The Compendious Book on Calculation by Completion and Balancing/Computation of returns

by Abū ʿAbdallāh Muḥammad ibn Mūsā al-Khwārizmī

Layout 2

COMPUTATION OF RETURNS.^[1]

On Marriage in Illness.

“A man, in his last illness, marries a wife, paying (a marriage settlement of) one hundred dirhems, besides which he has no property, her dowry being ten dirhems. Then the wife dies, bequeathing one-third of her property. After this the husband dies.”^[2] Computation: You take from the one hundred that which belongs entirely to her, on account of the dowry, namely, ten dirhems; there remain ninety dirhems, out of which she has bequeathed a legacy. Call the sum given to her (by her husband, exclusive of her dowry) thing; subtracting it, there remain ninety dirhems less thing. Ten dirhems and thing are already in her hands; she has disposed of one-third of her property, which is three dirhems and one-third, and one-third of thing; there remain six dirhems and two-thirds plus two-thirds of thing, the moiety of which, namely, three dirhems and one-third plus one-third of thing, returns as his portion to the husband.^[3] Thus the heirs of the husband obtain (as his share) ninety-three dirhems and one-third, less two-thirds of thing; and this is twice as much as the sum given to (99) the woman, which was thing, since the woman had power to bequeath one-third of all which the husband left;^[4] and twice as much as the gift to her is two things. Remove now the ninety-three and one-third, from two-thirds of thing, and add these to the two things. Then you have ninety-three dirhems and one-third equal to two things and two-thirds. One thing is three-eighths of it, namely, as much as three-eighths of the ninety-three and one-third, that is, thirty-five dirhems.

If the question is the same, with this exception only, that the wife has ten dirhems of debts, and that she bequeaths one-third of her capital; then the Computation is as follows:^[5] Give to the wife the ten dirhems of her dowry, so that there remain ninety dirhems, out of which she bequeaths a legacy. Call the gift to her thing; there remain ninety less thing. At the disposal of the woman is therefore ten plus thing. From this her debts must be subtracted, which are ten dirhems. She retains then only thing. Of this she bequeaths one-third, namely, one-third of thing: there remains two-thirds of thing. Of this the husband receives by inheritance the moiety, namely, one-third of thing. The heirs of the husband obtain, therefore, ninety dirhems, less two-thirds of thing; and this is twice as much as the gift to her, which was thing; that is, two things. Reduce this, by removing the two-thirds of thing from ninety, and adding them to two things. Then you have ninety dirhems, equal to two things and two-thirds. One thing is three-eighths of this; that is to say, thirty-three dirhems and three-fourths, which is the gift (to the wife).

If he has married her, paying (a marriage settlement of one hundred dirhems, her dowry being ten (100) dirhems, and he bequeaths to some person one-third of his property; then the computation is this:^[6] Pay to the woman her dowry, that is, ten dirhems; there remain ninety dirhems. Herefrom pay the gift to her, thing; then pay likewise to the legatee who is to receive one-third, thing: for the one-third is divided into two moieties between them, since the wife cannot take any thing, unless the husband takes the same. Therefore give, likewise, to the legatee who is to have one-third, thing. Then return to the heirs of the husband. His inheritance from the woman is five dirhems and half a thing. There remains for the heirs of the husband ninety-five less one thing and a half, which is equal to four things. Reduce this, by removing one thing and a half, and adding it to the four things. There remain ninety-five, equal to five things and a half. Make them all moieties; there will be eleven moieties; and one thing will be equal to seventeen dirhems and three-elevenths, and this will be the legacy.

“A man has married a wife paying (a marriage settlement of) one hundred dirhems, her dowry being ten dirhems; and she dies before him, leaving ten dirhems, and bequeathing one-third of her capital; afterwards the husband dies, leaving one hundred and twenty dirhems, and bequeathing to some person one-third of his capital.” Computation:^[7] Give to the wife her dowry, namely, ten dirhems; then one hundred and ten dirhems remain for the heirs of the husband. From these the (101) gift to the wife is thing, so that there remain one hundred and ten dirhems less thing; and the heirs of the woman obtain twenty dirhems plus thing. She bequeaths one-third of this, namely, six dirhems and two-thirds, and one-third of thing. The moiety of the residue, namely, six dirhems and two-thirds plus one-third of thing, returns to the heirs of the husband: so that one hundred and sixteen and two-thirds, less two-thirds of thing, come into their hands. He has bequeathed one-third of this, which is thing. There remain, therefore, one hundred and sixteen dirhems and two-thirds less one thing and two-thirds, and this is twice as much as the husband’s gift to the wife added to his legacy to the stranger, namely, four things. Reduce this, and you find one hundred and sixteen dirhems and two-thirds, equal to five things and two-thirds. Consequently one thing is equal to twenty dirhems and ten-seventeenths; and this is the legacy.

 

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On Emancipation in Illness.

“Suppose that a man on his death-bed were to emancipate two slaves; the master himself leaving a son and a daughter. Then one of the two slaves dies, leaving a daughter and property to a greater amount than his price.^[8]” You take two-thirds of his price, and what the other slave has to return (in order to complete his (102) ransom). If the slave die before the master, then the son and the daughter of the latter partake of the heritage, in such proportion, that the son receives as much as the two daughters together. But if the slave die after the master, then you take two-thirds of his value and what is returned by the other slave, and distribute it between the son and the daughter (of the master), in such a manner, that the son receives twice as much as the daughter; and what then remains (from the heritage of the slave) is for the son alone, exclusive of the daughter; for the moiety of the heritage of the slave descends to the daughter of the slave, and the other moiety, according to the law of succession, to the son of the master, and there is nothing for the daughter (of the master).

It is the same, if a man on his deatlı-bed emancipates a slave, besides whom he has no capital, and then the slave dies before his master.

If a man in his illness emancipates a slave, besides whom he possesses nothing, then that slave must ransom himself by two-thirds of his price. If the master has anticipated these two-thirds of his price and has spent them, then, upon the death of the master, the slave must pay two-thirds of what he retains.^[9] But if the master has anticipated from him his whole price and spent it, then there is no claim against the slave, since he has already paid his entire price.

“Suppose that a man on his death-bed emancipates a slave, whose price is three hundred dirhems, not having any property besides; then the slave dies, leaving three hundred dirhems and a daughter.” The computation is this:^[10] Call the legacy to the slave thing: He has to return the remainder of his price, after the deduction of the legacy, or three hundred less thing. This ransom, of three hundred less thing, belongs to the master. Now the slave dies, and leaves thing and a (103) daughter. She must receive the moiety of this, namely, one half of thing; and the master receives as much. Therefore the heirs of the master receive three hundred less half a thing, and this is twice as much as the legacy, which is thing, namely, two things. Reduce this by removing half a thing from the three hundred, and adding it to the two things. Then you have three hundred, equal to two things and a half. One thing is, therefore, as much as two-fifths of three hundred, namely, one hundred and twenty. This is the legacy (to the slave,) and the ransom is one hundred and eighty.

“Some person on his sick-bed has emancipated a slave, whose price is three hundred dirhems; the slave then dies, leaving four hundred dirhems and ten dirhems of debt, and two daughters, and bequeathing to a person one-third of his capital; the master has twenty dirhems debts.” The computation of this case is the following:^[11] Call the legacy to the slave thing; his ransom is the remainder of his price, namely, three hundred less thing. But the slave, when dying, left four hundred dirhems; and out of this sum, his ransom, namely, three hundred less thing, is paid to the master, so that one hundred dirhems and thing remain in the hands of the slave’s heirs. Herefrom are(first) subtracted the debts, namely, ten dirhems; there remain then ninety dirhems and thing. Of this he has bequeathed one-third, that is, thirty dirhems and one-third of thing; so that there remain for the heirs sixty dirhems and two-thirds of thing. Of this the two daughters receive two-thirds, namely, forty dirhems and four-ninths of thing, and the master (104) receives twenty dirhems and two-ninths of thing, so that the heirs of the master obtain three hundred and twenty dirhems less seven-ninths of thing. Of this the debts of the master must be deducted, namely, twenty dirhems; there remain then three hundred dirhems less seven-ninths of thing; and this sum is twice as much as the legacy of the slave, which was thing; or, it is equal to two things. Reduce this, by removing the seven-ninths of thing, and adding them to two things; there remain three hundred, equal to two things and seven-ninths. One thing is as much as nine twenty-fifths of eight hundred, which is one hundred and eight; and so much is the legacy to the slave.

