I'm trying to repair an old (1930's) Graybar electric fan model 7854-c
 I believe the fan uses an induction motor. The fan will not start on
its own, rather it makes a humming noise. If you give one if the
blades a push to get it started, it works perfectly - it's quiet and
all 3 speeds work. There does not seem to be much to the motor, what
makes this type of motor start? Is there a method of
cleaning/repairing this motor to get it to start on it's own. From the
motor to the speed control in the base there are three wires. from the
control resistor/transformer in the base to the standard AC plug there
are 2 wires.

Reminds me of my lawn mower, that won't start in tall grass, since the
direct drive electric motor cannot overcome the resisitance.
If the fan is 60 years old, maybe the motor or bearings are gummed up
a bit and cleaning them would eleviate the problem.

Single-phase induction motors are not self-starting in the way 3-phase
motors are. They invariably have an additional winding for starting
purposes, these days usually connected in series with a capacitor.  If
you have access to an Ohmmeter, you should test for continuity between
the three leads you say are coming from the motor (BE SURE THE MOTOR
IS NOT CONNECTED TO THE SUPPLY DURING TESTING). What I otherwise
suggest, is that you pay a visit to this site:-
http://www.uoguelph.ca/~antoon/Forum/Forum.html
and put your question there. There are some very knowledgeable and
practical people there who wil almost certainly help you further.

As boyo62 writes, induction motors have an extra winding to give that
initial push. It sounds like that's the problem. Some ideas:  1, the
most common way that the start winding is wired up is with a
capacitor. Old kinds of capacitors are prone to die after a while,
often by drying out. If there is a start capacitor --- maybe attached
to the motor, maybe inside its casing --- it might need to be
replaced. This would probably be pretty simple and cheap. 2, often
there is a centrifugal switch which disconnects the start winding
after the motor gets up to speed. If there is such a switch, it may
have gotten stuck in the 'off' position, and could just need some
cleaning. 3, the switch could have gotten stuck in the 'on' position
instead, and when the start winding wasn't disconnected after
starting, it could have burned out. (Some start windings are designed
to be energized the whole time; some must be disconnected to keep from
burning out, hence the centrifugal switch.)