A reprint from American Scientist the magazine of Sigma Xi, The Scientific Research Society This reprint is provided for personal and noncommercial use. For any other use, please send a request to Permissions, American Scientist, P.O. Box 13975, Research Triangle Park, NC, 27709, U.S.A., or by electronic mail to [email protected]. ©Sigma Xi, The Scientific Research Society and other rightsholders www.americanscientist.org 2004 May–June 1 ENGINEERING PYRAMIDS AS INCLINED PLANES Henry Petroski he last Egyptian hieroglyph is said to have than about 10,000 people working on the con- Tbeen inscribed late in the 4th century A.D., struction site during peak activity. In fact, Ed- but serious study of Egyptian culture by wards’s analysis is so convincing that it has led Westerners did not begin until the 17th century. It this reader to imagine that an even more efficient was then that the first relatively precise measure- process could have been followed, one that was ments of the Great Pyramid were made. Near the less onerous and more worker friendly than close of the 18th century, the young French re- those usually depicted. public sent to Egypt—under the command of Napoleon—a large expedition that included a Heave Ho to Ramps and Levers “scientific and artistic commission.” The expedi- Edwards begins his analysis by discrediting the tion not only resulted in volumes of scholarship ramp and lever theories. According to him, “the but also led to the accidental uncovering of the principal theory is that a massive ramp was built Rosetta Stone, which promised to be a key to de- against one full face of the pyramid, and was ciphering hieroglyphs. Thus the foundations of lengthened as construction proceeded.” With a Egyptology were laid, and for the past two cen- grade of 1 in 10, “considered the most practical” turies scholars and amateurs alike have built it according to Edwards, such a ramp would have into an edifice. Among the many intriguing open reached 1.5 kilometers in length and contained questions about the ancient culture has long more than three times the material in the pyra- been, how were the pyramids built? There have mid itself. Edwards’s skepticism should resonate been many answers, most of which raise more with anyone who has seen the ramp employed questions. at the World Trade Center site during the re- A recent issue of Technology and Culture, the in- moval of debris from atop the bedrock at ground ternational quarterly of the Society for the Histo- zero. The depth of the hole was only about one- ry of Technology, carried a remarkable research eighth the height of the Great Pyramid. Were the note on “probable construction methods em- hole as deep as the pyramid is high, it would ployed” in building the Great Pyramid at Giza. clearly have been impossible to reach its bottom According to James Frederick Edwards, a char- via a single straight ramp. It is for the same rea- tered consultant engineer from Manchester, Eng- son that deep open-pit mines are ringed with land, the ancient Egyptians probably did not con- spiral ramps. struct special ramps or use incremental levering Edwards is incredulous that a spiral ramp was techniques to raise large blocks of stone to their employed at Giza, however, since the relative final resting places. Rather, he proposes a “more narrowness of such a ramp would have present- logical and practical alternative methodology,” ed difficulties both for two-way traffic (teams in which the sides of the incomplete pyramid it- dragging stones on sledges up and others taking self were used as inclined planes up which the empty sledges down) and when negotiating the individual blocks were hauled on sledges. turns at each corner of the pyramid. Similarly, Using little more than empirical evidence and he finds fault with any scheme using levers, ar- elementary engineering calculations, Edwards guing that it would be slow and tricky, to say demonstrates that such a hauling system was not the least. He believes both ramps and levers only physically possible but also more probable “would have been inefficient in their deploy- than previously proposed methods. He con- ment of personnel, for in both cases the haulers cludes that, using the system he describes, the and lifters would have had to ascend and de- Egyptians could have completed the entire struc- scend the pyramid structure as part of each ele- ture of the Great Pyramid within the 23-year vating cycle.” He estimates that, when the pyra- reign of King Khufu with a force of no greater mid was half finished, “the elevating cycle for one core block would have been 40 minutes us- Henry Petroski is A. S. Vesic Professor of Civil Engineering and a ing a straight ramp and seven hours using professor of history at Duke University. Address: Box 90287, levers.” And he proposes methods that he be- Durham, NC 27708-0287. lieves to be easier and quicker. 