Thanks for your question. As you have noted, the history of the Computer can be divided cleanly into four key periods. These are as followed.
1. Manual Calculation.
This was spearheaded by Egyptian civilisation circa 500 BC. They were said to use these calculations to analyse patterns in Egypt, including the rise and fall of the Nile, the ebb and flow of buildings and monuments and other such analysis. During this time, the 'abacus' was invented to expedite the process. This is a contraption that involves beads strung on a wire.
2. Mechanical calculation
This began in France by mathematician Blaise Pascal circa 1642. This invention was a mechanical calculation, which was purpose build as a means of adding and subtracting.
3. Electromechanical calculation
German born Von Leibnitz 1679 was the first to invent a calculator which was capable of doing four basic operations.
4. The Analytical Engine.
English Mathematician Charles Babbage invented in 1864 the analytical engine. This was capable of solving only simple problems, but it was a great advancement for its time.
Hopefully this helps!