FRANK J. SWETZ
Surveying is the mathematical science that incorporates the application of geometric principles with concepts of measurement in order to delineate the forms, position, and extent of terrestrial features or man-made structures. It is an ancient activity that has been used by all urbanized societies. The establishment and maintenance of land boundaries, the construction of walls, and the lay out of irrigation systems and aqueducts rely on the use of some kind of surveying. Extent structures such as Stonehenge in England, the Great Wall of China, Machu Picchu in the Andes of Peru, the Great Pyramids of Giza, or the temples of Angkor in Cambodia testify to their builders' knowledge and use of surveying. However, all too often, in the ancient world, surveying activities were the closely guarded prerogative of an elite, members of a priestly or bureaucratic class. Knowledge of their mathematical discipline was usually transmitted orally from master to student. Such practices, combined with the ravishes of time, account for few existing documented records of old surveying practices and techniques. Therefore, much of what can be gleaned about surveying in early non-Western societies is speculative in nature and rests on extant archaeological and architectural evidence.
The most basic of surveying activities include the determination of straight horizontal and vertical lines and the establishment of a level plane upon which structures can be erected or reference slopes established. Instruments to achieve these goals are strikingly simple: straight horizontal lines can be obtained through the use of sighting poles or stakes and retained by the use of stretched cords or ropes; straight vertical lines are obtained by the use of a plumb bob, a weight suspended from a string, and a level plane can be constructed with the assistance of a leveling device such as a water-filled trench.
Herodotus (ca. 484–425 BCE), the Greek historian, attributed the origins of geometry to the Nile Valley of Egypt, where priest-surveyors stretched ropes to mark out land boundaries. These "rope-stretchers," or as Herodotus called them harpedonaptae, are the first historically recognized surveyors. Rope stretching activities also took place in Babylonia where ancient clay tablets mention the act of "stretching a field," that is using a rope to determine the dimensions of an agricultural field and denoting a particular individual as "the dragger of the rope," a surveyor. It appears that these early Egyptian and Babylonian land measuring activities were prompted by the need for royal levies. However, in the early Indus Valley civilization, rope stretching served another need. The Śulbasūtras (Rules of the Cord), compiled between the fifth to eighth centuries BCE, supply geometric prescriptions for the construction of ritual altars. These prescriptions were based on rope-stretching and were carried out by the Vedic priests. Later Pali literature makes mention of "rope holders" (surveyors) in reference to land measurements. In all early urbanized societies, rope stretching provided the basis for surveying.
Surveying activities in Egypt certainly preceded the fifth century BCE observations of Herodotus. An inscription on the Palermo Stone dating from the Old Kingdom period of Egyptian history (ca. 3000 BCE) notes the existence of land surveys. Tomb inscriptions of about the same period mention the existence of land registry offices. A wall scene in the tomb of Menna at Thebes depicts surveyors at work. The scene shows two men measuring a field of corn with a long cord on which knots are marked at intervals of about 4 or 5 cubits. A standard measuring cord or rope of this period was 100 cubits (52.5 m) long. To obtain accurate vertical lines as well as to sight over long distances, Egyptian surveyors use the merkhet. This instrument also existed in Egypt from the earliest times; it consisted of a short plumb-line and plummet hanging from a holder that contained a sighting slit. Thus alignments could be made on distant objects. Merkhets were employed in the orientation of temples in a process called the "stretching of the measuring cord." Egyptian surveyors also employed two types of levels, the water level via a water filled trench which was suitable for large scale leveling, and the plumb-bob level erected with the aid of a wooden, right isosceles triangular frame work. Modern surveys have affirmed that the ancient Egyptians obtained very accurate results using their simple tools: the foundation for the Great Pyramid of Giza is almost perfectly level; boundary markers on the sides of the Nile River are Page 2064 | Top of Articlealigned over long distances and a very accurate system of nilometers (flood gauges), were established along the Nile from its delta to the First Cataract, a marvelous feat of leveling.
