The Compendious Book on Calculation by Completion and Balancing/On legacies

by Abū ʿAbdallāh Muḥammad ibn Mūsā al-Khwārizmī

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ON LEGACIES.

 

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On Capital, and Money lent.

(65) “A man dies, leaving two sons behind him, and bequeathing one-third of his capital to a stranger. He leaves ten dirhems of property and a claim of ten dirhems upon one of the sons.”

Computation: You call the sum which is taken out of the debt thing. Add this to the capital, which is ten dirhems. The sum is ten and thing. Subtract one-third of this, since he has bequeathed one-third of his property, that is, three dirhems and one-third of thing. The remainder is six dirhems and two-thirds of thing. Divide this between the two sons. The portion of each of them is three dirhems and one-third plus one-third of thing. This is equal to the thing which was sought for.^[1] Reduce it, by removing one-third from thing, on account of the other third of thing. There remain two-thirds of thing, equal to three dirhems and one-third. It is then only required that you complete the thing, by adding to it as much as one half of the same; accordingly, you add to three and one-third as much as one-half of them: This gives five dirhems, which is the thing that is taken out of the debts.

If he leaves two sons and ten dirhems of capital and a demand of ten dirhems against one of the sons, and bequeaths one-fifth of his property and one dirhem to a stranger, the computation is this: Call the sum which is taken out of the debt, thing. Add this to the property; the sum is thing and ten dirhems. Subtract one-fifth of this, since he has bequeathed one-fifth of (66) his capital, that is, two dirhems and one-fifth of thing; the remainder is eight dirhems and four-fifths of thing. Subtract also the one dirhem which he has bequeathed; there remain seven dirhems and four-fifths of thing. Divide this between the two sons; there will be for each of them three dirhems and a half plus two-fifths of thing; and this is equal to one thing.^[2] Reduce it by subtracting two-fifths of thing from thing. Then you have three-fifths of thing, equal to three dirhems and a half. Complete the thing by adding to it two-thirds of the same: add as much to the three dirhems and a half, namely, two dirhems and one-third; the sum is five and five-sixths. This is the thing, or the amount which is taken from the debt.

If he leaves three sons, and bequeaths one-fifth of his property less one dirhem, leaving ten dirhems of capital and a demand of ten dirhems against one of the sons, the computation is this: You call the sum which is taken from the debt thing. Add this to the capital; it gives ten and thing. Subtract from this one-fifth of it for the legacy; it is two dirhems and one-fifth of thing. There remain eight dirhems and four-fifths of thing; add to this one dirhem, since he stated “less one dirhem.” Thus you have nine dirhems and four-fifths of thing. Divide this between the three sons. There will be for each son three dirhems, and one-fifth and one-third and one-fifth of thing. This equals one thing.^[3] Subtract one-fifth and one-third of one-fifth (67) of thing from thing. There remain eleven-fifteenths of thing, equal to three dirhems. It is now required to complete the thing. For this purpose, add to it four-elevenths, and do the same with the three dirhems, by adding to them one dirhem and one-eleventh. Then you have four dirhems and one-eleventh, which are equal to thing. This is the sum which is taken out of the debt.

On another Species of Legacy.

“A man dies, leaving his mother, his wife, and two brothers and two sisters by the same father and mother with himself; and he bequeaths to a stranger one-ninth of his capital.”

Computation:^[4] You constitute their shares by taking them out of forty-eight parts. You know that if you take one-ninth from any capital, eight-ninths of it will remain. Add now to the eight-ninths one-eighth of the same, and to the forty-eight also one-eighth of them, namely, six, in order to complete your capital. This gives fifty-four. The person to whom one-ninth is bequeathed receives six out of this, being one-ninth of the whole capital. The remaining forty-eight will be distributed among the heirs, proportionably to their legal shares.

If the instance be: “A woman dies, leaving her husband, a son, and three daughters, and bequeathing to a stranger one-eighth and one-seventh of her (67) capital;” then you constitute the shares of the heirs, by taking them out of twenty.^[5] Take a capital, and subtract from it one-eighth and one-seventh of the same. The remainder is, a capital less one-eighth and one-seventh. Complete your capital by adding to that which you have already, fifteen forty-one parts. Multiply the parts of the capital, which are twenty, by forty-one; the product is eight hundred and twenty. Add to it fifteen forty-one parts of the same, which are three hundred: the sum is one thousand one hundred and twenty parts. The person to whom one-eighth and one-seventh were bequeathed, receives one-eighth and one-seventh of this. One seventh of it is one hundred and sixty, and one-eighth one hundred and forty. Subtracting this, there remain eight hundred and twenty parts for the heirs, proportionably to their legal shares.

On another Species of Legacies,^[6] viz.

If nothing has been imposed on some of the heirs,^[7] and something has been imposed on others; the legacy amounting to more than one-third. It must be known, that the law for such a case is, that if more than one-third of the legacy has been imposed on one of the heirs, this enters into his share; but that also those on whom nothing has been imposed must, nevertheless, contribute one-third.

Example: “A woman dies, leaving her husband, a son, and her mother. She bequeaths to a person two-fifths, and to another one-fourth of her capital. She imposes the two legacies together on her son, and on her mother one moiety (of the mother’s share of the residue); on her husband she imposes nothing but one-third, (which he must contribute, according to the law).”^[8] Computation: You constitute the shares of the (69) heritage, by taking them out of twelve parts: the son receives seven of them, the husband three, and the mother two parts. You know that the husband must give up one-third of his share; accordingly he retains twice as much as that which is detracted from his share for the legacy. As he has three parts in hand, one of these falls to the legacy, and the remaining two parts he retains for himself. The two legacies together are imposed upon the son. It is therefore necessary to subtract from his share two-fifths and one-fourth of the same. He thus retains seven twentieths of his entire original share, dividing the whole of it into twenty equal parts. The mother retains as much as she contributes to the legacy; this is one (twelfth part), the entire amount of what she had received being two parts.

Take now a sum, one-fourth of which may be divided into thirds, or of one-sixth of which the moiety may be taken; this being again divisible by twenty. Such a capital is two hundred and forty. The mother receives one-sixth of this, namely, forty; twenty from this fall to the legacy, and she retains twenty for herself. The husband receives one-fourth, namely, sixty; from which twenty belong to the legacy, so that he retains forty. The remaining hundred and forty belong to the son; the legacy from this is two-fifths and one-fourth, or ninety-one; so that there remain forty-nine. The entire sum for the legacies is, therefore, one hundred and thirty-one, which must be divided among the two legatees. The one to whom two-fifths were bequeathed, receives eight-thirteenths of this; the one to whom one-fourth was devised, receives five-thirteenths. If you wish distinctly to express the shares of the two legatees, you need only to multiply (70) the parts of the heritage by thirteen, and to take them out of a capital of three thousand one hundred and twenty.

But if she had imposed on her son (payment of) the two-fifths to the person to whom the two-fifths were bequeathed, and of nothing to the other legatee; and upon her mother (payment of) the one-fourth to the person to whom one-fourth was granted, and of nothing to the other legatee; and upon her husband nothing besides the one-third (which he must according to law contribute) to both; then you know that this one-third comes to the advantage of the heirs collectively; and the legatee of the two-fifths receives eight-thirteenths, and the legatee of the one-fourth receives five-thirteenths from it. Constitute the shares as I have shown above, by taking twelve parts; the husband receives one-fourth of them, the mother one-sixth, and the son that which remains.^[9] Computation: You know that at all events the husband must give up one-third of his share, which consists of three parts. The mother must likewise give up one-third, of which each legatee partakes according to the proportion of his legacy. Besides, she must pay to the legatee to whom one-fourth is bequeathed, and whose legacy has been imposed on her, as much as the difference between the one-fourth and his portion of the one-third, namely, nineteen one hundred and fifty-sixths of her entire share, considering her share as consisting of one hundred and fifty-six parts. His portion of the one-third of her share is twenty parts. But what she gives him is one-fourth of her entire share, namely, thirty-nine parts. One third of her share is taken for both legacies, and besides nineteen parts which she must pay to him alone. The son gives to the legatee to whom two-fifths are bequeathed as much as the difference between two-fifths of his (the son’s) share (71) and the legatee’s portion of the one-third, namely, thirty-eight one hundred and ninety-fifths of his (the son’s) entire share, besides the one-third of it which is taken off from both legacies. The portion which he (the legatee) receives from this one-third, is eight-thirteenths of it, namely, forty (one hundred and ninety-fifths); and what the son contributes of the two-fifths from his share is thirty-eight. These together make seventy-eight. Consequently, sixty-five will be taken from the son, as being one-third of his share, for both legacies, and besides this he gives thirty-eight to the one of them in particular. If you wish to express the parts of the heritage distinctly, you may do so with nine hundred and sixty-four thousand and eighty.

 

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On another Species of Legacies.

“A man dies, leaving four sons and his wife; and bequeathing to a person as much as the share of one of the sons less the amount of the share of the widow.” Divide the heritage into thirty-two parts. The widow receives one-eighth,^[10] namely, four; and each son seven. Consequently the legatee must receive three-sevenths of the share of a son. Add, therefore, to the heritage three-sevenths of the share of a son, that is to say, three parts, which is the amount of the legacy. This gives thirty-five, from which the legatee receives three; and the remaining thirty-two are distributed among the heirs proportionably to their legal shares.

If he leaves two sons and a daughter,^[11] and bequeaths to some one as much as would be the share of a third son, if he had one; then you must consider, what (72) would be the share of each son, in case he had three: Assume this to be seven, and for the entire heritage take a number, one-fifth of which may be divided into sevenths, and one-seventh of which may be divided into fifths. Such a number is thirty-five. Add to it two-sevenths of the same, namely, ten. This gives forty-five. Herefrom the legatee receives ten, each son four-teen, and the daughter seven.

If he leaves a mother, three sons, and a daughter, and bequeaths to some one as much as the share of one of his sons less the amount of the share of a second daughter, in case he had one; then you distribute the heritage into such a number of parts as may be divided among the actual heirs, and also among the same, if a second daughter were added to them.^[12] Such a number is three hundred and thirty-six. The share of the second daughter, if there were one, would be thirty-five, and that of a son eighty; their difference is forty-five, and this is the legacy. Add to it three hundred and thirty-six, the sum is three hundred and eighty-one, which is the number of parts of the entire heritage.

