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TABLE OF CONTENTS.

PAGES.

PREFACE.
Respecting the Gradations in Euclid's Plane Geometry, Skeleton

Propositions, &c.

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INTRODUCTION.
SECTION I. Gradual Growth of Geometry and of the Elements of
Euclid

1-5
SECTION II. Symbolical Notation and Abbreviations that may be
used

5-7
SECTION III. Explanation of some Geometrical Terms...

7-10
SECTION IV. Nature of Geometrical Reasoning.....

10-20
SECTION V. Application of Arithmetic and Algebra to Geometry... 21 – 27
SECTION VI. On Incommensurable Quantities......

28 - 30
SECTION VII. On Written and Oral Examinations and Means of
Progress

30 - 35

BOOK I.

Geometry of Plane Triangles...

39 - 148

Definitions

40 – 46

Postulates and Axioms

47 - 49

Propositions i. – xlviii...

51 - 147

Remarks

148

208 - 211

>

III. Principles of Construction :
SECTION 1. For Geometrical Instruments to Measure Distances
and Angles

211 - 212
SECTION 2. For Geometrical Figures to exhibit representative
values of Magnitude and Space...

212 - 215 IV. Principles for accurately calculating Distances, Magnitudes,

and Areas :-
SECTION 1. Lines or Distances

215
SECTION 2. Angles

216 SECTION 3. Magnitudes or Areas

216 Remarks

217 - 218

APPENDIX.

I. Geometrical Synthesis and Analysis
II. Variety of Paths ...
III. Geometrical Exercises

219 - 223 223 – 225 226 - 233

SERIES 1°. Problems in Book I. 1-62

226 Theorems in Book I. 1-17

228 Problems in Book II. 1-14

229 Theorems in Book II. 1-6

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

“Even the Sleeping Geometrician," says old RALPH CUDWORTH, p. 160, "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.
SKELETON PROPOSITIONS TO EUCLID,

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.,
KEY TO EXERCISES

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.

"Mr. Henry Green, A.M., has published the first and second books of Euclid, for pupil. teachers and others. We recommend this as a very useful work introductory to plain geometry, and one which may be put into the hands of either schoolboys or adult mechanics with great advantage.”—Manchester Guardian, Dec. 2, 1858.

Also, Price 4s. 6d.,

EUCLID'S PLANE GEOMETRY, Books II. — VI., OR GRADATIONS IN EUCLID, PART II., WITH ALGEBRAICAL

AND ARITHMETICAL ILLUSTRATIONS, EXPLANATORY NOTES, AND A SYNOPTICAL INDEX- TO THE Sıx Books, SHOWING THE USES OF THE PROPOSITIONS, &c.

School Books published by John Heywood.

HOME EXERCISES.
WILLIE'S HOME EXERCISES.

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GRADE I. Embraces the First Half of Part I., adapted to Standards I. and II.

One Penny.

GRADE II.
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Threepence ; or Cloth Limp, Fourpence.

PART I.
This contains a systematic and graduated Series of Exercises in the

Simple Rules of Arithmetic, Spelling, and Composition, for Home
Practice. It is adapted to Standards I., II., and III. of the Revised

Code.
Stif Paper Covers, price Pourpence; Cloth, Sixpence. With 9 plain

Maps, Cloth, Tenpence; with 9 coloured Maps, strongly bound and gilt lettered, One Shilling and Fourpence.

PART II. This is a systematic and graduated

Series of Exercises in the Compound
Rules, Proportion, Practice, Spelling, Composition, Grammar, and
Geography, for Home Practice.

In Parts I. and II., Sixpence,
ANSWERS TO ARITHMETIC.

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School Books published by John Heywood.

SCIENCE

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