卷二·第 92–104 问　有形之数与四则、开方、分数

§92. What are the successive triangular numbers?

第九十二问　何谓连续三角数？

They are the sums of the successive numbers starting from one, and they are what is called in Indian saṅkalita. For example, one is a triangle potentially, since we have said that all things are in it. The second triangle is the sum of 1 2, that is 3; the third triangle is the sum of 1 2 3, that is 6; and the fourth triangle is the sum of 1 2 3 4, that is 10. From this figure the nature of triangles becomes clear.

它们是从一开始的连续诸数之和，就是印度话所称的“桑卡利塔”。例如，一在潜能上就是三角形，因为我们说过万物都在它之中。第二个三角形是一与二之和，即三；第三个三角形是一、二、三之和，即六；第四个三角形是一、二、三、四之和，即十。从这个图里，三角形的性质就清楚了。

§93. What are the successive square numbers?

第九十三问　何谓连续平方数？

They are the sums of the successive odd numbers. The first square is one; add 3 to it and the result is 4, the second square, which arises from multiplying 2 by 2. If 5 is added to the 4, the result is 9, the third square, which arises from multiplying 3 by 3. From this figure that becomes clear.

它们是连续诸奇数之和。第一个平方数是一；给它加上三，结果是四，是第二个平方数，由二乘二而来。给四加上五，结果是九，是第三个平方数，由三乘三而来。从这个图里，这一点就清楚了。

§94. What are cone numbers?

第九十四问　何谓锥数？

They are the sums of the successive triangles, and are called in Indian shaṅkalita. One is the first cone; the sum of 1 3, that is 4, is the second cone; and the sum of 1 3 6 is the third cone. From this figure they are easily conceived.

它们是连续诸三角数之和，印度话称为“珊卡利塔”。一是第一个锥数；一与三之和，即四，是第二个锥数；一、三、六之和是第三个锥数。从这个图里，很容易想见它们。

§95. What are pyramid numbers?

第九十五问　何谓塔数？

They are numbers heaped up in the manner of the weights of a balance placed one upon another, the smaller above the greater, so that they are like steps and stairs. When the steps are of equal thickness, the number is the sum of the successive squares, such as 1, then 4, then 9, then 16, and it is called in Indian varga-saṅkalita. When the steps are unequal, it is the sum of the successive cubes, such as 1, then 8, then 27, then 64, and it is called in Indian ghana-saṅkalita. The properties and names of numbers are almost without end, so let us turn to what is put to use.

它们是像天平的砝码那样一个叠在另一个上、小的在上大的在下堆起来的数，因而形同台阶与梯级。若各级厚度相等，这个数就是连续诸平方数之和，如一，然后四，然后九，然后十六，印度话称为“瓦尔加桑卡利塔”。若各级厚度不等，就是连续诸立方数之和，如一，然后八，然后二十七，然后六十四，印度话称为“伽那桑卡利塔”。数的性质与名称几乎无穷无尽，所以让我们转向实际用得着的东西。

§96. What is calculation?

第九十六问　何谓计算？

It is the handling of numbers and their properties in working out what is sought, by the two operations of joining and separating, that is, by additions and subtractions.

计算就是运用数及其性质来求出所要求的东西，靠的是合与分两种运算，也就是加法与减法。

§97. What is multiplication?

第九十七问　何谓乘？

It is taking one of two numbers as many times as there are units in the other. For example, if we want to multiply 5 by 7, either you take 5 seven times, which makes 35, or you take 7 five times, which also makes 35; for the meaning of our saying '5 by 7' is: how much are 5 seven times, or how much are 7 five times?

乘就是把两数中的一个，按另一个所含单位的数目取那么多次。例如我们要以五乘七：要么你取五七次，得三十五；要么你取七五次，也得三十五；因为我们说“五乘七”的意思就是：五取七次是多少，或七取五次是多少？

§98. What is division?

第九十八问　何谓除？

It is finding the share of one unit of the divisor out of the dividend; this share is called the quotient. For example, we divide 35, the dividend, by 7, the divisor. Let us call the dividend wealth and the divisor men: then the share of a single man of this wealth is 5, and that is the quotient.

除就是求出被除数中分给除数每一个单位的那一份；这一份称为商。例如我们以三十五为被除数，以七为除数来除。且把被除数称为财物，把除数称为人：那么这笔财物中一个人的那一份是五，这就是商。

§99. What are squaring and the extraction of roots?

第九十九问　何谓平方与开方？

Squaring is multiplying a number by itself, since the product of that is called a māl ('wealth'). For example, if 7 is multiplied by 7, the result is 49, and that is the māl. Root extraction is finding the number from whose multiplication by itself the māl arose, that is, finding the 7 from whose multiplication by itself the 49 arose. This number is called the root, that is, the origin, because the side of the māl is the origin to which it is referred.

平方就是一个数自乘，因为自乘之积称为“马勒”（财）。例如七乘七，结果是四十九，这就是财。开方就是求出那个自乘而生出财的数，也就是求出那个自乘而生出四十九的七。这个数称为根，即本原，因为财的边就是它所归属的本原。

§100. What are a rational root and a surd root?

第一百问　何谓有理根与聋根？

A rational root is one whose true value can be expressed. It is also called expressible, unrestricted and open, such as 3 for 9 and 4 for 16. A surd ('deaf') root is one that cannot be expressed, such as the root of 10: it is impossible for there to be a number which, multiplied by itself, makes 10. It is also called so because it does not answer the one who calls on it seeking it, so that he can find it only by approximation.

