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In the history of Rome there are numerous instances of legions being marched up and down Europe, from Spain to Rome and from Rome to Syria. I'm interested to know how quickly these armies could move? Did it take a half a year to move from the Alps to Rome or did it take a week?

Also how does that speed compare to the pace of the mobile cavalry reserve used to such great effect by the late roman emperors?

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The practical maximum march rate is what the Germans call the "10 kilo(meter) tempo (6mph). –  Tom Au Apr 3 '13 at 14:22
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This is not a direct answer to your question, but take a look at the ORBIS project, which calculates travel times and costs in the ancient Roman world: orbis.stanford.edu –  Robert Petermeier Apr 3 '13 at 16:32

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up vote 10 down vote accepted

This (fun) site assumes 10 miles per day, while adding the necessary disclaimer "it depends".

However, I was not able to confirm this number in the cited source: John Pebbie's The Roman War Machine seems to refer to "10 miles" only in specific relation to a march undertaken by Caesar's army on its final approach to the Battle of Sabis.

The same book contains several concrete numbers (including references to primary sources) for speeds perhaps achievable by the Roman postal service and by individuals traveling on important missions. Here is an example:

Plutarch relates that Julius Caesar on one occasion travelled 100 miles a day for eight days in succession, driving in a hired raeda.

It relates that cavalry ("widely deployed in a protective screen around the army on the march, and penetrating deeper into the surrounding countryside") could have covered 40 miles a day. It also quotes Vegetius (5th century CE) on the practice of training marches with complete armor over distances of ten miles from a camp plus return:

Decem milia passuum armati instructique omnibus telis pedites militari gradu ire ac redire iubebantur in castra ...

And FYI, here is a later account (again from Andrew Wheatcroft's The Enemy at the Gate: Habsburgs, Ottomans, and the Battle for Europe) that quantifies daily distances traveled by an Ottoman army in the 17th century:

The whole force could advance only at the pace of ox carts and the cannon, perhaps twelve miles a day.

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+1 I was worried you wouldn't mention ox carts. –  Nathan Cooper Apr 3 '13 at 16:51

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