Monday, March 28, 2011

criticism of John Dee on Euclid

12. Of Dee's prowess as a mathematician, Heilbron says (p. 17): "The fact of Dee's contemporary reputation is easier to ascertain than its basis. We can dismiss the suggestion that he was admired for 'profundity'. He quite rightly does not figure on van Roomen's list of the chief mathematicians of the later sixteenth century. Dee's contributions were promotional and pedagogical; he advertised the uses and beauties of mathematics, collected books and manuscripts, and assisted in saving and circulating ancient texts; he attempted to interest and instruct artisans, mechanics, and navigators, and strove to ease the beginner's entry into arithmetic and geometry. It is in this last role, as pedagogue, that Dee displayed his competence, and made his occasional small contributions (which he classed as great and original discoveries) to the study of mathematics." As a sample of the sort of thing Dee added to Euclid, Heilbron notes (p. 25) that Dee shows how to find lines x, y, z such that x/y = y/z = a/b and xyz = c3 , where a, b and c are given. (I have used anachronistic notation. Heilbron states the proportions as x:y::y:z::a:b.) What Dee does, in effect, though with techniques adapted to the use of proportions prevalent in his time, is set z = c, x = (a/b)/c and y = c/(a/b). With the advantage of algebra as we know it nowadays, one sees by multiplying x, y and z, that indeed xyz = c3 . The technique then available to Dee for handling proportions was a little more involved, and he it was necessary for him to work in terms of line segments directly, and not with lengths of line segments. However, the procedure involved were elementary and commonplace according to the standards of the time. What is interesting, though, in connection with evaluating Dee's status as a mathematician, as Heilbron observes, is that Dee connects this to the ancient problem of duplicating a cube, i.e. constructing a cube with volume equal to twice the volume of a given cube with only a straight edge and an ordinary compass, or more abstractly, using only the axioms and postulates given by Euclid in his Elements. Heilbron quotes Dee (p. 25): "Listen to this and devise, you couragious Mathematicians: consider how nere this creepeth to the famous Probleme of doubling the Cube." In fact, as Heilbron notes, the problem solved by Dee is useless in this regard. This can be seen by noting that presumably Dee has in mind taking a cube with length of edge equal c, and using his technique to find x such that x3 = 2c3 . But the 2, necessary to accomplish the doubling, is nowhere involved in Dee's construction. Again, this wouldn't have been so apparent to Dee, since algebra was in his time in the process of being developed into the system, or systems, we know today, and Dee was evidently working with traditional Euclidean geometrical constructions of line segments. In this regard, however, Heilbron notes that "although he [Dee] usually sets up his problems and manipulates his proportions geometrically, his treatment is strongly algebraic in spirit. The examples so far given show his tendency to set up equations (as proportions) and to juggle them until a solution emerges in the form of a constructible line."

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