Last year, British historian Simon Elliott published a book titled Alexander the Great versus Julius Caesar: Who was the Greatest Commander in the Ancient World?

I haven't read the book, but in my opinion the question is very easily settled. Julius Caesar was vastly superior. He won more battles, fought against more varied armies (Pontus, Gauls, Germanic tribes, Egyptians, other Roman legions), and was more often outnumbered.
One of the ultimate test of leadership and military genius is when you fight people with the exact same training and equipment - in this case other Roman legions. Caesar did it several times during the Civil War.
He even defeated Pompey the Great, one of the greatest Roman generals (obviously after Caesar himself) at the Battle of Pharsalus (48 BCE) where Pompey's army outnumbered Caesar's 2 to 1. Caesar not only won, but suffered about 30 to 60 times less casualties and losses than Pompey. According to Wikipedia, Caesar lost 200–1,200 soldiers while Pompey's army had 6,000–15,000 killed and 24,000 captured.
Julius Caesar then went on to defeat the leaders of the Optimates party at the Battle of Thapsus (46 BCE). The Optimates had about 20,000 more legionaries and 10,000 more cavalry, yet Caesar won with 10x less losses. The next year he crushed Pompey's sons and allies at the Battle of Munda (45 BCE), where Caesar had 40,000 troops (8 legions) against Pompey's 70,000 (13 legions). Once again it was an easy victory for Caesar, who lost 1000 men against 30,000 for Pompey's sons.
Alexander the Great only defeated one great enemy: the Persian Empire. It may seem very impressive, but the Achaemenids were already on the wane and had lost the two Greco-Persian Wars due to the superiority in Greek military equipment (hoplites, phalanges, ships), training (more professional army) and strategy. Alexander's father had unified Macedonia and Greece for the first time and his untimely demised didn't leave him time to invade Persia himself, which left the task to Alexander. But whoever was in charge among the Greco-Macedonians would probably have defeated the Persians at that time.
Alexander's greatest victories were the Battle of Issus (333 BCE) and the Battle of Gaugamela (331 BCE). According to modern estimates Alexander had 37,000 troops at Issus, mostly heavy cavalry and heavy infantry, and fought against 50,000 to 60,000 Persians, who were mostly light infantry (+ 10,000 Greek mercenaries whose loyalty to the Persians was dubious at best). Almost 2 to 1, but well-trained heavy hoplites and heavy cavalry can easily defeat a bigger army of light infantry. Similar situation at Gaugamela, where 47,000 heavily armed Macedonians were against 50,000–120,000 lightly armed Persians. Easy victory, as it should be.
Alexander feared the Thracians and the Celts to the north of Macedonia and made sure to sign non-agression pacts with them before embarking on his campaign against Persia. The Celts actually defeated the Greco-Macedonians in 279 BCE only a few decades after Alexander's death (he would have been 76 years old had he lived longer), sacked Delphi and went on to establish an independent kingdom in Galatia (central Anatolia) after defeating the Hellenistic Seleucid Empire.
The Romans also waged several wars against the Macedonians and Seleucids and always won, showing that Roman legions were superior to Hellenistic armies.
Add to that that Julius Caesar also singlehandedly defeated all the Gallic tribes, while only an offshoot of Gauls (the future Galatians) had defeated the Macedonians and Seleucids. In fact Caesar's greatest victory is assuredly the Battle of Alesia, where he vanquished a grand coalition of some 328,000 to 470,000 Gauls on two fronts with only about 60,000 men (so outnumbered about 7 to 1).
I haven't read the book, but in my opinion the question is very easily settled. Julius Caesar was vastly superior. He won more battles, fought against more varied armies (Pontus, Gauls, Germanic tribes, Egyptians, other Roman legions), and was more often outnumbered.
One of the ultimate test of leadership and military genius is when you fight people with the exact same training and equipment - in this case other Roman legions. Caesar did it several times during the Civil War.
He even defeated Pompey the Great, one of the greatest Roman generals (obviously after Caesar himself) at the Battle of Pharsalus (48 BCE) where Pompey's army outnumbered Caesar's 2 to 1. Caesar not only won, but suffered about 30 to 60 times less casualties and losses than Pompey. According to Wikipedia, Caesar lost 200–1,200 soldiers while Pompey's army had 6,000–15,000 killed and 24,000 captured.
Julius Caesar then went on to defeat the leaders of the Optimates party at the Battle of Thapsus (46 BCE). The Optimates had about 20,000 more legionaries and 10,000 more cavalry, yet Caesar won with 10x less losses. The next year he crushed Pompey's sons and allies at the Battle of Munda (45 BCE), where Caesar had 40,000 troops (8 legions) against Pompey's 70,000 (13 legions). Once again it was an easy victory for Caesar, who lost 1000 men against 30,000 for Pompey's sons.
Alexander the Great only defeated one great enemy: the Persian Empire. It may seem very impressive, but the Achaemenids were already on the wane and had lost the two Greco-Persian Wars due to the superiority in Greek military equipment (hoplites, phalanges, ships), training (more professional army) and strategy. Alexander's father had unified Macedonia and Greece for the first time and his untimely demised didn't leave him time to invade Persia himself, which left the task to Alexander. But whoever was in charge among the Greco-Macedonians would probably have defeated the Persians at that time.
Alexander's greatest victories were the Battle of Issus (333 BCE) and the Battle of Gaugamela (331 BCE). According to modern estimates Alexander had 37,000 troops at Issus, mostly heavy cavalry and heavy infantry, and fought against 50,000 to 60,000 Persians, who were mostly light infantry (+ 10,000 Greek mercenaries whose loyalty to the Persians was dubious at best). Almost 2 to 1, but well-trained heavy hoplites and heavy cavalry can easily defeat a bigger army of light infantry. Similar situation at Gaugamela, where 47,000 heavily armed Macedonians were against 50,000–120,000 lightly armed Persians. Easy victory, as it should be.
Alexander feared the Thracians and the Celts to the north of Macedonia and made sure to sign non-agression pacts with them before embarking on his campaign against Persia. The Celts actually defeated the Greco-Macedonians in 279 BCE only a few decades after Alexander's death (he would have been 76 years old had he lived longer), sacked Delphi and went on to establish an independent kingdom in Galatia (central Anatolia) after defeating the Hellenistic Seleucid Empire.
The Romans also waged several wars against the Macedonians and Seleucids and always won, showing that Roman legions were superior to Hellenistic armies.
Add to that that Julius Caesar also singlehandedly defeated all the Gallic tribes, while only an offshoot of Gauls (the future Galatians) had defeated the Macedonians and Seleucids. In fact Caesar's greatest victory is assuredly the Battle of Alesia, where he vanquished a grand coalition of some 328,000 to 470,000 Gauls on two fronts with only about 60,000 men (so outnumbered about 7 to 1).
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