word - Another another Add-A-Gram

This is based on the Add-A-Gram puzzles by wildBillMunson. You need to find a chain of words, such that each next word is an anagram of the previous word plus an additional letter.

Example: a doctor inside a rejection inside an elector; the solution is "vet", "veto", "voter".

Here is the chain for you to solve:

a root inside a symbol inside a division inside a landing inside dullness inside brightness inside a counter inside a liar inside a lie inside commanders inside substitutes inside instruments.

Each one is a single word. That's 12 different words in total. Can you identify them?

Answer

Here we go -

a root — i   ($i=\sqrt{-1}$)
a symbol — pi
a division — rip
a landing — pier
dullness — tripe

brightness — esprit
a counter — riposte
a liar — imposter
a lie — misreport
commanders — imperators
substitutes — temporaries
instruments — polarimeters or vaporimeters