I was reading an 1808 document from Mexico and it cites someone's birthplace as 34 latitude, 265 longitude. I had a feeling the person was born somewhere in the area of West Texas/ New Mexico, so the 34 latitude fits. The 265 on the other hand doesn't make sense considering the modern longitude system doesn't go above 180. Is there any chance the author is using an outdated system I'm unfamiliar with? If so, how does it work and what location is the document referring to?
I'm a cartographer who does a lot of work with historic maps, but I've never encountered such a system either. Such a problem does come up with some computer map databases, and can be converted to the way we're used to looking at longitudes by subtracting 360. That would give you -95, or 95º W, putting your point near Idabel, Oklahoma, several miles north of the Red River, which seems a rather unlikely place for anyone other than a Caddo Indian in 1808.
However, in the 19th century, longitudes in North America were often given in terms of the Washington Meridian, and the boundaries of several western states were set in relationship to it. (As the Wikipedia entry points out, there were in fact several Washington Meridians through the decades, with slightly different positions. The David Rumsey Collection of historic maps contains this 1820 map of Louisiana and Mexico that uses the Washington Meridian. I can't help but notice that coordinates of 34ºN, 26.5ºW (Washington Meridian) would put us close to Portales, New Mexico. But prior to computers, no one ever gave coordinates using decimals; they would have written 26° 5′ W. That (26.08ºW instead of 26.5ºW of Washington) doesn't move us too far, just a little closer to today's Texas-New Mexico boundary.
Unlike the Equator as a reference line for latitude, the earth doesn't have a natural line from which to count longitude, and various cartographers and nations made their own choices. The French, naturally, chose Paris and continued to hold out for it until 1911. The Spanish meridian, probably the one at Cádiz, would seem relevant to your inquiry if you think this person was born in New Spain. Cádiz is at 6.28º west of Greenwich, so if I've done the weird modulo math right, that could put us near Floydada, Texas (Lubbock area). On the right side of the Red River, at least—but still pretty inhospitable to a European in 1808.
An 1884 international conference (held in Washington, ironically) chose the Greenwich Meridian as the international Prime Meridian. The US officially adopted the Greenwich Meridian for all uses in 1912.
Do you have a link to a copy of the original document or can you give the verso information?
Measurements quoted in Twitchell's "Leading Facts of New Mexico's History" suggests that by the end of the 18th Century cartographic co-ordinates were given in a conventional manner in Mexico and the United States. It may be that your reference is a distance and a heading from a known point, or there may be more to examine.
It's a fascinating question which I'm genuinely looking forward to finding out more about!