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enunciations of Euclid's propositions are given in an Appendix.

It is not necessary, and perhaps not desirable, that on his first reading the student should work through every example in each section. He should in each case, however, write out a sufficient number to insure his mastery of the principles involved; the others will be found useful when he comes to revise.

The exercises have been gathered from all available sources, including examination papers and geometrical text-books, English and foreign.

The authors acknowledge valuable suggestions and assistance from Messrs. Butters, Clark, and Walker, Heriot's Hospital School; Mr. R. F. Davis; the Rev. W. F. Failes, Westminster School ; Mr. Hayward, Harrow School; Mr. Macdonald, Daniel Stewart's College; Dr. Mackay, Edinburgh Academy; Rev. J. J. Milne; Dr. Muir, Glasgow High School; Professor Raitt, Glasgow Technical College ; Mr. Robertson and Mr. Mackay, Edinburgh Ladies' College ; Professor Scott Lang, University of St. Andrews; Rev. G. Style and Mr. WynneEdwards, Giggleswick School; Mr. Tucker, University College School; Dr. Kolbe, Cape Town, and other friends.

Additional parts, corresponding to the remaining books of Euclid, are in preparation.

BOOK II.

CHAPTER I.

THEOREMS.

$ 31. (Bookwork, EUCLID, II. 1-3.)

1. If A, B, C, D be four points in order in the same straight line, the rectangle AC-BD shall be equal to the sum of the rectangles AD:BC and AB.CD.

(A Standard Theorem.) A

B

AC:BD=AC:BC+ AC CD,

[Euc. II. 1. =AC BC+AB.CD+BC.CD, [ =AC.BC+CD.BC+AB.CD, [By Transposition.

AD:BC +AB.CD. [Euc. II. 1. 2. Prove the above theorem by means of a Geometrical Construction.

=

[blocks in formation]

Draw CE and DGAD and equal to BD and BC respectively.

Complete the rectangles AG, AE and AM.

Prove that CG and KE are rectangles contained by equal lines, and are therefore equal.

Hence show that AE=AG+LM, and deduce the required result.

3. The square on any straight line is four times the square on half the line.

(A Standard Theorem.) E

B

H

Use Euc. II. 1, to show that

AB2=AB:AE+AB:EB, etc. ; or prove by Geometrical Construction. See also the proof given in $ 20, Ex. 3.

4. The square on any straight line is nine times the square on one-third of the line.

Use Euc. II. 1, A

D B

or prove by Geometrical Construction.

5. If AB be divided in D so that AD=2 DB, show that

4AB2=9 AD2.
Use the methods of Ex. 3 and 4.

6. If AB be trisected in C and D, show that

AB2= AD2+BC2+CD2. 7. If P be any point within a A ABC and PL, PM, PN be drawn I BC, CA, AB; show that

BC.BL+CA.CM+AB:AN=BC:LC+CA:MA+AB.NB.

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8. If P be any point in AB, and I be the mid-point of AP, show that

AB AQ=2 AQ2+QP-PB.
Use Euc. II. 3 to show that
AB.AQ=AQ?+AQ BQ,

=AQ2+QP.QB=etc. 9. The hypotenuse BC of a right-angled A ABC is divided in D, so that BC-BD=AB2; show that BC.DC=AC2.

B

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