396 
accelerations of a given body are as the forces pro- 
ducing them. This result combined with the fact 
that all bodies have the same gravitational accelera- 
tion corresponds to the form (F/w=a/g) recommended 
for elementary teaching and favoured by Sir George 
Greenhill. i 
The acknowledgment which Newton makes to 
Wren, Wallis, and Huygens for the discovery of the 
laws of impact is generally known in connexion with 
his description of his own experiments on impact. 
His attitude towards these experiments is different 
from that of the critical exposition of dynamics of 
to-day, in which the Third Law is placed in the position 
of honour from which the Second Law is derived by 
observation or experiment. With Newton, however, 
the Third Law requires justification, as shown by the 
conclusion of his description of his experiments on 
impact, “‘atque ideo actionem et reactionem esse 
aequales.”’ 
One other extract is worthy of attention. Under 
Definition III. of materiae vis insita Newton remarks, 
“ Per inertiam materiae fit ut corpus omne de statu 
suo vel quiescendi vel movendi difficulter deturbetur.”’ 
This objective view of inertia is better adapted for 
the general qualitative introduction to inertial mass 
than the innate view consequent on an initial state- 
mentofthe FirstLaw. This objective view frequently 
finds expression in elementary text-books, but might 
receive greater emphasis in view of the electromagnetic 
theory of inertia, and the initial discrimination 
between inertial mass and gravitational mass forced 
on us by the modern theory of relativity. The 
quantitative definition of mass as a measure of inertia 
merely interprets ‘‘ difficulter deturbetur ”’ in terms 
of acceleration. We may say then, as a preliminary 
to a more exact definition, mass is a measure of the 
difficulty of accelerating a body. 

F. E. HAckeEtT?. 
College of Science for Ireland, 
Dublin. 
The Resonance Theory of Hearing. 
I HAVE been reading with great interest various 
accounts of ingenious models made to illustrate the 
resonance theory of hearing, but I have been un- 
fortunate enough to miss any clear reference to any 
structure in the cochlea which could respond on a 
physical basis to all vibrations which are capable of 
being appreciated by the human ear, or rather nervous 
system. 
I have before me a pianoforte with a register of 
seven octaves, containing wires which vary from about 
150 cm. in length and more than o-4 cm. in diameter to 
wires 10 cm. long and tightly stretched. If the range 
were continued to the eleven possible octaves the 
extreme dimensions would be proportionately modi- 
fied, being lengthened in the one case and shortened 
in the other. This, I take it, is the best pianoforte 
manufacturers can do, and that if they could have 
used shorter or finer wires they would have done so. 
Let us turn then to the human cochlea and form 
some idea of its dimensions relative to such an instru- 
ment. It consists of a tube coiled two-and-a-half 
times, about 35 mm. in length, and varying from 
4 mm. to I mm. in diameter. The total cubic con- 
tents of the cochlea, according to Sir Arthur Keith, 
are 70 cubic mm. The third canal of the cochlea 
has a diameter varying from o-5 mm. too-8mm. The 
basilar membrane has, according to Keith, a diameter 
varying from 0-17 mm. to 0-4 mm., with an average 
area of 13-2 sq. mm. 
If the cochlea as a whole be considered, it can be 
likened in size to a stout silk thread 35 mm. in length. 
NO. 2786, VOL. 111] 
weal URE 
If the third canal of the cochlea alone be considered 



[Marcu 24, 1923 
it can be likened toa silk thread 35 mm. in length, with 
an average width of 0-5 mm. 
How is it possible to imagine structures of this order 
of magnitude capable of differential resonance to the 
vibrations of sound ? From the ability of the investi- 
gators who have been dealing with the problem, I 
am certain that such an elementary difficulty cannot 
have escaped them for a moment, but I shall be grate- 
ful to any physicist who will throw light on a problem 
which is as difficult as itis fascinating. Ifthe presence 
of anatomical resonators capable of responding to 
vibrations of the varying length indicated can be 
demonstrated, the resonance theory can well be con- 
sidered. Otherwise it must be abandoned. 
JAMES W. BARRETT. 
105 Collins Street, Melbourne, 
January 5. 

SIR JAMES BARRETT’s letter expresses a difficulty in 
the way of acceptance of the resonance theory which 
I believe to be more generally felt than perhaps any 
other, namely, the difficulty of conceiving that a 
structure so minute as the cochlea, which may be 
compared in size to a small split-pea, can contain a 
series of resonators capable of responding to some 
4000 separate tones extending over about 11 octaves. 
When we compare the suite of strings of a piano, 
which will respond only to 85 separate tones in 7 
octaves, although they occupy with their case a space 
of ro to 15 cubic feet, and weigh several hundred- 
weight, the whole conception seems indeed bizarre 
and absurd. 
This difficulty may be considered under two head- 
ings : 
(1) How to account for the minuteness of the scale. 
(2) How it is possible to have such a wide range of 
tones within so small a cubic space. 
(1) Scale.—If it be granted that we are to look for 
our resonating elements in the transverse fibres of the 
basilar membrane, the scale of the cochlea will be 
determined by the length of these fibres. This again 
will be determined by the formula 
Number of vibrations per sec. 
i I cae tension in dynes 
2x length of string mass of unit length of string 


or 
It is obvious that in this formula, for any particular 
value of », / can be given any value we choose by 
assigning suitable values to ¢ and m. Theoretically, 
there is no reason why the resonators should not be 
10 or even 1000 times smaller than they are in the 
cochlea. Practically, the limits of what is possible 
are set by the strength, fineness, and foe 
the materials available. The particular factor which 
renders this extraordinary reduction of scale possible 
is, that in the cochlea the factor m is large out of all 
proportions with what obtains in any of our stringed 
instruments. This result is attained by the beautiful 
mechanical device of loading the strings each with a 
definite mass of cochlear fluid. 
(2) Differentiation.—The fibres of the basilar mem- 
brane are differentiated for length, tension, and mass 
just as are the piano strings. Accepting Keith’s 
measurements, the differentiation for length is 
sufficient to account for 14 octaves; that for mass 
(as determined by the “ fluid load ’’) for about 2} 
The remaining six to seven octaves of the 
octaves. 
audible scale must be due to variations of tension, as 
applied by the spiral ligament. This means a pro- 
portion of something like 1 to 5000 or 10,000 between 


