The Indian Rope Trick: Rope vs Compass-Box C. K. Raju, Centre for Studies in Civilizations The compass-box The compass-box is so essential an aspect of a child’s school kit. The box strongly suggests the straight line as the basic notion of geometry. The geometry presupposed by the compass-box is metric rather than synthetic2 for the box has a scale (rather than an unmarked straight edge) and compasses (which are not “collapsible”), so that distances can be picked and carried, and it is meaningful to measure lengths. However, the rigid scale allows one to measure the length of only straight line segments. The compass-box does not provide any instrument to measure the length of a curved line segment. One cannot measure even the length of an arc of a circle, only the angles in degrees that various circular arcs might form—hence many children never understand the natural radian measure, which depends upon being able to measure the length of the arc. Hence, also, most students are more comfortable with 360_ than with 2p. In fact, students acquire only an operational understanding of the notion of a degree by using the protractor. Since an angle is defined not as the relative length of a circular arc, but as something connected with pairs of straight line segments, students remain woolly about the meaning of a degree. A degree is the 90th part of a right angle, but what is the entity which is being divided into 90 equal parts? This hangover presumably relates also to the practical value of navigation, for the compass-box also mimics the tools of the European navigator who, proceeding from an idealistic understanding of geometry, took the straight line as the primary geometric figure. The dependence of European navigators on the straight line became evident in the 16th c. CE when they started making long voyages across the sea. Over short distances, such as those in the Mediterranean sea, the surface of the sea could be regarded as approximately plane. On a plane surface a ship steering a constant course (set by, say, a magnetic compass or by the straight line joining two stars) should trace a straight line. However, on the globe, the ship traces a curved line, called a loxodrome (from loxos = oblique, and dromos = curve). Except in cardinal directions, this curve is not even a great circle as European navigational theorists like Nunes initially took it to be. Because European navigational techniques were so dependent upon the straight line, European navigators in the 16th c. CE could not navigate without charts which showed loxodromes5 as straight lines. Hence, the great value of the Mercator chart (common “map of the world”) in which loxodromes are straight lines. So great was the value of this chart to European navigators that subsequent British naval supremacy is put down to a better understanding of this chart! Because the compass box mimics the European navigator’s paraphernalia, though set squares and dividers are rarely used, they ritually remain part of the box. Indian rope geometry Now, India has had an old tradition of geometry from the days of the ´sulba s¯utra-s (−6th c. CE), which precede Greek geometry. The “´sulba” refers to a rope, and “rajju”, also meaning rope, or string, was a common part of the Indian school syllabus in pre-British times. It is still used by artisans, but is no longer taught in formal schools—such practical things are looked down upon from the (“Platonic”) point of view of geometry regarded as high metaphysical discourse. The introduction of British education in India, even in contexts where there was no noticeable colonial plot, sometimes made local people conceptually and technologically dependent on remote foreign sources, while teaching techniques that were inferior to local techniques. Let us see this in the case of the rope vs the compass-box. The rope or a string can be used to do a number of things. 1. By holding it taut (possibly by fastening one end) one can draw a straight line, so it can perform the function of a straight edge. 2. By choosing any appropriate unit, it can be made into a scale. Traditionally, knots were used for marks. This “primitive” technique, when combined with the two-scale principle (nowadays called the Vernier principle) as in early navigational instruments like the kam¯al, gives a remarkably high degree of accuracy. 3. By keeping one end fastened and moving the other end around, one can draw a circle. So the rope performs the function of a compass. 4. Most importantly, a rope can be used to directly measure the length of the arc, hence an angle in radians: simply lay the rope along the curve, and straighten it to compare with the length of a straight line segment. By measuring circular arcs, a rope also serves as a protractor which measures angles in radians. 5. By marking two points on it a distance can be picked and carried, so a rope (or string) can perform the function of a divider. 6. Using the “fish figure”, it is easy to construct a right angle, and by bisecting or trisecting it, it is easy to construct angles of 45_, 30_, and 60_, so it also performs the function of set squares. 