Mental Exercise Rotating Circles to Rectilinear Oscillations

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In the diagram below, determine the position of the planet at each eighth of a rotation if the equally sized circles rotate with:

1. equal speeds in opposite directions;

2. equal speeds in the same direction.

In addition to his famous Tusi couple, al-Tusi also devised a combination of equally sized rotating circles that produced a rectilinear oscillation. One circle's center rides on the circumference of the other circle, that circle rotating in the opposite direction and at half the speed of the first circle. In the above diagram, begin with the planet at Z carried around by circle ZE, whose center is H on circle HK. Rotate circle HK a quarter of a circle toward A, moving the center of the circle ZE from H to T (and the planet down onto line BA, beyond T toward A). Now rotate the circle (originally ZE) carrying the planet in the opposite direction and at twice the speed, completing half a circle of rotation. The planet, already rotated onto line BA, will move upward and then downward and in the direction of B, returning to the line BA. The center (originally H) of the circle (originally ZE) carrying the planet is now at T, placing the planet at E. Now repeat the process, rotating the center (originally H) of the circle (originally ZE) carrying the planet another quarter of a circle, from T to K. this moves the planet from E upward and to the right. Then the planet's circle rotates half a circle, carrying the planet to M. Repeating the process again, the center moves to L and the planet to E. When the center completes its rotation, returning to H, the planet moving at twice the speed completes two rotations, returning to Z. The net motion of the planet is up and down, and back and forth along the line GD between Z and M, with no motion right or left, toward A or B.

(p. 269). This suggestive note was generally ignored. Perhaps ethnocentric Western scholars were more comfortable with a generalization that, although Arabs had preserved Greek astronomy and transmitted it back to the West, they had contributed nothing to its advance.

Half a century after Dreyer's prescient but ignored footnote, a handful of surviving manuscripts of the work of the fourteenth-century Damascus astronomer Ibn al-Shatir were studied, and his lunar theory was recognized as being nearly identical with Copernicus's. Then a historical connection was forged from al-Shatir back to al-Tusi, and hence also a possible link between al-Tusi and Copernicus. Furthermore, both al-Shatir and Copernicus employed the Tusi couple to reproduce the motion of Mercury.

Traditionally, the origins of modern science have been traced to the Greeks. With the new realization of Arab contributions in astronomy, a few scholars are beginning to argue that dynamic revolutionary ideas in astronomy were developed in the Islamic world to refute the Greek astronomical tradition, and that the resulting Arabic mathematical techniques then made possible the astronomical revolution in the West associated with Copernicus and, more generally, the European Renaissance itself.

Claims of a dynamic revolution leading to the European Renaissance may go too far. A reasonable argument can also be advanced that the Tusi couple was yet another geometrical device to reproduce observed motions with combinations of uniform circular motions, as were deferents, epicycles, and equants. The Tusi couple, as clever as anything formulated by Apollonius, Hipparchus, or Ptolemy, resides comfortably within the ruling paradigm of Greek geometrical astronomy. We will need to look beyond Islamic planetary astronomy for additional causes of the scientific revolution of the sixteenth and seventeenth centuries.

Why did the scientific initiative shift from the Islamic world back to the West, as it had earlier shifted from the Greeks and Romans to the Arabs? Are there fundamental differences in attitudes toward science inherent in the natures of Islamic and Christian societies? Answers to these questions could encourage and direct current efforts to obtain the benefits of modern science for all the peoples inhabiting our planet.

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