## Mathematics and Physics National Examinations

James Booth delivered two lectures in 1856 which were published as

*How to learn and what to learn. Two lectures advocating the system of Examinations established by the Society of Arts; the former delivered on the 24th of September, at the Mechanics' Institution, Lewes, the latter on the 16th of October, 1856, at the Mechanics' Institution, Hitchin.*The published version of the lectures contains an appendix with the*Programme of Examinations for 1857*. Booth was an examiner for Mathematics, Physics and Geography. The programme set by the examiners for Mathematics and Physics is given below:**1. Mathematics.**

The Examiners in Mathematics recommend that the mathematical course should comprise:-

Arithmetic.

A knowledge of the principles of a decimal currency on the basis of the pound unit.

Book-keeping by double entry.

Algebra, including Quadratic Equations, and the Theory and Application of Logarithms. Elements of Euclid, Books I. II. III. and VI., XL, (first 21 propositions,) and XII., (first 10 propositions.)

Mensuration.

Plane and Spherical Trigonometry, with the solution of both plane and spherical triangles by the use of Logarithmic Tables.

Land Surveying.

Nautical Astronomy.

The properties of Conic Sections treated geometrically, or graphically by the method of projections.

The Examiners do not attach much importance to the selection of the manuals in which mathematics may be learned. Such text books as those of Colenso, or, Barnard Smith in Arithmetic and Algebra, the Trigonometry of the Society for the Promotion of Christian Knowledge, A knowledge of the principles of a decimal currency on the basis of the pound unit.

Book-keeping by double entry.

Algebra, including Quadratic Equations, and the Theory and Application of Logarithms. Elements of Euclid, Books I. II. III. and VI., XL, (first 21 propositions,) and XII., (first 10 propositions.)

Mensuration.

Plane and Spherical Trigonometry, with the solution of both plane and spherical triangles by the use of Logarithmic Tables.

Land Surveying.

Nautical Astronomy.

The properties of Conic Sections treated geometrically, or graphically by the method of projections.

*Mathematical Tables*(Chambers' Educational Course), and the

*Elements of Descriptive Geometry*, (J W Parker, Stand), may be recommended. The Examiners will consider of no moment a familiarity with technical rules got up as exercises of the memory. A knowledge of the principles involved in the subjects of examination and a facility in applying them to particular cases will alone be valued. They strongly recommend students who take an interest in the practical applications of Geometry, to master the principal propositions in the Eleventh and Twelfth Books of Euclid.

The Examiners desire to impress on the notice of the candidates the importance of making themselves acquainted with methods of investigation and principles. They believe that much valuable time is lost over curious problems, which have nothing to recommend them but their difficulty, and in solving Algebraical equations, which require nothing but expertness in manipulating symbols.

The Examiners are of opinion that a candidate by directing the course of his reading in this way, with no greater expenditure of time than at present is given to acquiring a knowledge of the rudiments of Mathematics, might obtain some acquaintance with the principle of Duality, learn something of the method of Poles and Polars, of Transversal lines, and acquire some familiarity with the elementary principles of Solid and Descriptive geometry, and of the methods of Projection applied to derive the principal proportions of the Conic Sections from the circle. To solve Algebraical equations or Geometrical problems is seldom required in the business of practical life. The Examiners will be prepared to test the candidates by an examination conducted in principles and methods which may be obtained by careful reading, rather than by setting curious puzzles to tax their ingenuity.

**2. Physics.**

The Examiners in Mechanics would particularly direct the attention of candidates to the importance of obtaining a clear knowledge of the general principles on which all machines must be constructed. They are of opinion that this knowledge, though at first sight it may seem abstract, will in the end prove of far more value to the intelligent engineer than any early familiarity with the construction or working of individual machines. When the mind once lays fast hold of a general principle, it becomes a very easy matter to apply it as circumstances may arise. The uninstructed workman, after the familiarity it may be of years with the working of machinery in motion, comes at last to gather the general principle from a life-long induction of particulars. He invents for himself a theory of mechanics. Had he been properly instructed, he might have started where he ends. The Examiners will endeavour to test the candidates' knowledge of general principles applied to the construction of ordinary and familiar machines. The Examiners in Physics desire to direct the attention of candidates to the study of the properties of Heat, a knowledge of which is of so much importance to the right understanding of the action of the great motive power Steam, the construction of furnaces, and the economical consumption of fuel. The principles involved in the theory of Hydrostatics are of daily practical application. Indeed there is no knowledge of a general principle or acquaintance with a particular physical fact that may not sometime or other be turned to account.

The Examiners in Mechanics and in the Natural Sciences recommend that the course should comprise:-

The elements of Statics and Dynamics.

The Principles of Mechanism.

Practical Mechanics.

Hydrostatics.

Pneumatics.

Electricity.

The theory of heat, especially with reference to the properties of steam.

The Examiners recommend the separate treatises by Galbraith and Haughton, Lardner's The Principles of Mechanism.

Practical Mechanics.

Hydrostatics.

Pneumatics.

Electricity.

The theory of heat, especially with reference to the properties of steam.

*Handbooks of Natural Philosophy*, and Brooke's edition of Golding Bird's

*Elements of Natural Philosophy*.

The Examiners desire it to be distinctly understood that they can assign very little value even to correct answers, when unaccompanied by a demonstration, where practicable.

They consider the habit of accurate inductive reasoning to be an important secondary advantage accruing from the study of Mathematics and Physics.

*** Candidates in Mathematics or Physics, who are not prepared to be examined in the whole of the above subjects, may take up a portion of them and receive Certificates of the Grade to which the Examiner shall consider them entitled.