1.  Marriage Record:

2.  1820 Census Index/Images (online at Ancestry.com, Image #1 of 7 — indexed "Bown"; list was roughly alphabetized; extracted by Diana Gale Matthiesen):

1820

NY

Essex Co.

Jay

Roll M33_69

p. 419

Ln. 31

Bruse Joseph

100100-20100-0-100

These data indicate:

No. & Sex	Age Class	Therefore Born	Individuals Inferred
1 male	9 or under	1810-1820	= Joshua Franklin (b. 1816/7)
1 male	19-25*	1794-1801	= Joseph (b. 1795/6)
2 females	9 or under	1810-1820	= Daughter B = Daughter A
1 female	16-25	1794-1804	= Rebecca (b. 1796/7)
1	person engaged in agriculture
*In the 1820 Census, the third column is age class 16-18 and the fourth column is age class 16-25, which means any individual in column three is duplicated in column four.  So, for an accurate representation of the household, the number in column three should be subtracted from the number in column four. If, after the subtraction, any individuals remain in column four, then they have to be age 19-25, thus creating a "new," more precise age class.

Also listed:  Joseph's brother, Jonah BRUCE (indexed "Bruer").  (The page image at Genealogy.com is sharper than the one at Ancestry.com, though Joseph and Jonah are mis-indexed there, too.)

3.  1830 Census Index/Images (online at Ancestry.com, Image #3 of 20; extracted by Diana Gale Matthiesen):

1830

NY

Essex Co.

Jay

Roll 114

p. 290

Ln. 25

Joseph Bruse

001 001 - 002 001

These data indicate:

No. & Sex	Age Class	Therefore Born	Individuals Inferred
1 male	10-14	1815-1820	= Joshua Franlin (b. 1816/7)
1 male	30-39	1790-1800	= Joseph (b. 1795/6)
2 females	10-14	1815-1820	= Daughter B = Daughter A
1 female	30-39	1790-1800	= Rebecca (b. 1796/7)

4.  1840 Census Index/Images (online at Ancestry.com, Image #11-2 of 26; extracted by Diana Gale Matthiesen):

1840

NY

Essex Co.

Jay

Roll 282

pp. 100-101

Ln. 22

Joseph Bruce

000 120 100 - 000 010 100

0400000

These data indicate:

No. & Sex	Age Class	Therefore Born	Individuals Inferred
1 male	15-19	1820-1825	= ?
2 males	20-29	1810-1820	= Joshua Franklin (b. 1816/7) = ?
1 male	40-49	1790-1800	= Joseph (b. 1795/6)
1 female	20-29	1810-1820	= Daughter A or B
1 female	40-49	1790-1800	= Rebecca (b. 1796/7)
4	persons employed in agriculture

It's hard to say who the extra males are, but they are not with our subjects in 1830, so are not their sons.  Listed next to Joseph's older brother, Jonah BRUCE (æ 50-60), who is next to Jonah's son, Benjamin BRUCE (æ 20-30).

5.  1850 Census Every-Name-Index/Images (online at Ancestry.com, Image #51 of 66):  Jay, Essex Co., NY, Roll M432_504, p. 213A, 3025/3099, enumerated 13 Sep 1850, official enumeration date 1 Jun 1850 (extracted by Diana Gale Matthiesen):

Joseph Bruce	54	M		Farmer	4000
Rebecca	33	F
Franklin	33	M		"
Lorrain	32	F
Louisa	8	F
Robert	3	M
Daniel Hopper	15	M
Hannah Felt	33	F

The blanks in the birthplace column are presumably intended to mean "NY," but if Joseph's parents didn't move to Jay until 1804, then Joseph was born in MA.  Rebecca's age has to be wrong if she's Franklin's mother (as it says in Joshua Franklin's bio, see below).  Earlier censuses say she was born by 1800, so she is most likely 53, not 33.  Listed next to Joseph's nephew, Nymphas BRUCE, son of his brother, Jonah BRUCE.

6.  1860 Census Every-Name-Index/Images (online at Ancestry.com):  can't find, and I even did a page-by-page search of Jay.  Also didn't find their son, Joshua Franklin, who should also be in Jay.  Were they missed by the enumerator?  Or did they slip off to Canada for a few years?

7.  H.P. Smith.  1885.  History of Essex County [New York].  D. Mason & Co., Publ., Syracuse, NY (online at Ancestry.com).  On p. 717:

Bruce.  The Bruce family is among the first families of Jay.  Their ancestor, Benjamin Bruce, first settled in Jay in 1804.  He was a shoemaker by trade and a soldier of the Revolution.  He died September 11th, 1839, leaving six children:  Joseph, Jonah, Anna, Polie, Phila, and Hannah.  Joseph was married to Rebecca Houghton, a native of Vermont, and the venerable Joshua F. Bruce, of Jay, is her only son...

8.  Messages posted to the BRUCE Surname Board (online at RootsWeb.com/Ancestry.com).