324 
We shall commence with quoting a few definitions, and 
simply italicising their most salient incongruities. Further 
comment will rarely be needed. 
“ Extension is the sface occupied by a body” (p. 1). 
“ A straight line is that which has all its Jozwés in the 
same direction” (p.1). Direction is, of course, not de- 
fined, nor is it stated whence the points of the line are 
supposed to be viewed. 
“A plane angle is the gveater or /ess inclination of two 
straight lines fo a common point” (p. 5). In a note 
hereto the author complacently observes that “ our defini- 
tion accords with Euclid, Bk. i., n. 8.” 
“The angle AO B increases continually zz proportion 
as the straight line OB takes the direction OC, OD, 
&c.” (p. 6). 
‘The name adjacent angles is given to those angles 
that have one side common” (p.6). The author appears 
to have felt that there was some insufficiency here, but in- 
stead of expunging the passage as wholly useless, he tries 
again in small print, and almost, though not quite, 
succeeds. That some haziness still clings around his 
conception of adjacent angles is obvious ; for when we 
come, in the Geometry of Space, to angles between two 
planes, dihedral angles, we are told (p. 90) that “ adjacent 
dihedral angles are those which have a common plane, 
and whose other two sides are in one plane.” The condi- 
tion here italicised is wholly unessential, and the really 
essential one is again omitted ; viz., that the two angles 
should be on opposite sides of their common plane. 
This misconception of the nature of adjacent dihedral 
angles leads the author, naturally, to the following 
absolutely false definition :—‘“If two adjacent dihedral 
angles are equal, each one is named a dihedral right 
angle.” We may here observe that there is deplorable 
confusion in this part of the work between polyhedra 
and polyhedral angles. Trihedral angles are generally, 
though not always, termed ¢vz/edra, dihedral angles occa- 
sionally crop up as @hedra, and uncouth entities such as 
polyhedrals, dihedral sides of a polyhedral angle (p. 94), 
and polyhedral triangles (p. 95), not unfrequently stop 
the way. 
Leaving definitions, however, let us glance at Mr. 
Morell’s enunciations of theorems, and the “im- 
proved shorter demonstrations” of them with which he 
supplies us. 
On p. 3 we find the following short paragraph, into 
which a theorem and its demonstration are supposed to 
be condensed :—‘ Any diameter whatever divides a 
circumference into two equal parts ; ¢herefore if on super- 
posing its two halves they did not agree, the radii of the 
same circumference would be unequal, which is absurd.” 
The word ¢herefore would imply that the assertion which 
precedes it is not itself the theorem to be demonstrated, 
but one which is to be employed.in demonstrating some- 
thing else. But what is this something else? Long 
reflection failed to furnish any answer to the question ; it 
seemed but to give to the last three words a more extended 
significance than the author could have contemplated. 
The entire paragraph remained a mystery to us, in fact, 
until we reached p. 25, when Mr. Morell’s happy arrange- 
ment enabled him to return tothe subject in these slightly 
modified words :—“ Every diameter divides the circum- 
ference into two equal parts ; for if on placing one kaif 

NATURE 


| Hed. 23, 1871 

on the other they did not coincide, the radii of one and 
the same circumference would be different in length, which 
is absurd.” Here the word for being substituted for theve- 
Sore, we can no longer doubt that what precedes it zs the 
theorem to be proved ; the moment one fart, however, 
is termed a /a/f the whole question is begged ; and even 
if this gross blunder had been avoided, the demonstration 
would still have been worthless, so long as the mode of 
placing one part on the other was left wholly unexplained. 
The enunciation and demonstration of the two funda- 
mental theorems of parallels are thus given on p, 21 :— 
“Tf two straight lines are parallel, and are cut by a 
secant, the alternate angles are equal, and also the corre- 
sponding angles. Conversely, if two straight lines cut 
by another form equal alternate or corresponding angles, 
the said lines will be parallel.” 
“For two alternate internal, or alternate external, or 
corresponding angles are both either acute or obtuse, 
and consequently equal in the case of parallels. But, if 
of two internal or external angles on the same side, one 
is acute, and the other obtuse, these angles are, ¢here- 
fore, supplementary.” 
Let the reader picture to himself for a moment the 
perplexity of the misguided student, “ preparing for his 
examination,” as he vainly strives to extract a meaning 
from this sheer nonsense. Distrustful of his own powers, 
the poor fellow will probably attribute his failure to his 
own incapacity, and in despair commit the precious 
passage to memory for the purpose of reproduction when 
his hour of trial shall come. The consequences of such 
rashness need not be stated. And yet Mr. Morell cites a 
respectable French geometer as an authority for this 
“shorter demonstration!” It was with no small curiosity 
that we turned to the pages of Amiot’s E/éments de Géo- 
metrie to see how an author, who usually writes with 
admirable clearness, though not always with desirable 
rigour, could have been made responsible for such absur- 
di y. The process was simpler than we expected; editorial 
scissors had simply clipped away everything worthy of 
the name of demonstration, and the editorial pen had 
garbled the feeble residue into the chaotic sentences above 
reproduced. We know of no epithet too severe to apply 
to edirorial transgression of the kind here exposed. 
To proceed with the painful task we have imposed upon 
ourselves, we have next to draw attention to two blunders, 
not in geometry, but in simple logic. On p. 20 we read as 
follows :— 
“It is often assumed as a self-evident proposition that 
through a given point only one parallel can be drawn toa 
straiyht line,” and, in a foot-note hereto, we are told that 
“this is the opposite of Euclid’s 12th axiom, similarly 
assumed to base upon it his theory of parallels.” Had 
Mr. Morell read Amiot, even, with any thing like attention 
he would have seen that this is not the case. In forming 
the opposite of a theorem, the new hypothesis and pre- 
dicate must be made to contradict, respectively, the 
original hypothesis and predicate. The opposite of 
Euclid’s 12th axiom, therefore, as Mr. Morell may now 
convince himself by trial, is very different from the one 
above quoted. 
On p. 44 again, we have this marvellously short enun- 
ciation and demonstration of Euclid’s Prop. 8, Book I. ; 
“Let A BC, D E F be two triangles, having A B = DE 
