March 3%, 1870] 
NATURE 5 
on 
MN 
work called ‘‘Sight and Touch,” 1864, p. 57, collates the 
opinions of Heraclitus, Aristotle, and Cicero, who all assign to 
the moon’s apparent diameter the length of a foot. He quotes, 
too, from Arthur Collier’s Cl/avis Universalis the expression, 
“the moon which I see is a little figure of light, no bigger than 
a trencher.” The old-fashioned trencher was about a foot in 
diameter. The late Sir William Rowan Hamilton pointed out 
to me this passage in Descartes’ ‘‘ Dioptricae,” cap. VI. § xx. :— 
**Abque hac patet ex eo quod Luna et Sol, qui sunt e numero 
corporum remotissimorum, quz contueamur, ... . fedales ut 
plirimum, vel ad summume bipedales nobis videantur,” &c. 
Bishop Berkeley, as Philonous, asks Hylas, ‘‘ Since, according 
to you, men judge of the reality of things by their senses, how 
can a man be mistaken in thinking the moon a plain, lucid sur- 
face, about a foot in diameter?” Many more such cases could 
be cited from ancients and moderns, all concurring in assigning a 
foot, or something between one foot and two feet, to the apparent 
diameter of the moon. Let me now cite two recent cases. A 
law-clerk, whose lay notions certainly owed very little to books, 
told me that the moon always appeared to him of the size of a 
door-handle. This would give, at most, a diameter of three 
inches. An eminent astronomer tried his daughter with the 
question. She replied that the moon looked to her about /a/fa 
degree. He said, ‘*Come, you learned that from astronomy ; 
but answer as a girl of common sense.” She now replied, ‘* A 
small saucer.” That would be some four or five inches in diameter. 
I suppose I was too early spoiled by trigonometry to enter into 
the merits of this style of estimate. Look at the moon as I may, 
I cannot compare her to anything definite, as a door-handle, 
a saucer, or a trencher. Judging by the distance at which we 
ordinarily see such things in the use of them, they all seem to me 
to be enormously too large. Looking at the moon through my 
window, sitting three or four yards from it, I should guess that a 
wafer stuck on it would eclipse her. My own conviction is that 
in the ordinary estimate (or rather comparison), there is no 
reference, the most covert or subconscious, to any standard of 
distance. ‘True, thirty-six or thirty-seven yards would be the 
distance at which Arthur Collier’s trencher would subtend the 
same angle as the moon. But who thinks of this in connection 
with a trencher, which is usually under a man’s nose or on the 
kitchen rack or shelf, at a distance not exceeding four or five 
yards? In another letter I will, if you will allow me, call 
attention to some of the vere cause which are probably concomi- 
tant in these popular estimates; and this I shall do in respect to 
the apparently augmented size of the moon’s disc on the horizon. 
Meanwhile, let me ask as a preliminary to that inquiry, is it a 
matter of fact that 40 the aked eye the moon does subtend the 
same angle at the horizon as at positions near the zenith? Iam 
unable to perform the measurement myself, not merely for want 
of a proper instrument, but by reason of the fact that I always 
see in the moon a multitude of discs partially overlapping each 
other, five of which I can distinctly count. It would be awkward 
to find that one was attempting to solve an imaginary problem, 
like the Royal Society over King Charles’s fish. 
Ilford, March 24 C. M. INGLEBY 
Tue ‘lurking idea” of Mr. G. C. Thompson, that the 
moon looks about the size of a fourpenny piece, seems to me to 
show that those views of it have made most impression on him 
which he has taken when standing a few feet from the win- 
dow, when it would cover some such space: while others, with 
the one foot or two feet idea, have been more wrought upon by 
unconscious measurement of it against trees in the garden, or 
house chimneys along the street. Ido not think we can get 
beyond this, in regard to a ‘‘personal equation.” As to the 
apparent difference between the moon near the horizon and the 
moon in mid-sky, your correspondents have not yet referred to 
the theory that the felt degree of convergence of the eyes is one 
help toward measuring distance; which, however, soon ceases 
_as the object is more remote, and the convergence insensible : 
and that, in looking at the moon along the earth’s surface, we 
feel that she lies beyond this limit by comparison with the 
objects which intervene, while in looking up through free air 
there is no such gradation to guide us; that, therefore, we 
assign, unconsciously, a greater distance to her, 7, a greater 
“lurking idea”’ of estimated magnitude for the same apparent 
surface, in the former case than in the latter. I write from 
a dim recollection of one of Sir Sidney Smith’s lectures on 
Moral Philosophy, but I suppose the notion is trite to experts. 