If he leaves three sons, and bequeaths to some onę as much as the share of one of his sons, less the share of a daughter, supposing he had one, plus one-third of the remainder of the one-third; the computation will be this:^[13] distribute the heritage into such a number of parts as may be divided among the actual heirs, and also among them if a daughter were added to them. Such a number is twenty-one. Were a daughter among the heirs, her share would be three, and that of a son seven. The testator has therefore bequeathed to, the (73) legatee four-sevenths of the share of a son, and one-third of what remains from one-third. Take therefore one-third, and remove from it four-sevenths of the share of a son. There remains one-third of the capital less four-sevenths of the share of a son. Subtract now one-third of what remains of the one-third, that is to say, one-ninth of the capital less one-seventh and one-third of the seventh of the share of a son; the remainder is two-ninths of the capital less two-sevenths and two-thirds of a seventh of the share of a son. Add this to the two-thirds of the capital; the sum is eight-ninths of the capital less two-sevenths and two thirds of a seventh of the share of a son, or eight twenty-one parts of that share, and this is equal to three shares. Reduce this, you have then eight-ninths of the capital, equal to three shares and eight twenty-one parts of a share. Complete the capital by adding to eight-ninths as much as one-eighth of the same, and add in the same proportion to the shares. Then you find the capital equal to three shares and forty-five fifty-sixth parts of a share. Calculating now each share equal to fifty-six, the whole capital is two hundred and thirteen, the first legacy thirty-two, the second thirteen, and of the remaining one hundred and sixty-eight each son takes fifty-six.

 

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On another Species of Legacies.

“A woman dies, leaving her daughter, her mother, and her husband, and bequeaths to some one as much as the share of her mother, and to another as much as one-ninth of her entire capital.”^[14] Computation: You begin by dividing the heritage into thirteen parts, two of which the mother receives. Now you perceive that the (74) legacies amount to two parts plus one-ninth of the entire capital. Subtracting this, there remains eight-ninths of the capital less two parts, for distribution among the heirs. Complete the capital, by making the eight-ninths less two parts to be thirteen parts, and adding two parts to it, so that you have fifteen parts, equal to eight-ninths of capital; then add to this one-eighth of the same, and to the fifteen parts add likewise one-eighth of the same, namely, one part and. seven-eighths; then you have sixteen parts and seven-eighths. The person to whom one-ninth is bequeathed, receives one-ninth of this, namely, one part and seven-eighths; the other, to whom as much as the share of the mother is bequeathed, receives two parts. The remaining thirteen parts are divided among the heirs, according to their legal shares. You best determine the respective shares by dividing the whole heritage into one hundred and thirty-five parts.

If she has bequeathed as much as the share of the husband and one-eighth and one-tenth of the capital,^[15] then you begin by dividing the heritage into thirteen parts. Add to this as much as the share of the husband, namely, three; thus you have sixteen. This is what remains of the capital after the deduction of one- eighth and one-tenth, that is to say, of nine-fortieths. The remainder of the capital, after the deduction of one-eighth and one-tenth, is thirty-one fortieths of the same, which must be equal to sixteen parts. Complete your capital by adding to it nine thirty-one parts of the same, and multiply sixteen by thirty-one, which gives four hundred and ninety-six; add to this nine thirty-one parts of the same, which is one hundred and forty-four (75). The sum is six hundred and forty. Subtract one-eighth and one-tenth from it, which is one hundred and forty-four, and as much as the share of the husband, which is ninety-three. There remains four hundred and three, of which the husband receives ninety-three, the mother sixty-two, and every daughter one hundred and twenty-four.

If the heirs are the same,^[16] but that she bequeaths to a person as much as the share of the husband, less one-ninth and one-tenth of what remains of the capital, after the subtraction of that share, the computation is this: Divide the heritage into thirteen parts. The legacy from the whole capital is three parts, after the subtraction of which there remains the capital less three parts. Now, one-ninth and one-tenth of the remaining capital must be added, namely, one-ninth and one-tenth of the whole capital less one-ninth and one-tenth of three parts, or less nineteen-thirtieths of a part; this yields the capital and one-ninth and one-tenth less three parts and nineteen-thirtieths of a part, equal to thirteen parts. Reduce this, by removing the three parts and nineteen-thirtieths from your capital, and adding them to the thirteen parts. Then you have the capital and one-ninth and one-tenth of the same, equal to sixteen parts and nineteen-thirtieths of a part. Reduce this to one capital, by subtracting from it nineteen one-hundred-and-ninths. There remains a (76) capital, equal to thirteen parts and eighty one-hundred-and-ninths. Divide each part into one hundred and nine parts, by multiplying thirteen by one hundred and nine, and add eighty to it. This gives one thousand four hundred and ninety-seven parts. seven parts. The share of the husband from it is three hundred and twenty-seven parts.

If some one leaves two sisters and a wife,^[17] and bequeaths to another person as much as the share of a sister less one-eighth of what remains of the capital after the deduction of the legacy, the computation is this: You consider the heritage as consisting of twelve parts. Each sister receives one-third of what remains of the capital after the subtraction of the legacy; that is, of the capital less the legacy. You perceive that one-eighth of the remainder plus the legacy equals the share of a sister; and also, one-eighth of the remainder is as much as one-eighth of the whole capital less one-eighth of the legacy; and again, one-eighth of the capital less one-eighth of the legacy added to the legacy equals the share of a sister, namely, one-eighth of the capital and seven-eighths of the legacy. The whole capital is therefore equal to three-eighths of the capital plus three and five-eighth times the legacy. Subtract now from the capital three-eighths of the same. There remain. five-eighths of the capital, equal to three and five-eighth times the legacy; and the entire capital is equal to five.and four-fifth times the legacy. Consequently, if you assume the capital to be twenty-nine, the legacy is five, and each sister’s share eight.

On another Species of Legacies.

“A man dies, and leaves four sons, and bequeaths to some person as much as the share of one of his sons; and to another, one-fourth of what remains after-the deduction of the above share from one-third.” You perceive that this legacy belongs to the class of those (77) which are taken from one-third of the capital.^[18] Computation: Take one-third of the capital, and subtract from it the share of a son. The remainder is one-third of the capital less the share. Then subtract from it one-fourth of what remains of the one-third, namely, one-fourth of one-third less one-fourth of the share. The remainder is one-fourth of the capital less three-fourths of the share. Add hereto two-thirds of the capital: then you have eleven-twelfths of the capital less three-fourths of a share, equal to four shares. Reduce this by removing the three-fourths of the share from the capital, and adding them to the four shares. Then you have eleven-twelfths of the capital, equal to four shares and three-fourths. Complete your capital, by adding to the four shares and three-fourths one-fourth of the same. Then you have five shares and two-elevenths, equal to the capital. Suppose, now, every share to be eleven; then the whole square will be fifty-seven; one-third of this is nineteen; from this one share, namely, eleven, must be subtracted; there remain eight. The legatee, to whom one-fourth of this remainder was bequeathed, receives two. The remaining six are returned to the other two-thirds, which are thirty-eight. Their sum is forty-four, which is to be divided amongst the four sons; so that each son receives eleven.

If he leaves four sons, and bequeaths to a person as much as the share of a son, less one-fifth of what remains from one-third after the deduction of that share, then this is likewise a legacy, which is taken from one-third.^[19] Take one-third, and subtract from it one share; there remains one-third less the share. Then return to it that which was excepted, namely, one-fifth of the one-third less one-fifth of the share. This gives one-third and one-fifth of one-third (or two-fifths) (78) less one share and one-fifth of a share. Add this to two-thirds of the capital. The sum is, the capital and one-third of one-fifth of the capital less one share and one-fifth of a share, equal to four-shares. Reduce this by removing one share and one-fifth from the capital, and add to it the four shares. Then you have the capital and one-third of one-fifth of the capital, which are equal to five shares and one-fifth. Reduce this to one capital, by subtracting from what you have the moiety of one-eighth of it, that is to say, one-sixteenth. Then you find the capital equal to four shares and seven-eighths of a share. Assume now thirty-nine as capital; one-third of it will be thirteen, and one share eight; what remains of one-third, after the deduction of that share, is five, and one-fifth of this is one. Subtract now the one, which was excepted from the legacy; the remaining legacy then is seven; subtracting this from the one-third of the capital, there remain six. Add this to the two-thirds of the capital, namely, to the twenty-six parts, the sum is thirty-two; which, when distributed among the four sons, yields eight for each of them.

If he leaves three sons and a daughter^[20], and bequeaths to some person as much as the share of a daughter, and to another one-fifth and one-sixth of what remains of two-sevenths of the capital after the deduction of the first legacy; then this legacy is to be taken out of two-sevenths of the capital. Subtract from two-sevenths the share of the daughter: there remain two-sevenths of the capital less that share. Deduct from this the second legacy, which comprises (79) one-fifth and one-sixth of this remainder: there remain one-seventh and four-fifteenths of one-seventh of the capital less nineteen-thirtieths of the share. Add to this the other five-sevenths of the capital: then you have six-sevenths and four-fifteenths of one-seventh of the capital less nineteen thirtieths of the share, equal to seven shares. Reduce this, by removing the nineteen thirtieths, and adding them to the seven shares: then you have six-sevenths and four-fifteenths of one-seventh of capital, equal to seven shares and nineteen-thirtieths. Complete your capital by adding to every thing that you have eleven ninety-fourths of the same; thus the capital will be equal to eight shares and ninety-nine one hundred and eighty-eighths. Assume now the capital to be one thousand six hundred and three; then the share of the daughter is one hundred and eighty-eight. Take two-sevenths of the capital; that is, four hundred and fifty-eight. Subtract from this the share, which is one hundred and eighty-eight; there remain two hundred and seventy. Remove one-fifth and one-sixth of this, namely, ninety-nine; the remainder is one hundred and seventy-one. Add thereto fivesevenths of the capital, which is one thousand one hundred and forty-five. The sum is one thousand three (80) hundred and sixteen parts. This may be divided into seven shares, each of one hundred and eighty-eight parts; then this is the share of the daughter, whilst every son receives twice as much.

