Harpedo tapes

Harpedonapten (Greek: “rope tensioner”; composition of harpedonä = rope and hapto = to touch, to tie on) were the surveyors in ancient Egypt . You alone were responsible for determining angles and measuring buildings and properties on behalf of the Pharaoh .

Their main measuring instrument was the twelve-knot cord . It was used to determine lengths and angles. The angle measurement was carried out by tracing back to a length measurement.

Cause of the activity

“In Egypt, metrology was strongly influenced by the Nile, since after the annual floods all farmland had to be re-measured. The art of surveying was therefore highly developed and, like measuring the water level of the Nile, extremely important. ”The survey also had the purpose of collecting the property tax.

The cords of the harpedonapten

To determine angles, the harpedo tapes used closed (ring) cords of various lengths. The three basic cords had lengths of 12 Meh (long cord) equal to 84 Shep, 72 Shep or Schesep (medium cord), and 60 Shep (short cord). The middle cord is 12 small cubits (12 x 6/7 meh) in length. According to the division of a Shep (hand width) into 4 Djeba (fingers), the three basic cords were proportionally reduced if necessary. The long line is divided into sections of 7 shep. The middle line is divided into 6 shep sections and the short 5 shep section. The 5 shep section on the short line is the length of the remen . With these cords, the harpedonapten determined all corners of the Egyptian world. The cords are based on the Pythagorean triple 3: 4: 5. By extending the short cord to a length of 70 Shep (10 Meh) you were given the further possibility of stretching the Pythagorean triple 20: 21: 29. As with the short cord the cathete stretched vertically with 20 Schep. This stretching is the direct transition between these two Pythagorean triples.

For the determination of right angles in a horizontal position, Mark Lehner suspects the use of a triangular dimension with a length of 84 Königsellen (43.9824 m). He justifies this with the discovery of post holes with an average spacing of 7 royal cells along the sides of the Great Pyramid. Mark Lehner compares the methods for measuring the field of the right angle. The assumed construction with the 3-4-5 triangle achieves the necessary precision. Doubts are reported when constructing with circular arcs. Neither the distance and the position of the post holes nor the achievable precision confirm a construction of the right angle by circular arcs.

The rope test

At the beginning of the measurement, the measuring cable was checked for correctness. This is also the symbol for the start of construction. This test is shown in many representations as a joint act of the Pharaoh and the goddess Seschat . The measuring rope is stretched as a loop between two batons ( flails ). With the long line, the distance between the batons must therefore be exactly 6 Meh. Markings are made on the string at the locations of the batons. The distance between these marks is halved. One of the markings forms the corner point of the right angle when clamping as a Pythagorean triple. Unfortunately, these markings (knots) on the rope have not yet been recognized in the representations of the rope test, so the tension of the rope as a loop is only an indication of the existence of twelve-knot cords in the old empire.

The "loop test" is a test of the twelve-knot cord (to determine the right angle) for correct length and not a measurement on the structure. The rope test is the calibration of the twelve-knot cord.

The invention of the merchet

The Merchet ( mrḥ.t ) is a measuring device for measuring ancient Egyptian slope angles . Its existence is known from a corresponding hieroglyph. In the case of merchants, the short leg of the measuring cord is replaced by a horizontal wooden strip and the long leg is replaced by a perpendicular plumb line. The line part of the hypotenuse is omitted. A merchet based on the short cord has a wooden bar of 42 shep. Half of the bar is graded in Shep. The plumb bob with an exact length of 20 Shep is attached to the end of the bar “with” graduation. Half of the bar "without" division is used to rest on the upper level of the embankment. The horizontal position of the bar must be maintained for a clean measurement. By moving the bar, the Egyptian slope angle is determined with a reading on the upper slope edge. The Merchet is a further development of the findings from practice with the cords of the harpedonapten.

The way the harpedonapten work

There were two methods of establishing angles in ancient Egypt. The first method was to measure the setback on a right-angled royal cell. The unit of measurement for these angles was called a seced . This method is the normalization of the angles and was used more by higher priests. The second method was used by the practitioners. They stuck to their measuring cords (or the merchet) and used the method of adding or subtracting to the baseline of the measuring cord stretched as a right-angled triangle. For the determination of angles smaller than 45 °, the measuring cord was stretched with the long leg and for angles from 45 ° with the short leg as the base line. But there are also cases with inverse stretching of the measuring cord. Both methods use the opposite cathetus (height) as the reference distance. Today, mathematics is usually normalized to the adjacent e.g. B. Incline information on traffic signs.

