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PRINCIPLES 3,\ 



OF 



'~^^J'J^ 



PLANE GEOMETRY 






( ^ 



J. W. MACDONALD 

PRINCIPAL OF THE STONEHAM (MASS.) HIGH SCHOOL 



Let hhn know a thing because he has found it out for himself, and not 
because you have told him. 7- J. J. Rousseau 



ALLYN AND BACON 
1894 



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Copyright^ 1889, 
By J. W. MacDonald. 



SnficTsitfi 9rt«s: 
John Wilson and Son, Cambridge. 



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"WWW. 



PREFACE. 



The most appropriate preface for a book of this 
kind would be an exposition of the principles of 
psychology pertaining to the development and train- 
ing of the reasoning and linguistic faculties. As 
such a preface, however, would be more pretentious 
than the book proper, and as these principles have 
been so well expounded by many eminent writers, I 
must content myself by merely urging studious in- 
vestigation upon all teachers who are ambitious to 
practise the best methods. If teachers will do this, 
of one thing I am confident, — they will grant that 
the purpose of this book is right in theory, even if in 
practice certain difficulties may seem to them insur- 
mountable. As a help, however, to teachers who 
may wish it, I have thought it advisable to publish 
as a companion to this book a monograph on teach- 
ing geometry,* illustrating actual class-work, and 
showing in detail how some of the most difficult 
topics may be mastered, — not, let me add, as a 

* Geometry in the Secondary School. Willard Small, Publisher, 
24 Franklin Street, Boston, Mass. 

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IV PREFACE. 



dogmatic declaration of the method of development, 
but only as a suggestive illustration of a method. 
The earnest teacher, who thoroughly understands 
the subject he is teaching, and who has a purpose 
clear in his mind, will make his own, and for him the 
best method. 

It cannot be said of this Geometry, as I have heard 
it said of others, that it is designed to aid inefficient 
teachers. The teacher who does not thoroughly un- 
derstand elementary geometry, — who is not sharp to 
detect inaccuracies in definitions and arguments, — 
who, in short, is dependent on the written text for 
what he teaches, — should not undertake to use this 
book. 

I have thought it best to publish the books in an 
inexpensive form, so that, where the free text-book 
system exists, it would be as economical to purchase 
them new each year as to transmit a more expen- 
sive volume from pupil to pupil, with much distasteful 
accumulation. They will be furnished in a more 
durable form, if desired. 

In conclusion, I wish to thank the publishers and 
the proof-reader, by whose suggestions and watch- 
fulness the text has been much improved and saved 
from numerous errors. 

J. W. M. 

Stoneham, August I, 1889. 



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CONTENTS. 



Pagb 

Definitions of Solids, Surfaces, etc i 

Locus OF A Point 4 

Position of Lines 4 

Plane Angles 4 

Axioms 5 

Postulates 5 

Propositions 6 

Symbols and Abbreviations . 6 

BOOK L 

Angles having Special Names 9 

Triangles 12 

Quadrilaterals 16 

Polygons of more than Four Sides 18 

Axis of Symmetry, etc 18 

Supplementary Propositions 20 

BOOK IL 

Ratio 22 

Proportions 22 

The Theory of Limits 25 

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VI CONTENTS. 



BOOK III. 

Pack 

The Circle 29 

Inscribed Angles and Polygons 30 

Supplementary Propositions 38 

BOOK IV. 

Similar Polygons 43 

Dividing Lines Internally and Externally ... 45 

Dividing Lines Harmonically 45 

Extreme and Mean Ratio 47 

Supplementary Propositions 48 

BOOK V. 

Measurement of Areas - 50 

Projection 55 

Special Problems in Triangles 56 

BOOK VI. 

Regular Polygons 58 

Special Problems in Inscribed Polygons .... 63 



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PLANE GEOMETRY. 



DEFINITIONS. 
I. Solids. — Surfaces. — Lines. — Points. 

1. What is a geometrical solid? 

a. What dimensions has it? 

b. How is it bounded? 

2. What is a surface ? Its dimensions ? 

a. A plane surface ? 

b. A curved surface ? 

c. How is a surface bounded? 

3. What is a line? Its dimension? 

a. What is a straight line or right line? 

b. What is a curved line? 

c. What is a broken line? 

Illustration. 

d. What is a mixed line ? 
Illustration. 



Note. The mixed line and the broken line have no practical valut 
in geometrical discussions. 

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PLANE GEOMETRY. 



4. What is a point? 

5. How may a line be generated? 

6. How may a surface be generated? 

7. How may a solid be generated ? 

& Wliat determines the position of a point ? 

9. What determines the position of a line ? 

10. What determines the position of a surface ? 

11. What determines the form of a surface ? 

12. What determines the form of a solid? 



13. What is a figure ? 

a. A plane figure? 

Illustrations. 




o 

b. A rectilinear figure? 

Illustrations. 

c, A curvilinear figure ? 

Illustrations. 




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PLANE GEOMETRY. 



d. A mixtHinear figure ? 

Illustration& 



c> 



e. Similar figures ? 



Illustrations. 



^ ^) [c::, ^ 



/, Equivalent figures ? 



lUastrations. 





^. Equal figures? 



Illustrations. 



'/4.5Q.\\ '4 5Q. 

inchW inch 




A. Solid figures ? 



Illustrations. 



c 




14. What is magnitude ? 
a. Of a line? 
d. Of a surface ? 
c. Of a solid? 



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PLANE GEOMETRY. 



15. How is magnitude measured? 

a. Of a line ? 

b. Of a surface ? 

c. Of a solid? 

16. What is geometry? 

a. Solid geometry? 

b. Plane geometry ? 

LOCUS. 

17. What is the locus of a point? 

18. Problem I. Find the locus of a point a given dis- 
tance from a given point. 

Problem II. Find the locus of a point a given distance 
from a given circumference. 

II. Positions of Lines. — Plane Angles. 

19. What are parallel lines ? 

20. What is a perpendicular line ? 

21. What is an oblique line ? 

22. What is an angle ? 

a, A right angle ? 



Illustration. 



b. An acute angle ? 
Illustration. 




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PLANE GEOMETRY. 



r. An obtuse angle? 
Illustration. ^ 

d. A straight angle ? 

Illustration. JS — 

e, A reflex angle ? 

Illustration. 



-C 



\d. 



"A 

23. What are complementary angles? 
84. What are supplementary angles ? 

III. Axioms. — Postulates. — Propositions. 

25. What is an axiom ? 

26. What axioms can be formed from the following data? 

a. Things equal to the same thing. 

'^. Adding equals. 

c. Adding unequals. 

d. Subtracting equals. 

e. Subtracting unequals. 
/. Multiplying equals. 
g. Dividing equals. 

h. The whole and a part 

/. The whole and all the parts. 

27. What axioms may be asserted as to the equality of 

a. Right angles ? 

b. Straight angles ? 

r. Complementary angles? 
//. Supplementary angles? 

28. What is a postulate ? 

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PLANE GEOMETRY. 



89. What must be granted as to the following : 

a. Drawing a straight line ? 

b. Prolonging a line ? 

c. Drawing a circumference ? 

d. Dividing lines, angles, etc. ? 

30. What is a proposition? 

a. A problem? 

b. A theorem? 

c. A corollary? 

d. A scholium ? 

31. Of what does every theorem consist? 

IV. Mathematical Symbols and Abbreviations. 

32. a. + , plus. 

b. — , minus. 

c. X , multiplied by. 

d, -7", divided by. 

e, ==, equal to. 

/. «<^, equivalent to. 

g. >, greater than. 

h. <, less than. 

u :, ::, :, signs of proportion. 

/ Z. , angle ; A, angles. 

>&. A, triangle; A, triangles. 

/. D, square ; [U, squares. 

m. Oy parallelogram ; ZI7, parallelograms. 

«. O, circle ; (D , circles. 

0, '^j arc ; ^, arcs. 



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BOOK I. 



Proposition I. A Theorem. 

33. If one straight line meets another so as to form 
two adjacent angles, the sum of these angles is equal to 
two right angles; that is, the angles are supplements of 
each other. 

