Prisca Sapientia

It's ironic that most of the men who participated in the "scientific
revolution", whose contributions seem (to us) so original and 
innovative, were themselves convinced that they were merely re-
discovering the vast body of pristine knowledge (prisca sapientia) 
that had been possessed by the ancients, but somehow lost and 
forgotten during the centuries that came to be called the "dark 
ages" of western civilization.  This was not an entirely unreason-
able belief, because the great works, both material and intellectual, 
of the classical civilizations were (and to some extent still are) 
very imposing.  The intellectual culture of Western Europe really did
decline during the fall of Rome, and the institutions for preserving
and passing along knowledge, as well as the inclination to do so, 
were severely diminished.  Then, after so long an absence, when
the ancient texts were re-discovered, the scholars of the Renasiance
and later periods were acutely aware of their intellectual inferiority 
vis-a-vis "the ancients".  Also, the fact that many of the ancient 
texts were now available only in fragmentary form, often in third-
hand translations, and many of the references were to works totally
unknown and presumably lost, contributed to the impression that the
ancients had known far MORE, if we could only find it out.

This attitude toward the past is, in some ways, the exact opposite
of our usual view today, which is of a totally ordered sequence of
eras progressing from less knowledge in the past to more knowledge
in the future.  It's hard for us to imagine, today, the intellectual
climate among people who believed (knew) they were scientifically 
and mathematically inferior to their ancestors in the distant past.

Interestingly, this peculiar historical circumstance undoubtedly
contributed to the unique flourishing of intellectual affairs in
Western civilization that occurred soon after the ancients had
been re-discovered.  Part of the psycological impetus came from
the great appreciation they felt for recorded knowledge, and the
esteem they had for the great thinkers of antiquity.  Also, the
enduring value of the recorded knowledge (if it was preserved), 
and the kind of immortality it gave to the authors, surviving a
millenium of neglect only to be more wondered at when finally
re-discovered, was a source of immense fascination, and inevitably 
tempted men to participate in the process, even if only (at first)
by translating and copying the great works.

Richard Feynman often pointed to the early 16th century discovery of 
the general solution of cubic polynomials as a great turning point in 
scientific history, because this was the first time a "modern" man
made a significant discovery that went BEYOND the ancient knowledge.
(Needless to say, there were acrimonious disputes between Cardano,
Ferro, Tartaglia about who deserved to be credited with this 
discovery.)  The tantalizing prospect of "bettering" the ancients 
was thus raised, and was an incredibly powerful incentive for making
intellectual discoveries.

Nevertheless, the belief that the ancients had possessed a vast body 
of knowledge, of which we have only fragments and scatterred hints, 
persisted.  As late as the 1600's men like Fermat were developing 
their original ideas in the form of speculative "reconstructions" of 
lost works from antiquity.  For example, Fermat completed a recon-
struction of Appolonius' lost work on "Plane Loci", and Fermat 
himself said that this effort led directly to his development of 
what we now call analytic geometry.  (Needless to say, there was 
an acrimonious dispute about whether Fermat or Descartes deserves 
credit for this discovery.)

Newton was convinced that "the ancients" had used analytical methods 
to arrive at their results, and then consciously concealed their 
methods by expressing the results in synthetic form.  Regarding the 
solution of the locus problem, Newton remarked

  ...they [the ancient geometers] accomplished it by certain 
  simple proportions, judging that nothing written in a different 
  style was worthy to read, and in consequence concealing the 
  analysis by which they found their constructions.

Even with regards to the calculus, Christianson's biography of Newton
tells us that

  ...in May 1694...Newton had recently completed his brilliant 
  mathematical treatise 'De quadratura' which introduced the now 
  familiar dot notation for fluxions, and he expressed the belief 
  [to David Gregory] that its contents were known to the Greeks, 
  who had destroyed all evidence of algebraic analysis in favor 
  of more elegant geometrical proofs.

In discussing the question of why Newton, the inventor of the 
calculus, avoided any explicit use of it in his Principia, 
Christianson says

 "Had he been more forthright, he would have simply admitted to
  his preference for classical geometry on the grounds that it was
  more elegant than the analytic algorithms of the fluxional
  calculus, and to his belief that it had enabled the ancients to
  discover what he was only rediscovering some two millennia later."

