May 1, 1902] 
NATURE 7 
met with the approval, among others, of Prof. Perry, London, 
and Prof. Malverd Howe, America. In revising this matter 
for the new edition of our ‘* Applied Mechanics,” I find that 
polygonal cylinders of uniform plates freely hinged at their 
edges and displacing their own weight of fluid and lying 
horizontally are also in equilibrium, provided the polygon be 
regular. 
In the diagram the square shell is shown just reaching the 
surface and rolled into three positions. The proof is the same 
as for the ordinary statical problems on festoons of rods hinged 
at the ends, only now there is the external fluid pressure in 
addition to the weights. The fluid is kept out by face plates at 
the ends, the face plates having the same density as the fluidand 
being quite smooth, so as to allow the shell freely to change its 
shape. If the shell be slightly compressed it will collapse, but 
the friction of the face plates and the confined air afford a slight 
degree of stability. The diagram shows the regular hexagonal 
shell, and by increasing the number of sides we arrive, as 
before, at the circular cylinder. In the polygonal shells there 
are bending moments on the sides as well as the thrust, but on 
the circular there is only hoop thrust, as it may be a plenum of 
joints. Submerging only adds a symmetrical load all round, 
and the shells are still balanced. As they are also balanced 
with the axis vertical it follows that they are in equilibrium in 
any position whatever. 
My first letter led to some correspondence, and I hope this 
may be of interest to your readers. TuHos, ALEXANDER. 
Trinity College, Dublin, April 19. 
Mycoplasm, 
Since 1889 a fungus hyphal layer has been known to exist 
in the nucellar remnants of the grains of the Darnel grass, 
Lolium temulentun:, and to these hyphze have been attributed 
the poisonous properties of the Darnel. Later investigations 
have shown that the fungus could be found in the growing point 
of developing plants, in the inflorescence, and finally in the 
ovular rudiments. The manner of entrance of the fungus 
had, however, escaped detection. Nestler (Ber. d. deutsch. 
Sot. Gesellsch., B. xvi., 1898, p. 210) and others failed to 
observe the fungus in the embryo in the mature grain. The 
hyphz in the growing point could not be observed before the 
eighth day of germination. 
Eriksson has recently! quoted the work of Nestler and 
others on the fungus of Lolewm temudentum in support of his 
theory on mycoplasma. According to Nestler, the embryo does 
not contain the hyphee, which appear in the seedling on the 
eighth day. In only one case was he able to see hyphz in the 
embryo. In view of the support which this work appears to 
give to Eriksson’s mycoplasma theory, an advance note on some 
of my results in the investigation of the fungus of Lolium 
zemulentum, which has been carried on in the laboratory of 
Prof. Marshall Ward at Cambridge University, may be of 
interest. In appropriately stained sections of the embryo taken 
from the mature seed of Loléwm temulentum, hyphe in great 
abundance may be seen in the growing point, sometimes but 
two cells from the tip; these hyphe may be traced to 
their point of entrance at the juncture of the coleorhiza and 
scutellum on the outer surface of the latter in the region of the 
median longitudinal plane of the scutellum. Previous investi- 
gators had entirely overlooked the presence of a considerable 
1 Eriksson, Aun. des Sc. Nat., T. xv., 1902, p. 73, says :—‘‘ Les tentatives 
infructeuses d’A. Nestler d’apprendre 4 connaitre de quelle maniére le 
champignon qu'on trouve presque toujours dans les fruits du Lodium 
temulentum est entré dans le cone végétatif de l'embryon du fruit 
amenent aussi la supposition d’un état mycoplasmatique latent.” 
NO. 1696, VOL. 66] 
amount of mycelium in that part of the grain which lies directly 
against the scutellum in the median basal region, where it has 
grown around the end of the aleurone layer. The infection 
takes place apparently before the grain has reached complete 
maturity, as the fungus is well established in the ripe grain. 
There can, therefore, be no question here of mycoplasm, since 
direct hyphal infection can very easily be demonstrated. There 
is no evidence to prove that the fungus is a Uredine. The 
detailed results, with other particulars of the nature and 
development of the fungus, will be published soon. 
April 20. E. M. FREEMAN. 
Rearrangement of Euclid I. 1-32. 
THE rearrangement outlined in my previous letter was devised 
to meet the difficulty which, as Prof. Bryan states, is the chief 
objection to Euclid’s Elements as an elementary course. 
Beginners cannot solve riders because 
(1) They do not grasp the reasons for Euclid’s limited postu- 
lates and axioms, and never fairly understand the ‘‘ rules of the 
game ” ; consequently their early attempts violate his conditions, 
and their rejection discourages. 
(2) Too much time is occupied by the propositions, with the 
result that they regard them, not as tools, but as models, and 
imitate Euclid’s methods of proof. There is nothing in 1-8 
worthy of imitation. 
(3) They do not distinguish between data and queesita unless 
they have drawn accurate figures. It is impossible to draw 
accurate figures by proved methods in Euclid’s scheme (e.g. I. 4), 
and we therefore have recourse to figures drawn on the prin- 
ciple of Artemus Ward’s horses. This is the great difficulty in 
working riders. Allow a boy to assume the mid-point of a line 
and he will assume the most impossible constructions. He 
should never be allowed to quote a construction which he 
cannot perform, and no construction should be shown him with- 
out proof. Freehand copies of blackboard figures are useless ; 
if he has drawn a dictated figure, there is no confusion between 
hypothesis and conclusion. There is also the additional advan- 
tage that the less intelligent feel that in drawing the figure they 
have accomplished something, and this frequently stimulates to 
further effort. 
To remove these difficulties we must extend the axioms and 
postulates, reduce the number of standard propositions, and 
introduce problems as early as possible. The advocates of a 
purely theoretical scheme have two courses open to them—either 
they must teach constructions first without proof (which is 
extremely illogical), or they must postpone them until the com- 
pletion of the theory, and therefore postpone riders indefinitely. 
Geometry without riders resembles arithmetic without examples. 
In the scheme which we have found most successful, riders 
commence with the definitions. Every standard proposition is 
treated as a rider and evolved by the class ; one proposition a 
fortnight is considered sufficiently rapid progress, the intervening 
lessons being devoted to riders. 
The circle gives a method of drawing equal lines, and, with 
the idea of angular measurement, a method of constructing equal 
angles. Of course we assume the shape of the circle. 
I. 15 and 32 give the fundamental fact of rotation and 
introduce easy theorems and numerical examples. 
I. 8 with its riders elicits I. 9, and I. 4 is followed by I. 10, 
locus of points equidistant from two given points, I. 11, 12, 5. 
Having reached this point, possible riders are endless, and the 
only difficulty lies in their selection ; many propositions of III. 
and IV. may be included in the riders. Every pupil can now 
draw an accurate figure from dictation, and knows exactly what 
data he has to work upon. The rate of progress may appear 
slow, but we are teaching Book VI. in the second year. It 
should be noted that I. 1 is a rider, 20 an axiom, and that 2, 3, 
7, 18, 19, 21, 24, 25 are not read. 
In teaching riders, theorems should, as a rule, be grouped on 
methods of proof; the required figure should be dictated and 
the class asked to prove any fact they can concerning it. A 
general enunciation should then be invented; in this way 
standard propositions for future proof are frequently suggested. 
It is a mistake to hurl a general enunciation at a class of beginners. 
Problems usually give more trouble, but if grouped on loci their 
difficulties vanish. 
There would be no examination difficulty if papers were set 
on riders only. Euclid’s Elements might then be reserved for 
university examinations—a geometrical ‘‘ Paley.” 
Leyton Technical Institute, April 25. T. Petcn. 
