A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding
From: Frank Reed
Date: 2017 Nov 30, 12:27 -0800
Star distances?? Nooooo! That's the long way around! :) All you need is one magic number. And that number is... drum roll, please... 3438! That converts angles as pure ratios to minutes of arc.
So what's an angle as a "pure ratio"?? They never taught us that in school. Yeah, they did, but it gets so buried in further details, and then it's thoroughly ruined when they teach trigonometry. A pure ratio as an angle is a very simple thing. Suppose you have some object in front of you that is one unit of distance "across" or one unit "wide". Then if that one unit object is, let's say, at a distance of 100 units, its angular size is a "pure ratio" of 1-to-100 or 1:100 or 1/100 or 0.01. Get it? It's just size over distance. There's one detail here: the distance "across" should technically be measured along the arc of a circle at a fixed distance from the observer. For handheld "paper sextants" like this, it makes no difference. More generally if the angle being measured is less than about 10°, you can just think of distance or size "across the line of sight" and not worry about the limitation to the arc of circle.
Clearly, you can easily set up angles as ratios, and you alluded to that when you mentioned a tape measure. As for using a computer graphics program for the scale, well, ok. How about a common index card? They're pre-ruled. Let's suppose the spacing between lines on an index card is 0.25 inches. This is very common on so-called three-by-five index cards printed in the USA, but one should check, and the numbers will certainly be different in most other countries. Whatever the line spacing is, multiply it by 100 to find the distance at which you should hold the card. For standard index cards, that's 25 inches, which is conveniently "arm's length" for most people. At this distance, the spacing between any pair of lines is an angle of 0.01 as a "pure ratio". While pure ratios are the most mathematically clean and "rigorous" way of dealing with angles, we as navigators much prefer things in degrees and minutes. And that's where the magic number comes in. You multiply the angle as a ratio by 3438 to get the angle in minutes of arc. At arm's length, then, the spacing between two lines on an index card (common US 3x5 card) is 34.4'. You can then easily extend this up the card, and in no time at all you have a paper sextant (kamal if you prefer --I don't myself since it's just empty jargon). See the pdf below for a short talk that I did not have time for at the Navigation Symposium.
Note that when you use a paper sextant like this, you can certainly measure some angles with an accuracy as good as +/-5 minutes of arc. This can have real navigational value. The biggest concern should be the distance from your eyes. Holding an index card "at arm's length", it's clear that the card is closer to your eye when you're looking up high than when you're looking out to the horizon because your arm pivots at your shoulder. You can handle this by pinning the card on the end of a 25-inch stick if you want to ensure accuracy, but that might be over-kill.
Five Minutes of Navigation: You Need a Paper Sextant. PDF generated from Original Presentation:
Clockwork Mapping / ReedNavigation.com
Conanicut Island USA
PS: In case it's not familiar, 3438 is just 180·60/pi. In engineering language, it's the number of minutes of arc in "one radian". In ratio terms, that's a "unit angle".