May 8, 1902] 
NATURE 
31 
sary and sufficient criterion for the existence of a definite integral 
was supplied by Cantor and Dedekind. 
Thus the great body of analysis had been built up long before 
the fundamental notion of a limit was completely established. 
A somewhat similar course might be traced in the develop- 
ment of modern ideas as to the basis of mechanics. 
In Prof. Perry’s method, especially in teaching the calculus, 
it is recognised that this is the natural way to approach the 
subject, not only for the science as a whole, but in the mind of 
the individual student, and its foundations are soundly laid on 
direct geometrical intuition and the notion of a rate of increase, 
full analytical treatment being left to a much later stage. 
This enables the calculus to be introduced at a much earlier 
stage than usual, and I may here quote the graphic advice of 
Prof. Burkhardt, all the more striking as it comes from a 
mathematician distinguished in pure mathematics :—‘‘ Dem 
angehenden Jiinger der Mathematik wiirde ich raten, sogleich 
mit beiden Fiissen in die Differential- und Integralrechnung zu 
springen.” F. M. SAXELBY. 
Royal Technical Institute, Salford. 
Rearrangement of Euclid'’s Propositions. 
I FEEL that Prof. Lodge’s proposal to change the order of 
Euclid Book I., 1-32, is the real solution of the present 
problem of the teaching of elementary geometry. The budding 
engineer has his practical mathematics, the embryo wrangler 
will absorb geometrical truths served up in almost any manner ; 
but the ninety per cent. to whom mathematics is a mere mental 
training must have their work put before them in an interest- 
ing, practical and yet logical manner. I should, however, like 
to put forward the following three points :— 
(a) That Prof. Lodge’s idea should be carried further, and 
Euclid, Books I.-VI., divided into four new books, as :— 
The straight line — Euclid I. 1-32 in some good order. 
The circle — Euclid III. 1-34, IV. 1-5 and escribed 
circles. 
Areas — Euclid I. 35-48, II., III. 35, 36, 37, IV. 
6 to end. 
Proportion — Euclid V. and VI. 
For Book I., I would suggest an order commencing with 
I. 32, cor. 2, which is the most general proposition for all 
rectilinear figures ; and also that certain well-accepted riders 
should be added, many of which form more powerful instru- 
ments for solving geometrical problems than the majority of 
Euclid’s propositions ; that, in the circle, tangents should be 
treated as limiting chords; that, in areas, the ‘‘ alternative” 
proofs of Euclid Book II. should be ¢e proofs; finally, 
that proportion be done semi-algebraically, using fractional 
notation = 
CD’ 
(4) That it is not necessary—I may say, not advisable—to 
teach a beginner the words of a strict definition ; but he should 
be given the idea alone, built up from practical use of a set of 
instruments, the verbal definition following when he is able to 
appreciate it. I would advocate that the following definitions 
be substituted for Euclid’s unsatisfactory ones. 
A straight line is one such that if any part be taken up and 
applied to any other part 27 ay manner, so that its extremities 
fall on that part, it will coincide altogether. 
The angle—the trigonometrical definition. 
Parallel straight lines—the converse of axiom XII. 
These would lead, for the student, to the ideas that a straight 
line can be drawn with a 7vz/er, an angle drawn or measured 
with a protractor, and parallel straight lines drawn with two 
set-squares, one fixed and the other movable. 
If these were accepted, I. 13, 14, 15, 27, 28, 29 follow 
almost axiomatically, and we are enabled to prove I, 32, cor. 2, 
by a supposititious construction, obviating such practical proofs 
as ‘‘ walking round the sides ” or Prof. Minchin’s better idea of 
placing a pin along a side and moving it round, substituting a 
purely geometrical proof. 
(c) That it is unreasonable to bar supposititious proof-con- 
structions—e.g. in the bisector proof of I. 5. For no exception 
is taken to the particular enunciations of I. q or I. 8, although 
at that stage we are unable to dvaw one triangle with its parts 
equal to those of the other. J. M. CuILp. 
Technical College, Derby. 
NO. 1697, VOL. 66] 
The Sweet Briar as a Goat Exterminator. 
