# NavList:

## A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding

**Re: Coordinates on Cook's maps**

**From:**Alexandre Eremenko

**Date:**2007 Apr 20, 01:43 -0400

Dear George, First, I give the precise citation: According to Bird, "...a right line cannot be cut on brass, so as accurately to pass through two given points, but a circle may be described from any centre, to pass accurately through a given point". This is cited in Rees Cyclopaedia. vol XVI, article "Graduation", and also in the paper by Saul Moskowitz, The world first sextants, (one of the very few good papers on the Inst of Navigation CD you recently bought. In general, Moskowitz is a rare exception among most of the authors on this CD; his papers are substantial and interesting). This sounds paradoxal, but if you really think carefully on how exactly would you draw a straight line passing through two given points, and actually TRY to do it with high precision, you will see that Bird is right. We are talking here of the accuracy of 0.0001 = one over ten thousand of an inch when graduating a decent 8 inch arc, by the way. Just try with a very sharp pencil on paper and then look at the result through a magnifying glass, if you don't like thought experiments:-) The pencil point never goes exactly along the straight edge, it always goes parallel to it on some unknown distance. I am not even speaking of aligning the straightedge with the given points: how exactly do you do this? By the way, according to the same British division masters of the late XVIII century, dividing a straight rule is even harder than dividing a circle. I think it was Ramsden who invented one of the first machines for dividing straight rules. Returning to "high school geometry", the iterative construction you propose, of course permits to divide approximately. In PRINCIPLE, to any given accuracy. But if you think on how would you actually perform this successive approximation procedure, you will disvover difficulties. For example, you will need a microscole through which you can see much smaller details than the accuracy of your division. I mean to draw the tiny arcs on the last steps of your procedure. There exists a method of dividing any arc into two equal parts by compass only. And EXACT metod, not requiring a microscope. The understanding of the method requires nothing but a solid background in "high school geometry". But I would not call it "simple"! It was discovered in XVIII century, and I don't know whether the discoverer had in mind a practical application to instrument division, or he was driven by curiosity of a pure mathematician. He actually proved that EVERY construction which is possible with a compass and a rule is ALSO possible with compass only. Neither I know whether this method was actually used. http://www.cut-the-knot.org/do_you_know/compass.shtml Alex --~--~---------~--~----~------------~-------~--~----~ To post to this group, send email to NavList@fer3.com To unsubscribe, send email to NavList-unsubscribe@fer3.com -~----------~----~----~----~------~----~------~--~---