Large scale construction projects existed in the Tigris-Euphrates region as early as 2300 BCE. The accomplishments of Gudea of Lagash, an engineer or architect of this time, is commemorated by statues which depict him holding a tablet containing scaled plans for a structure. These plans are superimposed on a rectangular grid. Both the use of scaling and a rectangular reference grid indicate the existence of a high level of surveying skill at this time. Further, extensive systems of irrigation channels relied on the establishment of adequate gradients or slope determined by leveling techniques. Fragments of Babylonian clay astrolabes dating back to the second millennium BCE have been found. With such instruments, observers of the time could determine angles of inclination; however, it is believed that these astrolabes were employed for astronomical sightings rather than terrestrial surveying.
Archaeological evidence from such sites as Mohenjo-Daro and Harappa, cities of the early Indus civilization (3500–2500 BCE) indicate that city planning principles were followed. Buildings were uniform in appearance, and roads were laid out at right angles to each other. The existence of sewerage systems as well as flowing aqueducts testify to a knowledge and use of leveling principles. Builders of the cities of Mohenjo-Daro and Harappa knew surveying. Linear measuring scales found at excavation sites in the region indicate the early Indus peoples employed a decimal system of measurement. This system was based on a "Mohenjo-Daro inch" of 0.67 cm. Further, at Lothal, the remains of a sighting instrument were found. The instrument consists of a hollow shell with four slits cut into its sides. These slits are situated at right angles to each other and allow for perpendicular sighting as would be necessary in surveying a rectangular road system. During the later Śulbasūtra period (800 BCE), bamboo poles, saṇku, were used for measuring and laying out circular regions, and a standardized chain or measuring rope, rajju, was employed. The Śulbasūtra texts describe an extensive mathematics supporting its rope stretching surveying techniques. Included in this mathematics was a theory of similar triangles.
While works of later Indian mathematical authors primarily concerned applications of mathematics to astronomy, some also included mathematical information for surveyors. Āryabhaṭa I (476-550) in his Āryabhaṭiya (ca. 499) discusses procedures for finding areas and volumes of plane figures and solids. Brahmagupta (ca. AD 628) in his Brahmasphutasid-dhānta (Correct Astronomical System of Brahma) provides much information relevant to the needs of surveyors including specific computation procedures necessary for working with a shadow gnomon, a sighting staff employed for inclined sightings, thus incorporating a concept of angle into surveying activities.
China and the Far East
Early Chinese society was river based. Settling along the banks of the Yangtze and Hwang Ho rivers, the Chinese people harnessed and controlled the rivers by a system of dikes, canals, and irrigation channels. These construction projects required a knowledge and use of land surveying. Surveying was openly recognized as an important societal activity; folk hero Fu Xi and legendary emperor-engineer Yu the Great were often depicted holding surveying instruments.
Discussions and illustrations in extant texts and reference works provide some knowledge of the instruments used by early Chinese surveyors. Calibrated sighting poles, biao, were used in conjunction with sighting tubes, wang tong, or sighting boards, ce shi pai. A water level, zhun, was employed for leveling and a bamboo measure tape, bu che, devised for chaining land. A primary surveying instrument was the L-shaped set square or gnomon, ju. The earliest documented reference to surveying is found in the Zhoubi Suanjing (The Arithmetical Classic of the Gnomon and the Circular Paths of Heaven, ca. 100 BCE–AD 100) where in a fanciful conversation between Zhou Gong, a duke of the Zhou dynasty (ca. 1030–221 BCE) and the Grand Prefect Shang Gao, the duke advises the Prefect in the use of the set square. More substantial information on the mathematics of surveying is supplied in Jiuzhang suanshu (The Nine Chapters on the Mathematical Art, ca. 100 BCE). The work contains nine chapters on specific applications of mathematics. Chap. 1, "Field Measurements"; Chap. 5, "Construction Consultations" and Chap. 9 "Gougu" (right-triangle) are directly concerned with surveying computations (Fig. 1).
