Volume 507 Biographies 294: Astronomers and Mathematicians 2

=李潢=李潢，字云門，鍾祥人。乾隆三十六年進士，由翰林官至工部左侍郎。博綜群書，尤精算學，推步律呂，俱臻微妙。著九章算術細草圖說九卷，附海島算經一卷，共十卷。

Li Huang, styled Yunmen, was a native of Zhongxiang. He received his jinshi degree in 1771 and advanced from the Hanlin Academy to the post of Left Vice Minister of Works. He was broadly learned, but excelled above all in mathematics; in calendrical astronomy and in the theory of pitch-pipes alike he attained a rare subtlety. He authored Detailed Explanatory Diagrams for the Nine Chapters on Mathematical Procedures in nine fascicles, with the Sea Island Mathematical Manual appended in one fascicle—ten fascicles in all.

其自序重差圖云：「圖九，望遠，海島舊有圖解，餘八圖今所補也。同式形兩兩相比，所作四率，二三率相乘，與一四率相乘同積。如欲作圖明之，第取一三率聯爲一邊，又取二四率聯爲一邊，作相乘長方圖之，自然分為四冪。又以斜弦界為同式句股形各二，則形勢驗矣。舊圖於形外別作同積二方，至兩形相去遼遠者，又必宛轉通之，皆可不必也。圖中以四邊形、五邊形立說，似與句股不類，然於本形外補作句股形，則亦句股也。四率比例法，在九章粟米謂之今有，一為所有率，二為所求率，三為所有數，四為所求數，在句股則統目之為率。劉氏注云：'句率股率，見句見股者是也。 '今祗云同式相比者，取省易耳，異乘同除則一也。」書甫寫定，潢即病。俟吳門沈欽裴算校，方可付梓。越八年，其甥程矞採家為之校刊，以成其志。

In his preface to the double-difference diagrams he writes: "There are nine figures in all. For distant sighting, the Sea Island Manual already had illustrated solutions; the other eight figures are additions of my own. When similar figures are paired for comparison, one forms four proportional terms: the product of the second and third equals that of the first and fourth. To illustrate this with a figure, join the first and third terms as one side and the second and fourth as the other, construct the rectangle of their product, and it naturally falls into four areas. Draw the diagonal as a boundary to form two pairs of similar right triangles, and the geometric relation is confirmed at a glance. Older diagrams drew two equal-area squares outside the figure and, when the shapes lay far apart, connected them with convoluted lines—none of which is really needed. The figures treat quadrilaterals and pentagons, which seem unrelated to right triangles; yet once auxiliary right triangles are added around the original shape, the problem is right-triangle geometry after all. The rule of four proportional terms—in the Millet chapter of the Nine Chapters it is called 'present quantity': the first is the given rate, the second the sought rate, the third the given number, and the fourth the sought number; in right-triangle problems all are simply termed rates. Liu Hui's commentary states: "Leg-rate and base-rate are the rates that appear as leg and base." I speak only of comparing similar figures"—that is for brevity; cross-multiplication with like division amounts to the same thing." No sooner was the manuscript complete than Huang fell ill. He waited for Shen Qinpei of Suzhou to verify the calculations before the work could go to press. Eight years later his nephew Cheng Yucai edited and published the work from the family papers, thereby fulfilling his uncle's intent.

九章初經東原戴氏從永樂大典中錄出，一刻於曲阜孔氏，再刻於常熟屈氏，悉依戴氏原校本刊刻。其時古籍甫顯，校訂較難，不無間有扞格，自是天下之習九章者，莫不家★L3一編，奉為圭臬。而劉徽九章亦從此有善本矣。潢又嘗因古算經十書中，九章之外最著者，莫如王孝通之輯古。唐制開科取士，獨輯古四條限以三年，誠以是書隱奧難通。世所傳之長塘鮑氏、曲阜孔氏、羅江李氏各刻本，又悉依汲古閣毛影宋本，祗有原術文而未詳其法，且復傳寫脫誤。雖經陽城張氏以天元一術推演細草，但天元一術創自宋、元時人，究在王氏後，似非此書本旨。爰本九章古義，為之校正，凡其誤者糾之，闕者補之，著考註二卷。以明斜袤廣狹割截附帶分並虛實之原，務如其術乃止。稿未成，潢歿後，為南豐劉衡授其鄉人，以西士開方法增補算草，並附圖解，刻於江西省中，喧賓奪主，殊亂其真。矞採取江西刻本削去圖草，仍以原考注刊布。

The Nine Chapters was first recovered by Dai Zhen of Dongyuan from the Yongle Encyclopedia, printed once by the Kong family of Qufu and again by the Qu family of Changshu—all editions following Dai's collated text. Ancient texts had only lately resurfaced, and collation was arduous; occasional discrepancies were inevitable. Thereafter every student of the Nine Chapters kept a copy at home and treated it as the authoritative text. Thanks to this effort, Liu Hui's commentary on the Nine Chapters likewise acquired a reliable edition. Huang also observed that among the Ten Mathematical Classics, the most renowned work after the Nine Chapters was Wang Xiaotong's Collected Ancient Problems. Under the Tang examination system, candidates faced only four problems from the Collected Ancient, with three years allowed to pass—such was the book's obscurity and difficulty. Circulating editions from the Bao family of Changtang, the Kong family of Qufu, and the Li family of Luojiang all followed Mao Jin's Jiguge facsimile of the Song text—they preserved the original procedures but not the underlying methods, and were further marred by transcription errors. Although Zhang of Yangcheng had elaborated detailed solutions using the celestial-element method, that technique arose only in the Song and Yuan—long after Wang Xiaotong—and can hardly reflect the book's original approach. Accordingly, on the basis of the Nine Chapters' ancient principles, he collated the text—correcting errors, supplying omissions—and produced two fascicles of critical commentary. His aim was to clarify the principles behind oblique dimensions, sectional cutting, attached division, and the handling of positive and negative quantities—stopping only when the procedures themselves were fully intelligible. The draft remained unfinished at Huang's death. Liu Heng of Nanfeng passed it to a fellow townsman, who padded the calculation drafts with Western root-extraction methods and added diagrams for an edition printed in Jiangxi—the additions overshadowing the original and badly obscuring its intent. Cheng Yucai took the Jiangxi edition, stripped out the supplementary diagrams and drafts, and published Huang's original commentary alone.

武進李兆洛為之序，曰：「輯古何為而作也？蓋闡少廣、商功之蘊而加精焉者也。商功之法，廣袤相乘，又以高若深乘之為立積，今轉以積與差求廣袤高深，所求之數，最小數也。曷為以最小數為所求數？曰，求大數，則實方廉隅，正負雜糅。求小數，則實常為負，方廉隅常為正也。觀台羨道，築堤穿河，方倉圓囤，芻甍輸粟，其形不一，概以從開立方除之何也？曰，一以貫之之理也。物生而後有像，象而後有滋，滋而後有數。斜解立方，得兩巉堵，一為陽馬，一為鱉臑。陽馬居二，鱉臑居一，不易之率也。今於平地之餘續狹斜之法，無論為巉堵、為陽馬、為鱉臑，皆作立積。觀其立積內不以所求數乘者為減積，以所求數一乘者為方法，再乘者為廉法，所求數再自乘為立方，即隅法也。從開立方除之，得所求數。若繪圖於紙，令廣袤相乘，以所求數從橫截之。剖平冪為若干段，又以截高與所求數乘之。分立積為若干段，若者為減積，若者為方，若者為廉，若者為隅，條段分明，歷歷可指。作者之意，不煩言而解矣。其云廉母自乘為方母，廉母乘方母為實母者之分，開方之要術也。先生於是書立法之根，如鋸解木，如錐劃地，又復補正脫誤，條理秩然，信王氏之功臣矣！爰述大旨，以告世之習是書者，無復苦其難讀雲。」

Li Zhaoluo of Wujin wrote a preface for the work, asking: "Why was the Collected Ancient Problems composed? It was written to unfold and refine the deeper principles of the Lesser Extension and Construction Works chapters. In the Construction Works procedures one multiplies breadth by length and then by height or depth to obtain volume; here the process is reversed—given volume and difference, one seeks breadth, length, height, and depth—and the root sought is always the smallest root. Why should the smallest root be the one sought? Because if one seeks the larger root, the dividend, square coefficient, edge coefficient, and corner term become entangled in positive and negative signs. If one seeks the smaller root, the dividend is consistently negative while the square, edge, and corner terms remain consistently positive. Consider the problems of terraces, embankments, river channels, square granaries, round silos, thatched sheds, and grain transport—the shapes differ, yet all are solved by cube-root extraction. Why is that? Because a single underlying principle governs them all. Things come into being and then take form; once formed they grow; once they grow, numbers arise to describe them. Slice a cube obliquely and one obtains two wedge-shaped solids—one called a yang horse, the other a turtle's foreleg. The yang horse counts for two parts, the turtle's foreleg for one—a fixed and invariable ratio. Beyond level ground Wang extends the method of narrow oblique solids; whether the figure is a wedge-block, a yang horse, or a turtle's foreleg, all are reduced to volumetric accumulation. Within the accumulated volume, terms not multiplied by the unknown are the subtractive volume; those multiplied once give the square coefficient, twice the edge coefficient, and the unknown squared twice gives the corner coefficient. Extract the cube root and divide accordingly to obtain the unknown. Draw the figure on paper, multiply breadth by length, and cut across at the value of the unknown; divide the planar area into segments, then multiply the sectional height by the unknown; and separate the solid volume into parts—subtractive volume here, square coefficient there, edge and corner terms elsewhere—each category distinct and plainly visible. The author's intent becomes clear without further words. His remark that the edge coefficient squared yields the square coefficient, and the edge times the square yields the dividend—that distinction is the essential technique of root extraction. In this book Huang laid bare the foundations of Wang's methods as cleanly as a saw through timber or an awl on earth; he corrected omissions and errors until the exposition stood in clear order—truly Wang Xiaotong's greatest champion! I have set forth these main points so that students of the book may no longer struggle with its difficulty."

=汪萊=汪萊，字孝嬰，號衡齋，歙縣人。年十五，補博士弟子。弱冠后，讀書於吳葑門外，慕其鄉江文學永、戴庶常震、金殿撰榜、程徵君易疇學，力通經史百家及推步曆算之術。嘉慶十二年，以優貢生入都，考取八旗官學教習，會御史徐國楠奏請續修天文、時憲二志，經大學士首舉萊與徐準宜、許澐入館纂修。十四年，書成。議敘，以本班教職用，選授石埭縣訓導。十八年，應省試，得疾歸，卒於官，年四十有六。先是十一年夏，黃河啟放王營減壩，正溜直注張家河，會六塘河歸海。兩江督臣奉上命，查量雲梯關外舊海口與六塘河新海口地勢高下，延萊測算，蓋其精算之名，久為官卿所知。曾制渾天、簡平、一方各儀器觀測。

Wang Lai, styled Xiaoying and known by the sobriquet Hengzhai, was a native of She County in Huizhou. At the age of fifteen he entered the Imperial Academy as a doctoral student. After reaching manhood he studied outside Suzhou's Feng Gate, emulating his fellow Huizhou scholars Jiang Yong, Dai Zhen, Jin Bang, and Cheng Yichou, and applied himself to the classics and histories, to the full range of traditional learning, and to calendrical astronomy and mathematics. In 1807 he came to the capital as an honors tribute student and passed the examination for instructor in the Eight Banners schools. When Censor Xu Guonan petitioned to revise the treatises on astronomy and the calendar, the Grand Secretary recommended Wang Lai together with Xu Zhunyuan and Xu Yun to serve on the editorial staff. The project was completed in 1809. When rewards were considered, he received a teaching appointment in his original class and was selected as Director of Studies in Shidi County. In 1813 he sat for the provincial examination, fell ill on the journey home, and died in office at the age of forty-six. Earlier, in the summer of 1806, the Wangying relief dam on the Yellow River was opened; the main channel poured directly into the Zhangjia River and thence through the Liutang River to the sea. The governor-general of the Two Jiangs, acting on imperial orders, surveyed the relative elevations of the old estuary outside Yunti Pass and the new estuary of the Liutang River, and engaged Wang Lai to perform the calculations—for his reputation in advanced mathematics had long been known among officials. He had constructed armillary sphere, simplified armillary, and equatorial instruments for astronomical observation.

與郡人巴樹穀最友善，客江、淮間，又與焦孝廉循、江上舍籓、李秀才銳，辯論宋秦九韶、元李冶立天元一及正、負開方諸法。天性敏絕，極能攻堅，不肯苟於著述。凡所言，皆人所未言，與夫人所不能言。

His closest friend was Ba Shugu of his home prefecture. While traveling in the Jianghuai region he also debated with Jiao Xun, Jiang Fan, and Li Rui the celestial-element algebra of Qin Jiushao and Li Ye and the methods of positive and negative root extraction. He was exceptionally quick by nature, relished the hardest problems, and refused to publish anything he had not fully mastered. Whatever he said was what others had not yet said—and what others could not yet say.

嘗以古書八線之制，終於三分取一，用益實歸除法求之，其一表之真數，僅得十之二。因悟得五分之一通弦與五分之三通弦交錯為三角形，比例立法，以取五分之一之通弦，而弦切之數益密。梅氏環中黍尺，有以量代算之術，惟求倚平儀外周之兩角，而縮於內半週之角未詳。其法較易，因立新術，量取不倚外周之角度，而三角之量法乃全。堆垛有求平三角、立三角、尖堆積法，不及三乘方以上，又復推而廣之，自三乘、四乘以上之尖堆，皆可由根知積。並及諸物遞兼之法，以補古九章所未備。

He once worked through the ancient eight-line trigonometric tables, which extend only to one-third of the arc, using the method of increased dividend and returning division—but recovered only about two-tenths of the true values in a single table. He then saw that the versed sines for one-fifth and three-fifths of the arc interlock as a triangle; by proportional construction he derived the versed sine for one-fifth, yielding chord and tangent values of far greater precision. In Mei Wending's Millet Measure within the Armillary Sphere there is a technique of replacing calculation with direct measurement, but it treats only the two angles on the outer circumference of the inclined leveling instrument, leaving the angles on the inner semicircle unexplained. Mei's method was the easier case; Wang devised a new technique by measuring angles without relying on the outer circumference, thereby completing the trigonometric measurement method. Stacked-summation problems already had methods for flat triangular, solid triangular, and pyramidal piles, but only up to the third power; Wang extended these so that pyramidal piles of the third power and higher could all have their volumes derived from the root term. He also developed methods for successive combinations, supplying what the ancient Nine Chapters had left incomplete.

又糾正梅文穆公句股知積術，及指識天元一，正、負開方之可知、不可知。其糾正句股知積術也，文穆赤水遺珍稱：「有句股積及股弦和較求句股，向無其術，苦思力索，立法四條。」其門人丁維烈又造減縱翻積開三乘方法，文穆許之。萊謂：「句股形等積、等弦和，帶縱立方形等基、等高闊和，皆有兩形互易。如句二十，股二十一，弦二十九，句弦和四十九，句股積二百一十。若句十二，股三十五，弦三十七，句弦積亦四十九，句股積亦二百一十。設問者暗執一形，則對者交盲兩數。梅、丁諸公法成而不可用，蓋兩句弦較，與一句弦和，恆為連比例之三率。其兩句弦較，即首、末二率；兩較減一和之餘，即中率；而句弦和必為三率𠊧。遂創立有兩積相等、兩句弦和相等、求兩句股形之法。以四倍句股積自乘，句弦和除之，為帶縱長立方積。以句弦和為縱，開得數為兩句弦較之中率，自乘為帶縱平方積。又以中率與句弦和相減為長闊和，求得長闊兩根為兩句股較，用求兩句股形各數。又同積之邊，彼此可互，三次之乘，先後可通，故四倍句股積自乘，即兩形之倍句相乘為底，兩形之股相乘為高，即猶以中末乘首。中化為中率，再乘為立方三率，𠊧為帶縱。由是推得立方形兩高數恆為首末二率，高闊和恆為三率，𠊧數與等積、等弦和之兩弦較及弦和絲毫無異。如高九闊十，高闊和十九，立方積九百。若高四闊十五，高闊和亦十九，立方積亦九百，其數莫不由兩形相引而出。故其法即命積為帶縱長立方積，以高闊和為所帶之縱。用帶縱長立方法開得本方根，為兩形高數之中率。與高闊和相減，餘為帶縱之平方長闊和。中率自乘，為帶縱平方積。用帶縱平方長闊和法開之，得長闊一根，為兩形之兩高數。兩高與和相減，為兩闊數。」

He also corrected Mei Wending's method for determining right-triangle areas from given products, and clarified the celestial-element method and which positive and negative roots are determinable and which are not. In correcting the right-triangle area method, Mei Wending wrote in his Red Water Legacy Treasures: "Given the product of leg and base together with the sum or difference of base and hypotenuse, to find the leg and base—no method formerly existed; after long effort I established four rules." His disciple Ding Weilie further devised a method of reducing the vertical term, reversing the accumulated product, and extracting the cubic root, which Mei approved. Wang Lai observed: "Right triangles of equal area and equal sum of leg and hypotenuse, and rectangular solids with attached vertical dimension of equal base area and equal sum of height and breadth—each admits two distinct figures that can be interchanged. For example: leg 20, base 21, hypotenuse 29, sum of leg and hypotenuse 49, product of leg and base 210. Yet if leg 12, base 35, hypotenuse 37, the sum of leg and hypotenuse is likewise 49 and the product of leg and base likewise 210. If the problem-setter secretly fixes one triangle, the solver is left blind to both numbers in the paired figure. The methods of Mei, Ding, and their followers were formally complete yet practically unusable, because the two leg-hypotenuse differences together with one leg-hypotenuse sum always form the three terms of a continued proportion. The two leg-hypotenuse differences are the first and last terms; the remainder when the two differences are diminished by one sum is the middle term; and the sum of leg and hypotenuse must be the product of the three proportional terms. He therefore devised a method for finding both right triangles when two products are equal and two sums of leg and hypotenuse are equal. Square four times the leg-base product and divide by the sum of leg and hypotenuse to obtain the attached-vertical long cubic volume. Take the sum of leg and hypotenuse as the attached vertical dimension; the root extracted is the middle term of the two leg-hypotenuse differences; squaring it gives the attached-vertical square volume. Subtract the middle term from the sum of leg and hypotenuse to obtain the sum of length and breadth; extract the two roots as the two leg-base differences, and from these derive all dimensions of both triangles. Moreover, sides yielding equal area can be interchanged, and cubic products can be linked in sequence; thus squaring four times the leg-base product is equivalent to multiplying the doubled legs of both figures as base and their bases as height—analogous to multiplying the first term by the middle and last. The middle term becomes the middle rate; multiplied again it yields the cubic three-term product, which becomes the attached vertical dimension. From this he inferred that in the rectangular solid the two heights are always the first and last proportional terms, the sum of height and breadth always the product of the three terms—and this product differs not at all from the two hypotenuse differences and hypotenuse sum in the equal-area, equal-sum right-triangle problem. For example: height 9, breadth 10, sum of height and breadth 19, cubic volume 900. If height 4 and breadth 15, the sum of height and breadth is likewise 19 and the cubic volume likewise 900—all values arising from the mutual relation of the two figures. Accordingly, the method designates the volume as the attached-vertical long cubic volume and the sum of height and breadth as the attached vertical dimension. Apply the attached-vertical long solid method to extract the cube root, which is the middle term of the two heights. Subtract this from the sum of height and breadth; the remainder is the attached-vertical square sum of length and breadth. Squaring the middle term gives the attached-vertical square volume. Apply the attached-vertical square method with the sum of length and breadth to extract one root, yielding the two heights of the paired figures. Subtract the two heights from the sum to obtain the two breadths."

其指識正、負開方也，「元李冶傳洞淵九容術，撰測圓海鏡、益古演段，以明天元如積相消，其究必用正、負開方，互詳於宋秦九韶數學九章。梅文穆公雖指天元一為西人借根方所由來，而正、負開方則未有闡明者。元和李秀才銳特為讎校，謂少廣一章，得此始貫於一。好古之士，翕然相從。萊獨推其有可知、有不可知。如測圓海鏡邊股第五問'圜田求徑二百四十步與五百七十六步共數'，而李仁卿專以二百四十為答。數學九章田域第二題'尖田求積二百四十步與八百四十步共數'，而秦道古專以八百四十為答。乃自二乘方以下，縷析推之，得九十五條。凡幾根數為帶縱長闊較則可知，為帶縱長闊和則不可知。又推得幾真數少，幾根數又多，幾平方與一立方積等多少雜糅，和較莫定。立法以審之，以幾平方數用幾立方數除之，得數乘幾根數，以較幾真數。若少於真數，則以幾平方為高闊較，是為可知。若多於真數，則或幾平方為通分法，三母總數、幾真數為三母維乘之共數，幾根數為通分之共子，如二、如六、如十二。設真數一百四十四，少二百八，根數多二十，平方積與一立方積相等，則三數皆同，是為不可知。」

In clarifying positive and negative root extraction he wrote: "Li Ye of the Yuan transmitted the Nine Containers method of Dongyuan and authored Measuring the Circle Sea Mirror and Augmenting the Ancient Segments to explain celestial-element elimination of like products; the method ultimately requires positive and negative root extraction, as Qin Jiushao also detailed in his Mathematical Treatise in Nine Chapters. Mei Wending traced the celestial element to the Western borrowed-root method, but no one had yet explained positive and negative root extraction. Li Rui of Yuanhe collated the texts and declared that only with this clarification did the Lesser Extension chapter become fully intelligible. Scholars devoted to ancient learning rallied to his view. Wang Lai alone went further, distinguishing which roots are determinable and which are not. For instance, in the fifth leg-and-base problem of Measuring the Circle Sea Mirror—"to find the diameter of a circular field: 240 bu combined with 576 bu"—Li Ye answered with 240 alone. In the second field-domain problem of Qin Jiushao's Mathematical Treatise in Nine Chapters—"to find the area of a pointed field: 240 bu combined with 840 bu"—Qin Jiushao answered with 840 alone. He then worked it out strand by strand from squared terms downward and arrived at ninety-five rules. Whenever a root term represents the attached-vertical length-breadth difference, the root is determinable; when it represents the attached-vertical length-breadth sum, it is not. He further found cases in which the constant term falls short, root terms abound, and square terms equal a cubic term in various combinations—so that neither sum nor difference can be fixed. He set up a test: divide one square term by one cubic term, multiply the result by one root term, and compare the product with the constant term. If the product falls short of the constant term, treating a square term as the height-breadth difference yields a determinable root. If it exceeds the constant term, a square term may serve as the common-denominator method: the sum of three denominators and the constant term form the common product of cross-multiplied denominators, while the root term becomes the common numerator—such as 2, 6, or 12. Suppose the constant is 144, short by 208, with twenty excess root terms and a square product equal to a single cubic product—then all three values coincide, and the root is indeterminate."

