The Descriptive Geometrical Stages

It was almost at the beginning of the 18th-century (C.E.) that the Greek letter π earned a new status in the writing of William Jones (A.D. 1675–1749), an English teacher of Mathematics. He used it for the ratio of the circumference (C) of a circle to its diameter (D) in his Synopsis Palmariorum Matheseos (London, 1706). Leonard Euler accepted the symbol in 1736 and its use in his ‘Analysis’ (published in 1748) made it widely publicized in the Western World. With the spread of Western mathematics and science, the notation gained ground in the international field of education. Before the acceptance of the symbol in Europe, the ratio had the name (i) Ludolph’s number in Germany after the name of Ludolph Van Ceulen (A.D. 1540–1610), a German scholar, and (ii) Paridhisaṁkhyā in India (vide Kriyākramakarī, (a Sanskrit Commentary of Līlāvatī of Bhāskara II) due to Śaṅkara Pāraśava (A.D. 1494–1570). But the nature of π as to what type of number it represents remained unknown until in 1882 Europe identified it as a transcendental number and thereby ending the long journey which started at in the 3500 B.C. or even earlier at the basin of the river Irāvatī (Ravi) in the Harappan region of the Indus Valley Civilization and continued through different parts of this Euro-Asian continent. Authors of History of mathematics (save for a few) described the results of investigations of ancient scholars mainly country-wise (as if the countries were different pockets in the continent) without caring for a historical coherence of the investigations regarding cultural exchanges among them. Similarities in the modes of their investigations, of strategies, and the results they obtained, are the first-hand indications of such cultural relations from a very early period, mostly validated in the light of later researches. Despite the non-availability of procedural details of the investigations (in which cases, we shall mention chronologically only the approximations), there is a marked methodological similarity in a good number of cases (so far as known). Based on the said types of criteria (i.e., the similarities of investigations including their results, one may view the entire history as consisted of the following seven stages.