The two numbers are identical; it suffices for you to multiply one of them by the issue. An example of this is a husband, three grandmothers, and three brothers. Its origin is six; the husband has three, the grandmothers have one share, and the brothers have two shares. You multiply one of the two numbers by the issue, and it becomes eighteen. The method of division in this is exactly the same as its method when the fractional remainder exists against one group. If there were six brothers, they would correspond to their share by a half; they would be reduced to three, and the procedure in it would be exactly as we mentioned. The second category is when the two numbers are proportional, meaning one of them relates to the other by one of its parts, such as its half, third, or any other part. It suffices for you to multiply the larger of the two numbers by the issue. An example of this is if the grandmothers in this issue were six, for the number of brothers is half the number of the grandmothers. Rely on the number of the grandmothers and multiply it by the origin of the issue; it becomes thirty-six, and from it, the issue becomes correct. If the number of brothers were six, their shares would correspond to them by a half, they would be reduced to three, and you would proceed according to what we have mentioned. The third category is when the two numbers are disparate, where one does not equal, relate, or correspond to the other, such as if the number of grandmothers is four and the brothers are three. You multiply one number by all the parts of the other, and whatever results, you multiply it by the issue. Whenever you multiply it here, it becomes twelve, and if you multiply it by the issue, it becomes seventy-two. If one of the two numbers corresponds to its shares and the other does not, take the concordant part of the one that corresponds and multiply it by that which does not correspond, and proceed according to what we have mentioned. If both correspond to their shares, reduce them both to their concordant parts, and perform the same calculation on the two concordant parts as you did with the original two numbers. The fourth category is that the two numbers agree by a half, a third, a fourth, or any other part. You reduce one of the two numbers to its concordant part, then multiply it by the entirety of the other; the result you then multiply by the issue. An example is if there are nine brothers and six grandmothers; they agree by a third. You reduce the grandmothers to their third, which is two, and multiply them by the number of brothers; it becomes eighteen. Then, you multiply that by the origin of the issue, and it becomes one hundred and eight, from which it becomes correct.
(5) In M: "the sisters".
العَدَدَانِ مُتَمَاثِلَيْنِ، فيُجْزِئُكَ ضَرْبُ أَحَدِهما في الْمَسْألةِ، ومِثَالُ ذلك، زَوْجٌ، وثلاث جَدَّاتٍ، وثلاثةُ إِخْوَةٍ، أَصْلُهَا مِنْ سِتَّةٍ، لِلزَّوْجِ ثلاثةٌ، ولِلْجَدَّاتِ سَهْمٌ، ولِلْإِخْوَةِ سَهْمَانِ، فَتَضْرِبُ أَحَدَ الْعَدَدَيْنِ فِي الْمَسْأَلَةِ، تَكُنْ ثَمَانِيَةَ عَشَرَ، وطَرِيقُ الْقِسْمَةِ فيها مِثْلُ طَرِيقها إِذا كان الْكَسْرُ على فَرِيقٍ وَاحِدٍ سَوَاءً. ولو كان الْإِخْوَةُ سِتَّةً، وَافَقُوا سَهْمَهم بِالنِّصْفِ، رَجَعُوا إِلى ثلاثةٍ، وكان الْعَمَلُ فيها كما ذَكَرْنَا سَوَاءً. الْقِسْمُ الثَّانِى، أَنْ يكونَ الْعَدَدَانِ مُتَنَاسِبَيْنِ، وهو أنْ يكُونَ أَحَدُهُما يَنْتَسِبُ إِلَى الْآخَرِ بِجُزْءٍ مِنْ أَجْزَائِهِ، كنِصْفِه وثُلُثِهِ، أَو غيرِ ذلك مِنَ الْأَجْزَاءِ، فيُجْزِئُكَ ضَربُ الْعَدَدِ الْأَكْثَرِ مِنهما في الْمَسْألةِ، ومِثَالُه ما لَوْ كان الْجَدَّاتُ في هذه الْمَسْأَلَةِ سِتًّا، فإِنَّ عَدَدَ الإِخْوَةِ (٥) نِصْفُ عَدَدِ الْجَدَّاتِ، فاجْتَزِئْ بِعَدَدِهِنَّ، واضْرِبْهُ في أَصْلِ الْمَسْأَلَةِ، تَكُنْ سِتَّةً وثلاثِينَ، ومنها تَصِحُّ. ولو كان عَدَدُ الْإِخْوَةِ سِتَّةً، وَافَقَتْهم سِهَامُهم بِالنِّصْفِ، وَرَجَعُوا إلَى ثلاثَةٍ، وعَمِلْتَ على مَا ذَكَرْنَاهُ. الْقِسْمُ الثَّالِثُ، أَنْ يكونَ الْعَدَدَانِ مُتَبَايِنَيْنِ، لا يُمَاثِلُ أحَدُهُما الْآخَرَ، ولا يُنَاسِبُهُ، ولا يُوَافِقُهُ، مِثْلُ أَنْ يكونَ عَدَدُ الْجَدَّاتِ أَرْبَعًا والْإِخْوَةِ ثَلَاثَةً، فإِنَّكَ تَضْرِبُ عَدَدَ أحَدِهما في جَمِيعِ الْأَجْزَاءِ، فما بَلَغَ ضَرَبْتَهُ في الْمَسْأَلَةِ، ومتى ضَرَبْتَهُ ههُنا كان اثْنَىْ عَشَرَ، فإِذا ضَرَبْتَهُ في الْمَسْأَلَةِ كانتْ اثْنَيْنِ وسَبْعِينَ. وإنْ وَافَقَ أَحَدُ الْعَدَدَيْنِ سِهَامَه دُونَ الْآخَرِ، أَخَذْتَ وَفْقَ الْمُوَافِقِ، وضَرَبْتَهُ فيما لم يُوَافِقْ، وعَمِلْتَ على مَا ذَكَرْنَا. وإِنْ وَافَقَا جَمِيعًا سِهَامَهما، رَدَدْتَهما إلى وَفْقِهِما، وعَمِلْتَ في الْوَفْقَيْنِ عَمَلَكَ في الْعَدَدَيْنِ الْأَصْلِيَّيْنِ. الْقِسْمُ الرَّابِعُ، أنْ يكونَ الْعَدَدانِ مُتَّفِقَيْنِ بِنصْفٍ، أو ثُلُثٍ، أو رُبُعٍ، أو غيرِ ذلك مِنَ الْأَجْزَاءِ، فإِنَّكَ تَرُدُّ أَحَدَ الْعَدَدَيْنِ إلى وَفْقِهِ، ثم تَضْرِبُهُ في جَمِيعِ الْآخَرِ، فما بَلَغ ضَرَبْتَهُ فِي الْمَسْأَلَةِ، ومِثَالُهُ، أَنْ تكُونَ الْإِخْوَةُ تِسْعَةً، وَالْجَدَّاتُ سِتًّا، فيَتَّفِقَانِ بِالثُّلُثِ، فتَرُدَّ الْجَدَّاتِ إلَى ثُلُثِهِنَّ اثْنَيْنِ، وتَضْرِبَهما فِي عَدَدِ الْإِخْوَةِ، تَكُنْ ثَمَانِيَةَ عَشَرَ، ثم تَضْرِبَ ذلك فِي أَصْلِ
(٥) في م: "الأخوات".