卷二·第 112–119 问　未知之物、双误法与字母记数

§112. What is the 'thing'?

第一百一十二问　何谓“物”？

'Thing' is the name for the unknown root; that is why, when it is multiplied by itself, it becomes a māl.

“物”是未知之根的名称；所以它自乘时就成为一个财。

§113. How are things multiplied?

第一百一十三问　物如何相乘？

A thing multiplied by a thing gives a māl; multiplied by a number it gives as many things as that number. If a thing is multiplied by 'less a thing', the result is a deficient māl, excepted in the form 'less a māl'. If 'less a thing' is multiplied by a number, the result is so many deficient things, in the form 'less so many things'. If 'less a thing' is multiplied by 'less a thing', the result is an added māl, for an exception is cancelled only by its like [corrected: MS has 'by a number'].

物乘物得一个财；物乘一个数，得与那个数同样多的物。若物乘“减一物”，所得是一个不足的财，以“减一财”的形式作除去之项。若“减一物”乘一个数，所得是那么多个不足的物，以“减若干物”的形式出现。若“减一物”乘“减一物”，所得是一个增加的财，因为除去之项只有遇到与它同类的除去之项才会消去〔抄本作“乘一个数”，据算改正〕。

§114. What is the reckoning of the dinar?

第一百一十四问　何谓第纳尔算法？

It is a reckoning derived from al-jabr and al-muqābala. Sometimes there is more than one unknown thing, and they need to be given names. Some name them dinar, dirham and fals; others give the things colours, like the Indians, who say a black thing, a yellow and a dusky.

这是从还原与对消之术派生出来的一种算法。有时未知之物不止一个，需要给它们起名。有些人把它们叫作第纳尔、迪拉姆和法尔斯；另一些人则给这些物配上颜色，如印度人，他们说一个黑物、一个黄物和一个暗褐之物。

§115. What is the reckoning of the two errors?

第一百一十五问　何谓双误算法？

For what is sought, one assumes any number that comes to hand. If testing it leads to the right answer, he has found it; if it leads to an error, he keeps note of the amount and repeats the process with another number, whatever it may be, which either passes beyond what is sought or falls short of it a second time. Then he works out what is sought from these two errors by a well-known method.

对于所求之数，先随手假定任意一个数。若检验它能得出正确的答案，就算找到了；若它导致误差，就记下误差的大小，再用另一个数——不论什么数——把这番运算重做一遍，这个数第二次或者超过所求，或者不及所求。然后用一种人所熟知的方法，从这两个误差算出所求之数。

§116. How are numbers written with the Arabic letters?

第一百一十六问　如何用阿拉伯字母记数？

This is a matter of agreement and convention. The order of numbers could have been set in the letters of the alphabet arranged as alif, bā, tā, thā, for they suffice for the 9 units, the 9 tens, the 9 hundreds and the thousand with them, their number being 28. But people set the order of these numbers in the jummal letters, because this order is the one current among the People of the Book who preceded the Arabs: abjad hawwaz ḥuṭṭī kalaman saʿfaṣ qarashat thakhadh ḍaẓagh. Their numerical values are set out in this table.

这是约定俗成的事。数的次序本来也可以排在按艾利夫、巴、塔、萨（ا ب ت ث）次序排列的字母上，因为这些字母共二十八个，足够九个个位、九个十位、九个百位，再加上一个千。但人们把这些数的次序排在“朱马勒”字母上，因为这种次序在阿拉伯人之前的有经之人中通行：艾卜杰德、哈瓦兹、胡蒂、凯莱曼、萨阿法斯、格赖沙特、塞哈兹、达扎格（ابجد هوز حطي كلمن سعفص قرشت ثخذ ضظغ）。它们的数值列在这张表里。

§117. Is there any disagreement about them?

