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In the large public schools of England, and in many of our other schools of established reputation, it has not, as a rule, been considered essential for all boys who are drawing near the end of their school career to pass any examinations in addition to those which are regularly held by the members of the school staff. In such schools the competition amongst the pupils is so keen, and the standard reached by the various forms so well known, that the fact that a pupil has reached a certain position in the school is a sufficient hall-mark for ordinary purposes. If, however, a boy wishes to complete his education at one of the Universities, or to enter one of the learned professions, he is required to pass some preliminary examination as a test of his general education.

In all these preliminary examinations a certain amount of mathematical knowledge is obligatory. In some of them "More advanced Mathematics" is one of the special subjects from which the candidate for admission is required to select a given number.

When we come to examine in detail the syllabuses of the several preliminary examinations for the different Universities, and for those professions in which a knowledge of Mathematics is not directly useful, it is rather difficult to determine from them what amount of Mathematics is generally considered to be necessary and sufficient for a liberal education. A certain amount of Arithmetic is required in all cases. A little Algebra and Geometry is also necessary for all, with the remarkable exception of Oxford University, where the candidate must take his choice between Algebra (to simple equations) and Elementary Geometry (substance of parts of Euclid, Books I., II., III.) The standard required for the Law is fixed by simple equations and four books of Euclid; for the medical profession quadratics and three books of Euclid; the requirements for the Army are frequently changing, and the examination is of a special type, which must be considered separately.

Of the Universities, the amount of Mathematics required for entrance at Oxford is practically fixed by the standard of Responsions as given above; that for Cambridge by the Previous Examination, in which the Algebra includes progressions, graphs, and logarithms, and the Geometry some knowledge of similarity. Corresponding to these two examinations, separate Matriculation Examinations are held by the Universities of London, Birmingham, and Durham, by the Joint Board of the Manchester Liverpool, Leeds, and Sheffield Universities, by the Welsh

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Scotch and Irish Universities, and by ges and Technical Colleges. Some idea it Matriculation Examinations may be hat no less than sixteen such examinathe regulations of the Institute of Civil ng, under stated conditions, from the sion as a student of that Institution. This te several Colonial University Examinaand other similar certificates which also Lexemptions. andard of the Elementary Mathematics deaminations is roughly the same as that of the

Examination given above. There are detail, such as the inclusion or omission of ces surds or proportion in Algebra, and simiThe questions set by the various examining fer so widely in character that this equality y more than nominal.

e the educational work carried on by the various teaching centres for higher education, they cal recognition of their duty to the sources ey derive their students by holding frequent

at various centres, and by examining and se stools in their various districts which desire At these examinations certificates are given to candidates, by means of which they can obtain c. most of the preliminary or matriculation examided that the range of subjects in which they have standard reached in them are satisfactory. ves the number of schools which avail themselves ces of the University Boards to examine the upper scus derably increased. In some cases the lower so examined, and junior or preliminary certificates to pupils who reach a high enough standard. over these examinations for lower forms, we may ce the jumble of pass examinations open to those

ed or are just leaving school into four classes. ege nominal standard required for a pass in Lematics.

Oond Higher Locals, the Cambridge Higher Locals es or which must have reached the age of 17.or have yo of the examinations in Class 2-and the Prelimion of the Institute of Civil Freineers

Wind Senior Locals the Cambridge Serior Locals. ge Pericus about sixteen different Matriculation some of the harder Preliminary Examinations, the Severtors 1st Class and the examinations for School

gher School Certificates. Serior School CertidSavol Leaving Certificates held by the Scotch Edues, the Intermediate Education Board of Ireland,

the Oxford and Cambridge Schools Examination Board, the Delegates of the Oxford Locals, the Cambridge Local Syndicate, and the Matriculation Boards of most of the other Universities and University Colleges.

3. The Junior Locals, Junior School Certificates, College of Preceptors 2nd Class, and the easier Preliminary Examinations. 4. Oxford Responsions.

In many of the above Certificate Examinations Algebra and Geometry are not compulsory subjects, but a certificate is given to any candidate who satisfies the examiners in a stated number of subjects. The certificate given states the subjects in which the candidate has passed, and the subjects, if any, in which distinction has been gained. Such certificate exempts the successful candidate from the various Matriculation and Preliminary Examinations, providing that it shows a sufficiently high standard in the subjects required for the particular examination which it is desired to escape. Certificates are also granted which exempt from various examinations candidates who have not obtained a full certificate for the examination which they have taken, e.g. in the examination held in July 1910 for the Higher Certificate of the Oxford and Cambridge Schools Examination Board, there were 1,706 candidates for full certificates, of whom 911 were successful, and 294 of these obtained distinction in one or more subjects; 440 candidates, who had already obtained certificates in a former examination, were now trying to gain "distinction" in one or more subjects; 158 were successful in this. Of those who failed to gain a full certificate 24 obtained certificates to exempt them from Responsions, and 103 were exempted from a part or parts of the Previous Examination.