If the heirs are the same, and he bequeaths to some person as much as the share of the daughter, and to another person one-fourth and one-fifth out of what remains from two-fifths of his capital after the deduction of the share; this is the computation:^[21] You must observe that the legacy is determined by the two-fifths. Take two-fifths of the capital and subtract the shares: the remainder is, two-fifths of the capital less the share. Subtract from this remainder one-fourth and one-fifth of the same, namely, nine-twentieths of two-fifths, less as much of the share. The remainder is one-fifth and one-tenth of one-fifth of the capital less eleven-twentieths of the share. Add thereto three-fifths of the capital; the sum is four-fifths and one-tenth of one-fifth of the capital, less eleven-twentieths of the share, equal to seven shares. Reduce this by removing the eleven-twentieths of a share, and adding them to the seven shares. Then you have the same four-fifths and one-tenth of one-fifth of capital, equal to seven shares and eleven-twentieths. Complete the capital by adding to any thing that you have nine forty-one parts. Then you have capital equal to nine shares and seventeen eighty-seconds. Now assume each portion to consist of eighty-two parts; then you have seven hundred and fifty-five parts. Two-fifths of these are three hundred (81) and two. Subtract from this the share of the daughter, which is eighty-two; there remain two hundred and twenty. Subtract from this one-fourth and one-fifth, namely, ninety-nine parts. There remain one hundred and twenty-one. Add to this three-fifths of the capital, namely, four hundred and fifty-three. Then you have five hundred and seventy-four, to be divided into seven shares, each of eighty-two parts. This is the share of the daughter; each son receives twice as much.

If the heirs are the same, and he bequeaths to a person as much as the share of a son, less one-fourth and one-fifth of what remains of two-fifths (of the capital) after the deduction of the share; then you see that this legacy is likewise determined by two-fifths. Subtract two shares (of a daughter) from them, since every son receives two (such) shares; there remain two-fifths of the capital less two (such) shares. Add thereto what was excepted from the legacy, namely, one-fourth and one-fifth of the two-fifths less nine-tenths of (a daughter’s) share.^[22] Then you have two-fifths and nine-tenths of one-fifth of the capital less two (daughter’s) shares and nine-tenths. Add to this three-fifths of the capital. Then you have one capital and nine-tenths of one-fifth of the capital less two (daughter’s) shares and nine-tenths, equal to seven (such) shares. Reduce this by removing the two shares and nine-tenths and adding them to the seven shares. Then you have one capital and nine-tenths of one-fifth of the capital, equal to nine shares of a daughter and nine-tenths. (82) Reduce this to one entire capital, by deducting nine fifty-ninths from what you have. There remains the capital equal to eight such shares and twenty-three fifty-ninths. Assume now each share (of a daughter) to contain fifty-nine parts. Then the whole heritage comprizes four hundred and ninety-five parts. Two-fifths of this are one hundred and ninety-eight parts. Subtract therefrom the two shares (of a daughter) or one hundred and eighteen parts; there remain eighty parts. Subtract now that which was excepted, namely, one-fourth and one fifth of these eighty, or thirty-six parts; there remain for the legatee eighty-two parts. Deduct this from the parts in the total number of parts in the heritage, namely, four hundred and ninety-five. There remain four hundred and thirteen parts to be distributed into seven shares; the daughter receiving (one share or) fifty-nine (parts), and each son twice as much.

If he leaves two sons and two daughters, and bequeaths to some person as much as the share^[23]. of a daughter less one-fifth of what remains from one-third after the deduction of that share; and to another person as much as the share of the other daughter less one-third of what remains from one-third after the deduction of all this; and to another person half one-sixth of his entire capital; then you observe that all these legacies are determined by the one-third. Take one-third of the capital, and subtract from it the share of a daughter; there remains one-third of the capital less one share. Add to this that which was excepted, namely, one-fifth of the one-third less one-fifth of the share: this gives one-third and one-fifth of one-third of (83) the capital less one and one-fifth portion. Subtract herefrom the portion of the second daughter; there remain one-third and one-fifth of one-third of the capital less two portions and one-fifth. Add to this that which was excepted; then you have one-third and three-fifths of one-third, less two portions and fourteen-fifteenths of a portion. Subtract herefrom half one-sixth of the entire capital: there remain twenty-seven sixtieths of the capital less the two shares and fourteen-fifteenths, which are to be subtracted. Add thereto two-thirds of the capital, and reduce it, by removing the shares which are to be subtracted, and adding them to the other shares. You have then one and seven-sixtieths of capital, equal to eight shares and fourteen-fifteenths. Reduce this to one capital by subtracting from every thing that you have seven-sixtieths. Then let a share be two hundred and one;^[24] the whole capital will be one thousand six hundred and eight.

If the heirs are the same, and he bequeaths to a person as much as the share of a daughter, and one-fifth of what remains from one-third after the deduction of that share; and to another as much as the share of the second daughter and one-third of what remains from one-fourth after the deduction of that share: then, in the computation,^[25] you must consider that the two legacies are determined by one-fourth and one-third. Take one-third of the capital, and subtract from it one share; there remains one-third of the capital less one share. Then subtract one-fifth of the remainder, namely, one-fifth of one-third of the capital, less one-fifth of the share; there remain four-fifths of one-third, less four-fifths of the share. Then take also one-fourth of the capital, and subtract from it one (84) share; there remains one-fourth of the capital, less one share. Subtract one-third of this remainder: there remain two-thirds of one-fourth of the capital, less two-thirds of one share. Add this to the remainder from the one-third of the capital; the sum will be twenty-six sixtieths of the capital, less one share and twenty: eight sixtieths. Add thereto as much as remains of the capital after the deduction of one-third and one-fourth from it; that is to say, one-fourth and one-sixth; the sum is seventeen-twentieths of the capital, equal to seven shares and seven-fifteenths. Complete the capital, by adding to the portions which you have three-seventeenths of the same. Then you have one capital, equal to eight shares and one-hundred-and-twenty hundred-and-fifty-thirds. Assume now one share to consist of one-hundred-and-fifty-three parts, then the capital consists of one thousand three hundred and forty-four. The legacy determined by one-third, after the deduction of one share, is fifty-nine; and the legacy determined by one-fourth, after the deduction of the share, is sixty-one.

If he leaves six sons, and bequeaths to a person as much as the share of a son and one-fifth of what remains of one-fourth; and to another person as much as the share of another son less one-fourth of what remains of one-third, after the deduction of the two first legacies and the second share; the computation is this:^[26] You subtract one share from one-fourth of the capital; there remains one-fourth less the share. Remove then (85) one-fifth of what remains of the one-fourth, namely, half one-tenth of the capital less one-fifth of the share. Then return to the one-third, and deduct from it half one-tenth of the capital, and four-fifths of a share, and one other share besides. The remainder then is one-third, less half one-tenth of the capital, and less one share and four-fifths. Add hereto one-fourth of the remainder, which was excepted, and assume the one-third to be eighty; subtracting from it half one-tenth of the capital, there remain of it sixty-eight less one share and four-fifths. Add to this one-fourth of it, namely, seventeen parts, less one-fourth of the shares to be subtracted from the parts. Then you have eighty-five parts less two shares and one-fourth. Add this to the other two-thirds of the capital, namely, one hundred and sixty parts. Then you have one and one-eighth of one-sixth of capital, less two shares and one-fourth, equal to six shares. Reduce this, by removing the shares which are to be subtracted, and adding them to the other shares. Then you have one and one-eighth of one-sixth of capital, equal to eight shares and one-fourth. Reduce this to one capital, by subtracting from the parts as much as one forty-ninth of them. Then you have a capital equal to eight shares and four forty-ninths. Assume now every share to be forty-nine; then the entire capital will be three hundred and ninety-six; the share forty-nine; the legacy (86) determined by one-fourth, ten; and the exception from the second share will be six.

 

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On the Legacy with a Dirhem.

“A man dies, and leaves four sons, and bequeaths to some one a dirhem, and as much as the share of a son, and one-fourth of what remains from one-third after the deduction of that share.” Computation: ^[27] Take one-third of the capital and subtract from it one share; there remains one-third, less one share. Then subtract one-fourth of the remainder, namely, one-fourth of one-third, less one-fourth of the share; then subtract also one dirhem; there remain three-fourths of one-third of the capital, that is, one-fourth of the capital, less three-fourths of the share, and less one dirhem. Add this to two-thirds of the capital. The sum is eleven-twelfths of the capital, less three-fourths of the share and less one dirhem, equal to four shares. Reduce this by removing three-fourths of the share and one dirhem; then you have eleven-twelfths of the capital, equal to four shares and three-fourths, plus one dirhem. Complete your capital, by adding to the shares and one dirhem one-eleventh of the same. Then you have the capital equal to five shareş and two-elevenths and one dirhem and one-eleventh. If you (87) wish to exhibit the dirhem distinctly, do not complete your capital, but subtract one from the eleven on account of the dirhem, and divide the remaining ten by the portions, which are four and three-fourths. The quotient is two and two-nineteenths of a dirhem. Assuming, then, the capital to be twelve dirhems, each share will be two dirhems and two-nineteenths. Or, if you wish to exhibit the share distinctly, complete your square, and reduce it, when the dirhem will be eleven of the capital.

If he leaves five sons, and bequeaths to some person a dirhem, and as much as the share of one of the sons, and one-third of what remains from one-third, and again, one-fourth of what remains from the one-third after the deduction of this, and one dirhem more; then the computation is this:^[28] You take one-third, and subtract one share; there remains one-third less one share. Subtract herefrom that which is still in your hands, namely, one-third of one-third less one-third of the share. Then subtract also the dirhem; there remain two-thirds of one-third, less two-thirds of the share and less one dirhem. Then subtract one-fourth of what you have, that is, one-eighteenth, less one-sixth of a share and less one-fourth of a dirhem, and subtract also the second dirhem; the remainder is half one-third of the capital, less half a share and less one dirhem and three-fourths; add thereto two-thirds of the capital, the sum is five-sixths of the capital, less one half of a share, and less one dirhem and three-fourths, equal to five shares. Reduce this, by removing the (88) half share and the one dirhem and three-fourths, and adding them to the (five) shares. Then you have five-sixths of capital, equal to five shares and a half plus one dirhem and three-fourths. Complete your capital, by adding to five shares and a half and to one dirhem and three-fourths, as much as one-fifth of the same. Then you have the capital equal to six shares and three-fifths plus two dirhems and one-tenth. Assume, now, each share to consist of ten parts, and one dirhem likewise of ten; then the capital is eighty-seven parts. Or, if you wish to exhibit the dirhem distinctly, take the one-third, and subtract from it the share; there remains one-third, less one share. Assume the one-third (of the capital) to be seven and a half (dirhems). Subtract one-third of what you have, namely, one-third of one-third;^[29] there remain two-thirds of one-third, less two-thirds of the share that is, five dirhems, less two-thirds of the share. Then subtract one, on account of the one dirhem, and you retain four dirhems, less two-thirds of the share. Subtract now one-fourth of what you have, namely, one part less one-sixth of a share; and remove also one part on account of the one dirhem; the remainder, then, is two parts less half a share. Add this to the two-thirds of the capital, which is fifteen (dirhems). Then you have seventeen parts less half a share, equal to five shares. Reduce this, by removing half a share, and adding it to the five shares. Then it is seventeen parts, equal to five shares and a half. Divide now seventeen by five and a half; the quotient is the value of one share, namely, three dirhems and one-eleventh; and one-third (of the capital) is seven and a half (dirhems). (89)

