NAVIGATION AND ASTRONOMY.

STAGE 3.

Results 1st Class, -; 2nd Class, 2; Failed, 3; Total, 5.

Of five candidates examined, one took Spherical Astronomy, the remainder confining their attention to Section A, which deals with Navigation and Nautical Astronomy. One candidate gained 53 per cent. of total marks, and a second 38 per cent. The other papers were failures. The candidate who answered questions in Stage 3, Spherical Astronomy, sent up one good answer, and attempted two other questions not wholly without

success.

HONOURS.

Results 1st Class, -; 2nd Class, 1; Failed, 1; Total, 2.

Two papers were sent in, one of which obtained 65 per cent. of the maximum. The other was a failure. Neither candidate took Spherical Astronomy.

Report on the Examinations in Practical Plane and Solid Geometry.

EVENING EXAMINATION.

STAGE I.

Results 1st Class, 649; 2nd Class, 518; Failed, 658; Total, 1,825. The number of candidates in this stage was 1,825, being five less than in 1905. The high standard of last year was maintained, and there are many schools in which the teaching is extremely efficient. Not only are the principles of the subject well understood, but care and accuracy in drawing is secured, the amount of distinctly careless work in these schools being very small. On the other hand, there are some schools in which the work is almost wholly bad, and these schools account for nearly the whole of the failures.

In all schools there is evidence that some portions of the subject do not receive adequate treatment, and that improvements are possible, more especially with students of less than average ability. The following points may be specially mentioned:

(a) There is a general lack of exact knowledge of the simpler facts relating to the tangency of lines and circles. See the detailed remarks under Question 3.

(b) In dealing with the important subject of vectors, the teaching often lacks precision. The simple graphical rules for the addition and subtraction of vectors have not been assimilated, and confusion results. In specifying direction, many candidates seem unable to distinguish East from West, and clockwise angles from counter-clockwise. Direction arrows are often omitted, a serious defect in vector work.

(c) The evidence afforded by the answers to Question 10, relating to a line in space, indicates that this branch of the subject is either badly taught or comparatively neglected. In a large number of cases the candidates were evidently quite unable to form any mental picture of the conditions of the problem, and their answers were absurd and worthless. The only successful method of dealing with lengths and angles in space is to provide students with a few simple models, devoid of construction lines, on which these quantities are directly measured, and the results verified by drawing projections of the models, and then determining the same lengths and angles by geometrical_construction. Suitable and inexpensive models will be suggested by a study of the questions set in the day and evening examination papers of the last few years.

(d) Teachers might, with advantage to themselves and to their students, occasionally ask for written answers to questions on certain parts of the subject which are of fundamental importance. In this way, misconceptions and difficulties could be detected and corrected, and, at the same time, useful practice would be provided for the students.

Q. 1. In working this question employ a decimal scale of 1 unit.

inch to

Draw a circular arc, radius 10 units, centre O. Mark a chord AB of this arc, 347 units long, and draw the radii OA, OB. Measure the angle AOB in degrees.

From B draw a perpendicular BM on OA, and at A draw a tangent to meet OB produced in N. Measure carefully BM and AN (on the above unit scale), and calculate the sine and the tangent of the angle AOB.

Give the correct values of the angle, the sine and the tangent, taken directly from the examination tables supplied.

The first two parts were well answered, indicating a widespread knowledge of trigonometrical ratios, but a rather common error was to make the chord 3:47 inches long instead of 3'47 half-inch units. With regard to the last part, comparatively few discovered from the tables that the angle whose chord is 0:347 is exactly 20°, and therefore that the correct values of the other functions were known.

Q. 2. The figure is the plan of a building site to a scale of 1 inch to 10 yards. Determine the area of the site in square yards.

Of the optional questions this was the favourite, and the answers were fairly satisfactory. The most usual though not the best method was division into triangles and calculation of their separate areas by measurements of bases and heights. Errors in scaling and measurements in vulgar fractions were few, and the arithmetic was not so bad as in some previous vears. Some, in attempting to reduce the figure to an equivalent triangle, went wrong over the re-entrant angle.

Q. 3. Draw the electric lamp to the dimensions given, which are in millimetres. Mark carefully the points of junction of the several arcs.

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N.B. A mere copy of the diagram will receive no credit. Attempted by about 60 per cent. of the candidates. Very few obtained correctly the points of junctions of the circular arcs, or determined properly the centres of the upper and lower arcs. Students seem to have a very scant knowledge of elementary tangential properties of circles. Teachers should give attention to this matter.

Q. 4. In arranging some elementary experiments in statics, cyclist trouser clips are used as spring balances. In order to be able to measure pulls, a series of weights are hung on a clip, and the corresponding openings AA are measured. The results being plotted, give the curve shown. For instance, when the pull is OM ounces, the opening is ON inches. On SS construct a decimal scale of ounces, which, being applied to the spring at AA, shall measure any pull up to 32 ounces.

Rarely attempted, not more than 10 per cent. of the candidates taking the question. There were a few full and complete answers, but the great majority got no marks. A scale of ounces in equal parts was generally shown, and even this was seldom sub-divided decimally.

