"After a classwide tangent about the nature of... " I haven't found the exact meaning of this word in dictionaries in this particular sentence
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In mathematical terms, a tangent is a line that meets a circle at precisely one point, before continuing into the distance. So it doesn't cross the circumference, but only grazes it.
So to "go off on a tangent" is a figure of speech that means to get distracted or sidetracked from the main topic and spend on something that is only loosely related to the matter at hand.
So if you started out talking about one thing, then got reminded of something, so you ended up talking about that, which reminded you of something else, and then something else...
You went off on a tangent.
There's not much info in your question but I'll give a try.
In mathematics a tangent is a curve or line that lies on or touches another line or curve. It comes from the Latin "tangere" which means to touch.
Here is an example of a tangent:
It can also mean the tangent function, but don't worry about that because I think your question is about the non-mathematical meaning.
In ordinary English a tangent is like a digression. Let's just say you're talking about a certain topic, and you make some brief side remarks about another topic, that would be "going off on a tangent" / digressing / changing subject.
- on a tangent at a tangent on a completely different or divergent course, esp of thought: to go off at a tangent.
source: Collins Dictionary
I don't have etymological evidence to back me up but I strongly believe that this usage has come from the metaphor of a line diverting from another course, as seen in maths.