PREFACE. It is scarce necessary to observe, that during the student's preparation for mathematical honors in the University of Cambridge, he is obliged to have frequent recourse to papers of examination questions, in order to satisfy himself and his instructors as to the degree of proficiency he may have attained. With the same object in view, it has now become the universal practice of those engaged in mathematical tuition, to draw up papers for the pupils under their charge; and in doing this, it is most important that the questions should be carefully chosen, and as much as possible of the same character as those usually proposed to students when they ultimately present themselves candidates for mathematical honors. The papers of questions which have actually been set from time to time, at the Senate-House examinations, have long been in use among the members of the university for the above purposes; it being justly argued, that the importance of the object for which they were originally formed, must ensure more judgment in the selection, and accuracy in the wording, than could be expected in those more hastily drawn up for the usual preparatory examinations. The Senate-House papers, however, so applied, labour under one great objection; namely, that as they have been formed for the examination of students, supposed to have a competent knowledge of the whole mathematical course pursued at the university, they contain questions in all the subjects, mingled together indiscriminately, so that a student endeavouring by means of them to examine himself in any particular branch, will find that a majority of those before him are foreign to his immediate purpose, or even depending upon investigations, of which, as yet, he may be entirely ignorant. To remedy this inconvenience, and to render so valuable a collection of problems and examples more generally available, they are in the present volume arranged in sections, corresponding to the various branches of science whence they are derived. These sections are placed, as much as possible, in the order in which the subjects naturally follow each other; and that it may be apparent what questions in each branch have been proposed in any one year, the dates are printed in the margin. Thus, at page 24, we see that the number of questions set in 1834, in "Arithmetic and Algebra, not including the general nature of Equations," was eighteen; and at page 60, that the number set in "Statics," in the same year, was sixteen. Such are the advantages it is hoped the present volume will be found to possess for the student; while the man of science, who devotes himself particularly to one or two branches of mathematical research, will be enabled by it to form a pretty just estimate of the attention applied at the university to his own favourite study, and the degree of proficiency which our most promising students are expected to attain in it. In this point of view, some description of the Senate-House examination may be useful to the reader. The students having resided the usual time in the university, and declaring themselves, through the tutors of their respective colleges, candidates for mathematical honors, are arranged in four classes, according to the degree of proficiency they are presumed to have attained. This arrangement, however, does not in any way influence the rank or class they may ultimately acquire; as the examination is, for the most part, the same for all the four classes; and the way in which they acquit themselves in it, is the only circumstance which determines their places in the list of mathematical honors, afterwards inserted in the public prints. The examination is conducted entirely by means of printed papers, prepared by the examiners; the particular questions to be asked being quite unknown to the students previous to their assembling at the appointed time to write their answers to them. When they are seated in the place appro priated to the examination, the examiner, whose turn it is to preside, delivers to each a copy of the questions he is to give his answers to, and generally from two to three hours are devoted to answering each paper. When the time allowed for answering the questions is past, each candidate delivers to the examiner the produce of his labours, leaving it to him and his fellow-examiners to affix to each a certain value dependent on its merit, which value is usually expressed in numbers called marks. The number of marks which each question, if correctly answered, entitles the candidate to, has been previously determined upon; and this number, in valuing the answers, is diminished, more or less, according to the degree of inaccuracy any particular one may betray. After a certain time intended for relaxation, the candidates return to the appointed place, when another paper is presented to them, and so on to the end of the examination, which on the whole employs them about twentyeight hours. The examiners having valued the answers, and summed up the number of marks gained by each candidate, arrange their names in the order of these numbers, and the list so formed is published in the Cambridge Calendar and in the newspapers. Ten papers of questions in succession are usually presented to each candidate, of which, four consist of original problems and questions, answers to which could not be directly obtained from any published books; these four being specially termed problem papers. The remaining six contain ques tions, the solutions of which the candidate is supposed to have acquired during his previous reading; and these, in contradistinction, are called book-work papers. The great responsibility of conducting such an examination, when so much depends upon its result, must at once be apparent to any one acquainted with the mathematical sciences; and the vast range which these sciences have at present attained, renders great care necessary in selecting the questions, so as to give no undue preference to any particular branch. The limited time allowed scarce admits of a full examination in each of the subjects, and still, limited as it is, the fatigue and anxiety endured by those meritorious students who aim at the highest honors is found in many instances to be so great, that any extension of it would probably produce evils which it is most important to avoid; constitutional strength would have too great a part in the contest, and might gain for mediocrity a reward intended only for superior acquirements. Our safeguard must be in judiciously choosing such questions as embrace as much as possible of the very essentials of science, and in so putting them as to exhibit clearly the intention of the examiner, and yet to give some exercise to the ingenuity of the candidates. There was a time when the scantiness of our mathematical resources, and the difficulty and uncertainty with which our investigations could be applied to the problems presented to our con |