祖暅原理说了什么？和卡瓦列里原理一样吗？
What does Zu Geng's principle say, and is it like Cavalieri's principle?
祖暅提出幂势既同则积不容异：夹在两平行面间的两个立体，若每一处的截面积都相等，则体积相等。这与西方卡瓦列里原理一致，早约1100年，用于求球体积。
Zu Geng stated that solids between parallel planes with equal cross-sectional area at every height have equal volume. This matches Cavalieri's principle, about 1100 years earlier, and was used to find the sphere's volume.
难度 专业 · 适合 高中(15-18)
CC BY 4.0 · 来源待核验 · Source pending review
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出处备注：著录仅给书名，无版次与年份；同一书名常对应多个版次，不臆测具体版本。
祖暅《缀术》
Zu Geng
#祖暅原理 #体积
祖冲之是怎么把圆周率算到小数点后7位的？
How did Zu Chongzhi compute pi to seven decimal places?
他用刘徽的割圆术，把圆内接正多边形的边数从6一直推广到24576边，反复用勾股定理开方，得到3.1415926小于π小于3.1415927，并给出约率22/7和密率355/113。
He used Liu Hui's circle-cutting method, extending the inscribed polygon to 24576 sides and repeatedly applying the Pythagorean theorem, obtaining 3.1415926 < pi < 3.1415927, plus the approximations 22/7 and the remarkably accurate 355/113.
难度 专业 · 适合 高中(15-18)
CC BY 4.0 · 来源待核验 · Source pending review
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出处备注：著录仅给书名，无版次与年份；同一书名常对应多个版次，不臆测具体版本。
祖冲之《缀术》
Zu Chongzhi
#圆周率 #祖冲之
唐代一行的大衍历有什么数学贡献？
What was the mathematical contribution of Yi Xing's Dayan calendar?
唐代一行编大衍历，用不等间距的二次内插法推算日食月食，是早期数值插值的重要应用；他还组织全国天文测量，体现大规模数据处理的思路。
Monk Yi Xing's Dayan calendar used quadratic interpolation with unequal intervals to predict eclipses, an important early use of numerical interpolation, and he organized a nationwide astronomical survey involving large-scale data.
难度 专业 · 适合 大众
CC BY 4.0 · 来源待核验 · Source pending review
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出处备注：著录仅给书名，无版次与年份；同一书名常对应多个版次，不臆测具体版本。
一行《大衍历》
Yi Xing
#大衍历 #插值
北宋沈括的“隙积术”怎样用“堆起来的东西”逼近高阶等差级数求和？
How did Shen Kuo's 11th-century 'xi ji shu' (gap-stacking) approximate sums of higher-order arithmetic series via stacked objects?
沈括在《梦溪笔谈》中提出“隙积术”，计算如酒坛层层堆成四棱台时“有空隙”的实体积：他用上底、下底、层数构造公式，处理的是二阶等差（方垛）求和，较此前“垛积术”更贴合实际堆叠。这是中国古代对高阶等差数列与体积近似的重要推进，早于欧洲同类讨论。
In 'Dream Pool Essays', Shen Kuo's 'xi ji shu' computes the real volume of objects stacked into a frustum (e.g., wine jars) with gaps, using top/base areas and layers to sum second-order arithmetic series—more practical than prior 'duo ji' stacking formulas. It advanced Chinese work on higher-order arithmetic progressions and volume approximation ahead of Europe.
难度 专业 · 适合 高中(15-18)
CC BY 4.0 · 已核验 · Verified 2026-11-02
来源 · Source
吴文俊 主编,《中国数学史大系》, 北京师范大学出版社
Wu, W.-T. (ed.), 'The Grand Series of the History of Chinese Mathematics', Beijing Normal University Press
独立核验依据 · Independent verification: 核验依据 1 ISBN 9787303049264（豆瓣书目条目: 中国数学史大系）
#沈括 #隙积术 #梦溪笔谈 #高阶等差
什么是垛积术？它和现代级数有什么关系？
What is duoji-shu and how does it relate to modern series?
