一些关于几何原本的问题

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热心网友时间：2024-07-13 07:45

BOOK II
Proposition 5
If a straight line is cut into equal and unequal segments, then the rectangle contained by the unequal segments of the whole together with the square on the straight line between the points of section equals the square on the half.

用代数的语言，xy={(x+y)/2}^2-{(x-y)/2}^2.

BOOK VI
Proposition 23
Equiangular parallelograms have to one another the ratio compounded of the ratios of their sides.

这是关于平行四边形面积的。This proposition is a generalization of the basic formula for the area of a rectangle, that is, the area of a rectangle is the proct of its length and width. Such a formula depends on predetermined units of length and area so that the unit area is the area of a square whose sides have length equal to the unit length. Euclid and other Greek mathematicians did not use predetermined units of length or area, so they expressed this formula as a proportion. We would state that proportion as saying the ratio of the area of a given rectangle to the area of a given square is the proct of the ratios of the lengths of the sides of the rectangle to the length of a side of the square. Of course, Euclid would say that without using the words 'area' and 'length' as follows: the ratio of the a given rectangle to a given square is the proct of the ratios of the sides of the rectangle to a side of the square. Note that his terminology for a proct of ratios involves "compounding the ratios." A natural generalization of the ratio of a rectangle to a square is the ratio of a rectangle to a rectangle. A broader generalization is the ratio of one parallelogram to another parallelogram having the same angles. That gives the generalization as stated in this proposition.

参考资料：Sir John Heath 注释的Euclid's Elements