If, on his sick-bed, he emancipates two slaves, besides whom he has no property, the price of each of them being three hundred dirhems; the master having anticipated and spent two-thirds of the price of one of them before he dies;^[12] then only one-third of the price of this slave, who has already paid off a part of his ransom, belongs to the master; and thus the master’s capital is the entire price of the one who has paid off nothing of his ransom, and one-third of the price of the other who has paid part of it; the latter is one hundred dirhems; the other three hundred dirhems: one-third of the amount, namely, one hundred and thirty-three dirhems and one third, is divided into two moieties among them; so that each of them receives sixty-six dirhems and two-thirds. The first slave, who has already paid two-thirds of his ransom, pays thirty-three dirhems and one-third; for (105) sixty-six dirhems and two-thirds out of the hundred belong to himself as a legacy, and what remains of the hundred he must return. The second slave has to return two hundred and thirty-three dirhems and one-third.

“Suppose that a man, in his illness, emancipates two slaves, the price of one of them being three hundred dirhems, and that of the other five hundred dirhems; the one for three hundred dirhems dies, leaving a daughter; then the master dies, leaving a daughter likewise; and the slave leaves property to the amount of four hundred dirhems. With how much must every one ransom himself?”^[13] The computation is this: Call the legacy to the first slave, whose price is three hundred dirhems, thing. His ransom is three hundred dirhems less thing. The legacy to the second slave of a price of five hundred dirhems is one thing and two-thirds, and his ransom five hundred dirhems less one thing and two-thirds (viz. his price being one and two-thirds times the price of the first slave, whose ransom was thing, he must pay one thing and two-thirds for his ransom). Now the slave for three hundred dirhems dies, and leaves four hundred dirhems. Out of this his ransom is paid, namely, three hundred dirhems less thing; and in the hands of his heirs remain one hundred dirhems plus thing: his daughter receives the moiety of this, namely, fifty dirhems and half a thing; and what remains belongs to the heirs of the master, namely, fifty dirhems and half a thing. This is added to the three hundred less thing; the sum is three hundred and fifty less half a thing. Add thereto the ransom of the other, which is five hundred dirhems less one thing and two-thirds; thus, the heirs (106) of the master have obtained eight hundred and fifty dirhems less two things and one-sixth; and this is twice as much as the two legacies together, which were two things and two-thirds. Reduce this, and you have eight hundred and fifty dirhems, equal to seven things and a half. Make the equation; one thing will be equal to one hundred and thirteen dirhems and one-third. This is the legacy to the slave, whose price is three hundred dirhems. The legacy to the other slave is one and two-thirds times as much, namely, one hundred and eighty-eight dirhems and eight-ninths, and his ransom three hundred and eleven dirhems and one-ninth.

“Suppose that a man in his illness emancipates two slaves, the price of each of whom is three hundred dirhems; then one of them dies, leaving five hundred dirhems and a daughter; the master having left a son.” Computation:^[14] Call the legacy to each of them thing; the ransom of each will be three hundred less thing; then take the inheritance of the deceased slave, which is five hundred dirhems, and subtract his ransom, which is three hundred less thing; the remainder of his inheritance will be two hundred plus thing. Of this, one hundred dirhems and half a thing return to the master by the law of succession, so that now altogether four hundred dirhems less a half thing are in the hands of the master’s heirs. Take also the ransom of the other slave, namely, three hundred dirhems less thing; then the heirs of the master obtain seven hundred dirhems less one thing and a half, and this is twice as much as the sum of the two legacies of both, namely (107) two things, consequently as much as four things. Remove from this the one thing and a half: you find seven hundred dirhems, equal to five things and a half. Make the equation. One thing will be one hundred and twenty-seven dirhems and three-elevenths.

“Suppose that a man in his illness emancipate a slave, whose price is three hundred dirhems, but who has already paid off to his master two hundred dirhems, which the latter has spent; then the slave dies before the death of the master, leaving a daughter and three hundred dirhems.”^[15] Computation: Take the property left by the slave, namely, the three hundred, and add thereto the two hundred, which the master has spent; this together makes five hundred dirhems. Subtract from this the ransom, which is three hundred less thing (since his legacy is thing); there remain two hundred dirhems plus thing. The daughter receives the moiety of this, namely, one hundred dirhems plus half a thing; the other moiety, according to the laws of inheritance, returns to the heirs of the master, being likewise one hundred dirhems and half a thing. Of the three hundred dirhems less thing there remain only one hundred dirhems less thing for the heirs of the master, since two hundred are spent already. After the deduction of these two hundred which are spent, there remain with the heirs two hundred dirhems less half thing, and this is equal to the legacy of the slave taken twice; or the moiety of it, one hundred less one-fourth of thing, is equal to the legacy of the slave, which is thing. Remove from this the one fourth of thing; then you have one hundred dirhems, equal to one thing and one-fourth. One thing is four-fifths of it, namely, eighty dirhems. This is the legacy; and the ransom is two hundred and twenty dirhems. Add the inheritance of the slave, which is three hundred, to two hundred, which (108) are spent by the master. The sum is five hundred dirhems. The master has received the ransom of two hundred and twenty dirhems; and the moiety of the remaining two hundred and eighty, namely, one hundred and forty, is for the daughter. Take these from the inheritance of the slave, which is three hundred; there remain for the heirs one hundred and sixty dirhems, and this is twice as much as the legacy of the slave, which was thing.

“Suppose that a man in his illness emancipates a slave, whose price is three hundred dirhems, but who has already advanced to the master five hundred dirhems; then the slave dies before the death of his master, and leaves one thousand dirhems and a daughter. The master has two hundred dirhems debts.”^[16] Computation: Take the inheritance of the slave, which is one thousand dirhems, and the five hundred, which the master has spent. The ransom from this is three hundred less thing. There remain therefore twelve hundred plus thing. The moiety of this belongs to the daughter: it is six hundred dirhems plus half a thing. Subtract it from the property left by the slave, which was one thousand dirhems: there remain four hundred dirhems less half thing. Subtract herefrom the debts of the master, namely, two hundred dirhems; there remain two hundred dirhems less half thing, which are equal to the legacy taken twice, which is thing; or equal to two things. Reduce this, by means of the half thing. Then you have two hundred dirhems, equal to two things and a half. Make the equation. You find one thing, equal to eighty dirhems; this is the legacy. Add now the property left by the slave to the sum which he has (109) advanced to the master: this is fifteen hundred dirhems. Subtract the ransom, which is two hundred and twenty dirhems; there remain twelve hundred and eighty dirhems, of which the daughter receives the moiety, namely, six hundred and forty dirhems. Subtract this from the inheritance of the slave, which is one thousand dirhems: there remain three hundred and sixty dirhems. Subtract from this the debts of the master, namely, two hundred dirhems; there remain then one hundred and sixty dirhems for the heirs of the master, and this is twice as much as the legacy of the slave, which was thing.

