(1) THE geometry of Mr. Buddon is a notable addition to the elementary text-books which owe their appearance to the freedom of the last few years. The subject is introduced by experimental work, very suggestive in character, and leading by induction to fundamental definitions and theorems. Thus from the sliding and folding of flat cards and the like the author arrives at his definition of a plane as “a surface, infinite in extent, which can be folded about any two points of the surface so that one part lies entirely on the other.” The definition of a straight line naturally follows as the infinitely extended fold of a plane. A plane angle is clearly and rationally defined. Parallel lines are those having the same direction in a plane, direction being measured by the angle made with any reference line. It is pointed out that a plane, a plane angle, and a straight line can in each case be reversed on itself, and thus symmetrical properties are satisfactorily established in which the two halves are alike but of opposite aspect. Then follow general cases of congruence. In dealing with ratio and proportion the idea of a continually subdivided decimal scale is introduced; this enables all numbers which can be expressed as continuous decimal fractions, e.g. 1.4142. …, to be included, and to any degree of approximation. In later chapters the subject-matter comprises a very full treatment of the properties of circles; elementary trigonometry; an introduction to projective geometry; conic sections treated by modern methods; and solid geometry with the mensuration of the simple geometrical solids. The book contains in profusion sets of graphical and deductive exercises. The figures are drawn with thick, thin, and dotted lines on a systematic plan to distinguish more readily between the data, the construction lines, and the result. The use of variable type serves to differentiate parts of greater or less importance. In fact, the book on every page bears witness to the great care and thought bestowed on its production. There is a stimulating freshness in the matter and its method of presentation. Some will doubt the wisdom of carrying on at school the study of pure geometry to the extent covered in the book; others may wish that the geometry of vectors had been included; but all will agree that the author has produced one of the most important of the new elementary text-books, and one that should be known to every teacher interested in the subject.
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