It should be remarked that Pascal, whilst he regarded geometry as affording the highest exercise of the powers of the human mind, held in very low estimation the importance of its practical results. Hence his speculations were irregularly turned to various unconnected subjects, as his curiosity might happen to be excited by them. The late creation of a sound system of experimental philosophy by Galileo had roused an irresistible spirit of inquiry, which was every day exhibiting new marvels; but time was wanted to develope the valuable fruits of its discoveries, which have since connected the most abstruse speculations of the philosopher with the affairs of common life. There is no doubt that his studious hours produced much that has been lost to the world; but many proofs remain of his persevering activity in the course which he had chosen. Amongst them may be mentioned his Arithmetical Triangle, with the treatises arising out of it, and his investigations of certain problems relating to the curve called by mathematicians the Cycloid, to which he turned his mind, towards the close of his life, to divert his thoughts in a season of severe suffering. For the solution of these problems, according to the fashion of the times, he publicly offered a prize, for which La Loubère and our own countryman Wallis contended. It was adjudged that neither had fulfilled the proposed conditions; and Pascal published his own solutions, which raised the admiration of the scientific world. The Arithmetical Triangle owed its existence to questions proposed to him by a friend respecting the calculation of probabilities in games of chance. Under this name is denoted a peculiar arrangement of numbers in certain proportions, from which the answers to various questions of chances, the involution of binomials, and other algebraical problems, may be readily obtained. This invention led him to inquire further into the theory of chances; and he may be considered as one of the founders of that branch of analysis, which has grown into such importance in the hands of La Place.