“Suppose that a man on his sick-bed emancipates a slave, whose price is five hundred dirhems, but who has already paid off to him six hundred dirhems. The master has spent this sum, and has moreover three hundred dirhems of debts. Now the slave dies, leaving his mother and his master, and property to the amount of seventeen hundred and fifty dirhems, with two hundred dirhems debts.” Computation:^[17] Take the property left by the slave, namely, seventeen hundred and fifty dirhems, and add to it what he has advanced to the master, namely, six hundred dirhems; the sum is two thousand three hundred and fifty dirhems. Subtract from this the debts, which are two hundred dirhems, and the ransom, which is five hundred dirhems less thing, since the legacy is thing; there remain then sixteen hundred and fifty dirhems plus thing. The mother receives herefrom one-third, namely, five hundred and fifty plus one-third of thing. Subtract now this and the debts, which are two hundred dirhems, from the actual inheritance of the slave, which is seventeen hundred and fifty; there remain one thousand (110) dirhems less one-third of thing. Subtract from this the debts of the master, namely, three hundred dirhems; there remain seven hundred dirhems less one-third of thing. This is twice as much as the legacy of the slave, which is thing. Take the moiety: then three hundred and fifty less one-sixth of thing are equal to one thing. Reduce this, by means of the one-sixth of thing; then you have three hundred and fifty, equal to one thing and one-sixth. One thing will then be equal to six-sevenths of the three hundred and fifty, namely, three hundred dirhems; this is the legacy. Add now the property left by the slave to what the master has spent already; the sum is two thousand three hundred and fifty dirhems. Subtract herefrom the debts, namely, two hundred dirhems, and subtract also the ransom, which is as much as the price of the slave less the legacy, that is, two hundred dirhems; there remain nineteen hundred and fifty dirhems. The mother receives one-third of this, namely, six hundred and fifty dirhems. Subtract this and the debts, which are two hundred dirhems, from the property actually left by the slave, which was seventeen hundred and fifty dirhems; there remain nine hundred dirhems. Subtract from this the debts of the master, which are three hundred dirhems; there remain six hundred dirhems, which is twice as much as the legacy.

“Suppose that some one in his illness emancipates a slave, whose price is three hundred dirhems: then the slave dies, leaving a daughter and three hundred dirhems; then the daughter dies, leaving her husband and three hundred dirhems; then the master dies.” Computation:^[18] Take the property left by the slave, which is three hundred dirhems, and subtract the ransom, which (111) is three hundred less thing; there remains thing, one half of which belongs to the daughter, while the other half returns to the master. Add the portion of the daughter, which is half one thing, to her inheritance, which is three hundred; the sum is three hundred dirhems plus half a thing. The husband receives the moiety of this; the other moiety returns to the master, namely one hundred and fifty dirhems plus one-fourth of thing. All that the master has received is therefore four hundred and fifty less one-fourth of thing; and this is twice as much as the legacy; or the moiety of it is as much as the legacy itself, namely, two hundred and twenty-five dirhems less one-eighth thing are equal to thing. Reduce this by means of one-eighth of thing, which you add to thing; then you have two hundred and twenty-five dirhems, equal to one thing and one-eighth. Make the equation: one thing is as much as eight-ninths of two hundred and twenty-five, namely, two hundred dirhems.

“Suppose that some one in his illness emancipates a slave, of the price of three hundred dirhems; the slave dies, leaving five hundred dirhems and a daughter, and bequeathing one-third of his property; then the daughter dies, leaving her mother, and bequeathing one-third of her property, and leaving three hundred dirhems.” Computation:^[19] Subtract from the property left by the slave his ransom, which is three hundred dirhems less thing; there remain two hundred dirhems plus thing. He has bequeathed one-third of his property, that is, sixty-six dirhems and two-thirds plus one-third of thing. According to the law of succession, (112) sixty-six dirhems and two-thirds and one-third of thing belong to the master, and as much to the daughter. Add this to the property left by her, which is three hundred dirhems: the sum is three hundred and sixty-six dirhems and two-thirds and one-third of thing. She has bequeathed one-third of her property, that is, one hundred and twenty-two dirhems and two-ninths and one-ninth of thing; and there remain two hundred and forty-four dirhems and four-ninths and two-ninths of thing. The mother receives one-third of this, namely, eighty-one dirhems and four-ninths and one-third of one-ninth of a dirhem plus two-thirds of one-ninth of thing. The remainder returns to the master; it is a hundred and sixty-two dirhems and eight-ninths and two-thirds of one-ninth of a dirhem plus one-ninth and one-third of one-ninth of thing, as his share of the heritage.

Thus the master’s heirs have obtained five hundred and twenty-nine dirhems and seventeen twenty-sevenths of a dirhem less four-ninths and one-third of one-ninth of thing; and this is twice as much as the legacy, which is thing. Halve it: You have two hundred and sixty-four dirhems and twenty-two twenty-sevenths of a dirhem, less seven twenty-sevenths of thing. Reduce it by(113) means of the seven twenty-sevenths which you add to the one thing. This gives one hundred and sixty-four dirhems and twenty-two twenty-sevenths, equal to one thing and seven twenty-sevenths of thing. Make the equation, and adjust it to one single thing, by subtracting from it as much as seven thirty-fourths of the same. Then one thing is equal to two hundred and ten dirhems and five-seventeenths; and this is the legacy.

“Suppose that a man in his illness emancipates a slave, whose price is one hundred dirhems, and makes to some one a present of a slave-girl, whose price is five hundred dirhems, her dowry being one hundred dirhems, and the receiver cohabits with her.” Abu Hanifah says: The emancipation is the more important act, and must first be attended to.

Computation:^[20] Take the price of the girl, which is five hundred dirhems; and remember that the price of the slave is one hundred dirhems. Call the legacy of the donee thing. The emancipation of the slave, whose price is one hundred dirhems, has already taken place. He has bequeathed one thing to the donee. Add the dowry, which is one hundred dirhems less one-fifth thing. Then in the hands of the heirs are six hundred dirhems less one thing and one-fifth of thing. This is twice as much as one hundred dirhems and thing; the moiety of it is equal to the legacy of the two, namely, three hundred less three-fifths of thing. Reduce this by removing the three-fifths of thing from three hundred, and add the same to one thing. This gives three hundred dirhems, equal to one thing and three-fifths and one hundred dirhems. Subtract now from three hun- dred the one hundred, on account of the other one hundred. There remain two hundred dirhems, equal to one thing and three-fifths. Make the equation with this. One thing will be five-eighths of what you have; (114) take therefore five-eighths of two hundred. It is one hundred and twenty-five. This is thing; it is the legacy to the person to whom he had presented the girl.

“Suppose that a man emancipates a slave of a price of one hundred dirhems, and makes to some person a present of a slave girl of the price of five hundred dirhems, her dowry being one hundred dirhems; the donee cohabits with her, and the donor bequeaths to some other person one-third of his property.” According to the decision of Abu Hanifah, no more than one-third can be taken from the first owner of the slave-girl; and this one-third is to be divided into two equal parts between the legatee and the donee. Computation:^[21] Take the price of the girl, which is five hundred dirhems. The legacy out of this is thing; so that the heirs obtain five hundred dirhems less thing; and the dowry is one hundred less one-fifth of thing; consequently they obtain six hundred dirhems less one thing and one-fifth of thing. He bequeaths to some person one third of his capital, which is as much as the legacy of the person who has received the girl, namely, thing. Consequently there remain for the heirs six hundred less two things and one-fifth, and this is twice as much as both their legacies taken together, namely, the price of the slave plus the two things bequeathed as legacies. Halve it, and it will by itself be equal to these legacies: it is then three hundred less one and one-tenth of thing. Reduce this by means of the one and one-tenth of thing. Then you have three hundred, equal to three things and one-tenth, plus one hundred dirhems. Remove one hundred on occount of (the opposite) one hundred; there remain two hundred, equal to three things and one-tenth. Make now the reduction. One thing will be as much as thirty-one (115) parts of the sum of dirhems which you have; and just so much will be the legacy out of the two hundred; it is sixty-four dirhems and sixteen thirty-one parts.