If he leaves four sons, and bequeaths to some person as much as the share of one of his sons, less one-fourth of what remains from one-third after the deduction of the share, and one dirhem; and to another one-third of what remains from the one-third, and one dirhem; then this legacy is determined by one-third.^[30] Take one-third of the capital, and subtract from it one share; there remains one-third, less one share; add hereto one-fourth of what you have then it is one-third and one-fourth of one-third, less one share and one-fourth. Subtract one dirhem; there remains one-third of one and one-fourth, less one dirhem, and less one share and one-fourth. There remains from the one-third as much as five-eighteenths of the capital, less two-thirds of a dirhem, and less five-sixths of a share. Now subtract the second dirhem, and you retain five-eighteenths of the capital, less one dirhem and two-thirds, and less five-sixths of a share. Add to this two-thirds of the capital, and you have seventeen-eighteenths of the capital, less one dirhem and two-thirds, and less five-sixths of a share, equal to four shares. Reduce this, by removing the quantities. which are to be subtracted, and adding them to the shares; then you have seventeen-eighteenths of the capital, equal to four portions and five-sixths plus one dirhem and two-thirds. Complete your capital by (90) adding to the four shares and five-sixths, and one dirhem and two-thirds, as much as one-seventeenth of the same. Assume, then, each share to be seventeen, and also one dirhem to be seventeen.^[31] The whole capital will then be one hundred and seventeen. If you wish to exhibit the dirhem distinctly, proceed with it as I have shown you.

If he leaves three sons and two daughters, and bequeaths to some person as much as the share of a daughter plus one dirhem; and to another one-fifth of what remains from one-fourth after the deduction of the first legacy, plus one dirhem; and to a third person one-fourth of what remains from one-third after the deduction of all this, plus one dirhem; and to a fourth person one-eighth of the whole capital, requiring all the legacies to be paid off by the heirs generally: then you calculate this by exhibiting the dirhems distinctly, which is better in such a case.^[32] Take one-fourth of the capital, and assume it to be six dirhems; the entire capital will be twenty-four dirhems. Subtract one share from the one-fourth; there remain six dirhems less one share. Subtract also one dirhem; there remain five dirhems less one share. Subtract one-fifth of this remainder; there remain four dirhems, less four-fifths of a share. Now deduct the second dirhem, and you retain three dirhems, less four-fifths of a share. You know, therefore, that the legacy which was determined by one-fourth, is three dirhems, less four-fifths of a share. Return now to the one-third, which is eight, and subtract from it three dirhems, (91) less four-fifths of a share. There remain five dirhems, less four-fifths of a share. Subtract also one-fourth of this and one dirhem, for the legacy; you then retain two dirhems and three-fourths, less three-fifths of a share. Take now one-eighth of the capital, namely, three; after the deduction of one-third, you retain one-fourth of a dirhem, less three-fifths of a share. Return now to the two-thirds, namely, sixteen, and subtract from them one-fourth of a dirhem less three-fifths of a share; there remain of the capital fifteen dirhems and three-fourths, less three-fifths of a share, which are equal to eight shares. Reduce this, by removing three-fifths of a share, and adding them to the shares, which are eight. Then you have fifteen dirhems and three-fourths, equal to eight shares and three-fifths. Make the division: the quotient is one share of the whole capital, which is twenty-four (dirhems). Every daughter receives one dirhem and one-hundred-and-forty-three one-hundred-and-seventy-second parts of a dirhem.^[33] If you prefer to produce the shares distinctly, take one-fourth of the capital, and subtract from it one share; there remains one-fourth of the capital less one share. Then subtract from this one dirhem: then subtract one-fifth of the remainder of one-fourth, which is one-fifth of one-fourth of the capital, less one-fifth of the share and less one-fifth of a dirhem; and subtract also the second dirhem. There remain four-fifths of the one-fourth less four-fifths of a share, and less one dirhem and four-fifths. The legacies paid out of one fourth amount to twelve two-hundred-and-fortieths (92) of the capital and four-fifths of a share, and one dirhem and four-fifths. Take one-third, which is eighty, and subtract from it twelve, and four-fifths of a share, and one dirhem and four-fifths, and remove one-fourth of what remains, and one dirhem. You retain, then, of the one-third, only fifty-one, less three-fifths of a share, less two dirhems and seven-twentieths. Subtract herefrom one-eighth of the capital, which is thirty, and you retain twenty-one, less three-fifths of a share, and less two dirhems and seven-twentieths, and two-thirds of the capital, being equal to eight shares. Reduce this, by removing that which is to be subtracted, and adding it to the eight shares. Then you have one hundred and eighty-one parts of capital, equal to eight shares and three-fifths, plus two dirhems and seven twentieths. Complete your capital, by adding to that which you have fifty-nine one-hundred-and-eighty-one parts. Let, then, a share be three hundred and sixty-two, and a dirhem likewise three hundred and sixty-two.^[34] The whole capital is then five thousand two hundred and fifty-six, and the legacy out of one-fourth^[35] is one thousand two hundred and four, and that out of one-third is four hundred and ninety-nine, and the one-eighth is six hundred and fifty-seven.

 

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On Completement.

“A woman dies and leaves eight daughters, a mother (93), and her husband, and bequeaths to some person as much as must be added to the share of a daughter to make it equal to one-fifth of the capital; and to another person as much as must be added to the share of the mother to make it equal to one-fourth of the capital.”^[36] Computation: Determine the parts of the residue, which in the present instance are thirteen. Take the capital, and subtract from it one-fifth of the same, less one part, as the share of a daughter: this being the first legacy. Then subtract also one-fourth, less two parts, as the share of the mother: this being the second legacy. There remain eleven-twentieths of the capital, which, when increased by three parts, are equal to thirteen parts. Remove now from thirteen parts the three parts on account of the three parts (on the other side), and you retain eleven-twentieths of the capital, equal to ten parts. Complete the capital, by adding to the ten parts as much as nine-elevenths of the same; then you find the capital equal to eighteen parts and two-elevenths. Assume now each part to be eleven; then the whole capital is two hundred, each part is eleven; the first legacy will be twenty-nine, and the second twenty-eight. If the case is the same, and she bequeaths to some person as much as must be added to the share (94) of the husband to make it equal to one-third, and to another person as much as must be added to the share of the mother to make it equal to one-fourth; and to a third as much as must be added to the share of a daughter to make it equal to one-fifth; all these legacies being imposed on the heirs generally: then you divide the residue into thirteen parts.^[37] Take the capital, and subtract from it one-third, less three parts, being the share of the husband; and one-fourth, less two parts, being the share of the mother; and lastly, one-fifth less one part, being the share of a daughter. The remainder is thirteen-sixtieths of the capital, which, when increased by six parts, is equal to thirteen parts. Subtract the six from the thirteen parts: there remain thirteen-sixtieths of the capital, equal to seven parts. Complete your capital by multiplying the seven parts by four and eight-thirteenths, and you have a capital equal to thirty-two parts and four-thirteenths. Assuming then each part to be thirteen, the whole capital is four hundred and twenty.

If the case is the same, and she bequeaths to some person as much as must be added to the share of the mother to make it one-fourth of the capital; and to another as much as must be added to the portion of a daughter, to make it one-fifth of what remains of the capital, after the deduction of the first legacy; then you constitute the parts of the residue by taking them out of thirteen.^[38] Take the capital, and subtract from it one-fourth less two parts; and again, subtract one-fifth of what you retain of the capital, less one part; then look how much remains of the capital after the deduction of the parts. This remainder, namely, three-fifths of the capital, when increased by two parts and three-fifths, will be equal to thirteen parts. Subtract two parts and three-fifths from thirteen parts, there remain ten parts and two-fifths, equal to three-fifths of capital. Complete the capital, by adding to the parts which you have, as much as two-thirds of the same. Then you have a capital equal to seventeen parts and one-third. Assume a part to be three, then the capital is fifty-two, each part three; the first legacy will be seven, and the second six.

If the case is the same, and she bequeaths to some person as much as must be added to the share of the mother to make it one-fifth of the capital, and to another one-sixth of the remainder of the capital; then the parts are thirteen.^[39] Take the capital, and subtract from it one-fifth less two parts; and again, subtract one-sixth of the remainder. You retain two-thirds of the capital, which, when increased by one part and two-thirds, are equal to thirteen parts. Subtract the one part and two-thirds from the thirteen parts: there remain two thirds of the capital, equal to eleven parts and one-third. Complete your capital, by adding to the parts as much as their moiety; thus you find the capital equal to seventeen parts. Assume now the capital to be eighty-five, and each part five; then the first legacy is seven, and the second thirteen, and the remaining sixty-five are for the heirs.

If the case is the same, and she bequeaths to some person as much as must be added to the share of the mother, to make it one-third of the capital, less that sum which must be added to make the share of a daughter equal, to one-fourth of what remains of the capital after the deduction of the above complement; then the parts are thirteen.^[40] Take the capital, and (96) subtract from it one-third less two parts, and add to the remainder one-fourth (of such remainder) less one part; then you have five-sixths of the capital and one part and a half, equal to thirteen parts. Subtract one part and a half from thirteen parts. There remain eleven parts and a half, equal to five-sixths of the capital. Complete the capital, by adding to the parts as much as one-fifth of them. Thus you find the capital equal to thirteen parts and four-fifths. Assume, now, a part to be five, then the capital is sixty-nine, and the legacy four.

