A characteristic of Greek social life is responsible both for the next
phase of the development of mathematical thought and for the
misapprehension of its nature by so many moderns. "When Archytas and
Menaechmus employed mechanical instruments for solving certain
geometrical problems, 'Plato,' says Plutarch, 'inveighed against them
with great indignation and persistence as destroying and perverting all
the good that there is in geometry; for the method absconds from
incorporeal and intellectual or sensible things, and besides employs
again such bodies as require much vulgar handicraft: in this way
mechanics was dissimilated and expelled from geometry, and being for a
long time looked down upon by philosophy, became one of the arts of
war.' In fact, manual labor was looked down upon by the Greeks, and a
sharp distinction was drawn between the slaves who performed bodily work
and really observed nature, and the leisured upper classes who
speculated, and often only knew nature by hearsay. This explains much of
the naive dreamy and hazy character of ancient natural science. Only
seldom did the impulse to make experiments for oneself break through;
but when it did, a great progress resulted, as was the case of Archytas
and Archimedes. Archimedes, like Plato, held that it was undesirable for
a philosopher to seek to apply the results of science to any practical
use; but, whatever might have been his view of what ought to be in the
case, he did actually introduce a large number of new inventions"
(Jourdain, _The Nature of Mathematics_, pp. 18-19). Following the Greek
lead, certain empirically minded modern thinkers construe geometry
wholly from an intellectual point of view. History is read by them as
establishing indubitably the proposition that mathematics is a matter of
purely intellectual operations. But by so construing it, they have, in
geometry, remembered solely the measuring and forgotten the land, and,
in arithmetic, remembered the counting and forgotten the things
counted.
Arithmetic experienced little immediate gain from its new association
with geometry, which was destined to be of momentous import in its
latter history, beyond the discovery of irrationals (which, however,
were for centuries not accepted as numbers), and the establishment of
the problem of root-taking by its association with the square, and
interest in negative numbers.
The Greeks had only subtracted smaller numbers from larger, but the
Arabs began to generalize the process and had some acquaintance with
negative results, but it was difficult for them to see that these
results might really have significance. N. Chuquet, in the fifteenth
century, seems to have been the first to interpret the negative numbers,
but he remained a long time without imitators. Michael Stifel, in the
sixteenth century, still calls them "Numeri absurdi" as over against the
"Numeri veri." However, their geometrical interpretation was not
difficult, and they soon won their way into good standing. But the case
of the imaginary is more striking. The need for it was first felt when
it was seen that negative numbers have no square roots. Chuquet had
dealt with second-degree equations involving the roots of negative
numbers in 1484, but says these numbers are "impossible," and Descartes
(_Geom._, 1637) first uses the word "imaginary" to denote them. Their
introduction is due to the Italian algebrists of the sixteenth century.
They knew that the real roots of certain algebraic equations of the
third degree are represented as results of operations effected upon
"impossible" numbers of the form _a_ + _b_ sqrt{-1} (where _a_ and _b_ are
real numbers) without it being possible in general to find an algebraic
expression for the roots containing only real numbers. Cardan calculated
with these "impossibles," using them to get real results
[(5 + sqrt{-15}) (5 - sqrt{-15}) = 25 - (-15) = 40], but adds that it is a
"quantitas quae vere est sophistica" and that the calculus itself "adeo
est subtilis ut est inutilis." In 1629, Girard announced the theorem
that every complete algebraic equation admits of as many roots, real or
imaginary, as there are units in its degree, but Gauss first proved this
in 1799, and finally, in his _Theory of Complex Quantity_, in 1831.
Geometry, however, among the Greeks passed into a stage of abstraction
in which lines, planes, etc., in the sense in which they are understood
in our elementary texts, took the place of actually measured surfaces,
and also took on the deductive form of presentation that has served as a
model for all mathematical presentation since Euclid. Mensuration
smacked too much of the exchange, and before the time of Archimedes is
practically wholly absent. Even such theorems as "that the area of a
triangle equals half the product of its base and its altitude" is
foreign to Euclid (cf. Cajori, p. 39). Lines were merely directions, and
points limitations from which one worked. But there was still dependence
upon the things that one measures. Euclid's elements, "when examined in
the light of strict mathematical logic, ... has been pronounced by C. S.
Peirce to be 'Riddled with fallacies'" (Cajori, p. 37). Not logic, but
observation of the figures drawn, that is, concrete symbolization of
the processes indicated, saves Euclid from error.
Roman practical geometry seems to have come from the Etruscans, but the
Roman here is as little inventive as in his arithmetical ventures,
although the latter were stimulated somewhat by problems of inheritance
and interest reckoning. Indeed, before the entrance of Arabic learning
into Europe and the translation of Euclid from the Arabic in 1120, there
is little or no advance over the Egyptian geometry of 600 B. C. Even the
universities neglected mathematics. At Paris "in 1336 a rule was
introduced that no student should take a degree without attending
lectures on mathematics, and from a commentary on the first six books of
Euclid, dated 1536, it appears that candidates for the degree of A. M.
had to give an oath that they had attended lectures on these books.
Examinations, when held at all, probably did not extend beyond the first
book, as is shown by the nickname 'magister matheseos' applied to the
_Theorem of Pythagoras_, the last in the first book.... At Oxford, in
the middle of the fifteenth century, the first two books of Euclid were
read" (Cajori, _loc. cit._, p. 136). But later geometry dropped out and
not till 1619 was a professorship of geometry instituted at Oxford.
Roger Bacon speaks of Euclid's fifth proposition as "elefuga," and it
also gets the name of "pons asinorum" from its point of transition to
higher learning. As late as the fourteenth century an English manuscript
begins "Nowe sues here a Tretis of Geometri whereby you may knowe the
hegte, depnes, and the brede of most what erthely thynges."
The first significant turning-point lies in the geometry of Descartes.
Viete (1540-1603) and others had already applied algebra to geometry,
but Descartes, by means of cooerdinate representation, established the
idea of motion in geometry in a fashion destined to react most
fruitfully on algebra, and through this, on arithmetic, as well as
enormously to increase the scope of geometry. These discoveries are not,
however, of first moment for our problem, for the ideas of mathematical
entities remain throughout them the generalized processes that had
appeared in Greece. It is worth noting, however, that in England
mechanics has always been taught as an experimental science, while on
the Continent it has been expanded deductively, as a development of _a
priori_ principles.