WildTrig3: Spread, angles and astronomy

Hi Prof. Wildberger, thanks for the awesome videos!

I've got a question about the "definition of an angle". I understand that "intuitive notions" such as line, plane, point, etc should not be defined because a definition breaks something difficult into smaller pieces, and in order to define the intuitive notions most complicated concepts are used. So i wonder if defining an angle is appropiate?

Your understanding, while common, is flawed. It is crucially important in mathematics to define all terms. The more fundamental, or `intuitive' a notion, the more important it is to define it well.

While line, point, plane, circle can be defined completely precisely in an elementary way, `angle' is really different---there is no elementary definition.

Please see my MathFoundations YouTube series for more about precise definitions. And thanks for the question.

ok so I watched it again, and, isnt this "spread" baically tan^2 ?

Actually the spread between two lines is sin^2 (angle). That's assuming you know what an angle is, what the function sin x is, and that you are working over the `real numbers'.

With spread, and its purely algebraic definitions, no transcendental concepts are required. Bottom line---its vastly simpler, more general, and a lot easier to compute with.

yeah sin^2... I guess after 1 month I am still sleepy :(

Sorry, I get excited.

Okay, I just watched again and I see now that slope and spread have little to no relation at all because you are using the hypontenuse for the denominator. hmm. Need to watch a few times more.

The slopes of two lines are quite easily related to the spread between them. That's one of the beautiful results, take the x axis, the spread between any positively sloped line heads towards one as the value of the slope increases.

I'm going to try to understand this better and will watch a few times. I understand for the most part except how to convert spread into degrees and weather or not spread is equal to slope. I'll watch again to see if I missed something. Thank you so much for these powerful videos!