Arithmetic rope

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The arithmetic rope, or knotted rope, was a widely used arithmetic tool in the Middle Ages that could be used to solve many mathematical and geometrical problems.

An arithmetic rope generally has at least 13 knots—therefore, it is often called thirteen-knot-rope—placed at equal intervals. More knots were beneficial, especially for multiplication and division.^[1]

In medieval architecture, the knotted rope was indispensable for architects, because it allowed the construction of equilateral and right-angled triangles, as well as circles.^[1]

In the depiction of the liberal arts in Hortus deliciarum, the allegory of arithmetics is a female figure with a knotted rope.^[1]

Arithmetics[1]
Addition	X + Y = Z	X knots are counted, then another Y. The total number of counted knots is Z.	e.g.: 5 + 4 = 9
Subtraction	X - Y = Z	X knots are counted, then Y knots are 'uncounted'. The total number of knots remaining counted is Z.	e.g.: 9 - 4 = 5
Multiplication	X * Y = Z	X knots are counted, and the resulting distance is put together Y times. The total number of counted knots is Z.	e.g.: 4 * 3 = 12
Division	X / Y = Z (remainder Q)	X knots are counted. From these knots, Y knots are taken and grouped together until all are used up. The number of groups is Z; the number of remaining knots represents the remainder, Q.	e.g.: 12 / 4 = 3
Geometrics[1]
Right angle		The two ends of the knotted rope are nailed together, and 5 knots are counted for the base. For the perpendicular side, 4 knots are required. The right-angled triangle is generated by pulling the sides taut.
Equilateral triangle		The two ends of the knotted rope are nailed together, and 5 knots are counted for each side. The sides are tautened to create an equilateral triangle.
Circle		One end is nailed down, and a stylus is attached at the desired distance. With the rope pulled taut, the stylus is moved around, forming a circle.

• ZDF Mediathek - Video showing the application of the arithmetic rope (in German)
• http://turba-delirantium.skyrocket.de/wissenschaft/rechenseil.htm (in German)