MODEL AIRPLANES, THE BERNOULLI
EQUATION, AND THE COANDA EFFECT © 1994
by Jef Raskin
aerodynamics, theory is what makes the invisible plain. Trying to
fly an airplane without theory is like getting into a fistfight with
"That we have written an equation
does not remove from the flow of fluids its
charm or mystery or its surprise."
sound theoretical understanding of lift had been achieved
within two decades of the Wright brothers' first flight (Prandtl's work
was most influential1),
but the most common explanation of lift
seen in elementary texts and popular articles today is problematical. Here
is a typical example of what is found:
The common explanation,
from The Way Things Work [Macaulay
reasoning--though incomplete--is based on the Bernoulli effect,
which correctly correlates the increased speed with which air
moves over a surface and the lowered air pressure measured at that
fact, most airplane wings do have considerably more curvature
on the top than the bottom, lending credence to this explanation.
But, even as a child, I found that it presented me with
a puzzle: how can a plane fly inverted (upside down). When I pressed
my 6th grade science teacher on this question, he just got mad,
denied that planes could fly inverted and tried to continue his
lecture. I was very frustrated and argued until he said, "Shut up,
Raskin!" I will relate what happened next later in this essay.
years later I carried out a calculation according to a naive
interpretation of the common explanation of how a wing works.
Using data from a model airplane I found that the calculated
lift was only 2% of that needed to fly the model. [See Appendix 1 for the calculation]. Given that Bernoulli's equation is
correct (indeed, it is a form of the law of conservation of energy),
I was left with my original question unanswered: where does
the lift come from?
next few sections we look at attempts to explain two related
phenomena--what makes a spinning ball curve and how a wing's
shape influences lift--and see how the common explanation of lift
has led a surprising number of scientists (including some famous
path of a ball spinning around a vertical axis and moving forward
through the air is deflected to the right or the left of a straight
path. Experiment shows that this effect depends both on the
fact it is spinning and that it is immersed in a fluid (air). Non-spinning
balls or spinning balls in a vacuum go straight. You might,
before going on, want to decide for yourself which way a ball
spinning counterclockwise (when seen from above) will turn.
see what five books say about this problem. Three are by
physicists, one is a standard reference work, and the last, just
for kicks, is from a book by my son's soccer coach. We'll start
with physicist James Trefil, who writes [Trefil 1984],
leaving the Bernoulli effect, I'd like to point out
one more area where its consequences should be explored,
and that is the somewhat unexpected activity of
a baseball. Consider, if you will, the curve ball. This
particular pitch is thrown so that the ball spins around
an axis as it moves forward, as shown in the top in
figure 11-4. Because the surface of the ball is rough,
the effect of viscous forces is to create a thin layer
of air which rotates with the surface. Looking at the
diagram, we see that the air at the point labeled A will
be moving faster than the the air at the point labeled
B, because in the first case the motion of the ball's
surface is added to the ball's overall velocity, while
in the second it is subtracted. The effect, then is
a 'lift' force, which tends to move the ball in the direction
Trefil's figure 11-4. It does
not agree with some other sources.
Baseball aficionados would
say that the ball curves toward third base.
Trefil then shows a diagram of a fast ball, shown as deflecting
downward when spinning so that the bottom of the ball is
rotating forward. It is the same phenomenon with the axis of rotation
shifted 90 degrees.
Physics of Baseball, Robert K. Adair
[Adair 1990] imagines a ball thrown toward
home plate, so that it rotates counterclockwise
as seen from above--as in Trefil's diagram. To the left
of the pitcher is first base, to his right is third base. Adair
can then expect the air pressure on the third-base
of the ball, which is travelling faster through the air,
to be greater than the pressure on the on the
side, which is travelling more slowly, and
ball will be deflected toward first base."
is exactly the opposite of Trefil's conclusion though they agree
that the side spinning forward is moving faster through the air.
We have learned from these two sources that going faster through
the air either increases or decreases the pressure on that side.
I won't take sides in this argument as yet.
