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11

14. Draw a rectangle whose sides are 2" and (3 – V5)". Prove graphically that it is equal to the square on a line of ( V5 – 1)".

[In practical work, the construction of Euclid II. 11, p. 110, is modified as follow3:-Draw BE = {AB and I to AB; join EA ; from EA mark off EL = EB; from AB mark off AH = AL; then AH? = AB.HB. Prove by showing that this is equivalent to Euclid's construction.

practical work, the construction of Add. Prop. 10, p. 152, is mudified as follows:-Draw BE = } AB and I to AB; join EA ; on AE produced mark off EL = EB ; on BA produced mark off AK = AL; then AK? = AB BK. Prove by showing that this is equivalent to the construction on p. 152.

lioto also the following construction ;-Bisect AB at P; draw BQ = AB and I to 1B; with centre P and radius PQ describe an arc cutting AB produced in R, then AB2 = AR.RB. Prove by Euclid II. 6 and I. 47.]

1: Divide a line of length 3" in medial section; also divide the larger portion in medial section. Measure the parts. [Cf. Add. Prop. 11, p. 152.]

16. Take a line of length 2" and divide it externally in medial section. Measure the produced part of the line. [Cf. Add. Prop. 10, p. 152.]

17. Using the figure and the result of Prop. 12, calculate the length of CD if AB = 6", BC= 3", CA = 4".

18. Using the figure and the result of Prop. 13, calculate the length of CD if AB = 4", BC = 5", CA = 6".

19. Draw a line AB of length 6 cm. Draw the locus of a point such that the sum of the squares of its distances from A and B is 25 sq. cm.

20. Draw a line AB of length 6 cm. Draw the locus of a point such that the difference of the squares of its distances from A and B is 12 sq. cm. [Cf. Add. Prop. 4, p. 152.]

21. Draw a line AB of length 6 cm. Find a point such that the square of its distance from A exceeds the square of its distance from B by 18 sq. cm., and such that its perpendicular distance from the line AB is 4 cm. Measure its distance from B.

22. Using II. 13 and I. 47 calculate the altitude of a triangle whose base is 21", and whose sides are 20" and 13".

23. Calculate the lengths of the medians of a triangle of sides 8", 12", 16". [Cf. Add. Prop. 5, p. 152.]

24. In a certain triangle the sum of the squares on the sides is 6800 square inches, and the median and altitude measure 50" and 40" respectively. Cal. culate the lengths of the sides.

EUC. I., II.

M

SYMBOLS AND ABBREVIATIONS.

The following Symbols are in common use :

Z angle

< is (or are) less than L's angles

* is (or are) not less than ... because

Il parallel O circle

o parallelogram Oce circumference

Il gram parallelogram is (or are) equal to

I perpendicular is (or are) not equal to

.. therefore is (or are) greater than

A triangle is (or are) not greater than

A's triangles = equal in every respect

ΠΗΛΑ

Signs for operations such as +, -, *, • , and indices should not be used.

Abbreviations of words may be used provided they do not lead to ambiguity. The following abbreviations are often used :adj. for adjacent

rect. for rectangle fig. figure

rt. opp. opposite

sq.

s square perp. » perpendicular

$99

» squares pt. point

st.

» straight quad. „ quadrilateral

» right

PROPOSITION I. 9 AS A SPECIMEN OF A WRITTEN PROPOSITION.

[blocks in formation]

В!

Constr.

...in the two A's DAF, EAF

AD = AE
and AF is common

and DF = EF
.. the < DAF the LEAF

.. the < BAC is bisected by AF.

Constr.

I. 8 Q. E. F.

NUMERICAL ANSWERS.

ANSWERS TO THE PRACTICAL EXERCISES ON BOOK I. 5. 2.16 cm. 6. 1.56". 7. First base 58.3 mm., second base = 74.3 mm. 12. 1.5 cm., 6 cm. 13. 1 cm.

14. 1.54". 15. 3". 23. (i):968 sq. in.; (ii) 1.5 sq. in.; (iii) 1.33 sq. in.; (iv) 6.5 sq. cm. 24. (i) •65 sq. in.; (ii) 726 sq. in. ; (iii) 10•4 sq. cm. ; (iv) 10•4 sq. cm. 25. A line || to the base at a distance 4 cm.

26. 3".

27, 1:6". 28. 5.31 cm. 29. 3•69 cm., 7.67 cm. 30. 37.5 mm. 32 (i) 24 ; (ii) 1.875; (iii) 6.72; (iv) 177 ; (v) 1.92; (vi) 7.82; (vii) 2.5 ; (viii) 1.74.

33. 46o. 34. 1•73". 35. 3:16". 37. 34.8 mm. 38. (i) 3.6"; (ii) 2.24"; (iii) 6.4 mm. ; (iv.) 5.39 mm. ; (v) 2.65" ; (vi) 4.8 cm. ; (vii) 5.74 cm. ; (viii) 6.56 cm.

39. 1.58". 40. 6.45 cm.

41. 4.898 mm.

ANSWERS TO THE PRACTICAL EXERCISES ON BOOK II. 3. 3.74 cm. 4. 1.82". 5. (i) 2.65" ; (ii) 3•16 cm. ; (iii) 4.36 cm.; (iv) 4.8 cm. ; (v) 2.12" ; (vi) 1.81". 6. 2:37", •63" 7. 5•3 cm., 107 cm. 8. 5.3, 1.7. 9. 15.5, 4.5. 10. 4.68, 4:28. 11. 1.94". 13. 20.8 mm. 15. 1:15", 1.15'', .7". 16. 3.24".

17. 1.83". 18. 4.5". 19. A circle of radius 1.87 cm. whose centre bisects AB. 20. Two lines I to AB and trisecting AB. 21. 4.27 cm.

22. 12". 23. 6.32", 11•1", 13•6". 24. 72.1", 40".

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UNIVERSITY TUTORIAL PRESS LD.

(University Correspondence College Press).
W. B. CLIVE, 157 DRURY LANE, LONDON, W.C.

CONTENTS.

MATHEMATICS AND MECHANICS .
CHEMISTRY
PHYSICS
BIOLOGY
MODERN HISTORY
ENGLISH LANGUAGE AND LITERATURE
ENGLISH CLASSICS
MENTAL AND MORAL SCIENCE
FRENCH
LATIN AND GREEK CLASSICS
LATIN AND GREEK GRAMMARS, ETC.
ROMAN AND GREEK HISTORY
THE UNIVERSITY CORRESPONDENT—DIRECTORIES
THE ORGANIZED SCIENCE SERIES

PAGB 3-5

5 6 7 7 3 9 10

11 12, 13

14 15 15 16

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A List of Books for London University Students, classified for the

various Examinations, List of Books for the Cambridge and Oxford Locals and the College of Preceptors Examinations, and also the Complete Catalogue of the University Tutorial Series, may be had post free on application to W. B. CLIVE, The University Tutorial

Press Warehouse, 157 Drury Lane, W.C. OCTOBER 1903.

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