Geometry and Surveying
This article provided by The Garden of Archimedes - A Museum for Mathematics.
Geometry becomes system
Euclidean geometry begins with the formalisation of surveyors' work, drawing straight lines and describing circles. Geometry shifts from the land to the table where the mathematician traces his diagrams, and from there to paper and books. Even the operating tools are different. What could be obtained on a large scale by pulling ropes, can now be drawn on a small scale with a ruler and compass, translated into mental objects and regulated by a complex system of definitions and axioms. The resulting reduction of scale takes the surveyor away from the land, and allows him to dominate, in all their complexity, many problems that the immensity of the horizon had excluded from his sight. Evidence for the use of geometry along the banks of the Nile dates to an extract from Herodotus, reported at the beginning, as well as a consistent tradition seeing the origins of Greek geometry in Egypt. To what extent this tradition is legendary is unclear, but it is reasonable to expect that evidence of the activity of the Egyptian land surveyors to be found in Greek mathematics and in particular in Euclid's Elements.
Faint evidence, since the Elements is a late work, where previous elaborations - now lost - converge, and one that is organised according to the axiomatic - deductive process, typical of Greek thinking and of all western mathematics. Evidence of early Egyptian geometry, then, won't be found in the general body of the work, nor in the demonstration of theorems. It is more likely that they appear in principles, definitions and postulates which are notable for having the function of translating natural phenomena into symbols and geometric shapes. In order that the abstract geometry of classical Greece describe the most profound properties of the real world, illuminate the most obscure relationships among the objects of the external world, what needs to be done is the following: First of all, it is necessary to precisely set the definitions of the objects taken in consideration to be totally sure of what is included or excluded in the dominion of geometry. Secondly, one must identify, through postulates, a system of primary properties and possible operations, starting from which the surveyors, aided only by logic, could extract the network of consistencies and interrelations that lie - often inaccessible to material investigation - in abstract concepts of geometry and, consequently, in ordinary objects.
If the definitions and postulates translate material objects from nature and the empiric procedures of the practice into the abstract figures and operations of geometry, it is in them that the evidence of a lost tradition can and must be found. They are conceptual evidences, since the main concepts echo the procedures and operations that were not formalised. These are linguistic clues, because the choice of terms is influenced by the operations on the defined objects.
Seen in this light, Euclid's Elements show a partial, but surprisingly clear correspondence, with the operations of the 'arpedonapti'. We have already discussed the right-angle. Early than that is the definition of the straight line -an always finite line, a segment- that recalls the operation of "pulling" a rope between two "points" ( literally "marks") that define its "limits". Its straightness is not dependent on it being the shortest distance between two points, but once again it brings us back to the uniformity of the tension, according to which "it lies uniformly in relation to its marks", a property that becomes even more evocative if read together with the definition preceding it: " the marks are the limits of the line".
Of the postulates, the first three reproduce exactly the land surveyors' operations:
to draw a line between two points:
one must draw a straight line from any mark to any other.
to prolong a given line:
and to make it straight, by rights, a finite line.
To describe a circumference:
with any centre and interval, describe a circle.
While the other two testify to, so to speak, the impossibility of demonstration. The fourth of them states that
all the right-angles are equal.
and the last is the famous "postulate of parallel lines":
If a line, falling on two other lines, forms internal angles smaller than right-angles on the same side, those two lines, if prolonged, will meet where the angles smaller than right-angles are.
The difference between these two postulates and the first three is evident. The first one, put into geometric terms, describes usual practical operations. What they ask is nothing but the translation into abstract form of the concrete process of land surveyors. The last two, on the contrary, have the nature of theorems. They do not express that which the surveyors can do, but the properties of the mathematic objects already introduced - essential, assumed properties in the demonstration of the theorems to follow.
In fact, these postulates are treated as theorems, and demonstrations attempted in different sources. Proclo, in his comment on the first book of the Elements, tried to prove the fourth postulate. The fifth was to be object of numerous attemps of demonstrations, which lead to the discovery of non Euclidean geometry at the beginning of the 19th century.