蓋以一答為可知，不止一答為不可知。故李秀才銳跋其書，括為三例以證明之。謂：「隅實同名者不可知；隅實異名，而從廉正負不雜者可知；隅實異名，而從廉正負相雜，其從翻而與隅同名者可知，否則不可知。隅實異名，即帶縱之長闊較也，較僅一答；隅實同名，即帶縱之長闊和也，和則不止一答。」銳以隅實同名、異名，明一答與不止一答；萊以長闊、和較，明可知、不可知，其義一也。著有衡齋算學七冊，考定通藝錄磬氏倨句解一冊。

In essence, a single answer means the root is determinable; more than one answer means it is not. Accordingly, Li Rui wrote a postface to Wang's book, condensing the argument into three rules by way of proof. He stated: "When the constant and leading coefficient share the same sign, the root is indeterminate; when constant and leading coefficient differ in sign and the intermediate coefficients do not mix positive and negative values, it is determinable; when constant and leading coefficient differ in sign and intermediate coefficients mix positive and negative, if the intermediate term is reversed and matches the constant in sign it is determinable; otherwise it is not. Constant and leading coefficient of opposite sign correspond to the attached-vertical length-breadth difference, which admits only one answer; constant and leading coefficient of the same sign correspond to the attached-vertical length-breadth sum, which admits more than one answer." Li Rui used same-sign and opposite-sign constant and leading coefficient to distinguish one answer from many; Wang Lai used length-breadth sum and difference to distinguish determinable from indeterminate roots—the underlying meaning is the same. He authored Hengzhai Mathematical Treatises in seven fascicles and Collated Explanations of the Ju Angle of the Bell-Clapper Family from the Tongyi Records in one fascicle.

=陳杰=陳杰，字靜弇，烏程諸生。考取天文生，任欽天監博士，供職時憲科兼天文科，司測量。累官國子監算學助教。道光十九年，謝病歸，卒於家。生平邃於算學，尤神明於比例之用。初著輯古算經細草一卷，後十餘年，又為之指畫形象，成圖解三卷；又博採訓詁，考正其傳寫之舛譌，稽合各本之同異，別成音義一卷。

Chen Jie, styled Jingyan, was a licentiate of Wucheng. He passed the examination for astronomical students, served as a Doctor at the Imperial Astronomical Bureau, and while on duty held posts in both the Calendar Section and the Astronomy Section, where he supervised measurements. He eventually rose to the post of Assistant Instructor in Mathematics at the Imperial Academy. In 1839 he resigned on grounds of illness, returned home, and died there. He was deeply versed in mathematics throughout his life and was especially masterful in the use of proportion. He first wrote Detailed Working for the Collected Ancient Mathematical Classic in one fascicle; more than ten years later he added illustrative figures, producing three fascicles of diagrammatic explanations; He also gathered philological glosses broadly, corrected copying and transmission errors, collated variant readings among editions, and separately produced one fascicle on pronunciation and meaning.

其自述比例言有曰：「比例之法，昉自九章，傳由西域，在古法曰異乘同除，在西法曰比例等。假如甲有錢四百，易米二斗，問乙有錢六百，易米幾何？答曰三斗。法以乙錢為實，甲米乘之，得數，甲錢除之，即得。錢與米異名相乘，與錢同名相除，故謂之異乘同除，此古法也。以甲錢比甲米，若乙錢與乙米。凡言以者一率，言比者二率，言若者三率，言與者四率。二三相乘，一率除之，得四率，此西法也。古法元、明時中土幾以失傳，不知何時流入西域。明神宗時，西人利瑪竇來中國，出其所著算書，中人矜為創獲，其實所用皆古法，但異其名色耳。茲以西人名色解王氏，固取其平近，亦以名中、西之合轍也。」

In his own discussion of proportion he writes: "The method of proportion originated in the Nine Chapters and was transmitted from the Western Regions. In the ancient Chinese method it is called 'multiply unlike terms and divide like terms'; in the Western method it is called proportional equality. Suppose A has 400 cash and buys 2 dou of rice; if B has 600 cash, how much rice can he buy? Answer: 3 dou. The method is to take B's cash as the dividend, multiply by A's rice, and divide by A's cash to obtain the answer. Cash and rice, being unlike in kind, are multiplied together, while like kinds are divided—hence the name 'multiply unlike and divide like.' This is the ancient method. As A's cash is to A's rice, so B's cash is to B's rice. In such a statement, 'as' marks the first proportional term, 'compared to' the second, 'so' the third, and 'with' the fourth. Multiply the second and third terms and divide by the first to obtain the fourth—this is the Western method. The ancient method was nearly lost in China by the Yuan and Ming periods; when it reached the Western world is unknown. During the reign of the Wanli Emperor, Matteo Ricci came to China and published his mathematical works; Chinese scholars hailed this as a fresh discovery, but in fact he was using the ancient method under different names. Here I explain Wang's method in Western terms—partly for clarity, and partly to show that Chinese and Western approaches follow the same path."

又有論曰：「二十一史律志無不用比例者，他如九章、緝古、十種算書，多用比例，無如古人總不言比例。如緝古第二問，求均給積尺，欲以本體求又一形之體，忽取兩面冪之數，一用以乘，一用以除，而得數。又第九問求員囤，第十問求員窖，忽以周徑乘除，即如方亭法求之，諸數悉得。走作圖解，審諦久之，而始知為比例，乃明言比例以揭之。嗣是而閱古算書者，罔弗比例矣。」

He also wrote a treatise stating: "In the calendrical treatises of the Twenty-One Histories, none fails to employ proportion; and works such as the Nine Chapters, the Reconstituted Ancient Mathematical Classic, and the Ten Mathematical Books use it constantly—yet the ancients never named it proportion. In the second problem of the Reconstituted Ancient Mathematical Classic, for example, when seeking equal distribution in accumulated chi and trying to find the volume of another solid from the original body, one suddenly takes the squared measures of two faces, multiplies by one and divides by the other, and obtains the result. Again, in the ninth problem for a circular granary and the tenth for a circular cellar, one suddenly multiplies and divides by circumference and diameter—just as in the square pavilion method—and all values follow. While preparing diagrammatic explanations, I studied the matter closely for a long time before recognizing it as proportion; only then did I state it explicitly as such. Since then, readers of ancient mathematical texts have found proportion everywhere."

又自道光以來，嘗親在觀像台督率值班天文生頻年實測黃、赤大距為二十三度二十七分，未經奏明，故當時未敢用。迨甲辰歲修儀象考成續編，監臣即取此數上之，而欽定頒行焉。

Moreover, from the Daoguang era onward he personally supervised the Observatory astronomers on duty in repeated actual measurements, finding the obliquity of the ecliptic to be 23°27′; because this had not yet been reported to the throne, he did not dare apply it at the time. When the Continuation of the Imperially Commissioned Instruments was compiled in the jiachen year, the bureau director submitted this value, and it was approved and promulgated by imperial decree.

晚年所譔為算法大成，上編十卷，首加、減、乘、除，次開方、句股，次比例、八線，次對數，次平三角、弧三角。門分類別，皆先列舊法，而以新法附之，圖說理解，不憚反覆詳明，

In his later years he authored Comprehensive Mathematical Methods. The upper compilation runs to ten fascicles, beginning with the four operations, then root extraction and right triangles, then proportion and the eight trigonometric lines, then logarithms, then plane and spherical trigonometry. Topics are arranged by category; in each he presents the old method first and appends the new, with diagrams and explanations repeated in painstaking detail,

專為引誘初學設也。下編十卷，則有目無書。其言曰：「算法之用多端，第一至要為治歷，故下編言在官之事，先治歷，次出師，次工程錢糧，次戶口鹽司，次堆積丈量；儒者則考據經傳，下及商賈庶民，則貲本營運，市廛交易，持家日用，凡事無鉅細，各設題為問答，以明算法之用，蓋如此之廣云。」下編似未成。其門人丁兆慶、張福僖均以算名。

all designed solely to guide beginners. The lower compilation of ten fascicles had a table of contents but no completed text. He wrote: "Computational methods have many uses, but calendrical work comes first and is most essential. The lower compilation was therefore to treat official affairs in this order: calendar reform, military campaigns, public works and revenue, household registers and the salt administration, and stockpiling and land survey; for scholars, textual investigation of the classics; for merchants and common people, capital and business operations, market transactions, and household daily needs—for every matter, great or small, he would set problems in question-and-answer form to show how widely computational methods apply." The lower compilation appears never to have been finished. His disciples Ding Zhaoqing and Zhang Fuxi were both celebrated for mathematics.

兆慶，字寶書，歸安人。沉潛好學，為項學正兩邊夾角迳求對角新法圖說，謂其講解明晰，戛戛獨造。

Zhaoqing, styled Baoshu, was a native of Gui'an. Deeply devoted to study, he wrote Diagrammatic Explanations of Academician Xiang's New Method for Finding the Opposite Angle through the Diameter of an Included Angle Between Two Sides; Chen praised its exposition as lucid and wholly original.

福僖，字南坪，烏程諸生。精究小輪之理，著有慧星考略。

Fuxi, styled Nanping, was a licentiate of Wucheng. He thoroughly investigated the theory of the minor epicycle and authored Brief Study of Comets.

=時曰淳=時曰淳，字清甫，嘉定人。精算術。發明古人術意，無不入微。咸豐末，與長沙丁取忠同客胡林翼幕府，每與商榷數理，見丁氏數學拾遺之百雞術，謂與二色方程暗合。因為廣衍，立二十八題，以「舊學商量加邃密、新知培養轉深沉」十四字識其上下，為十四耦。諸題皆借方程為本術，並述大衍求一術以博其趣，作百雞術衍二卷。

Shi Yuechun, styled Qingfu, was a native of Jiading. He excelled in mathematics. In recovering the intent of ancient methods, none eluded his subtle insight. At the end of the Xianfeng era he joined Ding Quzhong of Changsha as a guest in Hu Linyi's secretariat, where they often discussed mathematical reasoning. Reading Ding's Hundred-Fowl Method in Mathematical Relics from Picking up Oversights, he observed that it tacitly matched the two-unknown simultaneous equation method. He thereupon expanded the treatment, setting twenty-eight problems labeled above and below with the fourteen characters "Old learning discussed grows more refined; new knowledge cultivated turns deeper," forming fourteen paired sets. Every problem takes simultaneous equations as its fundamental method, while the Dayan remainder method is also presented to broaden the treatment; he produced Extension of the Hundred-Fowl Method in two fascicles.

自序略曰：「張丘建算經雞翁雞母題問，甄、李兩註及劉孝孫草，皆未達術意，不可通。近焦理堂所釋尤誤。讀吾友丁君果臣數學拾遺，設術與二色方程暗合，乃通法也。駱氏藝遊錄用大衍求一術，以大小較求中數，取徑頗巧，然遇較除共較實適盡者，則不可求。方程術則遇法除實得中數，不盡者以分母與減率相求而齊同之，無不可得。駱氏殆未知有方程本術耳。夫題祗本經一術，算理之微妙，不如孫子不知數一問，而術文各隱秘。彼則但舉用數，此亦僅著加減三率，於前半段取數之法皆闕如。豈古人不傳之秘，必待學者深思而自得乎？孫子求一術，至宋秦道古發之，獨是題襲謬傳訛，無借方程以問途者。曰淳蓄疑既久，今年春與果臣連榻鄂城，復一商榷，別後數月乃通之。怡然渙然，了無滯凝，亦窮愁中一快事也。因衍方程術為數學拾遺補，求負數法及加減率求答數法，附述求一術為藝遊錄補。以中小較求大數法，及大中較、大小較互求得中數、小數法，引伸鉤索，溫故知新，庶足以大暢厥旨乎！易翁、母、雛為大、中、小，設數不必以百，而統以百雞命之者，識斯術所自昉也。」

His preface briefly states: "In Zhang Qiujian's Mathematical Classic, the problem of the rooster, hen, and chicks—in the commentaries of Zhen and Li and the working notes of Liu Xiaosun—none grasped the method's intent, and none can be understood. The recent explanation by Jiao Litang is especially wrong. Reading my friend Ding Guochen's Mathematical Relics from Picking up Oversights, I found that the method he set forth tacitly matched the two-unknown equation method—it is the general solution. In Mr. Luo's Records of Artistic Pursuits, the Dayan remainder method finds the middle number from the large-small difference—a clever shortcut, but when the difference divisor and common-difference dividend divide evenly, no solution can be obtained. The simultaneous equation method, when the divisor divides the dividend, yields the middle number directly; when it does not divide evenly, one harmonizes divisor and reduction rate—nothing is unobtainable. Mr. Luo probably did not know that simultaneous equations were the original method. This problem rests on a single method in the original classic; its mathematical subtlety is less than that of Sun Zi's "I do not know the number" problem, yet both texts conceal their methods. That problem merely cites the numbers used; this one records only the three rates of addition and subtraction—the method for obtaining numbers in the first half is entirely absent. Were these secrets deliberately withheld by the ancients, to be grasped only through the student's deep reflection? Sun Zi's remainder method was not elucidated until Qin Jiushao of the Song; this problem alone perpetuated mistaken transmission, with no one showing the way through simultaneous equations. Yuechun had long harbored these doubts; this spring he shared lodgings with Guochen in Echeng and discussed the matter again, but only months after parting did he fully understand it. His doubts dissolved in joy and clarity—a rare pleasure amid poverty and care. He therefore extended the equation method as a supplement to Mathematical Relics from Picking up Oversights, setting forth methods for negative numbers and for finding answers through addition-reduction rates, and appended an account of the remainder method as a supplement to Records of Artistic Pursuits. Methods for finding the large number from the small-middle difference, and for finding middle and small numbers from the large-middle and large-small differences by mutual derivation—extending links and connections, reviewing the old to know the new—may these suffice to make the full intent clear! Substituting rooster, hen, and chick for large, middle, and small, the given numbers need not be one hundred, yet the work is universally titled Extension of the Hundred-Fowl Method to mark its origin."

=李銳=李銳，字尚之，元和諸生。幼開敏，有過人之資。從書塾中檢得算法統宗，心通其義，遂為九章、八線之學。因受經於錢大昕，得中、西異同之奧，於古歷尤深。自三統以迄授時，悉能洞澈本原。

Li Rui, styled Shangzhi, was a licentiate of Yuanhe. From youth he was quick and perceptive, with gifts beyond the ordinary. He found the Comprehensive Source of Computational Methods in his schoolroom, grasped its meaning at once, and went on to study the Nine Chapters and the eight trigonometric lines. Studying the classics under Qian Daxin, he mastered the parallels and divergences between Chinese and Western learning and was especially profound in ancient calendrical science. From the Triple Concordance through the Season-Granting system, he could penetrate the original principles of each.

嘗謂：「三統，世經稱殷術，以元帝初元二年為紀首，是年歲在甲戌。推而上之，一千五百二十歲而歲值甲寅為元首，又上四千五百六十年而歲复甲寅為上元。以此積年，用四分上推，太初元年得至朔同日，而中餘四分日之三，朔餘九百四十分之七百五，故太初術虧四分日之三，去小餘七百五分也。《漢書》載三統而不著太初，其實一月之日，二十九日八十一分日之四十三，是日法、月法與三統同。賈逵稱太初術鬥二十六度三百八十五分，是統法週天又與三統同。蓋四分無異於太初，而太初亦得謂之三統。鄭注召誥，周公居攝五年二月三月，當為一月二月，不云正月者，蓋待治定制禮，乃正言正月故也。江徵君聲、王光祿鳴盛以為據洛誥十二月戊辰逆推之，其說未核。今案鄭君精於步算，此破二月三月為一月二月，以緯候入蔀數，推知上推下驗，一一符合，不僅檢勘一二年間事也。」

He once observed: "The Triple Concordance, commonly called the Yin method, takes the second year of chuyuan in Emperor Yuan's reign as the era start—that year was jiaxu in the sexagenary cycle. Tracing upward, after 1,520 years the year value becomes jiayin as the era head; another 4,560 years upward and the year again reaches jiayin as the superior origin. Using this accumulated count and pushing upward by the Quarter-Remainder system, in the first year of Taichu new and full moons fall on the same day, but the median remainder is three-fourths of a day and the new-moon remainder 705/940ths—so the Taichu method falls short by three-fourths of a day, subtracting 705 from the small remainder. The Book of Han records the Triple Concordance but not Taichu; in fact one month is 29 and 43/81 days—the day divisor and month divisor are the same as in the Triple Concordance. Jia Kui states that the Taichu method gives the lodge at 26°385'—the concordance divisor and circuit of heaven again match the Triple Concordance. Thus the Quarter-Remainder differs not from Taichu, and Taichu may also be called the Triple Concordance. In Zheng Xuan's commentary on the Announcement to Duke Shao, when the Duke of Zhou served as regent, the fifth year's second and third months should read first and second month; he does not say 'first month' because he was awaiting the completion of regulations and ritual before speaking properly of the first month. Jiang Jun Sheng and Wang Guanglu Mingsheng argued by retrocalculating from the new moon on wuchen day in the twelfth month of the Announcement to Luo—their explanation has not been verified. Examining the matter now: Master Zheng was expert in astronomical calculation; reading second and third month as first and second, and using apocryphal calendrical and epoch numerology, one finds that pushing upward and testing downward match one by one—not merely in one or two years of events."

因據詩大明疏，鄭注尚書文王受命，武王伐紂時日皆用殷歷甲寅元，遂從文王得赤雀受命年起，以乾鑿度所載之積年推算，是年入戊午蔀，二十九年歲在戊午，與劉歆所說殷歷周公六年始入戊午蔀不同。歆謂文王受命九年而崩，崩後四年武王克殷，後七年而崩，明年周公攝政元年，較鄭少一年。又載召誥、洛誥俱攝政七年事，其年二月乙亥朔，三月甲辰朔，十二月戊辰朔，並與鄭不合。乃以推算各年及一月二月，排比干支，分次上下，著召誥日名考，此融會古歷以發明經術者也。

Accordingly, drawing on Kong Yingda's commentary on the Greater Brightness ode and Zheng Xuan's notes on King Wen's receipt of the mandate and King Wu's campaign against Zhou—both using the Yin calendar with jiayin as origin—he calculated from the year King Wen received the mandate when the red sparrow appeared, using the accumulated years in the Dry Channel Apocrypha: that year entered the wuwu obscuration, and in the twenty-ninth year the year was wuwu—differing from Liu Xin's claim that the Yin calendar under the Duke of Zhou entered the wuwu obscuration only in his sixth year. Liu Xin held that King Wen received the mandate and died after nine years; four years after his death King Wu conquered Yin; seven years later King Wu died, and the next year was the first year of the Duke of Zhou's regency—one year less than Zheng's reckoning. He also records that both the Announcement to Duke Shao and the Announcement to Luo belong to the seventh year of the regency, with new moons on yihai in the second month, jiachen in the third, and wuchen in the twelfth—all at odds with Zheng. He therefore calculated each year and the first and second months, arranged the stems and branches in sequence above and below, and composed A Study of the Day Names in the Announcement to Duke Shao—an effort to harmonize ancient calendars and illuminate the methods of the classics.

當是時，大昕為當代通儒第一，生平未嘗親許人，獨於銳則以為勝己。大昕嘗以太乙統宗寶鑑求積年術日法一萬五百歲，實三百八十三萬五千四十八分二十五秒為疑。銳據宋同州王湜易學，謂每年於三百六十五日二千四百四十分之外，有終於五分者，有終於六分者，有終於五六分之間者。終於五分者，五代王朴欽天歷是也，以七千二百為日法。終於六分者，近年萬分歷是也，以一萬分為日法。終於五六分之間者，景祐曆法載於太乙遁甲中是也，以一萬五百分為日法，此暗用授時法也。試以日法為一率，歲實為二率，授時日法一萬為三率，推四率，得三百六十五萬二千四百二十五分，即授時之歲實也。探本窮源，一言破的。

At that time Qian Daxin stood first among the learned scholars of the age. In his life he had never personally acknowledged anyone as his equal, yet of Li Rui alone he declared that Rui surpassed him. Daxin once questioned the day divisor of 10,500 used in the accumulated-years method of the Taiyi Tongzong Baojian, and the tropical year length given as 3,835,048 parts and 25 seconds. Rui, drawing on the Yixue of Wang Yan of Tongzhou in the Song, argued that beyond 365 days and 2,440 parts each year, some calendars end in five parts, some in six, and some between five and six. Those ending in five parts are Wang Pu's Qintian calendar of the Five Dynasties, which uses 7,200 as the day divisor. Those ending in six parts are the recent Ten-thousand-parts calendar, which uses 10,000 parts as the day divisor. Those ending between five and six parts are the Jingyou calendar method recorded in the Taiyi Dunjia, which uses 15,000 parts as the day divisor—covertly applying the Shoushi method. Taking the day divisor as the first ratio, the tropical year as the second, and the Shoushi day divisor of 10,000 as the third, one derives the fourth ratio and obtains 3,652,425 parts—the Shoushi tropical year. Tracing matters to their root and source, a single remark hit the mark.