有理根是其真值能够说得出来的根。它也称为可言的、不受限的、敞开的根，如九的三、十六的四。聋根是说不出来的根，如十的根：不可能有一个数自乘得十。它之所以这样称呼，还因为它不应答呼唤它、寻求它的人，所以那人只能用逼近的办法求得它。

§101. What are cubing and the extraction of the cube's side?

第一百零一问　何谓立方与求立方之边？

Cubing is multiplying a number by itself and the product again by that number, since the result of that is called a cube. For example, if 3 is multiplied by 3 and then by 3, the result is 27, and that is the cube. Extracting the side is finding the number from whose multiplication by itself, and of the product by it, the cube arose, that is, finding the 3 from whose multiplication by 3 twice the 27 arose. This number is called the side of the cube, and sometimes its kaʿb ('cube'). Some people, to be precise, call the cube itself the kaʿb, and so are obliged to call its kaʿb a 'side', to avoid confusion.

立方就是一个数自乘，再以其积乘那个数，因为这样所得的结果称为立方。例如三乘三，再乘三，结果是二十七，这就是立方。求边就是求出那个自乘、再以其积乘它而生出这立方的数，也就是求出以三乘了两次而生出二十七的那个三。这个数称为立方之边，有时也称为它的“卡阿卜”（立方）。有些人为了精确，把立方本身称为卡阿卜，于是为了避免混淆，就不得不把它的卡阿卜称为“边”。

§102. What is skipping?

第一百零二问　何谓跳位？

It is passing over one place or several in extracting square and cube roots. In root extraction one says 'it is, it is not, it is', and in extracting the side of a cube 'it is, it is not, it is not'. Sometimes 'plane' is said instead of 'it is', so that the last place given in the order may be known by it [text uncertain].

跳位就是在开平方与开立方时越过一位或几位。开平方时念作“有，无，有”，开立方时念作“有，无，无”。有时说“面”来代替“有”，好借此知道所给次序中的最后一位〔原文疑有讹误〕。

§103. What is the denominator?

第一百零三问　何谓分母？

It is the number of parts of the whole unit to which a fraction smaller than it is referred. Such is 3 for a third, for a third is one part of the whole when it is made into 3 parts, and likewise two-thirds are 2 parts of them; and such are 4 for a quarter and 5 for a fifth. The denominator is the smallest number that will serve for this, for 2 is also a fifth of 10, and 3 a fifth of 15; but the larger numbers have no limit, while the smallest is determinate, so it must be the one taken.

分母是整一所分成的份数，比整一小的分数就归属于它。如三之于三分之一：因为三分之一是整一被分成三份时的一份，三分之二同样是其中的两份；又如四之于四分之一，五之于五分之一。分母是能胜任此事的最小的数，因为二也是十的五分之一，三也是十五的五分之一；但较大的数没有限度，最小的数却是确定的，所以必须取最小的那个。

§104. What is reduction to one kind?

第一百零四问　何谓化为同类？

It is when there are whole numbers and fractions referred to a denominator: the denominator is multiplied by the whole numbers and the result added to the fraction, so that the total becomes of one kind. For example, 3 and a quarter is written as 3 wholes and 1 part of 4 parts of the whole. If we want to reduce it to one kind, we multiply the 3 wholes by the denominator, 4, which makes 12, and add it to the fraction, which makes 13 with 4 under it, that is, 13 quarters. Again, if there are different fractions, they are added together by reduction to one kind, such as two-sevenths and three-fifths. If we want to add them, we multiply denominator by denominator, that is, 7 by 5, which makes 35, the denominator for both together; of it the two-sevenths are 10 and the three-fifths 21, and their sum is 31, which is the total of the two-sevenths and the three-fifths in one kind. Again, if there are sexagesimal fractions, such as are used in astronomy, like minutes and seconds and what lies below them, and we want to reduce them to one kind, we multiply the highest by 60 and add the product to the one that follows it below; then we again multiply the sum by 60 and add the product to the one that follows it below, until we bring them down to the lowest place, so that the whole is in its kind. For example, we have 3 minutes, 4 seconds and 5 thirds, and want to reduce them to one kind. We multiply the 3 minutes by 60, which makes 180; this is added to the 4 seconds, and we multiply the sum by 60, which makes 11040 thirds; we add these to the 5 thirds, which makes 11045 thirds, and that is the whole of what we had, made into thirds.

当有整数，又有归属于某个分母的分数时，用分母乘整数，再把结果加到分数上，使全部成为同一类，这就是化为同类。例如三又四分之一，写作三个整数与整一的四份中的一份。我们若要把它化为同类，就用分母四乘三个整数，得十二，再把它加到分数上，得十三，下面写四，即十三个四分之一。又如有不同的分数，要通过化为同类把它们相加，如七分之二与五分之三。我们若要把它们相加，就用分母乘分母，即七乘五，得三十五，作为二者共同的分母；其中七分之二是十，五分之三是二十一，其和是三十一，这就是七分之二与五分之三在同一类中的总和。又如有六十进位的分数，如天文学中所用的分、秒及其以下各级，我们要把它们化为同类，就用六十乘最高的一级，把积加到其下紧接的一级；再用六十乘所得之和，把积加到其下紧接的一级，直到把它们降到最低一级，使全部成为同一类。例如我们有三分、四秒、五个三次分，要把它们化为同类。我们用六十乘三分，得一百八十；把它加到四秒上，再用六十乘其和，得一万一千零四十个三次分；把这些加到五个三次分上，得一万一千零四十五个三次分，这就是我们原有的全部，化成了三次分。