Hello Colin,
In the IoM birth registration was not compulsory until 1878, so few earlier births were recorded. Parish records of baptisms occasionally have a birth date as well as the bapt., especially when the child was unusually late being baptised, but these are rare.
An index to births can be found here: www.mannincloud.info/docs/BL_births.html
Otherwise you need to look up the person's baptism: familysearch.org/search/collection/1784428
Or see a wider selection at: familysearch.org/search/collection/list#page=1®ion=UNITED_KINGDOM_IRELAND
These are indexes. To see the actual entry you need to look at the filmed copy of the parish records either in the Manx Museum library or through a LDS Family History Centre worldwide (see under 'Church of Jesus Christ of Latter-Day Saints' in your local phone book).
Looking up a copy of the original entry is especially important if using Familysearch, where there are many submitted (guessed) entries - so always check the source.
Sue