Of course, Heiberg's 1906 discovery of "The Method", in which 
Archimedes described (in a private letter) analytical techniques - 
including what we would call an ituitive form of fluxional calculus - 
that he had used to discover his most important theorems, showed 
that Newton's suspicion had some validity.

Newton was clearly influenced by the hermetic tradition, which 
attributed all kinds of wisdom and secret knowledge to "the 
ancients", not just mathematical.  For example, he told Fatio in
1692 that the ancients knew the law of gravitation, and David 
Gregory noted that "Newton believed his natural philosophy was 
most consistent with the teaching of Thales, while the Egyptian 
Thoth 'was a believer in the Copernican system'."

Returning to mathematics, not everyone shared Newton's generous view
of the motives of the ancient sages in presenting their results in
the more elegant (i.e., synthetic) form.  Wallis (never having seen 
"The Method", of course) commented on the distinctly cryptical 
progression of many of Archimedes' presentations that they seemed 
to him

  "...as it were of a set purpose to have covered up the
   traces of his investigation, as if he had grudged
   posterity the secret of his method of inquiry, while
   he wished to extort from them assent to his results.

   Not only Archimedes, but nearly all the ancients so hid
   from posterity their method of Analysis (though it is
   clear that they had one) that more modern mathematicians
   found it easier to invent a new Analysis than to seek
   out the old."

Boyer says that Torricelli expressed similar sentiments.  These
men evidently believed that the motive of the ancients in presenting
their work in synthetic form was not the striving for elegance but
just a selfish effort to keep their methods secret.  (It's amusing
to speculate that the synthetic form of presentation so closely
associated with rigorous mathematics, and so widely used in education
for so many centuries (e.g., Euclid) may have been originally just
a cynical strategy to conceal the actual thought processes, like 
zero-information proofs!)

Going even further in impunging the not-so-prisca motives of the
ancients, not to mention disparaging the fullness of their sapientia, 
Descartes wrote (Regulae, Rule IV, quoted in Calinger) that

  We have sufficient evidence that the ancient geometers 
  made use of a certain "analysis" which they applied in the 
  resolution of their problems, although, as we find, they 
  grudged to their successors knowledge of this method...

  I could not but suspect they were acquainted with a 
  mathematics very different from that which is commonly 
  cultivated in our day.  Not that I imagined that they had 
  full knowledge of it; their extravagant exhultations, and
  the sacrifices they offered, for what are minor discoveries 
  suffices to show how rudimentary their knowledge must have 
  been... [!]

  ...[but] I am convinced that certain primary seeds of truth 
  implanted by nature in our human minds - seeds which in us 
  are stifled owing to our reading and hearing, day by day, so 
  many diverse errors - had such vitality in that rude and un-
  sophisticated ancient world that the mental light ... enabled 
  them to recognize true ideas in philosophy and mathematics, 
  although they were not yet able to obtain true mastery of 
  them...  These writers, I am inclined to believe, by a 
  certain baneful craftiness, kept the secrets of this 
  mathematics to themselves.  

  Acting as many inventors are known to have done in the case 
  of their discoveries, they have perhaps feared that their 
  method being so very easy and simple, would if made public, 
  diminish, not increase public estteem.  Instead they have 
  chosen to propound, as being the fruits of their skill, a
  number of sterile truths, deductively demonstrated with  
  great show of logical subtlety, with a view to winning an 
  amazing admiration, thus dwelling indeed on the results 
  obtained by way of their method, but without disclosing the 
  method itself - a disclosure which would have completely 
  undermined that amazement.

Here Descartes is claiming that the ancients not only kept their
true methods secret, but did so for the basest of reasons, to cover
up the fundamental simplicity of these results when approached
analytically.  Essentially Descartes accuses the ancient sages of
perpetrating a conscious fraud on the uninitiated - and on posterity.
This shows what a distance the western intellectual community had 
come from the wonder and awe that they once felt toward the ancient
writers.