Linear Linkage
Copyright 2011by Paul Niquette. All rights reserved.

ere are a couple of ordinary household accessories, a desk lamp and a vanity mirror.  They are both designed to be wall-mounted and each features a Linear Linkage that affords cantilevered support plus convenient positioning at variable distances away from the wall.   Some solvers may observe another common attribute in addition to the dozen hinges on scissors-like arms...
To maintain a vertical lamp or mirror while accommodating extension along a horizontal straight line, each configuration must be equipped with a sliding element at the base.
Below on the left is another application of a similar mechanism, this time designed for a child's protective gate, typically mounted at the top of a flight of stairs.  Again we see the equal links and sliding elements -- in this case taking the form of a telecoping handrail along the top. Below on the right is a sketch that applies a curious array of links having more than one size. The mechanism accomplishes the design objective of linear positioning with no sliding elements!  That exclamation point recognizes a singular historical significance..

inear mechanisms, which date back at least to Leonardo da Vinci (1452-1519), have dominated the design of uncountable inventions, reaching full flourish in the early 19th Century with the work of a most gifted designer of mechanisms, James Watt (1736-1819). For interested solvers, a magnificent collection of historical documents and engravings has been published by Project Gutenberg.  Particular relevance to the Linear Linkage puzzle will be found in Kinematics of Mechanisms from the Time of Watt
 

The sketch above appeared in the January 1968 edition of Product Engineering, where it was described as "straight-line carrier device" patented by J. A. Daniel, Jr. of Newton, NJ.  A certain puzzler clipped the article, thinking that its analysis would make an interesting challenge some day.  More than 42 years went by... 

Referring to the drawing below, we shall use Daniel's integer dimensions as follows:

Distance between supports (green) ~~~ 16
Long link (red) ~~~~~~~~~~~~~~~~~ 23
Short link (blue) ~~~~~~~~~~~~~~~~  9

 
Does it work?

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