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Forsyth was born in [[Glasgow]] on 18 June 1858, the son of John Forsyth, a marine engineer, and his wife Christina Glen.<ref name="Royal Society">{{cite book|title=Biographical Index of Former Fellows of the Royal Society of Edinburgh 1783–2002|date=July 2006|publisher=The Royal Society of Edinburgh|isbn=0-902-198-84-X|url=https://www.royalsoced.org.uk/cms/files/fellows/biographical_index/fells_indexp1.pdf|access-date=27 May 2016|archive-date=24 January 2013|archive-url=https://web.archive.org/web/20130124115814/http://www.royalsoced.org.uk/cms/files/fellows/biographical_index/fells_indexp1.pdf|url-status=dead}}</ref>
Forsyth studied at [[Liverpool College]] and was tutored by [[Richard Pendlebury]] before entering [[Trinity College, Cambridge]], graduating [[senior wrangler]] in 1881.<ref>{{acad|id=FRST877AR|name=Forsyth, Andrew Russell}}</ref> He was elected a fellow of Trinity and then appointed to the chair of mathematics at the [[University of Liverpool]] at the age of 24. He returned to Cambridge as a lecturer in 1884 and became [[Sadleirian Professor of Pure Mathematics]] in 1895.<ref>{{
Forsyth was forced to resign his chair in 1910 as a result of a scandal caused by his affair with Marion Amelia Boys, ''née'' Pollock, the wife of physicist [[C. V. Boys]]. Boys was granted a divorce on the grounds of Marion's adultery with Forsyth.
Forsyth became professor at the [[Imperial College of Science]] in 1913 and retired in 1923, remaining mathematically active into his seventies. He was elected a [[Fellow of the Royal Society]] in 1886<ref name="frs"/> and won its [[Royal Medal]] in 1897. He was a Plenary Speaker of the [[International Congress of Mathematicians|ICM]] in 1908 at Rome.<ref>{{cite book|author=Forsyth, A. R.|chapter=On the present condition of partial differential equations of the second order as regards formal integration|editor=G. Castelnuovo |title=Atti del IV Congresso Internazionale dei Matematici (Roma, 6–11 Aprile 1908)|series=ICM proceedings |year=1909|volume=1|pages=87–103|publisher=University of Toronto Press |chapter-url=https://babel.hathitrust.org/cgi/pt?id=miun.aag4063.0081.001;view=1up;seq=93}}</ref>
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