Brittanica  gives an explanation
which introduces the concept of drag into
drag of the side of the ball turning into the air
the direction the ball is travelling) retards the
whereas on the other side the drag speeds up
airflow. Greater pressure on the side where the
is slowed down forces the ball in the direction
low-pressure region on the opposite side, where a
increase in airflow occurs."
we have read that spinning the ball causes the air to move either
faster or slower past the side spinning forward, and that faster
moving air increases or decreases the pressure, depending on
the authority you choose to follow. Speaking of authority, it might
be appropriate to turn to one of the giants of physics of this
century, Richard Feynman. He opposes Trefil, and
uses a cylinder rather than a sphere [Feynman et al 1964. Italics
are theirs. The lift force referred to is shown pointing upwards.]:
flow velocity is higher on the upper side of a
[shown rotating so that its top is moving in
opposite direction to its forward travel] than on the lower
side. The pressures are therefore lower on
the upper side
than on the lower side. So when we have a combination
of a circulation around a cylinder and a net horizontal
flow, there is a net vertical force on
is called a lift force."
for my son's coach's book. The coach in this case is the world-class
soccer player, George Lamptey. There is almost no theory
given, but we can be reasonably sure that Lamptey has repeatedly
tried the experiment and should therefore report the direction
the ball turns correctly. He writes[Lamptey
banana kick is more or less an off-center instep
kick which adds a spin to the soccer ball. Kick
center to the right, the soccer ball curves to the
Kick off center to the left, the soccer ball
to the right... The amount the soccer ball curves depends
on the speed of the spin."
Lamptey, like Adair, has
the high pressure on the side moving into
the air. I will not relate more accounts, some having the ball
swerve one way, some the other. Some explanations depend on the
author's interpretation of the Bernoulli effect, some on viscosity,
some on drag, some on turbulence.
will return to the subject of spinning balls, but we are not
yet finished finding problems with the common explanation of lift.
explanation of how a wing works leads us to conclude,
for example, that a wing which is somewhat concave on the
bottom, often called an "undercambered" wing, will always generate less lift
(under otherwise fixed conditions) than a flat bottomed one. This conclusion
We then have to ask how
a flat wing like that of a paper airplane,
with no curves anywhere, can generate lift. Note that the
flat wing has been drawn at a tilt, this tilt is called "angle of
attack" and is necessary for the flat wing to generate lift. The
topic of angle of attack will be returned to presently.
A flat wing
can generate lift. This is a bit difficult
given the traditional mental model.
cross-sectional shapes of wings,like
those illustrated here, are called "airfoils." A
very efficient airfoil for small, slow-flying
models is an arched piece of thin sheet material, but it
is not clear at all from the common explanation how it can generate
lift at all since the top and bottom of the airfoil are the
If the common explanation
is all there were to it, then we should
be making the tops of wings even curvier than they now are. Then
the air would have to go even faster, and we'd get more lift. In
this diagram the wiggliness is exaggerated. More realistic lumpy
examples will be encountered in a few moments.
If we make the top of the
wing like this, the air on top has a
lot longer path
to follow, so the air
will go even faster than
with a conventional wing. You might
conclude that this kindof airfoil should have lots of lift.
In fact, it is a disaster.
examples. While Bernoulli's equations are correct, their
proper application to aerodynamic lift proceeds quite differently
than the common explanation. Applied properly or not, the
equations result in no convenient visualization that links the shape of
an airfoil with its lift, and reveal nothing about drag. This
lack of a readily-visualized mental model, combined with the prevalence
of the plausible-sounding common explanation, is probably
why even some excellent physicists have been misled.
friend Yesso, who works for the aircraft industry (though not
as a designer), came up with a proposed improved airfoil. Reasoning
along the lines of the common explanation he suggested that
you should get more lift from an airfoil if you restarted the top's
curve part of the way along:
An extra lump for extra lift?
This is just a "reasonable" version
of the lumpy airfoil that I presented
above. Yesso's idea was, of course, based on the concept that
a longer upper surface should give more lift. I was about to tell
Yesso why his foil idea wouldn't work when I happened to talk to Jörgen
He told me of a humped airfoil Albert Einstein4 designed
during WWI that was based on much the same reasoning Yesso
had used [Grosz 1988].
airfoil. It had no
This meant that instead
of telling Yesso merely that his idea wouldn't
work, I could tell him that he had created a modernized version
of Einstein's error! Einstein later noted, with chagrin, that he had goofed5.
were the case that airfoils generate lift solely because
the airflow across a surface lowers the pressure on that surface then,
if the surface is curved, it does not matter whether it
is straight,concave, or convex; the common
explanation depends only on flow parallel
to the surface. Here are some experiments
that you can easily reproduce to test this idea.