第一百一十七问　对这些字母有没有分歧？

The purpose of these letters is brevity in writing numbers in tables, and the people of tables are the astronomers, who have not disagreed among themselves. Only those for whom this is not their art have differed from them: in place of saʿfaṣ they put ṣaʿfaḍ, making the undotted ṣād 60 and the dotted one 90, and in place of qarashat they put qarasat, with undotted sīn. Some of them were led to this by linguistic word-patterns, and others by interpretations serving the ends of a creed, which is like raving. If the users had agreed on this order we would have allowed their practice, but it is a departure from custom.

用这些字母，是为了在表中简写数目，而表格家就是天文家，他们彼此之间并无分歧。只有那些不以此为本行的人与他们不同：他们把萨阿法斯换成萨阿法德，让无点的萨德（ص）作六十，有点的那一个（ض）作九十；又把格赖沙特换成格赖萨特，用无点的辛（س）。他们中有些人是受了语言上词形构造的引导，另一些人则是出于为某种信条服务的解释，那就像胡言乱语。假如用这些字母的人都一致采用这种次序，我们也就认可他们的做法了；但它是背离惯例的。

§118. How are they combined?

第一百一十八问　这些字母如何组合？

If a number has several places, such as units, tens and hundreds, begin with the greatest, that is, the hundreds, then the tens, then the units. For example, 115: write it qyh, and draw a line above it to show that it is a number and not a word. If the number is 105, write it qh; if it is 42, write it mb; if it is 1002, write it ghb. If we need to write 2000, write it bgh, because placing the smaller before the greater distinguishes it from ghb and shows that this is done for a purpose. It is customary with these letters not to give jīm its curve, to distinguish it from khāʾ; and khāʾ is seldom needed, because astronomical reckonings stop at 360. Bāʾ is curved towards dāl so that it does not resemble nūn with its dot. Kāf is written extended so as not to resemble lām. Nūn is made larger than zāy, and dotted, to distinguish them, for zāy is dotted and rāʾ undotted. The difference between sīn and shīn is that shīn is dotted, and all the letters after it are likewise dotted. When nūn and bāʾ are combined they may look alike, and then the dot of nūn tells them apart; but caution requires dotting the bāʾ as well. If a zero has to be written to fill an empty place, a circle is made with a line above it touching it, to distinguish it from hāʾ. In the Indian numerals the touching line is not needed, since there is no such letter there.

若一个数有好几位，如个位、十位、百位，就从最大的一位即百位开始，然后十位，然后个位。例如一百一十五，写作 قيه，并在上面画一道横线，表示它是数而不是词。若是一百零五，写作 قه；若是四十二，写作 مب；若是一千零二，写作 غب。若要写两千，就写作 بغ，因为把小的放在大的前面，就能把它与 غب 区分开，表明这样写是有意为之。用这些字母时，习惯上不给吉姆（ج）写出弯钩，以免与哈伊（خ）相混；而哈伊很少用到，因为天文的计算止于三百六十。巴（ب）向达勒（د）一侧弯曲，以免像带点的嫩（ن）。卡夫（ك）写得拉长，以免像拉姆（ل）。嫩写得比扎伊（ز）大，并且加点，以示区别，因为扎伊有点而拉（ر）无点。辛（س）与欣（ش）的区别在于欣有点，它后面的字母也全都有点。嫩与巴连写时可能相像，这时靠嫩的点来分辨；但谨慎起见，巴也要加点。若要写零来填空位，就画一个圆圈，上面加一道与它相接的横线，以与哈（ه）区别。印度数码则不需要这道相接的横线，因为那里没有这样的字母。

§119. Are these letters used for anything other than numbers?

第一百一十九问　这些字母除了记数还有别的用处吗？

They are sometimes made into signs for the zodiacal signs, and these too are originally derived from reckoning. They look like this:

它们有时被用作十二星座的记号，这些记号本来也出自计算。它们的样子如下：

What has gone before, in all that has been said of calculation, is enough for anyone who wants an introduction to it. Let us now turn to an account of the configuration of the heavens.

以上关于计算所说的一切，对于想要入门的人已经足够了。现在让我们转而讲述天的构形。