Shùshù jìyí 數術記遺
Records of Bequeathed Numerical Arts attributed (pseudepigraphically) to 徐岳 (Xú Yuè, late Hàn / Wèi); annotated by 甄鸞 (Zhēn Luán, Northern Zhōu)
About the work
A 1-juan compendium of unusual numerical-and-computational systems, traditionally attributed to Xú Yuè of the late Hàn / Wèi period with annotation by Zhēn Luán of the Northern Zhōu. The Sìkù 提要 mounts a thorough critical case against the traditional Hàn attribution, judging the work to be a Tang-period pseudepigraphic composition (tuōmíng 托名) attributed to Xú Yuè by an “ancient-loving private scholar” of the Tang for the purpose of meeting the imperial demand for ancient mathematical books that arose during the Suànjīng shíshū curricular canonization.
The 提要’s grounds for the pseudepigraphy judgment:
(1) Bibliographic absence: the Sui Jīngjí zhì lists Xú Yuè’s Jiǔzhāng suànjīng and Zhēn Luán’s Qīyào shùsuàn (and other works), but specifically does not list a Shùshù jìyí — making its first secure documentary appearance the Tang Yìwén zhì.
(2) Internal historical inaccuracies: the work claims that “[Xú Yuè] received teaching at Tàishān from Liú Huìjī” 劉會稽 (Liú Hóng, Kuàijī taishou) — but the HòuHànshū zhì citing the Yuán Shānsōng shū 袁山松書 records that Liú Hóng (Xú Yuè’s actual master) was Kuàijī Dūwèi (not Tàishǒu) and his Tàishǒu posting was at Dānyáng (not Kuàijī); furthermore, after Liú Hóng’s Kuàijī posting he never returned to Tàishān. So the work’s claim that Xú Yuè received teaching from “LiúKuàijī” at Tàishān is historically impossible.
(3) Title-anachronism: the work attributes Xú Yuè to the Hàn dynasty, but the Jìnshū records that Xú Yuè was active under the Wèi (Huángchū period, 220–227 CE), debating with the Tàishǐchéng Hán Yì on solar-lunar eclipse matters. Xú Yuè was a Wèi-period official, not a Hàn one.
(4) Anachronistic content: the work’s mathematical-substantive content — including Huángdì sānděng shù 黄帝三等數, Jīsuàn 積算, Tàiyī suàn 太一算 — has “absolutely no significant content”; the Tiānmén Jīnhǔ 天門金虎 (Heaven-Gate Gold-Tiger) terminology is “recondite Daoist deceptive language”, inconsistent with the technical mathematical-astronomical character of the genuine Xú Yuè / Zhēn Luán mathematical works.
(5) Multi-hand redaction: the annotation’s listed mathematical formulas often do not match the main text — suggesting the work was assembled by multiple hands rather than being the unified composition of a single author with consistent annotation.
The 提要 nonetheless preserves the work in the Sìkù — “to set out the false in order to dispel later people’s confusion” — both because the work has long circulated and may have continued utility as a curiosity, and because its critical annotation by Sìkù editors documents the methodology of late-imperial Chinese textual-criticism.
Tiyao
[Sub-classification: 子部, Tiānwén suànfǎ class 2, suànshū sub-category. Edition: WYG.]
Respectfully examined: Shùshù jìyí, 1 juàn, old title-attribution Hàn Xú Yuè zhuàn, Northern-Zhōu Zhēn Luán zhù. [Xú] Yuè was a man of Dōnglái. The Jìnshū Lǜlì zhì says: “the Wú Zhōngshūlìng Kàn Zé received Liú Hóng’s Qiánxiàng method from Xú Yuè of Dōnglái” — this [is the man].
The Suíshū Jīngjí zhì fully lists [Xú] Yuè’s and Zhēn Luán’s composed Jiǔzhāng suànjīng, Qīyào shùsuàn, and other titles — but uniquely [does] not [list] this book’s name. Reaching the Tang Yìwén zhì, [it] first appears in the bibliographic record.
Within the book, [it] says: “at Tàishān [Xú Yuè] saw LiúKuàijī, [who was of] broad-knowledge and many-heard, [his] knowledge spreading throughout the numerical arts. I therefore at the time of receiving teaching asked, ‘Do numbers have a limit?’ Kuàijī said, ‘I once roamed in the Tiānmù mountains and saw a hermit who said…’” and so on. In general it speaks of its transmission’s spirit-secret.
