His father was afflicted with a total deafness for so considerable a portion of his life, as never to have heard the sound of his son's voice; and was thus rendered incapable of communicating to him that instruction which he might otherwise have derived from a parent endowed with remarkable acuteness of understanding. He was, therefore, sent very early to school at Bury St. Edmunds. Here he continued, under the tuition of the Rev. Arthur Kinsman, till he was removed to Eton; on the foundation of which school he was afterwards placed. His studies having been completed with great credit to himself, under Doctor George, the head-master of Eton, in the year 1742 he succeeded to a scholarship of King's College, Cambridge, where his classical attainments were not neglected. He was admitted in 1745 to a fellowship of his college; and, in the next year, he took his degree of Bachelor of Arts. He now resided chiefly in the University, where his resistance to an innovation, attempted to be introduced into King's College, involved him in a dispute which occasioned the degree of Master to be refused him. That College had immemorially asserted for its members an exemption from the performance of those public exercises demanded of the rest of the University as a qualification for their degrees. This right was now questioned; and it was required of the Bachelor Fellows of King's, that they should compose and pronounce a Latin oration in the public schools. Such an infringement of privilege was not to be tamely endured. After some opposition made by Anstey, in common with the other junior Fellows, the exercise in dispute was at lenth exacted. But Anstey, who was the senior Bachelor of the year, and to whose lot it therefore fell first to deliver this obnoxious declamation, contrived to frame it in such a manner, as to cast a ridicule on the whole proceeding. He was accordingly interrupted in the recitation of it, and ordered to compose another; in which, at the same time that he pretended to exculpate himself from his former offence, he continued in the same vein of raillery. Though his degree was withheld in consequence of this pertinacity, yet it produced the desired effect of maintaining for the College its former freedom. While an under-graduate, he had distinguished himself by his Latin verses, called the Tripos Verses; and, in 1748, by a poem, in the same language, on the Peace; printed in the Cambridge Collection. His quarrel with the senior part of the University did not deprive him of his fellowship. He was still occasionally an inmate of the College, and did not cease to be a Fellow, till he came into the possession of the family estate at his mother's death, in 1754.