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rems or problems given by most writers on the same subject.
The Editors have to thank several friends for examining proof-sheets; in particular, Mr. A. F. Smith, B.A., The Grammar-School, Chester; Mr. H. Crofts, B.A., Parkfield School, Liverpool; Mr. G. M. Bates, B.A., Headmaster of the Harpur Trust Elementary School; and Mr. A. E. Field, B.A., of Trinity College, Oxford. They would be obliged to those teachers who use their work as a class-book for early notice of any errors that have escaped correction, and for such suggestions for its improvement as experience may provide.
EDWARD M. LANGLEY.
The Editors desire to call attention to the Reports of the Mathematical Board at Cambridge, and the Board of the Faculty of Natural Science at Oxford, upon the Memorial presented by the Council of the Association for the improvement of Geometrical Teaching to these Universities, praying for such changes in their Examinations in Elementary Geometry 'as would admit of the subject being studied from Text-books other than editions of Euclid, without the student being thereby placed at a disadvantage in those Examinations.'
The Cambridge Board reports :—
'The majority of the Board are of opinion that the rigid adherence to Euclid's texts is prejudicial to the interests of education, and that greater freedom in the method of teaching Geometry is desirable. As it appears that this greater freedom cannot be attained while a knowledge of Euclid's text is insisted upon in the examinations of the University, they consider that such alterations should be made in the regulations of the examinations as to admit other proofs besides those of Euclid, while following however his general sequence of propositions, so that no proof of any proposition occurring in Euclid should be accepted in which a subsequent proposition in Euclid's order is assumed.'
The Oxford Board, in nearly equivalent terms, reports :—
'I. That a rigid adherence to the ordinary text-books of Euclid should no longer be insisted on, but that a greater freedom of demonstration should be allowed, both in Geometrical teaching and in Examination.
'2. That, nevertheless, Euclid's method should be required in all Pass Examinations in Geometry, in so far as that no axioms other than those of Euclid shall be admitted, and that no proof of a proposition be allowed which assumes the truth of any proposition which does not precede it according to Euclid's order.'
1. A point is that which has no parts, or which has no magnitude.
2. A line is length without breadth.
3. The extremities of a line are points.
4. A straight line is that which lies evenly between its extreme points.
A straight line is sometimes spoken of as the join of its extreme points.
5. A superficies (or surface) is that which has only length and breadth.
6. The extremities of a surface are lines.
7. A plane superficies (flat surface) is that in which any two points being taken, the straight line between them lies wholly in that superficies.
9. A plane rectilineal angle is the inclination of two straight lines to one another, which meet together, but are not in the same straight line.
is pointed out in the Syllabus that the term angle is incapable of real definition. It would be better to say with the Syllabus, 'When two straight lines are drawn from the same point they are said to contain, or to make with each other, a plane angle,' and to indicate the nature of an angle by saying that the angle is greater or less, according as the amount of turning round the point that would bring either of the lines into coincidence with the other is greater or less. (See Syllabus)
When several angles are at one point B, any one of them is expressed by three letters, of which the letter which is at the vertex of the angle, that is, at the point at which the straight lines that contain the angle meet one another, is put between the other two letters, and one of these two letters is somewhere on one of those straight lines, and the other letter on the other straight lint. Thus the angle which is contained by the straight lines AB, CB is
named the angle ABC or CBA; the angle which is contained by the straigh lines AB, DB is named the angle ABD or DBA; and the angle contained by the straight lines DB, CB is named the angle DBC or CBD. But if there is only one angle at a point, it may be expressed by a letter placed at that point; thus the angle EFQ might be called simply the angle F. The latter method should always be employed when possible.
14. A figure is that which is enclosed by one or more boundaries.
A figure contained by three straight lines is called a
15. A circle is a plane figure contained by one line, which is called the circumference, and is such that all straight lines drawn from a certain point within the figure to the circumference are equal.
16. This point is called the centre of the circle, and the straight line drawn from the centre to the circumference is called the radius of the circle.