My 11-year-old so has read The Hobbit and has recently started FOTR and the questions are rolling in. He loves the military aspects of Middle-earth and asked me who was stronger militarily at their peak, Gondor or Arnor, and I could not give him an answer because I don't know. Did Tolkien give us the answer? or at least some hints?

That is an excellent question! For my answer I went to the book "Atlas of Middle-Earth by Karen Wynn Fonstad and her pages on the Kingdoms of the Dunedain, Arnor and Gondor. The author in my opinion has done an excellent job of using the books to compile maps to complement the ones done by Tolkien.

In the pages regarding Arnor and Gondor she references that at it's peak, Arnor sphere of influence totaled about 248,540 square miles while Gondor area under it's direct rule was 716,425 square miles. Gondor also had navel and cavalry forces that they used to come to Arnor's aid when Angmar and the Witch King attacked them.

I highly recommend the book, not only it shows the paths the different members of the Fellowship (and earlier Bilbo) but also has whole sections depicting battles of the three ages.

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"Even darkness must pass. A new day will come. And when the sun shines it will shine out the clearer."

I did not realize the size difference. I knew Gondor came to the aid of Arnor but it could be they faced a greater threat. But I think you are correct Gondor was likely stronger with navy/cal. Thanks for the info.