His principle, discovered while taking a bath and thinking about the correlation between the rise of water as he lowered himself into it and the increasing buoyant force: Buoyant force on an object = weight of fluid displaced by the object.  If you do not need to know why, you could skip the next two paragraphs.

Fluids (gases and liquids) have weight, and thus in a motionless fluid, any portion of it must be held up by an upward force equal to its weight. If you think about this a little, you will conclude that pressure increases with depth in such a way as to accomplish this. If the pressure on the surface of a liquid is P_o , it is found that the pressure at depth h is given by P = P_o + rgh, where r is density (mass/volume) of the liquid and g is the gravitational acceleration. Note that in change of depth Dh, pressure changes by D P = rgDh. This is the equation to calculate the change of pressure which you experience when you go up the elevator in a tall building or when you dive down in a lake. Down is the positive direction for h and Dh, so going up in the elevator is a negative, and pressure is dropping.

Let us check on that DP and see if it is the right stuff to hold up the fluid. Consider a portion of area A on top and bottom parallel to the surface and vertical height Dh. So its volume is ADh, its mass is rADh, and its weight is rgADh. The upward force on it due to the pressure difference is A times DP, or ArgDh, the same as the weight, so force up = force down.

Any shape portion of volume V of the fluid must be held up with force rgV, according to the above discussion, so an object of volume V must experience that much force also. V is the volume displaced by the object, so we say that the buoyant force is equal to the weight of the fluid displaced. A 10 ton boat displaces 10 tons of water. Note: r is mass density.  If you know weight density, just multiply it by the volume displaced to get the buoyant force.

A rock sinks because the water displaced by the rock weighs less than the rock. If you tie a string around a rock and lower it into water, there are two upward forces, the tension and the buoyant force; there is one downward force, the weight. No acceleration means force up = force down, or T + F_B = W.

Your volume is probably 0.05 to 0.10 m³ or so, so you displace that volume of air. Let us use the larger volume, 0.10 m³.  Air usually has density about 1.2 kg/m³, so the displacement is 0.12 kg of air.  A kg weighs about 2.2 lb, so the air's buoyant force on you is about a quarter of a pound (0.264 lb in this example). A helium-filled  balloon typically weighs less than the buoyant force, so if you want to keep it down, a downward force is needed.

Here is some more about buoyant force.

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