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Library/Paris Manuscript A/48

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Paris Manuscript A — 48

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Bibliothèque de l'Institut de France, Paris

Two sets of proportional triangles, including a smaller triangle each, for measuring the length of a long rope using ratios. Sides a b and f i of the larger triangles represent the first position of the rope, which is dragged to its second position to form part of the hypotenuse. The arcs described by the moving ends of the rope meet the hypotenuses at points n and l, respectively. The right and left triangles are labeled, respectively:

a m e n - b c d ; f m l K g h i

[I]

It is possible to know the length of any rope that is fixed at one of the ends and drawn by the other, without moving your feet.

Let a b be your rope. Fix the rope at point b, and set the square on part b m of the rope, with its turn1 at b c, and make right angle b. Then, move end b of the rope along to n, for the distance of one braccio, and set the square on part e n of the rope,2 and make a mark, and draw the mark down far enough that it comes to d, so that line b d and, likewise, e d touch each other. Then, with the square, make line e c one braccio long, and as many times as c d enters into c e, so many times will braccio b d go into rope a b.

To know the length of a fixed rope with a set-square

a m e n - b c d ; f m l K g h i. [I] It is possible to know the length of any rope that is fixed at one of the ends and drawn by the other, without moving your feet. Let a b be your rope. Fix the rope at point b, and set the square on part b m of the rope, with its turn at b c, and make right angle b. Then, move end b of the rope along to n, for the distance of one braccio, and set the square on part e n of the rope, and make a mark, and draw the mark down far enough that it comes to d, so that line b d and, likewise, e d touch each other. Then, with the square, make line e c one braccio long, and as many times as c d enters into c e, so many times will braccio b d go into rope a b.

Measuring a rope's length with proportional triangles

A set-square method to find the length of a fixed rope without pacing it

Geometry Arithmetic & Proportion

Two sets of proportional (similar) triangles show how to measure a rope fixed at one end and drawn by the other, without moving one's feet. With rope a b pinned at b, a set-square marks its swing: moving end b to n by one braccio and drawing the mark down to d builds a small triangle whose ratio to the whole gives the rope's length. As many times as c d fits into c e (one braccio), so many braccia b d go into rope a b.

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Proportional triangles for measuring a rope

The rope a b is fixed at b and the set-square laid on part b m, making right angle b. Moving the rope's end to n by one braccio and setting the square on part e n gives a mark drawn down to d, forming a small triangle similar to the large one whose hypotenuse the swinging rope traces.

The ratio c d : c e gives the rope's length

With the square, line e c is made one braccio long. As many times as c d enters into c e, so many times does braccio b d go into the whole rope a b, giving its length by simple proportion.

Reader's Edition — described with AI from the e-Leo transcription.