1825.) 
though writing rather for the experienced 
geometer than drawing, up,a mere nur- 
sery demonstration; I traced only the 
general outline of the process, leaving 
the more obyious. steps. to be supplied 
by, the reader, as he went on... Still I 
conceive, that those steps were traced 
with, ample, force. and distinctness, at 
least, forthe comprehension of any one 
who had studied the elements of geo- 
metry with common attention: and I 
think, it .will inthe end appear, that 
* A* has. been rather premature in his 
censure, unnecessarily officious in the 
assistance which he has given me, and 
that he, “ by striving to avoid one fault, 
has fallen into a.greater.” 
I. The theorem, of which the two 
first analogies in my paper were cases, 
and) which “A” has demonstrated, 
though not found in any of our elemen- 
tary works, is yet not new; and it is, 
moreover, so simple and so easy of dee 
monstration, that.“ obscurity’’. could 
searcely arise even from my passing it 
over as I did. Besides, the theorem is 
pretty generally known amongst mathe- 
maticians, and may, therefore, in a de- 
monstration (certainly not elementary 
and therefore not intended for the eye of 
elementary readers), be assumed as true, 
without any violation of scientific propri- 
ety: and, had I thought it necessary, I 
could have quoted at least half a dozen 
different places in which the theorem is 
_ to be found, or from which it could be 
derived without more than a single step 
of reducing analysis. ' 
II The third step in “ A’s” demon- 
stration is rather extraordinary ; itis to 
prove that parallels are divided into 
proportional segments by lines passing 
through the same point! Probably he 
may deem it necessary to amend 
HIS Own demonstration, with a view to 
prove that “ the three angles of a tri- 
angle are equal to two right angles,” or 
to'show us how “ to construct an equi- 
lateral triangle on a given finite right 
line.” \ ' 
TH. As we take the next two steps 
Para remark is required there. 
IV. Next tothe charge of “ obscu- 
rity” stands'that of coming to a “ con- 
clusion; geometrically, unsatisfactory ;” 
or, in-other. words, to a conclusion not 
warranted-by. the preceding arguments. 
I must: bespéak the reader’s patience — 
‘whilst Lexamine'this charge. __ 
We had. 
be i S EH: HC 2 H D : HF; 
but-here I paused, whilst my commen- 
proceeded together to the 
to 
Mathematical Problem Demonstrated, 
803 
tator performed eight distinct horse-in- 
the-mill operations—* permutando, coms 
ponendo, ,alternando, invertendo,” &¢. 
&c.—from which he ultimately obtamed 
OCs .OF 2 OF: OH Ton 
Let us now compare our relative posi- 
tions; perhaps we are not far apart, 
after all the seeming progress made by 
my obliging auxiliary. ee eee 
’ A” finds that the supposition of 
GK not passing through O involves the 
parallelism of that line to BF. roe 
I find, from the relation 
EH : HC :: WD: HF, 
that if H and H’ be not the same point, 
GF is parallel to BF. Where is the 
difference, then, between our respective 
analogies, and on what account is his’ 
conclusion more valid or more obvious 
than mine? The proportions 
OC : OF :: OH: OH’ and 
EH : HC:: H’D: HF, 
are, indeed, almost identical, and the 
conclusion is as clear from one as from 
the other. The eight intervening opera- 
tions are then, of course, so far from 
adding to the “ perspicuity and strict- 
ness” of the demonstration, that they 
‘are, in reality, so many redundant and 
ungeometrical applications of geometri- 
cal logic, which disfigure the proof that 
had previously been given. . 
V. My commentator contends that 
since the line GK cannot be parallel to 
BF, and, at the same time, intersect it in 
L, the line GK has no other alterna- 
tive than to pass through O: whilst I 
suppose my reader capable of tracing, 
for himself, the course of reasoning by 
whieh this very obvious conclusion is 
demonstrated. Such is the nature of 
my “ungeometrical” and “ unsatisfac- 
tory” conclusion—a very exalting com- 
pliment to the geometrical reader, most 
assuredly ! 
VI. The substitution of the term 
“ laterally” for “radially,” seems to 
me rather capricious than useful. I 
am the last man in the world who 
2 