“Suppose that some one emancipates a slave girl of the price of one hundred dirhems, and makes to some person a present of a slave girl, which is five hundred dirhems worth; the receiver cohabits with her, and her dowry is one hundred dirhems; the donor bequeaths to some other person as much as one-fourth of his capital.” Abu Hanifah says: The master of the girl cannot be required to give up more than one-third, and the legatee, who is to receive one-fourth, must give up one-fourth. Computation:^[22] The price of the girl is five hundred dirhems. The legacy out of this is thing; there remain five hundred dirhems less thing. The dowry is one hundred dirhems less one-fifth of thing; thus the heirs obtain six hundred dirhems less one and one-fifth of thing. Subtract now the legacy of the person to whom one-fourth has been bequeathed, namely, three-fourths of thing; for if one-third is thing then one-fourth is as much as three-fourths of the same. There remain then six hundred dirhems less one thing and thirty-eight fortieths. This is equal to the legacy taken twice. The moiety of it is equal to the legacies by themselves, namely, three hundred dirhems less thirty-nine fortieths of thing. Reduce this by means of the latter fraction. Then you have three hundred (116) dirhems, equal to one hundred dirhems and two things and twenty-nine fortieths. Remove one hundred on account of the other one hundred. There remain two hundred dirhems, equal to two things and twenty-nine-fortieths. Make the equation. You will then find one thing to be equal to seventy-three dirhems and forty-three one-hundred-and-ninths dirhems.

On return of the Dowry.

“A man, in the illness before his death, makes to some one a present of a slave girl, besides whom he has no property. Then he dies. The slave girl is worth three hundred dirhems, and her dowry is one hundred dirhems. The man to whom she has been presented, cohabits with her.” Computation:^[23] Call the legacy of the person to whom the girl is presented, thing. Subtract this from the donation: there remain three hundred less thing. One-third of this difference returns to the donor on account of dowry (since the dowry is one-third of the price): this is one hundred dirhems less one-third of thing. The donor’s heirs obtain, therefore, four hundred less one and one-third of thing, which is equal to twice the legacy, which is thing, or to two things. Transpose the one and one-third thing from the four hundred, and add it to the two things; then you have four hundred, equal to three things and one-third. One thing is, therefore, equal to three-tenths of it, or to one hundred and twenty dirhems, and this is the legacy.

“Or, suppose that he, in his illness, has made a present of the slave girl, her price being three hundred, her dowry one hundred dirhems; and the donor dies, after having cohabited with her.” Computation:^[24] Call the legacy thing: the remainder is three hundred less thing. The donor having cohabited with her, the dowry remains with him, which is one-third of the legacy, since the dowry is one-third of the price, or one-third of thing. Thus the donor’s heirs obtain three (117) hundred less one and one-third of thing, and this is twice as much as the legacy, which is thing, or equal to two things. Remove the one and one-third of thing, and add the same to the two things. Then you have three hundred, equal to three things and one-third. One thing is, therefore, three-tenths of it, namely ninety dirhems. This is the legacy.

If the case be the same, and both the donor and donee have cohabited with her; then the Computation is this:^[25] Call the legacy thing; the deduction is three hundred dirhems less thing. The donor has ceded the dowry to the donee by (the donee’s) having cohabited with her: this amounts to one-third of thing: and the donee cedes one-third of the deduction, which is one hundred less one-third of thing. Thus, the donor’s heirs obtain four hundred less one and two-thirds of thing, which is twice as much as the legacy. Reduce this, by separating the one and two-thirds of thing from four hundred, and add them to the two things. Then you have four hundred things, equal to three things and two-thirds. One thing of these is three-elevenths of four hundred; namely, one hundred and nine dirhems and one-eleventh. This is the legacy. The deduction is one hundred and ninety dirhems and ten-elevenths. According to Abu Hanifah, you call the thing a legacy, and what is obtained on account of the dowry is likewise a legacy.

If the case be the same, but that the donor, having cohabited with her, has bequeathed one-third of his (118) capital, then Abu Hanifah says, that the one-third is halved between the donee and the legatee. Computation:^[26] Call the legacy of the person to whom the slave-girl has been given, thing. After the deduction of it, there remain three hundred, less thing. Then take the dowry, which is one-third of thing; so that the donor retains three hundred less one and one-third of thing: the donee’s legacy being, according to Abu Hanifah, one and one-third of thing; according to other lawyers, only thing. The legatee, to whom one-third is bequeathed, receives as much as the legacy of the donee, namely, one and one-third of thing. The donor thus retains three hundred, less two things and two-thirds-equal to twice the two legacies, which are two things and two-thirds. The moiety of this, namely, one hundred and fifty less one and one-third of thing, must, therefore, be equal to the two legacies. Reduce it, by removing one and one-third of thing, and adding the same to the two legacies (things). Then you find one hundred and fifty, equal to four things. One thing is one-fourth of this, namely, thirty-seven and a half.

If the case be, that both the receiver and the donor have cohabited with her, and the latter has disposed of one-third of his capital by way of legacy; then the computation,^[27] according to Abu Hanifah, is, that you call the legacy thing. After the deduction of it, there remain three hundred less thing. Then the dowry is taken, which is one hundred less one-third of thing; so that there are four hundred dirhems less one and one-third of thing. The sum returned from the dowry is one-third of thing; and the legatee, who is to receive one-third, obtains as much as the legacy of the first, namely, thing and one-third of thing. Thus there remain four hundred dirhems less three things, equal to twice the legacy, namely, two things and two-thirds. (119) Reduce this, by means of the three things, and you find four hundred, equal to eight things and one-third. Make the equation with this: one thing will be forty eight dirhems.

“Suppose that a man on his sick-bed makes to another a present of a slave-girl, worth three hundred dirhems, her dowry being one hundred dirhems; the donee cohabits with her, and afterwards, being also on his sick-bed, makes a present of her to the donor, and the latter cohabits with her. How much does he acquire by her, and how much is deducted?”^[28] Computation: Take the price, which is three hundred dirhems; the legacy from this is thing; there remain with the donor’s heirs three hundred less thing; and the donee obtains thing. Now the donee gives to the donor part of thing: consequently, there remains only thing less part of thing for the donee. He returns to the donor one hundred less one-third of thing; but takes the dowry, which is one-third of thing, less one-third of part of thing. Thus he obtains one and two-thirds thing less one hundred dirhems and less one and one-third of part of thing. This is twice as much as part of thing; and the moiety of it is as much as part of thing, namely, five-sixths of thing less fifty dirhems and less two-thirds of part of thing. Reduce this by removing two-thirds of part of thing and fifty dirhems. Then you have five-sixths of thing, equal to one and two-thirds of part of thing plus fifty dirhems. Reduce this to one single part of thing, in order to know what the amount of it is. You effect this by taking three-fifths (120) of what you have. Then one part of thing plus thirty dirhems is equal to half a thing; and one-half thing less thirty dirhems is equal to part of thing, which is the legacy returning from the donee to the donor. Keep this in memory.

Then return to what has remained with the donor;

this was three hundred less thing: hereto is now added the part of thing, or one-half thing less thirty dirhems. Thus he obtains two hundred and seventy less half one thing. He further takes the dowry, which is one hundred dirhems less one-third thing, but has to return a dowry, which is one-third of what remains of thing after the subtraction of part of thing, namely, one-sixth of thing and ten dirhems. Thús he retains three hundred and sixty less thing, which is twice as much as thing and the dowry, which he has returned. Halve it: then one hundred and eighty less one-half thing are equal to thing and that dowry. Reduce this, by removing one-half thing and adding it to the thing and the dowry: you find one hundred and eighty dirhems, equal to one thing and a half plus the dowry which he has returned, and which is one-sixth thing and ten dirhems. Remove these ten dirhems; there remain one hundred and seventy dirhems, equal to one and two-thirds things. Reduce this, in order to ascertain what the amount of one thing is, by taking three-fifths of what you have; you find that one hundred and two are equal to thing, which is the legacy from the donor to the donee: and the legacy from the donee to the donor is the moiety of this, less thirty dirhems, namely, twenty-one.

On Surrender in Illness.

(121) “Suppose that a man, on his sick-bed, deliver to some one thirty dirhems in a measure of victuals, worth ten dirhems; he afterwards dies in his illness; then the receiver returns the measure and returns besides ten dirhems to the heirs of the deceased.” Computation: He returns the measure, the value of which is ten dirhems, and places to the account of the deceased twenty dirhems; and the legacy out of the sum so placed is thing; thus the heirs obtain twenty less thing, and the measure. All this together is thirty dirhems less thing, equal to two things, or equal to twice the legacy. Reduce it by separating the thing from the thirty, and adding it to the two things. Then, thirty are equal to three things. Consequently, one thing must be one-third of it, namely, ten, and this is the sum which he obtains out of what he places to the account of the deceased.

