An Alternative to Euclid's Book I Proposition 2 - Collapsible
- In the time of Euclid, it was considered improper mathematical etiquette
to create constructions by simply lifting the compass and striking
a mark somewhere else on the page. However, over time this perception
changed and we now can lift the compass and place it somewhere else
on the page. Of course, it is easy for modern students of geometry
to consider this process archaic as we now have software
on Mobile Phones that can perform
the same construction in a single step. Regardless of our opinions
of the collapsible compass construction,
Euclid chose to begin his elements with the most basic and elegant
construction - how to copy a segment from one point in space onto another.
- A construction to copy the segment is presented in Euclid’s
Proposition 2 of The Elements of Geometry. Mathematical historians question
whether Euclid was truly a mathematician or simply a librarian. Because
Euclid had access to many of the books available on the topic of Geometry,
he was able to develop the first organized text devoted to the topic.
How much of a contribution that Euclid made to The Elements would be
pure speculation. However, it cannot be disputed that the organization
of this text would take an understanding of Geometry that few students
or even teachers of the subject will ever reach.
- The construction as presented in Euclid’s Elements is presented
quite clearly at the web
page of David E. Joyce, Professor of Mathematics and Computer Science
at Clark University. Euclid’s construction requires the construction
of an equilateral triangle and two circles in order to copy a segment
to a random point.
- The construction I propose as an alternative to Euclid’s solution
to the collapsible compass problem involves the construction of three
circles and a midpoint. This construction I propose is the same for
any case. However, the proofs for each case are different. In practice,
this issue is really not a problem. If you are interested, you can download
the proof of my Collapsible Compass Construction
as a PDF file in worksheet format. To view the steps to construct this
alternative pictured to the right, click
here. You can also download a paper I wrote while in college on
the topic of the Collapsible Compass Construction
alternatives. I can provide assistance with any step of the construction
by contacting me at firstname.lastname@example.org.
An algebraic proof of each case is available upon request.
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