No, this isn't about belief, it's about measuring. You are under the misconception that NAD27 was precisely laid out in a global context akin to satellite based measurements across North America on the Clarke Ellipsoid in the early 1900's using chaining tables and triangulation.

You seriously believe this?

Despite me showing you shifts in opposite directions, those are peppered all throughout NAD27. If it was precisely laid out as you seem to believe, then why even bother going to NAD83. Just use the ellipsoid from 1866.

Moe, when you define the lat/lon of a point on an ellipsoid, Meades Ranch, and an azimuth to another point on that ellipsoid, Waldo, then you have fixed the ellipsoid relative to the earth and you have a datum.

Now when you define another ellipsoid with a different shape and center, then Meades Ranch and Waldo are going to have a different lat/lon on this other ellipsoid.

Points on the first ellipsoid are apples and points on the second one are oranges. Your distance comparisons have no basis. It's like subtracting the elevation of Helena from its population and claiming that the answer is meaningful.

There were several reasons why NAD27 was replaced. The ellipsoid height was not zero throughout and others, but the satellites also couldn't match its coordinates because they were orbiting around the earth's center and the Clarke 1866 ellipsoid was not earth centered.

Don't insult me, Moe. I know what I'm talking about and the distance calculation posted above is strong evidence that you're wrong.

Math teacher QW0554 isn't a measured point. QW0575 is a measured point. The distance comparison isn't meaningful cause it's calculated.

Two close together measured points are rare in this part of the world, I can't really think of two you might use for your calculation.

I understand you know this math, I respect that. And I have the utmost respect for the surveyors that laid out the geodetic control with boots on the ground. But they had technical limitations. They did an amazing job and were incredibly accurate. No one will care about a 100' error across the continent or what ever it might be in the context of the two ellipsoids. But there is error in NAD27, measurable error and surveyors that used it extensively realized it when more advanced equipment became available.

Indeed, there was error, a lot of it related to the assumption that the ellipsoid height was everywhere zero. When NGS did long precise levelling runs, it was apparent As a datum, NAD27 was flawed and as instruments and measurements improved, the errors became easier to find and harder to deal with.

Again, I think that when you say NAD27, you're talking about a datum in totality, including how well it conforms to the shape of the earth.

I am concerned mainly with how ground distances can be converted to grid distances and back again.

You have to take a much broader view because you deal with broader issues with real-world implications. I just like the math and hope that I can contribute meaningful conversation.

NAD83 has problems, too. It's not really ECEF and updating positions, as was done for Meades Ranch and Waldo in going from USSD to NAD27, is getting tedious and burdensome.

Change is inevitable.

It may be easier to find in NC, it's rare here that the NGS placed measured numbers on existing NAD27 triangulation stations. It's important to understand the datasheets often contain calculated positions instead of measured ones.

I have a point a few miles north of town that some of my engineering partners used. They couldn't understand why they were .8' from my value for the point since they took the datasheet number. I explained that my number was from CORS and they insisted that the "official" number from the datasheet was the one to use. The NAD83 number was calculated from the old 27 number.

Anyway, I know you like to play with numbers so I have three actual datasheets for you that are both NAD83 and NAD27 real numbers. If you want to work with them they are PW0138 a CBN, QV0300 a FBN and PW0217 a CBN. They are relatively close to each other and ready for some serious calculations. I would suggest transforming back and forth holding the datasheet values and inversing between them NAD83 and 27. I did that some years ago because we were using them and I tried to "get on" NAD27 as well as 83, I got frustrated with 27 and scrapped the Idea of a local projection, couldn't make it work.

I'll do that and thanks. You earlier gave me a point in AZ. It's a relative newcomer, set if I remember correctly in the 1970s. I went to this old publication:

to do a little more. The advantage here is that I can get both Clarke 1866 lat/lon and NAD27 plane coordinates that are pure, like the old data sheet you posted earlier.

Distances are given between many points and reading a bit of the text is interesting and informative:

Anyway, the two points I've selected are DV2104 DESIGNATION - MARICOPA NORTHWEST BASE and DA1442 DESIGNATION - TABLE. The NAD27 lat/lons on their data sheets duplicate the ones in Publication 224 described above. Elevations have to be taken from the data sheets or else hunted for in other C&GS publications. They're not close, so simple averages of combined factors may not work too well, but I'll cross that bridge when I get to it.

So the coordinates are verified and we need scale factors. NCAT is the source and the way I do that is to enter lat/lon, set the From and To both to NAD27 and get the results.

If the NCAT state plane coordinates match the Publication 224 coordinates, then I think I'm good to go with the scale factors.

Then I compute ground distances using NAD27 and whatever the data sheet has for its state plane coordinates.

If they're very, very close, then I think that the NAD27 numbers are reliable from a mathematical perspective.