7. By fastening two points, one can also draw an ellipse with the rope. This is impossible with the compass-box. So, a rope (or a piece of twine) can be used to do everything that can be done with a compass-box, and something more: it can measure the length of both straight and curved lines. The most important new capability is the fourth one, above, for it directly assigns a meaning to the length of the arc, or the length of a curved line. This is the sort of meaning that a child can easily grasp (with the hand as well as the mind). The same thing is not possible with a ruler, and assigning a meaning to a curved line starting from straight lines requires the calculus. The difficulty As already stated, students accustomed to the compassbox find it difficult to grasp the notion of the length of a curved line. The level of difficulty involved is made clearer by the reaction of a leading Western thinker, regarded as one of the founders of modern geometry, Ren´e Descartes, when he was first exposed to the notion of the length of a curved line. Descartes went so far as to assert that assigning a meaning to the length of a curved line, using straight lines, was beyond the capacity of the human mind! “The ratios between straight and curved lines are not known, and I believe cannot be discovered by human minds, and therefore no conclusion based upon such ratios can be accepted as rigorous and exact”.(Descartes, 1996, Book 2, p. 544). Descartes’ reaction was not an idiosyncratic one. Another great name of the times was Galileo, who privately raised similar objections about the calculus in his letters (Mancosu, 1996) to Cavalieri—hence Galileo allowed Cavalieri to publish on the calculus, but did not publish on it himself for he was unwilling to jeopardize his reputation. The basic difficulty noticed by Descartes, Galileo et al. is that to measure a curved line, using straight lines, one requires an infinity of infinitesimal line segments, and these thinkers thought that the concept of an infinity of infinitesimals brought in problems of the sort that were best left to the divine. Berkeley’s devastating critique of Newton and Leibniz, articulated a century later (Berkeley, 1734), proceeded on similar grounds, and could simply not be answered by his contemporaries (Jurin, 1735; Robins, 1735). It is on account of these epistemological difficulties that it took so long for the calculus to become epistemologically respectable in Europe, despite its obvious practical value. The epistemological basis of these difficulties is, in a way, captured by the differences between rope geometry and compass-box geometry: historically speaking, the notion of the length of a curved line followed the arrival of the calculus in Europe, while the notion preceded the development of the calculus in India—just because with a rope there is nothing mysterious about the length of a curved line. On the principle that phylogeny is ontogeny, one can expect the difficulties (raised e.g. by Descartes, Galileo and Berkeley) about the length of curved lines to be repeated innumerable times in the minds of students as history repeats in the classroom today. Other considerations Apart from the epistemological angle, we can also consider the situation from the economic angle, which is important if we want to take education to poorer people in India. In India, a person is defined as poor if that person cannot purchase two square meals (2400 Calories) a day. In this context, a compass-box or geometry set is expensive. It uses metals and plastics, and cannot be built locally. Most poorer children who cannot afford to purchase books do not purchase compass-boxes. If they do, and a piece goes missing it is not replaced. The compass-box is designed for use with pencil and paper (and sharpener and eraser) all of which add to the “running costs”. These costs might be trifling in the US, but they are non-trivial in India, and unaffordable for a large group of poor students for whom the free mid-day meal offered in schools is a major attraction. Finally, we can also look at the ecological angle. A rope or string can be used to draw figures even on the ground, although one could easily design it for use with pencil and paper if one wanted to, and also add markings, as in a measuring tape, to make it function like a scale. This re-usability (of the ground) with the string also makes it more eco-friendly than even acid free paper! Ironically, this is also appropriate to the conditions of many Indian schools, since, over the last half a century, the Indian government has consistently managed to provide ample luxuries for government officials but has not been able to provide classrooms for poor villagers, even though the Indian Constitution guarantees free education for all children, but does not guarantee luxuries for government officials. Conclusions The rope (or string) is flexible in more ways than one and can be used to do everything that can be done with a compass-box. It can further be used to measure the length of a curved line, impossible with the instruments in a compass box. This is helpful for the measurement of angles, and the subsequent transition to trigonometry and calculus. The rope is also inexpensive, locally-constructible, eco-friendly, and suited to conditions prevalent in countries like India. Hence, it is a superior replacement for the compass-box. To read more such articles, click on the link below: theengineersclub.net/Articles/451/1/The-Indian-Rope-Trick-Rope-vs-Compass-Box.html
Posted on: Fri, 16 Aug 2013 04:44:37 +0000
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