In AD 263, the scholar-official Liu Hui wrote a commentary on the Jiuzhang and revised much of its contents. He paid particular attention to the ninth chapter and extended its collection of problems to allow for more surveying situations which involved the obtaining of measurements to inaccessible points. Liu stressed a technique called chong cha (double difference) requiring two distinct sighting observations from separate locations. At the beginning of the Tang dynasty (AD 618–906), Liu's problems involving double differences were separated from the Jiuzhang and made into an independent mathematical work on Page 2065
Top of Article
surveying, the Haidao suanjing (Sea Island Mathematical Manual). In AD 656, the Royal Academy established an official curriculum to be used for the training of state officials. The Haidao was included among the ten mathematical works to be studied. Later, this curriculum was adopted in Japan and Korea where the instructions of the Haidao provided the basis for surveying. One of the most accomplished feats of early Chinese surveying was begun in the year 724 when the State Astronomical Bureau of the Tang dynasty initiated the first meridian survey in the ancient world. Under the supervision of the scholar Yixing, 13 observation stations were established near the meridian 114°E and between latitudes 29°N to 52°N. Observations were taken over a period of several years. The expedition determined the angular attitude of the north celestial pole above the horizon and recorded the length of shadows at noon for the summer and winter solstices and equinoxes.
The Haidao suanjing's problems became the basis of later works which considered surveying computations. In 1247, the mathematician Qin Jiushao published Shushu jiuzhang (Mathematical Treatise in Nine Sections). Three of its nine chapters concerned survey applications: Chap. 4 was entitled "Surveying"; Chap. 7, "Architecture"; and Chap. 8, "Military Matters" which concerned the use of surveying techniques for observing an enemy from a distance. His contemporary, Yang Hui wrote Tian mu bi lei cheng chu jie fa (Practical Rules of Arithmetic for Surveying). Toward the end of the Ming dynasty, Western influence began to penetrate China. In 1582, the Italian Jesuit Matteo Ricci arrived in Macao and eventually made his way to Beijing where he used his mathematical and scientific knowledge to win favor with the court. Ricci collaborated with the scholar Xu Guangqi to publish Celiang fayi (Essentials of Surveying) (1607–1608). This book introduced contemporary European survey and land measurement methodology to China. Xu himself was a skilled surveyor and wrote an appendix to the Essentials of Surveying. It appeared in 1608 under the title Celiang i tung (Similarities and Differences [Between Chinese and European] Surveying Techniques). From this period onward, surveying in China combined both European and traditional theories and practices.
The Islamic World
Islam did not emerge as an intellectual and political force until about AD 726 with the establishment of the ʿAbbasid Caliphate. From their capital in Baghdad, the early ʿAbbasid caliphs patroned the collection of scientific works and used this acquired knowledge to consolidate their religious and political empire. Their era was marked by the building of new canals, bridges and aqueducts and by the reconstruction of old Babylonian irrigation systems. Surveying was used in this work. While Muslims became the heirs of Page 2066 | Top of ArticleBabylonian, Egyptian and Greek surveying theory, they soon became accomplished practitioners and innovators in their own right. Religious prescriptions required daily prayers toward Mecca; in turn, mosques had to be constructed facing Mecca. Thus determining the qibla, or direction of Mecca from a given location, became an important task for surveyors.
Muslim surveyors used several methods of leveling: the plumb-bob level was employed as well as leveling poles. Al-Khāzinī (ca. 840) and Ibn-al-ʿAwwām wrote on the use of leveling poles. The latter wrote a handbook for farmers that included the layout of fields. Right angles were laid out with the use of an L-shaped square, kunija. Abu'l Wafā (940–998) wrote about the use of such a square. The most complete early Muslim treatise on surveying was written by Muḥammad ibn al-Hassan al-Hasib al-Karajī (ca. 1000). It was entitled Kitāb alʿuqūd wa'l abniyah (Of Vaults and Building). Al-Karajī discussed both the mathematical and practical aspects of surveying and provided specific instructions for the surveying of tunnels and underground aqueducts, qanat. Later writer Abū Saqr al-Qabisi introduced trigonometric methods into surveying computations.
The one area of surveying in which Muslim scientists and craftsman excelled was the design and utilization of measuring instruments. Their knowledge of the astrolabe was obtained from the works of Ptolemy. The first noted Muslim maker of astrolabes was al-Fazārī (d. ca. 777) who worked under the patronage of al-Mansur. By the end of the eighth century a number of scholars were producing works on the construction and use of the astrolabe. The most famous of these scholars was Māshā'allāh (762–ca. 815), a Jew who worked under Islamic patronage. His Kitāb sana ʿat al-asturlāb wa l-ʿanal bihā (Book of the Construction of an Astrolabe and Its Use) became the authoritative reference of its time. At a later date (ca. 1380), Chaucer used Māshā 'allāh's theories in his European introduction of the astrolabe as a scientific instrument. It was Muslim efforts that resulted in the astrolabe becoming a valued instrument in land surveying activities. Similarly, the Jacob's staff, a popular medieval instrument for determining planar angles, is believed to have reached Europe via Muslim sources and may have had its origins in navigational methods used by early Muslim traders. The oldest actual description of this instrument comes from a navigator's manual, the Mohit, written by Sidi al-Chelebri, captain of the Turkish fleet under Sultan Suleimannin 1554.
