TIO 
NATURE 
[APRIL 2, IQ14 
It seems probable that both classes of effects are 
involved in the actual magnetisation in question, 
though experiment has shown that any effect of 
class (a) is at least exceedingly minute unless the 
magnetic behaviour of the interior parts of the earth 
and sun is quite different from that of matter at 
ordinary temperatures on the surface of the earth. 
S. J. Barnett. 
The Ohio State University, Columbus, 
Ohio, U.S.A., March 12. 
A Triangle that gives the Area and Circumference of 
any Circle, and the Diameter of a Circle equal in 
Area to any given Square. 
Ir is not possible to measure exactly the intermin- 
able fraction required for the line BZ in the following 
figure, but it is quite easy to draw it so nearly that 
the error is practically immensurable. 
First Method.—Draw a line AB=44, 
BZ=22. 
AY will then be short, about 1 in 600,000. 
AX will also be short, about 1 in 300,000. 
and make 
ao a 
“2 =0-5227272, which is a little too long. 
Second Method (with the familiar ratio 333 Make 
I 
a circle with diameter AB=11-3, and let AX=82. Draw 
a perpendicular from X to cut the circle in Y. Join 
AY, and continue the line to Z. 
Then the error in AX, the + circumference, will be 
less than I in 11,780,000 in excess. 
The true angle for the line AZ lies between the lines 
found by the two methods, but the difference is too 
small for measurement, and in any accurate drawing 
the lines will appear to coincide. 
AX being found, equal to an are of 90°, a line for 
any other arc may be found; and the triangle once 
drawn on a sufficiently large scale, is true for all 
circles. 
Z 
ae 
ap if) 
® 
Let AB=1, and at right angles | oe eee 
BZ =0'§227232008 +, join AZ | Bngle A- 27" (35 Ova 
Then, any circle with diameter upon AB and one 
extremity at A, willcut the line AZ (or AZ produced) in 
a point Y, making AY the side of a square equal in 
area to the circle. 
Also, a line from Y perpendicular to AB will cut 
the diameter in a point X, making AX equal to 
x circumference of the circle. 
Again, any square with base upon AB and a corner 
at A, will, with its side opposite to A, cut AZ in a 
point Y, making AY the diameter of an equal circle. 
T. M. P. Huenes. 
5 The Croft, Tenby. 
Tue following remarks may help to explain Mr. 
Hughes’s constructions :— ; 
Let AY be a chord of a circle of which AD is a 
diameter, and let 2 DAY=6. Then if the square on 
AY equals the area of the circle, (2r cos 6)?=7r2, and 
therefore cos ?0=7/4, tan ?6=(4—7)/z, and 
tan 6=0-5227232, 
very nearly, as stated. Now if 
NO, '2318,. VOL. 192)| 
we express this 
approximate value of tan@ as an ordinary continued 
fraction we find the successive convergents, 
1/ Ip, 1/25. 10/20, tales, 23 Aneto, 
where the first of those not written has four digits in 
numerator and denominator. Hence, as Mr. Hughes 
has discovered, 23/44 is a very close approximation 
to the transcendent number /(4—7)/¥z. It seems 
absurd to speak of a mathematical accident, but we 
do seem to have something of the kind here. Suppos- 
ing that a/b is a _ rational approximation to 
V(4—7)/z, we should not expect beforehand a solu- 
tion correct within about 4-10-° for values of a, b, 
each less than 100. 
The second construction is obtained by putting, as 
an approximation, 
AX: AD =7/4=355/4:113,=710/ 5-119 —=65/00-3- 
It would be easy to make a set-square with its 
shorter sides in the ratio 23:44, and this could be 
used for the approximate quadrature and rectification 
of any given circle. It is interesting to see how the 
same figure solves both problems to the same degree 
of exactness (practically). I suppose the error in the 
set-square could be reduced to o-1 per cent., or less; 
the question is, what percentage of error is likely to 
occur in using it. For the rectification we have to 
draw the perpendicular YX; it seems to me that for 
the quadrature we are likely to obtain the most 
accurate results by using a straight edge as well as 
the set-square; that is to say, we should not try to 
adjust the set-square without first placing a straight 
edge along a diameter of the circle. If this is so, the 
graphical solutions of both problems are likely to be 
affected by the same percentage of error; because to 
obtain X, after marking Y, we have only to slide the 
set-square along the straight edge until a shorter side 
goes through Y; and if we repeat the manipulation 
several times, I do not think the error in finding X, 
regarded as a distance from the true position, can be 
so much as five times the error in finding Y, or con- 
versely. Of course, by ‘‘the same percentage of error” 
I mean here that the two errors, on the same scale, 
are of the orders +a:10-", +b-10-", where a, b both 
lie above 1, while neither of them is equal to, or 
exceeds, 5. G. B. MatHews. 
New Units in Aerology. 
WirtH reference to Prof. McAdie’s letter in NATURE 
of March 19, p. 58, I should like to point out that 
throughout my ‘‘ Thermodynamics,” published in 1878, 
a megadyne per square centimetre is used as the unit 
of pressure, and it is termed a c.g.s. atmosphere. 
Ever since 1888, when the B.A. committee (of which 
I was a member), adopted the barad, I have employed 
in my lectures the above pressure unit under the name 
of megabarad. The corresponding unit of work, and 
also of heat, adopted in the book is the megalerg. 
Megerg to my ear is too cacophonous for use. 
Ropert E. Baynes. 
Christ Church, Oxford, March 25. 
PROGRESS IN WIRELESS TELEPHONY. 
As: attention of telephonic engineers has of 
late years been very closely directed to the 
improvement of the line wire in ordinary tele- 
phony. Apart from the imperfections of the tele- 
phone transmitter and receiver, per se, a very 
considerable effect is produced on the transmitted 
speech by the line itself if it is at all long. This 
action, from an electrical point of view, consists 
in the distortion of the wave form of the current 
orca aoe OF 
