volitation, says in his Mathematical Magick (1648) that
it was related that ``a certain English monk called Elmerus,
about the Confessor's time,'' flew from a town in Spain for
a distance of more than a furlong; and that other persons
had flown from St Mark's, Venice, and at Nuremberg. Giovanni
Battista Dante, of Perugia, is said to have flown several
times across Lake Trasimene. At the beginning of the 16th
century an Italian alchemist who was collated to the abbacy
of Tungland, in Galloway, Scotland, by James IV., undertook
to fly from the walls of Stirling Castle through the air to
France. He actually attempted the feat, but soon came to
the ground and broke his thigh-bone in the fall--an accident
which he explained by asserting that the wings he employed
contained some fowls' feathers, which had an ``affinity''
for the dung-hill, whereas if they had been composed solely
of eagles' feathers they would have been attracted to the
air. This anecdote furnished Dunbar, the Scottish poet, with
the subject of one of his rude satires. Leonardo da Vinci
about the same time approached the problem in a more scientific
spirit, and his notebooks contain several sketches of wings
to be fitted to the arms and legs. In the following century
a lecture on flying delivered in 1617 by Fleyder, rector of
the grammar school at Tubingen, and published eleven years
later, incited a poor monk to attempt to put the theory into
practice, but his machinery broke down and he was killed.
In Francis Bacon's Natural History there are two
passages which refer to flying, though they scarcely
bear out the assertion made by some writers that he
first published the true principles of aeronautics.
The first is styled Experiment Solitary, touching Flying
in the Air --``Certainly many birds of good wing (as kites
and the like) would bear up a good weight as they fly; and
spreading leathers thin and close, and in great breadth, will
likewise bear up a great weight, being even laid, without
tilting up on the sides. The further extension of this
experiment might be thought upon.'' The second passage is
more diffuse, but less intelligible; it is styled Experiment
Solitary, touching unequal weight (as of wool and lead or
bone and lead); if you throw it from you with the light end
forward, it will turn, and the weightier end will recover to
be forwards, unless the body be over long. The cause is, for
that the more dense body hath a more violent pressure of the
parts from the first impulsion, which is the cause (though
heretofore not found out, as hath been often said) of all
violent motions; and when the hinder part moveth swifter (for
that it less endureth pressure of parts) that the forward
part can make way for it, it must needs be that the body
turn over; for (turned) it can more easily draw forward the
lighter part.'' The fact here alluded to is the resistance
that bodies experience in moving through the air, which,
depending on the quantity of surface merely. must exert a
proportionally greater effect on rare substances. The passage
itself, however, after making every allowance for the period
in which it was written, must be deemed confused, obscure and
unphilosophical. In his posthumous work, De Motu Animalium,
published at Rome in 1680-1681, G.A.Borelli gave calculations
of the enormous strength of the pectoral muscles in birds;
and his proposition cciv. (vol. i. pp. 322-326), entitled Est
impossibile ut homines pro priis viribus artificiose volare
possint, points out the impossibility of man being able by his
muscular strength to give motion to wings of sufficient extent
to keep him suspended in the air. But during his lifetime two
Frenchmen, Allard in 1660 and Besnier about 1678, are said to
have succeeded in making short flights. An account of some
of the modern attempts to construct flying machines will be
found in the article FLIGHT AND FLYING; here we append a
brief consideration of the mechanical aspects of the problem.
The very first essential for success is safety, which will
probably only be attained with automatic stability. The
underlying principle is that the centre of gravity shall
at all times be on the same vertical line as the centre of
pressure. The latter varies with the angle of incidence. For
square planes it moves approximately as expressed by Joessel's
formula, C + (0.2 + 0.3 sin a) L, in which C is the distance
from the front edge, L the length fore and aft, and a the
angle of incidence. The movement is different on concave
surfaces. The term aeroplane is understood to apply to
flat sustaining surfaces, but experiment indicates that
arched surfaces are more efficient. S. P. Langley proposed
the word aerodrome, which seems the preferable term for
apparatus with wing-line surfaces. This is the type to which
results point as the proper one for further experiments. With
this it seems probable that, with well-designed apparatus,
40 to 50 lb. can be sustained per indicated h.p., or about
twice that quantity per resistance or ``thrust'' h.p., and
that some 30 or 40 k of the weight can be devoted to the
machinery, thus requiring motors, with their propellers,
shafting, supplies, &c., weighing less than 20 lb. per
h.p. It is evident that the apparatus must be designed to
be as light as possible, and also to reduce to a minimum
all resistances to propulsion. This being kept in view,
the strength and consequent section required for each member
may be calculated by the methods employed in proportioning
bridges, with the difference that the support (from air
pressure) will be considered as uniformly distributed, and the
load as concentrated at one or more points. Smaller factors
of safety may also have to be used. Knowing the sections
required and unit weights of the materials to be employed,
the weight of each part can be computed. If a model has been
made to absolutely exact scale, the weight of the full-sized
apparatus may approximately be ascertained by the formula
$$W' = W\sqrt{\left({S'\over S}\right)}^3,$$
in which W is the weight of the model, S its surface, and
W' and S' the weight and surface of the intended apparatus.
