learning his lines

yup. I'm pretty sure that's true. Taken, I believe, from the concept that school children would have to learn the lines written on pieces of horn:




I hope that helps!

In googlebooks 1760 to 1900 here, here, here, here. here, here, and here.

My impression is the phrase is used very literally to mean lines in a book that must be memorized. Lines of a poem or play, for instance. In one case, lines of musical notation. In one case lines on a map. Here.

None of the 40-odd examples had to do with mathematics.

Ah!

Job beat me to it. I was puzzled by the mathematical connection but couldn't find my sources.

As far as I can check the expression has always referred to actually lines of writing. Hadn't seen the lines of music but that makes sense too as they are lines!

Mathematics, i.e. children's arithmetic, used to be written in squares, each number in one square, this was to keep the ones, tens and hundreds etc from wandering as the children wrote or calculated.

I always assosiated it with learning lines for a play or like, Job said, poetry or music.

I have done a bunch of 18th century research for England and Europe and I remember once coming across a book that said that referred to memorizing bible verses. I imagine its a broad statement that it can just be used generally or how ever you want