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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