Make a strip of writing paper about 5 cm X 25 cm. Hold it in
front of your lips so that it hangs out and down making a convex
upward surface. When you blow across the top of the paper, it
rises. Many books attribute this to the lowering of the air pressure
on top solely to the Bernoulli effect.
Now use your fingers to
form the paper into a curve that it is slightly
concave upward along its whole length and again blow along
the top of this strip. The paper now bends downward.
As per the diagrams below, build a box of thin plywood or cardboard
with a balsa airfoil held in place with pins that allow it
to flap freely up and down. Air is introduced with a soda straw.
That's one of the nice things about science. You don't have to take anybody's
word for a claim, you can try it yourself!6In this
wind tunnel the air flows only across the top of the shape. A student
friend of mine made another where a leaf blower blew on both
top and bottom and he got the same results, but that design takes
more effort to build and the airfoil models require leading and
trailing edge refinement. Incidentally, I tried to convince a company
that makes science demonstrators to include this in their offerings.
They weren't interested in it because "it didn't give the
right results.""Then how
does it work?" I asked. "I don't know," said
the head designer.
experiment may be difficult to interpret but, unless it is fraudulent,
it cannot give the wrong results.
These drawings are full size, but the exact size
and shape aren't important. I made a number of airfoils to test.
Here are drawings of the ones I made:
the straw is blown into, the normal airfoil
promptly lifts off the bottom and floats
up. When the blowing stops, it goes back
down. This is exactly what everybody expects. Now consider
the concave shape; the curve is exactly
the same as the first airfoil , though
turned upside down. If the common explanation
were true, then, since the length along the curve is the
same as with the "normal" example, you'd expect this one to rise,
too. After all, the airflow along the surface must be lowering
the pressure, allowing the normal ambient air pressure below
to push it up. Nonetheless, the concave airfoil stays firmly down;
if you hold the apparatus vertically, it will be seen to move away from
other words, an often-cited experiment which is usually taken
as demonstrating the common explanation of lift does not do so;
another effect is far stronger. The rest of the airfoils are for
fun--try to anticipate the direction each will move before you put
them in the apparatus. It has been noted that "progress in science
comes when experiments contradict theory" [Gleick 1992] although
in this case the science has been long known, and the experiment
contradicts not aerodynamic theory, but the often-taught
common interpretation. Nonetheless, even if science does not
progress in this case, an individual's understanding of it may.
Another simple experiment will lead us toward an explanation that
may help to give a better feel for these aerodynamic effects.
If a stream of water is flowing along
a solid surface which is curved slightly
away from the stream, the water will tend to follow the surface. This is
an example of the Coanda effect7and
is easily demonstrated by holding the
back of a spoon vertically under a thin
stream of water froma faucet. If you
hold the spoon so that it can swing, you
will feel it being pulled toward the stream
of water. The effect has limits: if you use a sphere instead
of a spoon, you will find that the water will only follow a
part of the way around. Further, if the surface is too sharply curved,
the water will not follow but will just bend a bit and break
away from the surface.
The Coanda effect works
with any of our usual fluids, such as air at
usual temperatures, pressures, and speeds. I make these qualifications
because (to give a few examples) liquid helium, gasses
at extremes of low or high pressure or temperature, and fluids
at supersonic speeds often behave rather differently. Fortunately,
we don't have to worry about all of those extremes with
of air, such as what you'd get if you
through a straw, goes in a straight line
of air alongside
surface still goes
in a straight line
of air alongside
a curved surface tends
the curvature of
the surface. Seems
a stream of air
alongside a curved
that bends away from it still
follow the curvature
of the surface.
thing we don't have to
wonder about is why the Coanda effect
works, we can take it as an experimental fact. But I
hope your curiosity is unsatisfied on this point and that you will
word often used to describe the Coanda effect is to say that
the airstream is "entrained" by the surface. One advantage of
discussing lift and drag in terms of the Coanda effect is that we can
visualize the forces involved in a rather straightforward way. The
common explanation (and the methods used in serious texts on aerodynamics)
are anything but clear in showing how the motion of the
air is physically coupled to the wing. This is partly because much
of the approach taken in the 1920s was shaped by the need for the
resulting differential equations (mostly based on the Kutta-Joukowski
to have closed-form solutions or to yield useful
numerical results with paper-and-pencil methods. Modern approaches
use computers and are based on only slightly more intuitive
constructs. We will now develop an alternative way of visualizing
lift that makes predicting the basic phenomena associated
with it easier.