“Suppose that some one on his sick-bed delivers to a person twenty dirhems in a measure worth fifty dirhems; he then repeals it while still on his sick bed, and dies after this. The receiver must, in this case, return four-ninths of the measure, and eleven dirhems and one-ninth.”^[29] Computation: You know that the price of the measure is two and a half times as much as the sum which the donor has given the donee in money; and whenever the donee returns anything from the money capital, he returns from the measure as much as two and a half times that amount. Take now from the measure as much as corresponds to one thing, that is, two things and a half, and add this to what remains from the twenty, namely, twenty less thing. Thus the heirs of the deceased obtain twenty dirhems and one (112) thing and a half. The moiety of this is the legacy, namely, ten dirhems and three-fourths of thing; and this is one-third of the capital, namely, sixteen dirhems and two-thirds. Remove now ten dirhems on account of the opposite ten; there remain six dirhems and two-thirds, equal to three-fourths of thing. Complete the thing, by adding to it as much as one-third of the same; and add to the six dirhems and two-thirds likewise one-third of the same, namely, two dirhems and two-ninths; this yields eight dirhems and eight-ninths, equal to thing. Observe now how much the eight dirhems and eight-ninths are of the money capital, which is twenty dirhems. You will find them to be four-ninths of the same. Take now four-ninths of the measure and also five-ninths of twenty. The value of four-ninths of the measure is twenty-two dirhems and two-ninths; and the five-ninths of the twenty are eleven dirhems and one-ninth. Thus the heirs obtain thirty-three dirhems and one-third, which is as much as two-thirds of the fifty dirhems.—God is the Most Wise!

---

But the reasons for reducing the question to these two equations are not given by the author, and seem to depend on the dicta of the sages of the Arabian law.

1. The solutions which the author has given of the remaining problems of this treatise, are, mathematically considered, for the most part incorrect. It is not that the problems, when once reduced into equations, are incorrectly worked out; but that in reducing them to equations, arbitrary assumptions are made, which are foreign or contradictory to the data first enounced, for the purpose, it should seem, of forcing the solutions to accord with the established rules of inheritance, as expounded by Arabian lawyers.
   The object of the lawyers in their interpretations, and of the author in his solutions, seems to have been, to favour heirs and next of kin; by limiting the power of a testator, during illness, to bequeath property, or to emancipate slaves; and by requiring payment of heavy ransom for slaves whom a testator might, during illness, have directed to be emancipated.
2. Lets $s$ be the sum, including the dowry, paid by the man, as a marriage settlement; $d$ the dowry; $x$ the gift to the wife, which she is empowered to bequeath if she pleases.
   She may bequeath, if she pleases, $d+x$; she actually does bequath ${\tfrac {1}{3}}[d+x]$; the residue is ${\tfrac {2}{3}}[d+x]$, of which one half, viz. ${\tfrac {1}{3}}[d+x]$ goes to her heirs, and the other half reverts to the husband
   $\therefore$ the husband’s heirs have $s-[d+x]+{\tfrac {1}{3}}[d+x]\;{\text{or}}\;s-{\tfrac {2}{3}}[d+x]$; and since what the wife has disposed of, exclusive of the dowry, is $x$, twice which sum the husband is to receive, $s-{\tfrac {2}{3}}[d+x]=2x\;\therefore \;{\tfrac {1}{8}}[3s-2d]=x$. But $s=100\;;d=10\;\therefore \;x=35\;;d+x=45\;;{\tfrac {1}{3}}[d+x]=15$. Therefore the legacy which she bequeaths is $15$, her husband receives $15$, and her other heirs, $15$. The husband’s heirs receive $2x=70$.
   But had the husband also bequeathed a legacy, then, as we shall see presently, the law would have defeated, in part, the woman’s intentions.
3. In other cases, as appears from pages 92 and 93, a husband inherits one-fourth of the residue of his wife’s estate, after deducting the legacies which she may have bequeathed. But in this instance he inherits half the residue. If she die in debt, the debt is first to be deducted from her property, at least to the extent of her dowry (sce the next problem.)
4. When the husband makes a bequest to a stranger, the third is reduced to one-sixth. Vide p. 137.
5. The same things being assumed as in the last example $s-[d+x]$ remains with the husband; $d$ goes to pay the debts of the wife; and ${\tfrac {x}{3}}$ reverts from the wife to the husband.
   $\therefore \;s-d-{\tfrac {2}{3}}x=2x\;\therefore \;{\tfrac {3}{8}}[s-d]=x$
   $\therefore \;$ if $s=100$, and $d=10$, $x=33{\tfrac {3}{4}}$; she bequeaths $11{\tfrac {1}{4}};11{\tfrac {1}{4}}$ reverts to her husband; and her other heirs receive $11{\tfrac {1}{4}}$. The husband’s heirs receive $2x=67{\tfrac {1}{2}}$.
6. This case is distinguished from that in page 133 by two circumstances; first, that the woman does not make any bequest; second, that the husband bequeaths one-third of his property.
   Suppose the husband not to make any bequest. Then, since the woman had at her disposal $d+x$, but did not make any bequest, ${\tfrac {1}{2}}[d+x]$ reverts to her husband; and the like amount goes to her other heirs.
   $\therefore \;s-[d+x]+{\tfrac {1}{2}}[d+x]=2x\;\therefore \;x{\tfrac {1}{5}}[2s-d]$
   and since $s=100$, and $d=10$; $x=38$; $d+x=48$; $d+x=48$;${\tfrac {1}{2}}[d+x]=24$ reverts to the husband, and the like sum goes to her other heirs; and $2x=76$, belongs to the husband’s heirs.
   Now suppose the husband to bequeath one-third of his property. The law here interferes with the testator’s right of bequeathing; and provides that whatever sum is at the disposal of the wife, the same sum shall be at the disposal of the husband; and that the sum to be retained by the husband’s heirs shall be twice the sum which the husband and wife together may dispose of.
   

