Herodotus traces the origins of geometry to the division of land reclaimed from marshes that were drained and channelled during the reign of Pharaoh Senusret II. Geometry was then understood in its etymological sense of land surveying, and the knowledge acquired was purely empirical. It took the genius of the Greeks to establish proofs. In the early 6 th century BCE, Thales of Miletus was unable to apply his mathematical techniques to a figure that was not bounded by straight lines. In the following century, known as the Age of Pericles, the orator Antiphon regarded the circle as a regular polygon with infinitely many sides. This view of infinity was too far ahead of its time and found no audience.
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Implicit infinity -----------------
As a matter of principle, the Greeks excluded infinity because it could not "enter" into any argument. The concept was limited to the mere possibility of continuing a procedure that could be repeated without limit. Thus, around 430 BCE, Hippias of Elis established a method for constructing the quadratrix, known as the quadratrix of Dinostratus, in infinitely many steps. Antiphon's idea was nevertheless taken up by Euclid, who bounded the circle using regular polygons circumscribed about and inscribed in it, and deduced that the area of a circle is proportional to the square of its diameter.
Astronomers and physicists, faced with the material world, may have been the first to conceive of infinitesimal quantities. Zeno of Elea envisaged the indefinite divisibility of bodies, while astronomers made approximations by disregarding certain quantities. Thus Aristarchus of Samos, the first heliocentrist according to Archimedes, treated the Sun's rays as parallel when modelling the boundary between light and shadow on the Moon as a great circle. In mathematics, Pythagoras's theory of incommensurable quantities, studied by Democritus, can be seen as introducing this concept. Plato's school did develop methods for constructing tangents, but these applied only to conics, and then only thanks to the special properties of their diameters (see FOCUS "The diameters of conics"). For Aristotle, infinity could exist only potentially (the concept of infinity could be conceived only as a possibility), not actually (that is, as a mathematical object in its own right, which could be manipulated and subjected to arithmetic operations…), and was synonymous with incompleteness. A significant advance came only with Eudoxus of Cnidus in the 4 th century BCE and his method of exhaustion, a term implying that everything is taken into account, down to the smallest part. The earliest known text to use this method is Book XII of Euclid's Elements.