For every construction, Émile Lemoine defines its "simplicity" and its "exactitude" to be the total cost of the performed operations, according to the following table:
|To place a compass end on a given point.||1||1|
|To place a compass end on a given line.||1||1|
|To draw a circle when the compass is placed.||1||0|
|To place a ruler edge through a given point.||1||1|
|To draw a line when the ruler is placed.||1||0|
On page 58 of , Coolidge notes:
"Lemoine's criteria are, perhaps, the simplest and easiest to apply; it is hard to imagine anything rougher."
He couldn't be more wrong: our complexity measure (number of lines + number of circles) is much simpler, easier and rougher!
Note that our measure can be given in terms of Lemoine's: it is simply the simplicity minus the exactitude.
|Back to The Complexity of Geometric Constructions|