1815 - May 14, 1882.

Superintendent USMA:  March 31, 1855 to September 8, 1856.

Publications: The USMA catalogue lists 16 items. I have only included those of the most interest.
 

• Analysis of rotary motion as aplied to the gyroscope. New York : D. Van Nostrand, [s.a.]. SPECIAL COLL-BSMT-B30 QA862.G9 B3. Description: [537]-560, [2] p. : ill ; 22 cm.

        Notes: "Reprinted from Barnard's American Journal of Education for June, 1957"
        Local note: SPEC copy inscribed : "Lieut F. E. Prince with compliments of Major Barnard."
 
• The phenomena of the gyroscope analytically examined, with two supplements, on the effects of initial gyratory velocities, and of retarding forces on the motion of the gyroscope.  New York: D. Van Nostrand; Hartford, F. C. Brownell, 1858

        USMA:   SPECIAL COLL-BSMT-B30  QA862.G9 B32 1858
        Description:   p. [537]-560, 530-536, [299]-304. illus., diagrs. 23 cm
        Notes:  Various pagings. From Barnard's American journal of education, no. 9, 11, 13, 1857-1858.

• Query, The Mathematical Monthly, vol. 1 (1859), pp. 49-50. Rickey has copy of this journal.
The query deals with why the value of an expression is real when it consists of the sum of the values of a function at conjugate complex numbers. An answer, involving Taylor expansions, is provided by the editor, J. D. Runkel.
 
• Letter to the editors of the National intelligencer, in answer to the charges against the United States Military academy in the report of the of the secretary of war, of July, 1861. By Major J. G. Barnard ..   New York, D. Van Nostrand, 1862

        USMA:  CIRCULATING BOOKS  U410.F7 B255
        USMA:  SPECIAL COLL-BSMT-B30  U410.F7 B255

• On the internal structure of the earth considered as affecting the phenomena of precession and nutation, supplementary to article under the above head, Smithsonian contributions to knowledge, vol. XIX, no. 240 being the third of the problems of rotary motion. By J. G. Barnard. [Washington, Smithsonian Institution, 1877]

        USMA:  COLL-BSMT-B30  Q11 .S68 vol. 23.  Description:  iii, 16 p. diagrs. 29 cm
        Series:  Smithsonian contributions to knowledge. [v. 23, art. 4]
                    Smithsonian Institution publication, 310