| a | = | a | · | b | = | a′ | · | b″ | , if | a | = | a′ | and | b | = | b″ | . |
| c | b | c | b′ | c″ | b | b′ | c | c″ |
The theorem in Prop. 23 is the foundation of all mensuration of areas. From it we see at once that two rectangles have the ratio of their areas compounded of the ratios of their sides.
If A is the area of a rectangle contained by a and b, and B that of a rectangle contained by c and d, so that A = ab, B = cd, then A : B = ab : cd, and this is, the theorem says, compounded of the ratios a : c and b : d. In forms of quotients,
| a | · | b | = | ab | . |
| c | d | cd |
This shows how to multiply quotients in our geometrical calculus.
Further, Two triangles have the ratios of their areas compounded of the ratios of their bases and their altitude. For a triangle is equal in area to half a parallelogram which has the same base and the same altitude.
§ 67. To bring these theorems to the form in which they are usually given, we assume a straight line u as our unit of length (generally an inch, a foot, a mile, &c.), and determine the number α which expresses how often u is contained in a line a, so that α denotes the ratio a : u whether commensurable or not, and that a = αu. We call this number α the numerical value of a. If in the same manner β be the numerical value of a line b we have
a : b = α : β;
in words: The ratio of two lines (and of two like quantities in general) is equal to that of their numerical values.
This is easily proved by observing that a = αu, b = βu, therefore a : b = αu : βu, and this may without difficulty be shown to equal α : β.