Shen Gua 沈括 (1031–1095) gives a formula for the sum of a geometric series and apparently derives it geometrically rather than algebraically. In this note I propose one possible reconstruction of his derivation.

The formula is one part of an jotting in Shen Gua’s Mengxi bitan 夢溪筆談 (‘Dream Brook jottings’) (Hu Daojing 1962, ch. 18, item no. 301, pp. 574–575). The entire jotting has been translated and analyzed by Andrea Bréard (1998; 2008), and I have elsewhere discussed a different part of the same jotting (Wagner 2023: 75–80). In this note I am concerned only with a narrow technical point, a possible geometric derivation of Shen Gua’s formula. The sum in question is

This can be represented as the volume of a stack of cubes, each 1 × 1 × 1 unit, as shown in Figure 1. 

In this note I refer to three of the basic solids of classical Chinese mathematics, chutong 芻童, qiandu 塹堵, and yangma 陽馬. These are shown in Figure 2. Their volumes were derived by Liu Hui 劉徽 (3rd cent. CE) in his commentary on the famous mathematical text Jiuzhang suanshu 九章算術, chapter 5 (Chemla & Guo 2004: 387–457; Guo Shuchun et al. 2013: 490–633; Wagner 1979). The red lines in Figure 1 delineate a chutong which I will refer to as the ‘inner chutong’. Shen Gua states in effect that the sum of the series, i.e. the volume of the stack, is the volume of the inner chutong, V_inner , plus a corrective term:

A modern algebraic derivation of this equation could be as follows, using the fact that h  =  a–c+1  =  b–d+1 and d  =  b–a+c:

This derivation would have been difficult for a pre-modern mathematician. Note the second and third terms in equation (*): Shen Gua would no doubt have been able to derive the sum of an arithmetic series,  , but deriving the sum of squares,  , algebraically would have been much more difficult. Bréard (1998: 157; 2008: 82) notes that this result was later known by Yang Hui 楊輝 and Zhu Shijie 朱世傑 in the 13th century and therefore may have been known by Shen Gua. But it is likely that a geometric construction was used to derive their result: perhaps a construction like the one I propose here, or the elegant construction posted by an anonymous person on Facebook. The latter construction was also given by Du Zhigeng 杜知耕 in the 17th century (Martzloff 1997: 302). In any case, the fact that Shen Gua states his calculation as a geometric result certainly suggests a geometric derivation. 

It is convenient, though not strictly necessary, to ‘push the cubes to a corner’, as in Figure 3. Here again the red lines delineate an ‘inner chutong’, while the blue lines delineate an ‘outer chutong’. These are not strictly chutong ’s after classical usage, in which the upper rectangle must be centred over the lower rectangle, but Shen Gua would surely have seen that they are equivalent to chutong’s. If he felt the need to prove this he could have used the same methods as Liu Hui, or the version of Cavalieri’s theorem stated by Zu Xuan 祖暅 (5th cent. CE, Wagner 1978).

The construction in Figure 4 shows the difference between the stack and the outer chutong and Figure 5 shows the difference between the outer and inner chutong’s. The difference between these differences is the difference between the stack and the inner chutong. The two figures are divided into corresponding sections, labelled A through E.

Section A in Figure 4 consists of h qiandu’s, each with dimensions 1 × 1 × c and with total volume ½hc. In Figure 5 Section A is equivalent to a qiandu with the same volume, and the difference here is therefore zero. The same reasoning gives a difference of zero for section E.

Section C of Figure 4 consists of h yangma’s, each with dimensions 1 × 1 × 1 and with total volume ⅓ h. In Figure 5 Section C is equivalent to a yangma with the same volume. Here again the difference is zero.

Each of sections B and D in Figure 4 consists of h qiandu’s which (because a–c = b–d) can be rearranged as in Figure 6. Their total volume is ½ h(a–c). Sections B and D in Figure 5 are equivalent to two yangma’s with total volume ⅔ h(a–c). Thus

QED.

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