Leaning beams and canes: geometry of the right angle
Finding how far a sliding beam's foot moves, by root-taking and proportion
Three worked problems on leaning beams and canes, illustrated with faint sketches. A beam five braccia long leaning on another has its top lowered one braccio; squaring gives 25 minus 16 equals 9, whose root 3 is how far its foot slides out. A second cane against a four-braccia wall is solved without measuring, by a plumb line or drafting square and the proportion of c b to a b. A third construction uses a compass at n and e to find both the cane's descent and its displacement from the wall.
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How far the foot of a leaning beam moves
A five-braccia beam leans on another of the same length; lowering its top by one braccio, the standing height becomes four. Since five times five is twenty-five and four times four is sixteen, nine remains, whose root three is how far the foot slides from the other beam.
A leaning cane's descent found without measuring
A cane as tall as a four-braccia wall leans against it; its foot is moved an unknown distance and the top drops an unknown amount. Dropping a plumb line or using a drafting square at a c, as many times as c b enters into a b, so many times does d b enter into the four-braccia cane.
Finding the cane's fall with a compass
Setting one compass foot at n and the other at e, line e b is drawn; the distance from b to d is how far the cane's top has descended. Then line b c gives, by how many times c d enters b d, how far the cane's foot stands from the wall.