the mind is strengthened and enlarged. In times marked by self-confidence and disdain of labour, it cannot be unseasonable to assert the necessity of regular advancement; and to assure the most undoubting genius, that he is not doomed to light, at one bound, on the top-most pinnacle of science.

To those who are unacquainted with the examinations which the Candidates for the degree of Bachelor of Arts must necessarily undergo, it may not be improper to state, that those examinations last for the space of five days: that, for the purpose of ascertaining the relative proficiency of the candidates, demonstrations of questions, involving the principles of the most important parts of pure and mixed mathematics, are expected to be compleatly drawn out: and that the papers under each year, in the following collection, constitute a very small part of the whole examination; three hours only, on each of the two first days, being appropriated to them.*

It will not be expected, by considerate persons, that all the ensuing propositions should be new and

* Let it be observed, that in this account of University discipline, the highest classes are chiefly referred to; and that mention of the examinations in Moral Philosophy and Metaphysics is purposely omitted.

important. Many of them present truths of real and intrinsic value. The purpose, however, for which they were constructed would probably be thought to be sufficiently attained, when an example was afforded of the combination and application of known principles.

Great discrimination and judgment must evidently be demanded in adapting the questions to the circumstances under which their solution is expected. Considerable diversity, it should appear, is desirable, both as to the sources from which they are derived, and to the difficulty which they involve. The intentions with which they are given will be equally frustrated when the principles of solution are too obvious and simple, and when they are too remote, or too difficult of application. Such, it must at the same time be admitted, is the influence of favourite pursuits on the human mind, that it would not be easy to select an instance in which, in the unanimous opinion of the learned, the practicable exercise of ingenuity had, through an entire paper of questions, been successfully kept in view.

Haste can never be accepted as an apology for incorrectness, except when revision is impossible.

This preface, which must now terminate, was undertaken at the request of the publisher, when the volume which it announces was on the verge of publication. Let the unavoidable rapidity, with which it was drawn up, atone, in some degree, for the many imperfections, both in sentiment and expression, which it must undoubtedly exhibit. Let it be considered as the sudden effusion of one who is more accustomed to thinking than to writing; who is not wholly unacquainted with the principles of just composition, and who does not wish to be quite inattentive to them. In full consciousness that rudeness was not designed, the hardihood of right intention has not been solicitously restrained. The motives which prompted the foregoing observations, will, it is confidently expected, be contemplated with unalloyed satisfaction, even in moments of the most serious reflection; and the manner in which they are offered will not, it is hoped, be surveyed with very troublesome discontent. To have vindicated the cause of good learning must always afford a subject for exultation; and bad indeed must be that method of procedure, which can overbalance all the merit arising from the exposure of misrepresentation.

CAMBRIDGE PROBLEMS,

1801.

1.

Morning Problems.-Mr. Hornbuckle.

FIRST AND SECOND CLASSES.

COMPARE the velocities acquired in

falling freely from different altitudes towards different centers of force, the law of force being the inverse square of the distance.

2. Investigate the equation to the reciprocal spiral, and thence determine the law of the force by which a body may describe the curve.

3. Let a given parabola be just immersed vertically in a fluid; at what distance from the vertex must a line be drawn parallel to its base, that the pressure on the upper part may be to that on the lower in the proportion of m to n.

4. A body falls freely by the force of gravity down AB, and uniformly describes the space BD, equal to twice AB, on the horizontal plane, with the velocity acquired. Determine, geometrically, the length and inclination of a plane drawn from A to BD, the time down which may be equal to the time down A B and along BD.

5. Divide a given angle into two angles, whose tangents shall be to each other in a given ratio.

6. On a given day in a given latitude the length of the shadow cast by a tower at 12 o'clock was (a) feet. Required the height of the tower.

7. Find the length of an arc of the meridian corresponding to any given latitude, according to Mercator's projection, and reconcile it to the construction given by Cotes.

8. Where must an eye be placed that an object, seen through a double convex lens, may appear of the same magnitude at all distances from the lens.

9. A body is projected from the top of a given inclined plane. Required the direction of projection in which the least velocity will bring it to the bottom of the plane. Required also this velocity.

10. Compare the values of the respective infinite series 1+2x+3x2+4x3+ &c. and 1-2x+3x2 — 4x3+ &c.

11. Compare the chances of throwing an ace in two trials with one die, and in one trial with two. 12. Transform the cubic x3-px2+qx—r=0, whose roots are a, b, c, into one whose roots are 1 1

1

13. Any two sides of a spherical triangle remaining constant, determine the ratio of the nascent increments of the angle included between those sides, and of either of the other angles.

14. Suppose a given sphere to be projected in a medium whose density is that of itself; compare the velocity of projection with that remaining in the