卷二·第 105–111 问　升位、数位与代数

§105. What is raising a number?

第一百零五问　何谓升位？

Whenever we get a number of more than 60 in one of the sexagesimal places, we raise it to the place above by dividing by 60, and we go on doing so until we reach the whole units, if that is possible. For example, we have 11045 seconds. No place may hold more than 59, so we divide them by 60, which gives 184 minutes with 5 seconds remaining. Since the minutes are more than 60, we again divide them by 60, which gives 3 whole units with 4 minutes remaining. Whole units, however many, are not raised. So by raising we have obtained 3 units, 4 minutes and 5 seconds.

每当六十进位的某一位上累积的数超过六十，我们就用六十除它，把它升到上一位；如果可能，就一直这样做，直到升到整数为止。例如我们有一万一千零四十五秒。任何一位上都不许超过五十九，所以我们用六十除它，得一百八十四分，余五秒。因为分仍超过六十，我们再用六十除它，得三个整数，余四分。整数不论多少都不再升位。这样，经过升位，我们得到三个整数、四分、五秒。

§106. What is reduction?

第一百零六问　何谓约简？

It is making two common numbers smaller by dividing each of them by their common measure and putting the quotients in their place. For example, we have 72 out of 360, and the common measure between them is 72. If we divide each of them by it, the first gives 1 and the second 5; 1 out of 5 is as 72 out of 360, so 1 and 5 take their place.

约简就是用两个共度数的公度去除它们各自，以所得之商代替它们，从而使这两个数变小。例如我们有三百六十分之七十二，二者之间的公度是七十二。我们用它除二者，前者得一，后者得五；五分之一如同三百六十分之七十二，所以一与五可以代替它们。

§107. What are the natural orders?

第一百零七问　何谓自然阶？

If you set down one and multiply it by some number, and go on multiplying each product by that number, you obtain proportional numbers. The first of them after the one set down is called the root, the second the māl, the third the cube, the fourth the māl-māl, the fifth the māl-cube, the sixth the cube-cube, and the rest are arranged by analogy. The ratio of one to the root is always as that of the root to the māl, of the māl to the cube, of the cube to the māl-māl, of the māl-māl to the māl-cube, and of the māl-cube to the cube-cube. An example of this in two sets of numbers, one obtained by multiplying by 2 and the other by multiplying by 3:

若你放下一，用某个数乘它，再用同一个数去乘每次所得之积，就得到一串成比例的数。放下的一之后的第一个称为根，第二个称为财，第三个称为立方，第四个称为财财，第五个称为财立方，第六个称为立方立方，其余依此类推。一与根之比，总是如同根与财之比、财与立方之比、立方与财财之比、财财与财立方之比、财立方与立方立方之比。下面以两组数为例，一组由乘二而得，另一组由乘三而得：

§108. What are the conventional places?

第一百零八问　何谓约定位？

They are the places used in the practice of calculation, and they are one of the kinds of conventional order, since the convention concerning the number assumed for multiplication has settled on 10; so the ratio between the places is tenfold. The first of these places is the units, holding from 1 to 9 and increasing by one; the second the tens, holding from 10 to 90 and increasing by ten each time; the third the hundreds, holding from 100 to 900 and increasing by a hundred; the fourth the thousands, holding from 1000 to 9000. This fourth takes the place of the units, since what follows it is tens of thousands, just as tens follow units, and it differs from the units in name only by being referred to thousands. Whenever a particular number is set in each of these places, one unit of the preceding place is always a tenth of one unit of the following place. If a place is empty of any number, a sign is set in its stead to show that it is empty. We make it a small circle and call it ṣifr ('empty'); the Indians make it a dot. This is the form of numbers in their places, for the sake of instruction. If we want to write it out and express it in words, we say: nine thousand thousand thousand thousand, and eight thousand thousand thousand, and six hundred and seventy-five thousand thousand, and thirty-four thousand, one hundred and two (9008675034102).

约定位就是计算实践中所用的数位，它们是约定次序的一种，因为人们对用来相乘的那个数约定为十，所以各位之间的比是十倍。这些位中，第一是个位，容纳一到九，每次增一；第二是十位，容纳十到九十，每次增十；第三是百位，容纳一百到九百，每次增一百；第四是千位，容纳一千到九千。这第四位又相当于个位，因为它后面是十千，正如个位后面是十位；它与个位在名称上的不同，只在于它归属于千。每一位上放了某个数时，前一位的一个单位总是后一位的一个单位的十分之一。若某一位上空无一数，就在那里放一个记号，表示它是空的。我们把这记号画成一个小圆圈，称为“隋弗尔”，意思是“空”；印度人则把它画成一个点。下面是诸数在各位中的样式，供教学之用。若我们要把它写出来、用话念出来，就说：九千千千千，又八千千千，又六百七十五千千，又三十四千，一百零二（九〇〇八六七五〇三四一〇二）。

§109. What are al-jabr and al-muqābala?

