Nice coin, Michael. These old beauties are highly sought by collectors, especially if they are in great numismatic shape, like the coin in the picture.
Two things strongly affect the value of your coin: (1) its numismatic grade or condition, and (2) whether or not it is 'slabbed' by a third party grading service.
The coin in the picture comes from eBay seller LoneStarNumismatics. CoinQuest thanks Lone Star for use of their picture. They have it on sale for $2149 US dollars or best offer, a fair price for a nice, almost uncirculated 1837 coin in a PCGS slab. Most of the coins in this series command high prices when they are in superb numismatic condition. They fall lower, however, when worn, scratched, or heavily circulated.
NEVER CLEAN A COIN. CLEANING RUINS VALUE.
A good way to estimate the value of these old gold pieces is to first compute their 'melt' value and then add value due to numismatic (coin collector) appeal. These coins contain:
QUARTER EAGLE 2 1/2 DOLLARS: 0.121 troy ounces of gold
HALF EAGLE 5 DOLLARS: 0.242 ounces gold
Using web sites such as kitco.com, find the current price of gold and multiply. At the time of this writing, the price of gold is around $1200 US dollars per ounce, so the melt value of a quarter eagle would be 0.121 x 1200 = $145 melt value. A half eagle would be twice the melt value. Melt value is also called base value (BV).
Here are some typical total values for these coins, that is, the base value plus the numismatic value.
QUARTER EAGLES 2 1/2 DOLLARS:
worn: BV + $100
average circulated: BV + $350
well preserved: BV + $600
fully uncirculated: BV + $3000
HALF EAGLES 5 DOLLARS:
worn: BV + $200
average circulated: BV + $300
well preserved: BV + $600
fully uncirculated: BV + $4000
The rapid jump in price between circulated and uncirculated coins shows the importance of authenticating, grading, and encapsulating rare coins -- the 'slab' makes a big difference.
Michael's coin, the 1836, is one of the common dates, and most coins in the series carry common dates. Look for the C or D mint marks under Miss Liberty's head and above the date.
QUARTER EAGLE BETTER DATES:
1838C: $3800 in well preserved condition
1839: $1400 well preserved
1839C: $3000 well preserved
1839D: $4000 well preserved
1839O: $2200 well preserved
HALF EAGLE BETTER DATES:
1838C: $8000 in well preserved condition
1838D: $5000 well preserved
The slab is important because there are plenty of fake gold coins on the market today. Certification by PCGS, NGC, ICG, or ANACS makes authenticity much more certain. Look them up on the Interent. Do not use other services.
If you have a half eagle $5 gold piece dated 1834, check the 4. If you have the crosslet 4, your coin is much more valuable than the plain 4. An 1834 crosslet 4 catalogs at $4000 in well preserved condition.
Finally, if your 1834 quarter eagle or half eagle looks like this picture, with E PLURIBUS UNUM above the eagle on the reverse side, zowie! You have a very valuable capped bust coin with catalog values as follows.
1834 QUARTER EAGLE WITH E PLURIBUS UNUM
1834 HALF EAGLE WITH E PLURIBUS UNUM
See this page for a discussion of counterfeits.