垛积术（沈括、杨辉等）研究堆叠的酒坛、层塔等体积，实质是求高阶等差级数求和，如三角垛、方垛，相当于现代的多项式求和与组合恒等式。
Duoji-shu studied stacked objects like wine jars and pagoda layers, essentially summing higher-order arithmetic series (triangular and square piles), equivalent to modern polynomial summation and combinatorial identities.
难度 专业 · 适合 高中(15-18)
CC BY 4.0 · 已核验 · Verified 2026-11-02
来源 · Source
Martzloff, J.-C., 《中国数学史》, Springer
Martzloff, J.-C., 'A History of Chinese Mathematics', Springer
独立核验依据 · Independent verification: 核验依据 1 OPENLIBRARY 标识 /works/OL32240873W：A History of Chinese M
#垛积术 #级数
秦九韶的正负开方术解决了什么问题？
What problem did Qin Jiushao's positive-negative root extraction solve?
它是一元高次方程的数值解法，记载于1247年《数书九章》，能求任意次方程的实根，原理相当于现代的霍纳法，比西方早数百年。
It is a numerical method for solving polynomial equations of any degree, recorded in Qin Jiushao's 1247 book, equivalent in principle to Horner's method and centuries ahead of the West.
难度 专业 · 适合 高中(15-18)
CC BY 4.0 · 来源待核验 · Source pending review
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出处备注：著录仅给书名，无版次与年份；同一书名常对应多个版次，不臆测具体版本。
秦九韶《数书九章》
Mathematical Treatise in Nine Sections
#秦九韶 #方程
什么是天元术？它和现代代数有什么关系？
What is the tianyuan method and how does it relate to modern algebra?
天元术是金元时期（如李冶）创立的列方程方法，用天元表示未知数，在筹式上布列多项式并令其为0，相当于现代的代数方程思想。
Developed in the Jin-Yuan period (e.g. by Li Ye), the tianyuan method used a symbol for the unknown and arranged polynomials on counting-rod layouts set equal to zero, an early form of algebraic equation solving.
难度 专业 · 适合 高中(15-18)
CC BY 4.0 · 来源待核验 · Source pending review
来源 · Source
出处备注：著录仅给书名，无版次与年份；同一书名常对应多个版次，不臆测具体版本。
李冶《测圆海镜》
Li Ye, Ceyuan Haijing (Jade Mirror on the Universe), 1248
#天元术 #代数
杨辉三角和帕斯卡三角是同一个东西吗？
Is Yanghui's triangle the same as Pascal's triangle?
内容相同：每行数字是上一行相邻两数之和，用来展开二项式系数。杨辉在1261年《详解九章算法》中记载此图，比帕斯卡早约400年，所以中国称杨辉三角。
They are the same object: each entry is the sum of the two above it, giving binomial coefficients. Yang Hui documented it in 1261, about 400 years before Pascal, so in China it is called Yanghui's triangle.
难度 进阶 · 适合 大众
CC BY 4.0 · 来源待核验 · Source pending review
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出处备注：著录仅给书名，无版次与年份；同一书名常对应多个版次，不臆测具体版本。
杨辉《详解九章算法》
Yang Hui's book
#杨辉三角 #二项式
为什么《九章算术》被认为是中国数学的源头？
Why is The Nine Chapters on the Mathematical Art seen as the source of Chinese mathematics?
它成书于汉代，系统收集了246个应用题，涵盖方田、粟米、衰分、少广、商功、均输、盈不足、方程、勾股九类，奠定了中国算法化、重实用的数学传统。
Compiled in the Han dynasty, it collects 246 applied problems in nine categories (fields, grain, proportions, engineering, taxation, excess-deficit, equations, and the Gougu theorem), establishing China's algorithmic and practical mathematical tradition.
