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to the circumference the proposed number of times, will form the polygon required.

PROBLEM XXIV.

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To make a triangle equal to a given trapezium ABCD.

Draw the diagonal DB, and through C draw CE || to DB, meeting AB, produced in E, and join DE; the AADE will beequal to thetrapezium ABCD. For the ADEB is = the ADCB, (Geo., prop. 26.)

B

PROBLEM XXV.

To make a triangle equal to any right lined figure. ABCDEFG.

Produce the side AB both. ways at pleasure, and draw the diagonals EA, EB, and by the last problem make the AEIA= the trapezium AGFE, and the

H AEHB=the trapezium BCDE. Draw IK || to AE, and HL || to EB; then join KE and EL, K and the AKEL

will be equal to the figure ABCDEFG.

A

B

PROBLEM XXVI.

m

n

To describe an oval or figure resembling an ellipse on a given straight line AB. Take any points, C, D,

at equal distances from A, B, and on CD describe two isosceles As CED, CFD, and produce the sides to the points p, 0, n, m; then from the centres C and D, with the radius AC or DB, describe arcs bounded by the sides of the isosceles As produced ; and from the centre F, with radius Fm or Fn, describe the arc mn, and from the centre E, with the radius Ep or Eo, describe the arc po, and the figure thus formed will be an oval.

A

GENERAL EXERCISES IN ALGEBRA.

1. Find the sum of 3a2 + 4ab +62, 4a2-4ab +6, -a? +4ab +462, 7a2-362, and 5a2 + 10ab + 56.

Ans. 18a2 + 14ab+862. 2. Find the sum of 16ac+16c2 + 4a?, 12c2 +7acta?, 6a2 + 4c?, 13ac + 144a2 +co, lla? +22ac +4c>, and 3a2+ 12ac+12c.

Ans. 169a2 +70ac +49c?. 3. Find the sum of 14a + vac—5c, 13a–5 Jac +8c, 7a14 Nac, 3c+50, 76–6Jac+5a, 13a + 14 Jac+3c, and 4 Nac6a+4c.

Ans. 510–6 ac +20c. 4. Find the sum of 4a+76—3c, a +4abc, 5a_3ab + 4c, 3a—ab+c+3d, 4a+7c44d, and 5dmc+40—25.

Ans. 21a+56+7c+4d. 5. Find the difference of 15a2 + 12ab + 1362, and 7a+ + 110%.

Ans. 8a? +12ab + 262. 6. From 17ac+5a2 +6c>, take 17ac5a2 +3c2_4d.

Ans. 100? +30° +4d. 7. From the sum of 3a + 2ax +2.", and 5a? 7ax-x", take the sum of 7a-7ax + x>, and 12ax—3a2_-4x2.

Ans. 4a" +4ax +48". 8. From a2 +62 +62 + 2ab+2ac+2bc, take 2ab+2ac25c-a2_62

Ans. 2a: +262 +2c +4bc. 9. Multiply a + b by c-d. Ans. ac+bc-adbd. 10. Multiply a-6+-d by a+b-c-d.

Ans. a?—62—c+d2—2ad+2bc. 11. Find the product of (x—10)(x+1)(x+4).

Ans. x35x-_-46x-40. 12. Find the product of a' Iac+c, and a'-ac+c".

Ans. attače + 13. Find the continued product of a+1, a+2, a +3, and a +4.

Ans. a4+100% +35a2 +50a +24. 14. Find the continued product of a—x, a+x, a’ + ax +x, and a--ax+x?.

Ans. a6_006. 15. Multiply x2 + 10xy +7, by x_6xy +4.

Ans. x++4x*y-60x2y2 +11x?-2xy +28. 16. Divide 6x -3x + 12.0" +14x, by 3x.

Ans. 2x3 — x2 + 4x +45. 17. Divide x4_81, by x—3. Ans. 3 3.x' +9x+27. 18. Divide a +63, by a+b.

Ans. a'-ab+b.

C

19. Divide x +y', by x+y.

Ans. 24—xoy + xoyz_XY® + y4. 20. Divide x + xy + xy + y', by x +yo. Ans. ** +y?. 21. Divide x5-px+ + 2x*—qx?+px-1, by x-1. Ans. x'—(P-1)x+(9-p+1)x2—(P-1)x+1. 3.75 77203 43.x2 33.0

23 22. Divide

,

-4x2 +

8

4

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23. Find the sum of the products (a+b+c)(a+b-c), (a+b+c)(ab+c), and (a+b+c)(-a+b+c).

Ans. a' +12+c+ 2ab +2ac+26c. 24. Find the sum of the products (a+w)(a +x), (a—2) (a—«), and (a+x)(a—«).

