Heron Of Alexandria Essay Research Paper Another

Heron Of Alexandria Essay, Research Paper

Another worker in applied mathematics belonging to the period under consideration was Heron of Alexandria. His much disputed date, with possibilities ranging from 150 BC to 250 AD, has recently been plausibly placed in the second half of the first century AD. His works on mathematical and physical subjects are so numerous and varied that it is customary to describe him as an encyclopedic writer in these fields. There are reasons to suppose he was an Egyptian with Greek training. At any rate his writings, which so often aim at practical utility rather than theoretical completeness, show a curious blend of the Greek and the Oriental. He did much to furnish a scientific foundation for engineering and land surveying. Fourteen or so treatises by Heron, some evidently considerably edited, have come down to us, and there are references to additional last works.

Heron.s works may be divided into two classes, the geometrical and the mechanical. The geometrical works deal largely with problems on mensuration and the mechanical ones with descriptions of ingenious mechanical devices.

The most important of Heron.s geometrical works in his Metrica, written in three books and discovered in Constantinople by R. Sch.ne as recently as 1896. Book 1 deals with the area mensuration of squares, rectangles, triangles, triangles, trapezoids, various other specialized quadrilaterals, the regular polygons from the equilateral triangle to the regular dodecagon, circles and their segments, ellipses, parabolic segments, and the surfaces of cylinders, cones, spheres, and spherical zones. (Eves, 146.) He is best remembered for having discovered how to find the area of a triangle in terms of the lengths of its sides and for having invented an early steam-powered machine. In fact he created many interesting mechanical devices besides the steam engine and wrote a treatise on surveying (Dioptrica). In his Mechanica, part of which is quoted by Pappus; he considers the mechanics of a bent lever. Pappus uses this principle of Heron to discuss the problems of the power (force) required to move a weight up an inclined plane. He imagines the weight as located at the center of a sphere being rolled up the inclined plane and balanced by a fictitious weight B on the surface of the sphere at the same elevation as the center and as close as possible to the plane (see Fig. 1). He takes the power required as the sum of the power required to move the two weights along a horizontal surface. (This reasoning uses Aristotelian principles of physics that we no longer accept. On the modern view, except to overcome rolling friction, no force at all is required to roll a ball along a horizontal surface once it has started to roll.) Thus, although the principle of the lever was well understood in Hellenistic times, that of the inclined plane was not. Since the modern laws involves only the proportions in a triangle, it seems strange that this simple principle was not discovered. (Cooke, 146,147.)

Figure 1: The law of the inclined plane according to Heron and Pappus.

Once again, Heron of Alexandria is best known in the history of mathematics for the formula, bearing his name, for the area of a triangle:

K = √s(s . a)(s . b)(s . b)

Where a, b, c are the sides and s is half the sum of these sides, that is, the semiperimeter. The Arabs tell us that .Heron.s formula. was known earlier to Archimedes, who undoubtedly had a proof of it, but the demonstration of it in Heron.s Metrica is the earliest that we have. Although now the formula usually is derived trigonometrically, Heron.s proof is conventionally geometric. The Metrica, like the Method of Archimedes, was long lost, until rediscovered at Constantinople in 1896 in a manuscript dating from about 1100. The word .geometry. originally meant .earth measure,. but classical geometry, such as that found in Euclid.s Elements and Apollonius. Conics, was far removed from the mundane surveying. Heron.s work, on the other hand, shows us that not all mathematics in Greece was of the .classical. type. There evidently were two levels in the study of configurations.comparable to the distinction made in numeral context between arithmetic (or theory of numbers) and logistic (or techniques of computation).one of which, eminently rational, might better be described as geodesy. The Babylonians lacked the type of mathematics that is found in Heron. It is true that in the Metrica an occasional demonstration is included, but the body of the work is concerned with numerical examples in mensuration of lengths, areas, and volumes. There are strong resemblances between his results and those found in ancient Mesopotamian problem texts. For example, Heron gave a tabulation of the areas An of regular polygons of n sides in terms of the square of one side sn, beginning with A3 = 13/30S32 and continuing to A12 = 45/4S122. As was the case in pre-Hellinistic mathematics, Heron also made no distinction between results that are exact and those that are only approximations. For A5, for example, Heron gave two formulas.5/3s52 and 12/7s52.the first of which agrees with a value found in a Babylonian table, but neither of which is precisely correct. For the hexagon Heron.s ratio of A6 to s62 is 13/5, the Babylonian is 2;37,30, whereas the true value lies between these and is of course irrational. In such calculations we should h