“A man dies, and leaves a son and five daughters, and bequeaths to some person as much as must be added to the share of the son to complete one-fifth and one-sixth, less one-fourth of what remains of one-third after the subtraction of the complement.”^[41] Take one-third of the capital, and subtract from it one-fifth and one-sixth of the capital, less two (seventh) parts; so that you retain two parts less four one hundred and twentieths of the capital. Then add it to the exception, which is half a part less one one hundred and twentieth, and you have two parts and a half less five one hundred and twentieths of capital. Add hereto two-thirds of the capital, and you have seventy-five one hundred and twentieths of the capital and two parts and a half, equal to seven parts. Subtract, now, two parts and a half from seven, and you retain seventy-five one hundred and twentieths, or five-eighths, equal to four parts and a half. Complete your capital, by (97) adding to the parts as much as three-fifths of the same, and you find the capital equal to seven parts and one-fifth part. Let each part be five; the capital is then thirty-six, each portion five, and the legacy one.

If he leaves his mother, his wife, and four sisters, and bequeaths to a person as much as must be added to the shares of the wife and a sister, in order to make them equal to the moiety of the capital, less two-sevenths of the sum which remains from one-third after the deduction of that complement; the Computation is this:^[42] If you take the moiety from one-third, there remains one-sixth. This is the sum excepted. It is the share of the wife and the sister. Let it be five (thirteenth) parts. What remains of the one-third is five parts less one- sixth of the capital. The two-sevenths which he has excepted are two-sevenths of five parts less two-sevenths of one-sixth of the capital. Then you have six parts and three-sevenths, less one-sixth and two-sevenths of one-sixth of the capital. Add hereto two-thirds of the capital; then you have nineteen forty-seconds of the capital and six parts and three-sevenths, equal to thirteen parts. Subtract herefrom the six parts and three-sevenths. There remain nine-teen forty-seconds of the capital, equal to six parts and four-sevenths. Complete your capital by adding to it its double and four-nineteenths of it. Then you find the capital equal to fourteen parts, and seventy (98) one hundred and thirty-thirds of a part. Assume one part to be one hundred and thirty-three; then the whole capital is one thousand nine hundred and thirty- two; each part is one hundred and thirty-three, the completion of it is three hundred and one, and the exception of one-third is ninety-eight, so that the remaining legacy is two hundred and three. For the heirs remain one thousand seven hundred and twenty-nine.

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Let of ${\tfrac {1}{13}}$ the residue $=v$
$1-{\tfrac {1}{9}}=13v\;\therefore \;{\tfrac {8}{9}}=15v$
$\therefore \;v={\tfrac {8}{135}}$ of the capital
A mother’s share= ${\tfrac {16}{135}}$


1. If a father dies, leaving $n$ sons, one of whom owes the father a sum exceeding an $n$th part of the residue of the father’s estate, after paying legacies, then such son retains the whole sum which he owes the father: part, as a set-off against his share of the residue, the surplus as a gift from the father.
   In the present example, let each son’s share of the residue be equal to $x$.
   ${\tfrac {2}{3}}[10+x]=2x\;\therefore \;1+x-3x\;\therefore \;10=2x\;\therefore \;x=5.$
   The stranger receives $5$; and the son, who is not indebted to the father, receives $5$.
2. ${\begin{aligned}{\tfrac {4}{5}}[10+x]-1=2x&\;\therefore \;{\tfrac {2}{5}}[10+x]-{\tfrac {1}{2}}=x\\\;\therefore \;3{\tfrac {1}{2}}={\tfrac {3}{5}}x&\;\therefore \;x={\tfrac {35}{6}}=5{\tfrac {5}{6}}\end{aligned}}$
   The stranger receives ${\tfrac {1}{5}}[10+{\tfrac {35}{6}}]+1=4{\tfrac {1}{6}}$
3. ${\tfrac {4}{5}}[10+x]+1=3x\;\therefore \;9=2{\tfrac {1}{5}}x\;\therefore \;{\tfrac {45}{11}}\;{\text{or}}\;4{\tfrac {1}{11}}=x$
   The stranger receives ${\tfrac {1}{5}}[10+{\tfrac {45}{11}}]-1=1{\tfrac {9}{11}}$
4. It appears in the sequel (p. 96) that a widow is entitled to ${\tfrac {1}{8}}$th, and a mother to ${\tfrac {1}{8}}$th of the residue; ${\tfrac {1}{8}}+{\tfrac {1}{6}}={\tfrac {14}{48}}$, leaving ${\tfrac {34}{48}}$ of the residue to be distributed between two brothers and two sisters; that is, ${\tfrac {17}{48}}$ between a brother and a sister; but in what proportion these 17 parts are to be divided between the brother and sister does not appear in the course of this treatise.
   

   Let the whole capital of the testator = $1$
   and let each 48th share of the residue = $x$
   ${\tfrac {8}{9}}=48x\;\therefore \;{\tfrac {1}{9}}=6x\;\therefore \;{\tfrac {1}{54}}=x$

   that is, each 48th part of the residue = ${\tfrac {1}{54}}$th of the whole capital.

5. A husband is entitled to ${\tfrac {1}{4}}$th of the residue, and the sons and daughters divide the remaining ${\tfrac {3}{4}}$ths of the residue in such proportion, that a son receives twice as much as a daughter. In the present instance, as there are three daughters and one son, each daughter receives ${\tfrac {1}{3}}$ of ${\tfrac {3}{4}},={\tfrac {3}{20}}$ of the residue, and the son ${\tfrac {6}{20}}$. Since the stranger takes ${\tfrac {1}{8}}+{\tfrac {1}{7}}={\tfrac {15}{56}}$ of the capital, the residue $={\tfrac {41}{56}}$ of the capital, and each ${\tfrac {1}{20}}$th share of the residue $={\tfrac {1}{20}}\times {\tfrac {41}{56}}={\tfrac {41}{1120}}$ of the capital, The stranger, therefore, receives ${\tfrac {15}{56}}={\tfrac {15\times 20}{56\times 20}}={\tfrac {300}{1120}}$ of the capital.
6. The problems in this chapter may be considered as belonging rather to Law than to Algebra, as they contain little more than enunciations of the law of inheritance in certain complicated cases.
7. If some heirs are, by a testator, charged with payment of bequests, and other heirs are not charged with payment of any bequests whatever: if one bequest exceeds in amount ${\tfrac {1}{3}}$d of the testator’s whole property; and if one of his heirs is charged with payment of more than ${\tfrac {1}{3}}$d of such bequest; then, whatever share of the residue such heir is entitled to receive, the like share must he pay of the bequest wherewith he is charged, and those heirs whom the testator has not charged with any payment, must each contribute towards paying the bequests a third part of their several shares of the residue.
8. If the bequests stated in the present example were charged on the heirs collectively, the husband would be entitled to $\scriptstyle {\frac {1}{4}}$, the mother to $\scriptstyle {\frac {1}{6}}$ of the residue $\scriptstyle {\frac {1}{4}}$+$\scriptstyle {\frac {1}{6}}$ = $\scriptstyle {\frac {5}{12}}$; the remainder $\scriptstyle {\frac {7}{12}}$, would be the son’s share of the residue; but since the bequests, ${\tfrac {2}{5}}+{\tfrac {1}{4}}={\tfrac {13}{20}}$ of the capital, are charged upon the son and mother, the law throws a portion of the charge on the husband.

   The Husband contributes	${\tfrac {1}{4}}\times {\tfrac {1}{3}}=20\times {\tfrac {1}{240}}$,	and retains	${\tfrac {1}{4}}\times {\tfrac {2}{3}}=40\times {\tfrac {1}{240}}$
   The Mother contributes	${\tfrac {1}{6}}\times {\tfrac {1}{2}}=20\times {\tfrac {1}{240}}$,	and retains	${\tfrac {1}{6}}\times {\tfrac {1}{2}}=20\times {\tfrac {1}{240}}$
   The Son contributes	${\tfrac {7}{12}}\times {\tfrac {13}{20}}=91\times {\tfrac {1}{240}}$,	and retains	${\tfrac {7}{12}}\times {\tfrac {7}{20}}=49\times {\tfrac {1}{240}}$
   Total contributed = ${\tfrac {131}{240}}$		Total retained = ${\tfrac {109}{240}}$

   ${\tfrac {2}{5}}+{\tfrac {1}{4}}={\tfrac {8}{20}}+{\tfrac {5}{20}}={\tfrac {13}{20}}$
   The Legatee, to whom the ${\tfrac {2}{5}}$ are bequeathed, receives ${\tfrac {8}{13}}\times {\tfrac {131}{240}}={\tfrac {8\times 11}{3120}}$
   The Legatee, to whom ${\tfrac {1}{4}}$ is bequeathed, receives ${\tfrac {5}{13}}\times {\tfrac {131}{240}}={\tfrac {5\times 131}{3120}}$

9. ${\tfrac {2}{5}}+{\tfrac {1}{4}}={\tfrac {8+5}{20}}={\tfrac {13}{20}}$
   The Husband, who would be entitled to ${\tfrac {1}{4}}$ of the residue, is not charged by the Testator with any bequest.
   The Mother who would be entitled to ${\tfrac {1}{6}}$ of the residue, is charged with the payment of to the Legatee A.
   The Son, who would be entitled to ${\tfrac {7}{12}}$ of the residue, is charged with payment of ${\tfrac {2}{5}}$ to the Legatee B.
   

