external hard drive not showin on iMac desktop

I got a dialog box from Time Machine today saying that it has not backed up in 10 days. I noticed that my LaCie external hard drive is not showing up on the desktop. It's connected, light is on, and I have checked the cables and connections and all are good. I cannot turn on Time Machine. It's asking me to browse other backup disks. When I click on that option, a box comes up with no disks available to use with Time Machine. I have no idea what is going on, because it was working fine up to 10 days ago. I know the external disk is probably full, or almost full, because I was getting messages that Time Machine was adding new backups and deleting the oldest backups. Worked fine like that for months. All of a sudden, there's no backups.

Double check:

- Finder
- Preferences
- then check all the devices (icons) you want to see on the desktop

* Nick

Thanks, I did that first. All are checked.