Euclid's elements proposition 15 book 3

[astrololo]

http://aleph0.clarku.edu/~djoyce/java/elements/bookIII/propIII15.html

I have understood the proof in general. It is only a small detail which I'm not sure. Maybe it's because english isn't my first language. Anyway, the part of the proof which says : "Then, since BC is nearer to the center and FG more remote, EK is greater than EH." I have no problem understanding what is said here. This is supported by this definition : "And that straight line is said to be at a greater distance on which the greater perpendicular falls." (Definition 5 of book 3) Now, this is where I'm unsure. From what I understand of it, it says that if I have a perpendicular that is bigger than the other, than my straight line is said to be at a greater distance. (This is how I understand it) Now, in the proof, we do the inverse. We know that one line is at a greater distance than the other and we conclude with the definition that one perpendicular is bigger than the other. How is this correct ? Unless the definition implies that the reverse is also ok, then this works. But if the definition implies only one direction, (The one which is defined) then how is the proof valid ?

By the way, you don't need to read all of the proof. Only the things at beginning are needed.

 

[HallsofIvy]

All definitions are "if and only if". To say that the "that straight line is said to be at a greater distance on which the greater perpendicular falls." is the same as saying "a straight line is at a greater distance if and only if it has the greater perpendicular."

 

[astrololo]

Yes I understand this. You mean that "if and only if" makes the inverse(converse) also true, right ?