SECTION I. Gradual Growth of Geometry and of the Elements of SECTION II. Symbolical Notation and Abbreviations that may be SECTION III. Explanation of some Geometrical Terms... SECTION IV. Nature of Geometrical Reasoning.... SECTION V. Application of Arithmetic and Algebra to Geometry... 21 – 27 SECTION VI. On Incommensurable Quantities...... I. Problems 5-16, for the Construction of Geometrical Figures stated, and proved in Books I. and II..... Subsidiary Problems, 17 – 31, in Books I. and II. II. Problems in Books III., IV., and VI. most intimately connected 208 - 211 " III. Principles of Construction : 211 - 212 212 - 215 IV. Principles for accurately calculating Distances, Magnitudes, and Areas :- 215 216 SECTION 3. Magnitudes or Areas 216 Remarks 217 - 218 APPENDIX. I. Geometrical Synthesis and Analysis 219 - 223 223 – 225 226 - 233 SERIES 1°. Problems in Book I. 1-62 Theorems in Book I. 1-17 226 228 229 230 SERIES 2°. Propositions not fully proved, or not inserted in the Gradations :Problems.-Book I. 1-13.. 230 Theorems.- Book I. 1-15... 231 Problems.-Book II. 1-10.. 232 Theorems.-Book II. 1-15.. 232 p. 160, “ Even the Sleeping Geometrician,” says old Ralph CUDWORTH, “ hath at that time, all his Geometrical Theorems and Knowledges some way on him : as also the Sleeping Musician, all his Musical Skill and Songs : and therefore why may it not be possible for the Soul to have likewise some Actual Energie on it, which it is not Expressly Conscious of ?" John Heywood, Printer, 141 and 143, Deansgate, Manchester. In Three Parts, 4to. at 6d. each Part. Books I. AND II. FOR PEN-AND-INK EXAMINATIONS. N.B.-The SKELETON PROPOSITIONS may be had either with or without References, stitched in a cover, or collected into a case in sheets. Price 4d., Exercises for the first and Second Books of Guclid; BEING THE APPENDIX TO THE GRADATIONS IN EUCLID : Containing GEOMETRICAL ANALYSES and GEOMETRICAL EXERCISES ; with an EXPLANATORY PREFACE respecting the SKELETON PROPOSITIONS. Price 6d., FOR THE FIRST AND SECOND BOOKS OF EUCLID, CRITICAL NOTICES. “Mr. Green's 'Gradations' is a valuable work, though it takes in only the first two books of Euclid. Without endorsing all the detail, which we have not had time to examine, we may unhesitatingly speak in praise of the method, which we have examined with care. No introductory Euclid which has fallen in our way contains anything like the help, in the shape of notes and arrangement, that this does." “The whole of the propositions are arranged after the manner recommended by Professor De Morgan, in his Treatise on the Study and Difficulties of Mathematics. To many of the propositions is appended an account of the practical uses to which the proposition may be applied. An Appendix is added, which contains much that is useful in the way of notes and exercises. We feel sure that schoolmasters will largely patronise Mr. Green's 'Gradations in Euclid,' as a very complete apparatus for teaching the First and Second Books.”—Papers for the Schoolmaster, July, 1858, p. 112. “We commend the work to the consideration of Teachers.”—Athenaun, July 17, 1858. “ By printing the distinct parts of the propositions separately a very great help is gained, and the principles more thoroughly learned.”—The Bookseller, July 24, 1858. “This is a useful book ; in fact, the best introduction to Euclid with which we are acquainted, and we heartily recommend it to the favourable notice of our readers. Its main object is to combine theory with practice, and so soon as a geometrical truth has been established, to point out its use and application.”—The School and the Teacher, August, 1858. 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