Corollary I. Any number of angles in the same plane, 
formed about a given point on one side of a straight line, 
are equivalent to two right angles. 

Corollary II. The sum of all the angles in the same 
plane, formed about a given point, is equal to four right 
angles. 

Proposition II. A Theorem. 

34. Conversely, if two angles whose sum equals two right 
angles are placed adjacent to each other, their exterior sides 
will form one straight line. 



Proposition III. A Theorem. 

35. If two straight lines intersect each other, the vertical 
angles are equal 

Sis page 5, §§ 24 and 27 d. 

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PLANE GEOMETRY. 



Proposition IV. A Theorem. 

36. At any point in a straight line there can be but one 
perpendicular on either side. 

Proposition V. A Theorem. 

37. From any point outside of a straight line there can 
be but one perpendicular to the line. 

Proposition VI. A Theorem. 

88. Two lines in the same plane perpendicular to a third 
line are parallel to each other. 

Proposition VII. A Theorem. 

39. If a straight line is perpendicular to one of two 
parallel lines, it is perpendicular to the other also. 

Proposition VIM. A Theorem. 

40. Angles having the sides of the one parallel to the sides 
of the other are either equals or supplements. 

Scholium. If both pairs of parallel sides extend in the 
same direction from the vertices, or both in opposite direc- 
tions, the angles are equal ; but if one pair extends in the 
same direction and the other pair in opposite directions, 
the angles are supplements. 

Proposition IX. A Theorem. 

41. Angles having the sides of the one perpendicular to 
the sides of the other are either equals or supplements. 

Scholium. How can it be determined whether the angles 
are equals or supplements? 



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BOOK I. 



V. Angles having Special Names. 

42. Let two straight lines be intersected by a third : 

a. What are the exterior angles? 

b. What are the interior angles ? 

c. What are alternate exterior angles ? 

d. What are alternate interior angles? 

e. What are external angles on the same side (of 

the intersecting line) ? 
/. What are internal angles on the same side? 
g. What. are opposite external-mtemal angles? 

Proposition X. A Theorem. 

43. If two parallel straight lines are intersected by a 
third : 

I. Alternate interior or exterior angles will be equal. 
11. Opposite external-internal angles will be equal. 
IIL Interior or exterior angles on the same side will 
be supplements. 

See Proposition VIII. 

Proposition XI. A Theorem. 

44. Two straight lines intersected by a third line will be 
parallel : 

I. If alternate interior or exterior angles are equal. 
II. If opposite external-internal angles are equal. 
III. If interior or exterior angles on the same side are 
supplements. 



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lO PLANE GEOMETRY. 

Proposition XII. A Theorem. 

45. Two lines parallel to a third are parallel to each other. 
See Proposition VII. 

Proposition XIII. A Theorem. 

46. The sum of two lines drawn from any point to the ex- 
tremities of a line is greater than the sum of any two lines 
similarly drawn from an included point. 

Proposition XIV. A Theorem. 

47. The shortest distance from any point to a given 
straight line is a perpendicular to that line. 

Proposition XV. A Theorem. 

48. Two oblique lines extending from any point in a per- 
pendicular to points in the base line equally distant from the 
foot of the perpendicular are equal. 

Proposition XVI. A Theorem. 

49. Of two oblique lines extending from any point in a 
perpendicular to points in the base line unequally distant 
from the foot of the perpendicular, the one extending to the 
farther point will be the longer. 

Corollary. There can be but two equal oblique lines 
drawn from any point in a perpendicular to the base line. 

Proposition XVII. A Theorem. 

80. If two oblique lines drawn from any point in a per- 
pendicular to the base line are equal, they extend to points 
equally distant from the foot of the perpendicular. 

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BOOK I. II 

Proposition XVI 11. A Theorem. 

51. If a perpendicular be erected at the middle point of a 
straight line : 

I. Any point in the perpendicular will be equally distant 
from the extremities of the line. 

II. Any point out of the perpendicular will be unequally 
distant from the extremities of the line. 

Corollary I. Conversely, all points equally distant from 
I he extremities of a line are in the perpendicular at its middle 
point. 

Corollary II. The perpendicular at the middle point of 
a line will cut the longer of two lines joining a point with its 
extremities. 

Proposition XiX. A Problem. 

52. To erect a perpendicular at the middle of a line. 
See page 4, §§ 17 and 18. 

Proposition XX. A Problem. 

53. To bisect a given line. 

Proposition XXI. A Problem. 

54. To erect a perpendicular at any point in a straight 
line. 

Proposition XXJI. A Problem. 

55. From a point outside of a straight line to draw a per- 
pendicular to the line. 

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12 



PLANE GEOMETRY. 



Proposition XXlll. A Theorem. 

56. If an angle be bisected by a straight line, every point 
in the bisector is equally distant from the sides. 



Proposition XXIV. A Theorem. 

57. Conversely, every included point equally distant from 
the sides of an angle is in the bisector of the angle. 

Scholium. What is the locus of a point equally distant 
from the sides of an angle ? 



VI. Triangles. 

58. What is a triangle? 

a. An equilateral triangle ? 

b. An isosceles triangle? 



Illustrations. 



c. A scalene triangle? 
Illustrations. 





d. A right-angled, or right triangle ? 

e. An obtuse-angled, or obtuse triangle ? 
/ An equiangular triangle ? 

59. What is the hypotenuse of a right triangle? 
a. What are the other sides called ? 



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BOOK I. 13 

60. When are triangles equal ? Equivalent ? 

a. What are homologous sides and angles of equal 
triangles ? 

61. What is the base of a triangle ? 

a. What the altitude? , 

b. What are its medians ? 

Proposition XXV. A Theorem. 

62. The sum of two sides of a triangle is greater than the 
third, and the difference is less. 

Proposition XXVI. A Tjieorem. 

63. If two triangles have two sid^s and the included angle 
of one equal to two sides and the included angle of the other, 
each to each, the other homologous parts are also equal, and 
the triangles are equal. 

Proposition XXVII. A Theorem. 

64. If two triangles Have two angles and the included side 
of one equal to two angles and the included side of the 
other, each to each, the other homologous parts are equal, 
and the triangles are equal. 

Proposition XXVIII. A Theorem. 

66. If two triangles have the three sides of one equal to 
the three sides of the other, each to each, the triangles are 
equal. 

Proposition XXIX. A Problem. 

66. Construct an equilateral triangle having sides equal 
each to a given line. 

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14 PLANE GEOMETRY. 

Proposition XXX. A Problem. 

67. Construct a triangle having sides equal to the sides of 
a given triangle. 

Proposition XXXI. A Theorem. 

68. If two triangles have two sides of the one respectively 
equal to two sides of the other and the included angles un- 
equal, the third side of the one having the greater angle will 
be longer than the third side of the other. 

Scholium. Three different cases may arise ; prove each. 

Proposition XXXII. A Theorem. 

69. Conversely, if two triangles have two sides of the one 
respectively equal to two sides of the other and the third 
sides unequal, the angle opposite the longer third side will 
be greater than the angle opposite the shorter. 

Proposition XXXIII. A Theorem. 

70. The sum of the angles of a triangle is equal to two 
right angles. 

Corollary. In a right triangle the two acute angles are 
complements of each other. 

Scholium. The exterior angle formed by prolonging one 
of the sides of a triangle is equal to the sum of the two 
opposite interior angles. 

Proposition XXXIV. A Theorem. 

71. In an isosceles triangle the angles opposite the equal 
sides are equal. 

Corollary. An equilateral triangle is also equiangular. 

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BOOK I. 15 

Proposition XXXV. A Theorem. 

78. If two angles of a triangle are equal, the sides opposite 
these angles are equal, and the triangle is isosceles. 

Corollary. An equiangular triangle is also equilateral. 

Proposition XXXVI. A Theorem. 

73. If two sides of a triangle are unequal, the angle op- 
posite the longer side is greater than the angle opposite the 
shorter side. 

Proposition XXXVII. A Theorem. 

74. If two angles of a triangle are unequal, the side op- 
posite the greater angle is longer than the side opposite the 



Proposition XXXVIII. A Theorem. 