难度 进阶 · 适合 大众
CC BY 4.0 · 已核验 · Verified 2026-11-02
来源 · Source
《九章算术》
The Nine Chapters
独立核验依据 · Independent verification: 核验依据 1 ISBN 9787101160802（豆瓣书目条目: 九章算术）
#九章算术 #古代数学
《周髀算经》的“日高术”是怎样用影子长短和相似三角形测出太阳高度的？
How did the 'sun-height method' in the Zhoubi Suanjing measure the sun's height using shadow lengths and similar triangles?
《周髀算经》用“髀”（八尺表竿）立地测影，在不同地点同时量得影差，借助相似直角三角形（今称日高术公式 h = 表高×间距÷影差 + 表高）反推日高与日远。它是最早把勾股定理与比例关系用于天文测量的记录，体现了“以影测天”的中国古代数理天文思想。
The Zhoubi Suanjing set an 8-chi gnomon, measured shadow differences at two sites simultaneously, and used similar right triangles (the formula h = gnomon·distance/shadow-difference + gnomon) to infer the sun's height and distance. It is among the earliest records applying the Pythagorean relation and proportion to astronomy—'measuring heaven by shadow.'
难度 进阶 · 适合 初中(12-15)
CC BY 4.0 · 已核验 · Verified 2026-11-03
来源 · Source
Cullen, C., 《中国古代的天文学与数学：周髀算经》, Cambridge University Press
Cullen, C., 'Astronomy and Mathematics in Ancient China: The Zhou Bi Suan Jing', Cambridge University Press
独立核验依据 · Independent verification: 核验依据 1 ARCHIVE 标识 astronomymathema0000cull：Astronomy and mathem
#周髀算经 #日高术 #相似三角形 #影差
为什么《周髀算经》里就有勾股定理？
Why does the Zhoubi Manual already contain the Pythagorean theorem?
《周髀算经》（约公元前1世纪）记载了勾三股四弦五的特例并用勾股计算日影，说明中国很早就掌握了直角三角形三边关系，用于天文测量。
Dating to about the 1st century BCE, the Zhoubi records the 3-4-5 special case and uses right-triangle relations to measure solar shadows, showing early Chinese mastery of the theorem for astronomy.
难度 进阶 · 适合 初中(12-15)
CC BY 4.0 · 来源待核验 · Source pending review
来源 · Source
出处备注：著录仅给书名，无版次与年份；同一书名常对应多个版次，不臆测具体版本。
《周髀算经》
Zhoubi Suanjing
#周髀算经 #勾股定理
为什么古中国用闰月来协调历法？这和数学有什么关系？
Why did ancient China use leap months, and what does that have to do with math?
一个朔望月约29.53天，12个月约354天，比回归年少约11天。通过19年7闰使历法与季节同步，背后是分数近似与同余思想的运用。
A lunar month is about 29.53 days, so 12 months miss the solar year by about 11 days. Inserting 7 leap months every 19 years keeps the calendar aligned, an application of fractional approximation and congruences.
难度 进阶 · 适合 大众
CC BY 4.0 · 已核验 · Verified 2026-08-23 · 2 条独立证据
来源 · Source
Morris Kline,《Mathematical Thought from Ancient to Modern Times》Oxford University Press,1972;中译《古今数学思想》上海科学技术出版社
M. Kline, Mathematical Thought from Ancient to Modern Times, Oxford University Press, 1972; Chinese ed. Shanghai Scientific & Technical Publishers
独立核验依据 · Independent verification: 核验依据 2 OPENLIBRARY 标识 /works/OL1947921W：Mathematical Thought fr
#闰月 #历法
什么是重差术？为什么它能测远处的山高？
What is the chongcha method and why could it measure distant heights?
重差术是刘徽在《海岛算经》中发展的测量术，利用两次观测的相似三角形之差来推算高远距离，是早期三角测量与比例几何的应用。
Chongcha, developed by Liu Hui in the Sea Island Mathematical Manual, uses the difference of two similar triangles from two observations to find heights and distances, an early form of trigonometric surveying.
