50 
[May 20, 1886 
NATURE 
beef-tea or Liebig’s extract, and lastly, xanthine can be 
obtained from guanine, so that it seems not improbable 
that the manufacture of caffeine for medicinal purposes 
from Peruvian guano may be looked for as a consequence 
of the researches already made on the chemistry of these 
substances. 
EUCLID REVISED 
Euclid Revised, containing the Essentials of the Elements 
of Plane Geometry as given by Euclid in his First Six 
Books, with numerous Additional Propositions and 
Exercises. Edited by R. C. J. Nixon, M.A., formerly 
Scholar of St. Peter’s College, Cambridge. (Oxford : 
Clarendon Press, 1886.) 
HE movement for greater freedom in the teaching of 
elementary geometry than is consistent with a 
rigid adherence to Euclid’s Elements, which may be 
regarded as having taken definite shape with the forma- 
tion of the Association for the Improvement of Geometri- 
cal Teaching in the year 1871, gains strength surely, if 
not rapidly. Of this Mr. Nixon’s book is one of many 
indications, notwithstanding his decision in favour of re- 
taining Euclid’s Elements as the basis of geometry. For 
this decision he assigns “ ¢wo substantial reasons of ex- 
pediency and convenience :— 
“(1) That an established order of geometric proof is 
convenient for examination purposes ; 
“(2) A recognised numbering of fundamental results is 
convenient for reference.” 
He adds, “‘ as co-operative reasons—the fact that there 
is no consensus of opinion among experts as to the supe- 
riority of any other scheme yet proposed ; and the senti- 
ment of repugnance at the thought of sweeping away an 
institution rendered venerable by the usage of more than 
2000 years.” 
It may be questioned whether Mr. Nixon’s first reason 
“of expediency and convenience” leads as a consequence 
to the retention of Euclid as a class-book. The experi- 
ence of the examinations of the University of London is 
held by many examiners as well as teachers to prove the 
contrary. But apart from this we would enter our protest 
against the subordination here, as so often elsewhere, 
assumed of teaching to examination, of teachers to ex- 
aminers. Examiners are doubtless strong, but teachers, 
if they will only combine and assert their convictions in 
practice, are stronger. We believe too that those who 
have most carefully considered the question of a rival 
order of sequence of geometrical propositions would agree 
that the best order in a logical arrangement does not 
seriously confizct with Euclid’s order, except by simpli- 
fying it. Rather, by bringing the proofs of each proposi- 
tion nearer to the fundamental axioms and definitions 
than Euclid does, it renders less assumption of previous 
propositions necessary for the proof of any given proposi- 
tion. It stretches the chain of argument straight instead 
of carrying it round one or many unnecessary pegs. 
Many instances of this may be found in Mr. Nixon’s 
own book. To mention one only— the proof which he 
judiciously gives of the fundamental proposition that 
“similar triangles are to one another in the duplicate 
ratio of their homologous sides” depends directly on the 
Ist Proposition only of the Sixth Book, instead of the 
chain being carried round the unnecessary peg of the 15th 
Proposition, as it is by Euclid himself. 
Waiving, however, farther discussion of these general 
considerations, and granting Mr. Nixon his postulates of 
expediency and sentiment without farther cavil, we have 
no hesitation in thanking him for having produced a good 
and useful book. The conditions under which he has 
worked are such as to make it unsatisfactory to one who 
seeks for a natural and symmetrical sequence and group- 
ing of propositions forming a well compacted whole, but 
all the materials are there for enabling the student, if he 
has sufficient patience, to make it for himself. The book 
is well furnished with important propositions not con- 
tained in the ordinary editions of Euclid, but various 
excrescences in the shape of addenda and lemmas have 
been necessary to accommodate them, and in these 
addenda those which are of real importance for after use 
are rather hidden amid a crowd of other consequences, 
interesting as results, but not necessary parts of the geo- 
metrical edifice. 
Mr. Nixon has, wisely as we think, distributed his 
axioms and definitions among the propositions, intro- 
ducing each one exactly when it is required, instead of 
commencing with the full series, but it seems to us a 
serious defect that they are neither numbered nor any- 
where collected together for reference. We are rather 
surprised that he has not taken the opportunity of 
revising Euclid’s editors, and reverting to Euclid’s 
division, into common notions and postulates, of what 
modern editions call the axioms and postulates; the 
common notions embracing those general axioms which 
are true for all magnitudes, while the postulates relate to 
geometrical magnitudes only and are the really essential 
basis of geometry. 
While retaining the order of Euclid’s propositions, Mr. 
Nixon has very freely revised his demonstrations both in 
substance and in form. Where he has introduced new 
demonstrations, they are in all cases, we believe, im- 
provements. The famous fos disappears in favour of a 
proof founded on turning the triangle about one of its 
equal sides till it falls again into its original plane. 
Philo’s proof of i. 8 is adopted, and consequently 1. 7 
omitted as useless. In Book II. the diagonals of the 
rectangles disappear. Euclid’s propositions about the 
correspondence of equal chords, arcs, and angles at the 
centre of a circle are proved directly by superposition, as 
recommended in the Syllabus of the Geometrical Associa- 
tion, to which here as elsewhere Mr. Nixon acknowledges 
his indebtedness, but he still retains the propositions 
about similar segments which we should have expected 
him to omit (as in the case of i. 7), as thereby rendered 
useless. Book V. contains the essentials of the theory of 
proportion, deduced from Euclid’s definition, in the form 
first suggested by De Morgan. In Book VI. superposi- 
tion is often employed, where Euclid makes a separate 
construction, but not invariably, as, we think, might have 
been done with advantage. 
We are less satisfied with the form of Mr. Nixon’s 
demonstrations than with their substance. He objects 
strongly to Euclid’s “‘ prolixity,” of which he goes so far 
as to say, after twenty years’ experience as a teacher, that 
“NOTHING is so great a hindrance to the learner.” We 
doubt this, speaking also not without experience, In 