75. Two right triangles are equal if the hypotenuse and 
one side of the one are equal to the hypotenuse and one 
side of the other. 

Proposition XXXIX. A Theorem. 

76. Lines bisecting the angles of a triangle meet at a point 
which is equally distant from the side of the triangle. 

See Propositions XXIII. and XXIV. 

Proposition XL. A Theorem. 

77. Perpendiculars bisecting the sides of a triangle meet 
at a point equally distant from the vertices of the angles. 

See Proposition XVIII. 

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1 6 PLANE GEOMETRY. 



VII. Quadrilaterals. 

78. What is a quadrilateral or quadrangle ? 
a. A trapezium? 

Illustrations. 




±. 



Illustration. ^^ 

g, A rhomboid ? 
Illustration. 



K \ 



:^ 



b, A trapezoid ? 

Illustrations. /_ _\^ \ ^ 

€. A parallelogram? 

d, A rectangle? 

e, A square? 
/. A rhombus ? 



8 

79. What is the diagonal of a quadrilateral? 

80. What are the upper and lower bases of a quadrilateral? 

81. What is the altitude of a parallelogram or trapezoid? 

82. What are the bases of a trapezoid ? 

a. What is its median ? 

Illustration. ^, 



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BOOK I. 17 

Proposition XLl. A Tlieorem. 

83. The opposite sides and angles of a parallelogram are 
equal. 

Corollary I. The diagonal divides a parallelogram into 
equal triangles. 

CorolIary IL The parts of parallel lines cut off between 
parallel lines are equal. 

Proposition XLII. A Theorem. 

84. If the opposite sides of a quadrilateral are equal, the 
figure is a parallelogram. 

Proposition XLl 1 1. A Theorem. 

85. If two sides of a quadrilateral are equal and parallel, 
the other two sides are also equal and parallel, and the figure 
is a parallelogram. 

Proposition XLIV. A Theorem. 

86. The diagonals of a parallelogram bisect each other. 

Proposition XLV. A Theorem. 

87. Two parallelograms are equal if they have two sides 
and the included angle of one equal to two sides and the 
included angle of the other, each to each. 

Proposition XLVI. A Theorem. 

88. Parallel lines are everywhere equally distant 

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1 8 PLANE GEOMETRY. 



VIII. Polygons of more than Four Sides. 

88. What is a polygon ? 

a. A pentagon? 

b. A hexagon? 

c. A heptagon? 

d. An octagon ? 

e. A nonagon? 
/. A decagon ? 

^. An undecagoni^ 
A. Aduodecagon? 

80. What are salient angles of a polygon? 

81. What are re-entrant angles? 

82. What is an equilateral polygon ? 

83. What is an equiangular polygon? 

84. What is a concave polygon? 

85. When are two polygons mutually equiangular? 

86. When are two polygons mutually equilateral? 

87. What are homologous sides or angles? 

88. What are equal polygons? 

88. When is a polygon symmetrical with reference to any 
dividing line? 

100. What is an axis of symmetry? 

101. What is a centre of symmetry? 

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BOOK I. 19 

* Proposition XLVII. A Theorem. 

102. Two equal polygons may be divided into the same 
number of equai triangles. 

Proposition XLVII I. A Theorem. 

108. The sum of the interior angles of a polygon is equal 
to as many right angles as twice a number two less than the 
number of its sides. 

Scholium. To how many right angles is the sum of the 
angles of figures from pentagons to duodecagons equal ? If 
equiangular, how large is each angle? 

Proposition XLIX. A Theorem. 

104. If each side of a polygon be produced in order, the 
sum of the exterior angles equals four right angles. 



OPTIONAL PROPOSITIONS. 

Proposition L. A Theorem. 

105. I. In a regular polygon having an odd number of 
sides, a line joining the vertex of an angle with the middle 
point of the opposite side is an axis of symmetry. 

11. In a regular polygon having an even number of sides, 
a line joining the vertices of opposite angles, or the middle 
points of opposite sides, is an axis of symmetry. 

Proposition LI. A Problem. 

106. Draw parallel lines a given distance apart 

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20 PLANE GEOMETRY. 

Proposition LM. A Theorem. • 

107. Any number of parallel lines equally distant from 
each other intercept equal parts on any transverse line 
crossing tliem. 

SUPPLEMENTARY PROPOSITIONS. 

1. What is the supplement to an angle of 35°? The 
complement? 

2. If three or more angles be formed at the same point 
on the same side of a straight line, any one of them will be 
a supplement to the sum of all the others. 

3. If two adjacent supplementary angles be bisected, the 
bisectors will form a right angle. 

4. A line bisecting one of two vertical angles will, if con- 
tinued, bisect the other. 

5 . The sum of any two sides of a triangle is greater than 
the sum of any two lines drawn from any point in the tri- 
angle to the extremities of the third side. 

6. An equiangular triangle is also equilateral. 

7. The bisector of the vertical angle of an isosceles trian- 
gle, if continued to the base, is an axis of symmetry. 

8. In a right triangle the two acute angles are comple- 
ments of each other. 

9. In a right triangle, if one of the acute angles is of 30®, 
the side opposite is one half the hypotenuse. 

10. Find the locus of a point equally distant from two 
points. 

1 1 . The opposite angles of a parallelogram are equal. 

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BOOK I. 21 

, 12. In a parallelogram, the angles adjacent to any one 
side are supplements. 

13. All the angles of a parallelogram are equal to four 
right angles. 

14. The diagonals of a rectangle are equal. 

15. If the diagonals of a quadrilateral bisect each other, 
the quadrilateral is a parallelogram. 

16. If a diagonal divides a quadrilateral into two equal 
triangles, the quadrilateral is a parallelogram. 

17. Through a given point to draw a parallel to a given 
line. 

18. If several parallel lines intercept equal parts on any 
transverse line, they are an equal distance apart. 

19. If a line drawn through a triangle parallel to one of 
the sides bisects one of the other sides, it will bisect both of 
them. 

20. A line bisecting two sides of a triangle is parallel to 
the third. 

21. A line connecting the middle points of two sides of a 
triangle is half the length of the third. 

22. The medians of a triangle intersect one another at 
the same point, which is distant from the vertex of each 
angle two thirds the length of its median. 

23. The median of a trapezoid is parallel to the bases and 
equally distant from them. 

24. The median of a trapezoid is equal to half the sum of 
the bases. 



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BOOK II. 

THE PRINCIPLES OF PROPORTION AND THE 
THEORY OF LIMITS. 



I. Ratio and Proportion. 

108. What is a ratio? 

a. How expressed ? Example, axb^oxy 

b. What names are given the terms? 

109. What is a proportion? 

a. How expressed ? Ex. a\b\\c\d,ox j = 2* 

b. What names are given the terms ? 

110. What is a fourth proportional? A mean propor- 
tional? A third proportional? 

11 1. Given a proportion a\b\\cid, what is changing it : 

a. By inversion ? Ex. b.awdic, 
^. By alternation ? Ex. a\c::b:d,or d-.b-.-.c-.a. 
C. By composition ? Ex. a-\-bib::c + d',d. 
//. By division ? Ex. a — bxbwc — d.d. 

112. What common divisor has 8 and 34? Their ratio? 
What common divisor has 3.6 and 54? Their ratio? 
What common divisor has 5 and V^? Their ratio? 

113. In each of the following problems how long a line 
will exactly divide both the given lines? 

a. Lines 10 and 25 inches, respectively. Their ratio? 

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BOOK II. 23 



b. Lines 8 J and 15^ inches, respectively. Their ratio ? 

c. Lines 6 and Vs inches, respectively. Their ratio? 

114. What are commensurable quantities? Incommen- 
surable quantities? 

115. How can a common measure and the ratio of two 
lines be found? 

116. What are equimultiples of two quantities? 

Proposition 1. A Tlieorem. 

117. If four quantities are in proportion, the product of 
the extremes equals the product of the means. 

Corollary. A mean proportional is equal to the square 
root of the product of the other two tqrms. 

Proposition 11. A Tlieorem. 