难度 进阶 · 适合 大众
CC BY 4.0 · 来源待核验 · Source pending review
来源 · Source
出处备注：著录仅给书名，无版次与年份；同一书名常对应多个版次，不臆测具体版本。
刘徽《海岛算经》
Sea Island Manual
#重差术 #测量
《孙子算经》里的“物不知数”问题，怎样成了中国剩余定理的雏形？
How did the 'unknown number of objects' problem in the Sunzi Suanjing become a prototype of the Chinese remainder theorem?
《孙子算经》（约公元 4 世纪）问：“今有物不知其数，三三数之剩二，五五数之剩三，七七数之剩二，问几何？”它给出“三人同行七十稀，五树梅花廿一枝，七子团圆正月半”的口诀，用 70、21、15（各模其余两两互素数的特解）拼出答案，正是最早的中国剩余定理实例。
The 'Sunzi Suanjing' (c. 4th century) asks for a number that leaves remainders 2,3,2 when divided by 3,5,7, and gives the rhyme using 70,21,15 (special solutions modulo the co-prime counterparts) to assemble the answer—the earliest instance of the Chinese remainder theorem.
难度 进阶 · 适合 初中(12-15)
CC BY 4.0 · 已核验 · Verified 2026-11-02
来源 · Source
Martzloff, J.-C., 《中国数学史》, Springer
Martzloff, J.-C., 'A History of Chinese Mathematics', Springer
独立核验依据 · Independent verification: 核验依据 1 OPENLIBRARY 标识 /works/OL32240873W：A History of Chinese M
#孙子算经 #物不知数 #中国剩余定理 #同余
为什么分数在中国古代特别发达？
Why were fractions especially well developed in ancient China?
农业赋税、度量衡和历法都需要精确分割，所以《九章算术》开篇就系统讲分数四则（合分、减分、乘分、经分），比欧洲早千年建立了完整分数算法。
Taxation, weights and calendars all needed precise division, so the Nine Chapters opens with a full treatment of fraction arithmetic (addition, subtraction, multiplication, division), a complete system a millennium ahead of Europe.
难度 进阶 · 适合 大众
CC BY 4.0 · 已核验 · Verified 2026-11-02
来源 · Source
Martzloff, J.-C., 《中国数学史》, Springer
Martzloff, J.-C., 'A History of Chinese Mathematics', Springer
独立核验依据 · Independent verification: 核验依据 1 OPENLIBRARY 标识 /works/OL32240873W：A History of Chinese M
#分数 #四则
什么是割圆术？为什么边数越多越接近圆？
What is the circle-cutting method and why does more sides mean closer to a circle?
割圆术是刘徽提出的方法：用圆内接正多边形逼近圆周。边数越多，多边形每条边与圆弧之间的空隙越小，其周长就越接近圆周长。
Liu Hui's method approximates a circle with an inscribed regular polygon. As the number of sides grows, the gap between each polygon edge and the arc shrinks, so the perimeter approaches the circle's circumference.
难度 进阶 · 适合 初中(12-15)
CC BY 4.0 · 来源待核验 · Source pending review
来源 · Source
出处备注：著录仅给书名，无版次与年份；同一书名常对应多个版次，不臆测具体版本。
刘徽《九章算术注》
Liu Hui's commentary
#割圆术 #刘徽
什么是盈不足术？古代怎么用它解难题？
What is the ying-bu-zu method and how was it used?
盈不足是《九章算术》一章，先假设两个数值代入，根据多余（盈）与欠缺（不足）的比例反推真值，相当于线性插值法，能解许多难以整除的应用题。
A chapter of the Nine Chapters, it assumes two trial values and uses the excess and deficit to interpolate the true answer, effectively linear interpolation for problems hard to solve by direct division.