Example of slope angle of the Great Pyramid : use of the long cord. The leg with 21 Shep (short leg) is stretched as a base line. There is the supplement of a hand (1 Shep) to the baseline. The angle is 21 plus 1 shep. In standardized notation, it is Seked. ${\ displaystyle 5 + {\ frac {1} {2}}}$

Example of slope angle of the Mykerinos pyramid : use of the short cord. The leg with 15 shep (short leg) is stretched as a base line. There is the supplement of a hand (1 Shep) to the baseline. The angle is 15 plus 1 shep. In standardized notation, it is Seked. ${\ displaystyle 5 + {\ frac {1} {2}} + {\ frac {1} {10}}}$

Example of slope angle of the Chephren pyramid : All cords are possible. There is no surcharge or deduction. The angle is 21 Shep with a long line. In standardized notation, it is Seked. ${\ displaystyle 5 + {\ frac {1} {4}}}$

Example of slope angle of the red pyramid : use of the short cord. The leg with 15 shep (short leg) is stretched as a base line. 6 hands (6 shep) are added to the baseline. The angle is 15 plus 6 shep. Or you can use the 10 Meh cord. The leg with 21 Shep (long leg) is stretched as a base line. There is no surcharge or deduction. The angle is 21 shep. The extension of the short cord to the 10 Meh cord is the direct transition to the Pythagorean triple 20: 21: 29. In normalized notation, it is Seked. The angle of the Red Pyramid is more than 7 seconds and therefore less than 45 °. ${\ displaystyle 7 + {\ frac {1} {3}} + {\ frac {1} {60}}}$

Deviations from the construction of the pyramids:

Cheops: Deviation (today) no longer detectable; Deviation less than 0.001 ° from the specified inclination of the pyramid.
Mykerinos: deviation (today) no longer detectable; Deviation less than 0.001 ° from the specified inclination of the pyramid.
Chephren: 0.037 °
Sneferu, Red Pyramid in Dahshur North: deviation less than 0.0003 °. Due to the soft subsoil, the Red Pyramid was already submerged during the construction period. This also led to a change in the built slope angle. To compensate for this, they continued to build a little steeper. The angle of repose is therefore not constant. "The layers of the pyramid sides are concave."

literature

Solomon Gandz: The Harpedonapten or rope tensioners and rope knotters. In: O. Neugebauer, Julius Stenzel, Otto Toeplitz (ed.): Sources and studies on the history of mathematics. Volume 1, Issue 3, Springer-Verlag, 1930, pp. 255-277.
Moritz Cantor : Lectures on the history of mathematics. First volume. From the oldest times to the year 1200 AD. 2nd edition. BG Teubner, Leipzig 1894, p. 64.
Moritz Cantor: About the oldest Indian mathematics. Archive of Mathematics and Physics. 3rd series, Volume 8 (1905) pp. 63-72

Individual evidence

↑ Tension ropes for right angles. In: FOCUS-SCHULE No. 3 (2008). Retrieved November 25, 2016 .
↑ Wolfgang Trapp : Small manual of the dimensions, numbers, weights and the time calculation. Komet-Verlag, 1998, ISBN 3-89836-198-5 , p. 18.
↑ Mark Lehner: Secret of the pyramids. Bassermann Verlag, 2004, ISBN 3-8094-1722-X , p. 213.
↑ Helmut Minow: Measuring tools and measures of length in ancient Egypt.
^ Anne Rooney: History of Mathematics, 83
↑ Frank Müller-Römer: The construction of the pyramids. Herbert Utz Verlag, 2011, ISBN 978-3-8316-4069-0 , p. 165.

[1] Tension ropes for right angles. In: FOCUS-SCHULE No. 3 (2008). Retrieved November 25, 2016 .

[2] Wolfgang Trapp : Small manual of the dimensions, numbers, weights and the time calculation. Komet-Verlag, 1998, ISBN 3-89836-198-5 , p. 18.

[3] Mark Lehner: Secret of the pyramids. Bassermann Verlag, 2004, ISBN 3-8094-1722-X , p. 213.

[4] Helmut Minow: Measuring tools and measures of length in ancient Egypt.

[5] Anne Rooney: History of Mathematics, 83

[6] Frank Müller-Römer: The construction of the pyramids. Herbert Utz Verlag, 2011, ISBN 978-3-8316-4069-0 , p. 165.