I've found plenty of primary and secondary sources on desertion and the San Patricios during the Mexican-American War, and I know that in reality, the deserters were not all Irish and not all Catholic. But the greater perception at the time seems to have been that they were mostly Irish Catholics, particularly within the San Patricios. Most of the memoirs from soldiers and books published in 19th century from on both sides talk about the Irish Catholic deserters.
I'm having more difficulty finding sources on nativism and the Know Nothing movement during and after the war. Newspapers played a large role in the public perception of the war back in the US, but there are only a few newspapers that have been scanned or otherwise uploaded online. There would have been some articles on the San Patricios and other deserters in the American Star, the American Republic, or other newspapers. I did find a blurb in a nativist 1856 Almanac from nearly ten years after the court martials that still talked about the Irish Deserters and John Riley, so I suspect there is more discussion to be found on the Mexican War. Due to the COVID closures, my access to sources is more limited. It seems as if I would need to visit the physical archives for certain sources, and most of them are closed.
I feel like I've hit a wall in my research, so I'm looking for other sources to either work towards or counter my line of inquiry. If anyone has some ideas to contribute to nativism and the Mexican-American War, I would appreciate additional insight.
One of the challenges in researching this topic is that it's often easier to find source material from after the Mexican-American War than before due to the brief prominence of the Know-Nothing/American Party in the mid-1850s. If you need secondary material, I would check the last couple of chapter's of Ronald Formisano's For the People, which is about populist movements in the late 18th through mid-19th centuries. I'd also suggest taking a look at the work of Tyler Anbinder, whose book Nativism and Slavery is one of the premier texts on this topic, and he's written several published articles on nativism/immigration related topics. I don't know what your access to online resources looks like, but the Formisano book and several of Anbinder's articles should be available on JSTOR. Additionally, you might find some information in secondary texts on temperance and other reform movements, as these were often linked to anti-Catholic nativism in the 19th century.
I'm guessing you're probably already using Chronicling America to comb through print sources, but if you aren't, it may be worth your while. Because of the sometimes secretive nature of nativism's political arm, finding openly nativist print sources can be tricky. I would recommend looking through Whig newspapers from this period, as many of the early adopters of political nativism were former members of the Whig Party, and it was common for Whig newspapers to switch to supporting Know-Nothings in the 1850s after the Whig Party imploded in 1852. Finding a complete run of any party newspaper from the 19th century can be a tall order, but it was extremely common for papers to run verbatim copies of material printed in other papers of the same party, so you can sometimes piece things together that way. If you can't find the paper itself online, you might be able to find another paper online that quotes it. As I'm sure you've already figured out, a lot of these papers are only available on currently inaccessible microfilm reels. Hope this helps.
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For anyone looking for a follow-up on sources for Saint Patrick's' Battalion or the bit I found on nativism, here is my bibliography. I haven't received a response from the mods on whether this is allowed or not, so here goes.
Primary Sources
Alcaraz, Ramón. The Other Side: or Notes for the History of the War Between Mexico and the United States. Translated and edited by Albert C. Ramsey. New York: John Wiley, 1850. Accessed 28 May, 2020. https://books.google.co.uk/booksid=HUUIAAAAQAAJ&source=gbs_navlinks_s
Ballentine, George. The Mexican War, by an English Soldier: Comprising Incidents and Adventures in the United States and Mexico with the American Army. New York: W. A. Townsend and Company, 1860. Accessed 28 May, 2020. https://babel.hathitrust.org/cgi/pt?id=uc2.ark:/13960/t76t0hk2n&view=1up&seq=11
Carpenter, William W. Travels and Adventures in Mexico: In the Course of Journeys of Upward of 2500 miles, Performed on Foot; Giving an Account of the Manners and Customs of the People, and the Agricultural and Mineral Resources of that Country. New York: Harper and Brothers, 1851. Access 28 May 2020. https://digital.library.pitt.edu/islandora/object/pitt%3A31735054855162
Chamberlain, Samuel E. My Confession: The Recollections of a Rogue. New York: Harper and Brothers, 1956. Access 28 May, 2020. https://archive.org/details/in.ernet.dli.2015.226260
Cong. Globe, 28th Cong., 2nd Sess., 32-35 (1844). Accessed 11 June 2020. https://memory.loc.gov/cgi-bin/ampagecollId=llcg&fileName=015/llcg015.db&recNum=16
Davis, George Turnbull Moore. Autobiography of the Late Col. Geo. T. M. Davis, Captain and Aid-de-Camp Scott’s Army of Invasion (Mexico), from Posthumous Papers. New York: Jenkins & McCowan, 1891. Access 09 June, 2020. https://books.google.com/booksid=1T5EAQAAMAAJ&printsec=frontcover&source=gbs_ge_summary_r&cad=0#v=onepage&q&f=false
“Interesting War Intelligence.” The New York Herald, 17 October 1847, Accessed 19 June 2020. https://chroniclingamerica.loc.gov/lccn/sn83030313/1847-10-17/ed-1/seq-1/
Jay, William. A Review of The Causes and Consequences of The Mexican War. Freeport, NY: Books for Libraries Press, 1970.
