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我国古代求弧田(弓形田)面积S的方法(旧术)是: “以弦(b)乘矢(h),矢又自乘,并之,二而一”,即 S=1/2(bh+h~2),載在《九章算术》方田章。刘徽指出:(a)弓形作半圓时,依上式計算面积,“失之于少”;(b)若不滿半圓者,益复疏闊。批評正确,无待辞費。刘徽批判了旧术:提出了新术——刘徽弧田术。如图:于所給弓形內以弦为底作等腰三角形,于所得的各較小弓形內又作等腰三角形,这样继续續作下去;再由一系列的弦b,b_1,b_2,…和相当的矢h,h_1,h_2,…分别求各等腰三角形的面积,并依次把它們加起来,所得的結果就逐漸逼近于所求的弓形面积。这就是刘徽所示:“割之又割,使至极細,但举弦矢相乘之数,则必近密率矣”。用算式表之:
In ancient China, the method (old technique) for determining the area of the arc (arc field) was: “Take the string (b) multiplicative vector (h), the vector is also self-plucking, and the two, and one,” ie S = 1/2 (bh+h~2), Fang Tianzhang, Chapter 9 Arithmetic. Liu Hui pointed out that: (a) When arched as a semicircle, calculate the area according to the above formula, “losing less;” and (b) if dissatisfied with the semi-circle, benefit and sparsity. Criticism is correct and there is no need for resignation. Liu Hui criticized the old technique: he proposed a new technique—Liu Hui’s arc technique. As shown in the figure, the isosceles triangle is used as the base of the given bow, and isosceles triangles are formed in the resulting smaller bows. This continues sequel to it. Then a series of strings b, b_1, b_2,... And the corresponding vectors h, h_1, h_2, ... find the area of each isosceles triangle, and add them in sequence, and the result is gradually approaching the desired arcuate area. This is what Liu Hui says: “The cut is cut again and again, so that it is very fine, but the number of multiplications of the chords must be close to the density.” Using the formula: