In Proportion

In the June 2009 issue of Popular Woodworking, Chris Schwarz had an article about the use of the Sector for divining relative sizes of parts based on whole number ratios. The idea for using whole number ratios, rather than Imperial Measurements is truly archaic; antediluvian. As a matter for design and construction, Vitruvius dictated a scheme of proportions in the 1st Century BC based on the proportions of “The Perfect Man.” Before him, good old Pythagoras purportedly set down a whole number proportioning system based on ‘his’ ideas of musical harmony.

Whichever system one uses, figuring the correct ratio of parts in a construct is actually pretty simple. The “hard” way means taking a chosen part of a construct to be the “rule” and then setting a pair of dividers to be one span of the part. Meaning, if I have a board that is going to be the front face of a chest, and I want the length to be five spans while the width of the board will be two spans, I set the dividers wide enough to walk the breadth of the board twice. Without changing the setting on the dividers I then walk off five steps and mark that point to be the end of the length. Once cut (squarely and accurately) the proportion of the width to length will have a ratio of 2:5. The difficulty of this is likely having to make multiple adjustments to the dividers to get the “2” from the breadth. There’s got to be a better way. There is.

Schwarz’s article was about a simple tool, essentially a ruled divider, that can be used to find any ratio based on a chosen piece. If there are, say twelve, evenly spaced marks on the legs of a divider we can find a correct proportion with a little bit of basic math. In the example above we wanted a length that is five units to a width that is two. So, we take this ruled divider, or sector, and set the “2” marks on either leg at the ends of the width of the board. We can then take our standard pair of dividers and set the legs to a width that matches the two marks of the “5” on the sector. One stop shopping, we now have a measurement that is in proportion at a ratio of 2:5. If our dividers aren’t that big, we can still step it out if we set the sector on the board to “2,” or any other even number on the sector, then set our dividers to whatever half of the selected mark. Now our divider is set to half the breadth of our board, having two swings of the divider to cover the distance. Going to the long edge, we walk off five steps and come to the same end point for the length of our board.

There are several sets of instructions on how to make a sector of this sort online and in various magazine articles, so I’ll leave that up to the reader’s own academics. The question I had about this whole mess was: Where did the sector come from?

Mr. Schwarz traces the history of the device invented by Galileo in 1602 that is called a Military Compass.

Galileo's_geometrical_and_military_compass_in_Putnam_Gallery_2009-11-24 Galileo’s geometrical and military compass, made circa 1604, photo from Wikimedia Commons

 

 

 

The Galileo Compass has seven scales – Arithmetic, Geometric (based on square roots), Stereometric (for computing volume), Metallic (comparative weights of spheres of common metals and stones), Polygraphic (for circle to polygon comparisons), Tetragonic (for squaring a circle), and “Additional” lines which are used in conjunction with other scales for specific mathematical computations. In short, this is a damned advanced piece of equipment. It is very uncommon in the course of human history that a device gets more simple following it’s invention. But, that is what we are led to believe…

Digging back further for other references for similar devices, I’ve come across illustrations from the 14th Century that identify set proportional dividers. That is, a pair of dividers which are hinged with legs extending from either end where the distance from the hinge on one side is a set ratio of the distance from the hinge on the other. The points of one end of the divider, when opened, are in a predetermined ratio of the other. I made one some years back where the set proportion was 5:3. This preset is suited to measure and mark dimensions of this ratio and no other. Whatever the span between points of the long end, the short end will be 3:5. If I want or need another proportion, I’d have to make another Proportional Divider for that ratio. Simpler tool, but to use different proportions I need another tool. Schwarz’s sector is that tool, but I still haven’t found where it originates.

The other day I went perusing through the Smithsonian’s online catalog and paged through Jacob Bessoni’s “Theatrum Instrumentum et Machinarium” (1578). The first two plates include some measuring devices that are sectors, but which predate Galileo’s by thirty-six years.

Ruled Divider

This is from the first plate of Bessoni’s measuring devices. My Latin is poor, but the text with this plate translates as “Several tools, geometric and mechanical reason, for the fundamental dimensions edition, which for the most part are based on the following findings of this book.”

 

 

The second tool is an adjustable proportional divider and looks like this:

Adjustable Sector

The accompanying text indicates it is “new and unique” for measuring any proportion or symmetry of a piece. Da Vinci sketched something similar a century or so earlier, and reported it an innovation on a similar device. I suspect Bessoni’s “new and unique” was the slide and hinge mechanism.

Regardless, both these tools fit the mold of the arithmetic sector that we’d use for whole number proportioning and dimensioning. Of the two, the ruled divider is “more simple” though ornate. I suppose my quest for an older verifiable reference to a ruled single hinged proportional divider (sector) will continue.

 

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