Ans. 3aP+x. 25. Find the sum and difference of (a+x)(a +x), and (a—) (a-x).

Ans. Sum 2a +2002, diff. 4ax. 26. Find the sum and difference of (a+b+c)(a+b+c), and (a+b-c)(a+b-c).

Ans. Sum 2a2 +262 +2c2 +4ab, and diff. 4ac+4bc. 27. Find the sum of the products of (a+b+c+d) (a+b+c—d), and (a+b-c+d) (a6+c+d), and also their difference. Ans. Sum 2a2 + 2ab + 2ac +46c+ 2ad, and their diff. 2ab+2ac2ad +262 +2c2—2d?.

28. How much does the product of (a+b+2c)(a + b +2c), exceed the product of (a——2c)(a6–2c).

Ans. 4ab+8ac. 29. Divide ax_x , by ał_x}. Ans. a+am,}+x. 30. Divide a—6 by ał_61.

Ans. ai+azli+ail? +6! 31. Find the greatest common measure of a2 + ab1262, and a25ab +66%.

Ans. 4–35. 32. Find the greatest common measure of 6a2 +7ab362, and 6a+ 1lab+362.

Ans. 2a +35. 33. Find the least common multiple of 23—a?, and ?-a?

Ans. 24 + xoa—xa:_a. 34. Find the least common multiple of am), a'-1, 2—2, and a2_4.

Ans. a1_5a2 +4. 35. Find the least common multiple of 2a1, 4a?-1, and 4a2 +1.

Ans. 1694-1. ab +63

a+b 36. Reduce to its lowest terms

Ans. 363-cb

36-0

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43. Prove that by Arts. (29-31), Alg. {12–

-82

5a?+5ab

5a 37. Reduce to its lowest terms

Ans. %.

ab 1402_7ax

7a 38. Reduce to its lowest terms

Ans. 10ac-500

5c

actax+bx+bc 39. Reduce to its lowest terms

af+2ay + 2by+bf

ctx Ans.

f+2y

a2 +62 +62+2ab+2ac+2bc 40. Reduce to its lowest terms

a-63-_2bc

a+b+c Ans.

a-b-c a2+62-2 41. Prove that it

(a+b+c)(a+bc) 2ab

2ab

a+62_2 (a+c-6)(6+c-a) 42. Prove that I

2ab

2ab
162 +62

2c

(a+b+c)(6+c—a)(a+cb)(a+b—c) can be reduced to the form

16

a+b+c 44. Prove that if, in the last example, s= the ex

2 pression may be reduced to the form s(sa)(5–6)(8—c).

b 1 1 45. Show that

72
+
.q?

+7, except a=6.
46. Show that a+12a+a, except a=1.
47. If a 2x, show that (v + a)3—28_7ax?.

ca 48. Multiply 2cm

Ans. aa 2c3d4 7 ca

2c3d4 7 ca 49. Multiply 3+

by

+
204239
23

20*23*
4c6d8 14c5d4 49c*
Ans.
28

a428 a2_9a+20 a’_13a+42 alla+28 50. Multiply

by

Ans.
a26a
a2—5а

a
2a

bc ca 51. Divide

+ +
bc d be d2

de

Ans. + a_ 52. Divide by

Ans. 3(a+x). at 2:0 3a+6x

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INVOLUTION AND EVOLUTION. 55. Required the square of x+1.

Ans. 21-2x + 3x2–2x+1. 56. Required the cube of 1+2x+3x2. Ans. 1+6x+21x* +44x8 +63x4 +54x' +27x6.

1

1 57. Show that the cube of a

as_

23 58. Extract the square root of 4x4—12x3 +25x2—24x +16.

Ans. 2x2-3x+4.

3a2 59. Extract the square root of a1_2a + - +1.

2

Ans. az-a+7

aco 3aRc5 60. Extract the cube root of

+3abc4_133. 63 b

aca Ans. -bc.

b 61. Extract the cube root of x6_605+15x4_203c3 +15x2 -6x+1.

Ans. x-2x+1. 62. If two numbers differ by 1, prove that the difference of their squares is the sum of the two numbers.

63. If two numbers differ by any number (a), prove that the difference of their squares is the sum of the numbers multiplied by their difference (a).

64. If the sum of two fractions =l, prove that their. difference is equal to the difference of their squares. 65. Prove that the

square

of
any

odd number diminished by 1 is divisible by 8.

66. Prove that the product of two odd numbers is odd; and that the product of two even numbers, or of an even and an odd number is even.

EXERCISES IN SIMPLE EQUATIONS.

67. Given 80—4=13--2x, to find X.

Ans. x=2. 3x 68. Given 2x+7+=6x—23, to find X. Ans. x=12.

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