P l a n e G e o m e t r y
An Adventure in Language and Logic
THE SIDES AND ANGLES OF A TRIANGLE
Book I. Propositions 18 and 19
THE STUDENT by now must have some appreciation of what is involved in a logical theory. Each proposition must wait its turn, for each depends on some previous proposition or first principle -- even the most obvious.
Consider for example the following:
Any two sides of a triangle are together
In other words, a straight line is the shortest distance between two points!
If anyone wanted to ridicule mathematics for its insistence on the axiomatic method of orderly proof, this theorem offers a wide target. In fact, the Epicureans (those Athenian free-thinkers, who defined philosophy as the art of making life happy) did exactly that. They said that this theorem required no proof, and was known even to an ass. For if hay were placed at one vertex, they argued, and an ass at another, the poor dumb animal would not travel two sides of the triangle to get his food, but only the one side which separated them.
Such is the scorn that the true philosopher must bear! And what can the mathematician do but to point out, patiently, that mathematics as a logical science relies on deduction from first principles. Those principles moreover should be as few in number as possible — whatever can be proved should be. That is the intellectual sport.
This proposition -- Any two sides of a triangle are together greater than the remaining side -- is Euclid's Proposition 20. And to prove it he first had to prove
Proposition 19. A greater angle of a triangle is opposite a greater side.
If angle A is greater than angle B, then side CB is greater than side CA.
And to prove that he first had to prove
Proposition 18. A greater side of a triangle is opposite a greater angle.
And to prove that he had to prove Proposition 16, which we just saw.
This is what one has to go through to form a proof!
(Proposition 17 was not needed: Any two angles of a triangle are together less than two right angles. That will be Problem 4.)
Here, then, are Propositions 18 and 19.
PROPOSITION 18. THEOREM
We have proved this proposition directly. But we must prove the next proposition indirectly, and to do that we have to cite the previous one.
PROPOSITION 19. THEOREM
The proof involves a tacit assumption which these days is called the Law of Trichotomy. Given two magnitudes of the same kind, the first must be either equal to, less than, or greater than the second.
Trichotomy thus provides a method for indirect proof.
Please "turn" the page and do some Problems.
Continue on to the next proposition.
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Copyright © 2012 Lawrence Spector
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