   The Husband contributes ${\tfrac {1}{4}}\times {\tfrac {1}{3}}=780\times {\tfrac {1}{9360}}$ ; retains ${\tfrac {1}{4}}\times {\tfrac {2}{3}}={\tfrac {1560}{9360}}$

   The Mother contributes ${\tfrac {1}{6}}[{\tfrac {1}{4}}+{\tfrac {8}{13}}\times {\tfrac {1}{3}}]=710\times {\tfrac {1}{9360}}$ ; retains ...... ${\tfrac {850}{9360}}$

   The Son contributes ${\tfrac {7}{12}}[{\tfrac {2}{5}}+{\tfrac {5}{13}}\times {\tfrac {1}{3}}]=2884\times {\tfrac {1}{9360}}$ ; retains ...... ${\tfrac {2576}{9360}}$

   Total contributed = ${\tfrac {4374}{9360}}$; Total retained = ${\tfrac {4986}{9360}}$

   $\left.{\begin{aligned}{\text{The Legatee A, to whom}}{\tfrac {1}{4}}\;{\text{is}}\\{\text{bequeathed, receives}}\end{aligned}}\right\}\;{\tfrac {5}{13}}\times {\tfrac {4374}{9360}}={\tfrac {5\times 4374}{964080}}$

   $\left.{\begin{aligned}{\text{The Legatee B, to whom}}{\tfrac {2}{5}}\;{\text{is}}\\{\text{bequeathed, receives}}\end{aligned}}\right\}\;{\tfrac {8}{13}}\times {\tfrac {4374}{9360}}={\tfrac {8\times 4374}{964080}}$

10. A widow is entitled to $\scriptstyle {\frac {1}{8}}$th of the residue; therefore $\scriptstyle {\frac {7}{8}}$ths of the residue are to be distributed among the sons of the testator. Let $x$ be the stranger’s legacy. The widow’s share = ${\tfrac {1-x}{8}}$; each son’s share =${\tfrac {1}{4}}\times {\tfrac {7}{8}}[1-x]$; and a son’s share, minus the widow’s share = $[{\tfrac {7}{4}}-1]{\tfrac {1-x}{8}}={\tfrac {3}{4}}\cdot {\tfrac {1-x}{8}}\;\therefore \;x={\tfrac {3}{4}}\cdot {\tfrac {1-x}{8}}\;\therefore \;x={\tfrac {3}{35}};1-x={\tfrac {32}{35}}$ A son’s share = $\scriptstyle {\frac {7}{35}}$; the widow’s share= $\scriptstyle {\frac {4}{35}}$.
11. A son is entitled to receive twice as much as a daughter. Were there three sons and one daughter, each son would receive $\scriptstyle {\frac {2}{7}}$ths of the residue. Let $x$ be the stranger’s legacy.
    $\therefore {\tfrac {2}{7}}[1-x]=x\;\therefore \;x={\tfrac {2}{9}}\;{\text{and}}\;1-x={\tfrac {7}{9}}$
    Each Son’s share $={\tfrac {2}{5}}[1-x]={\tfrac {2}{5}}\times {\tfrac {7}{9}}={\tfrac {14}{45}}$
    The Daughter’s share $={\tfrac {1}{5}}[1-x]\ldots \ldots ={\tfrac {7}{45}}$
    The Stranger’s legacy $={\tfrac {2}{9}}\ldots \ldots \ldots \ldots ={\tfrac {10}{45}}$
12. Let $x$ be the stranger’s legacy; $1-x$ is the residue. A widow’s share of the residue is $\scriptstyle {\frac {1}{6}}$th: there remains ${\tfrac {5}{6}}[1-x]$, to be distributed among the children.
    ${\begin{aligned}\left.{\begin{aligned}{\text{Since there are 3 sons, and 1 daughter,}}\\{\text{a son’s share is.......................}}\end{aligned}}\right\}\;{\tfrac {2}{7}}\times {\tfrac {5}{6}}[1-x]\\\left.{\begin{aligned}{\text{Were there 3 sons and 2 daughters, a}}\\{\text{daughter’s share would be...........}}\end{aligned}}\right\}\;{\tfrac {1}{8}}\times {\tfrac {5}{6}}[1-x]\\{\text{The difference}}={\tfrac {9}{56}}\times {\tfrac {5}{6}}[x-1]\\\end{aligned}}$
    ${\begin{aligned}\therefore x={\tfrac {45}{336}}[1-x]\quad \quad \quad \therefore x={\tfrac {45}{381}}\\1-x={\tfrac {336}{381}};\;{\text{the widow’s share}}={\tfrac {56}{381}}\\{\text{the daughter’s share}}={\tfrac {40}{381}}\end{aligned}}$
13. Since there are 3 sons, each son’s share of the residue=${\tfrac {1}{3}}$. Were there 3 sons and a daughter, the daughter’s share would be ${\tfrac {1}{7}}$.
    

    ${\tfrac {1}{3}}-{\tfrac {1}{7}}={\tfrac {4}{7}}$
    Let $x$ be the stranger’s legacy, and $v$ a son’s share
    Then $1-x=3v$
    but $x={\tfrac {4}{7}}v+{\tfrac {1}{3}}[{\tfrac {1}{3}}-{\tfrac {4}{7}}v]$
    and $1-x={\tfrac {2}{3}}+{\tfrac {1}{3}}-{\tfrac {4}{7}}v-{\tfrac {1}{3}}[{\tfrac {1}{3}}-{\tfrac {4}{7}}v]=3v$
    $\therefore \;{\tfrac {2}{3}}+{\tfrac {2}{3}}[{\tfrac {1}{3}}-{\tfrac {4}{7}}v]=3v$
    $\therefore \;{\tfrac {2}{3}}+{\tfrac {2}{9}}=3{\tfrac {8}{21}}\times v,\;{\text{or}}\;{\tfrac {8}{9}}={\tfrac {71}{21}}v$
    $\therefore \;{\tfrac {8}{3}}={\tfrac {71}{7}}v\;\therefore \;v={\tfrac {56}{213}}=$ a son’s share
    $x={\tfrac {45}{213}}=$ the stranger’s legacy.