218 American Scientist, Volume 92 © 2004 Sigma Xi, The Scientific Research Society. Reproduction with permission only. Contact [email protected]. Adam Woolfitt/Corbis Figure 1. Over time, various theories have been developed to explain the means by which Egyptians lifted stone blocks weighing approximately a metric ton each into position to form pyramids. According to engineer James Frederick Edwards, complex solutions such as ramps or levers are unnecessary. The pyramids themselves form inclined planes with steep an- gles, which when faced (as shown near the top of Khafre’s pyramid, rear) could have been used to hoist the blocks. The fundamental question Edwards poses is, building block and the stone face of the incom- “why build separate ramps when the pyramid plete pyramid. has four inclined planes as an integral part of its Friction always opposes motion, so workers structure?” To show that 52-degree slopes (faced hauling stones up inclines must overcome not as they rise to provide as regular a surface as only the proportion of the weight acting down possible) are not too steep for gangs of workers the slope but also the friction force between the to drag stones up, he deduces some physical pa- sledge and the incline. The ratio of the friction rameters from ancient and contemporary evi- force itself to the force bearing down squarely dence and performs an elementary calculation on the surface over which the load is being (using mathematics no more complicated than dragged is known as the coefficient of friction. trigonometry) relating to the forces involved. Edwards appeals to “recent experiments” at Among the critical parameters is the coefficient Karnak Temple, in which “it was found that of friction between a wooden sledge bearing a three men could pull a sledge-mounted block www.americanscientist.org © 2004 Sigma Xi, The Scientific Research Society. Reproduction 2004 May–June 219 with permission only. Contact [email protected]. weighing one tonne [1,000 kilograms] over a Edwards imagines each stone being hauled up stone surface that had been lubricated with wa- the incline by a 50-man team working on the ter to reduce the effects of friction.” To estimate plateau formed on the partially completed pyra- the coefficient of friction, Edwards makes an as- mid. By remaining atop the pyramid throughout sumption about how much force an adult male the workday (“where they may indeed have lived can exert on a hauling rope. He takes this to be during the more intensive periods of construc- 68 kilograms, or 90 percent of average body tion”), the workers did not have to waste time re- weight. A simple calculation then gives the co- turning empty-handed to the base of the pyramid efficient to be 3 × 68/1,000, or about 0.2. after each block was raised. He further posits that a To check the reasonableness of his result, Ed- number of teams would have been working on the wards looks to an ancient wall painting in the plateau simultaneously, each being assigned to a Twelfth Dynasty tomb at Deir el-Bersha, which “slipway” 5 meters wide so as not to interfere with depicts the hauling on a sledge of a massive stat- other teams working in adjacent slipways. When ue of the Egyptian nobleman and tomb-occupant the pyramid had reached a quarter of its height, Djehutihotep. Using the coefficient of friction de- the 37-meter high plateau would have been about rived from the experiment at Karnak and the 173 meters on a side, thus allowing for 35 slipways. known weight (58 tonnes) of the Djehutihotep Given the width of the plateau compared to the statue, Edwards concludes that it would have length of the incline (47 meters), two teams could taken some 174 men to pull the load. Since the have worked without interference, simultaneously wall painting shows a team of 172 men pulling hauling blocks up two opposite sides of the pyra- the statue, Edwards concludes that his assump- mid and placing them on the plateau from the cen- tions and results are reasonable. He then pro- ter out. As the pyramid rose, the working space ceeds to calculate how many men it would take would have diminished, of course, and so would to haul a building block on a sledge up the side have the number of teams that could simultane- of the pyramid at Giza. ously work atop it. Nevertheless, at the half height Calculating how much force it takes to pull an of the pyramid, Edwards estimates that by his object up an inclined plane is elementary, involving scheme it would have taken less than 3 minutes to only the weight of the load, the coefficient of fric- haul a block from ground to plateau.