When the trapezium is shifted along the lever, the balancing area lies between two comparison areas R and Q.
当梯形在杠杆上移位悬挂时，平衡面积 P 介于两个比较面积 R 与 Q 之间。
AOB 是以 O 为支点的水平杠杆；右侧三角形或梯形悬挂在 OB 上，左侧面积 P 与它平衡。
Place AOB as a lever supported at O and hang the given triangle or trapezium on the right side.
把 AOB 看作以 O 为支点的杠杆，并把给定三角形或梯形悬在右侧。
Replace the suspended figure by an equal weight acting through its center of gravity.
把悬挂图形等效为经过重心的一份重量；这一换位不改变平衡。
Compare the lever arms. The ratio of arms becomes the ratio between the suspended figure and the balancing area.
比较两侧力臂；力臂之比转化为悬挂图形与平衡面积之比。
The required equality or inequality follows from where the center line falls between the marked points.
重心竖线落在相应标记点之间，于是得到本命题需要的等式或不等式。
These mechanical propositions are not a modern force diagram for its own sake; they convert centers of gravity into area inequalities used later.
这些机械命题不是孤立的力学图，而是把重心和平衡换成后面需要的面积不等式。
不看完整证明，说明“命题 12 · 移位梯形的双边估计”这一命题的已知、要证和关键比较对象。
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- 先把几何对象命名，再说它们之间要比较什么量。
- 穷竭法命题要特别留意“若大于”和“若小于”两边。