14. In the former examples (p. 90) when a husband and a mother were among the heirs, a husband was found to be entitled to ${\tfrac {1}{4}}={\tfrac {3}{12}}$ and a mother to ${\tfrac {1}{6}}={\tfrac {2}{12}}$ of the residue. Here a husband is stated to be entitled to ${\tfrac {3}{13}}$, and a mother to ${\tfrac {2}{13}}$ of the residue.
15. ${\tfrac {1}{8}}+{\tfrac {1}{10}}={\tfrac {9}{40}}$
    A husband’s share of the residue is ${\tfrac {3}{13}}$
    $\therefore 1-{\tfrac {9}{40}}-3v=13v\;\therefore \;{\tfrac {31}{40}}=16v$
    $\therefore v={\tfrac {31}{640}}$; a husband’s share = ${\tfrac {93}{640}}$
    The stranger’s legacy = ${\tfrac {237}{640}}$
16. ${\tfrac {1}{9}}+{\tfrac {1}{10}}={\tfrac {19}{90}}$
    $1-3v+{\tfrac {19}{90}}[1-3v]=13v$
    $\therefore {\tfrac {109}{90}}[1-3]=13v$
    $\therefore {\tfrac {109}{90}}=[13+{\tfrac {109}{30}}]v$
    $\therefore v={\tfrac {109}{1497}}$
    The husband’s share $={\tfrac {327}{1497}}$
    The stranger’s legacy $={\tfrac {80}{1497}}$
17. When the heirs are a wife, and $2$ sisters, they each inherit ${\tfrac {1}{3}}$ of the residue.
    Let $x$ be the stranger’s legacy.
    ${\begin{aligned}{\tfrac {1}{3}}[1-x]={\text{a sister’s share}}\\{\tfrac {1}{3}}[1-x]-{\tfrac {1}{8}}[1-x]=x\\\therefore \;{\tfrac {5}{24}}[1-x]=x\;\therefore \;{\tfrac {5}{24}}={\tfrac {29}{24}}x\\\therefore \;x={\tfrac {5}{29}}\therefore \;1-x={\tfrac {24}{29}}\\{\text{and a sister’s share}}={\tfrac {8}{29}}\end{aligned}}$
18. Let the first bequest $=v$; and the second $=y$
    Then $1-v-y=4v$
    i.e. ${\tfrac {2}{3}}+{\tfrac {3}{4}}-v-{\tfrac {1}{4}}[{\tfrac {1}{3}}-v]=4v$
    $\therefore {\tfrac {2}{3}}+{\tfrac {3}{4}}[{\tfrac {1}{3}}-v]=4v$
    $\therefore {\tfrac {2}{3}}+{\tfrac {3}{12}}=[4+{\tfrac {3}{4}}]v\therefore {\tfrac {11}{12}}={\tfrac {19}{4}}v$
    $\therefore v={\tfrac {11}{57}}$; the 2d bequest $={\tfrac {2}{37}}$
19. $1-v+{\tfrac {1}{5}}[{\tfrac {1}{3}}-v]=4v$
    or ${\tfrac {2}{3}}\times {\tfrac {1}{3}}-v+{\tfrac {1}{5}}[{\tfrac {1}{3}}-v]=4v$
    or ${\tfrac {2}{3}}+{\tfrac {6}{5}}[{\tfrac {1}{3}}-v]=4v$
    $\therefore \;{\tfrac {2}{3}}+{\tfrac {2}{5}}=[4+{\tfrac {6}{5}}]v\;\therefore \;{\tfrac {16}{15}}={\tfrac {26}{5}}v$
    $\therefore \;v={\tfrac {8}{39}}$, and the stranger’s lcgacy $={\tfrac {7}{39}}$
20. Since there are three sons and one daughter, the daughter receives ${\tfrac {1}{7}}$, and each son ${\tfrac {1}{7}}$ths of the residue.
    If the 1st legacy=$v$, the $2d=y$, and therefore a daughter’s share=$v$,
    ${\begin{aligned}1-v-y=7v;{\tfrac {1}{5}}+{\tfrac {1}{6}}={\tfrac {11}{30}}\\\therefore \;{\tfrac {5}{7}}+{\tfrac {2}{7}}-v-{\tfrac {11}{30}}[{\tfrac {2}{7}}-v]=7v\\{\text{i.e.}}\;{\tfrac {5}{7}}+{\tfrac {19}{30}}[{\tfrac {2}{7}}-v]=7v\\\therefore \;{\tfrac {5}{7}}+{\tfrac {19}{15\times 7}}=[7+{\tfrac {19}{30}}]v\\\therefore \;{\tfrac {94}{7}}={\tfrac {229}{2}}v\;\therefore \;={\tfrac {188}{1603}}\\\end{aligned}}$
    The 2d legacy =$\;\therefore \;y={\tfrac {99}{1603}}$
21. ${\tfrac {1}{4}}+{\tfrac {1}{5}}={\tfrac {9}{20}}$
    Let the 1st legacy $=v=$ a daughter’s share
    Let the 2d legacy $=y$
    ${\begin{aligned}1-v-y=7v\\\therefore {\tfrac {3}{5}}+{\tfrac {2}{5}}-v-{\tfrac {9}{20}}\left[{\tfrac {2}{5}}-v\right]=7v\\\therefore {\tfrac {3}{5}}+{\tfrac {11}{20}}\left[{\tfrac {2}{5}}-v\right]=7v\\\therefore {\tfrac {3}{5}}+{\tfrac {11}{10\times 5}}=\left[7+{\tfrac {11}{20}}\right]v\\\therefore {\tfrac {41}{5}}={\tfrac {151}{2}}v\;\;\therefore v={\tfrac {82}{755}}\\\end{aligned}}$
    and the 2d legacy, $y,={\tfrac {99}{155}}$
22. v=${\tfrac {1}{7}}$ of the residue = a daughter’s share.
    $2v$ = a son’s share
    ${\begin{aligned}1-2v+{\tfrac {9}{20}}[{\tfrac {2}{5}}-2v]=7v\\{\text{i.e.}}\;{\tfrac {3}{5}}+{\tfrac {2}{5}}-2v+{\tfrac {9}{20}}[{\tfrac {2}{5}}-2v]=7v\\\therefore \;{\tfrac {3}{5}}+{\tfrac {29}{20}}[{\tfrac {2}{5}}-2v]=7v\\\therefore \;{\tfrac {3}{5}}+{\tfrac {29}{10\times }}=[7+{\tfrac {29}{10}}]v\;\therefore \;{\tfrac {99}{5}}=99v\end{aligned}}$
    $\therefore \;v={\tfrac {59}{495}}$; a son’s share = ${\tfrac {118}{495}}$
    and the legacy to the stranger = ${\tfrac {82}{495}}$
23. Since there are two sons and two daughters, each son receives ${\tfrac {1}{3}}$, and each daughter ${\tfrac {1}{6}}$ of the residue. Let $v$ = a daughter’s share. Let the 1st legacy $=x=v-{\tfrac {1}{5}}[{\tfrac {1}{3}}-v]$
    Let the 2d legacy $=y=v-{\tfrac {1}{3}}[{\tfrac {1}{3}}-x-v]$
    and 3d legacy $={\tfrac {1}{12}}$
    ${\begin{aligned}&1-{\tfrac {1}{12}}-x-y=6v\\&{\text{i.e.}}\;{\tfrac {2}{3}}-{\tfrac {1}{12}}+{\tfrac {1}{3}}-x-v+{\tfrac {1}{3}}[{\tfrac {1}{3}}-x-v]=6v\\&{\text{or}}\;{\tfrac {2}{3}}-{\tfrac {1}{12}}+{\tfrac {4}{3}}[{\tfrac {1}{3}}-x-v]=6v\\&{\text{i.e.}}\;{\tfrac {7}{12}}+{\tfrac {4}{3}}[{\tfrac {1}{3}}-v+{\tfrac {1}{5}}[{\tfrac {1}{3}}-v]-v]=6v\\&{\text{or}}\;{\tfrac {7}{12}}+{\tfrac {4}{3}}[{\tfrac {6}{5}}[{\tfrac {1}{3}}-v]-v]6v\\&{\text{or}}\;{\tfrac {7}{12}}+{\tfrac {8}{15}}=[6+{\tfrac {4\times 11}{3\times 5}}]v={\tfrac {134}{15}}v\\&{\text{or}}\;{\tfrac {7}{4}}+{\tfrac {8}{5}}={\tfrac {134}{5}}a\;\therefore \;v={\tfrac {67}{536}}={\tfrac {1}{8}}\\\end{aligned}}$
    The 1st Legacy $=x={\tfrac {1}{12}}$
    The 2d Legacy $=y={\tfrac {1}{12}}$
    A son’s share $={\tfrac {1}{4}}$
24. ${\tfrac {201}{1608}}={\tfrac {1}{8}}={\tfrac {3}{24}}=v;\;{\text{and}}\;{\tfrac {1}{12}}={\tfrac {2}{24}}=y$
    The common denominator 1608 is unnecessarily great.
25. Let $x$ be the 1st legacy; $y$ the 2d; $v$ a daughter’s share.
    ${\begin{aligned}&1-x-y=6v\\&x=v+3{\tfrac {1}{5}}[{\tfrac {1}{3}}-v]\\&y=v+3{\tfrac {1}{3}}[{\tfrac {1}{4}}-v]\\&{\text{Then}}\;1-{\tfrac {1}{3}}-{\tfrac {1}{4}}+{\tfrac {1}{3}}-v-{\tfrac {1}{5}}[{\tfrac {1}{3}}-v]+{\tfrac {1}{4}}-v-{\tfrac {1}{3}}[{\tfrac {1}{4}}-v]=6v\\&{\text{or}}\;{\tfrac {5}{12}}+{\tfrac {4}{5}}[{\tfrac {1}{3}}-v]+{\tfrac {2}{3}}[{\tfrac {1}{4}}-v]=6v\\&\therefore \;{\tfrac {5}{12}}+{\tfrac {4}{15}}+{\tfrac {2}{12}}=[6+{\tfrac {4}{5}}+{\tfrac {2}{3}}]v\\&\therefore \;{\tfrac {51}{60}}={\tfrac {112}{15}}v\;\therefore \;{\tfrac {51}{448}}={\tfrac {153}{1344}}\\&x={\tfrac {212}{1344}}\;;\;y={\tfrac {214}{1344}}\end{aligned}}$
26. Let $x$ be the legacy to the 1st stranger
    and $y$ be the legacy to the 2d stranger; $v$=a son’s share ${\begin{aligned}&1-x-y=6v\\&x=v+{\tfrac {1}{5}}[{\tfrac {1}{4}}-v]\;;\;y=v-{\tfrac {1}{4}}[{\tfrac {1}{3}}-x-v]\\&{\text{i.e.}}\;{\tfrac {2}{3}}+{\tfrac {1}{3}}-x-v+{\tfrac {1}{4}}[{\tfrac {1}{3}}-x-v]=6v\\&{\text{or}}\;{\tfrac {2}{3}}+{\tfrac {5}{4}}[{\tfrac {1}{3}}-x-v]=6v\\&{\text{or}}\;{\tfrac {2}{3}}+{\tfrac {5}{4}}[{\tfrac {1}{3}}-{\tfrac {1}{4}}+{\tfrac {1}{4}}-v-{\tfrac {1}{5}}[{\tfrac {1}{4}}-v]-v]=6v\\&{\text{or}}\;{\tfrac {2}{3}}+{\tfrac {5}{4}}[{\tfrac {1}{12}}+{\tfrac {4}{5}}[{\tfrac {1}{4}}-v]-v]=6v\\&\therefore \;{\tfrac {2}{3}}+{\tfrac {5}{4\times 12}}+{\tfrac {1}{4}}=[7+{\tfrac {5}{4}}]v\\&\therefore \;{\tfrac {8}{3}}+{\tfrac {5}{12}}+1=33v\;\therefore \;{\tfrac {49}{12\times 33}}={\tfrac {49}{396}}=v\\&\therefore \;x=v+{\tfrac {10}{396}},\;{\text{and}}\;y=v-{\tfrac {6}{396}}\end{aligned}}$
27. Let the capital=$1$; a dirhem=$d$; the legacy=$x$; and a son’s share = $v$
    ${\begin{aligned}&1-x=4v\\&x=v+{\tfrac {1}{4}}[{\tfrac {1}{3}}-v]+d\\&\therefore \;{\tfrac {2}{3}}+{\tfrac {1}{3}}-v-{\tfrac {1}{4}}[{\tfrac {1}{3}}-v]-d=4v\\&\therefore \;{\tfrac {2}{3}}+{\tfrac {3}{4}}[{\tfrac {1}{3}}-v]-d=4v\\&\therefore \;{\tfrac {2}{3}}+{\tfrac {1}{4}}-d=[4+{\tfrac {3}{4}}]v\\&\therefore \;{\tfrac {11}{12}}-d={\tfrac {19}{4}}v\end{aligned}}$
    $\therefore \;{\tfrac {11}{57}}$ of the capital $-{\tfrac {12}{57}}$ of a dirhem = $v$ and ${\tfrac {13}{57}}$ of the capital $+{\tfrac {48}{57}}$ of a dirhem=$x$, the legacy. If we assume the capital to be so many dirhems, or a dirhem to be such a part of the capital, we shall obtain the value of the son’s share in terms of a dirhem, or of the capital only.
    Thus, if we assume the capital to be 12 dirhems,
    $v={\tfrac {12}{57}}[11-1]d={\tfrac {120}{57}}d=2{\tfrac {2}{19}}dirhems,$
    $x={\tfrac {12}{57}}[13-4]d={\tfrac {204}{57}}d=3{\tfrac {11}{19}}dirhems.$
28. Let the legacy=$x$; and a son’s share $v$
    ${\begin{aligned}&1-x=5v\\&{\tfrac {2}{3}}+{\tfrac {1}{3}}-v-{\tfrac {1}{3}}[{\tfrac {1}{3}}-v]-d-{\tfrac {1}{4}}[{\tfrac {2}{3}}[{\tfrac {1}{3}}-v]-d]-d=5v\\&{\text{i.e.}}\;{\tfrac {2}{3}}+{\tfrac {2}{3}}[{\tfrac {1}{3}}-v]-d-{\tfrac {1}{4}}[{\tfrac {2}{3}}[{\tfrac {1}{3}}-v]-d]-d=5v\\&{\text{i.e.}}\;{\tfrac {2}{3}}+{\tfrac {2}{4}}[{\tfrac {2}{3}}[{\tfrac {1}{3}}-v]-d]-d=5v\\&\therefore \;{\tfrac {2}{3}}+{\tfrac {1}{6}}-{\tfrac {1}{2}}v-{\tfrac {7}{4}}d=5v\\&\therefore \;{\tfrac {5}{6}}-{\tfrac {7}{4}}d={\tfrac {11}{2}}v\\\end{aligned}}$
    $\therefore \;{\tfrac {10}{66}}$ of the capital $-{\tfrac {21}{66}}$ of a dirhem = $v$
    $\therefore \;{\tfrac {16}{66}}$ of the capital $+{\tfrac {105}{66}}$ of a dirhem = $x$, the legacy.
    If the capital ${\tfrac {45}{2}}$ dirhems, or ${\tfrac {1}{3}}$ of the capital=$7{\tfrac {1}{2}}$ dirhems, $v={\tfrac {34}{11}}$=dirhems=$3{\tfrac {1}{33}}$ dirhems.
29. There is an omission here of the words “less one third of a share.”
30. Let the 1st legacy be $x$, the 2d $y$; and a son’s share = $v$
    ${\begin{aligned}&1-x-y=4v\\&{\text{i.e.}}\;{\tfrac {2}{3}}+{\tfrac {1}{3}}-v+{\tfrac {1}{4}}[{\tfrac {1}{3}}-v]-d-{\tfrac {1}{3}}[{\tfrac {1}{3}}-v+{\tfrac {1}{4}}({\tfrac {1}{3}}-v)-d]-d=4v\\&{\text{i.e.}}\;{\tfrac {2}{3}}+{\tfrac {2}{3}}[{\tfrac {1}{3}}-v+{\tfrac {1}{4}}({\tfrac {1}{3}}-v)-d]-d=4v\\&{\text{i.e.}}\;{\tfrac {2}{3}}+{\tfrac {2}{3}}[{\tfrac {5}{4}}({\tfrac {1}{3}}-v)-d]-d=4v\\&\therefore \;{\tfrac {2}{3}}+{\tfrac {5}{18}}-{\tfrac {5}{6}}v-{\tfrac {5}{3}}d=4v\\&\therefore \;{\tfrac {17}{18}}-{\tfrac {5}{3}}d={\tfrac {29}{6}}v\\&\therefore \;{\tfrac {17}{87}}-{\tfrac {20}{58}}d=v\\&{\text{also}}\;{\tfrac {14}{87}}+{\tfrac {33}{58}}d=x\\&{\tfrac {5}{87}}-{\tfrac {47}{58}}d=y\end{aligned}}$
31. ${\text{Capital}}={\tfrac {87}{17}}v+{\tfrac {30}{17}}d\;\therefore \;{\text{if}}\;v=17,\;{\text{and}}\;d=17,\;{\text{capital}}=117$
32. Let the legacies to the three first legatees be, severally, $x,y,z$; the fourth legacy = ${\tfrac {1}{8}}$; and let a daughters’ share $v$
    ${\begin{aligned}&\therefore \;{\tfrac {7}{8}}-x-y-z=8v\\&x=v+d;y={\tfrac {1}{5}}[{\tfrac {1}{4}}-x]+d;z={\tfrac {1}{4}}[{\tfrac {1}{3}}-x-y]+d\\&{\text{Then}}\;{\tfrac {7}{8}}-{\tfrac {1}{3}}+{\tfrac {1}{3}}-x-y-{\tfrac {1}{4}}[{\tfrac {1}{3}}-x-y]-d=8v\\&\therefore \;{\tfrac {13}{24}}+{\tfrac {3}{4}}[{\tfrac {1}{3}}-x-y]-d=8v\\&{\text{but}}\;{\tfrac {1}{3}}-x-y={\tfrac {1}{3}}-{\tfrac {1}{4}}+{\tfrac {1}{4}}-x-{\tfrac {1}{5}}[{\tfrac {1}{4}}-x]-d\\&={\tfrac {1}{12}}+{\tfrac {4}{5}}[{\tfrac {1}{4}}-x]-d\\&={\tfrac {1}{12}}+{\tfrac {1}{5}}-{\tfrac {4}{5}}v-{\tfrac {9}{5}}d\\&={\tfrac {17}{60}}+{\tfrac {4}{5}}v-{\tfrac {9}{5}}d\\&\therefore \;{\tfrac {13}{24}}+{\tfrac {3}{4}}\times {\tfrac {17}{60}}-{\tfrac {3}{5}}v-[{\tfrac {3}{4}}\times {\tfrac {9}{5}}+1]d=8v\\&\therefore \;{\tfrac {181}{240}}-{\tfrac {47}{20}}d={\tfrac {43}{5}}v\;\therefore \;v={\tfrac {181}{2064}}-{\tfrac {564}{2064}}d,\;{\text{and}}\;1={\tfrac {2064}{181}}v+{\tfrac {564}{181}}d\\&x={\tfrac {181}{2064}}+{\tfrac {1500}{2064}}d;y={\tfrac {67}{2064}}+{\tfrac {1764}{2064}}d;z={\tfrac {110}{2064}}+{\tfrac {1248}{2064}}d\end{aligned}}$
33. $v={\tfrac {181}{2064}}$ of the capital $-{\tfrac {564}{2064}}$ of a dirhem. If we assume the capital to be the equal to 24 dirhems
    $v={\tfrac {181\times 24-564}{2064}}$dirhems= ${\tfrac {4344-564}{2064}}d={\tfrac {3780}{2064}}d=1{\tfrac {143}{172}}$ dirhems.
34. The capital=${\tfrac {2064}{181}}+{\tfrac {564}{181}}d$
    If we assume $v=362$, and $d=362$, the capital$=5256$
    Then $x=724$; $y=480$; $z=499$; ${\tfrac {1}{8}}$th of capital $=657$.
35. The text ought to stand “the two first legacies are” instead of “the legacy out of one-fourth is.”
    

    The first legacy is ...........	724
    The second ....................	480
    ∴ the first + second legacy =	1204

36. In this case, the mother has ${\tfrac {2}{13}}$; and each daughter has ${\tfrac {1}{13}}$ of the residue.
    ${\begin{aligned}&1-x-y=13v\\&{\text{i.e.}}\;1-{\tfrac {1}{5}}+v-{\tfrac {1}{4}}+2v=13v\\&\therefore \;{\tfrac {11}{20}}=10v\;\therefore \;v={\tfrac {11}{200}};x={\tfrac {29}{200}};y={\tfrac {28}{200}}\end{aligned}}$
37. ${\begin{aligned}&1-[{\tfrac {1}{3}}-3v]-[{\tfrac {1}{4}}-2v]-[{\tfrac {1}{5}}-v]=13\\&{\text{i.e.}}\;1-{\tfrac {1}{3}}-{\tfrac {1}{4}}-{\tfrac {1}{5}}=7v\\&\therefore \;{\tfrac {13}{60}}=70\\&\therefore \;v={\tfrac {13}{420}}\end{aligned}}$
38. ${\begin{aligned}&1-x-y=13v\\&x={\tfrac {1}{4}}-2v\;;y={\tfrac {1}{5}}[1-x]-v\\&1-x-{\tfrac {1}{5}}[1-x]+v=13v\\&{\tfrac {4}{5}}[1-x]=12v\;\therefore \;{\tfrac {4}{5}}[{\tfrac {3}{4}}+2v]=12v\\&\therefore \;{\tfrac {3}{5}}=[12-{\tfrac {8}{5}}]v={\tfrac {52}{5}}v\\&\therefore \;v={\tfrac {3}{52}}\;\therefore \;x={\tfrac {7}{52}},y={\tfrac {6}{52}}\end{aligned}}$
39. ${\begin{aligned}&1-x-y=13v\\&x={\tfrac {1}{5}}-2v\;;y={\tfrac {1}{6}}[1-x]\\&1-x-{\tfrac {1}{6}}[1-x]=13v\\&\therefore \;{\tfrac {5}{6}}[1-x]=13v\\&\therefore \;{\tfrac {5}{6}}[{\tfrac {4}{5}}+2v]=13v\\&\therefore \;{\tfrac {2}{3}}+{\tfrac {5}{3}}v=13v\\&\therefore \;{\tfrac {2}{3}}={\tfrac {34}{3}}v\;\therefore \;v={\tfrac {1}{17}}\;;x={\tfrac {7}{85}}\;;y={\tfrac {13}{85}}\end{aligned}}$
40. ${\begin{aligned}&1-x+y=13v\;;\;{\text{and}}\;x={\tfrac {1}{3}}-2v\;;y={\tfrac {1}{4}}[1-x]-v\\&\therefore \;1-x+{\tfrac {1}{4}}[1-x]-v=13v\\&\therefore \;{\tfrac {5}{4}}[1-x]=14v\;\therefore \;{\tfrac {5}{4}}[{\tfrac {2}{3}}+2v]=14v\\&\therefore \;{\tfrac {5}{6}}={\tfrac {23}{2}}v\;\therefore \;v={\tfrac {5}{69}}\;;x-y={\tfrac {4}{69}}\end{aligned}}$
41. Since there are five daughters and one son, each daughter receives ${\tfrac {1}{7}}$, and the son ${\tfrac {2}{7}}$ of the residue.
    ${\begin{aligned}&1-x=7v\;;{\tfrac {1}{5}}+{\tfrac {1}{6}}={\tfrac {11}{30}}\\&\therefore \;{\tfrac {2}{3}}+{\tfrac {1}{3}}-{\tfrac {11}{30}}+2v+{\tfrac {1}{4}}[{\tfrac {1}{3}}-{\tfrac {11}{30}}+2v]=7v\\&\therefore \;{\tfrac {2}{3}}-{\tfrac {1}{30}}+2v+{\tfrac {1}{4}}[{\tfrac {-1}{30}}+2v]=7v\\&\therefore \;{\tfrac {2}{3}}+{\tfrac {5}{4}}[{\tfrac {-1}{30}}+2v]=7v\\&\therefore \;{\tfrac {4}{6}}-{\tfrac {1}{24}}={\tfrac {9}{2}}v\\&\therefore \;{\tfrac {5}{8}}={\tfrac {9}{2}}v\;\therefore \;v={\tfrac {5}{36}},\;{\text{and}}\;x={\tfrac {1}{36}}\end{aligned}}$
42. From the context it appears, that when the heirs of the residue are a mother, a wife, and 4 sisters, the residue is to be divided into 13 parts, of which the wife and one sister, together, take 5: therefore the mother and 3 sisters, together, take 8 parts. Each sister, therefore, must take not less than ${\tfrac {1}{13}}$, nor more than ${\tfrac {2}{13}}$. In the case stated at page 102, a sister was made to inherit as much as a wife; in the present case that is not possible; but the widow must take not less than ${\tfrac {3}{13}}$; and each sister not more than ${\tfrac {2}{13}}$. Probably, in this case, the mother is supposed to inherit ${\tfrac {2}{13}}$; the wife ${\tfrac {3}{13}}$; each sister ${\tfrac {2}{13}}$. ${\begin{aligned}&x+5v={\tfrac {1}{2}}\;;1-x+{\tfrac {2}{7}}[{\tfrac {1}{3}}-x]=13v\\&\therefore \;{\tfrac {2}{3}}+{\tfrac {1}{3}}-x+{\tfrac {2}{7}}[{\tfrac {1}{3}}-x]=13v\\&\therefore \;{\tfrac {2}{3}}+{\tfrac {9}{7}}[{\tfrac {1}{3}}-x]=13v\\&\therefore \;{\tfrac {2}{3}}-{\tfrac {3}{14}}=[13-{\tfrac {45}{7}}]v\\&\therefore \;{\tfrac {19}{42}}={\tfrac {46}{7}}v\;\therefore \;{\tfrac {19}{276}}=v\\&\therefore \;x={\tfrac {29}{276}},\;{\text{and the residue}}\;={\tfrac {247}{276}}\\\end{aligned}}$
    The author unnecessarily takes $7\times 276=1932$ for the common denominator.