On the Equality, Proportion, Agreement, and Difference of two Numbers.

THE temáthul of two numbers is the equality of one to the other; the tedákbul is, when the smaller of two numbers exactly measures the larger, or exhaufts it; or we call it tedákhul, when the larger of two numbers is divided exactly by the fmaller; or we may define it thus, when the larger exceeds the fmaller by one number or more equal to it, or equal to the larger; or it is, when the smaller is an aliquot part of the larger, as three of nine. The tawafuk, or agreement, of two numbers is, where the smaller does not exactly measure the larger, but a third number measures them both, as eight and twenty, each of which is measured by four, and they agree in a fourth; fince the number measuring them is the denominator of a fraction common to both. The tabáyun of two numbers is, when no third number whatever measures the two difcordant numbers, as nine and ten. Now the way of knowing the agreement or disagreement between two different quantities is, that the greater be diminished by the fmaller quantity on both fides, once or oftener, until they agree in one point; and if they agree in unit only, there is no numerical agreement between them; but, if they agree in any number, then they are (Said to be) mutawáfik in a fraction, of which that number is the denominator; if two, in half; if three, in a third; if four, in a quarter; and fo on, as far as ten; and, above ten, they agree in a fraction; I mean, if the number be eleven, the fraction of eleven, and, if it be fifteen, by the fraction of fifteen. Pay attention to this rule.

On

On Arrangement.

IN arranging cafes there is need of seven principles; three, between the shares and the persons, and four between persons and perfons. Of the three principles the firft is, that, if the portions of all the claffes be divided among them without a fraction, there is no need of multiplication, as if a man leave both parents and two daughters. The fecond is, that, if the portions of one class be fractional, yet there be an agreement between their portions and their perfons, then the measure of the number of perfons, whose shares are broken, must be multiplied by the root of the case, and its increase, if it be an increased case, as if a man leave both parents and ten daughters, or a woman leave a husband, both parents, and fix daughters. The third principle is, that, if their portions leave a fraction, and there be no agreement between those portions and the persons, then the whole number of the perfons, whose shares are broken, must be multiplied into the root of the cafe, as if a woman leave her husband and five fifters by the fame father and mother. Of the four other principles the first is, that, when there is a fractional division between two claffes or more, but an equality between the numbers of the perfons, then the rule is, that one of the numbers be multiplied into the root of the cafe; as if there be fix daughters, and three grandmothers, and three paternal uncles. The fecond is, when fome of the numbers equally measure the others; then the rule is, that the greater number be multiplied into the root of the cafe; as, if a man leave four wives and three grandmothers and twelve paternal uncles. The third is, when fome of the numbers are mutawafik, or compofit, with others; then the rule is, that the measure of the first of the numbers be multiplied into the whole of the second, and the product into the measure of the third, if the product of the third be mutawafik, or, if not, into the whole of the third, and then into the fourth, and fo on, in the fame

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same manner; after which the product must be multiplied into the root of the case: as, if a man leave four wives, eighteen daughters, fifteen female ancestors, and fix paternal uncles. The fourth principle is, when the numbers are mutabáyan, or not agreeing one with another; and then the rule is, that the first of the numbers be multiplied into the whole of the fecond, and the product multiplied by the whole of the third, and that product into the whole of the fourth, and the last product into the root of the case; as, if a man leave two wives, fix female ancestors, ten daughters, and seven paternal uncles.

Section,

WHEN thou defireft to know the fhare of each clafs by arrangement, multiply what each clafs has from the root of the cafe by what thou haft already multiplied into the root of the cafe, and the product is the share of that class; and, if thou desirest to know the fhare of each individual in that clafs by arrangement, divide what each clafs has from the principle of the case by the number of the perfons in it, then multiply the quotient into the multiplicand, and the product will be the share of each individual in that clafs. Another method is, to divide the multiplied number by whichever clafs thou thinkeft proper, then to multiply the quotient into the fhare of that set, by which thou haft divided the multiplied number, and the product will be the share of each individual in that fet. Another method is by the way of proportion, which is the cleareft; and it is, that a proportion be ascertained for the share of each class from the root of the cafe to the number of perfons one by one, and that, according to fuch proportion from the multiplied number, a share be given to each individual of that class.

On

On the Divifion of the Property left among Heirs and among Creditors.

If there be a disagreement between the property left and the number arifing from the arrangement, then multiply the portion of each heir, according to that arrangement, into the aggregate of the property, and divide the product by the number of the arrangement, but, when there is an agreement between the arrangement and the property left, then multiply the portion of each heir, according to the arrangement into the measure of the property, and divide the product by the measure of the number arifing from the arrangement: the quotient is the portion of that heir in both methods. This rule is in order to know the portion of each individual among the heirs; but, in order to know the portion of each class of them, multiply what each class has, according to the root of the cafe, into the measure of the property left, then divide the product by the measure of the case, if there be an agreement between the property left and the cafe ; but, if there be a disagreement between them, then multiply into the whole of the property left, and divide the product by the whole number arising from the verification of the cafe; and the quotient will be the portion of that class in both methods. Now, as to the payment of debts, the debts of all the creditors ftand in the place of the arranging number.

On Subtraction.

WHEN any one agrees to take a part of the property left, fubtract his share from the number arifing by the proof, and divide the remainder of the property by the portions of those who remain; as if a woman leave

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her husband, her mother, and a paternal uncle: now fuppofe that the husband agrees to take what was in his power of his bridal gift to the wife; this is deducted from among the heirs: then what remains is divided between the mother and the uncle in thirds, according to their legal shares; and thus there will be two parts for the mother, and one for the uncle.

On the Return.

THE return is the converse of the increase; and it takes place in what remains above the shares of those entitled to them, when there is no legal claimant of it: this furplus is returned to the sharers according to their rights, except the husband or the wife; and this is the opinion of all the Prophet's companions, as ÂLI and his followers, may GOD be gracious to them! And our mafters (to whom GOD be merciful!) have affented to it: ZAID, the fon of THÁBIT fays, that the surplus doth not revert, but goes to the publick treasury; and to this opinion have assented ÛRWAH and ALZUHRÍ and MÁLIC and ALSHAFIÎ,

Now the cafes on this head are in four divifions: the first of them is, when there is in the cafe but one fort of kinfmen, to whom a return must be made, and none of those who are not entitled to a return: then settle the cafe according to the number of perfons; as, when the deceased has left two daughters, or two fifters, or two female ancestors; fettle it, therefore, by two. The fecond is, when there are joined in the cafe two or three forts of thofe, to whom a return must be made, without any of those, to whom there is no return: then settle the case according to their shares; I mean by two, if there be

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