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72 A HISTORY OE MATHEMATICS
tion 4; two triangles are equal if two sides and the included
angle are equal respectively. To bring the triangles into coincidence,
one triangle may have to be turned over, but Eachd
is silent on this point. " Can it have escaped his notice that
in plane geometry there is an essential difference between
motion of translation and reversion ? " '
Book v., on proportions of magnitudes, has been greatly
admired because of its rigour of treatment.^ Beginners find
the book difficult. It has been the chief battle-ground of discussion
regarding the fitness of the Eleraents as a text-book
for beginners.
Book X. (as also Books VII., VIII., IX., XIIL, XIV., XV.)
is omitted from modern school editions. But it is the most
wonderful book of all. Euclid investigates - every possible
variety of lines which can be represented by \ Va ± VJ,
a and b representing two commensurable lines, and obtains 2.5
species. Every individual of every species is incommensurable
with all the individuals of every other species. De
Morgan was enthusiastic in his admiration of this book.^
1 Engel and Stackel, p. 8, note.
2 For interesting comments on Books V. and VI., see Hankel, pp.
389-404,
3 See his articles "Euclides" in Smith's Die. of Greek and Boman
Biog. and 3Iyth. and "Irrational Quantity" in the Penny Cuclopirdiii
or in the English Cyclopaedia. See also Nesselmann, pp, 165-183. In
connection with this subject of irrationals a remark by Dedekind is of
interest. See Richard Dedekind, Was sind und loas sollen die Zahkn,
Braunschweig, 1888, pp. xii and xiii. He points out that all Euclid's
constructions of figures could be made, even if the plane were not continuous
; that is, even if certain points in the plane were imagined to lie
punched out, so as to give it the appearance of a sieve. All the points
in Euclid's constructions wotdd lie between the holes; no point of the
constructions would fall into a hole. Tlic explanation of all this is to 1)6
sought in llVi I'aci that Euclid deals with certain algebraic irrationals, to
the exclusion of the transcendental.