[1] He was one of three children and two sons of James Lockhart (1735–1814), a Scottish banker, and his wife Mary Harriot Gray or Grey, from a Quaker family; John Ingram Lockhart was his brother, and the daughter was Mary Harriett, married name Greenwollers.
[7] Taking a lease on shooting on the Scottish island of Raasay, Lockhart spent time from 1845 on the Isle of Skye.
[8] Lockhart proposed challenge problems on the separation of the roots of equations of degree five and six, some being published in 1841.
[12] Detailed analysis of a quintic using Budan's theorem for separation of roots was given in 1842, by James R. Christie;[13] it was noted by Young.
[14] In 1843 Young commented that Budan's approach, and Lockhart's own ideas, could now be simplified on the basis of recent developments, which had led to Sturm's theorem.
[15] Young also commended books by Lockhart in discussing a problem on roots proposed by John Pell to Silius Titus.