A MENTAL MODEL OF HOW
A WING GENERATES LIFT AND DRAG
is typical of physicists, I have often spoken of the air moving
past the wing. In aircraft wings usually move through the air.
It makes no real difference, as flying a slow plane into the wind
so that the plane's ground speed is zero demonstrates. So I will
speak of the airplane moving or the wind moving whichever makes
the point more clearly at the time.
next illustration , it becomes convenient to look at the air from the point
of view of a moving airplane.
the air molecules, attracted
surface, are pulled down.
of the wing moving to the left, with the air standing still. The
air moves toward the wing much as if it was attached to the wing
with invisible rubber bands. It is often helpful to think of lift
as the action of the rubber bands that are pulling the wing up.
detail is important: the air gets pulled along in the direction
of the wing's motion as well. So the action is really more
like the following picture.
air is pulled forward as well
as down by the motion of the
you were in a canoe and tried pulling someone in the water toward
you with a rope, your canoe would move toward the person. It
is classic action and reaction. You move a mass of air down and the
wing moves up. This is a useful visualization of the lift generated
by the top of the wing.
the diagram suggests, the wing has also spent some of its energy,
necessarily, in moving the air forward. The imaginary rubber
bands pull it back some. That's a way to think about the drag that
is caused by the lift the wing generates. Lift cannot be had
acceleration of the air around the sharper curvature near the
front of the top of the wing also imparts a downward and forward
component to the motion of the molecules of air (actually a
slowing of their upward and backward motion, which is equivalent)
and thus contributes to lift. The bottom of the wing is
easier to understand, and an explanation is left to the reader.
experiments with the miniature wind tunnel described earlier
are readily understood in terms of the Coanda effect: the downward-curved
wing entrained the airflow to move downward, and a force
upward is developed in reaction. The upward-curved (concave) airfoil
entrained the airflow to move upwards, and a force downward
was the result. The lumpy wing generates a lot of drag by moving
air molecules up and down repeatedly. This eats up energy (by
generating frictional heat) but doesn't create a net downward motion
of the air and therefore doesn't create a net upward movement of the wing.
It is easy, based on the Coanda effect, to visualize
why angle of attack (the fore-and-aft tilt of the wing, as
illustrated earlier) is crucially important to a symmetrical airfoil,
why planes can fly inverted, why flat and thin wings work,
and why Experiment 1 with its convex and concave strips of paper
works as it does.
has been presented so far is by no means a physical account
of lift and drag, but it does tend to give a good picture of
the phenomena. We will now use this grasp to get a reasonable hold
on the spinning ball problem.
WHY THE SPINNING BALL'S
PATH CURVES, IN TERMS OF THE COANDA EFFECT
Coanda effect tells us the air tends to follow the surface
of the ball. Consider Trefil's side A which is rotating in the
direction of flight. It is trying to entrain air with it as it spins,
this action is opposed by the oncoming air. Thus, to entrain
the air around the ball on this side, it must first decelerate
it and then reaccelerate it in the opposite direction.
On the B side, which is
rotating opposite the direction of flight, the
air is already moving (relative to the ball) in the same direction,
and is thus more easily entrained. The air more readily follows
the curvature of the B side around and acquires a velocity toward
the A side. The ball therefore moves toward the B side by reaction.
It is again time for a
simple experiment. It is difficult to experiment
with baseballs because their weight is large compared to
the aerodynamic forces on them and it is very hard to control the
magnitude and direction of the spin, so let us look at a case where
the ball is lighter and aerodynamic effects easier to see. I use
a cheap beach ball (expensive ones are made of heavier materials
and show aerodynamic effects less). Thrown with enough bottom
spin (bottom moving forward) such a ball will actually rise in
a curve as it travels forward. The lift due to spin can be so strong
that it is greater than the downward force of gravity! Soon,
air resistance stops both the spin and the forward motion of the
ball and it falls, but not before it has shown that Trefil's explanation
of how spin affects the flight of a ball is
lift due to spinning while moving through the air is usually
called the "Magnus9effect." Some
books on aerodynamics also describe the "Flettner
Rotor," which is a long-since abandoned
attempt to use the Magnus effect to make an efficient boat
sail. Many sources besides Trefil get the effect backwards including
the usually reliable Hoerner [Hoerner 1965]. College-level
texts tend to get it right [Kuethe and Chow 1976; Houghton and
Carruthers 1982]. I was relieved to see that the classic Aerodynamics [von
Kármán 1954] gets the lift force on a spinning
ball in the correct direction though the reasoning seems a
I could send this essay to the 6th grade science teacher
who could not take the time to listen to my reasoning. Here's
what happened: he sent me to the principal's office when I came
in the next day with a balsa model plane with dead flat wings.
