Re: 1959 Johnson 35 HP
There are some wiring diagrams floating around here, but I can't find one at the moment. Someone will come along soon and ask to to send them a private message and they'll email one to you.
In the time being, have you checked the big wire on the starter to see if you're getting 12 V to it when you turn the key?
Check the pins and sockets on the big plug on the side of your motor. It's very, very common for them to make poor contact and that will stop things. Use a spray on contact cleaner and small brushes and pipe cleaners to clean those up thoroughly.
Also, you'll want to remove and clean every connection on the engine, the box holding the solenoid, and the battery. Shine 'em up good. Even if they look OK from the outside, remove each one and clean it well.
If still no luck...wait until you have a wiring diagram. It is very easy to ruin the coils in your magneto if 12 v goes to them, so don't just mess around with the wiring without knowing what you're up to.