The Bridge of Asses

Image: Jenny Mealing, via Wikimedia Commons.

I don’t mean to be crude or inappropriate with my title.  After all a donkey used to be called an ass.  I don’t know what brought about taboo on the word, but in fact, it is by many considered a bad word.  I only used this title because it seemed a good attention grabber.  The phrase did, grab my attention after all.  But, what is “the Bridge of Asses,” and how does it have anything to do with Euclid or mathematics?  Why am I writing about “the bridge of Asses?” It may be because, “Ass” is the first swear word I ever used.  I was in a Shakespeare play called Much Ado about Nothing, and one of my character’s lines was, “You Are an ASS!”  Of course it was not so hard for me to accept the fact that I was swearing, as it was to accept the fact that my parents seemed to think it was funny and OK.  I was 12, but I digress. Let me tell you the real reason I am discussing, “The bridge of Asses.”

Sometimes we want to be able to tell if someone is really interested in something, or even if they are able to quickly grasp a concept.  Euclid’s fifth proposition in the first book of his elements was used to do just that.  Now before I proceed, lest I be accused of shaming people who have a hard time with math, I must say that I struggle very much with math and while reading about Euclid’s fifth proposition often felt like the “ass.” Don’t mock me! We all have our strengths and weaknesses. I am just trying to tell you about a something which I find interesting. Let’s talk about some history.

Around 1250 a man named Roger Bacon gave an alternate name to Euclid’s fifth proposition in the first book of his elements, which I will from here on out refer to as “the Bridge of Asses” or the fifth proposition.  The name he gave it was Elefuga, another word I will use freely to refer to the fifth proposition.  Elefuga, derived from Greek, means, “escape from misery.”  Medieval boys were presented with the Elefuga shortly before their “escape from misery.”  That is to say most medieval young men’s experience in geometry ended shortly after they encountered the fifth element, because it proved they simply did not want to go on or their mentor felt they should not.  They, like a donkey fears crossing a bridge, had a hard time grasping the fifth proposition or refused to grasp it. I personally believe they refused to try to grasp it or the mentor did not want to walk them through it well enough. This is because I think with time and patience people can overcome most barriers, but again I am digressing.

To better explain this, “the Bridge of Asses,” also known as the isosceles triangle theorem, is Proposition 5 of Book 1 of Euclid’s Elements.  But, also, pons asinorum, the Latin translation of “the Bridge of Asses,” became a metaphorical statement for a problem that will separate the confident from the unconfident. In other words it is a critical test, of the ability and understanding, of an individual. You see things like this all the time in movies. Usually someone has a sensei or master and they are trying to prove themselves. Eventually they come to the test that decides if they will continue with their training or not. For Bruce Wayne in Batman Begins it is, possibly, when he brings the flower to the League of Shadows high up in the mountain so that he can begin training with them.  Now we want to pass “the Bridge of Asses” for math, or proposition 5. Let’s see if you and I can manage to cross the bridge of elements together.

First, what is proposition 5?  Straight from Euclid’s elements, it is that, “In isosceles triangles the angles at the base equal one another, and if the equal straight lines are produced farther, then the angles under the base equal one another. Now, just hearing it makes sense, but to cross “the Bridge of asses” we must also prove proposition 5 and most importantly understand the proof.

Now I have read many blogs and articles proving the fifth proposition so I feel that I must make it clear that I am deriving this proof from an article, “the Bridge of Asses,” from [1].  Also to make the proof more clear, I am going to list our proof in steps.

  1. We need to draw an isosceles triangle. It will have points ABC. For review, because I had forgotten, we must recognize that isosceles means that the sides AB and AC are equal.
  2. Now we want to extend past AB and AC indefinitely.

Step 2

  1. Now we want to add two more points D and E. The line AD will pass through the point B. The line AE will pass through the point C. AD and AE will be equal.

Step 3

  1. Now that we are past this point we must notice that the angle at DAC and the angle at EAB are equal. This is a simple to believe since they are the same angle.
  2. From step four we say that the triangle DAC and the triangle EAB have equal angles when all the corresponding side’s angles are compared. We can use the side-angle-side theorem to prove this.  It says that two triangles are equal when the triangles have two sides of the same length and the angle of those two sides is the same.

Step 4-5

  1. From step five we can conclude that the angles ADC and AEB are the same as well as that the lines DC and EB are equal.
  2. Now if we subtract AB from AD and AC from AE we can show that BD = CE.
  3. IT now holds by side-angle-side theorem that the triangles DBC and CEB are equal. If they are equal then so are the angles DBC and ECB
  4. We have now proven that the angels ABC and ACB are equal because the angle ABC = 180 degrees – the angle DBC and the angle ACB = 180 degrees minus the angle ECB when the two angles being subtracted are equal.

I hope that you found the proof I presented sufficient.  I don’t claim it as my own since I had to get help to cross this bridge. Hopefully I was able to help you across also, if you even needed help.  If you are still unsure, I suggest a pen and paper.  After all, that is really how it came to make sense to me.

Well now that we have crossed “the bridge of asses” together we are ready to further our careers in mathematics.  Really though I think the concept of “the Bridge of Asses” has a significant meaning. We will continually come across bridges in our education and careers. Sometimes we will feel that the bridge we are presented with is scary and hard to cross.  When I first saw the proof of proposition five that is what I thought. But if we take the time, think about it, and cross the bridge we will be that much better. Just like you and I crossed this bridge we can cross others. Don’t hold back, break a problem into steps, study it, think about it, and together we will cross “the Bridge of Asses.”





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