.

105

107

Note 11. EUCLID'S PROPOSITION V. OF BOOK I.

Note 12. EUCLID'S PROPOSITION VI. OF BOOK I.

Note 13. EUCLID'S PROPOSITION VII. OF Book I.

Note 14. EUCLID'S PROPOSITION VIII. OF Book I.

Note 15. ANOTHER PROOF OF EUCLID I. 24

Note 16. EUCLID'S PROOF OF PROP. XXVI. OF BOOK I.

MISCELLANEOUS EXERCISES ON BOOKS I. AND II.

DEFINITION XIII.

EUCLID'S PROPOSITIONS XXI. AND XXII.

Note 3. ON THE METHOD OF SUPERPOSITION AS APPLIED

TO CIRCLES

EUCLID'S PROPOSITIONS XXVI. TO XXIX.

Note 4. ON THE SYMMETRICAL PROPERTIES OF THE CIRCLE

WITH REGARD TO ITS DIAMETER

151

153

155

159

EUCLID'S PROPOSITIONS XXX. TO XXXVII.

MISCELLANEOUS EXERCISES ON BOOK III.

EUCLID'S PROPOSITIONS XXIII. AND XXIV. OF BOOK III. 175

APPENDIX TO BOOKS I.—IV.

EUCLID PAPERS SET IN THE CAMBRIDGE MATHEMATICAL

TRIPOS FROM 1848 TO 1872.

198

EUCLID'S ELEMENTS-BOOK V.

SECTION I.-ON MULTIPLES AND EQUIMULTIPLES-Pp. 211 to 214.

SECTION II.-ON RATIO AND PROPORTION-Pp. 215 to 229.

SECTION III.-CONTAINING THE PROPOSITIONS MOST FRE-

QUENTLY Referred to IN BOOK VI.-Pp. 221 to 226.

SECTION IV.-ON PROPORTION BY INVERSION, ALTERNATION

AND SEPARATION-Pp. 227 to 230.

SECTION V.-CONTAINING THE PROPOSITIONS OCCASIONALLY

REFERRED TO IN BOOK VI.-Pp. 231 to 234.

SECTION VI.-CONTAINING THE PROPOSITIONS TO WHICH NO

REFERENCE IS MADE IN BOOK VI.-Pp. 235 to 242.

ELEMENTS OF GEOMETRY.

INTRODUCTORY REMARKS.

WHEN a block of stone is hewn from the rock, we call it a Solid Body. The stone-cutter shapes it, and brings it into that which we call regularity of form; and then it becomes a Solid Figure.

Now suppose the figure to be such that the block has six flat sides, each the exact counterpart of the others; so that, to one who stands facing a corner of the block, the three sides which are visible present the appearance represented in this diagram.

Each side of the figure is called a Surface; and when smoothed and polished, it is called a Plane Surface.

The sharp and well-defined edges, in which each pair of sides meets, are called Lines.

The place, at which any three of the edges meet, is called a Point.

A Magnitude is anything which is made up of parts in any way like itself. Thus, a line is a magnitude; because we may regard it as made up of parts which are themselves lines.

The properties Length, Breadth (or Width), and Thickness (or Depth or Height) of a body are called its Dimensions.

We make the following distinction between Solids, Surfaces, Lines, and Points:

A Solid has three dimensions, Length, Breadth, Thickness. A Surface has two dimensions, Length, Breadth.

A Line has one dimension, Length.

A point has no dimensions.

S. E.