The Collapse of Compasses that Do Not Copy Segments, and the Lengths We Go ToDate: 04/16/2011 at 09:51:35 From: Steve Subject: Since when do compasses copy lengths? Having studied both ancient Euclidean geometry and more recent treatments of the subject, I am wondering exactly when it was that compasses began to be used to copy lengths. I have read that Euclid disallowed compasses from doing this because in his geometry there was no motion. To me, this restriction makes a lot of sense both from a practical and a theoretical viewpoint: one cannot set the compass at a certain length and then truly claim the ability to duplicate that length anywhere other than at a segment radiating from its current center. Anyway, Euclid covers this problem in his second proposition. But in our current geometry textbook, my students and I encounter "Copying a segment is easy: we just set the compass to the length of the segment, and then copy it elsewhere on the page." Perhaps this concession comes only after centuries' worth of students saying, "Why can't we just measure the length and then copy it?" Obviously, this does make a certain type of sense, but to me it seems a sour departure from Euclid's impregnable logic. What really confuses me is that Euclid already HAD a solution to this problem. Granted, that construction is somewhat involved, even difficult -- certainly out of proportion for such a simple objective. But not only does it WORK, it maintains our ability to do consistent geometry without introducing this vague notion of "moving" lengths. My current work on this problem involves the simple observation that when we move compasses, they often shift slightly. Unless we have a drafting compass or something, the "pick-up-the-compass-and-move-it" part will usually involve some alteration in the setting of the compass, however imperceptible. Date: 04/16/2011 at 21:49:41 From: Doctor Peterson Subject: Re: Since when do compasses copy lengths? Hi, Steve. You appear to be aware of most of this, so let me just say by way of background that I have discussed this matter previously: Collapsible Compass http://mathforum.org/library/drmath/view/66052.html The key idea here is that, as far as I can see, Euclid put his second proposition where he did so that he could talk about everything depending only on his simple postulates, WITHOUT having to use a real collapsible compass to do his work. See also the Guide section here: http://aleph0.clarku.edu/~djoyce/java/elements/bookI/propI2.html I can't say that I know anything about the history of compasses. Many modern compasses (and likely many old ones) can hold their settings reasonably well. Since using them that way is simply a shortcut to something that CAN be done with the collapsible compass assumed in Euclid's postulates (a la Proposition 2, to which you refer), to require students not to trust them to do this would make constructions a terrible chore, and would not make math any more understandable, much less enjoyable. In a course built around postulates, I would certainly mention that the postulate does not in itself allow transferring a length using a compass, but that we can use that process as a shortcut to a longer procedure that could be done with the so-called collapsible compass. But I would not subject students to more than one exercise in which they actually had to perform the long procedure of copying a segment! Once they know it can be done, doing it would be a waste of time. I imagine that may have been true even in Euclid's time. Of course I should note that there are also many (cheaper) compasses that don't hold their settings well enough to make even a single circle accurately! In my math course for elementary teachers, most of them buy a compass of poor quality, and I have to take some time to teach them how to use it so that it will retain its setting (which involves holding the compass in such a way that all forces are perpendicular to the direction in which it opens). This is probably a good thing for teachers to have to learn! For high school students, good compasses would be a good investment. If you have any further questions, feel free to write back. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ Date: 04/27/2011 at 16:25:11 From: Steve Subject: Thank you (Since when do compasses copy lengths?) Hi Doctor Peterson, Thanks for your response! I certainly agree with your statement that it would be a terrible chore to explicitly depend on this construction any time we wanted to copy a segment. From the perspective of logic, it is preferable to have as few assumptions as possible, and so it does make sense (as you say) that Euclid would position Prop 2 so early on. That way, what might normally need to be an assumption is introduced as a "theorem" of sorts, although an early enough one that it enters practically at the same time as the assumptions. Of course in actual practice, we would always use a compass to copy lengths, since as you say, the alternative would make many constructions very tedious. I guess my frustration, and concern, comes down to seeing my students fail to appreciate the distinction between being able to copy a segment because a compass "gets us close enough" (which is a casual assumption of the material world) and being able to copy a segment because of Prop 2 (which is a rigorous theorem derived from reason alone). Distinguishing between "assumption" and "theorem" in this way likely means more to me than it does to my students. Perhaps I should wait for them to mature intellectually a little bit before trying to make them understand such an abstraction! Again, thanks. -SD |
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