Posted by Jonathan (68.14.212.239) on November 02, 2003 at 03:19:48:
In Reply to: Re: Analysis of the Apologia wheel drawing posted by Jonathan on November 02, 2003 at 03:11:29:
See my other 'WOW!'(above on index page), I can't believe it!
: A thought just occurred to me: What printing techniques did they use in his time? If they weren't wonderfully accurate, then maybe being off by a few degrees was a result of the printing process and has nothing to do with a hint. If this is the case, then the fact that it is 95 and not 90 degrees doesn't really matter, all that matters is what he meant to have them be, which then would likely be 90.
: : Yes the white triangles touch the inner circle but that's not the point. To me JB is not giving you the answer directly, he's making us think about it. So the (approximate) right angles indicate a circle with four quadrants. The black and white triangles add up to (approximately) 120 degrees, which would normally only fit in a three segmented circle - how do you fit them into a four segmented circle? In the way I've tried to show.
: : Forgetting the limited accuracy of the angles for a moment, you have a three or a six compartment circle, which might indicate that there were that many compartments in the wheel, except that there is evidence of a sort that seems to show that there were either four or eight such compartments. So the answer might lie, not in measuring the angles themselves, but taking on board the hints that he gives - i.e. the presence of right angles, and the 30 degree angles (which to me seem too small to be able to contain useful space for a mechanism)which, when totalled together add up to 120 degrees simply because there are three (or six) of them.
: : John Collins
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: : : OH! It's so obvious now! But didn't the triangles in the apologia drawing have their points touching the inner circle, not the outer?
: : : : Analysis of the Apologia wheel drawing
: : : : By John Collins
: : : : You need a triangle with a corner of 120 degrees and two other corners each of 30 degrees. Draw a circle and divided it into four equal segments by drawing two diameters at right angles to one another. Look at your drawing. Let's say that one of the two diameters you've drawn runs from 9 o'clock across to 3 o'clock. Place a protractor on the horizontal line at the point where the line touches the rim at say 9 o'clock, so that you can draw an angle which comes out from that point into the middle of the circle. Mark off 30 degrees (on the right side). Draw the new 30 degree line from the rim into the circle to about half way across to the adjacent radius. Repeat for the adjacent radius and then do the same thing for the other radii. Then repeat the whole process for the under sides of all the radii. That means you've drawn eight 30 degree angles. Now place the point of a pair of compasses in the centre and draw a circle through the four points where, within each segment, the two 30 degree angles meet in each case. If you've got it right it looks like a four pointed star inside a circle. You should now have what I believe to be the essential working area of the mechanisms for a one way wheel. They operate within a 120 degree segment placed four times, each within a right angle inside the circle.
: : : : It's possible that such a diagram was intended also to work with just three mechanisms and weights and it is easy to do the same excercise just dividing the circle into three segments instread of four. But I tend towards the four weight design because of the 90 degree inclusion in the Apologia drawing.
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