they were intended to promote. The difficulty of obtaining proper teachers, already sufficiently formidable, would be ten times augmented by the abolition of all distinctions for academic proficiency, since the public would have no means of judging who were best qualified to teach. Boys will not study mathematics from a sense of duty-that is, not one in a hundred-it is too up-hill work; nor will they indeed, from the same abstract motive, study classics in a sound, regular way, taking the dry matter with the interesting as it comes. They will be apt to work in a dilettante way, and pick out the titbits. The example of the German universities is not in point. The German students have been worked hard at their gymnasia, and have passed severe examinations on quitting those. They are, at the university, occupied immediately upon their professional studies, for those of them who will not be lawyers, doctors, or clergymen, will be “ordinary" professors or government functionaries, immediately after taking their degrees. The fruit of their study is close at hand. With regard to the envy and ill-will supposed to be excited by competition for honors, they certainly are not evils inseparable from the system. You see nothing of them at Cambridge. The two Medallists, or the two Smith's Prizemen, are often warm personal friends, reading with the same tutor, and passing much of their time together. Even with us the extent of it is greatly exaggerated; but so far as it does exist, it is justly chargeable not on the principle of emulation, but on that spirit of envy and impatience of superiority so general in our country, which is expressly generated by our democratic institutions, and must be taken as one of the evils of those institutions along with their blessings. According to more general considerations, it is tolerably evident that emula tion is one of the main springs of human progress in all departments of life; that individuals and nations become torpid and retrograde without it; that success attending on patient industry and talent combined is the usual rule in this world; that the Divine Law itself is sanctioned by rewards and punishments; that a government without rewards should also in common fairness be one without punishments-which would end in being no government at all-and this perhaps is what some people would prefer. Most of these things are truisms s; indeed all the arguments have been presented, or rather alluded to, as briefly as possible, because the common sense of mankind readily agrees to them; and the digression was made merely not to pass over in silence any question that has been started in reference to our subject. [It would not be doing justice to the progress of our colleges if I omitted all notice of the provision which some of them have made for the instruction in the Humanities of graduates (a thing utterly unknown when the first edition of this book was printed), such as the course at Yale for the Doctorate of Philosophy. These additions, in my opinion, stamp the character of univer sity on an institution much more unequivocally than the establishment of special schools for the Fine Arts or the Positive Sciences. I also think that the new system of elective studies, by which students, who desire it are allowed to drop their classics after the second year, is not so great an innovation or injury as it appears at first sight to be. The average age of entrance has been raised within the last quarter of a century nearly two years, so that the Junior of to-day stands about where his father's contemporaries did when they graduated. At the same time, I cannot consider it a change for the better.] APPENDIX. To form a correct idea of Cambridge examinations the reader must see some of the papers. I therefore republish those of the two most important; the great final University examinations for Classical and Mathematical Honors, popularly known in my time as "the Senate-House" and "Tripos." The Mathematical Examination has lately been increased by the addition of Heat, Magnetism and Electricity (in their mathematical aspects); and the Classical by the addition of a paper on Greek and Roman Philosophy and Rhetoric, and another on Greek and Latin Philology. SENATE-HOUSE EXAMINATION. [N. B. The Differential Calculus is not to be employed.] 1. IF one side of a triangle be produced, the exterior angle is greater than either of the interior opposite angles. a. In equal circles equal circumferences are subtended by equal straight lines. 3. A Turkey carpet, measuring 11 ft. 6 in. by 9 ft. 8 in., is laid down on the floor of a room measuring 14 ft. by 12 ft. 6 in., determine the quantity of floor-cloth necessary to complete the covering of the area, and its price at 6s. per square yard. B. State and explain the rule for the extraction of the square root of an algebraical expression. Determine the square root of 4.4 4x3+5x2-2x + 1. 5. When is one quantity said to vary directly as another? If a varies directly as y when z is constant, and inversely as z when y is constant, then if y and z both y vary, x will vary as If 3, 2, 1 be simultaneous values of x, y, z respectively in the preceding proposition, determine the value of a when y 2 and z = 4. 7. Solve the following equations: 7. Investigate a rule for transforming a number from one scale of notation to another. In what scale will the number 95 be denoted by 137? an expression for the area of a plane triangle in terms. of the sides. Simplify the expression in the case of an equilateral triangle. ε. Find the equation of the straight line which meets the axes of x and y respectively at distances a and b from the origin. If the axes are rectangular, what angle |