Part 1 of 2: Answer on 'to whose extent'
"Whose" is a way of referring to an item that belongs to someone else. Generally, in modern English, we use "which" to describe items that belong to a thing.
Belonging to or associated with which person
It would be appropriate to rewrite this as "to which extent" since there are no people involved in this statement.
The quote's use of extent means the following, a little different than the idiomatic attribution you're giving it:
The area covered by something
In the case of the idiom, it usually means:
The amount to which something is or is believed to be the case
Could you rewrite to whose extent as "the extent of which"?
Yes... but it still wouldn't mean quite the same thing as what you're talking about in your other question (to which you linked, in your OP above).
There (and NOT here), in the extent of which, "extent" is figurative, it's a degree.
Here, in this question's quote, "extent" is a literal extent, an expanse of open space that is (in our era) measurable through scientific means.
But you can't rewrite it as "to the extent of which". Why not?
If extent is "degree" the way it is in the idiomatic phrase, then it is fine to say:
To the degree of which...
But in this case, extent is a measurable distance (the extent of the oceans).
There's a big hole in my yard, to the area of which is five feet.
This makes no sense.
There's a big hole in my yard, the area of which is five feet.
I think you can also see here how "area" refers to the "hole".
Similarly, "extent" refers to "gulf".
Part 2 of 2: Gloss on the quote's meaning
What they are talking about in this section is the failure of science at the time to be able to assign an exact measurement to the distance between some unnamed "remotest orb" and the nearest star (excluding the sun). Let's look at the previous sentence:
We have attained, by delicate observations and refined combinations of theoretical reasoning, to a correct estimate, first, of the dimensions of the earth; then, taking that as a base, to a knowledge of those of its orbit about the sun; and again, by taking our stand, as it were, on the opposite borders of the circumference of this orbit, we have extended our measurements to the extreme verge of our own system, and by the aid of what we know of the excursions of comets, have felt our way, as it were, a step or two beyond the orbit of the remotest known planet.
He's showing how, through scientific reasoning, they've taken step-by step movements toward understanding the size of our solar system.
- the dimensions of the Earth
- the dimensions of the Earth's orbit around the sun
- the size of the solar system.
Then comes your sentence:
But between that remotest orb
(the remotest known planet)
and the nearest star
(excluding the sun)
there is a gulf fixed, to whose extent no observations yet made...
A gulf is:
A deep ravine, chasm, or abyss.
These are specific geographic terms but the word is used in this case to refer to the vast empty space between two visible spacial bodies (the planet and the star).
Fixed means the below but can also imply "exists"
Fastened securely in position
So, here's what he means:
The size or scale of the space that exists between our most distant planet and the closest star is unknown and they don't even have a close guess.
He then goes on to explain that the tools of their era were not sufficiently exact to allow those numbers to be determined.
This is a very old text on astronomy and I don't know that it's a good source of information if you're just trying to learn astronomy... and even if you're only reading it for interest, having some basic knowledge of astronomy might be a good idea. Much of what we know about astronomy is much more recently discovered than this essay was written... for example, our eighth planet, Neptune was discovered in 1846! So he didn't know it existed... his "remotest orb" may have been Uranus, though Ceres (currently a dwarf planet) was considered the 8th planet from 1801 to 1851, so he could mean Ceres.