118. If two sets of proportional quantities have a ratio in 
each equal, the other ratios will be in proportion. 

Corollary. If the antecedents or consequents are the 
same in both, the other terms are in proportion. 

Proposition 111. A Theorem. 

119. If the product of two quantities equals the product of 
two other quantities, either two may be made the means and 
the other two the extremes of a proportion. 

Proposition IV. A Theorem. 

120. If four quantities are in proportion, they will be in 
proportion by inversion. 

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24 PROPORTION. 



Proposition V. A Theorem. 

121. If four quantities are in proportion, they will be in 
proportion by alternation. 



Proposition VI. A Theorem. 

122. If four quantities are in proportion, they will be in 
proportion by composition. 



Proposition VII. A Theorem. 

123. If four quantities are in proportion, they will be in 
proportion by division. 



Proposition VI 11. A Theorem. 

124. If four quantities are in proportion, they will be in 
proportion by composition and division. 



Proposition IX. A Theorem. 

126. Equimultiples of two quantities are proportional to 
the quantities themselves. 

Corollary. Any equimultiples of the antecedents are 
proportional to any equimultiples of the consequents. 

Proposition X. A Theorem. 

126. If four quantities are in proportion, their like powers 
or like roots are in proportion. 

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BOOK II. 25 



Proposition XI. A Ttieorem. 

127. In a series of equal ratios the sum of all the ante- 
cedents is to the sum of all the consequents as any one 
antecedent is to its consequent. 

Corollary. The sum of any number of the antecedents 
is to the sum of their consequents as any one antecedent is 
to its consequent 

Proposition XII. A Theorem. 

128. If two or more proportions be multiplied together, 
term by term, the products are in proportion. 



II. The Theory of Limits. 

129. What is a variable ? A constant ? 

a. An increasing variable ? 

b, A decreasing variable ? 

130. Suppose a point x moving on the line A B m such a 

X X X X „ 

A o 

1 234 

way that it goes one half the, distance from A io B the first 
second, one half the remaining distance the next second, 
one half the remaining distance the third, and so on in- 
definitely : 

a. What two varying distances does it produce ? 

b. What distance is the distance A x approaching, 

and when will it reach it? 

c. What is the distance x B approaching, and 

when will it reach it? 



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26 THEORY OF LIMITS. 

13L Reduce the fraction ^ to a decimal : 

a. How is the value of the decimal affected by 

each division, and what is it approaching? 

b. How is the difference between \ and the deci- 

mal affected by each division, and what is it 
approaching? 

132. What is the Umit to a variable? 

a, A superior limit ? 

b. An inferior limit? 

133. How near may a variable be conceived to approach 
its limit? 

134. Suppose a right triangle to be continually changing 
by the shortening of one of its legs ; 

a. What lines would be variables? Their limits? 

b. What angles would be variables ? Their limits ? 

c. How would its area be affected ? Its limit ? 

136. Why could not the diminishing leg become zero ? 

136. Suppose a regular polygon, as a square or equilateral 
triangle, to be inscribed in a circle (see Book III. § 159), and 
that by bisecting the arcs and drawing chords it be changed 
to a regular inscribed polygon of double the number of sides, 
four times the number of sides, and so on indefinitely : 

a. What variables result ? 

b. What are their limits? 



L" 



187. Sometimes the variable does not indefinitely ap- 
proach limits, as, for example, suppose the process in § 136 
reversed. 



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by Google 



BOOK II. 27 

Proposition Xiil. A Ttieorem. 

138. If two variables as they indefinitely approach their 
limits have any constant ratio, their limits have the same 
ratio. 

Corollary. If two variables as they indefinitely approach 
their limits are constantly equal, their limits are equal. 

Scholium. In the above corollary, the variables have the 
I constant ratio i, as have also their limits. 

Proposition XIV. A Theorem. 

139. If the product of two variables as they indefinitely 
approach their limits is constantly equal to a third variable, 
the products of their limits wiU equal the limits of the third. 

ScHOUUM. Sometimes the product of an increasing and a 
decreasing variable is a constant. 
See Book IV., Proposition XVII. 

Proposition XV. A Theorem. 

140. If several parallel lines are crossed by an oblique 
line, the segments of the oblique line are proportional to the 
distances between the parallels. 

Case I. When the parallels are an equal distance apart. 

Case II. When the parallels are unequal distances apart. 
a* When the distances between them are com- 

mensiu^ble. 
b. When these distances are incommensurable. 

Corollary. The corresponding segments of two oblique 
transversals are in proportion. 

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28 PLANE GEOMETRY. 

Proposition XVI. A Theorem. 

141. If one or more parallel lines be drawn through a tri- 
angle parallel to one side, the other two sides will be divided 
proportionally. 

Corollary. The intersected sides are to each other as 
any two corresponding segments. 
See Book II., Proposition VI. 

Proposition XVII. A Theorem. 

142. If a straight line divide the sides of a triangle propor- 
tionally, it is parallel to the third side. 

Proposition XVIII. A Problem. 

143. To divide a given line into parts proportional to 
given lines, or given parts of a given line. 

Proposition XIX. A Problem. 

144. To find a fourth proportional to three given lines. 

Proposition XX. A Problem. 

145. To find a third proportional to two given lines. 



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BOOK III. 



I. The 


Circle. 




146. What is a circle? 






147. What is the circumference? 




148. What is the radius? 






149. What is a chord? 


Illustration. 


O 


160. What is a diameter? 






151. What is a secant? 


Illustration. 


O 


152. What is a tangent? 


Illustration. 


<:) 


163. What is an arc? 






154. What is a segment? 


Illustration. 


o 


156. What is a sector? 


Illustration. 


o 


a. A quadrant? 


Illustration. 


r^ 



156. When do circles touch each other internally? When 
externally? 

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30 PLANE GEOMETRY. 

157. When is an angle inscribed in a circle? 111. (/ V) 

158. When is an angle inscribed in a segment ? m. 



159. When is a polygon inscribed in a circle? m. 

a. What is the relation of the circle to the polygon } 

180. When is a circle inscribed in a polygon? in. 

a. What is the relation of the polygon to the circle ? 

161. What are concentric circles? Illustration. 

182. When will circles be equal? 

183. What is an angle at the centre? 

Proposition I. A Theorem. 

184. The diameter of a circle is an axis of symmetry. 

Corollary. The diameter bisects the circle and its cir- 
cumference. 

Proposition li. A Tlieorem. 

185. A straight line cannot intersect the circumference of 
a circle at more than two points. 

Su Book I., Proposition XVI., Corollary. 

Proposition III. A Theorem. 
188. The diameter is longer than any other chord. 

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BOOK III. 31 



Proposition IV. A Theorem. 

167. In the same circle, or in equal circles, equal arcs are 
subtended by equal chords ; and conversely, if the chords 
are equal, the arcs also are equal. 



Proposition V. A Theorem. 

168. In the same circle, or in equal circles, of two une- 
qual arcs each less than a semicircumference, the greater 
arc is subtended by the longer chord; and conversely, of 
two unequal chords, the longer subtends the greater arc. 

See Book I., Propositions XXXI. and XXXII. 

Proposition VI. A Theorem. 

169. A radius drawn perpendicular to a chord bisects both 
the chord and its arc. 

Scholium. A line drawn perpendicular to the middle 
point of a chord is a radius and bisects the arc. 

Proposition VII. A Theorem. 

170. Through three given points not in the same straight 
line one circumference can be drawn, and but one. 

Proposition VIII. Problems. 

171. I. Given three points not in a straight line, to draw 
a circumference through them. 

See Book I., Propositions XVIII. and XIX. 
II. Given a circumference, to find its centre. 

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32 PLANE GEOMETRY. 

Proposition IX. A Tlieorem. 

172. In the same circle, or in equal circles, equal chords 
are equally distant from the centre ; and conversely, chords 
equally distant from the centre are equal. 

5<?^Book I., Proposition XXXVIII. 

Proposition X. A Theorem. 

173. In the same circle, or in equal circles, of two unequal 
chords the shorter is the farther from the centre. 

See Book I., Proposition XIV. 

