Hello!
I'm afraid that Sumitra Shah was right, because the Old Style dates
(Julian calendar) were used in Britain till 1752. That means that June 5
of 1723 in Old Style equaled to June 16 of 1723 in New Style (Gregorian
calendar). Since nowadays we all use Gregorian calendar then we should
correct the date of Adam Smith's baptizing.
Sincerely Yours,
Dmitry Krutikov