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R. BALDWIN HAYWARD, M.A.,
SENIOR MATHEMATICAL MASTER IN HARROW SCHOOL
FORMERLY FELLOW OF ST. JOHN'S COLLEGE, CAMBRIDGE
In composing the treatise here offered to the student of Elementary Geometry, it has been my principal object to present the subject as a logically coherent body of propositions obtained by the simplest lines of deduction from postulates explicitly stated, and to extend its scope beyond the rather meagre range of the first twenty-one' propositions of Euclid's Eleventh Book, which has long been the customary limit of the young student's formal study of Solid Geometry. With this object I have found it necessary to deviate considerably from Euclid's sequence of propositions and largely to modify his demonstrations. At the same time, as the deviations will, I believe, be found to be in the direction of simplification by bringing the propositions nearer, along the line of deduction, to the fundamental postulates, I think that the student who gets up the subject from this book will, even as regards mere examination purposes, be at no disadvantage from not having studied the text of Euclid, as it appears in the usual editions.
Many of the demonstrations in Euclid's Eleventh Book, notably those of Props. 4, 6, and 8, are highly artificial; and I believe this to be mainly due to his dealing with perpendiculars before parallels and intersections, and treating the latter as dependent on the former. In Section I. of this work I have treated of Intersections and Parallels, and then in Section II. proceeded to deal with Normals and Obliques. In the latter I have assumed an axiom, which I have not thought it necessary to state, because it is a general axiom, applicable to magnitudes of all kinds, and not specially a postulate of Geometry, and because I hold that it is not logically necessary, but only a question of convenience, whether such general axioms or common notions (the Kolvài čvvolai of Euclid) should be explicitly stated, while logical coherence requires that every special postulate should be distinctly enunciated. The axiom may be thus stated : “If a magnitude be such that it cannot be made to vanish, there is a finite value, or a set of equal finite values, of that magnitude less than any other values." From this I establish the existence of a normal at every point of a plane, and the properties of normals then directly follow. I would invite attention to the simplicity attained by this mode of treatment, and to facilitate comparison I have added to each proposition a reference to the corresponding proposition in Euclid.