doubled. Now, when any of these shares occur in cafes fingly, the divifor for each fhare is that number which gives it its name (except half, which is from two), as a fourth denominated from four, an eighth from eight, and a third from three: when they occur by two or three, and are of the fame fort, then each integral number is the proper divifor to produce its fraction, and also to produce the double of that fraction, and the double of that, as fix produces a fixth, and likewife a third, and two thirds; but, when half, which is from the first fort, is mixed with all of the fecond fort or with fome of them, then the divifion of the estate must be by fix; when a fourth is mixed with all of the fecond fort or with fome of them, then the divi fion must be into twelve; and when an eighth is mixed with all of the fecond fort, or with fome of them, then it must be into four and twenty parts.

On the Increase.

ÂUL, or increase, is, when fome fraction remains above the regular divifor, or when the divifor is too fmall to admit one fhare. Know, that the whole number of divifors is feven, four

of which have no increase, namely, two, three, four, and eight; and three of them have an increafe. The divifor, fix, is, therefore, increased by the aul to ten, either by odd, or by even, numbers; twelve is raised to seventeen by odd, not by even, numbers; and twenty-four is raised to twenty-feven by one increase only; as in the cafe, called Mimberiyya (or a cafe anfwered by ALI when he was in the pulpit), which was this, "A man left a wife, two daughters, and both his parents." After this there can be no increase, except according to IBN MASUUD (may GOD be gracious to him!) for, in his opinion, the divifor twentyfour may be raised to thirty-one; as if a man leave a wife, his mother, two fifters by the fame parents, two fifters by the same mother only, and a fon rendered incapable of inheriting.

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

THE temátbul of two numbers is the equality of one to the other; the tedákhul is, when the fmaller of two numbers exactly measures the larger, or exhaufts it; or we call it tedákbul, when the larger of two numbers is divided ex

actly by the smaller; 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 ta-. wáfuk, 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 smaller 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 (faid to be) mu tawáfik 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 Arrangement.

IN arranging cafes there is need of feven principles; three, between the fhares and the perfons, and four between perfons and perfons. Of the three principles the first 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 cafe, 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 thofe portions and the perfons, 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 divifion 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 some of the numbers are mutawâfik, or compofit, with others; then the rule is, that the measure of the first of the numbers be multiplied into the whole of the fecond, 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 manner; after which the product must be multiplied into the root of the cafe: 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 firft 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 laft product into the root of the cafe; as, if a