case can arise in which it is necessary to retain a magnitude so closely that no alteration however small can be permitted. But in exact reasoning, where any error however small is to be avoided, it is obvious that the arithmetic of commensurable magnitudes, and the arithmetic (if there be such a thing) of incommensurable magnitudes, must not be confounded." A clear example of approximating nearer and nearer to a point which can never be reached, is supplied by drawing a tangent to the diameter of a circle, and from the centre drawing a line until it shall cut the tangent,-this line cutting the tangent being named the secant; as in the circle TAF, TS being the tangent, and OS the secant. If in TF, or TF produced, points be taken, E, F, G, H, &c., as centres of circles, to all of which TS shall be the common tangent; then each circle, as it cuts AS, shall approach nearer to S, as in the points B, C, and D; but no circle shall ever pass through the point S, inasmuch as TS being the common tangent, the circles cannot touch TS in any point except the point T; for if they did, a curved line and a straight line would coincide. In a similar way, if £20 be subscribed annually for the purchase of books, and at the end of each year the books of the previous year be sold at half-price, and the proceeds of the sale added to a new subscription of £20, and if the two sums thus added be expended in the purchase of more books, and so on, year after year, the half of the value of the preceding year's books being yearly added to the fixed subscription of £20, there will never be £40 worth of books purchased in any one year; though in each successive year the expenditure will approach nearer and nearer to £40. SECTION VII. OF WRITTEN AND ORAL EXAMINATIONS. Written Examinations may be considered as the suitable test of accuracy,-oral examinations of readiness; the one allows of the exercise of the reflective powers; the other brings into play quickness of perception and leads to promptness of action. The union of the two kinds, according to the nature of the subject under examination, should be aimed at in striving to ascertain progress, and actual knowledge and skill. 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 questions of a written examination ought to run, and on calculations, altogether out of place in an oral examination." For subjects not mathematical, however, a high place may be 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." 339 30 WRITTEN AND ORAL EXAMINATIONS. 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. For those who purchase 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. |