But where does the blood go?

I have encountered a number of different things involving amputees lately, and they have caused an old question to press on my mind.

People have things in their body, right?  Things like circulatory systems and lymphatic systems and sundry other things called systems...

A normal, fully limbed person has a network of veins, arteries and capillaries that go (quite literally) from head to toe.  The blood goes around and around these sundry tubes in a somewhat circular fashion - hence the term "circulatory system".

Under normal circumstances, the blood goes from your heart down to your toes (or out to your fingers) and back up to your heart again.

So, when you amputate something, where does the blood go?

I can accept that modern doctors would be able to stitch the ends of your internal tubing together so that the circulating things can still circulate, but microsurgery is a fairly recent development and people have been losing limbs for centuries.

All of those pirates and Civil War soldiers who had their limbs lopped off in the middle of battlefields or on the high sees would not have had someone carefully attaching their plumbing up the right way.  So where did the blood go?  How did they not just bleed out and die?

What strange magic is at work in the body that a continuous system can cease to continue, and somehow that isn't as much of a problem as you might expect it to be?