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Hi,
As others have said, you can measure the voltage BETWEEN any two nodes not just across one resistor at a time. You can actually measure the voltage across each resistor and use math to calculate the voltage across nodes 1 and 2 in your diagram, but that would not be as accurate and that is the theory from which the null approach stems from.
To begin, consider the lone voltages at node 1 and node 2. If we measure these two voltages we may read 1.2v at node 1 and 1.1v at node 2. The difference is what we want to measure, and that would be 1.2-1.1=0.1 volts. That's rather easy to figure out. In most cases though we are looking for zero volts there. We will probably adjust something such that both voltages are the same like 1.2v and 1.2v. When we do that, we have achieved a null and that would be the end of the measurement. The thing that we adjusted would show us what we really wanted.
The reason this is more accurate over more direct measurements of each node voltage is because when we measure each node voltage individually we may measure 1.2v and 1.2v but those measurements will come with a tolerance. If the tolerance was just 2 percent then we might measure 1.224v on node 1 and 1.176v on node 2. This would mean we would not recognize that we obtained the desired null and so we would go on to adjust something, like one of the resistors. By the time we did obtain the null with this meter tolerance, we would have adjusted to a false null which would mean the adjusted resistor would be the wrong value.
Now if we measure the voltages across node 1 and node 2 and we use a null type meter movement, the tolerance in the measurement could be incredibly small. Even if this meter was also 2 percent off, it would still have to indicate the same measurement as it did when there was no signal. That's because to get a null the meter pointer would have to return to the same physical location as it did when there was no signal which is also 0.000 volts. It works almost like a balance scale where the scale has to return to the same position as when it had no weights on it at all. With a sharp eye, we could get down to a very tiny tolerance which would be hard to match with any direct individual resistor voltage measurements. When we get down this low the tolerance starts to depend mostly on the needle bearing tolerance and sticking friction which is often incredible small. I would say we could easily get down to a tolerance of 0.001 percent or lower, and this would mean the other construction details of the instrument would dominate the tolerance of the overall measurement (of say an unknown resistor value).
Just one more little detail and that is often the null meter movement is a very low current meter with a null position in the center. It measures very, very small currents like maybe 1 microampere. With no signals, it measures 0 microamperes, and with signals that cause a null situation, it again measures 0 microamperes.
With the simplest resistance bridge, when we adjust the adjustable resistor to the same value as the unknown resistor, we obtain a null. That is how we measure an unknown resistance with a resistance bridge that has a null indicator.
Historical note:
Some of the old bridges used a small CRT tube instead of a meter to measure the null. It was sometimes called the "eye" tube or "magic eye" tube because it had a phosphor coated end inside that looked like a green eye.
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