The advantages of Written Examinations in Geometry and in kindred subjects, are well pointed out by S. F. Lacroix in his Essay, "On Teaching in general, and on the Teaching of Mathematics in particular." He says, p. 197, 198, "It has been proposed to substitute examination by writing, which gives to the candidate more time to collect his ideas,-which lessens the disadvantages of timidity,—and which, being carried on at the same time for all the pupils, permits the same questions to be asked of each, and renders their answers more suitable for comparison. This written examination may also be less troublesome for the Examiner, because, instead of the unremitting attention which he must give to oral answers, and the efforts of memory necessary to recall to his mind the impression which those answers make, he has only a labour capable of being divided and suspended when he experiences too much fatigue; and all the papers which serve as a basis for his judgment, are at the same time under his eye."

"It is principally on the applications of theories, that the ques-tions of a written examination ought to run, and on calculations, altogether out of place in an oral examination."

be

For subjects not mathematical, however, a high place may assigned to Oral Examinations. Of written examinations Lacroix afterwards says, p. 199, "But by this written examination alone we are never perfectly informed as to the readiness with which a scholar may express himself—a readiness which it is necessary to exercise and encourage, because it is useful at almost every moment of life, and because it is indispensable for men who will some day have projects to bring forward or to discuss in the presence of their companions or of their superiors, and it is only an oral examination which can make them appreciated in this respect."

The Advantages of Written Examinations in Geometry have suggested the "Skeleton Propositions ;" and these advantages will, it is hoped, be increased by the aid which such outline propositions afford for training to method and exactness. Lest, however, the assistance given, by placing references in the margin, should be greater than is good for more advanced learners, a Second Course of Examinations is recommended—if, indeed, it be not absolutely necessary, a Course in which no other aid is afforded to those under examination than the General Enunciations of the Propositions and a few vertical lines, within which learners are themselves to place the references to the truths already established, and on which the construction and the demonstration depend.

Among means of Progress, it is of great advantage to have stored

32

WRITTEN AND ORAL EXAMINATIONS, ETC.

in the memory the VERY words of the Definitions, Postulates, and Axioms, and of the more important Propositions; and to associate with the words the numbers, as Definition 15, Axiom 8, Propositions 4, 8, 26, &c., of Book I.

But in the construction of Geometrical figures, and in the demonstration and application of Geometrical Truths, the Reasoning Faculty should be chiefly employed. A youth may repeat cleverly by rote every letter and every line of a long demonstration, and adduce the various proofs at the proper places, and yet be ignorant of the principles of the science. In the study of Geometry, no aids are so effectual as the determination and the endeavour thoroughly to understand the process of reasoning, and the nature and force of the argument.

As an instance of the method recommended to the learner, let him take that important Proposition, the 32nd of Book I. Having well considered the meaning of the words, let him, by reference to Prop. 31, and also to Prop. 29, Ax. 2; Prop. 13, and Ax. 1, recall to mind what is required for the construction of the figure, and what for the demonstration of the theorem. He may say to himself, "Here are several undoubted truths and facts: I have already proved them and accepted them as principles of Geometrical reasoning; and they are now given me that I may arrive at other truths. By using them, can I not demonstrate the inference or conclusion of this proposition?"

He may reply, that by thus exercising his judgment, he will do more than by any mere effort of memory, for really understanding and retaining Mathematical Truths.

For those who possess only the "Gradations of Euclid," and who yet wish to know the Plan proposed for the "Pen and Ink Examinations," an Example is now added of both Series,—of the one that has the references printed in the margin, and of the other without any references.

First,—with all the references by aid of which the Examination is to be conducted :

PROP. 1.-PROB.

To describe an equilateral triangle upon a given finite straight line.

Second,-without any

of the references, to make the examination

strict and thorough.

PROP. 1.-PROB.

To describe an equilateral triangle upon a given finite straight line.

SOLUTION.

DEMONSTRATION.

Exp.

CONS.

DEM.

USE AND APPLICATION.

The Skeleton Proposition when filled up will appear as below, symbols and contractions being allowed :—

PROP. 1.-PROB.

To describe an equilateral triangle on a given finite straight line. SOLUTION.-Psts. 3 and 1.-Pst. 3. A circle may be described from any centre at any distance from that centre.

Pst. 1. A st. line may be drawn from any one point to another. DEMONSTRATION.-Def. 15, and Ax. 1.-Def. 15. A circle is a plane figure bounded by one continued line called its circumference, and having a certain point within it from which all st. lines drawn to the circumference are equal.

Ax. 1. Magnitudes which are equal to the same, are equal to each other.

USE AND APPLICATION.-This problem may be applied to the measurement of inaccessible lines, by drawing on wood or brass an equil. triangle, and using the instrument, by placing it at . A, and along the st. line A B, so that .s C and B may be seen; then, if it be carried along A B, until, at B, C and A can be seen along the edges of the instrument, the side A B will have been traversed, and side A B is equal to AC or BC. The side A B, being measured, will equal the other distances, A C or C B.