第一百零九问　何谓“贾卜尔”与“穆卡巴拉”？

If things of different kinds are equal to one another, they stand like the weights balanced in the two pans of a scale, when its tongues are level and its beam is even. It is clear that if you take some quantity of one of the kinds out of one of the pans, you must take the like of it, in kind and in amount, out of the other pan, so that the balance and the state remain as they were; and likewise, if you put something into one of them, you must put the like into the other. So too, when magnitudes are equal to other magnitudes and on one of the two sides there is an exception and a deficiency, it must be restored by completing it, and the like of it must be added to the other side; this is al-jabr ('restoration'). For example, on one side there are 100 dirhams less 14 dinars, and on the other 13 istārs of iron less 12 dirhams, the two being equal. The restoration here is that you complete the 100 dirhams by removing the exception, which is 14 dinars, and add it to the other side, which then holds 13 istārs of iron and 14 dinars less 12 dirhams. Then you complete this side too by removing the exception, which is 12 dirhams, and add it to the other side; so after restoration 112 dirhams are equal to 13 istārs of iron and 14 dinars. Al-muqābala ('balancing'), after restoration is finished, is that you look at the things of like kind on the two sides, and remove the lesser from one of them and its equal from the other. For example, on one side there are 112 dirhams and on the other 13 istārs of iron and 12 dirhams, the two being equal. Here the like kinds are the dirhams, of which there is a quantity on each side; the lesser is 12, so we drop it, and its like from the other side, and there remain 100 dirhams equal to 13 istārs of iron.

若不同种类的东西彼此相等，它们就好比天平两盘中称平的砝码，此时秤舌端正，秤杆平直。显然，若你从一盘中取出某一类的若干，就必须从另一盘中取出种类与数量都相同的东西，天平与原来的状态才能保持不变；同样，若你往其中一盘放进什么，也必须往另一盘放进同样的东西。同理，当一些量等于另一些量，而两边之一带有除去之项与不足时，就必须把它补足以还原，并在另一边加上同样的东西；这就是“贾卜尔”，意思是“还原”。例如，一边是一百迪拉姆减十四第纳尔，另一边是十三伊斯塔尔铁减十二迪拉姆，两边相等。这里的还原，就是你去掉除去之项，即十四第纳尔，把一百迪拉姆补足，再把它加到另一边，于是那一边成为十三伊斯塔尔铁加十四第纳尔减十二迪拉姆。然后你把这一边也补足，去掉除去之项，即十二迪拉姆，再把它加到另一边；于是还原之后，一百一十二迪拉姆等于十三伊斯塔尔铁加十四第纳尔。“穆卡巴拉”，意思是“对消”，是在还原完毕之后，看两边同类的东西，从一边去掉较少的，并从另一边去掉与之相等的。例如，一边是一百一十二迪拉姆，另一边是十三伊斯塔尔铁加十二迪拉姆，两边相等。这里同类的是迪拉姆，两边各有若干；较少的是十二，于是我们去掉它，并从另一边去掉与之相同的，剩下一百迪拉姆等于十三伊斯塔尔铁。

§110. What are the simple equations?

第一百一十问　何谓单纯方程？

The art of al-jabr turns on three pillars. The first is absolute number, not related to anything; the second is related number, that is, the root of the māl; the third is related number, that is, the māl of the root. Among them three pairings arise. The first is roots equal to a number, meaning: what māl is it whose root, or some number of its roots, is so much in number? The second is māls equal to a number, meaning: what māl or māls are so much in number? The third is māls equal to roots, meaning: what māl or māls are equal to one of their roots or to several roots?

代数之术以三根柱石为转轴：第一是纯数，不与任何东西相关；第二是相关之数，即财的根；第三是相关之数，即根的财。三者之间生出三种配对。第一种是根等于数，意思是：是哪一个财，它的根，或它的若干个根，等于这么多的数？第二种是财等于数，意思是：是哪一个或哪几个财，等于这么多的数？第三种是财等于根，意思是：是哪一个或哪几个财，等于它的一个根或若干个根？

§111. What are the compound equations?

第一百一十一问　何谓复合方程？

They arise when you join every two of the simple ones and equate them with the third, which yields three equations, the compound ones. The first is the one equal to number, namely māls and roots equal to a number, meaning: what māl is it which, when you add to it its root or some number of its roots, comes to so much in number? For example, a māl and 10 roots equal 39; this māl is 9 and its root 3. The second compound is the one equal to roots, namely māls and number equal to roots, meaning: what māl is it which, when you add to it so much in number, is equal to its root or to some number of its roots? This second compound sometimes has two forms, that is, the question admits two answers. For example, a māl and 30 in number equal 13 roots: this māl is either 100, with root 10, or 9, with root 3, so it admits both forms together. The third compound is the one equal to the māl, namely roots and number equal to a māl, meaning: what māl is it that is equal to a number when that number is added to its root or to some number of its roots? For example, 6 in number and 5 roots equal a māl; this māl is 36 and its root 6 [corrected: MS has '10 roots'].

把单纯方程每两种合在一起，使之等于第三种，就生出三种方程，即复合方程。第一种是等于数的那一种，即财与根等于数，意思是：是哪一个财，你给它加上它的根或若干个根，就等于这么多的数？例如一个财加十个根等于三十九；这个财是九，它的根是三。第二种复合方程是等于根的那一种，即财与数等于根，意思是：是哪一个财，你给它加上这么多的数，就等于它的根或它的若干个根？这第二种复合方程有时有两种形态，也就是说，这个问题容许两个答案。例如一个财加三十个数等于十三个根：这个财或是一百，根为十；或是九，根为三，所以它同时容许两种形态。第三种复合方程是等于财的那一种，即根与数等于财，意思是：是哪一个财，等于把一个数加到它的根或它的若干个根上所得？例如六个数加五个根等于一个财；这个财是三十六，它的根是六〔抄本作“十个根”，据算改正〕。