They’re both cheap to purchase and can be found in multiple locations.
Effects In Battle
I’ll go right ahead and spoil the surprise: neither item does anything at all in battle. The Ruler apparently “sizes things up” when used by a party member, and has infinite uses. No other effect comes from this though. The Protractor however, is used to “get an angle up on things”. It can also be used infinitely in battle, but also has no effect other than the display text.
Effects In The Overworld
This is where the Ruler really repays that $2 purchase price, as it has one solitary use in Earthbound. It’s a pretty big one too. Not in terms of actually finishing the game, but for players that like to collect every single available item. Where the Ruler redeems itself in the Monkey Cave, found in Dusty Dunes Desert. The gimmick of this place is that monkeys will block each available path. Each one requires a specific item to move out the way, and it just so happens that one Monkey will ask for the Ruler! This monkey blocks the path to a King Banana, so it’s up to you whether you want to carry around a Ruler just for this. Sadly, the Protractor doesn’t reach the same lofty heights of usefulness. This thing just flat out does nothing at all. Well, almost nothing. During the events in Fourside that lead up to the Moonside section of the game, you’ll need to speak to Everdred in the park. There is a kid blocking your path though, and won’t move until you give him something. It really doesn’t matter what you give him, as he will move regardless. If you’re still somehow carrying a Protractor at this point, now is the perfect time to give it away!