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What's the name of this circuit?

Hi,
This isn't strictly homework, i'm too old for school and haven't done a college course in years. It fits best here as it's beginner level stuff (it's self supported study).

I booked out a library book some years ago and found a circuit which is an improvement on this basic 1 transistor circuit. the LED has a harder leading edge and subsequently remains off for more of the time that variable resistor VR1 is decreased
 

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As the variable resistor resistance is decreased then the base current in the transistor will increase until the variable resistor and/or the base-emitter junction of the transistor will burn out.

There is no such thing as "an LED having a harder leading edge". When the transistor's base current is low then its collector current and the LED current will also be low therefore the LED will glow dimly. As the base current is increased by the variable resistor then the base current, collector current, LED current and LED brightness will increase.
 
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