Math Is Fun Forum

So, if I have three numbers such that : a/b=b/c

Then we have a/c which is a duplicate ratio of of a/b

If we have 4 numbers such that : a/b=b/c=c/d

Then we have a/d is triplicate ratio of a/b

We can continue this way with other proportions.

So, what does it mean exactly ?

Hi, there's something that have been bugging me.

If we have quantities A, C, E

And if we have quantities B, D, F

And if we take the equimultiples G, H, K from A, C, E

And if we take the equimultiples L, M, N from B, D, F

And we show that whenever G is superior to L, then H is going to be superior to M. And if equal, equal. And if less, then less.

And we show that whenever K is superior to N, then H is going to be superior to M. And if equal, equal. And if less, then less.

Then, how can you be sure that whenever G is superior L, K is going be superior to N. If equal, equal. And if less, then less.

Usually we have a situation of if X is equal to B and if Y is equal to B, then X is going to be equal to Y. (This is not hard to digest, obviously)

But in the situation just described, i'm having a hard time digesting this... I have no problem understanding the statement but the conclusion is hard to digest or to be accepted by my mind.

Thank you!

I have the following theorem : "In a circle the angle at the center is double the angle at the circumference when the angles have the same circumference as base."

(Figure is in the link) http://aleph0.clarku.edu/~djoyce/java/elements/bookIII/propIII20.html

English isn't my first language, so I just want to make sure that I understood something correctly. We prove the theorem by putting the two angles one on the other for the circumference. I was just wondering, can I assume that the angles do not need to be one on the other and they can have different portion of the circumference, as long as the circumference are of the same lenght ? (Will the proposition still work in this way?) I guess that Euclid did the proof by putting the angles one on the other for making the demonstration less wordy. (Less long to read)

Thank you!

geometry proof-verification euclidean-geometry

Hi,

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.

Thank yoU!

"Then, since on the circumference of each of the circles ABDC and ACK two points A and C have been taken at random, the straight line joining the points falls within each circle, but it fell within the circle ABCD and outside ACK, which is absurd.
Therefore a circle does not touch a circle externally at more points than"

There's only one little detail which i'm not sure of. We are trying to prove that circles which touch one another will only touch at one point. Fine. I understood the first part which treats of a circle in another one. It's only the case where one circle touches another one from the outside. By using proposition 2 of book 3, we prove that the line AC will be inside both of circles since the two points are on each circumference of the two circles. Now, this is where I get lost. We say that "but it (line AC) fell within the circle ABCD and outside ACK" and we prove this by using definition 3 of book 3 (Circles are said to touch one another which meet one another but do not cut one another.) In other words, this definition says that circles which touch another do not cut one another. In our situation, we have two circles which touch one another and are not supposed to cut one another. This is where I don't understand, how does this justify this : "but it fell within the circle ABCD and outside ACK." How do we get that conclusion from the definition?

Thank you!

http://aleph0.clarku.edu/~djoyce/java/elements/bookIII/propIII13.html

I have a small question regarding proposition 33 of the elements of Euclid. http://aleph0.clarku.edu/~djoyce/java/elements/bookI/propI33.html

We want to prove that two lines joining equal parallels at their ends are themselves equals and parallels. The only thing I don't understand is why does Euclid use the side-angle-side theorem to prove that the remaining angles are equal to one another ? Why doesn't he use the proposition 29 (A straight line falling on parallel straight lines makes the alternate angles equal to one another, the exterior angle equal to the interior and opposite angle, and the sum of the interior angles on the same side equal to two right angles.) to prove that all alternate angles are equal to one another?

In my opinion, I think it's because were considering only at the begining the "finite" straight lines BA-DC, so this limits us only to two angles ABC and BCD.

Thank you

I have small question regarding this proposition : http://aleph0.clarku.edu/~djoyce/java/elements/bookI/propI26.html

To prove that one side is equal to another, Euclid assumes that one side is bigger than the other. Finally, when Euclid arrives at a contradiction, he dismisses the assumption about the inequality of sides and considers them equal. What I was wondering is, if we assume that a side is unequal to another one (A is bigger than B) and arrive at a contradiction, shouldn't we also try the inverse, B being bigger than A and assure ourselves that we arrive also at a contradiction to conclude that finally, A is equal to B ? Thank you!