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4. Eureka!
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29 EUREKA! IV Were Archimedes to have used the word Eureka! each time he made a new discovery, he would have said it many times throughout his life. One of his most important discoveries involved the spiral, to which he directed an entire treatise entitled On Spirals. The spiral is a flat curve, characterized by infinite twists that coninuously increase in a mathematical progression around a point. Such a curve cannot be constructed with a compass. Rather, one can conceptualize it with the movement of a vector that grows constantly out of a single space with every rotation on a given angle; or, it can be approximated by the winding of a tube that recalls the curl of a hair, the bark of a tree, or a vortex [plate 4]. Archimedes uses numerous such elegant and refined descriptions to address this topic in his treatise. In the letter of dedication to the Alexandrian mathematician Dositheus, Archimedes says that some theorems had been entrusted to him many years before by Conon, but “before they had had sufficient 30 THE GREAT ARCHIMEDES IV time to be developed by Conon, he had passed to another life.” This specification allows us to place the work among the last writings of the scientist of Syracuse. In the opening section of his treatise, Archimedes explained the mechanical generation of the spiral as a geometric line, still known to us today as the “spiral of Archimedes.” Thus does Vitruvius, at the end of the first century BC, describe the Ionic capital in his treatise De architectura (On Architecture). As transmitted through Vitruvius, therefore, Archimedes would come to influence the architectural treatises of the Renaissance [fig. 9].1 A spiral proceeds from a point on a plane (called its origin or beginning) moving in uniform motion along a straight line, while rotating in a uniform circular motion around the point. To describe the spiral’s motion, Archimedes gives the definitions of straight uniform motion, of circular uniform motion, and of their interaction. In proposition 1, this is calculated vis-à-vis the passage of time: If a point is moved with constant speed on a line, and two segments are assumed on this line, they will have between them the same ratio as the times in which the point moved across them [fig. 10].2 Turning to more characteristically geometric properties , Archimedes reaches some surprising results on rings and on segments that are limited by the turns of the spiral, particularly in his proposition 27: Of the area comprised of the spiral and the straight lines in rotation, the third revolution is twice the second, the fourth is triple, the fifth is a quadruple of the second, and always the area following is a multiple of the second area by successive integers, while the area covered by the spiral in the first revolution is the sixth part of that of the second revolution. [34.237.245.80] Project MUSE (2024-03-19 07:09 GMT) EUREKA! 31 IV Figure 10 A clockwise spiral, one of the curves studied by Archimedes with great innovation. Figure 9 The development of the spiral studied by Archimedes in an Ionic capital of the Temple of Portunus (formerly known as Fortuna Virilis) in Rome, constructed during the second century BC and preserved mostly intact. 32 THE GREAT ARCHIMEDES IV Moreover, it seems that the scientist not only attached importance to the legitimacy of the objects with which he was concerned but reckoned that, however complicated they were, they could be measured. The more complex an object appeared to him, the more his intelligence was stimulated: On Spirals can perhaps be considered Archimedes’ finest work with regard to his inventiveness, of which he gave a demonstration, and to the brilliantly simple results that he presented, and the very fine organization that he orchestrated.3 Many of Archimedes’ discoveries in the field of statics and hydrostatics seem to have originated from a search for technical efficiency. Particularly noteworthy is the account of Archimedes’ being faced with a difficult problem to which he was able to discover an unforeseen solution (plate 5). This sudden discovery made him exclaim with enthusiasm “Eureka, Eureka!” which, in Greek means, “I have found it, I have found it!” It is an expression used cross-culturally to this day and often without the user’s knowledge of its origin in antiquity (one thinks, for example, of its association with the California Gold Rush). For Archimedes, it was first associated with the discovery...