Armaments

A broad term for all the equipment used by a military force. It's not just the "arms"—hand-held weapons—that we normally think of, but larger things like tanks, submarines, ships, etc.

The European nations

The participants in World War I, who were figuring out how to pay each other back for World War I—primarily France, the United Kingdom, Belgium, and Germany.

The great conflict

World War I. A.k.a. the Great War. A.k.a. that other world war without the Nazis.

The Great War

Remember, World War I wouldn't be regularly called that in 1925, because there hadn't been a World War II yet. You don't attach a "I" until the sequel comes out.

Larceny

Robbery, thievery…stealing stuff. Feel free to pick a synonym.

The Monroe Doctrine

You can read all about the Monroe Doctrine here, but basically in 1823 President James Monroe said that the U.S. would not get involved in other countries' (or their colonies') affairs, but also they won't let other countries interfere in the Americas any more. Messing with South or Central America would mean messing with the U.S.

The Old World

This is a reference to Europe, which is traditionally referred to as "the Old World" because when European nations established colonies in the Americas, starting in the late 1400s, these continents were commonly called "the New World."

The one hundred and fiftieth year

This is Coolidge's fancy way of referring to 1776, the year America declared its independence.

Parchment

The term is based on old-fashioned paper, which was made from animal skin before people figured out how to make plant-based paper. It's become a sort of classy synonym for paper.

Permanent Court of International Justice

Before we had the United Nations, we had the League of Nations, established as one of Woodrow Wilson's great projects after World War I. Despite Wilson's involvement, the U.S. never joined the League of Nations (isolationism ftw). However, it did participate in the Permanent Court of International Justice, which was the predecessor to today's International Court of Justice, commonly referred to as the Hague. Because it's based in the Hague, not just because it sounds cool.

Public ownership of railroads and electric utilities

Coolidge is probably referring to one of the planks in the platform of the Progressive Party candidate Robert La Follette in the 1924 election, which was pushing for (surprise surprise) public ownership of railroads and certain utilities (source). La Follette didn't do so well in that election though; the Progressive Era was on its way out.

Sophistries

Ever hear someone make a really convincing argument only to realize that it's based on wrong information? Or just straight-up lies? That's sophistry—false or misleading arguments and reasoning.

Subterfuges

You'll see this term a lot in spy novels or mystery novels. It involves using some kind of trick or disguise to escape a situation, or to avoid getting in trouble for something.

The Washington Conference

The Washington Naval Conference was held between 1921 and 1922. It consisted of nine countries getting together, led by the U.S., to talk about how everyone can calm down and reduce the size of their navies now that all that war business was over. It was an attempt to demilitarize the world and keep the peace after the war. A bunch of treaties came out of it between different groups of the nine countries.