4. The most important Propositions are,-three, namely, Props. 4, 8, and 26, containing the criteria, or conditions of equality between triangles ; one Prop. 32, the equality of the exterior angle to the two interior and opposite angles, and of the three interior angles of every triangle to two right angles ; one Prop. 41, the proportion of the parallelogram to the triangle on the same base and between the same parallels ; and one Prop. 47, the relation between the hypotenuse and the sides about a rt. angle. These propositions at least must be thoroughly mastered, not by committing them to memory, but by becoming so perfectly familiar with the principles contained in them, and with the connexions which exist between the arguments or reasonings employed, as never to feel at a loss for the demonstration, however diversified may be the figures constructed, nor even though no figure at all be drawn. The great aim should be to understand, and as a means to this, to follow up each proposition regularly through all its gradations, and verify it by its appropriate proofs. BY THE SAME AUTHOR, Price 6s. 6d., pp. about 680, Diagrams above 570; EUCLID'S PLANE GEOMETRY, Books I-VI., Practically Applied ; With an Explanatory Preface, Algebraical and Arithmetical Illustrations, Explanatory Notes, Synoptical Index, showing the Uses of the Propositions, &c. CRITICAL NOTICES, « This is a most valuable edition of Euclid, and for the purposes of the practical tutor will be found a very great aid.” “On the whole, we have met with no edition of Euclid (in a considerable experience) so likely to aid the teacher by its thorough grasp of the subject, and the great variety of its practical illustrations.”—The Spectator. “Of this edition of Mr. Green's we can speak very highly. 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It is invaluable to the selftaught."--The Kendal Mercury. ** A book very serviceable to all students, but of special value to the self-taught."-Dr. Beard's Self-Culture, p. 334. “No introductory Euclid, which has fallen in our way, contains anything like the help, in the shape of notes and arrangement, that this does." - Papers for the Schoolmaster. " 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.”—The School and the Teacher. “A very useful work"_" which may be put into the hands of either schoolboys or adult mechanics with great advantage."--Manchester Guardian. ALSO, EXERCISES for the first and second Books of Euclid, price 4d. MANCHESTER : JOHN HEYWOOD, PRINTER, 143, DEANSGATE. Works published by John Heywood, 143, Deansgate, Manchester. Drawing of a Condensing Steam Engine, Size, 19 by 17 inches, with all the principal parts named. 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