In the first place, how do you know that you can't have a fractional charge? To us today, it's obvious that atoms are made of electrons around a nucleus, and thus it only "makes sense" that atoms can have say 1 electrons'worth or 2 electrons'worth of charge, but not say 1 and a half electrons'worth of charge.
What Milikan did was ionize oil droplets (in modern terms strip electrons away from them), and then measure their charge - which is a story in itself. He found that their values were quantized, and at regular intervals!
Think of it this way: suppose you stood outside a candy store and looked at how much people were being charged (what a pun. heh heh.) for the sweets they bought. You don't get to look inside the shopping bag, though! The first person's bill is $1.80. The next person's bill is $1.20. The next, $2.40. The next, $3.00. After a few hours you find that everybody's bill comes in multiples of $0.60. It's pretty obvious, then, that on that particular day the candy shop only sold things with discrete prices of $0.60 - nothing cost anything else, or else you would've gotten someone with a bill which didn't add up to a multiple of $0.60.
So in the same way, Milikan discovered quantization, and interpreted his results to mean that atoms gain or lose charge in terms of electrons being pasted on or peeled off, and they went on one at a time - you can't get half an electrons'worth of charge.
Now, how to measure the mass of an electron? Well, suppose you have an electric field in space. The amount of force an electron experiences depends on its charge. However, how much acceleration it experiences depends both on the force and on its mass! (All other things being equal, it takes a lot more force to push a truck than it takes to push a small car.) And since acceleration determines velocity, maybe we could measure the velocity of an electron accelerating in a known field, and then find out its charge - basically, find out how fast a given field makes an electron move; the bigger an electron's mass, the more "push" you need to get it to a certain velocity.
In normal life you measure velocity by setting a starting point and an ending point, and then seeing how much time it takes to get between them. However, I think electrons would zip by far too fast for that to work. Enter magnetism! The magnetic force on an electron depends on its charge and its velocity; in particular, if an electron traveling at some speed enters a uniform magnetic field, it ends up turning a corner, and the radius of that bend allows you to calculate its mass and velocity.
And so that's how. =)
And this is good example that science is mere deduction. You think the candy bar was $0.60, but who is to say whether not 2 candy bars were sold for 1? And that maybe one had nuts in it and the other raisins?
Or whether the supposed 1 charge or mass of the electron was not composed itself of subcomponents.
What I am trying to get at is this: we deduct certain theories from what we observe. Nothing wrong with that. But we have to bear in mind that they are just that: theories.
Theories cannot be proven, only falsified, as Karl Popper so rightly said.
And here is where oftentimes scientists go wrong: they claim it is so, because of multiple proofs and what not. Instead of that they should say: we think it so. Or otherwise: we believe it is so. And now we're almost back to religion.....
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