Thus if the model has been made one-quarter size in its
homologous dimensions, the supporting surfaces will be sixteen
times, and the total weight sixty-four times those of the
model. The weight and the surface being determined, the three
most important things to know are the angle of incidence, the
``lift,'' and the required speed. The fundamental formula for
rectangular air pressure is well known: P=KV2S, in which P is
the rectangular normal pressure, in pounds or kilograms, K a
coefficient (0.0049 for British, and 0.11 for metric measures),
V the velocity in miles per hour or in metres per second, and
S the surface in square feet or in square metres. The normal
on oblique surfaces, at various angles of incidence, is given
by the formula P = KV2Se, which latter factor is given both
for planes and for arched surfaces in the subjoined table:--.
PERCENTAGES OE AIR PRESSURE AT VARIOUS ANGLES OF INCIDENCE
PLANES (DUCHEMIN FORMULA,
VERIFIED BY LANGLEY). WINGS (LILIENTHAL).
N = P(2sina/(1+sin2a)). Concavity 1 in 12
Angle. Normal. Lift. Drift. Normal. Lift. Drift. Tangential
a e ecosa esina e ecosa esina force a
-9 deg. 0.0 0.0 0.0 +0.070
-8 deg. 0.040 0.0396 -0.0055 +0.067
-7 deg. 0.080 0.0741 -0.0097 +0.064
-6 deg. 0.120 0.1193 -0.0125 +0.060
-5 deg. 0.160 0.1594 -0.0139 +0.055
-4 deg. 0.200 0.1995 -0.0139 +0.049
-3 deg. 0.242 0.2416 -0.0126 +0.043
-2 deg. 0.286 0.2858 -0.0100 +0.037
-1 deg. 0.332 0.3318 -0.0058 +0.031
0 deg. 0.0 0.0 0.0 0.381 0.3810 -0.0 +0.024
+1 deg. 0.035 0.035 0.000611 0.434 0.434 +0.0075 +0.016
+2 deg. 0.070 0.070 0.00244 0.489 0.489 +0.0170 +0.008
+3 deg. 0.104 0.104 0.00543 0.546 0.545 +0.0285 0.0
+4 deg. 0.139 0.139 0.0097 0.600 0.597 +0.0418 -0.007
+5 deg. 0.174 0.173 0.0152 0.650 0.647 +0.0566 -0.014
+6 deg. 0.207 0.206 0.0217 0.696 0.692 +0.0727 -0.021
+7 deg. 0.240 0.238 0.0293 0.737 0.731 +0.0898 -0.028
+8 deg. 0.273 0.270 0.0381 0.771 0.763 +0.1072 -0.035
+9 deg. 0.305 0.300 0.0477 0.800 0.790 +0.1251 -0.042
10 deg. 0.337 0.332 0.0585 0.825 0.812 +0.1432 -0.050
11 deg. 0.369 0.362 0.0702 0.846 0.830 +0.1614 -0.058
12 deg. 0.398 0.390 0.0828 0.864 0.845 +0.1803 -0.064
13 deg. 0.431 0.419 0.0971 0.879 0.856 +0.1976 -0.070
14 deg. 0.457 0.443 0.1155 0.891 0.864 +0.2156 -0.074
15 deg. 0.486 0.468 0.1240 0.901 0.870 +0.2332 -0.076
The sustaining power, or ``lift'' which in horizontal flight
must be equal to the weight, can be calculated by the formula
L=KV2Secosa, or the factor may be taken direct from the
table, in which the ``lift'' and the ``drift'' have been obtained
by multiplying the normal e by the cosine and sine of the
angle. The last column shows the tangential pressure on concave
surfaces which O. Lilienthal found to possess a propelling
component between 3 deg. and 32 deg. and therefore to be negative
to the relative wind. Former modes of computation indicated
angles of 10 to 15 as necessary for support with planes. These
mere prohibitory in consequence of the great ``drift''; but
the present data indicate that, with concave surfaces, angles
of 2 deg. to 5 will produce adequate ``lift.'' To compute the
latter the angle at which the wings are to be set must first be
assumed, and that of @ will generally be found preferable.
Then the required velocity is next to be computed by the formula
$$V = \sqrt{L\over KS\eta\cos\alpha};$$
or for concave wings at +3 deg. :
$$V = \sqrt{W\over 0.545KS}.$$
Having thus determined the weight, the surface, the angle of
incidence and the required seed for horizontal support, the
next step is to calculate the power required. This is best
accomplished by first obtaining the total resistances, which
consist of the ``drift'' and of the head resistances due to
the hull and framing. The latter are arrived at preferably
by making a tabular statement showing all the spars and parts
offering head resistance, and applying to each, the coefficient
appropriate to its ``master section,'' as ascertained by
experiment. Thus is obtained an ``equivalent area'' of resistance,
which is to be multiplied by the wind pressure due to the
speed. Care must be taken to resolve all the resistances at
their proper angle of application, and to subtract or add the
tangential force, which consists in the surface S, multiplied
by the wind pressure, and by the factor in the table, which
is, however, 0 for 3 and 32, but positive or negative at other
angles. When the aggregate resistances are known, the ``thrust
h.p.'' required is obtained by multiplying the resistance by the