Is there anything in it? Jen: 
Concomitant Sounds and Colours. 
THE investigation of the points of resemblance between two 
sciences, has its value and assists the development of both, 
Music gains by being thus raised from a mere sentimental 
recreation to the dignity of a science, but the science of colours 
may perhaps gain even more than music by the comparison, and 
this because the ear, in most persons, can distinguish with more 
precision a discord in sound, than the eye can in colour. 
In the most ancient times it was well known that con- 
comitant sounds produce a resultant whose vibrations are gene- 
rated by the interference of the sonorous waves of the primaries. 
This physical fact was not only known but employed in the 
construction of Gregorian Caztilenas, whose succession of 
intervals shows a deep penetration of this truth. 
The law of combination of the vibrations of concomitant sounds 
may be stated thus:—The resultant of two sounds has, asits num- 
ber of vibrations, the difference between those of its primaries. 
Also any number of sounds combined two and two together, 
the rst with the 2nd, the 2nd with the 3rd and so on, will form a 
series of resultants, which similarly combined two and two 
together form a second series of resultants ; so that (continuing 
this process) we finally arrive at a single resultant which is that 
of the original combination. This law has been tested experi- 
mentally by Hallstrém and Scheibler. I considered that it might 
be useful to express this law by a general formula, so I will give 
it in this place. 
If we have 7 sounds whose vibrations are 1, xy, %3,..+ %, 
all in ascending order as to fitch, then the resultant will be 
R = (x = 1)n=g (%_ = X), 
where the suffixes must be treated as indices, and (+ — I)n- 
expanded according to the binomial theorem. 
F is not, however, a resultant in the strict sense requiring the 
vanishing of the primary sounds; it might, perhaps, be better 
called the Residuant of the combination. It is thus the measure 
of the imperfection of the combination which is, more or less, 
a discord according as # is less or more nearly related to the 
primaries. 
If we apply this formula we shall easily see that the tonic and 
subdominant generate a note two octaves below the subdominant: 
Also CEG, CGC?, and CDE 
and all similar combinations in which the vibrations stand in 
arithmetical progression generate no residuant, hence, combina- 
tions of this class are perfectly consonant and are called by 
Boethius eguisonal concords, (The combination CDZ is dis- 
cordant enough on a modern instrument, but I mean CD£Z tuned 
perfectly without temperament. ) 
Supposing then that an impression is made upon the retina by 
two or more colours in juxtaposition, analogous to that produced 
on the auditory nerve by two or more simultaneous sounds. We 
shall perceive that two complementary colours placed side by side 
ought to increase in intensity that one whose vibrations are the 
most rapid, Red and green, for instance, should give intensity 
for example, C and eee 
to the green, since D and Ggenerate ~. Moreover, the colours 
corresponding to the eguzsonal concords ought to give us the 
most harmonious combinations ; these are they :— 
Violet placed between two yellows, 
Red 9 mD greens, 
Orange . +5 blues, 
Vellow es AB indigo-blues, 
Green oF oe violets, 
Blue A a reds. 
Yellow and indigo-violet ought always to be discordant, as they 
correspond with the discord F B or “fone, Again— 
Violet, orange, green 
Yellow, blue, violet 
Indigo-blue, red, yellow 
Violet, red, orange 
Yellow, green, blue 
Indigo-blue, violet, red. 
Tt will be noticed that these tints must be precisely of the same 
shade as those in Newton’s image ; the slightest variation would 
destroy the harmony of colour. Ihave no doubt if pigments 
were made of tints identical with the ring-colours, the beauty of 
these combinations would be appreciated by all who used them. 
All correspond to equisonal 
concords and should form per- 
fectly harmonious combina- 
tions. 
