Numerology that sorta kinda works!
One of my half-baked projects is to take apart an early 20th century Monroe mechanical calculator and reverse engineer it so that I have a full set of engineering diagrams of every part. This would enable anyone to recreate broken parts and fix their calculator.
Reverse engineering the design of an early 20th century mechanical part has a lot in common with numerology. If the numbers coincidentally fit, then they're probably right. If they almost, but not quite fit, then they're probably right anyway.
Here's a part that I scanned on an Epson Perfection V700 scanner. This scanner is based on a CCD, not LiDe, which means that it has non-zero depth of field. That means that you can scan a part that has height and it won't end up too blurry. I scanned the part at 1400 dpi so that I could optically measure it. The thing sticking out at the top is just a screwdriver that I used to hold the part horizontal.
I could pop this into Illustrator and use the pen tool to trace around the part, but all this would get me is an outline of this particular part with no insight into why it had that particular outline. This part was designed, not evolved. It was designed to work with other parts. So clearly its measurements and the relationships between one bit of the part and another are not arbitrary.
For example, take the hole at the top. It fits over a shaft. Now, "the ancients" probably didn't use shafts of arbitrary diameter. They were standard, and since this was an American design, that meant fractions of an inch, specifically inverse powers of two: 1/2, 1/4, 1/8, and so on. The hole at the top measures between 0.187" and 0.188" on my calipers. But 3/16" is 0.1875", so it makes sense that the engineers designed this hole to be exactly 3/16". This fits around the shaft that is 0.001" under 3/16", which I suppose is a standard undersized shaft.
In Illustrator, I pulled up the image and drew a circle of diameter 3/16", placing it so that it fit exactly into the hole in the image. Now I had the center of that hole, and I could draw more concentric circles. Because I could measure these diameters directly on the part, I used those diameters: 3/8" and 7/16".
The measurements in the image were done using VectorScribe.
Note that while the inner circle and middle circle (diameter 3/8") fit exactly, the outer circle (7/16") does not. The outer circle does not seem to be quite concentric, but numerology: if it's nearly right, it probably is. By moving the outer circle a few thous, I was able to get a good registration. Under high magnification, I was able to tell that the inner subpart was welded onto the sheet metal subpart, so all this indicates that there were several steps involved in manufacturing this part: first, turn the small subpart on a lathe. Then create the larger subpart from sheet metal. Then weld the two together. Welding the two together was apparently not an extremely exact procedure.
After moving the large circle to its new center, the centers no longer coincide.
Now for the rest of the part. Using SubScribe, I drew a circular arc on a circular-looking feature. Then I measured its radius.
0.283" x 2 = 0.566" is close enough to a diameter of 9/16" to say that this was the intent of the original engineer. I drew the circle, and then measured the distance between the centers.
0.568" is again close enough to 9/16". And not coincidentally, this second center coincides precisely with the location of another shaft on the machine. That certainly nails down the intent of the engineer.
I can now draw the inner tangent line between the two circles (done again using SubScribe):
The length and angle of this line in fact do not matter, since there is one and only one inner tangent line connecting these two circles in the right direction. Certainly 0.269 is close to 7/64, but that was not the design constraint. The line had to be tangent to the two circles, and drawing inner and outer tangent lines were geometric constructions that were familiar to the ancients.
We can now draw another concentric circle corresponding to the outer outline of the part, and draw an outside tangent line. Again, knowledge of design intent lets us set the outer circle's diameter at 7/8", which seems to fit precisely onto the part.
The more excitable among you may have noticed by now that the inner surface of the inner tab appears to be a circular arc, and you would be right. Drawing the circle freehand gives us a diameter of 0.439", which is close enough to 7/16" as to fix that measurement. But right now I won't analyze the tab, since I want to get the larger part done.
Near the bottom of the part, we can draw some tangent lines.