Kirkham, Ralph W. The Mexican War Journal and Letters of Ralph W. Kirkham. Edited by Robert Ryal Miller. College Station, TX, Texas A&M University Press, 1991. Accessed 28 May 2020. http://search.ebscohost.com/login.aspxdirect=true&db=nlebk&AN=18227&site=ehost-live.
“News of the Week.” O.U.A., 26 August 1848, Accessed 11 June 2020. https://curiosity.lib.harvard.edu/immigration-to-the-united-states-1789-1930/catalog/39-990072657280203941
Oswandel, J. Jacob. Notes of the Mexican War, 1846-1848. Knoxville: University of Tennessee Press, 2010. Accessed 19 June 2020. https://muse.jhu.edu/book/13437
Roa Bárcena, José María. Recuerdos de la Invasión Norte-Americana 1846-1848 por un Joven de Entonces. Mexico City: Imp. de V. Agüeros, 1902. Accessed 09 June 2020. http://www.cervantesvirtual.com/obra/recuerdos-de-la-invasion-norte-americana-1846-1848-por-un-joven-de-entonces/
“The Irish Legion.” Niles National Register, 13 March 1847, Accessed 19 June 2020. https://babel.hathitrust.org/cgi/pt?id=inu.30000117880546&view=1up&seq=46
The Know Nothing Almanac and the True American’s Manual for 1855. New York: De Witt & Davenport, 1855. Accessed 6 June, 2020. https://digital.library.villanova.edu/Item/vudl:15134#?c=&m=&s=&cv=&xywh=-1584%2C-1%2C6150%2C4376
The Know Nothing Almanac and the True American’s Manual for 1856. New York: De Witt & Davenport, 1856. Accessed 14 June, 2020. https://digital.library.villanova.edu/Item/vudl:15058#c=&m=&s=&cv=19&xywh=-164%2C1934%2C3249%2C3050
Zeh, Frederick. An Immigrant Soldier in the Mexican War. Translated by William J. Orr. College Station, TX: Texas A&M University Press, 1997. Accessed 28 May 2020. http://search.ebscohost.com/login.aspx?direct=true&db=nlebk&AN=18202&site=ehost-live.
Secondary Sources
Anbinder, Tyler. Nativism and Slavery: The Nothern Know Nothings and the Politics of the 1850s. New York: Oxford University Press, 1992.
Berger, Max. "The Irish Emigrant and American Nativism as Seen by British Visitors, 1836-1860." The Pennsylvania Magazine of History and Biography 70, no. 2 (1946): 146-60. Accessed June 22, 2020. www.jstor.org/stable/20087822.
Billington, Ray Allen. The Protestant Crusade 1800-1860: A Study of the Origins of American Nativism. Chicago: Quadrangle Books, 1938. Accessed 11 June 2020. https://archive.org/details/in.ernet.dli.2015.214564/mode/2up
Bauer, K. Jack. The Mexican War 1846-1848. New York: Macmillan Publishing Co, 1974.
Cogliano, Frank, Jacqueline Fear-Segal, and Thomas Rath,“The Mexican-American War.” Interview by Melvin Bragg. In Our Time. Podcast audio. June 28, 2018. https://www.bbc.co.uk/programmes/b0b7d18j
Eisenhower, John S. D. So Far From God: The U.S. War with Mexico 1846-1848. New York: Random House, 1989.
Guardino, Peter. "“In the Name of Civilization and with a Bible in Their Hands:” Religion and the 1846–48 Mexican-American War." Mexican Studies/Estudios Mexicanos 30, no. 2 (2014): 342-65. Accessed June 14, 2020. doi:10.1525/msem.2014.30.2.342. https://www.jstor.org/stable/10.1525/msem.2014.30.2.342
------. The Dead March: A History of the Mexican-American War. Cambridge: Harvard University Press, 2017.
Howe, Daniel Walker. What Hath God Wrought: The Transformation of America 1815-1848. Oxford: Oxford University Press, 2007.
Miller, Robert Ryal. Shamrock and Sword: The Saint Patrick’s Battalion in the U.S.-Mexican War. Norman, OK, University of Oklahoma Press, 1989.
Nevin, David. The Mexican War. Alexandria, VA: Time-Life Books, 1978. Accessed 19 May, 2020. https://archive.org/details/mexicanwar00time/mode/2up
Stevens, Peter F. The Rogue's March: John Riley and the St. Patrick’s Battalion, 1846-48. Washington D.C.: Potomac Books, 2005.
Wynn, Dennis J. The San Patricio Soldiers: Mexico’s Foreign Legion. El Paso: Texas Western Press, 1984.