近世曆算之學，首推吳江王氏錫闡、宣城梅氏文鼎，嗣則休寧戴氏震亦號名家。王氏謂土盤曆元在唐武德年間，非開皇己未；梅氏謂回回曆實用洪武甲子為元，而託之於開皇己未。其算宮分，雖以開皇己未為元，其查立成之根，則在己未元後二十四年，二說並同。

In recent calendrical and computational learning, the foremost were Wang Xichan of Wujiang and Mei Wending of Xuancheng; after them Dai Zhen of Xiuning was also counted among the masters. Wang held that the Tupan calendar's epoch fell in the Wude reign of Tang, not in Kaihuang jihai; Mei held that the Islamic calendar in actual use took Hongwu jiazi as its epoch, but ascribed it to Kaihuang jihai. In computing the solar year, although Kaihuang jihai was taken as the epoch, the root for consulting the ready-made tables lay twenty-four years after the jihai origin—the two views agreed on this point.

戴氏謂回回曆百二十八年閏三十一日，是每歲三百六十五日之外，又餘百二十八分日之三十一也。以萬萬乘三十一，滿百二十八而一，得二千四百二十一萬八千七百五十，地谷所定歲實三百六十五日二十三刻三分四十五秒，通分內子以萬萬乘之，滿日法而一，亦得二千四百二十一萬八千七百五十，與梅氏疑問所云合。是三家所論，未嘗不確知灼見，然均未得其詳。銳據明史曆志、回回本術，參以近年瞻禮單，精加考核，謂回回曆有太陽年，彼中謂為宮分；有太陰年，彼中謂為月分。宮分有宮分之元，則開皇己未是也；月分有月分之元，則唐武德壬午是也。自開皇己未至洪武甲子，積宮分年七百八十六，自武德壬午至洪武甲子，積月分年亦七百八十六，其惑人者即此兩積年相等耳，因著回回曆元考。有求宮分白羊一日入月分截元後積年月日法，以為不明乎此，雖有立成，不能入算也。稿佚未刊。

Dai held that in the Islamic calendar 128 years intercalate 31 days—that is, beyond 365 days each year there remains an additional 31/128 of a day. Multiplying 31 by 10,000 and dividing by 128 yields 24,218,750; Tycho Brahe's tropical year of 365 days, 23 quarters, 3 minutes, and 45 seconds, converted to a common denominator and multiplied by 10,000, then divided by the day divisor, likewise yields 24,218,750—matching what Mei's inquiry had stated. Thus the three schools' discussions were indeed sound and penetrating, yet none had grasped the matter in full. Rui, drawing on the Calendar Treatise in the History of Ming and the original Islamic methods, and comparing them with recent prayer almanacs, examined the matter closely and held that the Islamic calendar has a solar year, which in that tradition is called the gongfen; and also a lunar year, which they call the yuefen. The solar year has its own epoch, which is Kaihuang jihai; the lunar year has its own epoch, which is Wude renwu of Tang. From Kaihuang jihai to Hongwu jiazi the accumulated solar years total 786; from Wude renwu to Hongwu jiazi the accumulated lunar years also total 786—the very thing that misled people was that these two accumulated counts are equal—whereupon he wrote An Inquiry into the Epoch of the Islamic Calendar. He also worked out a method for finding the accumulated years, months, and days from the lunar epoch after the day Aries enters the solar year, arguing that without understanding this, even ready-made tables cannot be used in calculation. The manuscript was lost and never published.

梅氏未見古九章，其所著方程論，率皆以臆創補，然又囿於西學，致悖直除之旨。銳尋究古義，探索本根，變通簡捷，以舊術列於前，別立新術附於後，著方程新術草，以期古法共明於世。古無天元一術，其始見於元李冶測圓海鏡、益古演段二書，元郭守敬用之，以造授時曆草，而明學士顧應祥不解其旨，妄刪細草，遂致是法失傳。自梅文穆悟其即西法之借根方，於是李書乃得鄭重於世。其有原術不通，別設新術數則，更於梅說外辨得天元之相消，有減無加，與借根方之兩邊加減法少有不同。

Mei had never seen the ancient Nine Chapters; in his Treatise on Equations he generally filled gaps by conjecture, yet was also confined by Western learning, causing him to violate the principle of direct elimination. Rui investigated the ancient meaning, explored the root principle, and adapted the method to make it simpler; placing the old procedure first and appending a new one after it, he wrote Draft of a New Method for Equations, hoping that the ancient method might be made clear to the world. Anciently there was no tian yuan yi method; it first appears in Li Ye's Sea Mirror of Circle Measurement and Augmented Ancient Segments of the Yuan. Guo Shoujing used it to draft the Shoushi calendar, but the Ming scholar Gu Yingxiang did not understand its purport and rashly deleted the detailed working, so the method was lost. When Mei Wending realized that it was the same as the Western root-borrowing method, Li Ye's books then came to be held in high esteem. Where the original method did not work, he set out new methods; further, beyond Mei's view he distinguished the cancellation of tian yuan terms—subtraction without addition—which differs slightly from the Western method of adding and subtracting on both sides.

且不滿顧氏所著之句股、弧矢兩算術，謂：「弧矢肇於九章方田，北宋沈括以兩矢冪求弧背，元李冶用三乘方取矢度，引伸觸類，厥法綦詳。顧氏如積未明，開方徒衍，不亦傎乎？」爰取弧矢十三術，入以天元，著弧矢算術細草。並仿演段例，括句股和較六十餘術，著句股算術細草，以導習天元者之先路。

He was also dissatisfied with the right-triangle and arc-sagitta methods in Gu's works, writing: "Arc-sagitta methods originated in the square fields chapter of the Nine Chapters; in the Northern Song Shen Kuo used the product of two sagittas to find the arc back, and in the Yuan Li Ye used the cube to obtain the sagitta length—extending and analogizing, the method is exceedingly detailed. Gu did not understand the product-as-area method and merely carried out root extraction—is this not perverse?" He then took the thirteen arc-sagitta methods, applied tian yuan to them, and wrote Detailed Working for Arc-Sagitta Calculation. Following the example of Augmented Segments, he collected more than sixty methods of right-triangle sum and difference and wrote Detailed Working for Right-Triangle Calculation, to pave the way for those learning tian yuan.

又從同里顧千里處得秦九韶數學九章，見其亦有天元一之名，而其術則置奇於右上，定於右下，立天元一於左上。先以右上除右下，所得商數與左上相生，入於左下。依次上下相生，至右上末後奇一而止，乃驗左上所得以為乘率。與李書立天元一於太極上，如積求之，得寄左數與同數相消之法不同。因知秦書乃大衍求一中之又一天元，秦與李雖同時，而宋元則南北隔絕，兩家之術，無緣流通，蓋各有所授也。

He also obtained Qin Jiushao's Mathematical Treatise in Nine Chapters from Gu Qianli of the same district, and saw that it too had the name tian yuan yi, but its method placed the odd term at the upper right, fixed the lower right, and set tian yuan yi at the upper left. First divide the lower right by the upper right; multiply the quotient by the upper left and enter the product into the lower left. Multiply upward and downward in sequence until the final odd one at the upper right is reached, then verify the value obtained at the upper left as the multiplier. This differs from Li Ye's method of setting tian yuan yi on the Supreme Ultimate, seeking the product-as-area, obtaining the number held on the left and canceling like terms. Thus he knew Qin's book represented yet another tian yuan within the Dayan remainder method; although Qin and Li were contemporaries, the Song and Yuan were separated north and south, and the two schools' methods had no channel of exchange—they must each have had their own transmission.

銳嘗謂：「四時成歲，首載虞書，五紀明歷，見於洪範。歷學誠致治之要，為政之本。乃通典、通考置而不錄，邢云路雖撰古今律歷考，然徒援經史，以侈卷帙之多。梅氏祗有欲撰曆法通考之議，卒未成書。因更網羅諸史，由黃帝、顓頊、夏、殷、週、魯六歷，下逮元、明數十餘家，一一闡明義蘊，存者表而章之，缺者考而訂之，著為司天通志，俾讀史者啟其扃，治歷者益其智。」惜僅成四分、三統、乾象、奉天、佔天五術注而已。餘與開方說皆屬稿未全。

Rui once said: "The four seasons make a year, first recorded in the Book of Yu; the five reckonings that clarify the calendar appear in the Great Plan. Calendrical learning is truly essential to good government and the foundation of administration. Yet the Comprehensive Institutions and Comprehensive Examination omit it; although Xing Yunlu wrote An Examination of Ancient and Modern Pitch-Pipes and Calendars, he merely cited classics and histories to inflate the bulk of his volumes. Mei only proposed writing a Comprehensive Examination of Calendar Methods, but in the end never completed the book. He therefore gathered materials from the various histories, from the six calendars of Huangdi, Zhuanxu, Xia, Yin, Zhou, and Lu down to several dozen schools of the Yuan and Ming, clarifying the meaning of each in turn; where records survived he displayed and explicated them, where they were missing he investigated and corrected them, and composed Comprehensive Calendar Records, so that readers of history might open its gate and calendar-makers might increase their knowledge." Alas, he completed only commentaries on the five methods of Quarter-Remainder, Triple Concordance, Supernal Appearance, Fengtian, and Zhantian. The rest, together with the Treatise on Extraction of Roots, remained incomplete drafts.

開方說三卷，銳讀秦氏書，見其於超步、退商、正負、加減、借一為隅諸法，頗得古九章少廣之遺，較梅氏少廣拾遺之無方廉者，不可以道裡計。蓋梅氏本於同文算指、西鏡錄二書，究出自西法，初不知立方以上無不帶從之方。銳因秦法推廣詳明，以著其說。甫及上、中二捲而卒，年四十有五。其下卷則弟子黎應南續成之。

Treatise on Extraction of Roots in three fascicles: reading Qin's book, Rui saw that in the methods of overshooting steps, retreating the quotient, positive and negative terms, addition and subtraction, and borrowing one for the corner, he had largely recovered the legacy of the ancient Nine Chapters' lesser breadth—compared with Mei's Lesser Breadth Supplement, which lacked side and corner terms, the difference cannot be measured in li. For Mei's work was based on the Tongwen Computational Guide and Record of the Western Mirror, which ultimately derive from Western methods and at first did not know that above the cube there is always an accompanying square term. Rui extended and clarified Qin's method in detail to set forth his exposition. He had just completed the first and second fascicles when he died, at the age of forty-five. The third fascicle was then completed by his disciple Li Yingnan.

應南，字見山，號鬥一，廣東順德人。嘉慶戊寅順天經魁，以書館議敘，選浙江麗水縣知縣，調平陽縣知縣。海疆俸滿，加六品銜，卒於官。

Li Yingnan, styled Jianshan and known by the sobriquet Douyi, was a native of Shunde in Guangdong. In 1818 he ranked first in the Shuntian provincial examination; by memorial recommendation from the Hanlin Academy he was selected as magistrate of Lishui County in Zhejiang, then transferred to magistrate of Pingyang County. When his coastal-service salary term was complete, he received the sixth-rank insignia and died in office.

=駱騰鳳=駱騰鳳，字鳴岡，山陽人。嘉慶六年舉人，道光六年，大挑一等，用知縣。以母老不原仕，改授舒城縣訓導。未一年，告養歸，教授里中，學徒甚眾。二十二年八月，卒於家，年七十有二。性敏銳，好讀書，尤精疇人術。在都中從鍾祥李潢學，研精覃思，寒暑靡間。

Luo Tengfeng, styled Minggang, was a native of Shanyang. He received his provincial degree in 1801; in 1826 he was selected in the first class of the grand selection and appointed magistrate. Because his mother was old and he did not wish to serve, he was reassigned as director of studies of Shucheng County. In less than a year he requested leave to care for her and returned home, teaching in his district with many disciples. In the eighth month of the twenty-second year he died at home, aged seventy-two. By nature quick and sharp, he loved reading and was especially skilled in the methods of calendar-makers. In the capital he studied under Li Huang of Zhongxiang, investigating with deep refinement through summer and winter alike.

著開方釋例四卷，自序略謂：「天元一術，見宋秦九韶大衍數中，不言創於何人。元李冶測圓海鏡、益古演段二書，亦用此例。冶稱其術出於洞淵九容，今不可詳所自矣。是書自平方以至多乘，悉用一術，即芻童、羨餘諸形，亦可握觚而得，洵算術之秘鑰也。西法借根方實原於此，乃以多少代正負，徒欲掩其襲取之跡。不知正負以別異同，多少以分盈朒，毫釐千里，必有能辨之者。」

He wrote Explanatory Examples of Root Extraction in four fascicles; in his preface he briefly wrote: "The tian yuan yi method appears in Qin Jiushao's Dayan mathematics of the Song, without stating by whom it was created. Li Ye's two books Sea Mirror of Circle Measurement and Augmented Ancient Segments also use this pattern. Li Ye said his method came from the Nine Containers of the Abyss; today one can no longer trace its origin in detail. This book, from square roots up through higher powers, uses a single method throughout; even for the haystack, surplus, and other figures one can grasp the brush and obtain the result—it is truly the secret key of calculation. The Western root-borrowing method in fact derives from this, yet substitutes greater and lesser for positive and negative, merely wishing to conceal the traces of borrowing. They do not know that positive and negative distinguish sameness and difference, while greater and lesser distinguish surplus and deficit—a hair's breadth or a thousand li, and there must be those who can tell the difference."

又著遊藝錄二卷，自識云：「餘於正、負開方之例，既為釋例以明其法矣。至於衰分方程、句股等法，以及九章所未載，與夫古今算術之未能該洽者，輒為溯其源，正其誤。不敢掠前哲之美以為名，亦不為黯黮之詞以欺世也。隨所見而識之，彙為一編。」遺稿凡十餘萬言，即今傳本也。

He also wrote Record of Recreational Calculation in two fascicles; in his own note he wrote: "Having already written Explanatory Examples to clarify the method of positive and negative root extraction, as for methods of proportional division, equations, right triangles, and what the Nine Chapters did not record, and whatever ancient and modern calculation has not fully covered, I trace the source and correct the error wherever I encounter it. I dare not seize the merit of earlier sages for a name, nor write obscure words to deceive the world. I record whatever I see and compile it into one collection." The surviving draft totals more than a hundred thousand characters—the transmitted text of today.

南匯張文虎嘗與青浦熊戶部其光書論之曰：「承示駱司訓算書二種，讀竟奉繳。李四香開方說，詳於超步、商除、翻積、益積諸例，而不言立法之根，令初學者茫不解其所謂。駱氏於諸乘方、方廉、和較、加減之理，皆質言之，而推求各元進退、定商諸術，尤足補李書所未備，誠學開方者之金鎖匙。汪孝嬰創設兩句股同積同句股和一問，以兩句弦較中率轉求兩句弦較，立術迂迴。駱氏以正、負開方徑求得兩句，頗為簡易。衡齋亦當首肯也。」其為人所推服如此。

Zhang Wenhu of Nanhui once wrote to Xiong Qibu of Qingpu discussing them, saying: "You showed me the two mathematical works of Instructor Luo; having read them through I return them with thanks. Li Sixiang's Treatise on Extraction of Roots is detailed on the examples of overshooting steps, quotient division, inverted product, and augmented product, but does not state the root of the method's establishment, leaving beginners quite unable to understand what is meant. Luo plainly states the principles of the various powers, side and corner terms, sum and difference, and addition and subtraction, and in deriving the advance and retreat of each unknown and fixing the quotient, he especially supplements what Li's book did not provide—truly a golden key for those learning root extraction. Wang Xiaoying devised a problem of two right triangles with equal area and equal sum of legs, using the mean rate of the two hypotenuse differences to turn and seek the two hypotenuse differences—the method established is circuitous. Luo directly obtained the two legs by positive and negative root extraction, which is quite simple. Hengzhai too would surely approve." Thus he was admired by others in this way.

=項名達=項名達，字梅侶，仁和人。舉人，考授國子監學正。道光六年，成進士，改官知縣，不就，退而專攻算學。三十年，卒於家，年六十有二。著述甚富，今傳世者，但有下學庵句股六術及圖解，复附句股形邊角相求法三十二題，合為一卷。以句股和較相求諸題術稍繁難，爰取舊術稍為變通。分術為六，使題之相同者通為一術，釐然悉有以御之。第一、二、三術及第四術之前二題，悉本舊解，餘為更定新術，皆別注捷法，各為圖解，以明其意。第四、五、六術其原皆出於第三術，可釋之以比例。第三術以句弦較比股，若股與句弦和，以股弦較比句，若句與股弦和，是為三率連比例。凡有比例加減之，其和較亦可互相比例。故第四、五、六術諸題，皆可由第三術之題加減而得，即可因第三術之比例而另生比例。因比例以成同積，而諸術開方之所以然遂明。名達又創有弧三角總較術，求橢員弧線術，術定，未有詮釋，以義奧趣幽，難猝竟事，故六術獨先成雲。

Xiang Mingda, styled Meilü, was a native of Renhe. A provincial graduate, by examination he was appointed rectifier of the Imperial Academy. In 1826 he passed the jinshi examination and was assigned the post of magistrate, but did not take office; he withdrew and devoted himself exclusively to mathematics. In the thirtieth year he died at home, aged sixty-two. His writings were very numerous; what survives today is only the Six Methods of Right Triangles from the Xiaoxue'an, with diagrams, plus thirty-two problems on finding sides and angles of right triangles, together one fascicle. Because the methods for problems of right-triangle sum and difference were somewhat complex, he took the old methods and made slight adaptations. He divided the methods into six, so that problems of the same type share one method and each can be handled with clear order. The first, second, and third methods and the first two problems of the fourth method all follow the old solutions; the rest are newly revised methods, each with a separate note on the quick method and a diagram to clarify the meaning. The fourth, fifth, and sixth methods all derive originally from the third method and can be explained by proportion. The third method compares the shorter-leg hypotenuse difference to the longer leg; when the longer leg and shorter-leg hypotenuse sum are given, it compares the longer-leg hypotenuse difference to the shorter leg; when the shorter leg and longer-leg hypotenuse sum are given—this is a three-term continued proportion. Wherever there is proportion with addition and subtraction, the sum and difference can also be mutually compared in proportion. Thus the problems of the fourth, fifth, and sixth methods can all be obtained by adding and subtracting from the problems of the third method, and new proportions can be generated from the proportion of the third method. By proportion one obtains equal area, and the reason root extraction is used in the various methods becomes clear. Mingda also created a general method for spherical triangles and a method for finding elliptical arc length; the methods were fixed but had no commentary, because the meaning was abstruse and the topic remote, and the matter could not be completed at once—so the Six Methods alone were finished first.

名達與烏程陳杰、錢塘戴煦契最深，晚年詣益精進，謂古法無用，不甚涉獵，而專意於平弧三角，與傑意不謀而合。與傑論平三角，名達曰：「平三角二邊夾一角，迳求斜角對邊，向無其法，竊嘗擬而得之，君聞之乎？」傑曰：「未也。」錄其法以歸。蓋以甲乙邊自乘與甲丙邊自乘相加，得數寄左；乃以半徑為一率，甲角餘弦為二率，甲乙、甲丙兩邊相乘倍之為三率，求得四率，與寄左數相減，鈍角則相加，平方開之，得數即乙丙邊。

Mingda was closest in friendship to Chen Jie of Wucheng and Dai Xu of Qiantang; in his later years his skill advanced further, and he held that ancient methods were useless, scarcely studying them, but devoted himself to plane spherical triangles, in agreement with Jie's intent without prior consultation. Discussing plane triangles with Jie, Mingda said: "In a plane triangle, given two sides and the included angle, to find directly the opposite angle and opposite side—there has never been a method for this; I have tentatively devised one. Have you heard of it?" Jie said: "Not yet." He recorded the method and returned home. It takes the square of side jia-yi plus the square of side jia-bing, and holds the sum on the left; then takes the radius as the first ratio, the cosine of angle jia as the second ratio, and twice the product of sides jia-yi and jia-bing as the third ratio; obtaining the fourth ratio, subtract it from the number held on the left—for an obtuse angle add instead—extract the square root, and the result is side yi-bing.

又嘗謂泰西杜德美之割圜九術，理精法妙，其原本於三角堆，董方立定四術以明之，洵為卓見。惟求倍分弧，有奇無偶，徐有壬補之，庶幾詳備。名達嘗玩三角堆，歎其數祗一遞加，而理法像數，包蘊無窮，夫方圜之率不相通，通方圜者必以尖，句股，尖像也；三角堆，尖數也。古法用半徑屢求句股得圜週，不勝其繁。杜氏則以三角堆禦連比例諸率，而弧弦可以互通，割圜術蔑以加矣。然以此制八線全表，每求一數，必乘除兩次，所用弧線，位多而乘不便，董、徐二氏大、小弧相求法亦然。向思別立簡易法，因從三角堆整數中推出零數，但用半徑，即可任求幾度分秒之正餘弦，不煩取資於弧線及他弧弦矢。且每一乘除，便得一數，似可為製表之一助。

He also once said that the Western mathematician Du Demei's nine methods of circle division are refined in principle and subtle in method; their origin lies in the triangular pile, and Dong Fangli fixed four methods to clarify them—a truly outstanding insight. Only in seeking fractional arcs, where there are odd cases but no even ones, Xu Youren supplemented them, so that the treatment is nearly complete. Mingda once studied the triangular pile and marveled that its numbers increase by only one step at a time, yet in principle and method, form and number, the implications are inexhaustible; the rates of square and circle do not communicate—what communicates square and circle must be the point, and right triangles—the point is the image of the point; The triangular pile comprises the pointed numbers. The ancient method required repeatedly applying the radius to right triangles to obtain the circumference—a process unbearably tedious. Du instead employed the continued-proportion ratios of the triangular pile, so arc and chord could be converted freely, and the circle-division methods needed no further addition. Yet using this to compile the complete trigonometric table, obtaining each value required two rounds of multiplication and division; the arc lengths involved had so many digits that multiplication was awkward—and the methods of Dong and Xu for converting between large and small arcs were no better. Mingda separately devised a simplified method, deriving fractional values from the whole numbers of the triangular pile. Using only the radius, one could obtain the sine and cosine for any degree, minute, and second without relying on arc lengths or other arc-chord-sagitta values. Moreover, each round of multiplication and division yielded one value, which seemed a useful aid in compiling tables.

