In Dee's day far from diminishing they had rather, in some respects, increased and become more acute, they were encouraged by the all too easy misunderstanding of the approach to natural magic and theories of ceremonial magic adopted by many of the new school of "neo-Platonic" mathematicians. They survived hardily into the age of Francis Bacon, long after Dee's death, so that after the Savilian and Sedleian chairs had been established in 1619 at Oxford, Francis Osborne records that one effect of the University's encouragement of this study was that "not a few of our then foolish Gentry" thereupon refused to send their sons to it "lest they should be smutted with the Black Art." (13) It was a reputation kept alive by the inevitably somewhat isolated and individual activities of mathematicians, many of whom undeniably engaged in a variety of "magical" practices, in an attempt to attain wider benefits or more concrete results from their study, than it directly indicated.

Of theories recognising the importance of mathematics, it has been acutely observed "Les uns adoptent le mathematisme en considerant d'ou il vient; il ne faut pas les confondre avec ceux qui l'adoptent en considerant ou il permet d'aller." (14) In the sixteenth century the second consideration was perforce still largely a matter of faith, a faith which might be sustained by the promise held out by its supposed importance for ceremonial magic or belief in hidden spiritual knowledge contained in formal numbers, but one which insofar as it related to the physical or applied sciences could be based only on a particular interpretation of a restricted range of data, which opponents had no trouble in discounting. The general defence of mathematics could be far more effectively conducted, therefore, by appealing to the absolute certainty attendant on its demonstrations, its apparently a priori source, and the doctrine that the intellect of man was, in the sphere of pure reason, a reflection of the creative intellect of God (which together may be said, broadly, to make up the argument of Dee's Preface). Since moreover it rapidly became obvious that many conclusions reached in this field were in flagrant contradiction with accepted teachings, or views arising from seemingly natural modes of thought, some such philosophical justification as this was required if the results were to be accepted as a true picture of "reality"; otherwise they could only teach such a lesson in scepticism, with its accompanying lessening of interest, as Montaigne drew from the information that the "certain" logic of geometry had produced the impossible conclusion that lines could asymptotically approach each other (15). An imperfect understanding of mathematical operations, especially as regards their application to physical phenomena, posed questions which demanded answers, seemingly only to be found in non-mathematical fields - even in the solution of simple Archimedean problems of weights and balances difficulties would seem to have been encountered in conceiving these otherwise than as involving the intellectual performance of the, physically, clearly impossible operation of multiplying a "weight" by a "length" - indeed it is very striking how far in needless complication, techniques of calculation, based on "ratio" and proportion went as a rule to avoid such a suggestion when dealing with such problems. It is this inhibition against "mixing" different qualities in the same term that perhaps accounts for the fact that while in certain early thinkers, such as Leonardo, clear and exact statements of lever problems, even of those considering the bent arm lever, and requiring the concept of the "potential arm" are to be found, yet the idea of turning moments about a point, though to us it appears obviously to be already essentially involved in the correct expression, which they had arrived at, of the relationships they were considering and indeed to be only a transformed equivalent of this, is conspicuously absent. Another example is Dee's treatment of problems of medicine graduation in the Preface, in which both measures of volume and degrees of temperament were concerned, and in which he goes to some lengths, in working them out, to avoid multiplying coefficients of these two distinct "qualities" together.

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