Further to the other current thread, I have a different 'don't understand' issue for time dilation. I've thought it over and I might have a slight grasp of 'the answer', but it is still a bit murky for me: Two space ships of the future pass each other in opposite directions, each travelling at 0.5c relative to some arbitrary point. Each one observes the other to be a space ship going 0.7c away from them [if my presumption that relativistic speeds add up in quadrature?]. Both therefore see the clock of the other ship running slower than their own. How can both on-board clocks be running slower than each other?