the lord of the rings - How large were Saruman's and Sauron's armies?

I was pondering about Saruman's role as an opportunist ally of Sauron, when I realized that I had absolutely no idea how strong his army was compared to Sauron. We know Saruman fielded an impressive war economy in the relatively compact area of Isengard, and Sauron had behind him the might of the entire realm of Mordor, also implied to be heavily geared towards military production. Both have an obvious numerical superiority in the battles they fight.

However, I have no idea at all how large their armies were, overall. It is implied that Saruman's army was mostly gone after Helm's Deep, so I guess Sauron, who manages to keep fighting despite the crushing casualties at Pelennor fields, has by far more soldiers under his command. I'd like to see numbers, though. Is there anything in canon about the total amount of troops Saruman and Sauron could muster?

Answer

As noted by Richard in the comments, there isn't any definitive figure for Sauron's army, but we could guess anywhere upwards of 50,000 due to this bit of text, taken from Richard's link here:

Mordor's troops consisted of some 18,000 Easterlings and Haradrim,[2] several Haradrim war Oliphaunts, and tens of thousands of Orcs.

Again Saruman's army in the Battle of Hornburg is not specified, rather just an army "of great size". However the films mention 10,000 Uruk Hai. Not much is said about what is left of the army after that battle, and Saruman himself flees to Hobbiton with a group of men nearby and set up a thug gang calling himself "Sharkey".

Expanded analysis:

The important thing to note here, which you touched base with in the question, is that Tolkien implies technology, progress and military regime as bad and evil. The more natural Elves and Hobbits are depicted ultimately as good and innocent, and Man falls somewhere inbetween, which I personally feel is very fitting for the post-1900 world.

The idea of a huge "machine" of military is very much the depiction of evil Tolkien witnessed as a soldier in WWI, and his witnessing the industrialization of the world, the falling of trees and forests for "progress," is a direct influence on his writings.

Even though it's fairly obvious that big + aggressive = bad in Tolkien's world, and small + brave and courageous = good, it's still interesting to take it apart and theorize it's roots.