Which meaning of “naturally” fits in this context?

I came across this adverb (naturally) while reading a math book. In this book, a concept (extended complex plane) has been defined before and now the book says it can be motivated even more naturally by another mathematical concept (linear fractional transformations).

The extended complex plane…can now be motivated even more naturally by linear fractional transformations as follows.

I have searched in all available dictionaries and I have found some meanings but I don’t know which meaning is more appropriate.

These are what I have found:

  1. in a way that you would expect


  • After a while, we naturally started talking about the children.

  • His death has naturally come as a shock to us all.

  1. in a way that makes sense : in a logical and reasonable way


  • Her conclusions follow naturally from the theory.

  • Questions about the journalist’s sources arise naturally from such a
    controversial report.

  • When I saw that there were no lights on in the house, I naturally
    assumed you were asleep.

Here is an image of the page:
portion of the page from the math book


The definitions you cite both mean essentially the same thing: in a way you would expect, a way that is rational and logical, etc.

The other definition of "naturally" that comes to mind is "in accordance with nature, not artificial". Like, "We decided to move to the country, abandon all of our electronic gadgets, and live naturally."

In this case, either definition you give fits. I’d guess the writer just gave some way to derive a complex plane. Now he gives a second way that he says is more "natural", that is, more expected and logical.

  • Update *

A "setting" means a value assigned to some variable. This could be in a mathematical sense, like here: The settings are cw-a=0, z=infinity, etc. (I don’t know what "a/c be the image of z=infinity" means but I guess it’s not important that I understand the math here, so whatever.) "Settings" is also used in a more mechanical sense, like where you positioned the knobs on an electronic device.

Source : Link , Question Author : Arashrostami , Answer Author : Jay

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