P l a n e G e o m e t r y
An Adventure in Language and Logic
Book I. Propositions 39 - 41
THE NEXT TWO PROPOSITIONS are partial converses of the previous two; they show that if equal triangles are on the same base or on equal bases, then they are in the same parallels.
PROPOSITION 39. THEOREM
Now any triangle is half of a parallelogram; half the area, that is.
That was implied in I. 37 and 38. The following theorem confirms it.
PROPOSITION 41. THEOREM
From this theorem we derive the arithmetical formula that the area of a triangle is one-half the measure of the base times the measure of the height: A = ½bh.
Transformation of areas
Euclid's next four propositions began a fascinating problem in geometrical knowledge, which is to construct a simple figure, such as a rectangle or a square, which will be equal in area to a given figure. Proposition I. 45 solves the problem of constructing a rectangle equal to any rectilinear figure; and Proposition II. 14 solves the problem of constructing a square equal to any rectilinear figure. But an unsolved problem was to "square the circle," that is, to construct a square equal to a given circle. Not until the 19th century was it proved that that construction, using straightedge and compass alone, is impossible. The Pythagoreans were the ones who initiated and solved all these problems relating to rectilinear areas, and we shall have more to say about them when we prove Pythagoras's theorem.
The next proposition solves the problem of constructing a "square."
Please "turn" the page and do some Problems.
Continue on to the next proposition.
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