C _arithmetic_ meant 'number theory', and so the part concerned with the computation of "ambition, distraction, uglification, and derision" (Lewis Carroll) was _elementary arithmetic_.  (Before that it was _algorism_.)  When _higher arithmetic_ got its own name, the _elementary_ part was dropped in accordance with Grice's Maxim of Quantity ("be as informative as you can, giving as much information as necessary, but no more").  This did not happen to _algebra_, which still can mean either elementary or abstract algebra, still less to _geometry_.

In addition, from the teacher's viewpoint school mathematics is a continuum, including the elementary parts of arithmetic, algebra, geometry, trigonometry, and in recent times probability theory and

Hey that was 50 years ago! topics like Matrices, subjects like Algebra, Geometry, so things like Integration+ Differentiation, integration by parts, simple statistics etc.

Arithmetic was of the form "A customer buys 2 pairs of gloves at 1 and 6pence halfpenny per pair and a hat for a crown.  She pays with a guinea; what is the smallest number of coins in change you can give her."  (a guinea was 21shillings, 12 pence in a shilling, a crown was 5 shillings etc etc).  I think I would be better called mental arithmetic.  We had ounces and pounds, and stones and hundredweight.  Inches, hands, feet, yards, chains, furlongs, miles etc.  So perhaps arithmetic was a more required learning?