units thereof.-Sharifiyyah, page 60.

Explanation.

Reason.

Principle.

Example.

(f) Because, the number three is one-third part of nine, and, as such, it is an aliquot part of nine, and exhausts it by measuring thrice, and becomes equal thereto by the double addition of itself; so the number nine is divided by it without leaving a fraction as already shown. These are the illustrations of all the definitions or descriptions of tadákhul.-Ibid.

CI. The tawafuk (agreement) of two numbers is, when the smaller does not (exactly) measure the larger, but a third number measures (or exhausts) them both (g).* See ante, p. 201.

(g.) As eight and twenty, (the former of which does not measure the latter, butt) both of them are measured by four (which exhausts eight by measuring it twice, and twenty by measuring it five times;†) so they agree in a fourth; since the number measuring them is the denominator of a fraction common to both.-Sirájiyyah, page 26.

CII. The tabáyun or difference of two numbers is, when no third number measures the two (discordant) numbers (h).‡-Sirájiyyah, page 25.

(h.) As nine and ten, (for no number measures them both, except one, which, according to the author, is not a measuring number.†) —Ibid, page 26.

Now the way of knowing the agreement or disagreement between two different quantities is, that the greater is diminished by the smaller quantity on both sides, once or often until they agree in one point; and if they agree in unit only, there is no (numerical) agreement between them;

*They are said to be mutáwafik, or composit, where a third number measures them both.-Ibid, Princ. 72.

† Sharífiyyah, p. 60.

They are said to be mutabáyin, or prime, where no third number measures them both.-Ibid, Princ. 73. See ante, page 201.

V.

but if they agree in any other number, then they are (said LECTURE to be) mutwáfik (composit) in a fraction of which that number is the denominator (i).—If (they agree) in two, they are mutwáfik in half, (as in the instance of four and ten,*) if in three, (they are mutwáfik) in a third, (as in the instance of nine and twelve,*) if in four, (they are mutwáfik) in a quarter (as in the instance of eight and twelve,*) and so on as far as ten. Above ten, (they agree) in a fraction, I mean, if the common measure be eleven, they agree in a fraction of eleven (j);† if in fifteen, (the yagree) in a fraction of fifteen (k).—Sirájiyyah, page 26.

(i.) If you diminish a large number by the repetition of a small number, and if the large is exhausted by it, then they are mutadákhil (concordant); but if there remain one, then they are mutabáy in (prime), as no number except one measures them both.-Sharífiyyah, page 61.

(j) As twenty-two and thirty-three; for the number which measures or exhausts them is eleven only, the denominator, therefore, is a fraction of eleven, (that is, eleventh). Ibid, page 62.

(k.) As thirty and forty-five, both of which being measured by fifteen, are mutwáfik (composit) in a fraction thereof.-Sharifiyyah, page 62.

"The reason," says Sharif, "for limiting the relations of four sorts between numbers is, that when you compare one number with another, if they be the same in quantity, they are equal (mutamásil); if they do not, then, if the smaller measure or exhaust the greater, they are mutadákhil (concordant); if it (the smaller number) do not measure or exhaust (the other,) but a third number (excepting one) measure them, they are mutwáfik (composit); but if a third number also do not measure them, then they are mutabáy in (prime or discordant).-Sharífiyyah, pages 62, 63.

There are seven principles or rules of arrangement of cases respecting the division of shares. Of these rules,

* Sharífiyyah, page 62.

† By a fraction' is here meant the ordinal number of eleven, viz., 'the eleventh; they call it a fraction, because in the Arabic language there is no ordinal adjective of numbers exceeding ten, which deficiency, therefore, is supplied by the term a fraction' of that number.

LECTURE three concern persons and shares, and four persons and persons.*— Vide Sirájiyyah, page 27.

V.

Principle.

Example.

Expla

nation.

Of the three rules which concern shares and persons,

CIII. The first is-when the portions of all the classes (of heirs) are divided among them without a fraction, there is no need of multiplication.†-Sirájiyyah, page 28.

As (if a man leave) both parents and two daughters (1).-Ibid.

(1.) For the division in this case is by six, and a sixth part thereof, that is one (share), is for each of the parents; and two-thirds, that is four (shares, are) for the two daughters, two for each: thus, the portions of (all) the heirs are allotted without a fraction.-Sharífiyyah, page 63.

ANNOTATIONS.

ciii. Recourse is had to these in order that the shares may be (allotted and) received in the smallest possible number (or quantity) in a manner that there be no fraction upon (i.e., in the share of) any one of the heirs. If their portions agree with the number of their persons, then you must multiply the measure of (the number) of those persons by the root of the case, or by its increase, if it be an increased case. As in the case of one widow and six brothers (being left by a man) the brothers are entitled to three (out of four) shares, which (three) agree with their number by a third; consequently you must multiply two by four, and the case would be arranged by eight.-Durr-ul-Mukhtár, page 877.

* There are seven rules of distribution, the first three of which depend upon a comparison between the number of the heirs and the number of the shares; and the four remaining ones upon a comparison of the numbers of the different sets of heirs, after a comparison of the number of each set of heirs with their shares.-Macn. M. L., Chap. I, Sect. v, Princ. 74.

The first is when, on a comparison of the number of the heirs and the number of the shares, it appears that they exactly agree, there is no occasion for any arithmetical process. Thus where the heirs are a father, a mother, and two daughters, the share of the parents is one-sixth each, and that of the daughters, two-thirds. Here the division must be by six; of which each parent takes one, and the remaining four go to the two daughters.-Ibid, Princ. 75.

V.

CIV. The second is, that if the portions of one LECTURE class be fractional, yet there be an agreement between the portions and persons, then the measure of Principle. the number of those persons, whose shares are broken, must be multiplied by the root of the case, or by the increase thereof, if it be an increased one.* Sírájiyyah, page 27.

As (when a man leaves) both parents and ten Example. daughters (m);—or (when a woman leaves) a husband, both parents, and six daughters (n).—Ibid.

(m.) The first is the example of a case without increase, for (the number of) the root of the case is six, of which two-sixths, that is two, are for the parents, which are divided between them without a fraction, two-thirds, which are four, are for the ten daughters, but the same cannot be divided among them without fractions, there is, however, agreement in half between four and ten, for, the number which measures them both is two; consequently, the number of the persons that is ten, is reduced to half, that is five, and is multiplied by six, which is the root of the case, and the product is thirty, by which the case is arranged; inasmuch as, the two parents had two shares out of the root of the case, the same being multiplied by five, the multiplicand, amounted to ten, five of which is for each of them. The daughters had four (shares) out of the root of the case, which being multiplied by five amounted to twenty, so two go to each of them.

(n.) The second is the example of an increased case, for, here (the number of) the root of the case is twelve, by

* The second (rule) is when, on a comparison of the number of the heirs. and the number of sharers, it appears that the heirs cannot get their portions without a fraction, and that some third number measures them both, when they are mutwáfik or composit; as in the case of a father, a mother, and ten daughters. Here the division must be by six. But when each parent has taken a sixth, there remain only four to be distributed among the ten daughters, which cannot be done without a fraction, and on a comparison of the number of heirs who cannot get their portions without a fraction, and the number of shares remaining for them, they appear to be composit, or agree in two. In this case, the rule is, that half the number of such heirs, which is 5, must be multiplied into the number of the original division 6 thus 5×6=30; of which the parents take ten or five each, and the daughters twenty or two each.-Macn. M. L., Chap. I, Sect. v,

Princ. 76.