EXHIBIT 32(b)

                           SECTION 1350 CERTIFICATION

                    CERTIFICATION OF CHIEF FINANCIAL OFFICER
                       PURSUANT TO 18 U.S.C. SECTION 1350,
                             AS ADOPTED PURSUANT TO
                  SECTION 906 OF THE SARBANES-OXLEY ACT OF 2002

         In connection with the Quarterly Report on Form 10-Q of The
Sherwin-Williams Company (the "Company") for the period ended June 30, 2003 as
filed with the Securities and Exchange Commission on the date hereof (the
"Report"), I, Sean P. Hennessy, Senior Vice President - Finance and Chief
Financial Officer of the Company, certify, pursuant to 18 U.S.C. Section 1350,
as adopted pursuant to Section 906 of the Sarbanes-Oxley Act of 2002, that:

         (1)      The Report fully complies with the requirements of Section
         13(a) or 15(d) of the Securities Exchange Act of 1934; and

         (2)      The information contained in the Report fairly presents, in
         all material respects, the financial condition and results of
         operations of the Company.

Dated:  August 14, 2003                /s/ Sean P. Hennessy
                                       --------------------------------
                                       Sean P. Hennessy
                                       Senior Vice President - Finance and Chief
                                       Financial Officer

A signed original of this written statement required by Section 906, or other
document authenticating, acknowledging, or otherwise adopting the signature that
appears in typed form within the electronic version of this written statement
required by Section 906, has been provided to The Sherwin-Williams Company and
will be retained by The Sherwin-Williams Company and furnished to the Securities
and Exchange Commission or its staff upon request.