I have a copy of Coltons General Atlas of 1859 which will answer your questions re Wards in Manhattan in the afore mentioned time period.
        I don't know what you know about Manhattan Island so I will give you some minimal info. The island is narrow and runs South to North which are the narrow dimensions of the island. The Hudson River (a/k/a as the North River) parallels Manhattan on the South. The East River parallels the island on its east side.
         Broadway bisects the island pretty much in its middle. Delancy St. runs west to  the east towards the East River and is about six blocks from Broadway. Delancy is bordered on the north from Rivington St. On the South side of Delancey there are a number of intersecting streets that run towards the East river, namely, Christie St, Forsyth St., EldredgeSt., Allen St., Orchard St., Ludlow, Essec,, Norfolk, Suffolk, Clinton St., Attorney, Ridge, Pitt St., Willet St., Sherriff St., Columbia St., Cannon (?)St.,
        Lewis St., Goericts (?) St., Margin St., Tomkins St., East St. (Ends at East River).
        Now the answer to your question. In 1859 Ward 10 went from Christie St. to Essec St. Ward 13 went from Suffolk St. to East St. 
        A fairly recent map shows the Williamsburg Bridge ( The Bridge was not in existence at the time frame you are interested in) which spans the East River from the Williamsburg section of Brooklyn and connects to the area of what I give as East St. In addition all the streets I show on the south side of Delancey have been changed to Broome St. A street called Kenmare connects Broadway to Delancey. Rivington remains the same.

From: BROADWAY
Subject: Re: [NY IRISH] (NY IRISH) NYC Streets
Date: Wed, 9 Jan 2002 21:13:40 EST

Hi...just a few corrections to the email re: the street names......Eldridge 
St........Essex St.  Goerck St....Mangin St. and yes there was a Cannon St.