Proposition XI. A Theorem. 

174. Conversely, of two chords unequally distant from the 
centre, the farther one will be the shorter. 

Proposition XII. A Theorem. 

175. A tangent is perpendicular to the radius drawn to 
the point of tangency. 

See Book I., Proposition XIV. 

Corollary. Conversely, a straight line perpendicular to a 
radius at its termination in the circumference is a tangent to 
the circle. 

Proposition XIII. A Theorem. 

176. If from a point outside of a circle two tangents to 
the circle be drawn, and also a straight line to the centre of 
the circle : 

I. The tangents will be equal. 

II. The line drawn to the centre bisects the angle formed 
by the tangents. 

See Book I., Proposition XXIV. 

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BOOK III. 33 



Proposition XIV. A Theorem. 

177. Parallels intercept on a circumference equal arcs. 
Case I. When both parallels are tangents. 

Case II. When one is a tangent and the other a secant. 
Case III. When both are secants. 

Proposition XV. A Theorem. 

178. I. If two circles cut each other, the line joining their 
centres will be perpendicular to their common chord. 

II. If two circles touch each other externally or internally, 
the line joining their centres will be perpendicular to their 
common tangent 

Proposition XVI. A Theorem. 

179. In the same circle, or in equal circles, radii forming 
equal angles at the centre intercept equal arcs on the circum- 
ference ; and conversely, if the arcs intercepted are equal, the 
angles at the centre are equal. 

Proposition XVII. A Theorem. 

180. In the same circle, or in equal circles, angles at the 
centre are to each other as their arcs. 

Case I. When they are commensurable. 

Case II. When they are incommensurable. 

Scholium I. The angle may be measured by the arc ; why ? 

Scholium II. Explain the division of the arc into de- 
grees, etc. 

Scholium III. What arc measures a right angle? An 
acute angle? An obtuse angle? 

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34 PLANE GEOMETRY. 

Proposition XVIII. A Theorem. 

181. An inscribed angle is measured by half the inter- 
cepted arc. 

Case I. When one of the chords forming the angle is the 
diameter. 

See Book I., Proposition X. 

Case II. When the chords are on opposite sides of the 
centre. 

Case III. When both chords are on the same sifle of the 
centre. 

Corollary I. All angles inscribed in the same segment 
are equal. 

Corollary II. All angles inscribed in a semicircle are 
right angles. 

Corollary III. An angle inscribed in a segment smaller 
than a semicircle is obtuse; one inscribed in a segment 
greater Aan a semicircle is acute. 

Proposition XIX. A Theorem. 

182. The angle formed by two chords which intersect each 
other is measured by half the sum of the included arcs. 

Proposition XX. A Theorem. 

183. The angle formed by two secants is measured by 
half the difference of the included arcs. 

Proposition XXI. A Theorem. 

184. The angle formed by a tangent and a chord is meas- 
ured by half the intercepted arc. 

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BOOK III. 35 



Proposition XX 11. A Theorem. 

185. An angle formed by two tangents is measured by 
half the difference of the intercepted arcs. 

Note. In circles that are not equal, radii forming equal angles at 
the centre intercept arcs whose absolute length is not the same, but 
they contain the same number of degrees, and may be called homolo- 
gous. This can be easily shown by means of concentric circles. 
Hence, in any circles, {a) radii forming equal angles at the centre, 
{d) equal inscribed angles, {c) equal angles formed by intersecting 
chords, by secants, or by tangents, intercept homologous arcs ; and, 
conversely, if the arcs intercepted are homologous, the angles are 
equal. It is on this principle that Supplementary Propositions 13, 
14, and 15 of this Book depend. 

Proposition XXIll. A Problem 

186. To erect a perpendicular at the end of a given line. 
See Book III., Proposition XVIII., Corollary II. 



Proposition XXIV. A Problem. 

187, At a point on a line, to construct an angle equal te 
a given angle. 

See Book III., Proposition XVII., Scholium I., and Propo- 
sition IV. 

Proposition XXV. A Problem. 

188. To bisect a given arc. 
See Book III., Proposition VI. 



Proposition XXVI. A Problem. 
189. To bisect a given angle. 

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36 PLANE GEOMETRY. 

Proposition XXVII. A Problem. 

190. Through a given point to draw a line parallel to a 
given line. 

See Book I., Proposition XL 

Proposition XXVIII. A Problem. 

191. Two angles of a triangle being given, to find the 
third. 

Proposition XXIX. A Problem. 

192. To construct a triangle when two of its sides and the 
angle included by them are given. 

Proposition XXX. A Problem. 

193. Given a side and two angles, to construct the triangle. 

Proposition XXXI. A Problem. 

194. Given two sides of a triangle and the angle opposite 
one of them, to construct the triangle. 

Proposition XXXII. A Problem. 

195. Given the three sides, to construct the triangle. 
See Book L, Proposition XXX. 

Proposition XXXIII. A Problem. 

196. To construct a parallelogram when its adjacent sides 
and their mcluded angle are given. 

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BOOK III. 37 



Proposition XXXIV. A Problem. 

197. From a given point to draw a tangent to a given 
circle. 

See Book III., Proposition XII., and Proposition XVIII., 
Corollary II. 

Proposition XXXV. A Problem. 

198. At a given point in the circumference of a circle to 
draw a tangent. 

See Book III., Proposition XXIII. 

Proposition XXXVI. A Problem. 

199. In a given triangle to inscribe a circle. 
See Book I., Proposition XXXIX. 

Proposition XXXVII. A Problem. 

200. The chord being given, to construct a circle such 
that any angle inscribed in one of the segments will be equal 
to a given angle. 

See Book III., Proposition XXL, Proposition XVIII., Co- 
rollary I., and Proposition XII. 

Proposition XXXVIII. A Problem. 

201. Two arcs or two angles being given, to find their 
common measure. 

Proposition XXXIX. A Theorem. [Optional,) 

202. The side of an inscribed equilateral triangle and the 
radius perpendicular to it bisect each other. 



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38 PLANE GEOMETRY. 



SUPPLEMENTARY PROPOSITIONS. 

1. From any point in a circle the shortest distance to the 
circumference will be on the radius passing through the 
point. 

2. From any point in a circle the farthest distance to 
the circumference will be on the line passing through the 
centre. 

3. If a circle is touched internally by another circle having 
half the diameter, any chord of the larger circle drawn from 
the point of contact is bisected by the circumference of the 
smaller circle. 

• 4. The shortest chord that can be drawn through any 
point in a circle is the one drawn at right angles to the 
radius passing through the point. 

5. The opposite angles of an inscribed quadrilateral are 
supplements of each other. 

6. If the opposite angles of a quadrilateral are supple- 
ments of each other, a circumference can be circumscribed 
about it. 

7. The sides of an inscribed equilateral triangle are half 
the length of the sides of a similar circumscribed triangle. 

See Book I., Supplementary Propositions 19, 20, and 21. 

8. If two circles intersect each other, the distance be- 
tween their centres is less than the sum and greater than 
the difference of their radii. 

9. The sum of the opposite arcs intercepted by two 
chords crossing each other at right angles equals a semi- 
circumference. 



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BOOK III. 39 



10. If two equal circles intersect each other, parallel 
secants passing through the points of intersection cut off 
reciprocally equal arcs and segments. 

11. If two equal intersecting circles are cut by two se- 
cants passing through the points of intersection, chords 
subtending the exterior arcs intercepted by these secants 
will be parallel. 

Case I. When the secants do not cross. 

Case II. When the secants cross each other in one of 
the circles. 

12. If two equal circles touch each other, two secants 
passing through the point of contact, will intercept equal 
arcs ; and the chords subtending these arcs will be parallel. 

13. If two unequal circles intersect each other, two 
parallels passing through the points of intersection and ter- 
minated by the exterior arcs, will be equal. 

See Note, page 35. 

14. If two unequal intersecting circles are cut by secants 
passing through the points of intersection, chords subtending 
the exterior arcs intercepted are parallel. 

Case I. When the secants do not cross. 

Case II. When the secants cross each other in one of the 
circles. 

See Note, page 35. 

