What is the logic behind this ?
As the larger instruments have larger gaps between the strings, I would have expected a larger bow to accomodate more easily continuous bowing while changing strings.
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It actually has to do with the physics of sound production for the bowed string instruments.
The sound is produced on the viol family of instruments by the string "slipping across" the bow. That is, the bow catches (by friction) the string, displaces it a certain distance, until the restorative force from the tension in the string overcomes the friction and snaps back to original position (here using the common fact that static friction is stronger than dynamic friction, so once the string starts moving, it will essentially return to starting position before being caught by the bow again).
Now, the pitch produced by the string is given by the frequency of the string vibration. For higher pitched instruments the frequency is higher: the string must vibrate faster. To build up resonance, you must pull the bow at the same speed as the vibration speed of the string (else you may set up destructive interference making the bowing ineffective). To attain this faster vibration, the bow needs to displace the string at a higher rate.
Therefore the typical bowing speed is higher for higher pitched string instruments.
Because you bow faster on a violin, to hold the note for the same length of time without returning the bow requires a longer bow than on, say, the cello. This is why the higher pitched string instruments have longer bows. (Note that the viola bow is only marginally shorter than the violin one; the two instruments have three strings in common.)
Beside Willie Wong's nice answer, a double-bass player needs more pressure on the bow than a violin player. The longer the distance between your hand and the tip of the bow, the greater the force your wrist would need to apply.
In other words, playing the double-bass with a bow as long as a violin's may require too much wrist strength for playing with the tip of the bow.
Another aspect: the bow is also wider for larger instruments (more strength is needed to get the thicker strings to vibrate). A wider bow would easily get too heavy if it had the same length as the narrower ones.
It has more to do with the physical position of the arm and the instrument. A violin perched on a shoulder allows a longer bow stroke than a cello in front of the player's sternum. A cello player simply doesn't have a long enough arm to play quick notes on the A string with the tip of a 27.5" bow.