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Benvenuti in queste pagine dedicate a scienza, storia ed arte. Amelia Carolina Sparavigna, Torino

Showing posts with label Kha. Show all posts
Showing posts with label Kha. Show all posts

Thursday, October 13, 2011

How to measure the stair angle

According to the site of Duradek http://duradekmidatlantic.com/durarailchecklisthelp.html you can
use a protractor (available at most hardware stores) to determine this angle.


Lay a straightedge across a minimum of 3 stair nosings. Put your universal protractor on the straightedge and measure the angle. Note that the more stair nosings you have under the straightedge, the more accurate the angle will be. In the case that the run of stairs is very long, measure the stair angle at 2 or 3 places along the run to check that the angle remains constant along the full run.


I am proposing that Kha the architect at Deir El-Medina, Egypt, supervisor of the projects of three kings of the 18th Dynasty used his protractor to check the inclination of stairs for instance. You can see it at the Egyptian Museum of Torino. Read more at http://stretchingtheboundaries.blogspot.com/2011/08/khas-protractor.html


A modern universal protractor for building is that shown in the following figure.




Saturday, October 8, 2011

Kha


Kha, the architect
Museo Egizio, Torino 

Back to Kha's protractor


In the post "Kha's protractor", I have discussed an object found in the tomb of  an ancient egyptian architect, Kha.  It can be used to measure angles.Plums, levels, protractors , compasses and so on are instruments for surveying. See http://en.wikipedia.org/wiki/Surveying and the table http://en.wikipedia.org/wiki/File:Table_of_Surveying,_Cyclopaedia,_Volume_2.jpg
In this table there is the image of a compass rose. We can start from it and go back to the Kha's rose.

Monday, August 1, 2011

Kha's protractor

Kha was an architect at Deir El-Medina, Egypt, supervisor of some projects completed during the reigns of three kings of the 18th Dynasty (approximately 1440-1350 BC). Buried with his wife Merit, the items of their tomb are exposed at the Egyptian Museum, Torino. The papers at http://arxiv.org/abs/1107.4946 describes some devices of the ancient Egypt masonry (cubits, cords, plumb, levels and squares), that Kha used during his activity. An object among those found in his tomb could be used a protractor.


The figure shows an object from the Kha’s Tomb supposed to be the case of a balance scale, or the scale itself as reported by the corresponding label. In a previous preparation of the items of Kha’s Tomb, it was possible to see the front and back of the object. They are the same, with the same complex decoration.


The tool has a complex decoration, that suggested me the case could be used as a protractor, to determine directions and measure angles. The detail of decoration is shown in the following figure. 


 We see the 16-fold symmetry of a compass rose with 16 leaves as an inner decoration. Outside this rose there is a polygonal line with 18 corners and then 36 sides. In the inner decoration we can see the fraction 1/16 corresponding to one leaf. The Egyptian knew  and used the fractions as the sum of distinct unit fractions. That is, a fraction was written as a sum of fractions, each fraction having a numerator equal to 1 and a denominator equal to a positive  integer. Every positive rational number can be represented by an Egyptian fraction. In this ancient system of calculus, the Eye Of Horus defined the Old Kingdom number one, such as, 1 =  1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64, rounded-off at the six-term. A 1/64 is needed to have the exact value 1. The separated parts of the Eye of Horus were used to write describe the fractions. May be, Kha used the balance case for calculations, or simply as a protractor when he was using a plumb, a level or a scale, to find vertical or equilibrium positions, but also to measure the deviation from vertical or horizontal directions. 
Is there anything we can tell about number 36, the number of corners of the external decoration of the architect’s case? Let us remember that the Egyptians has the Decans, 36 groups of stars which rise in succession from the horizon due to the earth rotation. The rising of each decan marked the beginning of hours of the night. In the ancient Egypt, the decans were used as a sidereal clock beginning by at least the 9th or 10th Dynasty (ca 2100 BC). Probably, the object found in the Kha’s tomb had simply a geometrical decoration, such as that on another object of the tomb, which seems to be a “rose of direction”. 



The use of 1/16 fraction, the coincidence of the number of corners with that of decans, and the fact that the decoration was engraved on the instrument of an architect, suggest me that this object had been used as a protractor instrument with two scales, one based on Egyptian fractions, the other based on decans. Assuming this tool used to evaluate angles, let me show a possible set-up to measure the angle of an inclined plane. Note that the case has a lid. Let us imagine to remove the lid; the case has a perfectly linear side, that can be put on a smooth surface, as shown in the following figure.
 


When the surface is horizontal, using a plumb to have the vertical direction, one of the directions of the rose of the case coincides with the direction of the plumb (the line between leaves 1-16 or 8-9 seems to be perpendicular to the side of the case). If the surface is inclined, the direction of the rose is inclined forming a certain angle with respect to the vertical. This angle has the same value of the angle of the inclined plane.  Geometry originated as a practical science, to measure lengths, surfaces and volumes. In my opinion, Kha could have used his tool, with the contemporary use of a plumb to have a reference direction, to create a set-up able to provide a practical measurement of inclination. The Kha’s protractor could be one of the first devices to measure angles. 

References 
Turin Egyptian Museum: the tomb of Kha 
Deir El Medina: The Painted Tombs, Christine Hobson, Exploring the World of the Pharaohs: A 
complete guide to Ancient Egypt, Thames & Hudson, 1993. 
Building in Egypt; Pharaonic Stone Masonry, Dieter Arnold, Chapter 6, Tools and their 
applications, New York and Oxford, 1991 
E. Robens and R.S.H. Mikhail, The ancient Egyptian balance, Thermochimica Acta, Volume 
82(1), 1 December 1984, Pages 63-80. 7. Egypt Plumb Bobs, Wolfgang Ruecher, Wolf’s Plumb Bob News, Issue 02, Feb.2010,