Yes I can do that Bob, but it might be a bit more than you were asking about since its not exactly as simple as it might seem. I'm also trying hard not to throw Krypto's thread off topic. And I didn't add what I did as any criticism towards yourself Krypto, I think you did a very fine job. I was simply mentioning it as additional information for those that might not know such as Bob asking about it.
So fwiw, properly aligned lathes are designed so they face with a slight inward bias, or concave. It's a minimal amount, but still very important. And its quite easy to see on a finely finished faced surface. If a light is shining on that faced surface from the right direction, you'll see two tapered or triangular shapes of reflected light at 180 degrees to each other running from the parts center line and out to it's O.D. getting wider the more it progresses towards that O.D. Ok, so why is that type of alignment desirable or required? That way when two faced parts are joined together they sit flat and don't rock. The lathe will also wear in use towards facing flat and not immediately towards facing convex or where those parts would rock against each other. On the better lathes, the head stocks are aligned to point very slightly upwards as compensation for work piece weight. Depending on the manufacturer, it may also point towards the operator for that concave facing, or its done on the fixed dovetail of the cross slide.
The tail stocks on good lathes are also set up similar, it will or should point uphill again as compensation for work piece weight, and towards the operator as a slight compensation of those cutting forces. As I said, those minor misalignment's aren't very much, roughly a couple of 10ths over a 12" test length. With decent metrology equipment, it can be measured and verified well enough.
Proper machine alignments and some of there subtle misalignment's are far from being as intuitive as some seem to think. You have to visualize at the macro level to understand the concepts of and exactly the reasons behind why its being done in that way. For example, lathe leveling or tramming a mill head is just about the last step that would be done before verifying quite a few other items first on any new or used machine and before those would be done. Yet very seldom are those additional checks mentioned on these forums. As another example, how many posts even mention checking a mill table for just how square the X,Y axis travels are to each other? Most seem to just assume those would already be dead square to each other. From my limited checks on so far 3 different sized, brands, country of origin and quite a large difference in costs, I can assure you that most certainly aren't. And if you don't check, you have no idea what any machine is even capable of as far as accurate coordinate machining. Or hope to make any compensation if you don't know how much might be required. Fortunately most of the necessary checks only need to be done once and then a long time later just to access the amount of wear that's taken place. And after those initial checks, then the usual leveling or mill head tramming is fine. They also don't take that long to do once you've done so a few times.
And yes I well understand that for most parts, a very high degree of accuracy isn't that important in most home shops. Until it is, to use a model engine as another example, the overall part fit and/or surface finish is more important than hitting any exact size. But because of the scale effect, model engines develop little horse power. Exterior combustion models run on air even less just to rotate themselves. So the overall engines parts and the exact squareness they have to each other is in fact pretty important. The same is true for individual part location. Any binding in model engines of any type will affect how they operate and perform. So the general point on these forums about high accuracy isn't a requirement is in reality only partially true. And again your certainly correct Krypto, I have a collection of Model Engineer magazines dating back to 1998, many engines were certainly built using not much more than a steel scale for reference, inside and outside calipers that on there own can't display any measurement at all. Experience and hand fitting parts to each other got the job done.
In the 1930's Georg Schlesinger literally wrote the initial book about machine tool alignments. Multiple editions have be updated and printed since. In one way or another, all the better machine tool manufacturer's still base there machine tool certificate of accuracy either directly or slightly modified for the machine type on his work. A PDF of his book can be found here.
Testing Machine Tools (Dr.Schlesinger) - Free Download PDF Just remember, the minimum and maximum numbers he lists would be for high quality industrial machines. But they at least give you the proper test methods and some ball park numbers for any that might want to double check or access what they already have. Schlesinger's test methods aren't the only way any of these tests can be done, there's also multiple other ways various others have come up with. Some maybe better or worse depending on your perspective.
For a very good visualization of possible inaccuracy in any slide equipped machine tool, you need to properly understand the industry concept of the possible multiple axii of where and how that inaccuracy can be present. There's 6 possible degrees of inaccuracy that may be present in just one axis or even various amounts and combinations in multiple axii on any one single slide. The hard copy book is quite expensive from Moore Tools, but a PDF copy is here.
https://ia800104.us.archive.org/20/...curacy/Foundations_of_Mechanical_Accuracy.pdf Yes Moore delves into the millionths of an inch accuracy levels that are multiple times outside what most with a hobby level shop would ever need. But the information and concepts within it are just as useful for us as basic information. Plus you can't beat free.
I don't know of any online copy of the Connelly book Machine Tool Reconditioning, and it's again fairly expensive. (roughly $100)
Machine Tool Reconditioning for Machine Tools by Machine Tool Publications From my perspective, I think a great deal of it was taken from Schlesinger's book and expanded upon. But it's something that also helps to properly visualize the importance of the 3 dimensional alignments that are non optional. Indirectly all three sources of information I've given form the basics of 3 dimensional coordinate machining that are still in use today for both manual and cnc.
There is one I think very good Youtube channel that gets into machine tool alignments, rebuilding, improvements and modifications for any that might be interested.
https://www.youtube.com/@jansverrehaugjord9934 Watching how its done also helps to understand the written information a whole lot better.
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