Many scientists, including Liouville, Jacobi, Darboux, Poincaré, Kolmogorov, and Arnold, have extended Hamilton's work, thereby expanding our knowledge of mechanics and differential equations. While Hamiltonian mechanics is based on the same physical principles as the mechanics of Newton and Lagrange, it provides a powerful new technique for working with the equations of motion. More importantly, both the Lagrangian and Hamiltonian approaches, which were initially developed to describe the motion of discrete systems, have proven critical to the study of continuous classical systems in physics, and even quantum mechanical systems. Indeed, the techniques find use in electromagnetism, quantum mechanics, quantum relativity theory, and quantum field theory. In the Dictionary of Irish Biography David Spearman writes:  Hamilton's mathematical studies seem to have been undertaken and carried to their full development without any assistance whatsoever, and the result is that his writings do not belong to any particular ""school"". Not only was Hamilton an expert as an arithmetic calculator, but he seems to have occasionally had fun in working out the result of some calculation to an enormous number of decimal places. At the age of eight Hamilton engaged Zerah Colburn, the American ""calculating boy"", who was then being exhibited as a curiosity in Dublin. Two years later, aged ten, Hamilton stumbled across a Latin copy of Euclid, which he eagerly devoured; and at twelve he studied Newton's "Arithmetica Universalis". This was his introduction to modern analysis. Hamilton soon began to read the "Principia", and by age sixteen he had mastered a great part of it, as well as some more modern works on analytical geometry and the differential calculus. Around this time Hamilton was also preparing to enter Trinity College, Dublin, and therefore had to devote some time to classics. In mid-1822 he began a systematic study of Laplace's "Mécanique Céleste". From that time Hamilton appears to have devoted himself almost wholly to mathematics, though he always kept himself well acquainted with the progress of science both in Britain and abroad. Hamilton found an important defect in one of Laplace's demonstrations, and he was induced by a friend to write out his remarks, so that they could be shown to Dr. John Brinkley, then the first Royal Astronomer of Ireland, and an accomplished mathematician. Brinkley seems to have immediately perceived Hamilton's talents, and to have encouraged him in the kindest way. Hamilton's career at College was perhaps unexampled.