又著像數原始一書，未竟，疾革時，囑戴煦。後煦索稿於名達子錦標，校算增訂六閱月而稿始定，都為七卷。原書之四，僅六紙，並第七卷皆煦所補也。卷一曰整分起度弦矢率論，卷二曰半分起度弦矢率論，卷三、卷四曰零分起度弦矢率論，皆以兩等邊三角形明其像，遞加法定其數，末乃申論其算法。卷五曰諸術通詮，取新立弧弦矢求他弧弦矢二術、半徑求弦矢二術及杜、董諸術，按術詮釋之。卷六曰諸術明變，雜列所定弦矢求八線術，開諸乘方捷術，算律管新術，橢員求週術，以明皆從遞加數轉變而得。卷七曰橢員求週圖解，原術以袤為徑，求大員周及週較，相減而得周，補術則以廣為徑，求小員周，週較相加而得周，末係以圖解。徐有壬巡撫江蘇，郵書索煦寫定本梓行，刻甫就而有壬殉難，書與板皆毀焉。

He also began a book, Origins of Form and Number, but did not finish it; on his deathbed he entrusted the manuscript to Dai Xu. Later Xu obtained the draft from Mingda's son Jinbiao. After six months of collation, calculation, and revision the manuscript was settled—seven fascicles in all. Book four of the original amounted to only six pages, and the seventh fascicle too was entirely Xu's supplement. The first fascicle treats chord-sagitta ratios from whole minutes as the initial measure; the second treats those from half minutes; the third and fourth treat those from fractional minutes. All use isosceles triangles to clarify the underlying form, fix the values by successive addition, and finally expound the computational methods. The fifth fascicle, General Commentary on the Methods, takes the two newly established methods for finding other arc-chord-sagitta values from a given arc-chord-sagitta, the two methods for finding chord and sagitta from the radius, and the methods of Du, Dong, and others, and explains each in turn. The sixth fascicle, Clarifying the Transformations of the Methods, miscellaneously lists the established methods for finding the eight trigonometric lines from chord and sagitta, shortcut methods for various powers, a new method for calculating pitch pipes, and the method for finding an ellipse's circumference, to show that all are derived by transformation from successive-addition numbers. The seventh fascicle, Diagrammatic Explanation of Finding Ellipse Circumference, presents the original method, which takes the long axis as diameter, finds the larger circle's circumference and the circumference difference, and obtains the ellipse's circumference by subtraction; and a supplementary method, which takes the short axis as diameter, finds the smaller circle's circumference, and obtains the circumference by adding the circumference difference—the fascicle closes with a diagrammatic explanation. Xu Youren, as governor-general of Jiangsu, wrote requesting Xu's fair copy for printing. The blocks had scarcely been cut when Youren died in the uprising, and both book and printing blocks were destroyed.

有王大有者，字吉甫，仁和諸生。翰林院待詔。窮究天算，問業於處士戴煦。凡煦所著述，皆錄副本去，名達見之，因與煦訂交。大有嘗校割圜捷術合編。後殉於杭州。

Wang Dayou, styled Jifu, was a licentiate of Renhe. He held the post of awaiting-edict scholar in the Hanlin Academy. He pursued astronomy and mathematics to their depths and studied under the recluse Dai Xu. He copied duplicate drafts of everything Xu had written and took them away. Mingda saw them and thus struck up a friendship with Xu. Dayou once collated the Combined Compilation of Shortcut Circle-Division Methods. He later died a martyr at Hangzhou.

=丁取忠=丁取忠，字果臣，長沙人。研究像數，不求聞達，刻算書二十有一種，為白芙堂叢書。光緒初，卒於家，年逾七十。所自譔者為數學拾遺一卷，以所演算草較詳，可便初學，又意在拾遺，故未暇詳其義之出自何人。

Ding Quzhong, styled Guochen, was a native of Changsha. He studied form and number and sought no fame or advancement, engraving twenty-one mathematical works as the Baifu Hall Collectanea. In the early Guangxu reign he died at home, aged over seventy. His own composition was Mathematical Collectanea, one fascicle. Because the worked computational drafts are comparatively detailed, it serves beginners well; and since his aim was to gather overlooked material, he did not take time to trace each principle to its original author.

又譔粟布演草二卷，自序曰：「道光壬辰，餘始習算，友人羅寅交學博洪賓以難題見詢，久無以應。同治初元，始獲交南豐吳君子登太史，馭以開屢乘方法，餘始通其術，然未悉其立法之根也。後吳君遊嶺表，餘推之他題，及展轉相求，仍多窒礙。又函詢李君壬叔，蒙示以廉法表及求總率二術，而其理始顯。後吳君又示以指數表及開方式表，李君復為之圖解以闡其義。由是三事互求，理歸一貫。餘因取數題詳為演草，並捷法圖解，都為一卷。質之南海鄒君特夫，君復為增訂開屢乘方法，並另設題演草，補所未備。即算家至精之理，如圜內容各等邊形，皆可藉發商生息以明之，誠快事也！」

He also compiled Grain-and-Cloth Computational Drafts in two fascicles. In his preface he wrote: "In the renchen year of the Daoguang reign I first began to study computation. My friend Luo Yin, through the licentiate Hong Bin, presented me with a difficult problem, and for a long time I had no answer. At the beginning of the Tongzhi reign I finally met Wu Zideng, Hanlin compiler, of Nanfeng, who taught me the method of repeated multiplication. I then mastered the technique, but did not yet fully grasp the root of its establishment. Later, when Wu traveled to Lingnan, I extended the method to other problems, but in mutual conversion and repeated seeking I still encountered many obstacles. I also wrote to Li Renshu, who graciously showed me the discount-rate table and the two methods of finding the aggregate rate—and only then did the principle become clear. Later Wu also showed me the index table and the root-extraction table, and Li provided diagrammatic explanations to clarify their meaning. From this the three matters could be converted into one another, and the principle resolved into a single thread. I therefore worked out numerical problems in detail as computational drafts, together with shortcut methods and diagrammatic explanations, all comprising one fascicle. I submitted it to Zou Tefu of Nanhai, who revised and augmented the method of repeated multiplication and added separate problems with worked drafts to supply what had been lacking. Even the most refined principles of mathematicians—such as regular polygons inscribed in a circle—can all be clarified through merchant interest calculations. A truly delightful thing!"

後又譔演草補一篇，序云：「餘前年與左君壬叟共輯粟布演草，原為商賈之習算者設，或一例而演數題，或一題而更數式。或用真數，或用代數。其式或橫列，或直下，雜然並陳，無非欲學者比類參觀，易於領悟也。乃初學習之，猶謂茫無入門處，蓋商賈所習算書，大都詳於文而略於式。況代數又古算術所無，宜其卒然覽之而不解也。茲更擬一題附後，特仿數理精蘊借根方體例，專詳於文，庶初學讀之，可因文知義。算理既明，則全書各式，可渙然冰釋，或兼可為習代數者之先導乎？」其鄉人李錫蕃，亦以演算名。

He later composed a supplement to the computational drafts. The preface states: "The year before last I jointly compiled the Grain-and-Cloth Computational Drafts with Zuo Rensou, originally for merchants learning computation—sometimes one pattern applied to several problems, sometimes one problem worked in several forms. Sometimes actual numbers were used, sometimes algebraic notation. The forms were sometimes arranged horizontally, sometimes vertically, all mixed together—solely so that learners might compare like cases side by side and grasp the meaning easily. Yet beginners still found no point of entry, for the computational books merchants study are mostly detailed in prose and scant in formal notation. Moreover, since algebra was absent from ancient Chinese mathematics, it is no wonder they could not understand it at a glance. I now append one problem, specially following the borrowed-root format of the Essence of Numbers and Principles, concentrating on detailed prose, so that beginners may grasp the meaning through the text alone. Once the computational principle is clear, all the forms throughout the book should dissolve like melting ice—might this also serve as a preliminary guide for those learning algebra?" His fellow townsman Li Xifan was also famed for working out computations.

錫蕃，字晉夫。早卒，著有借根方句股細草一卷，衍為二十有五術，取忠刊入叢書。

Li Xifan, styled Jinfu. He died young. He authored Borrowed-Root Right-Triangle Detailed Drafts, one fascicle, developing twenty-five methods; Quzhong published it in the collectanea.

=謝家禾=謝家禾，字和甫，錢塘舉人。與同學戴氏兄弟熙、煦相友善。少嗜西學，點線面體四部，靡不淹貫。已，復取元初諸家算書，幽探冥索，悉其秘奧。乃輯平時所得析通分加減，定方程正負，以標舉立元大耍，撰演元耍義一卷。其自序云：「元學至精且邃，而求其要領，無過通分加減，凡四元之分正負，及相消法，互隱通分法，大致原於方程。方程者，即通分之義。方程不明，由於正負無定例，加減無定行，以譌傳譌，如梅宣城精研數理，未暇深究，他書可知矣。九章算經正負術甚明，而釋者反以意度，古誼之不明，可勝道哉！唯以衍元之法正方程之義，由是方程明而元學亦明。著演元要義，綜通分方程而論列之，附以連枝同體之分等法。通乎此，則四元庶可窺其涯涘耳。」

Xie Jiahe, styled Hefu, was a provincial graduate of Qiantang. He was on friendly terms with his fellow students, the Dai brothers Xi and Xu. From youth he was devoted to Western learning and thoroughly mastered all four branches—point, line, surface, and solid. Thereafter he took up the mathematical works of the Yuan masters, probing deeply until he had mastered all their hidden subtleties. He then compiled his findings on common-fraction addition and subtraction and on fixing positive and negative signs in equations, marking out the great essentials of establishing the yuan, and composed Working Out the Essentials of the Yuan, one fascicle. In his own preface he wrote: "Yuan learning is most refined and profound; yet in seeking its essentials, nothing surpasses common-fraction addition and subtraction. All the positive and negative divisions of the four yuan, the cancellation method, and the mutually hidden common-fraction method derive in the main from equations. Equations are precisely the meaning of common fractions. Equations remain unclear because there are no fixed rules for positive and negative and no fixed procedures for addition and subtraction—errors passed on as errors. Even Mei Wending of Xuancheng, who finely studied numbers and principles, had no leisure to probe deeply; the same can be said of other books. The positive-and-negative method in the Nine Chapters on the Mathematical Art is very clear, yet commentators instead measure by conjecture—how can the obscurity of ancient meaning be fully told! Only by using the method of working out the yuan to rectify the meaning of equations can equations become clear—and yuan learning with them. I compose Working Out the Essentials of the Yuan, synthesizing common fractions and equations and discussing them in order, appending methods of equal division for linked branches and identical forms. Master this, and one may perhaps glimpse the farthest shore of the four yuan."

又以劉徽、祖沖之之率求弧田，求其密於古率者，撰弧田問率一卷。同里戴煦為之序曰：「古率徑一周三，徽率劉徽所定，徑五十週一百五十七也。密率乃祖沖之簡率，徑七週二十二也。諸書弧田術皆用古率，郭太史以二至相距四十八度，求矢亦用古法。顧徽、密二率之週既盈於古，則積亦盈於古，試設同徑之圓，旁割四弧，其中兩弦相得之方三率皆同，知三率圓積之盈縮，正三率弧積之盈縮也。徽、密二率弧田古無其術，惟四元玉鑑一睹其名，而設問隱晦，莫可端倪。穀堂得其旨，因依李尚之孤矢算術細草設問立術，亦足發前人所未發也。」

He also used the ratios of Liu Hui and Zu Chongzhi to find arc-field area, seeking ratios finer than the ancient one, and composed Inquiring into Arc-Field Ratios, one fascicle. His fellow townsman Dai Xu wrote a preface for it, saying: "The ancient ratio: diameter one, circumference three. The Hui ratio was fixed by Liu Hui: diameter fifty, circumference one hundred fifty-seven. The fine ratio is Zu Chongzhi's simplified ratio: diameter seven, circumference twenty-two. Arc-field methods in various books all use the ancient ratio. Grand Astrologer Guo Shoujing, taking the forty-eight degrees between the two solstices, also used the ancient method in finding the sagitta. Yet since the circumferences of the Hui and fine ratios both exceed the ancient, their areas also exceed the ancient. Suppose circles of equal diameter with four arcs cut on the sides; the squares obtained from pairs of chords within them share the same three ratios—knowing the excess or deficit of the three-ratio circle area is precisely the excess or deficit of the three-ratio arc area. For arc fields under the Hui and fine ratios the ancients had no method; only in the Jade Mirror of the Four Yuan does one glimpse the name, yet the problem statements are obscure and no clue can be found. Jiatang grasped its intent and, following Li Shangzhi's Detailed Drafts of Arc-Sagitta Computation, set out problems and established methods—developing what predecessors had not developed."

又以直橫與句股弦和較辗轉相求，撰直積回求一卷，其自序云：「始戴諤士著句股和較集成，予亦著直積與和較求句股弦之書，然二書為義尚淺，且直積與句弦和求三事，用立方三乘方等，得數不易，而又不足以為率，其書遂不存。近見四元玉鑑直積與和較回求之法，多立二元，嘗與諤士思其義蘊，有不必用二元者。蓋以句弦較與句弦和相乘為股冪，股弦和與股弦較相乘為句冪，而直積自乘，即句冪股冪相乘也。如以句弦較乘股弦較冪，除直積冪，即為句弦和乘股弦和冪矣。句弦和乘股弦和冪，即弦冪和冪共內少半個黃方冪也。蓋相乘冪內去一弦冪，所餘為句股相乘者一，句弦相乘者一，股弦相乘者一，此三冪合成和冪，則少一半黃方冪。半黃方冪，即句弦較股弦較相乘冪也。加一半黃方冪，即為弦冪和冪共矣。加二直積，即二和冪也。減六直積，即二較冪也。又句弦和乘股弦較冪，為句冪內少個句股較乘股弦較冪也。股弦和乘句弦較冪，為股冪內多個句股較乘句弦較冪也。減一句股較乘股弦較冪，尚餘一句股較冪矣。術中精意，皆出於此。其他之參用常法者，可不解而自明耳。草中既未暇論，恐習者不知其理，因揭其大旨於簡端，見演段之不可不精也。」

He also took length and breadth together with the sums and differences of right-triangle legs and hypotenuse and sought them in mutual conversion, composing Direct-Area Reverse-Seeking, one fascicle. In his preface he wrote: "At first Dai Eshi authored the Integration of Right-Triangle Sums and Differences; I also authored a book on seeking legs, hypotenuse, and chord from direct area and sums and differences. Yet both books were still shallow in meaning, and seeking the three quantities from direct area and the sum of leg and hypotenuse required cube and triple-power methods—the values were hard to obtain and insufficient to serve as ratios, so the books were not preserved. Recently, seeing the method in the Jade Mirror of the Four Yuan for reverse-seeking from direct area and sums and differences, which mostly establishes two yuan, I once pondered its meaning with Eshi and found cases where two yuan need not be used. For the difference of leg and hypotenuse multiplied by their sum gives the square of the other leg; the sum of the other leg and hypotenuse multiplied by their difference gives the square of the leg; and the direct area squared is precisely the product of the square of the leg and the square of the other leg. If one multiplies the difference of leg and hypotenuse by the square of the difference of the other leg and hypotenuse, and divides by the square of the direct area, the result equals the sum of leg and hypotenuse multiplied by the square of the sum of the other leg and hypotenuse. The sum of leg and hypotenuse multiplied by the square of the sum of the other leg and hypotenuse equals the sum of the square of the hypotenuse and the square of the sum, lacking half a yellow-square power within. Within the product power, removing one hypotenuse power, the remainder consists of one leg times the other leg, one leg times the hypotenuse, and one other leg times the hypotenuse—these three powers together form the sum power, which is then lacking half a yellow-square power. Half a yellow-square power is precisely the product of the difference of leg and hypotenuse and the difference of the other leg and hypotenuse. Adding half a yellow-square power gives precisely the sum of the hypotenuse power and the sum power together. Adding twice the direct area gives precisely two sum powers. Subtracting six times the direct area gives precisely two difference powers. Again, the sum of leg and hypotenuse multiplied by the square of the difference of the other leg and hypotenuse equals the square of the leg, lacking within it one product of the difference of the legs and the difference of the other leg and hypotenuse. The sum of the other leg and hypotenuse multiplied by the square of the difference of leg and hypotenuse equals the square of the other leg, with one product of the difference of the legs and the difference of leg and hypotenuse in excess within. Subtract one product of the difference of the legs and the difference of the other leg and hypotenuse, and one square of the difference of the legs still remains. The refined intent within the methods all proceeds from this. As for the other parts that borrow ordinary methods, they need no explanation to become clear of themselves. Since there was no leisure to discuss this in the draft, fearing that students would not know the principle, I set forth its main intent at the beginning of the fascicle, to show that working out sections cannot but be precise."

家禾歿後，戴熙搜遺稿，囑其弟煦校讎而授諸梓。煦精算，見忠義傳。著有補重差圖說，句股和較集成消法簡易圖解，對數簡法，外切密率，假數測圓，及船機圖說等。

After Jiahe's death, Dai Xi searched out his remaining drafts and entrusted his younger brother Xu to collate and proof them for printing. Xu was skilled in computation; see the Biographies of the Loyal and Righteous. He authored Supplementary Diagrammatic Explanation of Double Differences, Simple Diagrammatic Explanation of the Cancellation Method in the Integration of Right-Triangle Sums and Differences, Shortcut Logarithmic Methods, Fine Ratio of External Tangency, Measuring the Circle by False Numbers, Diagrammatic Explanation of Ship Machinery, and others.

=吳嘉善=吳嘉善，字子登，南豐人。進士，改翰林院庶吉士，散館授編修。與徐有壬同治算學。同治改元，避粵匪亂遊長沙，識丁取忠。逾年，客廣州，因鄒伯奇又識錢塘夏鸞翔。三人誌同道合，相得益彰。光緒五年，奉使法蘭西，駐巴黎。後受代還，旋卒。

Wu Jiashan, styled Zideng, was a native of Nanfeng. A jinshi graduate, he entered the Hanlin Academy as a bachelor; upon leaving the academy he was appointed compiler. He studied mathematics together with Xu Youren. At the beginning of the Tongzhi reign, fleeing the Taiping turmoil he traveled to Changsha and met Ding Quzhong. The following year, staying in Guangzhou, through Zou Boqi he also met Xia Luanxiang of Qiantang. The three men shared the same aspiration and path, each enhancing the other. In 1879 he was sent on mission to France and was stationed in Paris. Later, relieved of his post and returned home, he soon died.

所譔算書，首述筆算。次九章翼，曰今有術，曰分法，曰開方，曰平方平員各術。推演方田者，曰立方立員術，推演商功者，曰句股，曰衰分術，曰盈不足術，曰方程術。於句股術後，次附平三角、弧三角測量高遠之術。又次則專述天元四元之書，為天元一術釋例，為名式釋例，為天元一草，為天元問答，為方程天元合釋，為四元名式釋例並草，為四元淺釋。自序曰：「算學至今日，可謂盛矣。古義既彰，新法日出，前此所未有也。餘與丁君果臣皆癖此，既忘其癖，更欲以癖導人。嘗苦近世津逮初學之書無善本，梅文穆公所刪之算法統宗，今亦不傳。因商榷述此，取其淺近易曉，以為升高行遠之助雲。」

Among the mathematical books he authored, he first treats written calculation. Next comes Wings to the Nine Chapters: the present-quantity method, the division method, root extraction, and the various methods of squaring and leveling circles. Developing field measurement: the methods of cubing and standing circles; developing merchant work: right triangles, the proportional-decrease method, the excess-and-deficit method, and the equation method. After the right-triangle methods, he appends methods of plane and spherical trigonometry for measuring height and distance. Next he treats books on the celestial yuan and four yuan: Exemplifying the Single Celestial-Yuan Method, Exemplifying Named Forms, Celestial-Yuan Draft, Celestial-Yuan Questions and Answers, Combined Explanation of Equations and the Celestial Yuan, Exemplifying Named Forms of the Four Yuan with Draft, and Elementary Explanation of the Four Yuan. In his preface he wrote: "Mathematics down to the present day can be called flourishing. Ancient meaning is already manifest, new methods arise daily—what never existed before this. Mr. Ding Guochen and I are both obsessed with this subject; having lost sight of the fact that it is an obsession, we wish all the more to lead others into it. We once lamented that recent introductory books lacked reliable editions, and the Comprehensive Source of Computational Methods edited by Lord Mei Wenmu is no longer extant. Accordingly we discussed and composed this work, selecting what is plain and easy to grasp, to serve as a stepping-stone toward more advanced study."

=羅士琳=羅士琳，字茗香，甘泉人。以監生循例貢太學，嘗考取天文生。咸豐元年，恩詔徵舉孝廉方正之士，郡縣交薦，以老病辭。三年春，粵匪陷揚州，死之，年垂七十矣。少治經，從其舅江都秦太史恩复受舉業，已乃棄去，專力步算，博覽疇人書，日夕研求數年。

Luo Shilin, styled Mingxiang, was a native of Ganquan. A student of the Imperial University, he entered the Grand Academy by the usual procedure and once passed the examination for astronomical clerk. In 1851, when an imperial edict called for recommendations of men of filial piety and upright character, the local officials jointly recommended him, but he declined on account of old age and illness. In the spring of 1853, when the Taiping rebels took Yangzhou, he died defending the city, aged nearly seventy. In youth he studied the classics under his maternal uncle Qin Enfu, a historiographer of Jiangdu, in the art of examination essays; afterward he abandoned this and devoted himself exclusively to step calculation, reading widely among the works of calendar mathematicians and pursuing research day and night for several years.

