Folding rules and proportions

A Treatise of Mathematical Instruments by John Robertson

Last fall at the WIA (Woodworking in America) conference in Valley Forge I had the chance to talk to Don McConnell and Larry Williams from Clark and Williams. Don and I have been exchanging e-mails for some time and he was excited to show me an antique ivory sector he had purchased. For those of you unfamiliar with a sector, it’s a precursor to a slide rule and was used in the 18th century for solving a wide array of math problems. It looks like a six inch folding rule and has a series of graduated scales radiating out from the center of the pivot where the two sides are joined together. Used in concert with a pair of dividers, it’s essentially an adjustable triangle that was used to solve geometry problems related to surveying, architecture, even naval navigation. What really piqued my interest is when Don introduced me to a book which describes the use of the sector – A Treatise of Mathematical Instruments by John Robertson. The book has a several sections on using a sector to manipulate proportions and how to utilize it to draw the classic orders. The sector has a pair of lines radiating from the pivot point called the “Line of Lines”. These lines are evenly divided into ten equal parts and are used to manipulate proportions. If you place compass points across any two identical increments on the line of lines, for example 9 and 9, the distance across the other increments will be proportional. Reset your compass across the 1 and 1 and it’s one ninth of the distance between the 9 and 9. This brings up several questions for me. Sectors were expensive scientific instruments most likely used by professional surveyors, scientists, etc. I’ve never seen one listed in a tool inventory from a cabinet maker. However, you can do these simple proportional exercises with a folding rule.

I was playing with an early 12” Stanley rule which had a set of inch scales on the inside faces that radiate right from the pivot point. I wonder if pre-industrial artisans commonly used a folding rule to transfer quick accurate proportional measurements? An example would be laying out the position of drawer hardware on a bank of drawer fronts. If you look at these closely they were seldom centered but usually offset. Let’s say you wanted to mark where to drill for the pulls and you wanted the holes to be 5/9 up from the bottom of each drawer. Simply pluck the drawer height from the opening in the case with a set of dividers. Adjust the folding rule until the points touch the rule 9” out from the center on both top and bottom scales. Reset the dividers to grab the distance 5” out from the center on your rule and you have the height you need. If you have a bank of graduated drawers all different heights you could use this to set all the hardware very quickly and with virtually no calculations and little chance of transferring the wrong measurement because they are pulled right from the case. Pretty slick.

George Walker