   $\therefore \;s-{\tfrac {1}{2}}[d+x]-x=4$
   

   $\therefore \;{\tfrac {1}{11}}[2s-d]=x\;{\text{if}}\;s=100,\;{\text{and}}\;d=10;x={\tfrac {190}{11}}=17{\tfrac {3}{11}};d+x=27{\tfrac {3}{11}};{\tfrac {1}{2}}[d+x]=13{\tfrac {7}{11}}$ reverts to the husband, and the like sum goes to the other heirs of the woman; $17{\tfrac {3}{11}}$ is what the husband bequeaths; and $69{\tfrac {1}{11}}=4x$ goes to the husband’s heirs.
7. Let $c$ be the property which the wife leaves, besides $d$ the dowry, and $x$ the gift from the husband. She bequeaths ${\tfrac {1}{3}}[c+d+x];$ ${\tfrac {1}{3}}[c+d+x]$ goes to her husband; and ${\tfrac {1}{3}}[c+d+x]$ to her other heirs. The husband leaves property $s$, out of which must be paid the dowry, $d$; the gift to the wife, $x$; and the bequest he makes to the stranger, $x$; and his heirs receive from the wife’s heirs ${\tfrac {1}{3}}[c+d+x]$
   $s-d-2x+[c+d+x]=4x$, according to the law of inheritance.
   $\therefore \;3s+c-2d=17x,\;{\text{and}}\;x={\tfrac {3s+c-2d}{17}}$
   ${\text{If}}\;s=120,c=10,\;{\text{and}}\;d=10,x={\tfrac {350}{17}}=20{\tfrac {10}{17}}$
   $c+d+x=40{\tfrac {10}{17}}\;;{\tfrac {1}{3}}[c+d+x]=13{\tfrac {9}{17}}$
   The wife bequeaths $13{\tfrac {9}{17}}$; $13{\tfrac {9}{17}}$ go to her husband, and $13{\tfrac {9}{17}}$ to her other heirs.
   The husband bequeaths to the stranger $20{\tfrac {10}{17}}$; he gives the same sum to the wife ; and $4x=82{\tfrac {6}{17}}$ go to his heirs.
8. From the property of the slave, who dies, is to be deducted and paid to the master’s heirs, first, two-thirds of the original cost of that slave, and secondly what is wanting to complete the ransom of the other slave. Call the amount of these two sums $p$; and the property which the slave leaves $\alpha$.
   Next, as to the residue of the slaves’ property:
   First. If the slave dies before the master, the master’s son takes ${\tfrac {1}{2}}[\alpha -p]$; the master’s daughter ${\tfrac {1}{4}}[\alpha -p]$, and the slave’s daughter ${\tfrac {1}{4}}[\alpha -p]$.
   Second. If the slave dies after the master; the master’s son is to receive ${\tfrac {2}{3}}p$, and the master’s daughter ${\tfrac {1}{3}}p$; and then the master’s son takes ${\tfrac {1}{2}}[\alpha -p]$, and the slave’s daughter ${\tfrac {1}{2}}[\alpha -p]$.
9. The slave retains one-third of his price; and this he must redeem at two-thirds of its value; namely at ${\tfrac {2}{3}}\times {\tfrac {1}{3}}={\tfrac {2}{9}}$ of his original price.
10. Let the slave’s original cost be $a$; the property which he dies possessed of, $\alpha$; what the master bequeaths to the slave, in emancipating him, $x$. Then the net property which the slave dies possessed of is $\alpha +x-a.{\tfrac {1}{2}}[\alpha +x-a]$ belongs, by law, to the master; and ${\tfrac {1}{2}}[\alpha +x-a]$ to the slave’s daughter. The master’s heirs, therefore, receive the ransom, $a-x$, and the inheritance, ${\tfrac {1}{2}}[\alpha +x-a]$; that is, ${\tfrac {1}{2}}[\alpha +a-x]$; and on the same principle as the slave, when emancipated, is allowed to ransom himself at two-thirds of his cost, the law of the case is that $2$ are to be taken, where $1$ is given.

    $\therefore \;{\tfrac {1}{2}}[\alpha +a-x]=2x\;\therefore \;x={\tfrac {1}{5}}[\alpha +a]$
    The daughter’s share of the inheritance	${\tfrac {1}{5}}[3\alpha -2a]$
    The master’s heirs receive .............	${\tfrac {2}{5}}[\alpha +a]$

    If, as in the example, $\alpha =a$, $x={\tfrac {2}{5}}a$; the daughter’s share = ${\tfrac {1}{5}}$a; the heirs of the master receive ${\tfrac {4}{5}}$a.

11. Let the slave’s original cost=$a$; the property he dies possessed of=$\alpha$; the debt he owes = $\epsilon$
    He leaves two daughters, and bequeaths to a stranger one-third of his capital.
    The master owes debts to the amount $\mu$; where $a=300;\alpha =400;\epsilon =10;\mu =20$.
    Let what the master gives to the slave, in emancipating him $=x$.
    Slave’s ransom=$a-x$; slave’s property—slave’s ransom=$\alpha +x-a$
    

    Slave’s property—ransom—debt=$\alpha +x-a-\epsilon$
    Legacy to stranger	$={\tfrac {1}{3}}[\alpha +x-a-\epsilon ]$
    Residue ...............	$={\tfrac {2}{3}}[\alpha +x-a-\epsilon ]$

    The master, and each daughter, are, by law, severally entitled to ${\tfrac {1}{3}}\times {\tfrac {2}{3}}[\alpha +x-a-\epsilon ]$
    The master’s heirs receive altogether $a-x+{\tfrac {2}{9}}[\alpha +x-a-\epsilon ]$ or ${\tfrac {7}{9}}[a-x]+{\tfrac {2}{9}}[[\alpha -\epsilon ]$, which, on the principle that $2$ are to be taken for $1$ given, ought to be made equal to $2x$.
    But the author directs that the equation for determining $x$ be
    ${\begin{aligned}&{\tfrac {7}{9}}[a-x]+{\tfrac {2}{9}}[\alpha -\epsilon ]-\mu =2x\\&\therefore \;x={\tfrac {1}{25}}[7a+2[\alpha -\epsilon ]-9\mu ]&=108\\&{\text{Hence the slave receives, the debts which he owes,}}\;\epsilon &=10\\&{\text{+the legacy to the stranger}}={\tfrac {1}{25}}[9[\alpha -\epsilon ]-6a-3\mu ]&=66\\&{\text{+the inheritance of 1st daughter}}={\tfrac {1}{25}}[6[\alpha -\epsilon ]-4a-2\mu ]&=44\\&{\text{+the inheritance of 2d daughter}}={\tfrac {1}{25}}[6[\alpha -\epsilon ]-4a-2\mu ]&=44\\&{\text{Total}}\;={\tfrac {1}{25}}[21\alpha -4\epsilon -14a-7\mu ]&=164\end{aligned}}$
    And the master takes $\mu +2x={\tfrac {1}{25}}[4\alpha -4\epsilon +14a-7\mu ]=236$
    Had the slave died possessed of no property whatever, his ransom would have been $200$.
    His ransom, here stated, exclusive of the sum which the master inherits from him, or $a-x,=192$.

12. Were there the first slave only, who has paid off two-thirds of his original cost, the master having spent the money, that slave would have to complete his ransom by paying two-ninths of his original cost, that is $66{\tfrac {2}{3}}$ (see page 141).
    Were there the second slave only, who has paid off none of his original cost, he would have to ransom himself at two-thirds of his cost; that is by paying $200$ (see also page 141).
    The master’s heirs, in the case described in the text, are entitled to receive the same amount from the two slaves jointly, viz. $266{\tfrac {2}{3}}$, as they would be entitled to receive, according to the rule of page 141, from the two slaves, separately; but the payment of the sum is differently distributed; the slave who has paid two-thirds of his ransom being required to pay one-ninth only of his original cost; and the slave who has paid no ransom, being required to pay two-thirds of his own cost, and one-ninth of the cost of the first slave.
13. Let $A$. be the first slave; his original cost $a$; the property he dies possessed of $\alpha$; and let $B$. be the second slave; and his cost $b$. Let $x$ be that which the master gives to A. in emancipating him.
    A.’s ransom is $a-x$; and his property, minus his ransom, is $\alpha -a+x$.
    A.’s daughter receives ${\tfrac {1}{2}}[\alpha -a+x]$, and the master’s heirs receive ${\tfrac {1}{2}}[\alpha -a+x]$
    Hence the master receives altogether from A.,
    $a-x+{\tfrac {1}{2}}[\alpha -a+x]={\tfrac {1}{2}}[\alpha +a-x.]$
    B.’s ransom is $b-{\tfrac {b}{a}}x$
    The master’s heirs receive from A. and B. together ${\tfrac {1}{2}}[\alpha +a+2b]-{\tfrac {1}{2a}}[a+2b]x$; and this is to be made equal to twice the amount of the legacies to A. and B., that is, ${\tfrac {1}{2}}[\alpha +a+2b]-{\tfrac {1}{2a}}[a+2b]x=2{\tfrac {a+b}{a}}x$
    $\therefore \;x=a{\tfrac {\alpha +a+2b}{5a+6b}}={\tfrac {1700}{15}}=113{\tfrac {1}{3}}$
    The master’s heirs receive from A., ${\tfrac {2a[\alpha +a+b]+3\alpha b}{5a+6}}=293{\tfrac {1}{3}}$
    A.’s daughter receives $[a+b]{\tfrac {3\alpha -2}{5a+6}}a=800\times {\tfrac {600}{4500}}=106{\tfrac {2}{3}}$
    The legacy to B. is $b\;{\tfrac {\alpha +a+2b}{5a+6b}}=188{\tfrac {8}{9}}$; his ransom $b{\tfrac {4a+4b-\alpha }{5a+6b}}=311{\tfrac {1}{9}}$
    The master’s heirs receive from A. and B. together $2[a+b]{\tfrac {\alpha +a+2b}{5a+6b}}=604{\tfrac {4}{9}}$.
14. The first slave is A.; his cost $a$; his property $\alpha$; he leaves a daughter.
    The second slave is B.; his cost $b$.
    Then (as in page 147) ${\tfrac {1}{2}}[\alpha -a+x]$ goes to the daughter ; and $x=a{\tfrac {\alpha +a+2b}{5a+6b}}$
    The daughter receives $[a+b]{\tfrac {3\alpha -2a}{5a+6b}}$
    The master receives from A.${\tfrac {2a[a+\alpha +b]+3\alpha b}{5a+6b}}$
    and the master receives from A. and B. together $2[a+b]{\tfrac {\alpha +a+2b}{5a+6b}}$
    

