When you push on the plunger of the movable syringe, the arm rises; when you pull on the plunger, the arm descends.
Pushing on the plunger applies pressure on the water in the movable syringe. Since the water is confined and incompressible, Pascal’s principle comes into play, telling us that the pressure is transmitted undiminished to all parts of the water and to the walls of its container. Since the plunger of the fixed syringe at the other end of the tube forms part of the “container” for the water, and is the only part of the container that can expand, the pressure causes the plunger in the fixed syringe to move.
Pascal’s principle and a little mathematics can be used to show that—if the syringes are identical—the force you apply to one plunger is transmitted in full to the other plunger (see Math Root, below). Additionally, as you can observe, each plunger moves the same distance.
With the 1-mL syringe, you need to push with less force than with the 10-mL syringe, but the arm is not lifted nearly as far. In accordance with Pascal’s principle, the pressure on the plunger of the 10-mL syringe is the same as the pressure on the plunger of the 1-mL syringe. However, since the area of the 10-mL plunger is far larger than the area of the 1-mL plunger, the force exerted on the 10-mL plunger is far larger than the force you push with (remember, F = pA). The good news is that you have obtained a force advantage, but the bad news is that you’re paying for it with a distance penalty. Mechanical work is the product of force times the distance the force moves through (W = Fd), and this product remains constant.
In the right-side-up position, the plunger pushes on the arm to raise it. But when you turn the whole assembly upside down, the syringe pulls on the arm to raise it, just like your muscles do with your own arms. The muscle that allows your forearm to lift things, called the biceps, is attached near your shoulder and just below your elbow. When the biceps contracts, it has the same effect on your arm as the syringe has on the hydraulic arm when the assembly is upside down. In both cases, a large force is exerted so that a small weight can be lifted, but the weight can be lifted a large distance compared to the distance the force moves (the distance the syringe plunger moves, or the distance your muscles contract).