Before the days of blinkenlights and fancy vacuum tubes, humans still found creative ways to compute.
This list is in the range fingers/mental < ancientcomputing < eniac, so fingers, pure brainpower, and charts aren't included, nor are any electronic computers past tabulators and the bombe. i.e. no NewMath Computers.
The dictionary definition of computing (i.e. http://dictionary.reference.com/search?q=computing) has "To determine an amount or number.", which would include measuring devices, so maybe our definition of computing is limited by assuming algorithms. "Computing" might very well include sundials and thermometers. Simple devices/computers have their own category.
"The Egyptians improved upon the sundial with a merkhet, the oldest known astronomical tool. It was developed around 600 B.C. and uses a string with a weight on the end to accurately measure a straight vertical line (much like a carpenter uses a plumb bob today). A pair of merkhets were used to establish a North-South line by lining them up with the Pole Star. This allowed for the measurement of nighttime hours as it measured when certain stars crossed a marked meridian on the sundial."
But note that, due to precession, the pole star is not fixed:
"For the ancient Egyptians, the Pole Star was not Polaris, but Thuban in Draco, while observers in the far future will see yet other stars at the Northern Celestial Pole"