It would fly with either side up depending on how an aluminum
foil elevator adjustment was set. I used it to demonstrate
that the explanation the class had been given must have
been wrong, somehow. The principal, however, was informed that
my offense was "flying paper airplanes in class" as though done
with disruptive intent. After being warned that I was to improve
my behavior, I went to my beloved math teacher who suggested
that I go to the library to find out how airplanes fly--only
to discover that all the books agreed with my science teacher!
It was a shock to realize that my teacher and even the library
books could be wrong. And it was a revelation that I could trust
my own thinking in the face of such concerted opposition. My playing
with model airplanes had led me to take a major step toward
intellectual independence--and a spirit of innovation that later
led me to create the Macintosh computer project (and other, less-well-known
inventions) as an adult.
1 - Ludwig Prandtl (1875-1953), a German physicist, often
called the "father of aerodynamics." His famous book on the theory
of wings, Tragflügeltheorie, was published in 1918. back
2 - The surface roughness is not essential. The effect is
observed no matter how smooth the ball. back
3 - Mr. Skögh worked on aircraft design for Saab in Sweden
and for Lockheed in the United States. back
4 - Albert Einstein [1879-1955], a German-American physicist,
was one of the greatest scientists of all time. His small error in wing design
does not detract from the massive revolution his thinking brought about in
5 - Jörgen Skögh writes, "During the First
World War Albert Einstein was for a time hired by the LVG (Luft-Verkehrs-Gesellshaft)
as a consultant. At LVG he designed an airfoil with a pronounced mid-chord
hump, an innovation intended to enhance lift. The airfoil was tested in the
Göttingen wind tunnel and also on an actual aircraft and found, in
both cases, to be a flop." In 1954 Einstein wrote "Although it
is probably true that the principle of flight can be most simply explained
in this [Bernoullian] way it by no means is wise to construct a wing in such
a manner!" See [Grosz, 1988] for the full text. back
6 - In some fields, e.g. the study of sub-atomic particles,
you might need megabucks and a staff of thousands to build an accelerator
to do an independent check, but the principle is still there. back
7 - In the 1930's the Romanian aerodynamicist Henri-Marie
Coanda(1885-1972) observed that a stream of air (or other fluid) emerging
from a nozzle tends to follow a nearby curved or flat surface, if the curvature
of the surface or angle the surface makes with the stream is not too sharp. back
8 - Discovered independently by the German mathematician
M. Wilheim Kutta (1867- 1944) and the Russian physicist Nikolai Joukowski
9 - H. G. Magnus (1802-1870), a German physicist and chemist,
demonstrated this effect in 1853. back
QUANTITATIVE APPLICATION OF THE COMMON (INCORRECT) EXPLANATION
the pressure, in Newtons per square meter (Nm-2 =
on the top of a wing is notated ptop
, the pressure on the bottom pbottom ,
the velocity (ms-1)
on the top of the wing vtop, and the velocity
on the bottom vbottom ,and where ρ
is the density of air (approximately 1.2 kgm-3),
then the pressure difference across the
wing is given by the first term of Bernoulli's
1/2 ρ (vtop2 -
A rectangular planform (top view) wing
of one meter span was measured as having
a length chordwise along the bottom of 0.1624 m while
the length across the top was 0.1636 m. The ratio of the lengths
is 1.0074. This ratio is typical for many model and full-size
aircraft wings. According to the common explanation which has two
adjacent molecules separated at the leading edge mysteriously meeting
at the trailing edge, the average air velocities on the top
and bottom are also in the ratio of 1.0074.
speed for a model plane of 1m span and 0.16m chord with a mass of 0.7 kg
(a weight of 6.9 N) is 10 ms-1, which
Given these numbers, we find a pressure
difference from the equation of about 0.9 kgm-1.
The area of the wing is 0.16 m2 giving
3 ms-1 to
calculation is, of course, an approximation since Bernoulli's
equation assumes nonviscous, incompressible flow and air
is both viscous and compressible. But the viscosity is small and
at the speeds we are speaking of air does not compress significantly.
