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I am studying Schilling-Belove Electronic Circuits, 3rd Ed. where in its second chapter under the "Maximum Symmetrical Swing" section it gives an example where the base current is kept constant and the beta is varied from 200 to 400 which makes the collector current reach the edge of the saturation value almost... and then it says.. "... (hence) the transistor is biased with a constant-emitter rather than a constant-base current".
What I don't understand is how can the emitter current stay constant while we can vary the base current to our will to satisfy design requirements.. It isn't that Ib and Ic can adjust between themselves keeping Ie constant.. Ib and Ic themselves are constrained by beta.. Or is there something hidden in the language I am not getting through..

This is how "constant-base current" circuit will look like:

^{simulate this circuit – Schematic created using CircuitLab}

The base current will be fairly constant as long as $V_CC >> V_BE$.

$$I_B = fracV_CC - V_BER_B1 approx fracV_CCR_B1$$

And due to the fact that $I_C = beta cdot I_B$ and $V_CE = V_CC - I_C cdot R_C1 $.
Any variations in $beta$ bale will have a huge effect on collector current and Vce voltage.

For example, if $ V_CC = 10V$ and $ beta $ changes from $beta = 200 $ to $beta = 400$ will will have:

Case 1 ($beta = 200 $)

$$I_B = frac10V - 0.6V400kOmega approx 25mu A$$ and

$$V_CE = 10V - 200 cdot 25mu A cdot 1kOmega = 5V $$

Everything looks good, the transistor in active mode

Case 2 ($beta = 400 $)

$$I_B = frac10V - 0.6V400kOmega approx 25mu A$$ and

$$V_CE = 10V - 400 cdot 25mu A cdot 1kOmega = 0V $$

In this case, we get $V_CE = 0V $ which is impossible and in fact, the transistor will be in saturation mode. And there will be some small voltage drop across BJT.

More about saturation here:
A question about Vce of an NPN BJT in saturation region

But we can bias the transistor in a different way to get "constant-emitter" current.
In this case, we fixed the emitter current at $I_E = fracV_ER_E$ and any change in $beta$ value will only change the base current $I_B = fracI_Ebeta +1$ because the emitter current will be fixed by the external voltage source and emitter resistance.

See the example:

^{simulate this circuit}

As you can see the emitter current will be $beta$ independent as long as we have an ideal voltage source at the base terminal.

$$I_E = fracV_B - V_BER_E approx frac1V200Omega = 5mA $$

And if the $beta$ changes from 200 to 400 the only thing that will change is the base current from $25mu A$ to $12.5mu A$.

In real life instead of a voltage source, we are using "stiff" voltage divider instead. Which means that the base current only slightly affects the output voltage of the voltage divider. And we can achieve this if we pick the voltage divider current much larget then the maximum base current.

See some examples

BJT Amplifier with Emitter Bypass Capacitor Design

BJT amplifier (Vce) voltage!