15. If two unequal circles touch each other, two secants 
passing through the point of contact will intercept homolo- 
gous arcs, and the chords subtending these arcs will be 
parallel. 



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40 PLANE GEOMETRY. 

1 6. Construct an angle of 60° ; of 120° ; of 30° ; of 15°. 

17. Construct an angle of 45°. Divide it into three equal 
angles. 

18. Divide a right angle into three equal angles. 

19. Find a point equidistant from three given points. 

20. Find a point equidistant from two given points, and a 
given distance from a third given point. 

21. Construct a perpendicular from the vertex of one 
angle of a triangle to the opposite side. 

22. Divide a line into three equal parts. 
See Book I., Supplementary Proposition 18. 

23. Given the radius and two points in the circumference, 
to construct the circle. 

24. A chord and a point in the circumference given, to 
construct the circle. 

25. To lay off on a given circumference an arc of 180° ; 
of 90°; of 60°; of 30°; of 120°. 

26. The base, the altitude, and one of the angles at the 
base given, to construct the triangle. 

27. Given one side, the diagonal, and the included angle, 
to construct a parallelogram. 

28. In a given circle to inscribe an equilateral triangle. 

29. About a given circle to circumscribe an equilateral 
triangle. 

30. The radius is two thirds the altitude of an inscribed, 
and one third the altitude of a circumscribed equilateral 
triangle. 

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BOOK III. 41 



31. Find in a given circumference two points such that 
tangents passing through them will meet at an angle of 30°. 

32. Find in a given circumference two points such that 
two tangents passing through them will meet at an angle 
of 90^ 

33. Given the perimeter and altitude of a triangle, and 
the point on the perimeter where the perpendicular from 
the opposite angle, which equals the altitude, would fall, to 
construct the triangle. 

34. To construct a triangle, the base, altitude, and angle 
at the vertex being given. 

See Book III., Proposition XXXVII. 

35. To construct a triangle, the base, angle at the vertex, 
and median connecting them being given. 

36. From a given point draw tangents to a given circle; 
connect these tangents by a line drawn tangent to the smaller 
intercepted arc ; a triangle will be formed, the sum of whose 
sides will be constant at whatever point on the arc the con- 
necting tangent be drawn. 

See Book III., Proposition XIII. 

37. If, with the conditions as given in 36, lines be drawn 
from the centre of the circle to the extremities of the con- 
necting tangent, the angle at the centre will remain constant 
through all positions of the tangent. 

38. To construct a right triangle, when given : 

a. Hypotenuse and one side. 

b. Hypotenuse and altitude on the hypotenuse. 
(, One si4e and altitude on the hypotenuse. 



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42 PLANE GEOMETRY. 

39. To construct a scalene triangle, when given : 

a. The perimeter and angles. 

b. One side, an adjacent angle, and sum of the 

other sides. 
€. The sum of two sides and the angles. 

d. The angles and the radius of an inscribed circle. 

e. An angle, its bisector, and the altitude from the 

given angle. 

40. To construct a rectangle, when given : 

a. One side and the angle formed by the diagonals. 

b. The perimeter and a diagonal. 

41. To construct a rhombus, when given : 

a. One side and the radius of the inscribed circle. 

b. One angle and the radius of the inscribed circle. 

42. To construct a rhomboid, when given : 

a. One side and the two diagonals. 

b. The base, the altitude, and one angle. 

43. To construct a trapezoid, when given : 

a. The bases, the altitude, and one angle. 

b. One base, the adjacent angles, and one side. 

c. One base, the adjacent angles, and the median. 



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BOOK IV. 



1. Similar Polygons. 

203. When are polygons similar? * 
804. What are their homologous parts ? 

205. What is meant by their ratio of similitude? 

Proposition I. A Theorem. 

206. Two mutually equiangular triangles are similar. 
See Book II., Proposition XVI. 

Corollary. Triangles having two angles mutually equal, 
or an angle* in each equal and the sides including it in pro- 
portion, are similar. 

Proposition II. A Theorem. 

207. If triangles have their sides taken in order in propor- 
tion, they are similar. 

Proposition III. A Problem. 

208. The ratio of the homologous sides being equal to the 
ratio of two given lines, to construct a triangle similar to a 
given triangle. 

* Form what proportions you can from two similar triangles ; from 
two similar quadrilaterals ; from two similar pentagons. 

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44 PLANE GEOMETRY. 



Proposition iV. A Theorem. 

209. Two triangles whose sides are parallel or perpendicu- 
lar are similar. 

Proposition V. A Theorem. 

210. Two similar polygons may be divided into the same 
number of triangles^ similar each to each. 

Proposition VI. A Theorem. 

211. If two polygons can be divided into the same num- 
ber of triangles, similar each to each, and similarly placed, 
the two polygons are similar. 

Proposition VII. A Problem. 

212. A polygon being given, on a line corresponding to 
one of its sides, to construct a similar polygon. 

Proposition VIII. A Theorem. 

213. The perimeters of two similar polygons have the same 
ratio as any two homologous sides. 

Proposition IX. A Theorem. 

214. Any number of straight lines intersecting at a com- 
mon point intercept proportional segments on two parallels. 

Cask I. When the parallels are on the same side of the 
common point. 

Case II. When they are on opposite sides. 

Note. This principle may be used in finding the sides in Fropo- 
sition VII. 



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BOOK IV. 45 



Proposition X. A Theorem. 

215. Conversely, all non-parallel lines intercepting pro- 
portional segments on two parallel lines intersect at a 
common point. 



II. Division of Lines. 

216. What is dividing aline internally? 

Example, a 



a. What are the segments? 

217. What is dividing a line eictemaUy? 
Example. ^. 



A 

a. What are the segments? 

SPECIAL PROBLEMS. 

218. a. To divide a line internally in the ratio of 2 : 3 ; 

of 3 : 5 ; of 2 : 7 ; etc. 
See Proposition IX., or Book II., Proposition XVIII. 

d. To divide a line externally in the ratio of 2 : 3 ; 
of 3 : s ; of 2 : 7 ; etc. 

219. What is dividing a line harmonically? 

Example. 

c 

Divided exteraally as 



SPECIAL PROBLEMS. 

220. To divide a given line harmonically in the ratio of 
3-4; of 3:5; of 2:7; etc. 



Divided internaUy as 


j4 


l^ 


t05 
1 1 1 1 


1 1 


3 


to 5 


1 










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46 PLANE GEOMETRY. 



Proposition XI. A Theorem. 

221. A line bisecting an angle of a triangle divides the 
opposite side into segments proportional to the adjacent 
sides including the angle. 

Proposition XII. A Theorem. 

222. A line bisecting an exterior angle of a triangle divides 
the opposite side externally into segments proportional to the 
other two sides. 

Proposition XIII. A Theorem. 

223. Lines bisecting adjacent interior and exterior angles 
of a triangle divide the opposite side harmonically. 

Proposition XIV. A Problem. 

224. To divide a line harmonically. 

Proposition XV. A Theorem. 

225. In a right triangle, if a perpendicular is drawn from 
the vertex of the right angle to the hypotenuse : 

I. The right triangle is divided into two triangles 

similar to itself and to each other. 
11. The perpendicular is a mean proportional be- 
tween the segments of the hypotenuse. 
III. Each side of the right angle is a mean propor- 
tional between the whole hypotenuse and the 
adjacent segment. 
Corollary I. The squares of the sides of the right angle 
are proportional to the adjacent segments of the hypotenuse. 

Corollary II. The square of the hypotenuse of a right 
triangle is equivalent to the sum of the squares of the other 
two sides. 



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BOOK IV. 47 



Proposition XVI. A Problem. 

226. To find a mean proportional between two given 
lines. 

Proposition XVII. A Theorem. 

227. The products of the two segments of all chords 
drawn through any fixed point in a circle are constant.* 



Proposition XVIII. A Theorem. 

228. From a point without a circle, in whatever direction 
a secant is drawn, the product of the whole secant by its 
external segment is constant. 

Proposition XIX. A Theorem. 

