Every hydraulic system — from your car's brakes to a 10,000-ton forging press — relies on a single principle stated by Blaise Pascal in 1653: pressure applied to a confined fluid is transmitted equally in all directions throughout the fluid.

Mathematically: P = F / A, where P is pressure (Pa or psi), F is force (N or lbf), and A is the piston area. Because pressure is constant throughout the fluid, two connected cylinders of different areas give us a force multiplier:

• F₁ / A₁ = F₂ / A₂
• Rearranging: F₂ = F₁ × (A₂ / A₁)

The ratio A₂/A₁ is your mechanical advantage — identical in concept to a lever or gear ratio, but transmitted through fluid instead of solid linkages.

Concrete example: a shop press. You have a hand pump with a piston diameter of 1 inch (area = 0.785 in²) and a ram cylinder with a 6-inch diameter (area = 28.27 in²). You push on the hand pump with 50 lbf. What force does the ram produce?

• Mechanical advantage = 28.27 / 0.785 = 36:1
• Ram force = 50 × 36 = 1,800 lbf (about 0.9 tons from a one-hand push)

The catch — and there's always a catch with mechanical advantage — is displacement. To move the ram 1 inch, you must pump 28.27 in³ of fluid, but each pump stroke only displaces 0.785 in³. That means roughly 36 pump strokes per inch of ram travel. You gain force but lose speed, and energy is conserved (minus friction losses).

Practical considerations engineers care about:

• Fluid incompressibility is what makes this work. Hydraulic oil compresses about 0.5% per 1,000 psi — close enough to incompressible for most applications. Air in the lines ruins this assumption, which is why you bleed brake systems.
• Seal integrity is the most common failure mode. High-pressure hydraulic leaks (3,000+ psi in industrial systems) can penetrate skin — never run your hand over a hose to check for leaks.
• System pressure rating determines your limits. A typical hydraulic shop press runs at 10,000 psi. The tonnage capacity is simply: Tons = PSI × cylinder area (in²) / 2,000.

Rule of thumb: to double the force output of a hydraulic cylinder without changing system pressure, increase the piston diameter by a factor of 1.41 (√2), since force scales with area, not diameter.