Not quite, I'm afraid!
The '45' actually refers to the 'depth' of the tyre, expressed as a percentage of the width; so 45 represents a very low profile tyre.
Oh, and I have a feeling that a certain Mr Dunlop had something to do with the early days of tyres too...
I've always assumed that mixed dimensions were used a) to avoid confusion, and b) to make the width sound more impressive: "My X is fitted with 240s!" sounds so much better than "9½s"
However, way back in the early days of my youth, I do have a fant recollection that tyre widths used at one time to be expressed in inches too; presumably they were metricated at some point in order to foster European harmonization...
http://en.wikipedia.org/wiki/Tire_code