Q. 5. The figure is the plan of a corner of a landing. Sis a stone, on which a stove is to rest. Show the corner of the landing carpet folded over along the line LL. On this fold draw the hole which must be cut in the carpet, so that when the latter is turned back into position the hole shall just fit the stone S.

This question was, perhaps, too easy. It was frequently attempted and very well answered.

Q. 6. Two pieces of sheet material are hinged together at A

One of them is pinned or hinged to the drawing board at B, and the point P on the other piece is moved in a straight line from L to M. Find the paths described by the points Q and R of the second piece.

Neither a favourite question nor well answered. The results were very inaccurate, and few discovered the geometrical character of the loci. It might, perhaps, have been better to have stated explicitly in the question that AB, AP, AQ, and AR were equal, and that P, A, Q, were in a straight line.

Q. 7. A and B are two points in a body which is moving in a plane. At a certain instant the positions of A and B are given by the

O being a fixed point of reference, and angles being measured anti-clockwise from a line drawn along the tee square to the right (eastwards).

Plot the points O, A, and B to a scale of 1" to 1'. Measure the magnitude and direction of the vector AB, that is, the position of B relatively to A.

This was usually well done, though some gave the vector sum OA+OB instead of the vector difference OB-OA. Also occasionally West was substituted for East, and clockwise rotation for counter-clockwise.

Q. 8. The vertices A, B, C of a triangle, referred to a point O as in Q. 7, have the positions defined by the vectors

Determine the centre of area, G, of the triangle, having given

Seldom attempted and rarely well done. Very few obtained G in any way. Some made the vector addition OA+OB+OC but did not plot G from the result. The vector equation given was in fact seldom understood. Q. 9. A piece of card is pinned temporarily to a drawing board, and forces P and Q are applied to the card as shown. It is desired to balance these two forces by a force of 30 ounces acting as A, so that the card shall remain at rest on removal of the pins. What must be the line of this third force, and what the original magnitudes of ' and Q?

Not very frequently attempted. Candidates often failed to see that the line of the required force must pass through the intersection of P and Q. But having assumed an arbitrary line through A, they were generally able to draw a corresponding triangle of forces, and to measure results consistent with their initial error.

Q. 10. A thin metal plate ABC, resting on the ground, is shown in plan A piece of thin wire AD, 2.5" long, with one end soldered to the plate at A, is also shown in plan at Ad.

(a) What is the distance of the end D of the wire from the plate?

(b) What angle does AD make with the plate?

(c) Draw the elevation of the plate and wire on xy.

There were many good answers, but far too many worthless ones, and, on the whole, the results were disappointing. A frequent mistake was to draw the elevation of AD on xy 2" long, and to give in answer to (b) the

angle between this elevation and xy. Sometimes (a) was correctly answered, and the angle just mentioned given in answer to (b). The given xy was often incorrectly transferred; and the elevations a', ', c' were often shown above the xy line.

Q. 11. The roof of a house is rectangular in plan, and two adjacent surfaces are each inclined at 32° to the horizontal. What is the inclination to the horizontal of the "hip," or line where the two surfaces meet?

Not very frequently attempted. The question was not well understood. Q. 12. The hopper shown in the diagram is required to be lined on its inner surface with sheet metal bent out of one piece. Find the shape of this piece, the joint being along AA.

Many good answers. With the poorer candidates, a common mistake was to give the shape of the elevation as the shape of the development. There was some want of accuracy in extending the development, after having obtained the true shape of one face.

Q. 13. Two elevations of a kitchen salt-box are given.

Copy the

elevation A, and from it project a plan. From the plan project an elevation on x'y'.

N.B.-The view B need not be drawn.

A favourite question and satisfactorily answered. The principal defect was inaccuracy in transferring dimensions.

Q. 14. The figure shows a portion of a timber joint. Represent this in pictorial projection, in a manner similar to that used for the cube, where lines parallel to oy and oz are drawn horizontally and vertically, and to a scale of full size, and lines parallel to ox are drawn by using the 45° set-square, and to a scale of half size.

N.B. The figure need not be copied, dimensions being taken directly from the diagram.

Fairly often attempted, and with much success. This kind of projection is attractive to students, and they are able to apply it well.

STAGE 2.

Results 1st Class, 253; 2nd Class, 436; Failed, 218; Total, 907.

The

In this stage there was an increase of 8 per cent., the numbers of candidates for 1905 and 1906 being respectively 843 and 907. The work was very satisfactory and the proportion of failures unusually small. previous remarks under (b), (c) and (d) apply in this stage, though not quite to the same extent. Further, more care should be given to the accurate determination of the lengths of curves, as detailed in the special remarks on Questions 21 and 32. And in some schools the use of blunt soft pencils is still permitted.

Q. 21. In working this question employ a decimal scale of inch to 1 unit.

Draw a circular arc, radius 10 units, centre O. Mark a chord AB of this arc, 347 units long, and draw the radii OA, OB. Measure the angle AOB in degrees.

From B draw a perpendicular BM on OA, and at A draw a tangent to meet OB produced in N. Measure carefully BM, AN, and the arc AB (on the above unit scale) and calculate the sine and tangent of the angle, and the angle in radians.

Give the correct answers for the degrees, sine, tangent, and radians, the numbers being taken directly from the examination tables supplied.