Level-triggered flip-flops in shift register

There are flip-flops and there are flip-flops.

The RTL (resistor-transistor logic) schematic you show is a simple bistable multivibrator that is either set or reset by pulses on the E1 and E2 inputs. For exmaple, pulsing E1 high will cause A1 to go low and A2 to go high.

"Elements of Computing Systems" is talking about a different kind of flip-flop: the master-slave edge-triggered flip-flop. Rather than being driven by pulses, this kind of flip-flop reacts to the rising edge of a (typically) square-wave clock signal. The output of the flip-flop immediately after such a clock edge matches the input right before that same clock edge. This is where the t and t-1 notation comes from.

In its easiest to understand form, the D-type master-slave flip-flop consists of eight NAND gates (or eight NOR gates in RTL) and two inverters. As you might guess, this gets cumbersome to draw as a schematic using resistors and transistors. It's much easier to draw the schematic for one gate, and then use a symbol to represent that logical function in higher-order structures.

However, in the days when computers were really built using discrete transistors, master-slave logic was relatively rare. Instead, multi-phase clocks were generated so that the simpler pulse-driven flip-flops could be used, keeping the overall circuit complexity down.

One reason we clock flip flops so that there isn't any chaos when the outputs of flip flops are fed through some logic functions and back to their own inputs.

If a flip-flop's output is used to calculate its input, it behooves us to have orderly behavior: to prevent the flip-flop's state from changing until the output (and hence the input) is stable.

This clocking allows us to build computers, which are state machines: they have a current state, and calculate their next state based on the current state and some inputs.

For example, suppose we want to build a machine which "computes" an incrementing 4 bit count from 0000 to 1111, and then wraps around to 0000 and keeps going. We can do this by using a 4 bit register (which is a bank of four D flip-flops). The output of the register is put through a combinatorial logic function which adds 1 (a four bit adder) to produce the incremented value. This value is then simply fed back to the register. Now, whenever the clock edge arrives, the register will accept the new value which is one plus its previous value. We have an orderly, predictable behavior which steps through the binary numbers without any glitch.

Clocking behaviors are useful in other situations too. Sometimes a circuit has many inputs, which do not stabilize at the same time. If the output is instantaneously produced from the inputs, then it will be chaotic until the inputs stabilize. If we do not want the other circuits which depend on the output to see the chaos, we make the circuit clocked. We allow a generous amount of time for the inputs to settle and then we indicate to the circuit to accept the values.

Clocking is also inherently part of the semantics of some kinds of flip flops. A D flip flop cannot be defined without a clock input. Without a clock input, it will either ignore its D input (useless!), or simply copy the input at all times (not a flip-flop!) An RS flip-flop doesn't have a clock, but it uses two inputs to control the state which allows the inputs to be "self clocking": i.e. to be the inputs, as well as the triggers for the state change. All flip flops need some combination of inputs which programs their state, and some combination of inputs lets them maintain their state. If all combinations of inputs trigger programming, or if all combinations of inputs are ignored (state is maintained), that is not useful. Now what is a clock? A clock is a special, dedicated input which distinguishes whether the other inputs are ignored, or whether they program the device. It is useful to have this as a separate input, rather than for it to be encoded among multiple inputs.