444 
NATURE 

1 
matters they would assuredly derive greater pleasure and 
profit from their pursuit, and do much more towards the 
progress of science. Mr. Young himself, however, not- 
withstanding the good advice he gives, is not always care- 
ful in drawing conclusions, geological evidence being some- 
times quite overlooked. Thus, we find him stating that 
the coal-measures (meaning, of course, the whole series 
of strata above the Millstone Grit) are ‘‘ evidently of land 
and fresh-water origin,” because they have yielded no 
marine organisms, save in one thin local bed near the top 
of the series, The occurrence of this stratum with its 
marine remains, indicates, as he believes, the return for a 
short time of the sea, which had for a very long period 
“been completely shut out by barriers.” Mr. Young is 
welcome to his belief. If every bed or series of beds in 
which no marine organisms occur must necessarily be of 
fresh-water origin, the lakes of old must have been some- 
thing worth seeing. There are several points suggested 
by the catalogue that we should like to have taken up, but 
our space is exhausted, and we can only conclude by 
strongly recommending Mr. Armstrong’s work to the 
notice of our geological readers, Io Ge 


LETTERS TO THE EDITOR 
The Editor does not hold himself responsible for opinions expressed 
by his Correspondents. No notice is taken of anonymous 
communications. | 
On the Solution of a Certain Geometrical Problem 
A WRITER in the number of Nature for September 21, Mr. 
Rk. A. Proctor, in the course of a letter on the state of geometrical 
knowledge in the university, alludes manifestly to the solution of 
a problem which I have adopted in my edition of Euclid. The 
matter is of small importance in itself, but nevertheless as some 
points of interest are incidentally involved, I request you to 
allow me the opportunity of offering a few remarks. 
The problem is this: to describe a circle which shall pass 
through a given point and touch two given straight lines. Your 
correspondent considers that in giving a solution which depends 
on the sixth book of Euclid, instead of one which depends only 
on the third book, I exhibit signal geometrical weakness. 
The problem, I need scarcely say, is very old ; indeed, so old 
that a writer who had been long engaged in teaching could not 
pretend to solve it afresh, for he would certainly have in his 
memory one or more solutions which had become quite familiar 
to him. The solution by the aid of the third book is well 
known, for it occurs in several of the collections of geometrical 
exercises. The solution which I have adopted is also old, but 
seems not so well known. It is, I think, conspicuous for sim- 
plicity, elegance, and completeness. The demonstration is of 
the best and most impressive kind, requiring no laborious effort 
to understand and retain it, but being almost self-evident from 
the diagram. Even if the problem be treated as an isolated exer- 
cise, the solution which I have preferred will sustain a favourable 
comparison with that which more commonly occurs. 
But the determining cause of my choice was the position 
which the solution occupies as one of a connected series. I have 
just before treated a similar problem by the third-book method, 
so that if the same method had been used for the present problem, | 
there would have been only repetition without any substantial 
increase of knowledge; whereas by the course adopted the | 
student is introduced to fresh and valuable matter. The principle | 
of similarity and the notion of a centre of similitude are most 
instructively involved, and the student is prepared for a subse- 
quent investigation, which is similar but more complex. To 
sum up, the third-book method would have constituted no advance 
in the subject, where the sixth-book method takes a step im- 
portant in itself and in its consequences ; and therefore, following 
the example of an eminent geometer, I adopted the latter method. 
I may perhaps venture on the strength of my own experience as 
to the utility of the solution, to recommend it to the attention of 
other teachers. 
It is very important to bear in mind the distinction between 
what I may call absolute and relative merit which I have just 
exemplified. The solution of a single problem furnished by a 
candidate under examination, or by a contributor to a mathema- 


