衡曰平衡，谓欲得其平也。此器有小有大，总名曰衡，小者曰等，大者曰秤，古文作称。（称去声。）
虞书曰律度量衡，言衡不言权；论语曰谨权量，言权不言衡。盖权衡合德而相须为用，
举其一则可以互见矣。吴韦氏曰∶衡有斤两之数，生于黄钟。黄钟之管，容 黍千二百粒，
是为一龠，二龠为合，合重一两，故律度量衡，于是乎生，而三代之制，权衡之起，信乎出于
律矣。夫一龠所容千二百黍之重，是为半合，即古之半两也；两龠所容，三千四百黍之重，
是为一合，即古之一两也。然则一升之重为十两，一斗之重为百两，一斛之重为千两矣。
故一斤之重为一升六合，一均之重乃四斗八升，一石之重乃一斛九斗二升也。权量相合，
未有得其量而不得其权者。今考羊头山 黍，以时制等子秤之，其中者百粒得二分五厘
整，积至两龠二千四百粒，秤重六钱。可见今之六钱为古一两，今之六斤为古十斤，其余可
以类推，大率古之于今，乃五分之三耳，先儒以为三分之一非也。置今求古则用六归，以古
求今则用六因。求度量亦如之，但率法不同耳。度以八为率，今之八寸，即古之一尺；量以
三为率，今之三斗，即古一斛；权以六为率，今之六钱，即古一两也。凡度量衡，以今求古，
皆置今为实而用归，以古求今，皆置古为实而用因，则得之矣。
Heng (balance) refers to equilibrium, meaning the pursuit of a balanced state. This instrument varies in size but is collectively named Heng; the smaller one is called Deng (small balance), and the larger one is called Cheng (steelyard scale). In ancient texts, it is written as Cheng (pronounced with the departing tone).
The Book of Documents states "Lü (pitch-pipes), Du (length), Liang (capacity), and Heng (weight)," mentioning Heng but not Quan (counterpoise); the Analects states "Be careful with the Quan and Liang," mentioning Quan but not Heng. This is because Quan and Heng possess complementary virtues and are mutually dependent for use; mentioning one allows the other to be inferred.
Wu Wei-shih stated: The Heng has standards of Jin and Liang (weights), originating from the Huangzhong pitch-pipe. The Huangzhong pipe can contain 1,200 grains of millet, which constitutes one Yue (a unit of capacity); two Yue constitute one He (a unit of capacity), and one He weighs one Liang (tael). Thus, Lü, Du, Liang, and Heng originated here, and in the institutions of the Three Dynasties, the origin of Quan and Heng indeed stemmed from Lü (pitch-pipes).
The weight of the 1,200 grains of millet contained in one Yue is equivalent to half a He, which is the ancient Banliang (half tael); the weight of the 3,400 grains of millet contained in two Yue is equivalent to one He, which is the ancient one Liang (tael). Therefore, the weight of one Sheng (sheng) is ten Liang, the weight of one Dou (dou) is one hundred Liang, and the weight of one Hu (hu) is one thousand Liang.
Thus, the weight of one Jin (jin) is equivalent to one Sheng and six He; the weight of one Jun (jun, 16 jin) is four Dou and eight Sheng; the weight of one Shi (dan, 10 dou) is one Hu and nine Dou two Sheng. Quan and Liang are combined; there is no case where one obtains the capacity without obtaining the corresponding weight.
Now, examining the millet from Yangtou Mountain, weighed using an equal-arm balance made according to seasonal standards, 100 medium-sized grains weigh exactly 2.5 fen (0.025 qian). Accumulating to two Yue, totaling 2,400 grains, the weight is 6 qian. It can be seen that 6 qian in modern times equals 1 Liang in ancient times, and 6 Jin in modern times equals 10 Jin in ancient times. The rest can be deduced by analogy. Generally, the ratio of ancient to modern measures is three-fifths; the earlier Confucians' belief that it was one-third is incorrect.
To convert modern units to ancient ones, divide by six; to convert ancient units to modern ones, multiply by six. Calculating length and capacity follows the same principle, differing only in the ratio coefficient. For length (Du), the ratio is eight: 8 cun (inches) now equals 1 chi (foot) then. For capacity (Liang), the ratio is three: 3 Dou now equals 1 Hu then. For weight (Quan), the ratio is six: 6 qian now equals 1 Liang then. For all Du, Liang, and Heng, to find ancient values from modern ones, place the modern value as the dividend and divide; to find modern values from ancient ones, place the ancient value as the multiplicand and multiply, and the result will be obtained.