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.

III

CONTEMPORARY THOUGHT IN ARITHMETIC AND GEOMETRY