Pascal's principle and hydraulics

Squeeze an enclosed liquid anywhere and the pressure increase appears everywhere in it, undiminished. That is Pascal's principle, and it is the whole basis of hydraulics. Since pressure is force divided by area, applying the same pressure to a large piston yields a proportionally larger force: a small effort on a narrow cylinder becomes a big push on a wide one, which is how a car is lifted by a hand pump and stopped by a foot pedal.

The multiplication has to be paid for, and it is paid in distance. Push the small piston down a long way and the large one rises only a little, in exactly the ratio that keeps the volume of liquid constant. Force times distance — work — comes out the same on both sides, minus friction. Nothing is created; the lever principle has simply been rebuilt in fluid.

Two practical points. The ratio that matters is area, not diameter, and area goes as the square of the radius — so doubling a piston's diameter quadruples its force, a factor of four that is easy to miss. And the fluid must be effectively incompressible for any of this to work. A liquid barely changes volume under pressure and passes the push straight along; a gas compresses instead, swallowing the movement and never delivering it. That is exactly what a bubble of air in a brake line does, and why brake systems are bled until no gas remains.