I will state, despite my respect of
@Susquatch 's opinion, that a tool height gage is a waste of time, except as a project to familiarize yourself with operating the lathe (like there are not thousands to choose from!).
I have dealt with too many circumstances where the height of the tool was required to be adjusted a little up or down, to fixate myself upon that the tool "Must Be Perfectly On Center". In most cases, it can be pretty far off perfect, and will still work very well!
When you need to, or at least can, face across the end, you can quickly adjust up or down to get that perfect pip-free center. But if there is already a hole in there, the, it's not all that important!
When you cannot easily line up on a tailstock center, for example, (I used to deal with parts that needed you to use a crane to sling them into the chuck, so pulling the tailstock out to set height was, well, dumb), simply pinching a piece of flat stock gently between the work and the tool, and seeing how close to vertical that piece was supported, was great! Try that. Pinch a ruler, or a slip of shim, or some flat stock of almost any flavour, between the work and the tool tip. If it is really close to vertical, you are really close to being exactly on center. And the math (not my strongest field!) shows that the errors that may be produced, are pretty insignificant, well within most folks ability to judge by eye, if that slip of material is actually straight vertical!
As credited to Voltaire, don't let "Perfect" be the enemy of "Good"!
As an aside, sometimes you just find that a particular tool gives better results when it is higher or lower than it seems it should be. Go with the results, rather than the theory! Always!