tical periodical, is very different from the investigation of one out 
of a chain of propositions in a mathematical treatise. In the 
former case there are no antecedent or subsequent conditions to 
regard ; in the latter case we have to consider what agrees best 
with the whole scope of the work, with what is to follow as well 
as with what has gone before. A writer, after arranging a para- 
graph or a chapter in what seems the best manner, may find 
himself constrained at a subsequent stage to make changes which 
would have been unnecessary, perhaps even undesirable, if the 
earlier portion had stood alone. Then, if a reader opens the 
book at random and criticises a passage without any regard to the 
author’s sense, the criticism may very naturally be quite inap- 
propriate. 
There is, however, a very important consideration of another 
kind which has been frequently disregarded, but which is pressed 
upon our notice by the interest at present felt in geometrical 
studies. Let us determine the reason which leads us in some, or 
in many, cases, to prefer a solution which involves only the third 
book of Euclid to a solution which depends on the sixth book ; 
this, I apprehend, is merely a persuasion that Euclid’s order is a 
natural order, so that in a well-arranged system the propositions 
of the third book ought to precede those of the sixth book. I 
am of this persuasion myself; I think that no scheme can be 
perfect, and, on the whole, I am well satisfied with Euclid’s. 
But there are places where Euclid is strong, and there are places 
where Euclid is weak ; and the position which he has assigned 
to the last three propositions of his third book, must rather be 
classed with the latter than with the former. His object, of 
course, must have been to lead up to his construction of a regular 
pentagon, and we cannot be surprised at the introduction of that 
remarkable process. But I have always envied the advantage in 
this respect to be claimed for the non-Euclidean systems, which 
transfer these propositions and place them after the doctrine of 
similar triangles ; thus the long and rather artificial treatment 
which they receive from Euclid is superseded, and the proposi- 
tions become almost intuitive. Hence, in fact, if we have re- 
course to the sixth book of Euclid when we might have accom- 
plished our end by the aid of the first thirty-four propositions of 
the third book, we may be fairly liable to the charge that we have 
not adopted the simplest and most natural method ; but the last 
three propositions of the third book are quite different in kind 
from the others, and instead of using them, it may be really as 
simple and as natural in many cases to use the principle of 
similar triangles. 
I shall be obliged to any person who may be skilled in prac- 
tical geometry if he will state what he considers the best method 
of actually solving the problem, supposing that both circles are 
to be determined which satisfy the conditions. I assume that 
we have the aid of compasses and also of one of the ordinary 
contrivances for drawing parallel lines. This is a matter of some 
interest, though of course unconnected with the theoretical solu- 
tion of the problem. 
I should be glad to make some remarks on the general subject 
which led to the notice of the particular problem I have discussed, 
but at present I have not sufficient leisure. I must content my- 
self with having shown that the course into which I am supposed 
to have drifted by geometrical incapacity, was adopted deliberately 
under the guidance of reasonable geometrical knowledge. 
I, TODHUNTER 
St. John’s College, Cambridge, Oct. 2 

Structure of Fossil Cryptogams 
Ir was unfortunate that at the recent meeting of the British 
Association, Prof. Williamson’s paper had to be discussed in a 
very hurried manner, and he is, no doubt, justified in taking care 
‘*that there shall be no misunderstanding as to the real point at 
issue.” I do not think that he has brought it out very plainly 
in his paper in NATURE, and perhaps, as he mentions me as an 
opponent of his views, I may be allowed to state precisely in 
what respects I differ from him. 
First, as to matters of fact. Prof. Williamson speaks of the 
central structure of the stems of the extinct Lycopodiaceze asa 
“‘vascular medulla,” by which he explains that he means a 
“structure containing vessels,” and that there shall be no mis- 
apprehension he adduces Vefenthes as possessing it ; the instance 
is a well-known one, and leaves no room for doubt as to Prof 
Williamson’s meaning. Now from the examination of specimens, 
and of the drawings of them published by Mr. Carruthers (the 
accuracy of which I believe Prof. Williamson does not dispute) 
I am quite satisfied that the central structure consists wholly of 

[ Oct. 5, 18971 
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