    But if $b=a$	$x={\tfrac {1}{11}}[\alpha +3a]=127{\tfrac {3}{11}}$
    The daughter receives	${\tfrac {2}{11}}[3\alpha -2a]=163{\tfrac {7}{11}}$
    The master receives from A.	${\tfrac {1}{11}}[5\alpha +4a]=336{\tfrac {4}{11}}$
    The master receives from B.	${\tfrac {1}{11}}[8a-\alpha ]=172{\tfrac {8}{11}}$
    The master receives from A and B.	${\tfrac {4}{11}}[\alpha +3a]=509{\tfrac {1}{11}}$

    If $b=0$,
    The daughter receives ${\tfrac {1}{5}}[3\alpha -2a]$
    The master ${\tfrac {2}{5}}[\alpha +a]$, as in page 142.

15. The slave A. dies before his master, and leaves a daughter. His cost is $a$, of which he has redeemed $a^{\prime }$, which the master has spent; and he leaves property $\alpha$.
    

    Then the daughter receives	${\tfrac {1}{2}}[\alpha +a^{\prime }-a+x]$
    The master receives altogether	${\tfrac {1}{2}}[\alpha +a^{\prime }+a-x]$
    The master’s heirs receive	${\tfrac {1}{2}}[\alpha -a^{\prime }+a-x]$
    And ${\tfrac {1}{2}}[\alpha -a^{\prime }+a-x]=2x\;\therefore \;x={\tfrac {1}{5}}[\alpha -a^{\prime }+a]$
    Hence the daughter receives	${\tfrac {1}{5}}[3\alpha +2a^{\prime }-2a]=140$
    The master’s heirs	${\tfrac {1}{5}}[2\alpha -2a^{\prime }+2a]=160$
    The master receives, in toto,	${\tfrac {1}{5}}[2\alpha +3a^{\prime }+2a]=360$

    If the slave had not advanced, or the master had not spent $a^{\prime }$,
    the daughter would have received ${\tfrac {1}{5}}[3\alpha +3a^{\prime }-2a]=180$
    and the master would have received ${\tfrac {1}{5}}[2\alpha +2a^{\prime }+2a]=320$

16. A.’s price is $a$; he has advanced to his master $a^{\prime }$; he leaves property $\alpha$. He dies before his master, and leaves a daughter.
    The master’s debts are $\mu$; $x$ is what A. receives, in being emancipated; $a-x$; is the ransom; ${\tfrac {1}{2}}[\alpha +a^{\prime }-a+x]$ is what the daughter receives.
    Then $\alpha -{\tfrac {1}{2}}[\alpha +a^{\prime }-a+x]$ is what remains to the master; and $\alpha -{\tfrac {1}{2}}[\alpha +a^{\prime }-a+x]-\mu$ is what remains to him, after paying his debts; and this is to be made equal to $2x$.
    

    Whence $x={\tfrac {1}{5}}[\alpha +a-a^{\prime }-2\mu ]$
    ${\text{Hence the daughter receives}}$	$={\tfrac {1}{5}}[3\alpha -2a+2a^{\prime }-\mu ]=640$
    ${\begin{aligned}\left.{\begin{aligned}&{\text{The mother receives,}}\\&{\text{inclusive of the debt }}\end{aligned}}\right\}\;\end{aligned}}$	$={\tfrac {1}{5}}[2\alpha +2a-2a^{\prime }+\mu ]=360$
    ${\begin{aligned}\left.{\begin{aligned}&{\text{The master receives,}}\\&{\text{exclusive of the debt }}\end{aligned}}\right\}\;\end{aligned}}$	$={\tfrac {1}{5}}[2\alpha +2a-2a^{\prime }-4\mu ]=160$

    
    

    If the mode given in page $142$ had been followed, it would have given $x={\tfrac {1}{5}}[\alpha +a+a^{\prime }-2\mu ]$
    and the daughter’s portion $={\tfrac {1}{5}}[3\alpha =2a+3a^{\prime }-\mu ]=740$.

17. A. dies before his master, and leaves a mother. His price was $a$; he has redeemed $a^{\prime }$, which the master has spent. The property he leaves is $\alpha$. He owes debts $\epsilon$.
    The master owes debts $\mu$.
    ${\tfrac {1}{3}}[\alpha +a^{\prime }-a+x-\epsilon ]$ is the mother’s.
    $\alpha -{\tfrac {1}{3}}[\alpha +a^{\prime }-a+x-\epsilon ]-\epsilon$ is the master’s.
    $\alpha -{\tfrac {1}{3}}[\alpha +a^{\prime }-a+x-\epsilon ]-\epsilon -\mu =2x=$ the master’s, after paying his debts.
    

    Hence	$x={\tfrac {1}{7}}[2\alpha +a-a^{\prime }-2\epsilon -3\mu ]=300$
    Mother’s	$={\tfrac {1}{7}}[3\alpha -2a+2a^{\prime }-3\epsilon -3\mu ]=650$
    Master’s, without $\mu$	$={\tfrac {1}{7}}[4\alpha +2a-2a^{\prime }-4\epsilon -6\mu ]=600$
    Mother’s, with $\mu$	$={\tfrac {1}{7}}[4\alpha +2a-2a^{\prime }-4\epsilon +\mu ]=900$
    A. receives, inclusive of $\epsilon$	$={\tfrac {1}{7}}[3\alpha -2a+2a^{\prime }+4\epsilon -\mu ]=850$.

18. A. is emancipated by his master, and then dies, leaving a daughter, who dies, leaving a husband. Then the master dies.
    A.’s price$=a$; his property a. What he receives from the master $=x$.
    The daughter’s property $=d$
    A.’s ransom $=a-x$. The daughter inherits ${\tfrac {1}{2}}[\alpha -a+x]$, and ${\tfrac {1}{2}}[\alpha -a+x]$ goes to the master.
    ${\tfrac {1}{2}}[d+{\tfrac {1}{2}}[\alpha -a+x]]$ goes to the daughter’s husband and ${\tfrac {1}{2}}[d+{\tfrac {1}{2}}[\alpha -a+x]]$ to the master.
    Hence, according to the author, we are to make
    $a-x+{\tfrac {1}{2}}[\alpha -a+x]+{\tfrac {1}{2}}[d+{\tfrac {1}{2}}[\alpha -a+x]]=2x$
    

    $\therefore \;x={\tfrac {1}{9}}[3\alpha +a+2d]=200$
    Daughter’s share	$={\tfrac {1}{9}}[6\alpha -4a+d]=100$
    Husband’s	$={\tfrac {1}{9}}[3\alpha -2a+5d]=200$
    Master’s	$={\tfrac {1}{9}}[2\alpha +6a+4d]=400.$

