nearly with the law of this curve. It is therefore 
se x the most natural as well as the most convenient 
to be assumed, as ing the state of an undula- 
tion in general ; and the name of these harmonic sliders 
‘is very properly deduced from the harmonic curve. 
By means of this instrument, the process of nature, 
in the combinations of motion which take place in va- 
rious cases of the junction of undulations, is rendered 
visible and intelligible, with great ease, in the most 
‘complicated cases. It is unnecessary to explain here, 
‘how accurately both oa espace and motions of the 
particles of air, in sound, may be represented by the 
ordinates of the curve at different points : it is sufficient 
‘to consider them as merely indicating the height of the 
water constituting a tide, or a wave of any kind, which 
exists at once in its whole extent, and of which each 
point passes also in succession through any given place 
of observation. We have then to examine what will be 
the effect of two tides, produced by different causes, 
wheh united. In order to represent this effect, we 
must add to the elevations or depressions in consequence 
of the first tide, the elevations or depressions in conse- 
quence of the second, and subtract them when they 
couriteract the effect of the first: or we may add the 
whole height of the second above any given point or 
line, and aa subtract, from all the sums, the distance 
of the point assumed below the medium. 
To do this mechanically is the object of the harmonic 
sliders. The surface of the first tide is nted by 
" the curvilinear termination of a single Plate 
xxvii: CCLXXXVIII. Fig. 1. The second tide is also repre- 
sented by the termination of another surface; but, in 
order that the height at each point may. be added to the 
into a number of separate pieces or sliders, which are 
eonfined within a groove or frame, and tightened by a 
‘serew, Fig. 2. Their lower ends are situated originally 
‘in a right line; but, by loosening the screw and mo- 
ving the sliders, they may be made to assume any 
other form: thus they may be applied to the surface 
representing the first tide; and if the similar parts of 
each correspond, Fig. 3, the combination will represent 
‘a tide of twice the magnitude of the simple tides. 
The more the corresponding are separated, the 
“weaker will be the joint effect, Fig. 4. ; and, when they 
are furthest removed, the whole tides, if equal, will be 
annihilated, Fig. 5. ‘Thus, when the general tide of the 
ocean arrives two different channels at the same 
port, at such intervals of time that the high water’ of 
one would happen at the same instant with the low 
water of the other, the whole effect is destroyed, exce 
so far as the partial tides differ in magnitude. T. 
principle being once understood, it may easily be a 
plied to a multiplicity of cases: for instance, where 
undulations differ in their dimensions with regard to 
extent. Thus, the series of sliders being extended to 
three or four alternations, the effect of combining un- 
dulations in the ratio of 2 to 1, of 3 to 1, of 2 to 3, of 
to 4, may be ascertained, by making a fixed surface, 
terminating in a series of curves, that bear to those of 
the shi surface the ratio required: and, by making 
them differ but slightly, the phenomenon of the beating 
of an imperfect unison in music may be imitated, where 
the joint undulation becomes alternately redoubled and 
evanescent. In Fig. 6. the ion is that of 17 to 
18, and the curvilinear outline represents the progress 
of the joint sound from the st degree of intensity 
to the least, and a little it.” 
HARMONICAL Megan, isa term used by arithmetical 
HARMONICS. 
‘mentioned, * 
height of the first tide, the surface is cut transversely _ 
‘rean or harmonical.curve, 
639 
and algebraical. writers to express Certain relations of Harmonies 
numbers and quantities: but with which musical cal. “Vv 
culators will find, that they need have little to do; any 
more than with the harmonical or musical ion 
and progressions, of the same writers. two quan. 
tities @ and & are given, then es is said to be the hare 
monical mean between them ; for le, between 2 
and 6, the harmoni¢al mean is f= 3: een 5 and 
9, it is $¢=6}. Dr Smith in his Harmonics, 2d edit. 
p: 125, shews, how to find the harmonical mean, among 
fhe ea of quantities: see also p. 141, Tbid. 
ARMONICAL or Musicat Proportion, of arith- 
metical and algebraical writers, is said to obtain between 
three quantities, as a, b and c, when a: c:: a—b: b—e; 
and between four quantities, as d, e, f and g, when 
d:g::d—e:f—g; and so 2, 3 and 6; and 1, 3, 2 
and "6 are said to be in musical wy ; And se- 
veral of these writers say, that if to A ree terms aboy4 
ional terms are continued, there 
will arise an harmonical progression” or series: Put in- 
pogo cases, the aes armonical and musica’, seem 
ively applied. 
MECAEMONTCS! dn susie? besides being used to de- 
signate the science or philosophy of musical sounds, as 
Dr Robert Smith uses this term, in making it the title 
of his justly famous work on this subject, imply also 
certain derivative or dependent new sounds, which, 
under favourable circumstances, are erated and 
heard, along with every single musical sound, or ac- 
companying every consonance of two such sounds, but 
with less intensity or loudness than the original, or 
‘generators of these new sounds ; and in the latter case, of 
their production by a consonance, when they are called 
grave Harmonics, (see that article,) such new sounds 
are further reat mene by not having a fixed direc- 
tion, towards (or from) the sounding body, but, like the 
sensation called “ a singing in the ear,” they are alike 
heard in any direction to which the ear is turned; a. 
roperty of these derivative sounds, which Mr John 
ough first explained, we believe, in Nicholson’s Jour- 
nal, 8vo. vol. iv. p. 2. The other kind of these new 
sounds, derived tbo a single sound, are called acute 
Harmonics, which see. (¢) 
HARMONICS, Acute, are phenomena attending a. 
sounding string or pipe, &c. which were first noticed 
by Gali and subsequently Peter Marsenne, 
M. Sauveur, M. Tartini, &c.: but Daniel Bernoulli first 
discovered the reason, and explained the theory of the 
acute harmonics, by shewing, that a sounding string, 
at the same time that its whole length vibjated a given 
note, might maintain aren tet ak paced of 7 half, 
its third , its fourth, and its in : 
each oF ack a ie parts hipressing on the ne 
rounding air, independent pulses, the times of whose 
single vibrations are in the ratios 1, $, 4,4, $; and by 
which the original sound or ator, W' ac- 
companied by its octave or VIII, its major twelfth or 
XII, its double octave or XV, and by its or seven- 
teenth or XVII; although only the XITth XVIIth, 
or octaye of the fifth and double octave of the major 
third, had yet been described, among the acute harmo- 
nics attending a sound. And this great mathematician, 
although le to contrive any experiment, by which 
the Vibeptidine of the 4 and ith Ele the string might 
be evidently shewn to subsist, with the whole vi- 
brations, yet he shewed, from the nature of the Taylo- 
that these subdivisions of a 
sounding string, were not only alike possible, and even 
