Cryptic crossword
Cryptic crossword from Contours magazine 201415 Edition
1  2  
Maths  
3  
Cryptic  
No 1  
4 
Across

Rings from void one (6)

Round decrepit relic found at the end of a field (6)

Maclaurin has such views about the Royal Institution (6)
Down

Function for confused car companies (6)

Principles: they're not necessarily right! (6)
Solutions
Across

Rings from void one (6) ANNULI
Quite simply, annuli are rings  they are the space you find between two concentric circles:
(the rest of the clue works because void=ANNUL and one=I in Roman numerals).

Round decrepit relic found at the end of a field (6) CIRCLE
As you probably know, a circle is round! The rest of the clue works because CIRCLE is C followed by an anagram of “relic”  and in maths, \(\mathbb{C}\) is the symbol used to represent the set of complex numbers. The complex numbers are an example of a “field”, which is a kind of object studied in abstract algebra, and they include all the imaginary numbers such as the famous \(i=\sqrt{1}\).

Maclaurin has such views about the Royal Institution (6) SERIES
Maclaurin series are a special kind of infinite sum; they are named after the mathematician Colin Maclaurin (who was a professor at the University of Edinburgh). The rest of the clue comes from views=SEES surrounding (“about”) RI=Royal Institution.
Down

Function for confused car companies (6) ARCCOS
This is an anagram of CAR COS (= car companies), while “arccos” in maths is an example of a trigonometric function. Here is a graph of the function:

Principles: they're not necessarily right! (6) IDEALS
Ideals are objects which are studied in ring theory, a branch of abstract algebra. They come in two types  left and right.