初精西法，自譔言曆法者曰憲法一隅。又思句股、少廣相表裡，而方田與商功無異，差分與均輸不殊。按類相從，摘九章中之切於日用者，悉以比例馭之，彙為十二種。以各定率冠首，以藉根方繼後，以諸乘方開法附末，凡四卷，曰比例匯通，雖悔其少作，實便初學問途。

At first he mastered Western methods and himself composed a work on calendrical methods called A Corner of the Fundamental Law. He also reflected that right triangles and lesser breadth are complementary aspects of the same subject, while field measurement and merchant work are essentially the same, and proportional division and equal transport differ only in name. Grouping related topics together, he selected from the Nine Chapters what bears on daily use and governed all of it by proportion, compiling twelve categories. Each category opens with fixed rates, followed by root extraction, with the various methods of multiplication, powers, and root extraction appended at the end—in four fascicles, entitled Comprehensive Proportions; though he regretted its slightness, it was in fact a convenient guide for beginners.

後見四元玉鑑，服膺嘆絕，遂壹意專精四元之術。士琳博文強識，兼綜百家，於古今算法尤具神解，以硃氏此書實集算學大成，思通行發明，乃殫精一紀，步為全草，並有原書於率不通及步算傳寫之譌，悉為標出，補漏正誤，反覆設例，申明疑義，推演訂證。就原書三卷二十有四門，廣為二十四卷，門各補草。

Later, upon reading the Four Yuan Jade Mirror, he admired it with rapt wonder and then devoted himself exclusively to mastering the method of the four yuan. Shilin was broadly learned with a prodigious memory, synthesizing the hundred schools; in ancient and modern computational methods he possessed especially penetrating insight. Because Zhu's book truly gathers the great completion of mathematics, he wished to make it widely understood and elucidated; he then spent ten years in exhaustive refinement, working out complete drafts, and wherever the original book's rates were inconsistent or step calculations had transmission errors, he marked them all, supplementing omissions and correcting mistakes, repeatedly setting examples, clarifying doubtful points, and deriving proofs. Starting from the original book's three fascicles and twenty-four sections, he expanded it to twenty-four fascicles, each section with a supplementary draft.

嘗為提要鉤元之論，謂：「是書通體弗出九章範圍，不獨商功修築、句股測望、方程正負已也。如端匹互隱、廩粟回求寓粟布，如意混和寓借衰，茭草形段、果垛疊藏，如像招數寓商功中之差分，直段求源、混積問元、明積演段、撥換截田、鎖套吞容寓方田、少廣諸法。他若分索隱之為約分命分，方員交錯、三率究員、箭積交參之為定率兼交互。至於或問歌彖、雜範類會，以其各自為法，不能比類。故一則寄諸歌詞，一則編成雜法，均似補遺。大旨皆以加、減、乘、除、開方、帶分六例為問，每門必備此例，略簡易而詳繁難，尤於自來算書所無者，必設二問以明之。如混積問元中既設種金田及句三股四八角田為問。撥換截田中復設半種金田，鎖套吞容中復設方五斜七八角田為問。又果垛疊藏兩設員錐垛，雜範類會既設徽率割員，又設密率割員是矣。更有一門專明一義者，如和分索隱之分開方，三率究員兩儀合轍之反覆互求是矣。是書但云如積求之，如積有用定率為同數相消者，有如問加減乘除得積為同數相消者。祖序謂：'平水劉汝諧撰如積釋鎖，惜今不傳。 '意者其釋此例歟？」

He once set forth a thesis on extracting essentials, saying: "The book as a whole does not depart from the scope of the Nine Chapters—not only merchant work in construction, right triangles in surveying, and positive and negative equations. Such as end-and-span mutual concealment and granary grain reverse seeking, which employ grain-and-cloth methods; such as wish-fulfilling mixture, which employs proportional decrease; such as reed-and-grass shape segments and fruit-pile stacking and storage; such as image summoning numbers, which employ proportional division within merchant work; such as straight-segment source seeking, mixed accumulation questioning the unknown, bright accumulation derivation segments, exchange-and-cut fields, and lock-and-sheath swallowing and containing, which employ field measurement and lesser-breadth methods. Others such as division seeking concealment become simplified fractions and assigned fractions; square-and-circle interlacing, three-rate circle investigation, and arrow-pile cross combination become fixed rates combined with mutual exchange. As for the oracular questions, song formulas, and miscellaneous norm categories, because each has its own method, they cannot be classified by analogy. Therefore one is entrusted to song lyrics and one is compiled into miscellaneous methods—both appear to supplement omissions. The main intent is that all problems use the six cases of addition, subtraction, multiplication, division, root extraction, and mixed fractions; each section must include these cases, brief for the easy and detailed for the difficult, and especially where previous mathematical books had no precedent, two problems must be set to clarify the method. For example, in mixed accumulation questioning the unknown, problems of seed-gold fields and right-three-leg-four-corner fields are already set. In exchange-and-cut fields a half seed-gold field is again set; in lock-and-sheath swallowing and containing a square-five-slant-seven-eight-corner field is again set as a problem. Again, in fruit-pile stacking and storage two circular cone piles are set; in miscellaneous norm categories both fine-rate circle cutting and dense-rate circle cutting are set. Further, there is one section that specially clarifies a single principle, such as division root extraction in division seeking concealment, and the repeated mutual seeking in three-rate circle investigation's two-standards combined track. The book only says 'seek by accumulated products'; in accumulated products there are cases that use fixed rates as like numbers to cancel, and cases that use addition, subtraction, multiplication, and division to obtain the product as like numbers to cancel. Zhu's preface says: 'Ping Shui Liu Ruxie composed Unlocking Accumulated Products, but alas it is no longer extant today. Perhaps he explained this case?"

道光中，得硃氏算學啟蒙於京師廠肆，士琳復加斠詮刊布之。此書總二十門，凡二百五十九問，其名術義例多與玉鑑相表裡。士琳為之互斠，始於天元，終於四元，義主精邃，所得甚深。考大德四年莫若序，計後此書四年。此書首列乘除布算諸例，始於超徑等接之術，終於天元如積開方，由淺近以至通變，循序漸進，其理易知。名曰啟蒙，實則為玉鑑立術之根，此一證也。玉鑑原本十行，行十九字，「今有」低一格，「術曰」又低二格，與此書同，此二證也。玉鑑斗斛之「鬥」別作「」，此假借字，本漢書平帝紀及管子乘馬篇，尚雜見於唐以前之孫子、五曹、張丘建諸算經，鈞石之「石」，說文本作「柘」，玉鑑作「碩」，「碩」「石」古雖互通，然假「碩」為「石」，則僅見於毛詩甫田疏引漢書食貨志，而算書罕見，又玉鑑田之「」，雖見李籍九章音義，為字書所無，此書並同，此三證也。玉鑑雖亦三卷，而門則為二十四，問則二百八十八，較多此書四門二十九問，然以四字分類，其體裁同。且如商功、修築、方程、正負之屬，則又二書互見，此四證也。玉鑑如意混和第一問，據數知一秤為十五斤，適與此書之斤秤起率合，此五證也。玉鑑鎖套吞容第九問，方五斜七八角田左右逢元第六、第十三、第二十諸問，有小平小長，皆向無其術。此書卷首明乘除段，即載平除長為小長，長除平為小平之例。其田畝形段第十五問，复載方五斜七八角田求積通術，此六證也。他如玉鑑或問歌彖第四問，與此書盈不足術第七問，又玉鑑果垛疊藏第十四問，與此書堆積還原第十四問，又玉鑑方程正負第四問，與此書方程正負第五問，題皆約略相同，此七證也。知系硃氏原書佚而復出，併其算法一則，亦為附列，間有魚豕，悉仍其舊，但各標識於誤字旁，別記刊誤於卷末。

During the Daoguang era he obtained Zhu's Introduction to Mathematical Studies in a shop in Beijing's factory district; Shilin again added collation and commentary and published it. This book has twenty sections in all, with two hundred fifty-nine problems; its names, methods, and exemplary cases largely correspond with those of the Jade Mirror. Shilin made a comparative collation, beginning with the celestial yuan and ending with the four yuan; the meaning emphasizes refinement and depth, and what he obtained was very profound. Examining Mo Ruo's preface of 1300, this book came four years after that. This book first lists examples of multiplication, division, and written calculation, beginning with the method of exceeding diameter and equal joining, and ending with celestial-yuan accumulated-product root extraction—from the elementary to the advanced, advancing step by step, its principles easy to grasp. Though named Introduction, in fact it establishes the foundation of the methods of the Jade Mirror—this is the first proof. The original Jade Mirror has ten lines per page, nineteen characters per line; "Now suppose" is indented one space, and "The method says" is indented two spaces—the same as this book; this is the second proof. In the Jade Mirror the word for peck and picul, "dou," is written with a variant form; this is a borrowed character, originally found in the Annals of Emperor Ping in the Book of Han and the "Riding Horses" chapter of Master Guan, and still sporadically seen in the Sunzi, Wucao, and Zhang Qiujian mathematical classics before the Tang. For the word "shi" in jun and shi, the Shuowen originally writes "zhe"; the Jade Mirror writes "shuo." Though "shuo" and "shi" were interchangeable in antiquity, to borrow "shuo" for "shi" is seen only in Mao's commentary to "Fu Tian" citing the Treatise on Food and Money in the Book of Han, and is rarely seen in mathematical books. Again, the field character in the Jade Mirror, though seen in Li Ji's pronunciation glosses to the Nine Chapters, is absent from character dictionaries; this book likewise uses it—this is the third proof. Though the Jade Mirror is also three fascicles, its sections number twenty-four and its problems two hundred eighty-eight—four sections and twenty-nine problems more than this book; yet classified by four-character headings, its form is the same. Moreover such categories as merchant work, construction, equations, and positive and negative appear in both books—this is the fourth proof. The first problem of wish-fulfilling mixture in the Jade Mirror, from the numbers deducing one steelyard to be fifteen jin, exactly matches the initial rate of jin and steelyard in this book—this is the fifth proof. The ninth problem of lock-and-sheath swallowing and containing in the Jade Mirror, and the sixth, thirteenth, and twentieth problems of encountering the unknown on left and right in square-five-slant-seven-eight-corner fields, involve xiao ping and xiao chang—for which there had previously been no method. The section on clear multiplication and division at the head of this book immediately records the example of dividing the longer by the shorter to obtain xiao chang, and dividing the shorter by the longer to obtain xiao ping. Its fifteenth problem on field shape segments again records the general method for finding the area of a square-five-slant-seven-eight-corner field—this is the sixth proof. Others such as the fourth problem of oracular questions and song formulas in the Jade Mirror and the seventh problem of excess-and-deficit in this book; again the fourteenth problem of fruit-pile stacking and storage in the Jade Mirror and the fourteenth problem of pile accumulation restoration in this book; again the fourth problem of positive and negative equations in the Jade Mirror and the fifth problem of positive and negative equations in this book—the topics are all roughly the same; this is the seventh proof. Knowing that it is Zhu's original book, lost and then reappearing, one computational method is also appended; where there are textual errors, all are kept as in the original, but each is marked beside the mistaken character, with corrections separately recorded at the end of the fascicle.

又嘗以乾隆間明氏捷法校得八線對數表，一度十三分二十秒正切第五字「０」誤「一」；又六度四十一分十秒正切第五字「０」誤「六」；又十二度五十分正弦第六字「七」誤「五」；又十六度三十二分十秒正切第七字「九」誤「０」；又四十二度三十二分四秒正切第九字「五」誤「四」。可見西人所能，中人亦能之。

He also once collated the Ming family's Quick Method against the eight-line logarithm table of the Qianlong era: at one degree thirteen minutes twenty seconds, tangent, the fifth character "0" was mistaken for "one"; again at six degrees forty-one minutes ten seconds, tangent, the fifth character "0" was mistaken for "six"; again at twelve degrees fifty minutes, sine, the sixth character "seven" was mistaken for "five"; again at sixteen degrees thirty-two minutes ten seconds, tangent, the seventh character "nine" was mistaken for "0"; again at forty-two degrees thirty-two minutes four seconds, tangent, the ninth character "five" was mistaken for "four". Thus it can be seen that what Westerners can do, Chinese can also do.

又因會通四元玉鑑如像招數一門，更取明氏捷法，禦以天元，知密率亦可招差，其弧與弦矢互求之法，與授時曆之垛積招差一一符合。且以祖氏綴術失傳，其法廑見於秦書，即大衍之連環求等遞減遞加，亦與明氏捷法相近。爰融會諸家法意，撰綴術輯補二卷。

Also, because he harmonized the Four Yuan Jade Mirror's section on image summoning numbers, he further took the Ming family's Quick Method and governed it by the celestial yuan, showing that the dense ratio can also summon differences; the methods of mutual seeking between arc, chord, and versed sine accord one by one with the pile-summation summoning differences of the Season-Granting Calendar. Moreover, because Zu Chongzhi's Method of Interpolation was lost in transmission, its method is seen only sparingly in Qin's book—that is, the linked-ring seeking of equalities by successive decrease and increase in the Great Extension—which is also close to the Ming family's Quick Method. Thereupon blending the intent of the various schools' methods, he composed Interpolation: Collected Supplements in two fascicles.

又甄錄古今疇人，仍阮氏體例為列傳，採前傳所未收者，得補遺十二人，附見五人，續補二十人，附見七人，合共四十有四人，次於前傳四十六卷之後。

He also selected calendar mathematicians ancient and modern, following Ruan's biographical format, gathering those not collected in the previous biographies—obtaining twelve supplementary figures, five appended mentions, twenty continued supplements, and seven appended mentions, forty-four in all, placed after the previous biographies' forty-six fascicles.

集所校著都為觀我生室匯十二種。如四元玉鑑細草二十四卷，釋例二卷，校正算學啟蒙三卷，校正割圜密率捷法四卷，續疇人傳六卷，皆別有單行本。

His collated and authored works together form the Guan Wo Sheng Studio Collection of twelve kinds. Such as Detailed Draft of the Four Yuan Jade Mirror in twenty-four fascicles, Exemplifying Cases in two fascicles, Collated Introduction to Mathematical Studies in three fascicles, Collated Quick Method for Cutting the Circle and Dense Ratio in four fascicles, and Continued Biographies of Calendar Mathematicians in six fascicles—all have separate editions.

外已刻者尚得七種，曰句股容三事拾遺三卷，附例一卷，本繪亭監副博啟法補其遺，取內容方邊員徑垂線交互相求，一以天元馭之。曰三角和較算例一卷，取斜平三角形中兩邊夾一角術鎔入天元法，用和較推演成式。曰演元九式一卷，括玉鑑中進退消長諸例，借無數之數，以正負開方式入之。曰台錐積演一卷，以玉鑑茭草、果垛二門可補少廣之闕，爰取台錐形段引而伸之。曰周無專鼎銘考一卷，以四分週術佐以三統漢術，推得宣王十有六年九月既望甲戌，與銘辭正合。曰弧矢算術補一卷，以元和李四香原術未備，為增補二十七術，合成四十術。曰推算日食增廣新術一卷，推廣正升斜升橫升之算法，以求太陰隨地隨時之明魄方向分秒，复推其術，以求交食限內之方向，及所經歷之諸邊分。

Besides what was already printed, seven more kinds remain: Supplementary Remainders of the Three Cases of Right-Triangle Containment in three fascicles, with one fascicle of examples—based on the Supplementary Method of the Assistant Director of the Painting Pavilion Bo Qi, taking inner containment of square side, circle diameter, and perpendicular line for mutual seeking, all governed by the celestial yuan. Entitled Examples of Sum and Difference of Triangles in one fascicle—taking the method of two sides enclosing one angle in an oblique plane triangle and fusing it into the celestial-yuan method, using sum and difference to derive formulas. Entitled Nine Forms of Deriving the Unknown in one fascicle—embracing the advance, retreat, increase, and decrease cases in the Jade Mirror, borrowing the number without number, and entering them by positive and negative root-extraction methods. Entitled Derivation of Frustum Accumulation in one fascicle—because the reed-and-grass and fruit-pile sections of the Jade Mirror can supplement the gaps in lesser breadth, he took frustum shape segments and extended them. Entitled Examination of the Zhou Wu Zhuan Ding Inscription in one fascicle—using the four-part circumference method assisted by the Han method of the Triple Concordance, he deduced that on the day jiaxu after the full moon of the ninth month in the sixteenth year of King Xuan of Zhou, it exactly matches the inscription. Entitled Supplement to Arc-and-Versed-Sine Calculation in one fascicle—because Yuan Li Sixiang's original method was incomplete, he added twenty-seven methods, together forming forty methods. Entitled New Extended Method for Calculating Solar Eclipses in one fascicle—extending the algorithms for upright ascent, oblique ascent, and horizontal ascent, to seek the direction in minutes and seconds of the moon's bright and dark portion according to place and time on earth; again extending the method to seek the direction within the limits of intersection and eclipse, and the various edge divisions traversed.

餘若春秋朔閏異同考、綴術輯補交食圖說舉隅、句股截積和較算例、淮南天文訓存疑、博能叢話，凡若干卷，未有刻本。其同縣友有易之瀚者，亦以算名。

The rest, such as Examination of Differences in Spring-and-Autumn New and Intercalary Months, Interpolation Collected Supplements with Diagrams and Illustrations of Eclipses, Examples of Sum and Difference in Right-Triangle Intercepted Accumulation, Doubts Preserved from the Astronomical Instructions of Huainan, and Miscellaneous Talk on Broad Capacity—several fascicles in all—have not yet been printed. Among his friends of the same county was Yi Zhihan, who was also famed for calculation.

易之瀚，字浩川。知士琳有四元玉鑑補草，因從問難，為撰四元釋例一卷。凡開方例二十九則，天元例十一則，四元例十三則。

Yi Zhihan, styled Haochuan. Learning that Shilin had a supplementary draft of the Four Yuan Jade Mirror, he questioned him and composed Exemplifying the Four Yuan in one fascicle. In all twenty-nine cases of root extraction, eleven cases of the celestial yuan, and thirteen cases of the four yuan.

=顧觀光=顧觀光，字尚之，金山人。太學生，三試不售，遂無志科舉，承世業為醫。鄉錢氏多藏書，恆假讀之。博通經、傳、史、子、百家，尤究極天文曆算，因端竟委，能抉其所以然，而摘其不盡然。時復蹈瑕抵隙，蒐補其未備。如據周髀「笠以寫天，青黃丹黑」之文及後文「凡為此圖」云云，而悟篇中周徑里數皆為繪圖而設。天本渾員，以視法變為平員，則不得不以北極為心，而內外衡以次環之，皆為藉象，而非真以平員測天也。

Gu Guanguang, styled Shangzhi, was a native of Jinshan. A student of the Grand Academy, he failed the examination three times and then lost interest in the civil service examinations; he inherited the family profession as a physician. The Qian family of his district had many collected books, and he constantly borrowed them to read. Broadly versed in the classics, commentaries, histories, masters, and the hundred schools, he especially pursued astronomy, calendrical calculation, and mathematics to the utmost, tracing matters from beginning to end and able to probe their reasons, while also picking out where they fell short. From time to time he also seized on flaws and gaps, gathering and supplementing what was not yet complete. For example, based on the Zhou Bi's text "The bamboo hat writes heaven—azure, yellow, cinnabar, and black" and the later passage "In all, for this diagram" and so on, he understood that the circumference and diameter numbers in li in the chapter were all set up for drawing the diagram. Heaven is originally perfectly round; by the method of viewing it is transformed into a flat circle, so one cannot but take the north pole as center, with inner and outer balances encircling in order—all are borrowed images, and not truly measuring heaven with a flat circle.

開元占經魯歷積年之算不合，因用演積術，推其上元庚子至開元二年歲積，知佔經少三千六十年。又以佔經顓頊歷歲積考之史記秦始皇本紀，知其術雖起立春，而以小雪距朔之日為斷。蓋秦以十月為歲首，閏在歲終，故小雪必在十月，昔人未及言也。李尚之用何承天調日法考古曆日法朔餘強弱不合者十六家，以為未能推算入微。爰別立術，以日法朔餘展轉相減，以得強弱之數。但使日法在百萬以上皆可求，惟朔餘過於強率者不可算耳。授時術以平定立三差求太陽盈縮，梅氏詳說未明其故。讀明志乃知即三色方程之法。謂凡兩數升降有差，彼此遞減，必得一齊同之數。引而伸之，即諸乘差，則八線、對數、小輪、橢員諸術，皆可共貫。讀佔經所載瞿曇悉達九執術，知回回、太西曆法皆源於此。其所謂高月者即月孛，月藏者即月引數，日藏者即日引數，特稱名不同，亦猶回曆稱歲實為宮日數，朔策為月分日數也。

The calculation of accumulated years in the Lu calendar in the Kaiyuan Occupations Classic does not agree; therefore using the method of deriving accumulation, he calculated the accumulated years from the upper origin gengzi to 714 CE, finding the Occupations Classic short by three thousand six hundred years. Again, comparing the accumulated years of the Zhuanxu calendar in the Occupations Classic with the Basic Annals of Qin Shihuang in the Records of the Historian, he found that though its method begins from the Establishment of Spring, it takes the day of the new moon's distance at Lesser Snow as the cutoff. This is because Qin took the tenth month as year-head and placed intercalation at year's end, so Lesser Snow must fall in the tenth month—what earlier scholars had not yet stated. Li Shangzhi used He Chengtian's day-adjustment method to examine ancient and modern calendars where day divisor and new-moon remainder strong and weak did not agree—sixteen houses in all—and held that they had not been able to calculate with sufficient precision. Thereupon he separately established a method, using day divisor and new-moon remainder in successive mutual subtraction to obtain the numbers of strong and weak. So long as the day divisor is above one million, all can be sought; only when the new-moon remainder exceeds the strong rate can it not be calculated. The Season-Granting method uses the mean and fixed to establish three differences to seek solar excess and deficiency; Mei's detailed explanation did not clarify the reason. Reading the Ming History, he then knew it is precisely the method of three-color equations. He said: whenever two numbers rise and fall with a difference, each decreasing the other in turn, one must obtain a number of equal sameness. Extending this, it is the various product differences; then the eight lines, logarithms, epicycles, and ellipse methods can all be threaded on one string. Reading the Nine Executions Method of Gautama Siddha recorded in the Occupations Classic, he knew that Islamic and Western calendrical methods all originate from this. What it calls gaoyue is the moon's apogee, yuecang is the moon's anomaly number, and ricang is the sun's anomaly number—only the names differ; it is also like the Islamic calendar calling the tropical year gongri number and the synodic month yuefenri number.

