David Hall:
The 1875 $10 is the lowest mintage regular issue United States coin. A mere 100 pieces were minted. Survival estimates range from David Akers "5-6 business strikes" to as high as 12 to 15. Recent auction plating research done by Ron Guth shows that many appearances of what was thought to be different coins were in fact same coins appearing every 5 or 10 years. In my 40 years of attending coin shows and auctions I have seen only a handful of coins. I believe with certainty that less than 10 survivors exist in all grades, and the real number is probably 7 or 8. This makes the 1875 $10 a true ultra rarity.
The reason that this great rarity hasn't received the attention afforded other ultra rarities is unclear. Perhaps it's the fact that not many people actually collect $10 Liberties by date. Perhaps it's the fact that the coin is so rare and difficult to come by that dealers were never inclined to "promote" the issue. In my opinion, it is certainly easier to buy an 1894-S dime or 1870-S Liberty Seated dollar than it is to buy an 1875 $10.
Ron Guth:
In 1875, the Philadelphia Mint produced a mere 100 $10 gold pieces, making this the lowest mintage of any U.S. coin made for circulation. The number of survivors has been the source of much speculation in the past, much of it based on auction records and "gut instinct". In 1980, David Akers wrote that "...no more than 5-6 business strikes (including two of horrendously low quality) are known along with perhaps 7-8 proofs." Recent estimates have ranged from a dozen to possibly 15 known examples, making the 1875 $10 circulation strike at least as rare as the 1804 Silver Dollar. Despite, its extreme rarity, the 1875 $10 is one of the most affordable coins in its class, only recently having broken the $100,000 barrier, yet falling far short of more recognized coins such as the 1804 Silver Dollar, the 1913 Liberty Nickel, and the 1894-S Dime. The 1875 $10 simply has not received the same amount of publicity as the afore-mentioned rarities.
According to Breen, the date on both circulation strikes and proofs is low in the space between Liberty's bust and the edge of the coin. On circulation strikes, the left base of the 1 is right of center of a denticle; on proofs, the left base of the 1 is over the left edge of a denticle and the top of the second stripe is weak on the reverse. An additional differentiating mark may be a small spike coming out of the denticles below the 7 of the date on proofs. This spike does not seem to appear on circulation strikes, but more coins should be examined in person before this spike is determined to be diagnostic.
Most 1875 eagles feature prooflike luster in the protected areas, but virtually all show moderate to heavy bagmarks.
The National Numismatic Collection at the Smithsonian Institution contains a single circulation strike which would probably grade choice Extremely Fine.
No Uncirculated example has ever been seen or rumored to exist and at least two very well worn examples are known.