What are they

[Canus]

OK, I think I know but would appreciate my suspicions being confirmed.

 

[dgehricke]

my guess would be a Steady rest and a tailstock.mounting is a little strange

 

[mikey]

Steady rest and follow rest.

 

[Mitch Alsup]

Steady rest and follow rest.

Yes, exactly!

 

[brino]

.....and the follow rest has sacrificial wooden pads.
-brino

 

[Canus]

My suspicions are confirmed. however I'm quite sure these are not for a lathe but are for a cylindrical grinder of some make.

 

[RobertB]

The second one looks like a crankshaft grinder steady rest

 

[benmychree]

They are a steady and backrest for a cylindrical grinder, do not recognize the make.

 

[Canus]

These are of no use to me so if these are of use to anyone I'll let them go cheap.

 

[Suzuki4evr]

The second one looks like a crankshaft grinder steady rest

It does look like a crankshaft grinder steady,but not quite. I have ground so much cranks in my automotive years,it poured out of my ears. For it to be a crankshaft steady,it would need a second set screw knob and it doesn't seem to have one,but I could be wrong. If I could have inspected it from closeup,l could have said for sure, but good eye Robert.