Although no written records exist to document the surveying knowledge of early native American civilizations, archaeological sites testify to these peoples' application and understanding of surveying techniques and principles. The city planning and construction carried out by the Olmec, Maya, Teotihuacan, Toltec, and Aztec peoples of South and Central America indicate that they undertook some surveying activities. For example, the Aztec capital of Tenochtitlan was laid out according to a grid system. It contained markets, palaces and streets and canals and held a population of approximately 200,000 people in the year 1521. Temples and ball courts were oriented to the four cardinal directions. Aqueducts brought fresh water into the city from many miles distant. Some of the Aztec surveying instruments are known by name; plumb line, temetzlepilolli; the water level, atezcath; set square, tlanacazanimi; measuring scale, tlahuahuanoloni and the construction compass, tlayolloanaloni.
Amma, T. A. Sarasvati. Geometry in Ancient and Medieval India. Delhi: Motlal Banarsidass, 1979.
Belyea, Barbara. Mapping the Marias: The Interface of Native and Scientific Cartographies. Great Plains Quarterly 17 (1997): 165–84.
Berggren, J. L. Episodes in the Mathematics of Medieval Islam. New York: Springer-Verlag, 1986.
Burnett, D. Graham. The History of Cartography and the History of Science. Isis 90 (1999): 775–80.
Cartographic Encounters: Perspectives on Native American Mapmaking and Map Use. Ed. G. Malcolm Lewis. Chicago: University of Chicago Press, 1998.
Chakravarti, Ranabir, et al. The Creation and Expansion of Settlements and Management of Hydraulic Resources in Ancient India. Nature and the Orient: The Environmental History of South and Southeast Asia. Oxford: Oxford University Press, 1998, 1998. 87–105.
Guerra, Francisco. Aztec Science and Technology. History of Science 8 (1969): 32–52.
Hedquist, Bruce. On the History of Land Surveying in China. Surveying and Mapping 35 (1975): 251–4.
Joseph, George Gheverghese. The Crest of the Peacock: Non-European Roots of Mathematics. London: I.B. Tauris, 1991.
Kiely, Edmond. Surveying Instruments: Their History and Classroom Use. New York: Teacher's College Press, Columbia University, 1947.
Labate, Donato. Strumenti Agrimensori Nel Modenese: Gnomoni, Meridiane e Compassi. Pondera 2001. 321–6.
Létolle, René and Hocine Bendjoudi. Histoires d'une Mer au Sahara: Utopies et Politiques. Paris: Harmattan, 1997.
Mundy, Barbara E. The Mapping of New Spain: Indigenous Cartography and the Maps of the Relaciones Geográficas. Chicago: University of Chicago Press, 1996.
Nakamura, Tsuko. Acceptance and Adaptation of Octants and Sextants in Japan during the Eighteenth and Nineteenth Centuries. Journal of Astronomical History and Heritage 5 (2002): 9–20.
Rayner, W. H. Surveying in Ancient Times. Civil Engineering 9 (1939): 612–4.
Sarma, Sreeramula Rajeswara. Katapayadi Notation on a Sanskrit Astrolabe. Indian Journal of History of Science 34 (1999): 271–87.
Swetz, Frank J. The Sea Island Mathematical Manual: Surveying and Mathematics in Ancient China. University Park: The Pennsylvania State University Press, 1992.
Vogel, Kurt. A Surveying Problem, Travels from China to Paris. From China to Paris: 2000 Years Transmission of Mathematical Ideas. Proceedings of a Conference Held in Bellagio, Italy, May 8–12, 2000. Stuttgart, Germany: Steiner, 2002. 1–7.