Accounting for these details changes the outcome at most
a percent or so. This treatment also ignores the second term (not
shown) of the Bernoulli equation--the static pressure difference
between the top and bottom of the wing due to their trivially
different altitudes. Its contribution to lift is even smaller
than the effects already ignored. The use of an average velocity
assumes a circular arc for the top of the wing. This is not
optimal but it will fly. None of these details affect the conclusion
that the common explanation of how a wing generates lift--with
its naďve application of the Bernoulli equation--fails quantitatively.
There are many fine
books and articles on the subject of model
airplane aerodynamics (and many more on aerodynamics
in general). Commendably accurate and readable are books
and articles for modelers by Professor Martin Simons [e.g. Simons
1987]. Much can be learned from Frank Zaic's delightful, if not
terribly technical, series [Zaic 1936 to Zaic 1964] (Available from
the Academy of Model Aeronautics in the United States), and no
treatments are more professional or useful than those of Professor
Michael Selig and his colleagues [e.g. Selig et. al. 1989].
All of these authors are also well-known modelers. The other
references on aerodynamics, e.g. Kuethe and Chow  and Houghton
and Carruthers  are graduate or upper-level undergraduate
texts, they require a knowledge of physics and calculus
including partial differential equations. Jones  is an
informal treatment by a master and Hoerner  is a magnificent
compendium of experimental results, but has little theory--practical
designers find his work invaluable.
- Adair, Robert K. The Physics of
Baseball, Harper and Row, NY, 1990.
- Feynman, R. et. al. Lectures on Physics,
Vol II, Addison-Wesley 1964
pg. 40-9, 40-10, 41-11
- Gleick, J. Genius. Pantheon
Books, NY 1992 pg. 234
- Grosz, Peter M. "Herr Dr Prof Albert Who? Einstein the Aerodynamicist,
That's Who!" WWI Aero No. 118, Feb. 1988 pg. 42 ff
- Hoerner, S.F. Fluid-Dynamic Drag,
Hoerner Fluid Dynamics, 1965 pg. 7-11
- Houghton and Carruthers.Aerodynamics
for Engineering Students, Edward
Arnold Publishers, Ltd. London, 1982
- Jones, R.T. Modern Subsonic Aerodynamics.
Aircraft Designs Inc., 1988. pg.36
- Lamptey, George. The Ten Bridges
to Professional Soccer, Book 1: Bridge
of Kicking. AcademyPress,
Santa Clara CA, 1985.
- Levy, Steven. "Insanely Great." Popular Science, February, 1994. pg.
- Linzmayer, Owen. The Mac Bathroom
Reader, Sybex 1994
- Kuethe and Chow.Foundations of
Aerodynamics, Wiley, 1976
- Macaulay, David. The Way Things
Work. Houghton Mifflin Co. Boston,
1988. pg. 115
- Selig, M. et. al. Airfoils at Low
Speeds. Soartech 8. Herk Stokely,
1504 Horseshoe Circle, Virginia Beach VA 23451, 1989
- Simons, M. Model Aircraft Aerodynamics,
2nd ed.. Argus Books Ltd., London, 1987.
- Skögh, Jörgen. Einstein's
Folly and The Area of a Rectangle,
- Thornburg, Dave. Do You Speak Model
Airplane? Pony X Press, 5 Monticello
Drive, Albuquerque NM 87123, 1992
- Trefil, James S. A Scientist At
The Seashore. Collier Books, Macmillan
Publishing Co., 1984, pp 148-149
- von Kármán, T. Aerodynamics.
Oxford Univ. Press 1954 pg. 33
- Zaic, Frank. Model Aeronautic Yearbooks. Published from the 30's to
- Zaic, Frank. Circular Airflow.
Model Aeronautic Publications, 1964.
am very appreciative of the suggestions I have received from a number
of careful readers, including Dr. Bill Aldridge, Professors Michael
Selig, Steve Berry, and Vincent Panico, Linda Blum, and Galen Panger.
They have materially improved both the content and the exposition, but
where I have foolishly not taken their advice my own errors may
yet shine through.
Raskin was a professor at the University of California at San
Diego and originated the Macintosh computer at Apple Computer Inc
[Levy 1994; Linzmayer 1994]. He is a widely-published writer, an
avid model airplane builder and competitor, and an active musician