19. A. is emancipated, and dies, leaving a daughter, and bequeathing one-third of his property to a stranger.
    The daughter dies, leaving a mother, and bequeathing one-third of her property to a stranger.
    A.’s price is $a$; his property is $\alpha$
    The daughter’s property is $d$.
    A.’s ransom is $a-x$; $\alpha -a+x$ is his property, clear of ransom.
    ${\tfrac {1}{3}}[\alpha -a+x]$ goes to the stranger; and the like amount to A.’s daughter, and to the master.
    ${\tfrac {1}{3}}[3d+\alpha -a+x]$ is the property left by the daughter.
    ${\tfrac {1}{9}}[3d+\alpha -a+x]$ is the bequest of the daughter to a stranger.
    ${\tfrac {2}{9}}[3d+\alpha -a+x]$ is the residue, of which ${\tfrac {1}{3}}d$,
    viz. ${\tfrac {2}{27}}[3d+\alpha -a+x]$ is the mother’s,
    and ${\tfrac {4}{27}}[3d+\alpha -a+x]$ is the master’s;
    Hence, according to the author, we are to make
    $a-x+{\tfrac {1}{3}}[\alpha -a+x]+{\tfrac {4}{27}}[3d+\alpha -a+x]=2x$
    

    Therefore	$x={\tfrac {1}{68}}[13\alpha +14a+12d]=210{\tfrac {5}{17}}$
    The daughter’s share	$={\tfrac {1}{68}}[27\alpha -18a+4d]=136{\tfrac {13}{17}}$
    The daughter’s bequest	$={\tfrac {1}{68}}[9\alpha -6a+24d]=145{\tfrac {10}{17}}$
    The mother’s share	$={\tfrac {2}{68}}[3\alpha -2a+8d]=97{\tfrac {1}{17}}$
    The master’s	$={\tfrac {2}{68}}[13\alpha +14a+12d]=420{\tfrac {10}{17}}$

20. The price of the slave-girl being $a$; and what she receives on being emancipated $x$, her ransom is $a-x$.
    If her dowry is $\alpha$, he that receives her, takes $\alpha +x$. Hence, according to the author, we are to make
    $a-x=2[\alpha +x]$; whence $x={\tfrac {a-2\alpha }{3}}$
    And her ransom is ${\tfrac {2}{3}}[a+\alpha ]$
    But if a male slave be at the same time emancipated by the master, the donee must pay the ransom of that slave.
    If his price was $b\;,b-{\tfrac {b}{a}}x$ is his ransom.
    Hence, according to the author, we are to make the sum of the two ransoms, viz. $a-x+b-{\tfrac {b}{a}}x=2[\alpha +x]$
    $\therefore \;a+b-2\alpha =[3+{\tfrac {b}{a}}]x\;\therefore \;x=a{\tfrac {a+b-2\alpha }{3a+b}}=125$
    The donee pays ransom, in respect of the slave-girl $(a-x)=375$ and he pays ransom for the male slave $b-{\tfrac {b}{a}}x=75$.
21. The same notation being used as in the last example, the equation for determining $x$, according to the author, is to be
    ${\begin{aligned}a-x+b-{\tfrac {b}{a}}x-x=2[\alpha +2x]\\\therefore \;x={\tfrac {a}{6a+b}}[a+b-2\alpha ]=64{\tfrac {16}{31}}.\end{aligned}}$
22. The same notation being used as in the two former examples, the equation for determining $x$, according to the author, is
    $a-x+b-{\tfrac {b}{a}}x-{\tfrac {3}{4}}x=2[\alpha +1{\tfrac {3}{4}}x]$
    Whence $x={\tfrac {4a}{21a+4b}}[a+b-2\alpha ]=73{\tfrac {43}{100}}$
    
23. Let a be the slave-girl’s price $-\alpha$her dowry. Then, according to the author, we are to make
    $a-x+\alpha -{\tfrac {\alpha }{a}}x=2x$
    Therefore $x={\tfrac {a}{3a+\alpha }}[a+\alpha ]={\tfrac {3}{10}}\times 400=120$
    The donee is to receive the girl’s dowry, worth $400$, for $280$.
24. If the donor has cohabited with the slave-girl, the donor’s heirs are to retain the dowry, but must allow the donee, in addition to the legacy $x$, the further sum of ${\tfrac {\alpha }{a}}x$;
    The ransom is then $a-x-{\tfrac {\alpha }{a}}x$ which according to the author is to be made equal to $2x$.
    Whence $x={\tfrac {a^{2}}{3x+a}}=90$
    The donee is to receive the girl, worth $300$, for $210$.
25. If the donor has previously cohabited with the slave-girl, it appears from the last example, that the donee is entitled to ransom her for $a-x-{\tfrac {\alpha }{a}}x$.
    If the donee cohabits with the slave-girl, it appears from the last example but one, that he is entitled to redeem the dowry, $\alpha$, for $\alpha -{\tfrac {\alpha }{a}}x$
    The redemption of the girl and dowry is
    $a-x-{\tfrac {\alpha }{a}}x+\alpha -{\tfrac {\alpha }{a}}x$,
    which, according to the author, is to be made equal to $2x$.
    That $a+\alpha -{\tfrac {a+2\alpha }{a}}x=2x$
    Whence $x={\tfrac {a}{3a+2\alpha }}\times [a+\alpha ]=109{\tfrac {1}{11}}$
    The donee is to receive the girl and dowry, worth $400$, for $290{\tfrac {10}{11}}$.
26. The second case is here solved in a different way.
    ${\begin{aligned}a-x-{\tfrac {\alpha }{a}}x=2[x+{\tfrac {\alpha }{a}}x]\\\therefore \;x={\tfrac {a^{2}}{3[\alpha +a]}}\end{aligned}}$
    This being halved between the legatee and donee becomes
    ${\begin{aligned}&&&{\tfrac {\alpha ^{2}}{6[\alpha +a]=37{\tfrac {1}{2}}}}\end{aligned}}$
    The donee receives the girl, worth $300$, for $262{\tfrac {1}{2}}$.
27. According to the author’s rule, which is purely arbitrary,
    $a-2x+\alpha -{\tfrac {3\alpha }{a}}x=4[1+{\tfrac {\alpha }{a}}]x$
    Whence $x=a{\tfrac {a+\alpha }{6a+7\alpha }}=48$
    The donee will have to redeem the girl and dowry, worth $400$, for $352$.
28. We have here the only instance in the treatise of a simple equation, involving two unknown quantities. For what the donee receives is one unknown quantity; and what the donor receives back again from the donee, called by the author “part of thing,” is the other unknown quantity.
    Let what the donee receives $=x$, and what the donor receives $=y$.
    Then, retaining the same notation as before, according to the author, the donee receives, on the whole
    $x-y-[\alpha -{\tfrac {\alpha }{a}}x]+{\tfrac {\alpha }{a}}[x-y]=2y$
    and the donor receives, on the whole
    $a-x+y+[\alpha -{\tfrac {\alpha }{a}}x]-{\tfrac {\alpha }{a}}[x-y]=2[x+{\tfrac {\alpha }{a}}[x-y]]$
    Whence $x={\tfrac {1}{2}}{\tfrac {a}{4a^{2}+5a\alpha -\alpha ^{2}}}[3a^{2}+3a\alpha -2\alpha ^{2}]=102$
    $y={\tfrac {1}{2}}{\tfrac {a}{4a^{2}+5a\alpha -\alpha ^{2}}}[a^{2}-2\alpha ^{2}]=21$
29. Let a be the gift of money; and the value of the measure $m\times a$.
    It appears from the context that the donee is to pay the heirs ${\tfrac {2}{3}}ma$.
    It is arbitrary how he shall apportion this sum between the money capital and the measure.
    If he pays on the money capital $p\cdot a$
    and on the measure $q\cdot ma$
    we have the equation $p\cdot a+q\cdot ma={\tfrac {2}{3}}ma$
    or $p+q\;m={\tfrac {2}{3}}m$
    The author assumes $p={\tfrac {m}{2}}\cdot q$
    Whence $q={\tfrac {4}{9}}$, and $p={\tfrac {5}{9}}$, and therefore the donee pays
    on the money capital ${\tfrac {5}{9}}a=11{\tfrac {1}{9}}$
    and on the measure ${\tfrac {4}{9}}ma=22{\tfrac {2}{9}}$
    Total $33{\tfrac {1}{3}}$