In my earlier post “Accuracy vs Precision” I mentioned an issue that crops up from time to time. What do you do when you have to equally divide a line with an awkward, or unknown, length (say 39-5/16”) – into a difficult number of parts (say 7)? No problem in this modren age, right? Just whip out the old Construction Master™ tap a few keys and bingo there’s your number 5-5/8”. In order not to have to calculate (in your head_ 10-10/8” = 11-2/8” = 10-1/4”. . . 15-15/8” = 17-7/8. . . . . . you set a compass/divider to exactly 5-5/8” and set about stepping off your seven divisions. Except when you get to the last one you discover you’re almost 1/8” out. WTF?
Don’t get me wrong. I LOVE my Construction Master™. Could not live without it. But, it is lying to you (for your own good.) In actuality, seven EQUAL divisions of 39-5/16” is precisely 5-81/132”. In order not to make you crazy with a number you can’t possibly measure; it rounds it off to the nearest 1/16”. The more divisions you have, the worse the problem gets; because the rounding error accumulates with each repetition. It even tells you it’s gonna do that; in the manual.
There is a gag to beat this problem. I saw one of my heroes, Tom Silva, demonstrate the basic principle on an episode of “Ask this Old House” the other day, but I can’t find it on the website. Like me, he hates to do arithmetic. So, when he wants to find the center of a piece of ply of an awkward width, he runs his tape diagonally anywhere across the board until it reads a number easy to divide in half (for our 39-5/16” piece he would take a diagonal that is say 42”,) He then makes a mark at 21” along the diagonal with a caret to locate the exact point. A perpendicular from the base line through that point (a SpeedSquare® will work just fine in this situation) is exactly half.
As I’m certain Tommy will tell you, there is a “Greater Power” that lies within this gag. You can use it to precisely mark out any number of equal divisions of your odd length line. You can also mark out unequal divisions as long as they’re all multiples of the smallest one. Let me step you through it using our example above.
At one end of the line erect a perpendicular of a suitable length.
From the other end look for a diagonal that is easily divisible by the number of divisions you want; as it passes through the perpendicular.
Tick off the the chosen increments along the diagonal.
Perpendiculars, from the base line, which pass through these ticks gives you a precise division of the line into seven exactly equal parts.
This drawing may help make things clear.
I had thought I’d gotten this gag out of the Backstage Handbook; but The Backstage Handbook shows a different construction method for the equal division of a line.
The Backstage Handbook method is to draw a line at any angle to the one being divided of any length (as long as it can be easily divided into uniform increments of the number required.) Step off the increments on the diagonal. Connect the last point with the end of the line being divided. Lines parallel to this line which pass through the increment marks on the diagonal divide the original line.
Mathopenref.com shows a method which constructs a reverse mirror image, of the arbitrary stepped diagonal; from the opposite end of the line. Lines connecting the step marks of the two diagonals create the dividing lines
I prefer the method described above. Often there isn’t enough room on my layout for for the reverse angle. I also find it much easier to erect perpendiculars than to repeatedly construct parallels. That’s because I’m cheating. I’m employing the principles of construction; but I’m using my framing square.