I did this in SubScribe by picking a point on the straight section of the part, then drawing a line tangent to the circle. Then I extended the line outwards. Now those lines could have started anywhere on the circle. Why these particular points? Let's draw some lines intersecting the centers. I'll also rotate the diagram so that the inter-center segment is horizontal.
The angle that the rightmost tangent line forms with the horizontal, 67.35 degrees, is irrelevant, since the constraint for that intersection point was based on an outer tangent line. But consider the angle formed between that angle and the next intersection: 125.71 - 67.35 = 58.36 degrees. This is close to 60 degrees, a nice round angle. For the intersection in the inner circle, the angle is 171.76 - 67.35 = 104.41, which is very close to 105 degrees, which is 60 + 45, more round angles. So the design intent seems clear: the outer intersection is 60 degrees from the outer tangent line, while the inner intersection is 45 degrees away from that. Let's move the intersections and construct the tangent lines so these relations become exact.
As mentioned above, this pawl fits into a hole on a gear located on a shaft. There are three shafts so far, let's call them A, B, and C. The pawl fits on shaft A, goes around shaft B, and the gear it locks is on shaft C. We know that the distance between shaft A and shaft B is 9/16". I also know from direct measurement that the distance between shaft C and shaft B is also 9/16". However, the distance between shaft A and shaft C is irregular: 0.977", not close to any fraction at all. This may be due to some constraint that we do not yet know about.
However, let's pretend that 0.977" is eventually determined through some constraint, and place the location of shaft C on the diagram.
(Update, 17 Dec 2012: It turns out that a line drawn from B perpendicular to A-C has a length of very close to 9/32", which makes A-C tangent to the 9/16" diameter circle around B. Maybe that's why the shafts are where they are: A-B is 9/16", B-C is 9/16", and A-C is tangent to the 9/16" diameter circle around B.)
I also put a circle of diameter 3/16" (shaft C's diameter), and another of diameter 3/8" around shaft C's center, which corresponds to the size of shaft C's bushing where the hole is. You can imagine the pawl fitting into a hole on the bushing by looking at the diagram.
It seems fairly clear that the pawl's end is designed to fit into the hole. The end also isn't square; it is tapered. Remember that inner tab? There is a cam which that inner tab rides on. The large diameter for the cam measures 17/32", and the small diameter measures 29/64". When the cam is rotated so that the large diameter pushes the inner tab, the pawl lifts out of the hole. When the cam small diameter is against the tab, the pawl is inside the hole. I can add the two cam diameters and then rotate the image of the pawl to simulate the two states.
In the hole (locked state):
Out of the hole (unlocked state)
Clearly one design criteria we can deduce is that the width of the pawl's end at the outside of the hole when the pawl is in the locked state must be equal to the width of the hole, and the pawl must thereafter taper. Measuring the width of the pawl in the locked state at the hole gives 0.081". Perhaps not surprisingly, this is the diameter of the hole as measured with calipers. In fractions, this is near enough to 13/16", a drill size that any mechanical engineer would have had.
Here I've drawn the outline of the hole along with its centerline. I've made the depth just deep enough for the pawl's end. We can see that the tapered pawl end does indeed fit in the hole.
Resetting the part to its design position, I found that I could draw a line along the outer outline of the pawl tangent to shaft C's outline:
The angle formed between the C-B line and the beginning of the construction line is 129 degrees. It is not a very round angle, and if the angle were not too important, it would make more sense to have it be round, perhaps divisible by 5.
Another possibility is to look at the angle formed by a radial line with the intersection:
Relative to our reference angle at the right, this is 91.57 degrees. Too far away from 90 degrees; a line placed at 90 degrees intersects the upper outline nowhere near the right place. The radial line is also 31.57 degrees from the 60-degree line. This could be significant, since 31.5 is exactly 7/10 of 45 degrees, and placing a radial line at 31.5 degrees produces an intersection very nearly at the drawn intersection.
I don't know enough about early 20th century mechanical design techniques to know if this would be reasonable: angles measured in tenths of 45 degrees.
If you'd like to have a try at figuring this out, here's the Illustrator file.