Similarly in the examination for School Certificates held by the same body in July 1910 we find that, besides those who gained full certificates of types which would exempt from Responsions or the Previous Examination, there were 14 candidates who obtained certificates to exempt them from Responsions, and 15 who secured exemption from parts of the Previous Examination. Thirty-three gained School Certificates qualifying for the Army, and 60 obtained Leaving Certificates for Army purposes.

So far we have considered the various examinations from the point of view of the student who must pass some examination for the definite object of entering a profession or one of the centres of higher or specialized education. We have found that he has in general a great variety of examinations to choose from which are nominally of the same standard. But these examinations differ in certain details, so that a complicated system of requirements is insisted upon in order that any one of these examinations may exempt the successful candidate from passing any other of the same class. (There are six pages of about 500 words each in the Calendar of the University of London devoted

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- from which most st be regarded. As or anyone to possess success of his educapurposes previously who leave school for escape any direct test s but there are always or their parents or the nature of a ceras been and how they pecially strong amongst es, but have not had the comparing themselves with This leads many students, ducation at the University one of the Matriculation certificate as they can at one ool examination held by one Many schools also find it didates as possible at these erable competition amongst the best show in them. such candidates, for whom the merely stepping stones to enable but represent the goal itself to have some system of classification the successful candidates may be by dividing those who have passed distinction in the subjects in which en shown and recording the fact upon combination of these two methods. It with all the alternatives, but the following Matriculation Examination the names of the published in two divisions, each diviorder. Subjects taken are entered upon the edit is given for distinction in any subject. compulsory subject. "More atics is one of the optional subjects of which kon. It includes the binomial theorem (positive metry to solution of triangles, and the straight in Analytical Geometry. Mechanics stands by the optional subjects. sprosporating examination held by the Joint MatriNand of the four younger Universities the names of

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published in alphabetical order in two

divisions, but supplementary lists are published of those who gain distinction in certain subjects. In order to gain distinction higher alternative papers must be taken. These higher papers in Mathematics, which may be taken instead of the compulsory pass papers, include questions on the binomial theorem, Trigonometry to solution of triangles, Analytical Geometry to the end of the ellipse, and Pure Solid Geometry of the straight line and plane. Mechanics or Physics ranks as one of the optional subjects of which two must be taken, and the corresponding higher alternative paper includes both Mechanics and Physics.

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Those who pass the Cambridge Previous Examination are also arranged in classes: but this examination is only open to those who have arranged to come into residence as members of the University, and is not available for the purpose now under consideration.

To gain a first-class certificate from the College of Preceptors a candidate must pass in six subjects, of which Arithmetic and English are compulsory, and may offer twelve. Honours are gained by reaching a certain aggregate of marks, and distinction may be gained in any subject by getting three-quarters of the maximum for the paper in that subject. Algebra, Geometry, Trigonometry, and Mechanics are separate subjects. The Algebra includes progressions; Geometry the standard of Euclid VI. 1-20; Trigonometry solution of triangles.

In the Oxford Senior Locals the subjects in which a candidate may be examined are divided into twenty-three sections, and a certificate is granted to any candidate who passes in one examination in five sections. No section is compulsory, but there are certain restrictions as to choice. The title of Associate of Arts is conferred upon successful candidates who are under 19 years of age. "Over age" certificates are also awarded. Subject to the above limit of age the successful candidates are arranged in three honour classes and a pass list. Those in the first class are in order of merit; the second class is arranged in not more than five divisions; the pass list is in two divisions. Special lists in order of merit give the names of those who have gained distinction in any of the sections. The holder of a certificate can enter for any sections at a subsequent examination and gain supplementary certificates, which may include distinction if the candidate is still under 19. Of the above twentythree sections the three which interest us are Arithmetic, Mathematics, and Higher Mathematics. In Arithmetic distinction is not awarded. In Mathematics Algebra to the binomial theorem (positive integral index) and Geometry to the end of the circle, suffice for a pass, but Trigonometry to the solution of triangles must also be successfully offered in order to gain distinction. In Higher Mathematics there are three divisions : (a) Higher Algebra and Plane Trigonometry (including some differential), (b) Higher Geometry including analytical conics

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