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THE

FIRST SIX

AND THE

ELEVENTH AND TWELFTH BOOKS

OF

EUCLID'S ELEMENTS;

WITH

THE ELEMENTS OF PLANE TRIGONOMETRY,

AND

AN APPENDIX IN FOUR BOOKS.

WITH NOTES AND ILLUSTRATIONS.

BY JAMES THOMSON, LL.D.

PROFESSOR OF MATHEMATICS IN THE UNIVERSITY OF GLASGOW.

THIRD EDITION.

LONDON:

SIMMS AND MINTYRE,

ALDINE CHAMBERS, PATERNOSTER ROW; AND DONEGALL-STREET, BELFAST.

EASY EXERCISES

ON THE

FIRST PART OF THIS WORK.

EXERCISES ON THE FIRST BOOK OF EUCLID.

1. PROVE Euc. I. 9, I. 11, and I. 12, without using I. 8.

2. If there be two unequal parallel straight lines, the two straight

lines, joining their extremities which are towards the same parts,

will meet if produced through the extremities of the shorter par-

allel.

3. The three straight lines drawn from a point within a triangle

to the angles, are together less than the perimeter, but greater than

its half.

4. The diagonals of a parallelogram divide it into four equal

parts.

5. If a quadrilateral be bisected by both diagonals, it is a paral-

lelogram.

6. Prove Euc. I. 32, by the construction of Euc. I. 16.

7. Given the perpendicular of an equilateral triangle, to con-

struct it.

8. Given the diagonal of a square, to construct it.

9. Prove Euc. I. 17, without producing any side.

10. Prove the first and second parts of Euc. I. 28, independ-

ently of I. 27, and of one another: and from each, when proved, de-

rive I. 27, and the remaining part of I. 28.

11. Given the sum of the side and diagonal of a square; to con-

12. Given the difference of the side and diagonal of a square;

to construct it.

13. Given the sum of the diagonal and two sides of a square; to

construct it.

14. Given the sum of the side and perpendicular of an equila-

teral triangle; to construct it.

15. Given the difference of the side and perpendicular of an equi-

lateral triangle; to construct it.

16. Prove Euc. I. 36, without joining BE, CH, by producing

BA, FE, and CD, GH till they meet. When will this fail, and how will the proposition then be proved?