From: spamless@Nil.nil Subject: Re: ruler and compass constructions Date: 6 Jan 2000 04:09:47 -0500 Newsgroups: sci.math Summary: Classic Greek compasses were "floppy" David Petry wrote: > spamless@Nil.nil wrote >>However, the greeks couldn't. You have to remember that the classical >>compass (Euclidean compass) collapses as soon as you lift it from the >>page. You cannot use it to take one one length, move the compass and >>duplicate the length > That's news to me, and probably to a lot of others as well. I always > thought you could use the compass to draw two circles of the same > diameter. > What did the Greek's compass look like? > BTW, I feel silly responding to a post from someone who uses no > name other than "spamless". I get enough spam The greek compass had a hinge at the top (like the compasses you may see teachers use on blackboards). One can draw a circle with the same radius as a given circle with another center. But, it is not so simple as picking up the compass and moving it to the other center (it collapses once lifted). One can construct, say, a rectangle with one edge being the line segment joining the two centers and the radius being the other side (at the given circle). Then moving to the second center and using the corner of the rectangle to set the compass there, draw the translated circle (all this involves is drawing perpendiculars to a line at a given point on the line ... do that at the center of the first circle, perpendicular to the line joining the centers ... then at the second point ... then at the point where the first line intersects the original circle and perpendicular to that line. Where that final line intersects the second line is a point at the proper distance from the second point ... now you can put the compass on that point and the second center and draw the circle without lifting the compass ... constructing the perpendiculars is not difficult either). Since you can manage to do this with a collapsing compass, using a modern, non-collapsing, compass does not lead to new contructions (beyond the classical) unless you do resort to tricks such as using a fixed separation to convert a straight edge to a ruler for a trisection "construction." ============================================================================== From: hrubin@odds.stat.purdue.edu (Herman Rubin) Subject: Re: ruler and compass constructions Date: 7 Jan 2000 11:27:06 -0500 Newsgroups: sci.math In article , David Petry wrote: >spamless@Nil.nil wrote >>David Petry wrote: >>> you can simulate a marked >>> ruler by holding the points of the compass next to the unmarked >>> ruler. >>However, the greeks couldn't. You have to remember that the classical >>compass (Euclidean compass) collapses as soon as you lift it from the >>page. You cannot use it to take one one length, move the compass and >>duplicate the length >That's news to me, and probably to a lot of others as well. I always >thought you could use the compass to draw two circles of the same >diameter. Euclid's second proposition takes care of this situation. The assumed compass could draw a circle with a given center and passing through a given point. However, with the aid of a straightedge, it is possible to construct a circle with center O and radius AB using only compasses which cannot be lifted. -- This address is for information only. I do not claim that these views are those of the Statistics Department or of Purdue University. Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399 hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558 ============================================================================== From: mathwft@math.canterbury.ac.nz (Bill Taylor) Subject: Re: ruler and compass constructions Date: 11 Jan 2000 05:35:31 GMT Newsgroups: sci.math Just a little followup to relay some standard historical info about this topic. As mentioned, the Euclidean ideal for ruler-&-compass constructions does insist that the compasses be "floppy", so they can't be used as dividers to transfer a line segment around; but also as mentioned, this turns out to be no essential restriction. More interestingly, it turns out that the *opposite* extreme in compasses, namely "rusty compasses", are also sufficient. That is, a pair of compasses that is rusted into one unmovable radius. The Arabs showed that all proper Euclidean constructions could still be effected with these. Quite a tour de force. Later it was shown in moderately recent times, C19 I think, that even just ONE fixed circle, somewhere on the plane, was *already* enough to do all Euclidean constructions! If your stuff is a way off from it, you have to straight-line stuff over to intersect the circle, do stuff there, then straight line it back. Now that's *really* painful! Even more, one doesn't really need the *whole* of that one circle, just an (arbitrarily small) arc will do! Even an arc so tiny of a circle so huge that it looks like a straight line segment. Clearly we are a long way from the practicality of Egyptian field marking by now!! In the other direction, Mascheroni, in Napoleonic times, showed that compasses alone *without* straight edges was enough to do any Euclidean construction! Of course, one needs to re-define the various straight-line results appropriately, but that is no problem in fact. ====== As an exercise, you amateur geometers might like to ponder this one. What subset of Euclidean constructions can be done with only THICK RULERS. i.e. You may use straight-edges, and a device to draw a line parallel to a given line and a fixed distance away from it. With a struggle, you may find you can then construct perpendiculars, bisections, general parallels through given points, ... . How far can you go? Happy hunting! ------------------------------------------------------------------------------- Bill Taylor W.Taylor@math.canterbury.ac.nz ------------------------------------------------------------------------------- If we knew what we were doing it wouldn't be called research. -------------------------------------------------------------------------------