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Euclid's Elements of geometry, the first six books, chiefly from the text of Dr. Simson, with explanatory notes; a series of questions on each book ... Designed for the use of the junior classes in public and private schools online

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Online LibraryEuclidEuclid's Elements of geometry, the first six books, chiefly from the text of Dr. Simson, with explanatory notes; a series of questions on each book ... Designed for the use of the junior classes in public and private schools → online text (page 7 of 38)
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squares, or equal to any multiple of a given square ; or equal to the
difference of two given squares.

The truth of this proposition may be exhibited to the eye in some
.particular instances. As in the case of that right-angled triangle whose
three sides are 3, 4, and 5 units respectively. If through the points of
division of two contiguous sides of each of the squares upon the sides, lines
be drawn parallel to the sides (see the notes on Book ii.), it will be ob-
vious, that the squares will be divided into 9, 16 and 25 small squares,
each of the same magnitude ; and that the number of the small squares
into which the squares on the perpendicular and base are divided is equal
to the number into which the square on the hypotenuse is divided.

Prop. XLViii is the converse-of Prop, xlvii. In this Prop, is assumed
the Corollary that *' the squared described upon two equal lines are
equal," and the converse, which properly ought to have been appended
to Prop. xLvi.

The First Book of Euclid's Elements, it has been seen, is conversant
with the construction and properties of rectilineal figures. It first lays
down the definitions which limit the subjects of discussion in the First
Book, next the three postulates, which restrict the instruments by which
the constructions in Plane Geometry are eflbcted ; dnd thirdly, the twelve
axioms, which express the principles by which a comparison is made
between the ideas of the thinsrs defined.



QUESTIONS ON BOOK 1. 59

This Book may be divided into three parts. The first part treats of
the origin and properties of triangles, both with respect to their sides and
angles ; and the comparison of these mutually, both with regard to equality
and inequality. The second part treats of the properties of parallel lines
and of parallelograms. The third part exhibits the connection of the
properties of triangles and parallelograms, and the equality of the squares
on the base and perpendicular of a right-angled triangle to the square
on the hypotenuse.

When the propositions of the First Book have been read with the
notes, the student is recommended to use different letters in the diagrams,
and where it is possible, diagrams of a form somewhat different from those
exhibited in the text, for the purpose of testing the accuracy of his know-
ledge of the demonstrations. And further, when he has become suffici-
ently familiar with the method of geometrical reasoning, he may dis-
pense with the aid of letters altogether, and acquire the power of express-
ing in general terms the process of reasoning in the demonstration of any
proposition. Also, he is advised to answer the following questions
before he attempts to apply the principles of the First Book to the so-
lution of Problems and the demonstration of Theorems.

QUESTIONS ON BOOK L

1. What is the name of the Science of which Euclid gives the Ele-
ments? What is meant by Solid Geometry? Is there any distinction
between Plane Geometry y and the Geometry of Planes ?

2. Define the terra magnitude^ and specify the different kinds of
magnitude considered in Geometry. What dimensions of space belong
to figures treated of in the first six Books of Euclid ?

3. Give Euclid's definition of a "straight line.** What does he
really use as his test of rectilinearity, and where does he first employ it ?
What objections have been made to it, and what substitute has been
proposed as an available definition? How many points are necessary to
fix the position of a straight line in a plane? When is one straight
line said to cut, and when to meet another ?

4. What positive property has a Geometrical point? From the
definition of a straight line, shew that the intersection of two lines is a
point.

•5. Give Euclid's definition of a plane rectilineal angle. What are
the limits of the angles considered in Geometry ? Does Euclid consider
angles greater than two right angles ?

6. When is a straight line said to be drawn at right angles^ and when
perpendicular y to a given straight line ?

7. Define a triangle ; shew how many kinds of triangles there are ac-
cording to the variation both of the angles^ and of the sides.

8. What is Euclid's definition of a circle ? Point out the assumption
involved in your definition. Is any axiom implied in it? Shew that
in this as in all other definitions, some geometrical fact is assumed as
somehow previously known,

9. Define the quadrilateral figures mentioned by Euclid.

10. Describe briefly the use and foundation of definitions, axioms,
and postulates : give illustrations by an instance of each.