Now whether the marks on the ground are where the lat/lon says they are is another issue. That has to be verified in the field.

Thanks again for the data and the great discussion and have a great weekend.

Those points will be inappropriate for your task. They are converted points not measured points. What you need to find are points measured in NAD27 which they both are. Then you need to find points that have 27 measurements that were later measured in NAD83. They are out there, one that you first used was a point like that, neither of these are.

The three I sent to you are duel measured monuments, look for CBN, FBN and HARN points. Many HARN points will be old Bench Marks but sometimes they used tri monuments. DV2104 is a really interesting monument, read about in the comments. It was set in 1899 and I'd imagine it's one of the two baseline points for the region, very important point. However, I don't believe it survived to today which is sad, but there won't be any NAD83 data for it. Basically doing math using it will produce a conformation bias calculation. Same with Table, although it's still there, but it only has converted NAD83 positions. Not what you need.

Here are two AZ points that you might try out, FQ0759 and GP0641, I'm not sure what type of NAD27 they were I don't see the order for them but they both and 27 and measured 83 numbers.

So this statement on a data sheet

DV2104.The horizontal coordinates were established by classical geodetic methods

DV2104.and adjusted by the National Geodetic Survey in August 1993.

means that the classical geodetic methods were used to get the NAD27 coordinates and the NAD 83 coordinates were derived from those earlier coordinates. Whatever measurement error is in the 27 coordinates is brought forward to the 83 coordinates. The correlation between the two is near perfect. and the original measurement error is not detectable.

Measured 83 positions are assumed to be very accurate and will expose the 27 errors.

Thank you for explaining. it's a good lesson: The accuracy of derived points depends on the accuracy of the source point. Any errors in the source point will also be in the derived point.

LOL! I have work to do.

Well, when I calculate the distance between PW0138 and PW0217, the 27 and 83 distances are within 3 decimeters for both the grid distance and the ground distance.

the NAD 83 coordinates were derived from those earlier coordinates

I think the NAD83 coordinates were derived from the old measurements plus new measurements as were available around the network, not the old coordinates.

Also, I'm guessing the computerized best-fit process was better than the original manual one because the old one had to work in smaller blocks and iterate the merging of the blocks whereas the computer could fit larger networks and iterate more.

So yes, the error in any measurement was the same but the end effect might not be the same.

Three decimeters in 45,575 meters is about 6.6 ppm. Both PW138 and PW217 were established in 1957 and the NAD27 lat/lon for both contain 4 decimal places in seconds. This distance is too long to accurately calculate without a midpoint scale factor for each datum, but the comparison of the two somewhat inaccurate numbers might not be affected.

Apparently, there were changes in procedures between the 1930s and the 1950s for NAD27 locations.

It's still interesting, though, that calculated distances using the two datums are so close, given that state plane is deemed to be so inaccurate in general and in particular for NAD27.

t - T correction is the arc-to-chord correction.

Surprised that no one has referenced James Stem's SPCS 83 manual located here:

https://geodesy.noaa.gov/library/pdfs/NOAA_Manual_NOS_NGS_0005.pdf

Note that Stem has a nice illustration of this on printed page 19.

He also treats (print page 50) the issue of the most accurate determination of a line grid scale factor. The equation involves the use of the grid scale(k) at each end (labeled 1 and two respectively) as well as the midpoint (labeled m).

k(1 to 2) = ( k1 + 4km + k2)/6. Obviously we are talking about long distances between points.

For a good summary of NAD27 and related issues, I recommend Joseph Dracup's

https://geodesy.noaa.gov/library/pdfs/NOAA_TR_NOS_0088_NGS_0019.pdf

For comprehensive discussion of why NAD27 was replaced see https://geodesy.noaa.gov/library/pdfs/NOAA_PP_NOS_0002.pdf

As for the issue of Clarke 1866 and GRS80 reference ellipsoids as the bases for NAD27 and NAD83 respectively, NAD27 was a regional datum; NAD83 a global datum. See the link below to a 25 Feb 2010 lecture.

As indicated in Dracup's TR, the choice of Clarke 1866 was to maintain consistency with past practice. Note as well that the PP mentions how after the nationwide adjustment additional work was "grafted onto" NAD27 leading to distortions.

Which reminds me of Charlie Schwarz's paper The Trouble with Constrained Adjustments available from NGS or a better copy here:

http://geodesyattamucc.pbworks.com/w/file/52782335/ConstrainedAdjustments1.pdf

An old presentation discussing this issues is available here: http://geodesyattamucc.pbworks.com/f/25Feb2010_NAD.pdf

I blame whatever errors of biases in this information on my two-week recovery from my two-week vacation in tropical isles.

Just out of curiosity, how closely is this conversation tracking the intent of your original post? .. what WAS the intention of your original post?