More men were drawn to the Roman army than to Alexander's army. The Alexandrian army was made up of Greek and Macedonian warriors. Alexander's troops fought for Greece and dignity. The Greek warriors' dignity provided them an advantage in those important fights. The Greeks had been fighting the Persians for so long that they had developed a loathing for them. Some of Alexander's warriors were not as powerful because, unlike the Romans, he had to seek support wherever he could find it. Alexander had no primary spot to which he returned after fights.
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They came to escape religious persecution. :)