Henry Ernest Dudeney/Modern Puzzles/124 - Drawing a Straight Line/Solution

From ProofWiki
Jump to navigation Jump to search

Modern Puzzles by Henry Ernest Dudeney: $124$

Drawing a Straight Line
If we want to describe a circle we use an instrument that we call a pair of compasses,
but if we need a straight line we use no such instrument --
we employ a ruler or other straight edge.
In other words, we first seek a straight line to produce our required straight line,
which is equivalent to using a coin, saucer of other circular object to draw a circle.
Now, imagine yourself in such a position that you cannot obtain a straight edge --
not even a piece of thread.
Could you devise a simple instrument that would draw your straight line,
just as the compasses describe a circle?


Solution

Prime.mover suggests:
You can always fold a piece of paper.


However, Dudeney's suggestion is more elaborate, as follows.

Dudeney-Modern-Puzzles-124-solution.png

The diagram shows a linkage of rods which are free to rotate at the points at which they are joined.

The geometry is such that:

the longer rods are both the same length
the shorter rods are all the same length
the distance $AB$ is the same as the distance $BC$.

The points $A$ and $B$ are pinned to the drawing board so as to be able to rotate around those pins.

A writing instrument is attached to $D$ so as to draw on the surface when it is moved.

When $D$ is moved, a straight line is drawn.


Note that it is not strictly necessary for the rods themselves to be accurately straight, merely that the distances between the hinges are correct.


Proof

It remains to be proved that the line drawn by the writing instrument at $D$ is straight.


This theorem requires a proof.
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof.
To discuss this page in more detail, feel free to use the talk page.
When this work has been completed, you may remove this instance of {{ProofWanted}} from the code.
If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.



Historical Note

Martin Gardner, in his $1968$ repackaging 536 Puzzles & Curious Problems, mentions 1877: A.B. Kempe: How to Draw a Straight Line, in which a number of other linkages are described, all doing the same job.


Sources

Retrieved from "https://proofwiki.org/w/index.php?title=Henry_Ernest_Dudeney/Modern_Puzzles/124_-_Drawing_a_Straight_Line/Solution&oldid=560756"