78 
minimum seems to have been less than the later; while in 
Geneva the earlier minimum is the deeper. 
The facts above given may be usefully compared with Dr. 
Lockyer’s recent important researches, pointing to a cycle of 
about thirty-five years in the sun-spot variations. It may be 
doubted if the annual mean temperature of these European 
stations shows any good evidence of being ruled by the eleven- 
year cycle of sun-spots ; and if it did, the method of smoothing 
here adopted might even obscure such an effect somewhat. 
This, however, does not seem to affect the validity of evidence 
from other orders of data. ALEX. B. MACDOWALL. 
Resultant Tones and the Harmonic Series. 
IN reply to Prof. Thompson’s criticism of the plan of recover- 
ing differential resultant tones by means of the harmonic series, 
may I say that my position is that of a road-maker, not a 
discoverer—a Macadam rather than a Columbus. 
So long as authorities teach that resultant tones have a 
vibration frequency which is equal to the difference between the 
vibration frequencies of their generators, so long will the har- 
monic series afford an easier means to the same end. 
This applies also, of course, to summational tones. 
The question whether these latter are only ‘‘ one of the myths 
of science”’ or not I leave to abler heads than mine to decide. 
Meanwhile, the fact that the perfect fourth, the minor third 
and the minor sixth give, as the sum of their vibration frequen- 
cies, a vibration frequency which is intermediate between two 
notes, thus exactly agreeing with the harmonic series, 3 + 4 
=7,5+6=~srand 5 + 8 = 23, is at least interesting. 
MARGARET DICKINS. 
Tardebigge Vicarage, Bromsgrove, May 9. 
* Magic Squares. 
HAVING attempted some years ago to determine the number 
of magic squares of five having a nucleus forming a magic 
square of three, I was interested to find that further progress 
towards a solution of the problem has been made by your 
correspondent Mr. C. Planck, who seems to have found fifty- 
one solutions more than I from the same twenty-six nuclei, 
whereas I have only in one case, namely for the nucleus R 
(5, 7), found one more solution than he. The twenty-one solu- 
tions for this nucleus are appended in the following table, from 
which both the equations and the numbers forming the first 
row and the first column of the border may be read off without 
difficulty, if the first dotted number be put at the head of the 
column, and at the foot of the same the complement of the 
second. Thus, from the first row of the table, 
i 416 S18 0 
we gather that the first row of five minors (numbers less than 
13) may be converted into a normal row with sum 5 x 13 by 
replacing the three barred numbers by their complements, since 
2+4+13=1+6+ 12, whilst the remaining three minors, 
together with the dotted pair, furnish a normal column when 4 
and 3 are replaced by their complements, since here again 
4+3+13=2+8+10. The border with nucleus, accord- 
ingly, when completed, is 
a b UY a 
w | 2) 25) 20) 14,| 4 
b euler 18 
LOM) (OS SRE 1'6 
b' | 93 19 3) | 15) 5 
a’| 9214 | 16 [Balsa 
NO. 1699, VOL. 66| 
NATURE 
[May 22, 1902 
aS 4 eee ee eli 
2st. 2, ag eae Caen 
Veoui id, 2 | A. SG Meese 
leas 3 2 eeipa aes ear 
Salad. 2 6. Sassen Seen 
2) || SAMS i SOMES red je) oe ouceed 
al OREM et ee cilia fel ne 
Sale?) 3... 6. .aSeelots alee een 
Out, 2” '8.. Weae tee eee 
Os 2... 3. 4: Gana nee aaa 
He 2.) 3... 8) LO alee me 
27 4 6 Rees ta 
1333 4 6 (seo eee 
aa Tt 4 8 Ope 
Mis 1. 2 4% (Gta ae ten ame 
Michie: 2 4. SiO games mere 
hay |. 2 6 Seu ctr onan 
18-2 °3 6° \SagetON ee eeuneme 
193. 4. -6 “Sa elOn Teme 
20 lel. 2 6 (Sq wee Seneaneet 
on 3. 3 6 Se ae 
When the number 603 is multiplied by 288 we get 173,664 
for the number of such nuclear squares. When we proceed to 
inquire as to the number of all types of magic squares of five, 
we must begin by doubling the above number, since every 
magic square with odd root may be varied by permuting the 
rows above the mid-row, together with the rows below the 
same, and at the same time the columns on either side of the 
mid-column, so that the above square may be transformed by 
reversing the order of the marginal letters a, 6 anda’, 6’, as 
follows : — 
b a a bf 
6-11 | Saree ees 
a|.25| 2) 20) 4) 14 
| 9 | 10 Sete | 17 
a’| 1/22! 6 | 94) 42 
O19 | 23) 5) Ss 
If now we add to the number 347,328 thus obtained the 
squares in which each row and each column contains all the © 
units I. 2..5 increased by the four increments 5. 10. I5 . 20 
without repetitions of either, of which there are at least 
21,376, we get 368,704 without considering other types, 
probably some hundreds of thousands in number, which would 
certainly bring the minimum to more than half a million. 
Shipley, Yorks. J. Wits. 