THE introduction of the sweet briar into Australia, in many 
parts of which it is naturalised, affords a striking illustration of 
the mode in which the balance of nature may be disturbed in a 
wholly unforeseen way. 
The fruit of the sweetbriar consists of a fleshy receptacle 
lined with silky hairs which contains the seed-like carpels. 
I extract from the Agricultural Gazette of New South Wales, 
vol. xili., No. 3, March, 1902, p. 313, the following note by 
Mr. E. A. Weston, a well-known veterinary surgeon of 
Launceston, Tasmania :— 
“With reference to Rosa rudsiginosa, I thought it might 
interest you to know that the hairy linings of the fruit 
caused the death of a number of goats here by forming 
hairy calculi, which mechanically occluded the lumen of the 
bowels. These goats were put on the land with the idea that 
they would eat down the briars and ultimately eradicate them, 
but the briars came out best and eradicated the goats. The 
cattle running on the land are also very fond of the briar berries, 
and from time to time one will die, and on fost-/ortem no patho- 
logical changes can be found in any of the organs, nor do the 
hairy calculi appear in them, although their various stomachs 
are one mass of the briar seeds.” 
Kew. W. T. THISELTON-DYER. 
Stopping down the Lens of the Human Eye. 
In photography, if the lens is affected with spherical aber- 
ration or cther defects, or if the aperture is too large for good 
definition, the operator usually gets over the difficulty by using 
a smaller aperture or stop. This improves the definition and 
makes the picture sharp even to the corners of the plate. This 
process is technically called ‘‘stopping down the lens.” In 
amateur landscape work I generally use an aperture or stop with 
a diameter of one-fiftieth of the focal length of the lens, or 
Oo, 
But the human eye has defects, especially as we get old. For 
instance, the curvature of the crystalline lens becomes too flat, 
&c., and we have to use spectacles to enable us to read. Reason- 
ing by analogy, diminishing the aperture of the eye by ‘‘stopping 
down the lens” ought to improve defective definition and make 
the vision sharper, and experiment proves that such is actually 
the fact. I find that the best effect is obtained by holding a thin 
metal plate close to the eye, with an aperture in it one-fiftieth of 
an inch in diameter. This arrangement resembles the old single 
landscape lens used in photography. The small stop is in front, 
the lens in the middle and the sensitive plate or retina at the 
back. I use convex spectacles myself for reading, but with a 
stop of that size I can easily read small print within 4 inches 
of the eye (or even less) in a good light without spectacles. I 
have tried the experiment with séveral of my elderly friends, and - 
in every case with success. Anyone can try the experiment by - 
means of a pinhole in a card. 
I do not know exactly what is the focal length of the lens of 
the human eye, but supposing it to be half an inch, then with a 
stop of one-fiftieth of an inch the technical expression for the size 
of the stop would be //25, or double the diameter of the one I use 
in landscape photography. I enclose a metal disc with such an 
aperture. By looking through it I can read the smallest type 
in NATURE at 4 inches from the eye. Won. ANDREWS. 
Steeple Croft, Coventry, April 25. 
Prisms and Plates for Showing Dichromatism, 
DiIcHROMATISM, or the change of colour of an absorbing 
medium with increasing thickness, is usually shown with plates © 
of coloured glass. It is not always easy to obtain the right kind 
of glass, and only a few of the aniline dyes are suitable for the 
purpose. The medium should transmit two distinct regions of 
the spectrum, the absorption coefficient for one being greater 
than for the other. I have found that it is better to give the 
medium the form of a prism, for then the transmitted colours 
are separated, and the more rapid fading of one as the eye is 
moved from the refracting edge to the base can be followed. A 
number of years ago I found a small amount of an unlabeled 
dye which transmitted a red band and a green band, that is, it 
had a strong absorption band in the yellow and the blue. Thin 
layers of this dye were bright green, thick layers were blond red. 
I have never been able to find the dye again, though I have 
examined a large number of dyes, but I have found that a mixture 
of commercial ‘* brilliant green ” with a little naphthol yellow has 
