[HorstFue]

Chapter IV in Earth not a Globe http://www.sacred-texts.com/earth/za/za22.htm
great R. deduces distances to the north pool by measuring the circumference of earth using "eternities" or similar.
The first example looks quite promising. Numbers appear to be reasonable.

The superficial extent or magnitude of the earth from the northern centre to the southern circumference, can only be stated approximately. For this purpose the following evidence will suffice. In laying the Atlantic Cable from the Great Eastern steamship, in 1866, the distance from Valencia, on the south-western coast of Ireland, to Trinity Bay in Newfoundland, was found to be 1665 miles. The. longitude of Valencia is 10° 30´ W.; and of Trinity Bay 53° 30´ W. The difference of longitude between the two places being 43°, and the whole distance round the earth being divided into 360°. Hence if 43° are found to be 1665 nautical, or 1942 statute miles, 360° will be 13,939 nautical, or 16,262 statute miles; then taking the proportion of radius to circumference, we have 2200 nautical, or 2556 statute miles as the actual distance from Valencia, in Ireland, to the polar centre of the earth's surface.

^{In RET distance to north pole at 52°N should be 2280 nm (nautical miles, 60*(90-52))}

Some paragraphs later:

In the Australian Almanack for 1871, page 126 2, the distance from Auckland (New Zealand), to Sydney, is given as 1315 miles, nautical measure, which is equal to 1534 statute miles. At page 118 of the Australian Almanack for 1859, Captain Stokes, H.M.S. Acheron, communicates the latitude of Auckland as 36° 50´ 05″, S., and longitude 174° 50´ 40″, E.; latitude of Sydney, 33° 51´ 45″, S., and longitude 151° 16´ 15″, E. The difference in longitude, or time distance, is 23° 34´ 25″, calculating as in the case of Valencia to Newfoundland,

p. 94

we find that as 23° 34´ 25″ represents 1534 statute miles, 360° will give 23,400 statute miles as the circumference of the earth at the latitude of Sydney, Auckland, and the Cape of Good Hope. Hence the radius or distance from the centre of the north to the above places is, in round numbers, 3720 statute miles.

^{In RET distance to north pole at 33°S should be 7380 nm (nautical miles, 60*(90+33)), circumferenc at 33°S should be 18115 nm = 20851 statute miles}

But the issue is not alone the calculation of the radius from the circumference:
Didn't great R. know that there's no direct course possible from Auckland to Sydney? As Auckland is on the east side of New Zealand, You have to round the north of New Zealand, which gives an extra millage of about 200 nm (nautical miles).
using 1125 nm (1315-200): 23°34' represent 1125nm, so 360° are 17185nm or 19781 statute miles, according his calculation Sydney is 3148 statute miles from north pole...
So Valencia/Ireland is only 600 statute miles (3148-2556) north of Cape Town... You would not even cross the equator at that distance.
I found another good example: The Trans-Australian Railway
This railway is known for world longest dead-straight track 487km, 297 statue miles, 258.1nm
between
Loongana   Coordinates: 30°56′46″S 127°02′17″E
Ooldea         Coordinates   30°27′33″S 131°50′08″E

Difference in longitude is 4.795° for 258.1nm, gives a total circumference derived from the railway track at 30.7°S: 19377nm (22303 statute miles)
It's not exactly what RET would predict but close, about 5% error: 18572nm

[Rushy]

No ALL CAPS in titles, please. I've edited it out of your post.