50 SKELETON PROPOSITIONS &C. OF EUCLID * BOOKS I AND II, WITH REFERENCES, FOR PEN AND INK EXAMINATIONS: ; BY HENRY GREEN, A.M. PART III. MANCHESTER: JOHN HEYWOOD, 170, DEANSGATE. LONDON: Kent & Co., Paternoster Row; SIMPKIN & Co., Stationers' Hall Court. EDINBURGH; BELL & BRADFUTE. GLASGOW: W. HAMILTON. 1858. REMARKS ON THE SKELETON PROPOSITIONS, &c. THE SKELETON PROPOSITIONS, &o., for pen-and-ink examinations, are arranged in two Series,--one with the references in the margin; the other without those ' references. The first Series is intended for beginners; the second, for those who may be reasonably supposed to be prepared for a strict examination. The two Series will be found well adapted to test the Progress of the Learner, and to ascertain how far his knowledge of geometrical. principles, and his power to apply them, really extend. The object is, in the first Series, to furnish the Learner, step by step, with the truths from which other truths are to be evolved, but to leave him to work out the results, and from the results, as they arise, to aim at more advanced conclusions: in the second Series, where there are references in the margin, the object is to make the examinations strict and thorough, yet so as to be conducted on one uniform plan. This uniformity will be found greatly to assist Examiners, when they compare the examination papers together for the purpose of deciding on their respective merits. no The Skeleton propositions may be used either simultaneously with the Gradations, or, after the first and second books have been read in any of the usual editions of Simsox's Euclid, as a recapitulation of the ground already gone over: if used simultaneously, the Learner must first study the Definitions and Propositions irr their order, and then, laying the Gradations aside, reduce his knowledge to a written form, as the references indicate in the vertical columns of the Skeleton Propositions; but if used as a recapitulatory exercise, a course in some respects different is recommended. In the recapitulatory excercise, the following plan is recommended for adoption :- first, that the Learner should give in writing a statement of the meaning of various Geometri. cal Terms, of the nature of Geometrical Reasoning, and of the application of Algebra and Arithmetic to Geometry; secondly, that he should fill in, not by copying from any book, but from the stores of his own mind and thought, trained by previous study of the 'GRADATIONS or of some similar work,—the Definitions, Postulates, and Axioms of which the leading words are printed; and thirdly, that he should proceed to take the Propositions in order, and write out the proofs at large, as the printed forms and references in the margin indicate: this should be done systematically in all the propositions, beginning with those truths already established which are required for the Construction and Demonstration, and then taking in order the Exposition, the Data and Quæsita, or the. Hypothesis and Con 81 SKELETON PROPOSITIONS, &c., with REFERENCES. Воок ІІ. . DefinitioNS. Every right-angled parallelogram, or rectangle, is said to be contained Prop. I. Theor. If there be two st. lines, one of which is divided into any number of parts, the rectangle contained by the two st. lines is equal to the rectangles contained by the undivided line and the several parts of the divided line. Cox, P. 11. 3, 31, I. Den. 31 I, Ax. 8, Exp. | 1 | Hyp. 2 Concl. |