Heron of Alexandria

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. Herons 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 Herons geometrical works in his Metrica, written in three books and discovered in Constantinople by R. Schne 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:

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 Herons formula was known earlier to Archimedes, who undoubtedly had a proof of it, but the demonstration of it in Herons Metrica is the earliest that we have. Although now the formula usually is derived trigonometrically, Herons 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 Euclids Elements and Apollonius Conics, was far removed from the mundane surveying. Herons 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 configurationscomparable 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 formulas5/3s52 and 12/7s52the first of which agrees with a value found in a Babylonian table, but neither of which is precisely correct. For the hexagon Herons 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 have expected Heron to use trigonometric tables such as Hipparchus had drawn up a couple of hundred years before, but apparently trigonometry was at the time largely the handmaid of the astronomer rather than of the practical man.

The gap that separated classical geometry from Heronian mensuration is clearly illustrated by certain of the problems set and solved by Heron in another of his works, the Geometrica. One problem calls for the diameter, perimeter, and area of a circle, given the sum of these magnitudes. The axiom of Eudoxus would rule out such a problem from theoretical consideration, for the three magnitudes are of unlike dimensions, but from an uncritical numerical point of view the problem makes sense.

Moreover, Heron did not solve the problem in general terms but, taking a cue again from pre-Hellenistic methods, chose the specific case in which the sum 212; his solution is like the ancient recipes in which steps only without reasons, are given. The diameter14 is easily found by taking the Archimedean value for ? and using the Babylonian method of completing the square to solve a quadratic equation. Heron simply gives the laconic instructions, Multiply 212 by 154, add 841, take the square root and subtract 29, and divide by 11.

This is scarcely the way to teach mathematics, but Herons books were intended as manuals for the practitioner. Heron paid as little attention to the uniqueness of his answer as he did to the dimensionality of his magnitudes. In one problem he called for the sides of a right triangle if the sum of the area and perimeter is 280. This is, of course, and indeterminate problem, but Heron gave only one solution, making use of the Archimedean formula for area of a triangle. In modern notation, if s is the semiperimeter of the triangle and r the radius of the inscribed circle, then rs + 2s = s(r + 2) = 280.

Following his own cookbook rule, Always look for the factors, he chose r + 2 = 8 and s = 35. Then the area rs is 210. But the triangle is a right triangle, hence the hypotenuse c is equal to s r or 35 6 or 29; the sum of the two sides a and b is equal to s + r or 41. The values of a and b are then easily found to be 20 and 21. Heron says nothing about other factorizations of 280, which of course would lead to other answers. Heron was interested in mensuration in all its formsin optics and mechanics, as well as in geodesy.

The law of reflection for light had been known to Euclid and Aristotle (probably also to Plato); but it was Heron who showed by a simple geometrical argument, in a work on Catoprics (or reflection), that the equality of the angles of incidence and reflection is a consequence of the Aristotelian principle that nature does nothing the hard way. That is, if light is to travel from a source S to a mirror MM and then the eye E of an observer (Fig. 2), the shortest possible path is SPE is that in which Figure 2 The angles SPM and EPM are equal.

That no other path SPE can be as short as SPE is apparent on drawing SQS perpendicular to MM, with SQ = QS, and comparing the path SPE with the path SPE. Since paths SPE and SPE as are equal in length to paths SPE and SPE respectively, and inasmuch as SPE is a straight line (because angle MPE is equal to angle MPS), it follows that SPE is the shortest path. Heron is remembered in the history of science as the inventor of a primitive type of steam engine, described in his Pneumatics, of a forerunner of the thermometer, and of various toys and mechanical contrivances based on the properties of fluids and on the laws of the simple machines.

He suggested in the Mechanics a law (clever but incorrect) of the simple machine whose principle had eluded even Archimedesthe inclined plane. His name is attached also to Herons algorithm for finding square roots, but this method of iteration was in reality due to the Babylonians of 2000 years before his day. Although Heron evidently learned much of Mesopotamian mathematics, he seems not to have appreciated the importance of the positional principle for fractions had become the standard tool of scholars in astronomy and physics, but it is likely that they remained unfamiliar to the common man.

Common fractions were used to some extent by the Greeks, at first with numerator placed below the denominator, later with the positions reversed (and without the bar separating the two), but Heron, writing for the practical man, seems to have preferred unit fractions. In dividing 25 by 13 he wrote the answer as 1 + +1/3 + 1/13 + 1/78. The old Egyptian addiction to unit fractions continued in Europe for at least a thousand years after the time of Heron. (Boyer, 190-193)


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