What is the mathematical constant Apéry's constant?

Because it is a mathematical concept and because it does not change - whether it is defined as the ratio of the circumference to diameter of a circle or as the limiting value of a large number of...

Pi is a mathematical constant because it is the same for every perfect Circle. Pi = Circumference/Diameter of any Circle. It is constant because no matter the size of the circle that ratio always...

Jacob Bernoulli. See the Web Links to the left for more information.

weight is not constant because it is dependent to the gravitational pull on the referrence...?

No. Velocity includes a direction vector, which speed does not have.