I'm pretty sure you shouldnt need to set a position:
$("#dialog").dialog();
should centre by default.
I did have a look at the article, and also checked what it says on the official jquery-ui site about positioning a dialog : and in it were discussed 2 states of: initialise and after initialise.
Code examples - (taken from jQuery UI 2009-12-03)
Initialize a dialog with the position option specified.
$('.selector').dialog({ position: 'top' });
Get or set the position option, after init.
//getter
var position = $('.selector').dialog('option', 'position');
//setter
$('.selector').dialog('option', 'position', 'top');
I think that if you were to remove the position attribute you would find it centres by itself else try the second setter option where you define 3 elements of "option" "position" and "center".
Let us know what happens.
Cheers