难度 进阶 · 适合 大众
CC BY 4.0 · 已核验 · Verified 2026-11-02
来源 · Source
Martzloff, J.-C., 《中国数学史》, Springer
Martzloff, J.-C., 'A History of Chinese Mathematics', Springer
独立核验依据 · Independent verification: 核验依据 1 OPENLIBRARY 标识 /works/OL32240873W：A History of Chinese M
#盈不足 #插值
《九章算术》里的方程章其实讲的是方程组吗？
Does the 'equations' chapter of the Nine Chapters actually deal with systems of equations?
是的，它用算筹摆出增广矩阵，通过直除法（类似高斯消元）解线性方程组，是世界上最早的线性方程组系统解法记录之一。
Yes: it laid out augmented matrices with rods and solved linear systems by a 'direct-division' procedure akin to Gaussian elimination, one of the earliest systematic records of solving linear equations.
难度 进阶 · 适合 初中(12-15)
CC BY 4.0 · 已核验 · Verified 2026-11-02
来源 · Source
Martzloff, J.-C., 《中国数学史》, Springer
Martzloff, J.-C., 'A History of Chinese Mathematics', Springer
独立核验依据 · Independent verification: 核验依据 1 OPENLIBRARY 标识 /works/OL32240873W：A History of Chinese M
#方程 #消元法
算盘和算筹有什么关系？算盘为什么取代算筹？
How is the abacus related to counting rods, and why did it replace them?
算盘是算筹记数的便携化：用珠子代替筹棍，按档定位。它更快、更省空间、便于商业计算，自元明以后逐渐取代算筹。
The abacus made rod arithmetic portable: beads replaced rods on positioned columns. Faster and more compact for commerce, it gradually replaced rods from the Yuan-Ming period onward.
难度 入门 · 适合 小学低段(6-8)
CC BY 4.0 · 已核验 · Verified 2026-11-02
来源 · Source
Martzloff, J.-C., 《中国数学史》, Springer
Martzloff, J.-C., 'A History of Chinese Mathematics', Springer
独立核验依据 · Independent verification: 核验依据 1 OPENLIBRARY 标识 /works/OL32240873W：A History of Chinese M
#算盘 #计算工具
古代中国怎么表示零？
How did ancient China represent zero?
早期用空位表示零（算筹留空），后来在书写中用圆圈○表示空缺的数位。虽然没有独立的零符号参与运算，但十进位值制已经隐含了零的作用。
Early on, an empty space among the rods stood for zero; later a circle symbol marked the empty place. There was no independent operational zero at first, but the decimal place-value system already implied its role.
难度 入门 · 适合 小学低段(6-8)
CC BY 4.0 · 已核验 · Verified 2026-11-02
来源 · Source
Martzloff, J.-C., 《中国数学史》, Springer
Martzloff, J.-C., 'A History of Chinese Mathematics', Springer
独立核验依据 · Independent verification: 核验依据 1 OPENLIBRARY 标识 /works/OL32240873W：A History of Chinese M
#零 #记数
什么是算筹？古人怎么用它算数？
What are counting rods and how did ancients calculate with them?
算筹是汉代以前普遍使用的小竹棍，用纵式横式表示不同数位，所谓一纵十横、百立千僵，通过摆弄筹棍完成加减乘除与开方。它是世界上最早的十进位值制记数工具之一。
Counting rods were thin bamboo sticks used in ancient China: vertical and horizontal placements encoded digits in a decimal place-value system, enabling addition, subtraction, multiplication, division and root extraction by rearranging the rods.
难度 入门 · 适合 大众
CC BY 4.0 · 已核验 · Verified 2026-11-02
来源 · Source
Martzloff, J.-C., 《中国数学史》, Springer
Martzloff, J.-C., 'A History of Chinese Mathematics', Springer
独立核验依据 · Independent verification: 核验依据 1 OPENLIBRARY 标识 /works/OL32240873W：A History of Chinese M
#算筹 #记数法