229. If from a point without a circle a secant and a tan- 
gent be drawn, the tangent is a mean proportional between 
the whole secant and its external segment. 

How can this be proved by the Theory of Limits ? 



III. Extreme and Mean Ratio. 

230. What is dividing a line in extreme and mean ratio? 

Examples. 

C 

Internally. A B 

ABiAC::ACi CB. 

Externally. A 



CBiCA :: CA :AB, 



* Find a line proportional to three given lines by this principle. 

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48 



PLANE GEOMETRY. 



Proposition XX. A Problem. 

231. To divide a given line in extreme and mean ratio. 
.S*^^ Proposition XIX., and Book II., Proposition VII. 

Proposition XXI. A Theorem. 

232. In any triangle, the product of any two sides is equal 
to the product of the perpendicular to the third side from the 
opposite angle by the diameter of the circumscribed circle. 

Proposition XXII. A Theorem. 

233. If an angle of a triangle be bisected by a line termi- 
nating in the opposite side, the product of the segments of 
this side plus the square of the bisector equals the product of 
the other two sides. 

Proposition XXIII. A Theorem. 

234. Homologous altitudes of similar triangles are propor- 
tional to any two homologous sides. 

SUPPLEMENTARY PROPOSITIONS. 




1. The chord 
A B bisects the 
common tangent 
CD. 

2. The com- 
mon tangent CD 
is a mean pro- 
portional between 
the diameters of 
the circles. 



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BOOK IV. 49 



3. If two circles intersect each other, the common chord 
produced bisects the common tangent. 

4. If the common chord of two intersecting circles be 
produced, tangents drawn from any point in it to the circles 
are equal. 

5. To inscribe a square in a given triangle. 

6. To inscribe a square in a semicircle. 

7. To inscribe in a given triangle a rectangle similar to a 
given rectangle. 

8. To circumscribe about a circle a triangle similar to a 
given triangle. 

9. To construct a circle whose circumference will be tan- 
gent to a given line and pass through two given points. 

10. To construct a circle whose circumference will be tan- 
gent to two given lines and pass through one given point. 



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BOOK V. 

MEASUREMENT AND COMPARISON OF 
RECTILINEAR FIGURES. 



I. Area. 

236. What is area? 

a. How measured? 

Proposition I. A Theorem. 

236. The area of a rectangle is equal to the product of its 
base and altitude. 

Case I. When the base and altitude are commensurable. 

Case II. When they are incommensurable. 

Proposition II. A Theorem. 

237. The area of a parallelogram is equal to the product 
of its base and altitude. 

Corollary. Parallelograms having equal bases and alti- 
tudes are equivalent. 

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BOOK V. 51 



Propotition III. A Theorem. 

238. Parallelograms are to each other as the products of 
their respective bases and altitudes. 

Corollary. Parallelograms having equal altitudes are to 
each other as iheir bases ; those having equal bases are to 
each other as their altitudes. 

Proposition IV. A Theorem. 

239. The area of a triangle is equal to half the product of 
its base and altitude. 

Corollary. Triangles having the same base and altitude 
are equivalent 

Proposition V. A Theorem. 

240. Triangles are to each other as the products of their 
respective bases and altitudes. 

Corollary. Triangles having the same altitudes are to 
each other as their bases, and those having the same bases 
are to each other as their altitudes. 

Proposition VI. A Theorem. 

241. The area of a trapezoid is equal to the product of its 
altitude by half the sum of its bases, or by its median. 

Proposition VII. A Theorem. 

242. The areas of two triangles having an angle in each 
equal are to each other as the products of the sides including 
the equal angle. 



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52 PLANE GEOMETRY. 

Propotition VI 11. A Theorem. 

2431 The square described on the sum of two Imes is 
equal to the sum of their squares plus two rectangles con- 
tained by the lines. 

Scholium. Compare (tf + ^)^ = «* + 2ab + b^. 

PropositioR IX. A Theorem. 

244. The square described on the difference of two lines 
is equal to the sum of their squares minus two rectangles 
contained by the lines. 

ScHOUUM. Compare {a — by = a^ — 2 ab -^ ^*. 

Proposition X. A Theorem. 

245. The rectangle contained by the sum and difference 
of two lines is equal to the difference of their squares. 

Scholium. Compare {a + ^) (« — ^) = a* — ^. 

Proposition XI. A Theorem. 

246. The square described on the hypotenuse of a right 
triangle is equal to the sum of the squares of the other 
two sides. 

Corollary. The square described on either side forming 
the right angle is equal to the square of the hypotenuse 
minus the square of the other side. 

Proposition XII. A Problem. 

247. To construct a square equal to the sum of two given 
squares. 

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BOOK V. S3 



Proposition XIII. A Problem. 

248. To construct a square equal to the difference of 
two given squares. 

Proposition XIV. A Problem. 

SMS. To construct a square equal to the sum of any given 
number of given squares. 

Proposition XV. A Problem. 

250. I. To construct a square equivalent to a given 
rectangle. 

See Book IV., Proposition XVI. 

II. Equivalent to a given triangle. 

Proposition XVI. A Problem. 

251. The sum of the base and altitude given, to construct 
a parallelogram equivalent to a given square. 

Proposition XVII. A Problem. 

252. The difference between the base and altitude given, 
to construct a parallelogram equivalent to a given square. 

Proposition XVIII. A Theorem. 

25& The areas of similar triangles are to each other as 
the squares of their homologous sides. 
See Proposition VII. 

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54 PLANE GEOMETRY. 

Proposition XIX. A Tlieoram. 

254. The areas of any similar polygons are proportioBal to 
the squares of their homologous sides. 

Proposition XX. A Problem. 

255. I. To construct a polygon similar to two given poly- 
gons but equal to their sum. 

See Proposition XI L 

II. Equal to their difference. 

Proposition XXI. A Problem. 

256. I. To construct a triangle equivalent to a given 
polygon. 

XL To construct a square equivalent to a given polygon 
of five or more sides. 

Proposition XXII. A Problem. 

257. To construct a square having a given ratio to a 
given square. 

Proposition XXIII. A Problem. 

258. In a given ratio between their areas, to construct a 
polygon similar to a given polygon. 

Proposition XXIV. A Problem. 

259. To construct a polygon similar to one given polygon 
but equivalent to another. 

See Proposition XXI. 

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BOOK V. 



55 



II. Projection. 

260. What is the projection of a point on a line? 
Illustrations. 



The point. A 



The point 



261. What is the projection of a line on another line ? 
Illustrations. 





Proposition XXV. A Theorem. 

262. In a triangle, the square of a side opposite an acute 
angle is equal to the sum of the squares of the other two 
sides minus twice the rectangle formed by one of these sides 
and the projection of the other on it. 



Proposition XXVI. A Theorem. 

263. In a triangle, the square of the side opposite an ob- 
tuse angle is equal to the sum of the squares of the other 
two sides plus twice the rectangle formed by one of these 
sides and the projection of the other on it. 

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S6 PLANE GEOMETRY. 



Proposition XXVIi. A Tlieorem. 

264. In a triangle, if a median line is drawn from the ver- 
tex of any angle : 

I. The sum of the squares of the sides including the angle 
is equal to twice the square of the median plus twice the 
square of half the side it bisects. 

II. The difference of the squares of the two sides includ- 
ing the angle is equal to twice the rectangle formed by the 
third side and the projection of the median on it 

Corollary I. In any quadrilateral (not a parallelogram) 
the sum of the squares of the four sides is equal to the sum 
of the squares of the diagonals plus four times the square of 
the line joining the middle points of the diagonals. 

Corollary II. In a parallelogram the sum of the squares 
of the four sides is equal to the sum of the squares of the 
diagonals. 

SPECIAL PROBLEMS. 

1. Express the altitude of an equilateral triangle in terms 
of its sides. 

Suggestion. Let a be the length of one side, and x the altitude. 

2. Express the area of an equilateral triangle in terms of 
its sides. 

Suggestion. Let A represent the area. 

3. Express the altitude of any triangle in terms of its sides. 

Suggestion. Let a, b, and c represent the lengths of the different 
sides. 