其論婺源江氏冬至權度，推劉宋大明五年十一月乙酉冬至前以壬戌丁未二日景求太陽實經度，而後求兩心差，乃專用壬戌。今用丁未求得兩心差，適與江氏古大今小之說相反。蓋偏取一端，其根誤在高衝行太疾也。西法用實朔距緯求食甚兩心實相距，術繁而得數未確。改以前後兩設時求食甚實引徑得兩心實相距，不必更資實朔，較本法為簡而密矣。

In his discussion of the Jiang clan of Wuyuan's winter-solstice weighted measure, he calculated the sun's true ecliptic longitude before the winter solstice of the fifth year of the Great Ming, eleventh month, day yiyou, from the shadow lengths of days renxu and dingwei, and afterward sought the two-center difference—yet used only renxu exclusively. Now using dingwei to obtain the two-center difference, it exactly contradicts the Jiang clan's doctrine that the ancient was large and the present small. This is because he took one side only; the root of the error lies in the high apogee moving too fast. Western methods use true new moon distance from latitude to seek the true distance between the two centers at greatest eclipse—the method is complex and the number obtained is not exact. Changing to use two set times before and after to seek the true anomaly radius at greatest eclipse and obtain the true distance between the two centers, without needing to rely on true new moon—it is simpler and more precise than the original method.

西人割圜，止知內容各等邊之半為正弦，而不知外切各等邊之半為正切。乃依六宗、三要、二簡諸術，別立求外切各等邊之正切法，以補其缺。杜德美求員周術，用員內容六邊形起算，巧而降位稍遲，謂內容十等邊之一邊，即理分中末線之大分，距週較近。且十邊形之邊與周同數，不過遞進一位，而大分與全分相減即得小分，則連比例各率，可以較數取之。入算尤簡易，可用弧度入算，不用弧背真數。然猶慮其難記，仍不能無藉於表，因又合兩法用之，則術愈簡，而弧線、直線相求之理始盡。錢塘項氏割圜捷術，止有弦矢求餘線術，以為可通之割、切二線，因補其術。西人求對數，以正數屢次開方，對數屢次折半，立術繁重。李氏探原以尖錐發其覆，捷矣，而布算術猶繁。且所得者皆前後兩數之較，可以造表而不可徑求。戴氏簡法及西人數學啟蒙，又有新術，而未窮其理。乃變通以求二至九之八對數，因任意設數，立六術以御之，得數皆合。復立還原四術，並推衍為和較相求八術，為自來言對數者所未有也。又謂對數之用，莫便於八線，而西人未言其立表之根，因冥思力索，仍用諸乘方差，迎刃而解，尤晚歲造微之詣也。其它凡近時新譯西術，如代數、微分、諸重學，皆有所糾正，類此。

Westerners in cutting the circle know only that half of each equal side of the inscribed polygon is sine, but do not know that half of each equal side of the circumscribed polygon is tangent. Accordingly, relying on the Six Origins, Three Essentials, and Two Simplifications methods, he separately established a method for seeking the tangent of the circumscribed equal sides, to supplement the lack. Du Demei's method for seeking the circumference of the circle, starting calculation from the hexagon inscribed in the circle, is ingenious but the lowering of place is somewhat slow; he said that one side of the inscribed decagon is the major part of the extreme minute line in li fen, closer to the circumference. Moreover the side of the decagon and the circumference are of the same number, advancing only one place; and subtracting the major part from the whole part yields the minor part—then the successive proportional rates can be taken by comparing numbers. Entering calculation it is especially simple, and arc measure can be used in calculation without using the true number of arc and chord. Yet he still feared it would be hard to remember and still could not do without relying on tables; therefore he again combined the two methods in use—then the method became simpler, and the principle of mutual seeking between arc lines and straight lines was at last complete. In the Qiantang Xiang family's swift method of circle division, there were only techniques for finding the remaining lines from chord and versed sine; he judged that the secant and tangent lines could also be linked, and therefore supplemented the method. Western methods of finding logarithms relied on repeatedly extracting roots of positive numbers and repeatedly halving logarithms, making the established procedures laborious. In his Investigation of Origins, Li used the pointed cone to uncover the underlying principle—a clever advance—yet the procedures for written calculation remained cumbersome. Moreover, what one obtained was always the difference between two successive values—enough to build tables, but not to find a value directly. Dai's Simplified Method and the Western Mathematical Primer introduced new techniques as well, yet neither fully penetrated the underlying principles. He then devised flexible methods to obtain logarithms of the numbers two through nine; by choosing numbers at will and establishing six methods to govern the process, he obtained results that all agreed. He further set forth four restoration methods and extended them into eight techniques for mutual seeking by sums and differences—achievements that no previous writer on logarithms had ever possessed. He also held that among all uses of logarithms, none was more convenient than the eight trigonometric lines, yet Western writers had never explained the foundation of their tables; after long reflection and effort, he solved the problem using differences of successive powers as easily as splitting bamboo—an especially subtle mastery of his later years. Likewise, he corrected errors in other recently translated Western sciences—algebra, calculus, and the various branches of mechanics.

所著曰算賸初、續編凡二卷。曰九數存古，依九章分為九卷，而以堆垛、大衍、四元、旁要、重差、夕桀、割圜、弧矢諸術附焉，皆採古書而分門隸之。曰九數外錄，則隱括四術為對數、割圜、八線、平三角、弧三角各等面體、員錐三曲線、靜重學、動重學、流質重學、天重學，凡記十篇。曰六歷通考，則據佔經所紀黃帝、顓頊、夏、殷、週、魯積年而加以考證。曰九執歷解，曰回回曆解，皆就原法疏通證明之。曰推步簡法，曰新歷推步簡法，曰五星簡法，則就原術改度為百分，省迂迴而歸簡易，蓋於學實事求是，無門戶異同之見，故析理甚精，而談算為最雲。其友人韓應陛，亦以表章算書顯。

His works included Mathematical Surplus, Initial Compilation and Continued Compilation, two volumes in all. One was Nine Numbers Preserving Antiquity, arranged in nine volumes after the Nine Chapters, with pile stacking, Great Extension, four yuan, supplementary essentials, double differences, evening gnomon, circle division, arc-sagitta, and related methods appended—all culled from ancient texts and classified by topic. Another was External Record to the Nine Numbers, which gathered four fields into ten treatises on logarithms, circle division, eight trigonometric lines, plane and spherical triangles, regular polyhedra, the three conic sections, static, dynamic, and fluid mechanics, and celestial mechanics. His Comprehensive Investigation of the Six Calendars examined the accumulated years of the Yellow Emperor, Zhuanxu, Xia, Yin, Zhou, and Lu as recorded in the Classic of Divination and subjected them to critical verification. His Explanation of the Nine Celestial Calendars and Explanation of the Huihui Calendar both clarified and demonstrated the original methods. His Simplified Methods of Astronomical Calculation, Simplified Methods for the New Calendar, and Simplified Methods for the Five Planets reworked the original procedures by expressing degrees in hundredths, cutting away roundabout steps in favor of simplicity. Unburdened by partisan loyalties, he pursued truth from facts; his analyses were exceptionally keen, and in mathematical discourse he stood at the fore. His friend Han Yingbi likewise won renown for promoting mathematical books.

應陛，字對虞，婁縣人。舉人，官內閣中書舍人。少好讀週、秦諸子，為文古質簡奧，非時俗所尚。既而從同里姚椿遊，得望溪、惜抱相傳古文義法。西人所創點、線、面、體之學，為幾何原本，凡十五卷，明萬曆間利譯止前六卷。咸豐初，英人偉烈亞力續譯後九卷，海寧李壬叔寫而傳之。應陛反覆審訂，授之剞劂，亞力以為泰西舊本弗及也。外若新譯重、氣、聲、光諸學，應陛推極其致，往往為西人所未及雲。

Han Yingbi, courtesy name Duiyu, was a native of Lou County. A provincial graduate, he served as a Secretariat Drafter in the Grand Secretariat. From youth he loved the masters of Zhou and Qin; his prose was plain, archaic, and abstruse—not the sort contemporary taste favored. Later he studied with his fellow townsman Yao Chun and mastered the transmitted principles of classical prose associated with Wang Xishui and the Xibao school. The Western science of points, lines, surfaces, and solids was Euclid's Elements, fifteen books in all; during the Wanli reign of the Ming, Ricci translated only the first six. In the early Xianfeng period, the Englishman Wylie translated the remaining nine books, and Li Renshu of Haining copied and circulated them. Yingbi reviewed and corrected the text again and again before sending it to the block cutters; Wylie judged that even the original Western edition fell short of this one. In newly translated sciences of weight, air, sound, and light as well, Yingbi carried investigation to its limits, often surpassing what Westerners themselves had achieved.

=左潛=左潛，字壬叔，大學士宗棠從子。補縣學生。於詩、古文辭無不深造，尤明算理。長沙丁取忠引為忘年交。早卒，士林惜之。所學自大衍、天元及借根方、比例諸新法，無不貫通。且能自出己意，變其式，勘其誤，作為圖解，往往突過先民。嘗增訂徐有壬割圜綴術，既成，忽悟通分捷法析分母、分子為極小數，根同者去之，凡多項通分，頃刻立就。因演數草，為通分捷法一帙。

=Zuo Qian= Zuo Qian, courtesy name Renshu, was a nephew of Grand Secretary Zuo Zongtang. He entered the county school as a supplemental student. He achieved deep mastery in poetry and classical prose, and was especially adept at mathematical principles. Ding Quzhong of Changsha befriended him despite their age difference. He died young, to the grief of learned circles. From Great Extension and the celestial yuan to root-borrowing and proportional methods, he mastered every new technique without exception. He could also devise methods of his own, revise existing forms, correct errors, and compose illustrated explanations, often surpassing earlier masters. He once revised and expanded Xu Youren's Continuation Methods of Circle Division; upon finishing, he suddenly grasped a swift method for finding common denominators—breaking numerators and denominators into infinitesimal parts and canceling like roots—so that multi-term reductions were completed in an instant. He then worked out numerical drafts and compiled one fascicle on the Swift Method of Common Denominators.

所譔綴術補草四卷，自序曰：「自泰西杜德美創立割圜九術，以屢乘屢除通方圜之率，我朝明氏、董氏各為之說，而杜書之義，推闡靡遺。顧八線互求，尚無通術，未足以盡一圜之變，非明氏、董氏之智力，不能因法立以盡其變也。其能窮杜氏之義也，資於借根方；其不能廣杜氏之法也，亦限於借根方。蓋借根方即天元一之變術，究不如元術之巧變莫測也。是書祖杜宗明，又旁參以董氏之法，八線相求，各立一式，因式立法，因法入算。鄉之不可立算者，今皆能馭之以法，即有不能立法布算者，而其式存，則能濟法之窮；而度圜諸線，一以貫之矣。推其立式之由，所謂比例術，即明氏定半徑為一率，所有為二率或三率之法也。所謂還原術，即明氏弧背求正矢，又以正矢求弧背之法也。所謂借徑術，即明氏借十分全弧通弦率數求百分全弧通弦率數，求千分全弧通弦率數諸法也。所謂商除法，又即還原術之變法。是故綴術胎於明氏，而又足以盡明氏之變。明氏之未立式者，以藉根方取兩等數，其分母、分子雜糅繁重，既不可通，其多號、少號，辗轉互變，又不可約。試取明氏書馭之以綴術，其遞降各率，頃刻可求。則是書也，其真能因法立法，別樹幟於明、董之後者歟？書為徐君青先生所作，吳君子登成之，顧詳於式而略於草。敬考其立法之原，不可遽得，學者難焉，潛因於暇日為補草四卷，因綴數語於簡端雲。」

In his four-volume Supplementary Drafts to Continuation Methods, he wrote in the preface: "Since the Westerner Du Demei established the nine methods of circle division, using repeated multiplication and division to link the ratios of squares and circles, Ming and Dong of our dynasty each offered interpretations, and the meaning of Du's book was fully expounded. Yet mutual derivation among the eight trigonometric lines still lacked a general method, not enough to exhaust all transformations within a single circle; only minds as acute as Ming's and Dong's could derive new methods from old rules to capture every variation. Their success in exhausting Du's meaning owed much to root-borrowing; their failure to extend Du's methods was likewise bounded by root-borrowing. Root-borrowing is only a variant of the single celestial-yuan method, and in the end it cannot match the celestial yuan's unpredictable ingenuity. This book takes Du as its founder and Ming as its guiding patriarch, while also drawing on Dong's methods; for mutual derivation among the eight lines, it establishes a separate formula for each case, deriving rules from formulas and carrying them into calculation. Problems that once resisted calculation can now all be governed by method; and even where no rule can yet be laid out for written calculation, if the formula survives, it can rescue a method at its limit; thus all lines for measuring the circle are unified in one coherent scheme. Tracing how the formulas were established, the proportional method is precisely Ming's practice of fixing the semi-diameter as the first ratio and the given quantity as the second or third ratio. The restoration method is Ming's procedure of finding versed sine from arc length and then arc length from versed sine. The diameter-borrowing method comprises Ming's techniques of using the rate for a ten-part whole-arc chord to find those for hundred-part and thousand-part whole-arc chords. The quotient-division method is simply a variant of the restoration method. Thus continuation methods were born from Ming's work, yet were fully adequate to exhaust every variation within it. In cases where Ming had not yet established formulas, root-borrowing produced paired equal numbers whose numerators and denominators were entangled and unwieldy—neither reducible to common terms nor simplifiable through the shifting of positive and negative signs. Apply continuation methods to Ming's book, and every successive descending ratio can be found in an instant. Is this book, then, one that truly derives new rules from existing methods and raises its own banner after Ming and Dong? The book was written by Master Xu Junqing and brought to completion by Master Wu Zideng, but it was rich in formulas and sparse in worked examples. Because the origins of its rules could not readily be grasped, students found it difficult; in his spare time Qian therefore composed four volumes of supplementary drafts, adding a few words at the end of this note."

又譔綴術釋明二卷，湘鄉曾紀鴻為之序，略曰：「易系云：'極其數遂定天下之象。 '則綜天下難定之像以歸有定，莫數若矣。在昔聖神，制器尚象，利物前民，於數理必有究極精微，範圍後世者，代久年湮，漸至失傳。近三百年，泰西猶能推闡古法，而中國才智之士，或反率其成轍。孔子曰：'天子失官，學在四夷。 '正今日數學之謂也。中國舊有弧矢算術，而未標角度八線鈐表，則雖有用其理以入算者，而無表可檢。則每求一數，必百倍其功，而所得數仍非密率。明代譯出泰西八線表及八線對數表，覈其立法得數之原，甚屬繁難，而成表之後，一勞永逸。大至無外，細及極微，莫不以此表測之，則其用之廣大可想。然得表之後，雖無事於再求，而任舉一數，無從較其訛誤。若仍用舊術，則非★月經旬，不能得一數，此明靜菴、董方立推演杜德美弧矢捷術之所以可貴也。向來求八線者，例用六宗、三耍、二簡各法，若任言一弧，必不能考其弦矢諸數。至杜氏創立屢乘屢除之法，則但有弧徑，而八線均可求。董方立解杜術，先取其線之極微者，令與與弧線合，而後用連比例以推至極大。又考諸率數與尖錐理相合，故用尖錐以釋弧矢，而弧矢之數理以顯。明靜菴解杜術，先取四分弧與十分弧之通弦直線之極大者，用連比例以推至千分、萬分弧通弦之極微者，考其乘除之率數，與杜術乘除之原理合，故用綴術以釋弧矢，而弧矢之數理亦出。董、明二氏，均為弧矢不祧之宗，無庸軒輊。邇百年中繼起者，如戴、徐、李三氏所著書，雖自出心裁，要皆奉董、明為師資也。吾友左君壬叟，於數學尤孜孜不倦，遇有疑難，必窮力追索，務洞澈其奧窔。嘗謂方員之理，乃天地自然之數，吾之宗中宗西，不必分畛域，直以為自得新法也可。曾釋君青徐氏綴術，又釋戴鄂士求表捷術，茲又釋明靜菴弧矢捷術，而一貫以天元寄分之式，於員率一道三致意焉，可謂勤矣。孰意天厄良才，壬叟竟於甲戌秋不永年而逝，凡在同人，無不嘆惜！況餘與之為兩世神交，安能無愴切耶！」

He also wrote two volumes, Explication of Continuation Methods, prefaced by Zeng Jihong of Xiangxiang, who wrote in part: "The Book of Changes says: "Exhaust the numbers, and thereby fix the images of all under Heaven. Then to gather all the unstable images under Heaven into what is fixed—nothing serves better than number. In antiquity the sages fashioned tools and honored symbolic forms to benefit the people; they must have attained ultimate subtlety in number and principle to guide later ages—yet as centuries passed, much was gradually lost. Over the last three centuries, the West has still been able to develop ancient methods, while some gifted Chinese scholars have merely followed paths already worn smooth. Confucius said: "When the Son of Heaven lost his officers, learning resided among the four quarters beyond. That is exactly what mathematics today amounts to. China long possessed arc-sagitta arithmetic, but without tables keyed to angular degrees and the eight trigonometric lines; even when its principles were applied in calculation, there were no tables to consult. Each value required a hundred times the labor, and even then the result was not a close approximation. In the Ming, Western tables of the eight trigonometric lines and their logarithms were translated; the methods behind them were immensely difficult to establish, but once the tables existed, the labor was done for all time. From the vastest scale to the finest detail, nearly everything could be measured with these tables—so one may imagine how far-reaching their use was. Yet once the tables existed, although one need not derive values anew, there was no way to check the error in any number chosen at random. If one still relied on the old methods, not even months or ten-day periods would suffice to obtain a single value—this is why the arc-sagitta swift methods of Du Demei, as developed by Ming Jing'an and Dong Fangli, were so precious. Formerly, seekers of the eight lines relied on the methods of the six origins, three essentials, and two simplifications; name any arc at random, and one could not determine its chord, sagitta, and related values. Once Du devised his method of repeated multiplication and division, all eight lines could be found given only an arc and a diameter. Dong Fangli explained Du's method by first taking infinitesimal line segments, making them coincide with the arc, and then extending by continued proportion to the largest scale. Finding that the rate numbers accorded with the principle of the pointed cone, he used that model to explain arc and sagitta, thereby revealing their underlying mathematics. Ming Jing'an explained Du's method by first taking the largest straight lines—the common chords of four-part and ten-part arcs—and extending by continued proportion down to the infinitesimal common chords of thousand-part and ten-thousand-part arcs; the resulting rates of multiplication and division matched Du's underlying principles, so he used continuation methods to explain arc and sagitta, and their mathematics emerged likewise. Dong and Ming were both undisputed patriarchs of arc-sagitta studies; there is no need to rank one above the other. Over the last century, successors such as Dai, Xu, and Li produced original works, yet all still took Dong and Ming as their teachers. My friend Master Zuo Rensou was especially tireless in mathematics; whenever he met a difficulty, he pursued it to the end until he had mastered its innermost workings. He once said that the principles of square and circle are numbers natural to Heaven and Earth; whether one honors Chinese or Western learning, there is no need to draw boundaries—one may simply treat the result as a new method of one's own. He had already explicated Xu Junqing's continuation methods and Dai Eshi's swift method for deriving tables; now he explicated Ming Jing'an's arc-sagitta swift methods as well, consistently applying the celestial-yuan format of positional fractions—three times over he devoted himself to the theory of circular ratios; his diligence was remarkable. Who could have expected Heaven to afflict so fine a talent—Rensou died before his time in the autumn of Jiaxu, to the grief of all who knew him! And as for me, who had enjoyed with him a spiritual friendship across two generations—how could I not mourn him deeply!"

曾紀鴻，字栗誠，大學士國籓少子。恩賞舉人。早卒。紀鴻少年好學，與兄紀澤並精算術，尤神明於西人代數術。銳思勇進，創立新法，同輩多心折焉。謂大衍求一術亦可以代數推求，依題演之，理正相通，撰對數詳解五卷，始明代數之理，為不知代數者開其先路。中言對數之理，末言對數之用，明作書之本意。其於常對、訥對，辨析分明。先求得各真數之訥對，復以對數根乘之，即為常對數。級數朗然，有條不紊，雖初學循序漸進，無不可相說以解焉。

Zeng Jihong, courtesy name Licheng, was the youngest son of Grand Secretary Zeng Guofan. He received provincial graduate status by imperial grace. He died young. Jihong loved learning from youth; he and his elder brother Jize were both adept at mathematics, and he was especially brilliant in Western algebra. Bold and inventive, he devised new methods that won the admiration of his peers. He held that the Great Extension method of seeking unity could also be derived by algebra; working through problems, he showed the principles aligned; his five-volume Detailed Explanation of Logarithms began by setting forth Ming-dynasty algebraic theory, opening a path for those who did not yet know algebra. The middle treats the theory of logarithms and the end their applications, making clear the author's original purpose. His distinctions between common and Napierian logarithms were lucid. First find the Napierian logarithm of each true number, then multiply by the logarithmic base to obtain the common logarithm. The series are lucid and orderly; even a beginner, progressing step by step, can follow the explanation throughout.

