----- Original Message ----- From: "Frank King" <frank.k...@cl.cam.ac.uk> To: "Roger Bailey" <rtbai...@telus.net> Cc: <sundial@uni-koeln.de> Sent: Sunday, August 09, 2009 12:36 PM Subject: Re: Sorry but.....Square Roots.....Shatir Sundial
> Dear Roger, > > Thank you for pointing the list at this > YouTube video > > http://www.youtube.com/watch?v=cFH1lz0212o > > I found that very compelling and, indeed, all > the associated 10-minute clips. > > I particularly noted Al-Battani's expression > for the radius of the Earth: > > R = h.cos(phi)/(1-cos(phi)) > > where h = height of some mountain > > phi = dip to the horizon from the summit > > With John Carmichael in mind, I comment that this > is the kind of thing that 14-year olds could derive > in the 1950s but Mathematics graduates have to > struggle over today :-)) > > The value of phi in the clip was about 0.5 degrees > and the cosine of 0.5 is 0.99996 or so close to 1 > that you need 6-figure tables to make progress (or > knowledge of the series expansion for the cosine > function, or you could rearrange the expression). > > I know very little about the history of mathematical > tables. There was something in one of the clips > about the Arabs developing tables of sin and cos > but Al-Battini would need high precision to make > use of the quoted expression. > > The whole procedure seems terribly sensitive to > errors in the measurements to me and if it is > true that Al-Battini determined the radius of > the Earth to 0.1% this way he was very lucky > as well as very clever!! > > Thank you also for reminding the list about > God's Longitude and Simon Cassidy's work. > > It is one of God's better jokes that He should > have drawn His Longitude through Washington!!! > > Best wishes > > Frank > Dear Frank, I haven't yet seen that particular video, but if phi is about 0.5 degrees, that's about 0.0087 radians. For such small angles sin(phi)=phi and cos(phi)=1 - (phi^2)/2 So, R=2h.(1-(phi^2)/2) / (phi^2) or: R+h=2h/phi^2 Not too tricky. For phi=0.0087, I presume the 'mountain' was about 240 metres high, giving R+240=6,340,000 metres, approximately, without needing any tables of sin or cos. As for sensitivity to errors, I agree that 0.1% error sounds amazing, since he'd have had to measure the angle to within 0.05%. 0.05% of 0.5 degrees is 9 arc seconds - could you really measure dip that accurately? Best wishes Chris --------------------------------------------------- https://lists.uni-koeln.de/mailman/listinfo/sundial