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BOOK V. 57 



4. Express the area of a triangle in terms of its sides. 

5. Express a median of a triangle in terms of its sides. 

6. Express the bisector of an angle of a triangle in terms 
of its sides. 

7. Express the radius of a circle circumscribed about a 
triangle in terms of the sides of the triangle. 



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BOOK VI. 



I. Regular Polygons. 

265. What is a regular polygon? 

a. The apothem? 

Proposition I. A Theorem. 

266. A circle may be circumscribed about and one in- 
scribed within a regular polygon. 

Corollary. The radius drawn to the vertex of an angle 
of a regular inscribed polygon bisects the angle. 

Proposition II. A Theorem. 

267. An equilateral polygon inscribed in a circle is regular. 

Proposition III. A Theorem. 

268. If a circumference be divided into equal arcs : 

I. The chords subtending these arcs form a regular 
polygon. 

II. The tangents drawn at the points of division form a 
regular polygon. 

See Book III., Proposition XIIL 

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BOOK VI. 59 



Proposition IV. A Problem. 

269. I. Given a regular inscribed polygon, to construct 
one having double the number of sides. 

II. Given a regular circumscribed polygon, to construct 
one having double the number of sides. 

Proposition V. A Problem. 

270. In a given circle to construct a square. 



Proposition VI. A Theorem. 

271. The side of a regular inscribed hexagon is equal to 
the radius of the circumscribed circle. 



Proposition VII. A Problem. 

272. To inscribe a regular hexagon in a given circle. 

Proposition VIII. A Problem. 

273. To inscribe a regular decagon in a given circle. 
See Book IV., Proposition XX. 

ScHOUUM. Inscribe a regular pentagon. 

Proposition IX. A Problem. 

274. To inscribe a regular pentedecagon in a given circle. 

Proposition X. A Theorem. 

275. Two regular polygons of the same number ofsidesi 
are similar. 



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60 PLANE GEOMETRY. 

Proposition XI. A Theorem. 

276. The perimeters of two regular polygons of the same 
• number of sides are to each other : 

I. As their sides. 

II. As the radii of circumscribed circles. 

III. As the radii of inscribed circles. 

Corollary. If in two circles all possible regular polygons 
be drawn, the perimeters of those in one circle will have to 
the perimeters of the similar ones in the other circle a con- 
stant ratio. 

Proposition XII. A Theorem. 

277. The circumferences of circles are to each other as 
their radii or their diameters. 

See Book IL, §§ 120-128. 

Corollary. The ratio of circumferences to their radii or 
to their diameters is constant. 

Scholium L The constant ratio of the circumference to 
the diameter is represented by the Greek letter «•, and it will 
be hereafter one of our objects to ascertain its numerical 
value. 

Scholium II. Let 2 R represent the diameter ; the cir- 
cumference will be 2'irR, 

Proposition XIII. A Theorem. 

278. The areas of two regular polygons of the same num- 
ber of sides are to each other : 

r I; As the squares of their sides. 

: { \ll» As the squares of the radii of circumscribed circles. 

ni. As the squares of the radii of inscribed circles. 

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BOOK VI. 6 1 



Proposition XIV. A Theorem. 

279. The areas of circles are to each other as the squares 
of their radii or of their diameters. 

Corollary. The areas of similar sectors or segments are 
to each other as the squares of the radii or of the diameters. 

Proposition XV. A Theorem. 

280. I. The difference between the perimeters of regular 
inscribed and circumscribed polygons of the same number of 
sides is indefinitely diminished as the sides are indefinitely 
multiplied. 

II. The difference between their areas is indefinitely 
diminished as the sides are indefinitely multiplied. 

Proposition XVI. A Theorem. 

281. The area of a regular polygon is equal to half the 
product of its perimeter by its apothem. 

Proposition XVII. A Theorem. 

282. The area of a circle is equal to half the product of 
the circumference by the radius. 

Scholium. If 2 w^ (see Proposition XII., Scholium II.) 
represents the circumference, what will be the area of the 
circle ? 

Corollary. The area of a sector is equal to half the 
product of its arc and the radius. 



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62 PLANE GEOMETRY. 



Proposition XVIII. A Problem. 

Given the radius and a chord, to compute the chord 
of half the arc subtended. 

Scholium. This principle can be used, when the side of a 
regular inscribed polygon is known, to find the side, and 
therefore the perimeter, of a regular polygon of double the 
number of sides. 

Proposition XIX. Problems. 

284. I. To find the ratio between the perimeter of a regu- 
lar inscribed hexagon and the diameter of the circle. 

II. Between the perimeter of a regular inscribed duodeca- 
gon and the diameter of the circumscribed circle. 



Proposition XX. A Problem. 

285. To compute the numerical value of ir. 

OPTIONAL PROPOSITIONS. 

Proposition XXI. A Problem. 

286. The perimeters of a regular inscribed and a similar 
circumscribed polygon being known, to compute the perime- 
ters of the regular inscribed and circumscribed polygons of 
iJettble the number of sides. 

Proposition XXII. A Problem. 

287. To compute the numerical value of w from the pre- 
ceding problem. 

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BOOK VI. 63 



SPECIAL PROBLEMS. 
Proposition XXIII. A Problem. 

288. L Express the side of an inscribed equilateral tri- 
angle in terms of the radius. 

II. The same of a regular inscribed hexagon. 
III. The same of a regular inscribed duodecagon. 
Note. Continue this as far as desirable. 
Scholium. What would the perimeters be in each case? 

Corollary. Express the areas of the above in terms of 
the radius. 

Proposition XXIV. Problems. 

289. I. Express the side of an inscribed square in terms 
of the radius. 

II. The same of a regular inscribed octagon. 
Note. Continue as far as desirable. 

Scholium. What would the perimeters be in each case? 

Corollary. Express their areas in terms of the radius. 

Proposition XXV. Problems. 

290. I. Express the side (and perimeter) of a regular 
inscribed decagon in terms of the radius. 

See Proposition VIII. 

II. Express the side of a regular inscribed pentagon in 
terms of the radius. 

Corollary. Express their areas in terms of the mdius. 

Proposition XXVI. A Problem. 

291. Compute the numerical value of ir from one of the 
above problems. 

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64 PLANE GEOMETRY. 

SUPPLEMENTARY PROBLEMS. 

Geometrical^ Construction of Algebraic Equations. 

Note. In these problems, the first letters of the alphabet express 
known, or given lines ; in performing operations with them, the fol- 
lowing points should be kept in mind : 

1 . The product of two lines, or the square of a line is a surface. 

2. The product of three lines is a solid. 

3. A surface divided by a line, is a line. 

4. The square root of the product of two lines is a line. 

The letter x represents the element to be constracted 
and may be a line, surface, or solid as the case requires. 
Problem I. Construct x=^a-\- b. 
Problem IL Construct x^.a-^d. 
Problem III. Construct x^ab. 
Problem IV. Construct xz=iabc. 

Problem V. Construct a: = — . 

c 

See Book IL, Proposition XIX. 

Problem VI. Construct ^ = -r- . 

See Book IL, Proposition XX. 



Problem VII. Construct ^ = V«* + ^*. 
See Book V., Proposition XL 



Problem VIIL Construct x ^ \/ a^ — b^. 
Problem IX. Construct x = V^- 

See Book IV., Proposition XVI. 



Problem X. Construct x=z\/ a^-^ab. 
See Book IV., Proposition XIX. 

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SUPPLEMENTARY PROBLEMS. 65 



Problem XL Construct ^ = a ± V^^ — ^'. 

SuG. Construct a line equal to a, and at one extremity 
construct a perpendicular equal to ^. With the remote end 
of ^ as a centre and a radius equal to a, draw an arc cutting 
a and a prolonged. 

Problem XIL Form the equation for the larger seg- 
ment of a line a divided in extreme and mean ratio. 
See Book IV., Proposition XX. 

Problem XIIL Form the equation for the side of a 
square inscribed in a triangle whose base and altitude are 
given. 

Problem XIV. Given the radii and the distance be- 
tween the centres of two unequal circles, form the equation 
for the distance to the point where their common tangents 
will meet. 



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