=夏鸞翔=夏鸞翔，字紫笙，錢塘人。以輸餉議敘，得詹事府主簿。為項梅侶入室弟子。講究曲線諸術，洞悉員出於方之理。匯通各法，推演以盡其變，譔洞方術圖解二卷，自序略曰：「自杜氏術出，而求弦矢得捷徑焉。顧猶煩乘除，演算終不易，思一可省乘除之法而迄未得。丁巳夏，客都門，細思連比例術者，尖堆底也。尖堆底之比例，與諸乘方之比例等。以之求連比例術，必合諸乘方積而並求之。設不得諸乘方積遞差之故，方積何能並求？且並求方積而欲以加減代乘除，又必得諸較自然之數而後可，誠極難矣。既而悟曰，方積之遞加，加以較也。較之遞生，生於三角堆也。較加較而成積，亦較加較而成較。且諸乘方積之數與諸乘尖堆之數，數異而理同。三角堆起於三角形，故屢次增乘，皆增以三角。方積起於正方形，故累次增乘，皆增以正方。三角之較數，增一根則增一較；方積之較數，增一乘則增一較，理正同也。累次相較，較必有盡，惟其有盡，乃可入算。相連諸弦矢所以愈相較而較愈均者，正此理矣。諸較之理，皆起於天元一，而生於根差。遞加根一，諸乘方根差皆一。一乘之數不變，故可省乘。若增其根差，非復單一，則乘不能省。弦矢弧背之差，或一秒，或十秒，即以一秒、十秒弧線當根差，按根遞求，即可盡得諸乘方之較。以較加較，即盡得所求弦矢各數矣，豈不捷哉！爰演為求弦矢術，俾求表者得以加減代乘除。並細繹立術之義，以俟精於術數者採擇。」

=Xia Luanxiang= Xia Luanxiang, courtesy name Zisheng, was a native of Qiantang. Through meritorious service in transporting provisions, he was appointed Recorder in the Household Administration of the Heir Apparent. He was an inner disciple of Xiang Meilü. He studied methods of curves and thoroughly grasped the principle that the circle arises from the square. Synthesizing various methods and developing them to exhaust their possibilities, he wrote Illustrated Explanation of Cave-Square Methods in two volumes; his preface states in part: "Since Du's method appeared, finding chord and sagitta had a swift path. Yet multiplication and division remained tedious, and calculation was still difficult; he sought a way to avoid them but had not yet found one. In the summer of the dingsi year, while staying in the capital, he reflected carefully: continued proportion rests on the base of the pointed pile. The proportion at the base of the pointed pile equals that of successive powers. To derive continued proportion from this, one must combine the products of successive powers and solve them together. Without knowing why the products of successive powers differ in succession, how could square products be solved together? And to solve square products together while replacing multiplication and division with addition and subtraction, one must first obtain the natural difference numbers—a truly formidable task. Then he realized: the successive building of square products proceeds by adding differences. Differences arise in succession from the triangular pile. Differences added to differences yield a product; differences added to differences also yield further differences. Moreover, the figures for various power products and for various pointed piles differ numerically, yet their underlying principle is the same. The triangular pile originates from the triangle; hence each successive multiplication adds by triangular increments. The square product originates from the square; hence each successive multiplication adds by square increments. For triangular difference numbers, each increment of one root adds one difference. For square-product difference numbers, each increment of one multiplication adds one difference—the principle is precisely the same. With successive comparison, differences must eventually terminate; only because they terminate can they be brought into calculation. This is precisely why, as connected chord and versed sine values are compared further and further, the comparisons grow ever more uniform. The principles governing all differences arise from the celestial element one and are generated from root differences. When the root is increased by one at each step, the root differences for all powers are one. Because the first-multiplication numbers do not change, multiplication can be dispensed with. If the root difference is increased and is no longer uniform, multiplication can no longer be omitted. Whether the difference between chord, versed sine, and arc length is one second or ten seconds, one may take one-second or ten-second arc segments as the root difference and seek by root in succession to obtain fully the differences of all powers. By adding difference to difference, one fully obtains all the sought chord and versed sine values—how swift indeed! He then developed this into a method for finding chord and versed sine, enabling table-makers to use addition and subtraction in place of multiplication and division. He also explained in detail the rationale of the established method, to await adoption by those skilled in computation."

又譔致曲術一卷，曰平員，曰橢員，曰拋物線，曰雙曲線，曰擺線，曰對數曲線，曰螺線，凡七類。類皆自定新術，參差並列，法密理精。复著致曲圖解一卷，謂天為大員，天之賦物，莫不以員。顧員雖一名，形乃萬類。循員一匝，而曲線生焉。西人以線所生之次數分為諸類，一次式為直線；二次式有平員、橢員、拋物線、雙曲線四式；三次式有八十種；四次式有五千餘種；五次以上，殆難以數計矣。今但二次式四種，溯其本源，並附解諸乘方。拋物線形雖萬殊，理實一貫。諸曲線式備具於員錐體，員錐者，二次曲線之母也。橢員利用聚，拋物線利用遠，雙曲線利用散，其理皆出於平員。苟會其通，則制器尚象，仰觀俯察，為用無窮矣。今為一一解之，其目為諸曲線始於一點終於一點第一，諸式之心第二，準線第三，規線第四，橫直二徑第五，兌徑亦名相屬二徑第六，兩心差第七，法線切線第八，斜規線又名曲率徑第九，縱橫線式第十，諸式互為比例第十一，八線第十二。

He also compiled one fascicle on methods for curves: plane circle, ellipse, parabola, hyperbola, cycloid, logarithmic curve, and spiral—seven categories in all. Each category employed its own newly devised method, set out in parallel columns with dense procedures and refined principles. He also wrote one fascicle of Illustrated Explanations of Curves, arguing that Heaven is the great circle and that all things Heaven endows take circular form. Though "circle" is but one term, its forms are myriad. Trace a circle once around, and curves arise. Westerners classify curves by the order of the generating line; a first-order expression is a straight line. Second-order expressions comprise four forms: plane circle, ellipse, parabola, and hyperbola. Third-order expressions have eighty kinds. Fourth-order expressions number more than five thousand kinds. From the fifth order upward, they are nearly beyond reckoning. Here he treated only these four second-order forms, traced them to their origins, and appended explanations of various powers. Though parabolic forms are infinitely varied, their underlying principle is in fact one continuous thread. All curve formulas are fully embodied in the circular cone; the circular cone is the mother of second-order curves. The ellipse serves concentration, the parabola distance, and the hyperbola dispersion; all their principles derive from the plane circle. Once one grasps their common principle, whether in making instruments, honoring symbols, or observing heaven and earth, the applications are inexhaustible. He now explained each in turn under twelve headings: that all curves begin at one point and end at one point; the centers of the various forms; the directrix; the latus rectum; the horizontal and vertical diameters; the conjugate diameters, also called associated diameters; the difference of the two centers; the normal and tangent; the oblique latus rectum, also called the radius of curvature; the horizontal and vertical line formulas; mutual proportions among the forms; and the eight lines.

又嘗立捷術以開各乘方，不論益積、翻積，通為一術，俱為坦途，可徑求平方根數十位，成少廣縋鑿一卷。

He also devised a swift method for extracting roots of all powers, whether positive or negative products, unified into a single procedure on level ground, enabling direct extraction of square roots to ten places—forming one fascicle called Shao Guang by Rope and Chisel.

鸞翔同治三年卒。因方積之較而悟求求弦矢之術，駸駸乎駕西人而上之，然微分所棄之常數，猶方積之方與隅也。所求之變數，猶兩廉遞加之較也。其術施之曲線，無所不通，鸞翔猶待逐類立術，是則不能不讓西人以獨步。然西法開方，自三次式以上，皆枝枝節節為之，不及中法之一貫。鸞翔又於中法外獨創捷術，非西人所能望其項背雲。

Luanxiang died in the third year of the Tongzhi reign. Having grasped from the differences of square products the method for finding chord and versed sine, he was rapidly overtaking Westerners; yet the constants discarded in differential calculus remain like the square and corner of a square product. The variable sought is still like the difference of successive lateral additions. Applied to curves, his method is universally applicable; yet Luanxiang still had to establish methods category by category—thus he could not but yield to Westerners' unrivaled advance. Yet Western methods of root extraction, from the third order upward, proceed branch by branch and node by node, falling short of the Chinese method's unified thread. Luanxiang also independently created swift methods beyond the Chinese tradition—beyond what Westerners could hope to match.

=鄒伯奇=鄒伯奇，字特夫，南海諸生。聰敏絕世，覃思聲音文字度數之源。尤精天文曆算，能薈萃中、西之說而貫通之，靜極生明，多具神解。嘗作春秋經傳日月考，謂：「昔人考春秋者多矣，類以經、傳日月求之，未能精確。今以時憲術上推二百四十二年之朔閏及食限，然後以經、傳所書，質其合否，乃知有經誤、傳誤及術誤之分。」又謂：「尚書克殷年月，鄭玄據乾鑿度，以入戊午蔀四十二年克殷，下至春秋，凡三百四十八年。劉歆三統術以為積四百年，近人錢塘李銳皆主其說。今以時憲術上推，且以歲星驗之，始知鄭是劉非。」其解孟子「由周而來，七百有餘歲」句，謂閻百詩孟子生卒年月考據大事記及通鑑綱目，以孟子致為臣而歸在周赧王元年丁未，逆數至武王有天下，歲在己卯，當得八百有九年。然週共和以上年數，史遷已不能紀，可考者魯世家耳，此為劉歆歷譜所據。然將歆譜與史記比對，歆於煬公、獻公等年分多所加，共計五十二。若減其所加，則歆所謂八百有九年者，實七百五十七年耳。

=Zou Boqi= Zou Boqi, courtesy name Tefu, was a licentiate of Nanhai. Of intelligence surpassing his age, he pondered deeply the sources of sound, writing, and measures. He was especially skilled in astronomy and calendrical calculation, able to gather Chinese and Western theories and connect them; in deep stillness clarity arose, and many of his insights seemed divinely inspired. He once wrote An Examination of the Sun and Moon in the Spring and Autumn Classic and Commentary, stating: "Former scholars who examined the Spring and Autumn are many; most sought the sun and moon in the classic and commentary, yet could not achieve precision. Now, using the Shixian calendar to project upward the intercalations and eclipse limits of two hundred forty-two years, and then testing against what the classic and commentary record, one learns that errors may lie in the classic, in the commentary, or in the method itself." He also stated: "For the year and month of King Wu's conquest of Shang in the Documents, Zheng Xuan relied on the Qian Zao Du, placing the conquest in the forty-second year of the wuwu sequence; down to the Spring and Autumn, this totals three hundred forty-eight years. Liu Xin's Triple Concordance system holds it to be four hundred accumulated years; the recent scholar Li Rui of Qiantang upheld this view. Projecting upward by the Shixian calendar and verifying by the year star, one finds Zheng was correct and Liu was wrong." His explanation of Mencius's line "from the Zhou down, more than seven hundred years" holds that Yan Baichuan's examination of Mencius's birth and death years, based on the Record of Major Events and the Comprehensive Mirror, places Mencius's departure after serving as minister and return home in the first year of King Nan of Zhou, dingwei; counting back to Wu Wang's possession of the realm, the year was jimao, yielding eight hundred nine years." Yet for years above the Gonghe era of Zhou, Sima Qian could no longer record them; what can be verified is only the Annals of Lu—this is what Liu Xin's chronological tables relied upon. Yet comparing Liu Xin's tables with the Records of the Historian, Liu added to the year assignments of Duke Yang, Duke Xian, and others, totaling fifty-two. If what he added is subtracted, then what Liu called eight hundred nine years is in fact seven hundred fifty-seven years.

又謂向來注經者，於算學不盡精通，故解三禮制度多疏失，因作深衣考，以訂江永之謬。作戈戟考，以指程瑤田之疏。以文選景福殿賦「陽馬承阿」證古宮室阿棟之制。以體積論樐氏為量，以重心論懸磬之形，皆繪圖立說，援引詳明。

He also observed that commentators on the classics hitherto were not fully versed in mathematics, so their explanations of the institutions of the Three Rites were often loose and erroneous; therefore he wrote An Examination of the Deep Garment to correct Jiang Yong's errors. He wrote An Examination of the Dagger-axe to expose Cheng Yaotian's errors. Using the Jingfu Palace Rhapsody's "yang ma bearing the eaves" from the Selections of Refined Literature, he verified the eaves-and-purlin system of ancient palace architecture. Discussing the measure of the Lei clan by volume and the form of suspended bells by center of gravity, he drew diagrams and set forth theories, citing sources in detail.

又嘗謂群經註疏引算術未能簡要，甄鸞五經算術既多疏略，王伯厚六經天文篇博引傳注，亦無辨證。因即經義中有關於天文、算術，為先儒所未發，或發而未闡明者，隨時錄出，成學計一得二卷。

He also observed that commentaries on the classics cited mathematical methods without sufficient brevity; Zhen Luan's Mathematics of the Five Classics was largely loose, and Wang Bo's astronomical chapter in the Six Classics cited traditional commentary without verification. Therefore, wherever passages in the classics bearing on astronomy and mathematics had points not yet raised by earlier scholars, or raised but not clarified, he recorded them as they arose, forming two fascicles called One or Two Gains from Study.

天象著甲寅恆星表、赤道星圖、黃道星圖各一卷，自序略曰：「甲寅春，制渾球，以考證經史恆星出沒歷代歲差之故。然制器必先繪圖，繪圖必先立表，此恆星表之所由作也。史、漢、晉、隋諸志，於恆星但言部位，至唐、宋始略有去極度數，蓋舊傳新圖，大抵據步天歌意想為之，與天象不符。國朝康熙初，南懷仁作靈臺儀象志，然後黃、赤經、緯各列為表。乾隆九年，增修儀象考成，補正缺誤。道光甲辰，再加考測，為儀象考成續編，入表正座一千四百四十九星，外增一千七百九十一星，洵為明備。今逾十載，歲漸有差，故复據現時推測立表，庶繪圖制器密合天行也。」

On celestial phenomena he wrote the Jiayin Fixed Star Table, Equatorial Star Chart, and Ecliptic Star Chart, each one fascicle; his preface states briefly: "In the spring of jiayin, I made an armillary sphere to verify the rising and setting of fixed stars in the classics and histories and the causes of successive precession. Yet making instruments requires first drawing charts, and drawing charts requires first establishing tables—this is why the fixed star table was made. The histories of Han, Jin, and Sui speak of fixed stars only by position; from Tang and Song there were slightly degrees of distance from the pole, yet old transmitted new charts were mostly imagined according to the Song of Pacing Heaven and did not accord with celestial phenomena. In the early Kangxi reign of our dynasty, Ferdinand Verbiest wrote the Treatise on the Armillary Sphere and Celestial Globe, after which ecliptic and equatorial longitude and latitude were each set out in tables. In the ninth year of Qianlong, the Revised Instrument Star Catalogue was enlarged and corrected for omissions and errors. In jiachen of the Daoguang reign, further observation and measurement produced the Continuation of the Revised Instrument Star Catalogue, entering one thousand four hundred forty-nine stars in the main table and adding one thousand seven hundred ninety-one stars—truly complete and thorough. Now more than ten years have passed and the years gradually differ; therefore he again established tables by current projection, so that drawing charts and making instruments may closely accord with celestial motion."

又謂：「繪地難於算天，天文可坐而推，地理必須親歷。近人不知古法，故疏舛失實。因考求地理沿革，為歷代地圖，以補史書地誌之缺。」

He also stated: "Drawing the earth is harder than calculating heaven; astronomy can be pursued seated, but geography must be personally traversed. Recent men do not know ancient methods, hence their looseness, errors, and lack of reality. Therefore he investigated geographical changes through the ages, making historical maps to supplement gaps in the geographical gazetteers of historical books."

又手摹皇輿全圖，自序略曰：「地圖以天度畫方，至當不易。地球經緯相交皆正角，而世傳輿圖，至邊地竟成斜方形，殊失繪圖原理，其蔽在以緯度為直線也。昔嘗為小總圖，依渾蓋儀，用半度切線，以顯跡象。然州縣不備，且內密外疏，容與實數不符，故復為此圖。其格緯度無盈縮，而經度漸狹，相視皆為半徑與餘弦之比例。橫九幅，縱十一幅，合成地球滂沱四頹之形，欲使所繪之圖與地相肖也。

He also hand-copied the Complete Map of the Imperial Domain; his preface states briefly: "Maps use celestial degrees to draw squares—most correct and unchanging. Earth's longitude and latitude intersect at right angles, yet maps handed down in the world at border regions become oblique squares, greatly losing the principle of map drawing—the flaw lies in taking latitude as straight lines. Formerly he once made a small general map according to the armillary sphere, using half-degree tangents to display traces and phenomena. Yet prefectures and counties were not complete, and it was dense within and sparse without, tolerating discrepancy from real numbers; therefore he made this map again. Its grid latitudes have no excess or deficiency, while longitudes gradually narrow, each viewed as the proportion of radius to versed sine. Horizontally nine panels and vertically eleven panels compose the sprawling four-sloping form of the earth, seeking to make the drawn map resemble the land.

又變西人之舊，作地球正變兩面全圖，其序略曰：「地形渾員，上應天度，經緯皆為員線。作圖者繪渾於平，須用法調劑，方不失其形似。然視法有三，其一在員外視員，法用正弦，則經圈為橢員，緯圈為直線，其形中廣旁狹，作簡平儀用之。其一在員心視員，法用正切，則經圈為直線，緯圈為弧線，其形中曲旁殺，內密外疏，作日晷用之。斯二者，線無定式，量算繁難。且經緯相交，不成正角。其邊際或太促褊，或太展長，以畫地球，既昧方斜本形，复失修廣實數，所不取也。其一在員周視員，法用半切線，經緯圈皆為平員，雖亦內密外疏，而各能自相比例，西人以此作渾蓋儀，最為理精法密。今本之為地球圖，分正背兩面。正面以京師為中線，其背面之中，即為京師對沖之處，尊首都也。旁分二十四向，審中土與各國彼此之勢，定準望也。經緯俱以十度為一格，設分率也。」

He also altered the Westerners' old method, making full maps of the earth in both orthographic and transformed projections; his preface states briefly: "Earth's form is perfectly round, corresponding to celestial degrees above; longitude and latitude are all circular lines. Those who make maps draw the round on the flat and must use methods of adjustment, only then not losing its resemblance. Yet there are three methods of viewing: the first views the circle from outside the circle, using sine; then meridian circles are ellipses and parallel circles are straight lines, wide in the middle and narrow at the sides—used for the simplified plane instrument. The second views the circle from the circle's center, using tangent; then meridian circles are straight lines and parallel circles are arcs, curved in the middle and tapered at the sides, dense within and sparse without—used for sundials. These two methods have no fixed line formulas and are difficult to measure and calculate. Moreover, longitude and latitude do not intersect at right angles. Their margins are either too cramped and narrow or too extended and long; for drawing the earth they both obscure the original square and oblique form and lose the true length and breadth—what is not taken. The third views the circle from the circle's circumference, using half-tangent; meridian and parallel circles are all plane circles—though also dense within and sparse without, each can compare in proportion to itself; Westerners use this for the armillary sphere—it is most refined in principle and dense in method. Now following this as an earth map, it is divided into front and back faces. The front face takes the capital as the central meridian; the center of its back face is precisely the point opposite the capital—honoring the capital. To the sides it divides into twenty-four directions, examining the positions of the central domain and various countries relative to one another, establishing bearings. Longitude and latitude each take ten degrees as one grid, setting the scale."

因推演其法，著測量備要四卷，分備物致用、按度考數二題。備物致用其目四：一丈量器，曰插標、曰線架、曰指南尺、曰曲尺、曰丈竹、曰竹籌、曰皮活尺、曰蕃紙簿、曰鉛筆；二測望儀，曰指南分率尺、曰立望表、曰三腳架、曰矩尺、曰地平經儀、曰平水準、曰紀限儀、曰迴光環、曰折照玻璃屋、曰千里鏡、曰象限儀、曰秒分時辰標、曰行海時辰標、曰析分大日晷、日風雨針、曰寒暑針；三檢覈書，曰志書、曰地圖、曰星表、曰星圖、曰度算版、曰對數尺、曰八線表、曰八線對數表、曰十進對數表，曰現年行海通書、曰清蒙氣差表、曰太陽緯度表、曰日晷時差表、曰句陳四遊表、曰大星經緯表、曰對數較表、曰對數較差表；四畫圖具，曰大小幅紙、曰硯、曰墨、曰硃、曰顏色料、曰筆、曰五色鉛筆、曰筆殼、曰指南分率矩尺、曰長短界尺、曰平行尺、曰分微尺、曰機翦、曰交連比例規、曰玻璃片、曰橡皮。

Thereupon he deduced and developed its method, writing Essentials of Surveying in four fascicles, divided into two topics: preparing instruments for use, and examining numbers by degrees. Preparing instruments for use has four sections: first, measuring instruments—insertion markers, line frames, compass rulers, curved rulers, bamboo rods, bamboo tallies, leather flexible rulers, foreign paper notebooks, and pencils. Second, observation instruments—compass scale rulers, vertical observation poles, tripods, set squares, horizontal meridian instruments, levels, limit-recording instruments, reflection rings, refracting glass houses, telescopes, quadrant instruments, second-minute-hour markers, sea-travel hour markers, large divided sundials, wind-and-rain needles, and temperature needles. Third, verification books—gazetteers, maps, star tables, star charts, degree-calculation boards, logarithmic rulers, eight-line tables, eight-line logarithmic tables, decimal logarithmic tables, current Nautical Almanac, refraction and atmospheric correction tables, solar latitude tables, sundial equation tables, circumpolar four-motion tables, major star longitude and latitude tables, logarithmic comparison tables, and logarithmic difference tables. Fourth, drawing implements—large and small sheets of paper, inkstones, ink, cinnabar, color pigments, brushes, five-color pencils, pen cases, compass scale set squares, long and short boundary rulers, parallel rulers, minute-dividing rulers, scissors, proportional compasses, glass plates, and erasers.

