Taylor - Theoretic Arithmetic.pdf - Platonic Philosophy

INTRODUCTION
works, like the remains of Grecian art, are the models by
which the unhallowed genius of modern times has been
formed, to whatever mathematical excellence it may possess.
Newton himself, as may be conjectured from what he says of
Euclid, was convinced of this when it was too late, and commenced
his mathematical career with the partial study only of
these geometrical heroes. "For he spoke with regret, says Dr.
Hutton,* of his mistake at the beginning of his mathematical
studies, in applying himself to the works of Des Cartes, and
other algebraic writers,+ before he had considered the Elements
of Euclid with that attention, which so excellent a
writer deserves."
Having premised thus much, I shall in the next place present
the reader with some observations on the essence of mathematical
genera and species, on the utility of the mathematical
science, and on the origin of its name, derived from
* See the article Newton in Hunon's Mathematical Dictionary.
t Dr. Halley also, who certainly as a mathematician ranks amongst the greatest
of the moderns, appears to have had the same opinion of the transcendency of
the Grecian genius in the mathematical sciences. For in the preface to his translation
of Apollonius de Sectiolw Rationis, (for which work he conceived so great
an setcem, that he was at the pains to learn Arabic, in order to accomplish its
translation into Latin) he says: "Methodus hzc cum algebra speciosa facilitate
contendit, evidentia vero et demonstrationum elegantia eam longe superare
videtur: ut abunde constabit, si quis conferat hanc Apollonii doctrinam de Sectione
Rationis cum ejusdem Problematis Analysi Algebraica, quam exhibuit clarissimus
Wallisius, tom. 2. Operum Math. cap. 54. p. 220." i.e. "This method contends
with specious algebra in facility, but seems far to excel it in evidence and elegance
of demonstrations; as will be abundantly manifest, if this doctrine of Apollonius De
Sectione Rationis, is compared with the algebraic analysis of the same problem which
the most celebrated Wallis exhibits in the second volume of his mathematical works,
chap. 54. p. 220." And in the conclusion of his preface, he observes, "Verum
perpendendurn est, aliud esse problema aliqualiter resolutum dare, quod modis
variis plerumque fieri potest, aliud methodo elegantissima ipsum efficere: analysi
brevissima et simul perspicua; synthesi concinna, et minima operosa. i.e. "It is
one thing to give the solution of a problem some how or other, which for the most
part may be accomplished in various ways, but another to effect this by the most
elegant method; by an analysis the shortest, and at the same time perspicuous;
by a synthesis elegant, and in the smallest degree laborious."