INTRODUCTION.

THE Report of the Mathematical Board annually calls the attention of the Mathematical Students of this University to the importance of the Examples appended to the book-work questions in the Senate-House Examination Papers. The Board conceives that these Examples, or Riders, to use the more familiar term, afford a searching test of the merits of the candidates, and are peculiarly adapted to call forth an exhibition of style, which it must be allowed indicates the mathematician far more than a mere knowledge of books; and so high does it estimate their importance, that it has repeatedly recommended a diminution of the book-work questions in the Senate-House papers, in order to allow the admission of a larger number of Riders.

"To obtain," it observes, "a surer test of the acquaintance of the candidates with the subjects of their reading, examples and deductions have been attached to many of the propositions from books. The Board, however, having had before them an analysis of the answers to the questions proposed in 1846, 1847, 1848, and 1849, find that the number of answers to the examples and deductions has fallen below the amount which it is desirable to secure. They are of opinion that such a result may in a great measure be prevented by diminishing the number of questions, and they have agreed to recommend, that the papers containing the questions from books be shortened, in order to enable the candidates to give more time to Examples and Deductions."—Report of Mathematical Board for 1849.

The Report for 1850, with the same object in view, recommended a still further curtailment of the papers. The Report of the present year, while it again draws attention to the defect complained of in the former ones, acquaints us with the opinion of the Moderators aud Examiners, that the shortening of the papers has not had the desired effect. "It is obvious," it goes on to state, "that the only remedy lies in the previous practice and exercise of those who are to be examined," and in the students themselves "giving increased attention to the practical application of their reading. It is unnecessary to say anything in proof of the great importance of this portion of a mathematical examination, testing as it does very effectually the degree in which a student has really made himself master of the subjects which he professes to have read; and it is almost equally unnecessary to state, that a corresponding weight is attributed to it by the Moderators and Examiners, in estimating the relative merits of Candidates for Honours.'

A few observations therefore on the principles of the solution of this class of questions, exemplified by the solution of those actually proposed in the Senate-House, will not, it is hoped, be altogether useless to those who may feel the want of direction in a branch of their studies which forms so essential a preparation for the Examination for Honours.

Riders we define to be original questions arising either directly or indirectly out of the propositions to which they are appended. For distinctness' sake, we may divide them into the three following classes :

(1). The first and simplest kind are direct examples of a certain class of propositions; such, for instance, as investigate general rules for the various operations in different subjects. Examples of this kind are merely particular applications of the general rule which the proposition establishes, and must be answered by rigidly following out the method investigated in the foregoing book-work. It cannot be too carefully borne in

mind by the student, that the value of his solution of an example of this class is in exact proportion to the strictness with which it corresponds with the proposition.

(2). Another kind consists of those questions in which some fact or property enunciated in a theorem immediately preceding has to be applied; e.g.

PROP. "If a quantity vary directly as (a) when (b) is invariable, and inversely as (6) when (a) is invariable, prove that it

α

will vary as when both (a) and (b) are variable.

Ex. If 5 men and 7 boys can reap a field of corn of 125 acres in 15 days; in how many days will 10 men and 3 boys reap a field of corn of 75 acres, each boy's work being of a man's? (p. 24).

Under this head must be placed also all direct applications of formula. The point to be kept in mind is, that any solution independent of the formula or of the property enunciated in the proposition, however elegant in itself, and however excellent a solution of the question regarded as a problem, is altogether valueless as a solution of the rider.

(3). The third class consists of all those questions which are suggested by the proposition to which they are appended, or arise out of some particular part of the book-work investigation. This kind partakes more of the problematical character than the two former; but still we may in this case also apply the general observation, that the leading idea of the proposition or the method of its investigation should be the chief guide in the solution of the rider, and afford a pattern for its style.

It has been the aim of the following pages to follow out as closely as possible these principles. In all cases, the proposition has been given to which the question solved is a rider; and in several, a few observations have been made upon the proposition, which appeared necessary in order to connect it with

the question appended. At the end of the book will be found a short collection of Examples for practice.*

We will conclude with a word of practical advice to the student. Let him not consider any proposition, or piece of book-work, to have been thoroughly mastered till he has diligently practised examples connected with it, so as to be able, when called upon in an Examination, to apply it readily to any required purpose. In this way his knowledge of mathematics will become sound and practical, and the science itself will become interesting and attractive. To commit to memory a number of theorems, and then to reproduce them in examination without the power of exemplifying their use, is a process no less dry than useless; but he who makes himself, in the true sense of the word, familiar with them by an intelligent observation of their different uses and applications, and by acquiring a readiness in illustrating their utility, receives the full benefit from the wise system according to which this University appoints mathematics as the basis of her training-requiring of her members to study this branch of science, not so much for the purpose of acquiring the knowledge of it, as of disciplining their own minds by mastering it.

* It will be perceived that no Examples are solved under the head of Astronomy; the reason being, that this subject, as treated in the earlier part of the Examination, consists almost entirely of popular explanations, and not of propositions which can be applied to Examples.

1849.

EUCLID.

(A). Describe an equilateral triangle upon a given finite straight line. (Euc. I. 1.)

(B). By a method similar to that used in this problem, describe on a given finite straight line an isosceles triangle, the sides of which shall be each equal to twice the base.

Let AB (fig. 1) be the given finite straight line.

With centre A and radius equal to 2AB describe a circle CDF; and with centre B and radius equal to 2AB describe a circle CEF. Let the circles intersect in C. Join AC, BC. Then AC, BC being radii of the two circles are each equal to 2AB; and therefore ABC is the triangle required.

1850.

(A). The opposite sides and angles of parallelograms are equal to one another; and the diameter bisects them. (Euc. I. 34.)

(B). If the opposite sides or the opposite angles of any quadrilateral figure be equal, or if its diagonals bisect one another, the quadrilateral is a parallelogram.*

1848. (C). If the two diameters be drawn, shew that a parallelogram will be divided into four equal parts.

* This question must be considered as belonging to the third class of 'Riders.'

(See Introduction.)

B