按度考數其目四：一明數，曰尺度考、曰畝法、曰里法、曰方向法、曰經緯里數；二步量，曰量田計積、曰步地遠近、曰記方向曲折、曰認山形、曰準望所見；三測算，曰測量方向遠近法、曰測地緯度法、曰論平陽大海地平界角、曰測地經度法、曰經緯方向里數互求法；四布圖，曰正紙幅、曰定分率、曰縮展、曰識別設色。

Examining numbers by degrees has four sections: first, clarifying numbers—examination of scale, mu method, li method, direction method, and longitude-latitude li numbers. Second, pacing measurement—calculating field area, pacing distances, recording direction bends and turns, recognizing mountain forms, and aligning visible landmarks. Third, survey calculation—methods for measuring direction and distance, methods for measuring terrestrial latitude, discussion of the horizon angle at level plains and the open sea, methods for measuring terrestrial longitude, and mutual methods for seeking longitude, latitude, direction, and li numbers. Fourth, layout drawing—correct paper format, setting scale, reduction and enlargement, and identification coloring.

又因修改對數表之根求析小術，是開極多乘方法，可徑求自然對數，即訥對數，以十進對數根乘之即得十進對數，著乘方捷術三卷。

He also revised the root-seeking and decomposition method of logarithmic tables—this is the method of opening extremely many multiplications, enabling direct extraction of natural logarithms, that is Napierian logarithms; multiplied by the decimal logarithm root one obtains decimal logarithms, writing Swift Methods of Powers in three fascicles.

又創對數尺，蓋因西人對數表而變通其用，畫數於兩尺，相並而伸縮之，使原有兩數相對，而今有數即對所求數。一曰形制，二曰界畫，三曰致用，四曰諸善，五曰圖式，為記一卷。

He also created the logarithmic scale, adapting the Western logarithmic table by inscribing numbers on two rulers placed together and extended or contracted, so that when two given numbers face each other, the sought number faces the present number. The work comprises five sections—form and structure, boundary drawing, practical use, various excellences, and chart patterns—forming one fascicle of record.

又嘗撰格術補一卷，同郡陳澧序之，略曰：「格術補者，古算家有格術，久亡，而吾友鄒徵君特夫補之也。格術之名，見夢溪筆談，其說云：'陽燧照物，迫之則正，漸遠則無所見，過此則倒，中間有礙故也。如人搖艫，臬為之礙，本末相格，算家謂之格術。 '又云：'陽燧面窪，向日照之，則光聚向內，離鏡一二寸，聚為一點，著物火發。 '筆談之說，皆格術之根源也。宋以前蓋有推演為算書者，後世失傳，遂無有知此術者。徵君得筆談之說，觀日光之景，推求數理，窮極微眇，知西人製鏡之法皆出於此。乃為書一卷，以補古算家之術。蓋古所謂陽燧者，鑄金以為鏡也，西洋鐵鏡，即陽燧，玻璃為鏡，亦同此理。故推陽燧之理，可以貫而通之。有此書而古算家失傳之法復明，可知西人制器之法，實古算家所有，此今世之奇書也。至若古算失傳，如此者當復不少，吾又因此而感慨系之矣！」

He also compiled one fascicle, Supplement to the Ge Method; Chen Li of the same commandery prefaces it, stating briefly: "The Supplement to the Ge Method—ancient mathematicians had the ge method, long lost, and my friend Gentleman Zou the Recluse Tefu restored it. The name ge method appears in Dream Pool Essays, which explains: "When the burning mirror illuminates an object, held close the image is upright; as it moves farther nothing is seen; beyond that point it inverts—because an obstruction lies in the middle. Like a man swaying a boat, with the standard as the obstruction, root and tip mutually obstruct one another—mathematicians call this the ge method. It also states: "The burning mirror's surface is concave; when turned toward the sun, light gathers inward; one or two inches from the mirror it converges to a point, and touching an object kindles fire. The explanations in the Essays are all the roots of the ge method. Before the Song there were probably treatises that developed it fully; later ages lost the transmission, and thus none remained who knew this method. The Gentleman obtained the Essays' explanation, observed images cast by sunlight, pursued numerical principles to the utmost subtlety, and saw that Western methods of making mirrors all derive from this. He then wrote one fascicle to restore the ancient mathematicians' method. What the ancients called the burning mirror was cast metal fashioned into a mirror; Western iron mirrors are burning mirrors, and glass mirrors follow the same principle. Therefore by extending the principle of the burning mirror, one can connect and understand them all. With this book the lost method of ancient mathematicians is restored to clarity; one sees that Western methods of making instruments were in fact already possessed by ancient mathematicians—this is a marvelous book of the present age. Ancient mathematics lost in transmission must include many such cases; I am again moved by this to deep reflection!"

同治三年，郭嵩燾特疏薦之，堅以疾辭。曾國籓督兩江日，欲以上海機器局旁設書院，延伯奇以數學教授生徒，亦未就。八年五月，卒，年五十有一。

In the third year of Tongzhi, Guo Songtao specially memorialized recommending him; he firmly declined on grounds of illness. When Zeng Guofan governed the Two Jiangs, he wished to establish an academy beside the Shanghai Machinery Bureau and invite Boqi to teach mathematics to students—this also did not come to pass. In the fifth month of the eighth year he died, aged fifty-one.

=李善蘭=李善蘭，字壬叔，海寧人。諸生。從陳奐受經，於算術好之獨深。十歲即通九章，後得測圓海鏡、句股割圜記，學益進。疑割圜法非自然，精思得其理。嘗謂道有一貫，藝亦然。測圓海鏡每題皆有法有草，法者，本題之法也；草者，用立天元一曲折以求本題之法，乃造法之法，法之源也。算術大至躔離交食，細至米鹽瑣碎，其法至繁，以立天元一演之，莫不能得其法。故立天元一者，算學中之一貫也。並時明算如錢塘戴煦，南匯張文虎，烏程徐有壬、汪曰楨，歸安張福僖，皆相友善。咸豐初，客上海，識英吉利偉烈亞力、艾約瑟、韋廉臣三人，偉烈亞力精天算，通華言。善蘭以歐幾里幾何原本十三卷、續二卷，明時譯得六卷，因與偉烈亞力同譯後九卷，西士精通幾何者尟，其第十卷尤玄奧，未易解，譌奪甚多，善蘭筆受時，輒以意匡補。譯成，偉烈亞力歎曰：「西士他日欲得善本，當求諸中國也！」

=Li Shanlan= Li Shanlan, courtesy name Renshu, was a native of Haining. He was a licentiate. He studied the classics under Chen Huan and loved arithmetic with especial depth. At ten he already mastered the Nine Chapters; later obtaining Sea Mirror of Circle Measurement and Record of Right Triangles and Circle Division, his learning advanced further. He doubted that the circle-cutting method was not natural and, thinking deeply, obtained its principle. He once stated that the Way has one thread, and the arts are likewise. In Sea Mirror of Circle Measurement each problem has both a method and a draft; the method is the method of the original problem. The draft uses the celestial element one in twists and turns to seek the method of the original problem—it is the method of making methods, the source of methods. Arithmetic ranges from the great—planetary motion and eclipses—to the small—rice, salt, and trivial matters; its methods are extremely numerous, yet deployed through the celestial element one, none fail to yield their method. Therefore the celestial element one is the one thread within mathematics. Contemporary masters of calculation such as Dai Xu of Qiantang, Zhang Wenhu of Nanhui, Xu Youren and Wang Yuezhi of Wucheng, and Zhang Fuxi of Guian were all on friendly terms with him. In the early Xianfeng reign he was a guest in Shanghai and met three Englishmen—Alexander Wylie, Joseph Edkins, and William Muirhead; Wylie was skilled in astronomy and calculation and understood Chinese. Shanlan, regarding Euclid's Elements in thirteen fascicles with two continuations—six fascicles translated in the Ming—thereupon co-translated the latter nine fascicles with Wylie; Western scholars skilled in geometry were few, and the tenth fascicle was especially abstruse and not easy to understand, with many errors and omissions; when Shanlan received it by brush he often corrected and supplemented according to his understanding. When the translation was complete, Wylie sighed and said: "Western scholars hereafter wishing to obtain a good edition must seek it in China!"

偉烈亞力又言美國天算名家羅密士嘗取代數、微分、積分合為一書，分款設題，較若列眉，復與善蘭同譯之，名曰代微積拾級十八卷。代數變天元、四元，別為新法，微分、積分二術，又藉徑於代數，實中土未有之奇秘。善蘭隨體剖析自然，得力於海鏡為多。

Wylie also stated that the American master of astronomy and calculation Elias Loomis once combined algebra, differential calculus, and integral calculus into one book, dividing sections and setting problems as clearly as aligned eyebrows; he again co-translated it with Shanlan, entitled Step by Step through Algebra, Differential and Integral Calculus in eighteen fascicles. Algebra transformed the celestial element and four elements into a new method; differential and integral calculus, the two techniques, also borrow their path through algebra—truly marvelous secrets not previously possessed in the central domain. Shanlan analyzed naturally according to the subject, drawing much strength from the Sea Mirror.

粵匪陷吳、越，依曾國籓軍中。，用巡撫郭嵩燾薦，徵入同文館，充算學總教習、總理衙門章京，授戶部郎中、三品卿銜。課同文館生以海鏡，而以代數演之，合中、西為一法，成就甚眾。光緒十年，卒於官，年垂七十。

When Cantonese bandits overran Wu and Yue, he attached to Zeng Guofan's army. By recommendation of Governor Guo Songtao, he was summoned to the Tongwen Guan, serving as chief instructor of mathematics and a clerk of the Zongli Yamen, and was granted the rank of Director in the Ministry of Revenue with the honorary title of Third Rank. He taught Tongwen Guan students with the Sea Mirror, deploying it through algebra, merging Chinese and Western into one method, and achieved many disciples. In the tenth year of Guangxu he died in office, aged nearly seventy.

善蘭聰彊絕人，其於算，能執理之至簡，馭數至繁，故衍之無不可通之數，抉之即無不可窮之理。所著則古昔齋算學，詳藝文志。世謂梅文鼎悟借根之出天元，善蘭能變四元而為代數，蓋梅氏後一人云。

Shanlan's intelligence surpassed others; in calculation he could grasp the simplest principles and command the most complex numbers; therefore extended, no number was beyond reach, and extracted, no principle was beyond exhaustion. His works are the mathematics of the Zeguxi Studio, detailed in the Treatise on Literature. The age says Mei Wending understood that borrowed roots derive from the celestial element; Shanlan could transform the four elements into algebra—he was perhaps the one man after the Mei clan.

=華衡芳=華衡芳，字若汀，金匱人。能文善算，著有行素軒算學行世。其筆談一書，猶為生平精力所聚。凡十二卷，第一卷論加、減、乘、除之理；第二卷論通分之理；第三卷論十分數；第四卷論開方之理；第五卷論看題、馭題之法，以明加、減、乘、除、通分、開方之用；第六卷論天元及天元開方；第七卷論方程之術，已寓四元之意，末乃專論四元；第八卷論代數釋號及等式；第九卷論代數中助變之數及虛代之法；第十卷論微分；第十一卷論積分，分十六款以明之；第十二卷一論各種算學不外乎加、減、乘、除，二論一切算稿宜筆之於書，三論算學中可以著書之事，四論學算與著書並非兩事，五論繙算學之書，六論疇人傳當再續。綜計自加、減、乘、除、通分以至微分、積分，由淺入深，術本繁難，而括之以簡易之旨；理本艱深，而寫之以淺顯之詞。

=Hua Hengfang= Hua Hengfang, courtesy name Ruoting, was a native of Jin Gui. Able in letters and skilled in calculation, he authored the Xingsuxuan Mathematics circulated in the world. His Brush Talks was still where the energy of his lifetime was concentrated. In all twelve fascicles: the first fascicle discusses the principles of addition, subtraction, multiplication, and division. The second fascicle discusses the principles of common fractions. The third fascicle discusses decimal fractions. The fourth fascicle discusses the principles of root extraction. The fifth fascicle discusses methods of reading and handling problems, to clarify the uses of addition, subtraction, multiplication, division, common fractions, and root extraction. The sixth fascicle discusses the celestial element and celestial-element root extraction. The seventh fascicle discusses the method of equations, already embodying the idea of four elements, and at the end specially discusses four elements. The eighth fascicle discusses algebraic notation and equations. The ninth fascicle discusses auxiliary variable numbers in algebra and the method of virtual substitution. The tenth fascicle discusses differential calculus. The eleventh fascicle discusses integral calculus, divided into sixteen sections for clarity. The twelfth fascicle: first, that all kinds of mathematics are nothing beyond addition, subtraction, multiplication, and division; second, that all calculation drafts should be written in books; third, matters in mathematics on which one may write books; fourth, that studying calculation and writing books are not two separate things; fifth, on translating mathematical books; sixth, that the Biographies of Calculators should be continued. Taken together, from addition, subtraction, multiplication, division, and common fractions through differential and integral calculus, proceeding from shallow to deep—the methods are inherently complex, yet encompassed under a principle of simplicity. The principles are inherently difficult and deep, yet written in plain and accessible language.

又於，與英士傅蘭雅共譯代數術二十五卷，衡芳序之曰：「代數之術，其已知、未知之數，皆代之以字，而乘、除、加、減各有記號，以為區別，可如題之曲折以相赴。迨夫層累已明，階級已見，乃以所代之數入之，而所求之數出焉。故可以省算學之工，而心亦較逸，以其可不假思索而得也。雖然，代數之術誠簡便矣，試問工此術者，遂能不病其繁乎？則又不能也。夫人之用心，日進而不已，苟不至昏眊迷亂，必不肯終輟。故始則因繁而求簡，及其既簡也，必更進焉，而復遇其繁，雖迭代數十次，其能免哉？自是知代數之意，乃為數學中鉤深索隱之用，非為淺近之算法設也。若米鹽零雜之事，而概欲以代數施之，未有不為市儈所笑者也。至於代數、天元之異同優劣，讀此書者自能知之，無待餘言也。」

He also, together with the Englishman John Fryer, co-translated Algebra in twenty-five fascicles; Hengfang prefaces it, stating: "In the method of algebra, known and unknown numbers are all substituted with letters, and multiplication, division, addition, and subtraction each have symbols for distinction, so one may follow the twists of the problem to meet them. When the layers are clear and the stages appear, one then inserts the substituted numbers, and the sought number emerges. Therefore one can save labor in calculation, and the mind is also more at ease, because one may obtain results without relying on reflection. Although the method of algebra is indeed convenient, ask those skilled in this method—can they truly avoid finding it burdensome? They cannot. Human effort of mind advances daily without cease; unless one reaches blindness and confusion, one will surely not stop forever. Therefore at first one seeks simplicity because of complexity; once it is simple, one must advance further and again encounter complexity—even iterating dozens of times, how can one escape? From this one knows the intent of algebra is for probing depth and seeking hidden principles in mathematics, not established for shallow and near-at-hand algorithms. If for rice, salt, and miscellaneous trifles one wishes to apply algebra universally, there is none who is not laughed at by market peddlers. As for the similarities, differences, strengths, and weaknesses of algebra and the celestial element, readers of this book can know them themselves—no need for further words."

又與傅蘭雅共譯微積溯源八卷，序之曰：「吾以為古時之算法，惟有加、減而已。其乘與除乃因加減之不勝其繁，故更立二術以使之簡易也。開方之法，又所以濟除法之窮者也。蓋學算者自有加、減、乘、除、開方五法，而一切簡易淺近之數，無不可通矣。惟人之心思智慮日出不窮，往往以能人之所不能者為快，遇有窒礙難通之處，輒思立法以濟其窮，故有減其所不可減，而正負之名不得不立矣；除其所不受除，而寄母通分之法又不得不立矣。代數中種種記號之法，皆出於不得已而立者也。惟每立一法，必能使繁者為簡，難者為易，遲者為速，而算學之境界，藉此得更進一層。如是屢進不已，而所立之法，於是乎日多矣。微分、積分者，蓋又因乘、除、開方之不勝其繁，且有窒礙難通之處，故更立此二術以濟其窮，又使簡易而速者也。試觀圜徑求週、真數求對數之事，雖無微分、積分之時，亦未嘗不可求，惟須乘、除、開方數十百次，其難有不可言喻者。不如用微積之法，理明而數捷也。然則謂加、減、乘、除、代數之外，更有二術焉，一曰微分，一曰積分可也。其積分猶微分之還原，猶之開方為自乘之還原，除法為乘法之還原，減法為加法之還原也。然加與乘，其原無不可還，而微分之原，有可還有不可還者，是猶算式中有不可還原之方耳，又何怪焉！如必曰加減乘除開方已足供吾之用，何必更求其精？是捨舟車之便利，而必欲負重遠行也。其用力多而成功少，蓋不待智者而辨矣。又代數術中末卷之中，載求平員周率簡捷法式，為猶拉所設。未有此法之時，曾有算學士固靈用平員內容外切之多等邊形，費極大工夫，算得三十六位之數。設徑為一，週為三一四一五九二六五三五八九七九三二三八四六二六四三三八三二七九五零二八八。其臨死之時，囑其家以此數刻於墓碑，蓋平時得意之作，恐其磨滅，故欲傳之永久，亦猶亞基默得之墓，刻一球形與員柱形也。」

He also co-translated with Fryer Calculus Tracing Origins in eight fascicles, prefacing it: "I hold that ancient algorithms had only addition and subtraction. Multiplication and division arose because addition and subtraction could not bear the burden of complexity; therefore two further methods were established to make them simple. The method of root extraction is again what rescues division when it reaches its limit. Generally students of calculation have the five methods of addition, subtraction, multiplication, division, and root extraction, and all simple and near-at-hand numbers can be mastered. Yet human thought and intelligence are inexhaustible day by day; often one delights in what others cannot do; encountering obstructed and difficult passages, one then thinks to establish methods to rescue the limit—therefore there is subtracting what cannot be subtracted, and the names positive and negative must be established. Dividing what cannot be divided, and the method of borrowed denominators and common fractions must again be established. The various symbolic methods in algebra were all established from necessity. Yet each time a method is established, it must make the complex simple, the difficult easy, and the slow swift—and the realm of mathematics, by this means, advances one layer further. Advancing thus again and again without cease, the methods established daily grow more numerous. Differential and integral calculus—again because multiplication, division, and root extraction cannot bear the burden of complexity and have obstructed and difficult passages—therefore these two methods were further established to rescue the limit, again making matters simple and swift. Consider finding circumference from diameter or true numbers from logarithms: even without differential and integral calculus it is not impossible to seek, yet it requires multiplication, division, and root extraction dozens or hundreds of times—the difficulty is beyond words. Better to use the methods of differential and integral calculus—principles clear and numbers swift. Thus one may say that beyond addition, subtraction, multiplication, division, and algebra there are two further methods: one called differential calculus, one called integral calculus. Integral calculus is the restoration of differential calculus, just as root extraction is the restoration of self-multiplication, division the restoration of multiplication, and subtraction the restoration of addition. Yet addition and multiplication have origins that can all be restored, while the origins of differential calculus have some that can be restored and some that cannot—this is like formulas in which there are irreducible powers; what is strange about it! If one must say addition, subtraction, multiplication, division, and root extraction are already sufficient for our uses, why seek further refinement? This is abandoning the convenience of boats and carts yet insisting on carrying heavy loads and traveling far on foot. Much effort and little success—this needs no wise man to distinguish. Also in the middle of the final fascicle of Algebra is recorded a swift formula for seeking the ratio of plane circle circumference, set by Euler. Before this method existed, the mathematician Ludolph van Ceulen once used many-sided figures inscribed in and circumscribed about a plane circle, expending enormous effort to calculate a number of thirty-six places. Setting diameter as one, circumference as 3.14159265358979323846264338327950288. At the time of his death he instructed his family to inscribe this number on his tombstone—his usual proud achievement, fearing it would fade, therefore wishing to transmit it permanently—just as Archimedes had a sphere and cylinder inscribed on his tomb."

又與傅氏共譯三角數理，此書為英士海麻士所譔。海麻士專精三角、八線之學，著書十有二卷，皆言三角數理，即用為名。首明三角用比例之理；次論兩角或多角諸比例數；次論造八線比例表之法；次解平三角諸形；次論諸角比例乘約變化之理；紀彼國算士棣弗美創例也，附以專論對數術及諸三角形設題一百則，為書三卷，以引學者；次總說球上各圈及弧三角形之界；次解正弧斜弧三角形之法；次雜論求弧三角數種特設之表；終以弧三角形設題二十七則焉。然書中說解過於煩費，仍不能變外角和較與垂弧、次形、總較諸舊法，故自海氏書出，益覺徐有壬拾遺三術難能可貴，超越西人。

He also co-translated with Fryer Trigonometry; this book was composed by the Englishman Thomas Hymers. Hymers specialized in trigonometry and the eight lines, writing twelve fascicles of books, all treating trigonometry—using that as the title. First clarifying the principle of using proportions in triangles. Next discussing proportional numbers of two angles or multiple angles. Next discussing methods for constructing tables of eight-line proportions. Next solving plane triangular forms. Next discussing the principles of multiplication, reduction, and transformation of angular proportions. It records the models established by that country's mathematician Dipeimei, together with a specialized discussion of logarithms and one hundred problems on triangles, in three fascicles, to guide students. Next comes a general account of the circles on the sphere and the boundaries of spherical triangles. Next, methods for solving right-arc and oblique-arc spherical triangles. Next, a miscellaneous discussion of several specially constructed tables for spherical trigonometry. The work concludes with twenty-seven problems on spherical triangles. Yet the book's explanations are excessively prolix, and it still cannot displace the older Chinese methods of exterior-angle sum and difference, perpendicular arc, secondary form, and total difference. Since Hymers' treatise appeared, Xu Youren's three supplementary methods have seemed all the more rare and admirable—surpassing the Western approach.

又與傅氏共譯代數難題解法十六卷。

He also collaborated with Fryer on a sixteen-fascicle translation entitled Solutions to Difficult Problems in Algebra.

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其弟世芳，字若溪。亦通算術，著有近代疇人著述記。

His younger brother Shifang, styled Ruoxi. He too was skilled in mathematics and authored A Record of Writings by Recent Mathematical Scholars.

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