But examining the HòuHànzhì commentary cites Yuán Shānsōng shū saying: “Liú Hóng of Tàishān Méngyīn, in the Yánxī period [158-167 CE], with the Wèi commander Yīng Tàishǐ summoned, was appointed Lángzhōng. Later [he] became Kuàijī Dōngbù Dūwèi 都尉. Summoned-back, [he] reached the post but did not arrive — [he] received the office of Dānyáng Tàishǒu 太守, dying in office”. This [shows] [Liú] Hóng [after his] Kuàijī appointment [never] subsequently dwelt at home; [he] could not have at Tàishān seen [Xú Yuè]. Furthermore, [Liú] Hóng at Kuàijī was [serving as] Dūwèi; his Tàishǒu post was actually at Dānyáng. But the [present work’s] annotation calls [him] “Kuàijī Tàishǒu” — the inconsistency is extreme.
Furthermore, the old recension all titles “Hàn Xú Yuè zhuàn”; according-to-what the Jìnshū records, [Xú] Yuè belongs to Wèi Huángchū [220-227 CE], with the Tàishǐchéng Hán Yì disputing-debating five matters of solar-and-lunar-eclipse — then [Xú] Yuè had already served under Wèi, [and we] cannot tie [him] to Hàn. Examining-the-ancient is rather sparse-and-erroneous.
As to within [the work] listing the Huángdì sānděng shù and Jīsuàn Tàiyī suàn and similar things — all [are] absolutely without significant content. Its Tiānmén Jīnhǔ and similar phrases [are] then Daoist deceptive-and-fanciful sayings, very recondite-and-non-canonical. The annotated arithmetic-formulas and number-positions, examining the main text, [are] often not mutually-fitting — probably not by one hand composed.
The Tang dynasty’s xuǎnjǔ (selection) institution, the Suànxué, [had] Jiǔzhāng and Wǔcáo outside [of which it] also practiced this book. This must be [that], at the time, [the Tang court was] purchasing-and-seeking ancient arithmetic [books]; fond-of-the-strange [scholars] therefore by reliance pseudepigraphically composed [it], and falsely-attributed-the-name to [Xú] Yuè.
However, [its] transmission having long [continued], scholars often take the ancient recension as doubtful. Therefore [we have] still recorded-and-preserved [it], while in detail expounding its falseness, in order to dispel later-people’s confusions.
Abstract
Composition window: c. 600–700 CE (the Tang period when imperial demand for ancient mathematical books motivated pseudepigraphic composition). The traditional attribution to Xú Yuè (late Hàn / Wèi) is rejected by the Sìkù editors with substantial textual-historical argumentation. The Zhēn Luán annotations may have a more genuine grounding (Zhēn Luán’s annotation activity is well documented), but the present work’s annotations are also suspected of multi-hand composition.
The work’s significance:
(a) Documentation of Tang-period pseudepigraphy practice: the 提要’s argumentation establishes the work as an example of the Tang-period practice of pseudepigraphic mathematical-text composition under the spur of imperial collecting demand. This is methodologically interesting for the history of Chinese book-culture.
(b) The Sìkù critical-textual methodology: the 提要 itself is one of the more thoroughgoing exercises in kǎojù (evidential research) methodology in the entire Sìkù tíyào corpus — combining bibliographic absence-evidence, internal historical contradiction, character-attribution challenges, and content-incoherence to build a multi-pronged case for pseudepigraphy.
(c) Preservation despite criticism: the Sìkù editors’ decision to preserve the work despite their critical judgment exemplifies the high-Qīng editorial principle of preserving received texts even when their attributions are doubted — provided their critical assessment is also preserved alongside.
For the related works of the Suànjīng shíshū see KR3f0032 Jiǔzhāng suànshù, KR3f0033 Sūnzǐ suànjīng, KR3f0035 Hǎidǎo suànjīng, KR3f0036 Wǔcáo suànjīng, KR3f0037 Wǔjīng suànshù. For the principal commentator, see 甄鸞. For the (probably falsely) attributed author, see 徐岳.
Translations and research
- Limited substantial secondary literature — most history-of-Chinese-mathematics treatments mention the work only briefly, focusing on the more substantive Suàn-jīng shí-shū members.
- Martzloff, Jean-Claude. A History of Chinese Mathematics, Berlin: Springer, 1997.
- Needham, Joseph (with Wang Ling), Science and Civilisation in China, vol. 3.
Other points of interest
The 提要’s identification of “Daoist deceptive-and-fanciful sayings” within the work — Tiānmén Jīnhǔ and similar — suggests the pseudepigraphic author may have had Daoist-religious affiliations and was attempting to attach occult-cosmological terminology to a fundamentally mathematical exposition for added-cultural prestige. This pattern of Daoist-cosmological framing of mathematical material is paralleled in some other Tang-period mathematical compilations.