11. What objection may be made to the method and order in which
Euclid has laid down the elementary abstractions of the Science of Geo-
metry ? What other method has been suggested ?



60 Euclid's elemetsts.

12. What distinctions may be made between definitions in the
Science of Geometry and in the Physical Sciences ?

13. What is necessary to constitute an exact definition ? Are defini-
tions propositions ? Are they arbitrary ? Are they convertible ? Does
a Mathematical definition admit of proof on the principles of the Science
to which it relates ?

14. Enumerate the principles of construction assumed by Euclid.

15. Of what instruments may the use be considered to meet approxi-
mately the demands of Euclid's postulates ? Why only approximately ?

16. "A circle may be described from any center, with any straight
line as radius." How does this postulate differ from Euclid's, and
which of his problems is assumed in it ?

17. What principles in the Physical Sciences correspond to axioms
in Geometry?

18. Enumerate Euclid's twelve axioms and point out those which
have special reference to Geometry. State the converse of those which
admit of being so expressed.

19. What two tests of equality are assumed by Euclid? Is the
assumption of the principle of superposition (ax. 8.), essential to all
Geometrical reasoning ? Is it correct to say, that it is " an appeal,
though of the most familiar sort, to external observation" ?

20. Could any, and if any, which of the axioms of Euclid be turned
into definitions ; and with what advantages or disadvantages r

21. Define the terms, Problem, Postulate, Axiom and Theorem.
Are any of Euclid's axioms improperly so called ?

22. Of what two parts does the enunciation of a Problem, and of a
Theorem consist? Distinguish them in Euc. i. 4, 5, 18, 19.

23. When is a problem said to be indeterminate ? Give an example,

24. When is one proposition said to be the cenverse or reciprocal of
another? Give examples. Are converse propositions universally true?
If not, under what circumstances are they necessarily true ? Why is it
necessary to demonstrate converse propositions ? How are they proved ?

25. Explain the meaning of the woxd proposition. Distinguish between
converse and contrary propositions, and give examples.

26. State the grounds as to whether Geometrical reasonings depend
for their conclusiveness upon axioms or definitions.

27. Explain the meaning of enthymeme and syllogism. How is the
enthymeme made to assume the form of the syllogism ? Give examples.

28. What constitutes a demonstration? Statethe laws of demonstration.

29. What are the principle parts, in the entire process of establishing
a proposition ?

30. Distinguish between a direct and indirect demonstration.

31. What is meant by the term synthesis, and what, by the term,
analysis ? Which of these modes of reasoning does Euclid adopt in his
Elements of Geometry ?

32. In what sense is it true that the conclusions of Geometry are
necessary truths ?

33. Enunciate those Geometrical definitions which are used in the
proof of the propositions of the First Book.

34. If in Euclid i. 1, an equal triangle be described on the other side
of the given line, what figure will the two triangles form ?

35. In the diagram, Euclid i. 2, if DB a side of the equilateral tri-
angle DAB be produced both ways and cut the circle whose center is B
and radius BC in two points G and H ; shew that either of the dis-



QUESTIONS ON BOOK I. 61

tances DG, DH may be taken as the radius of the second circle ; and
give the proof in each case.

36. Explain how the propositions Euc. i. 2, 3, are rendered necessary
by the restriction imposed by the third postulate. Is it necessary for
the proof, that the triangle described in Euc. i. 2, should be equilateral?
Could we, at this stage of the subject, describe an isosceles triangle on a
given base ?

37. State how Euc. i. 2, may be extended to the following problem :
"From a given point to draw a straight line in a given direction equal to
a given straight line."

38. How would ^'■ou cut off from a straight line unlimited in both
directions, a length equal to a given straight line ?

39. In the proof of Euclid i. 4, how much depends upon Definition,
how much upon Axiom ?

40. Draw the figure for the third case of Euc. i. 7, and state why it
needs no demonstration.

41. In the construction Euclid i. 9, is it indifferent in all cases on
which side of the joining line the equilateral triangle is described?

42. Shew how a given straight line may be bisected by Euc. i. 1.

43. In what cases do the lines which bisect the interior angles of
plane triangles, also bisect one, or more than one of the corresponding
opposite sides of the triangles ?

44. ♦♦ Two straight lines cannot have a common segment." Has this
corollary been tacitly assumed in any preceding proposition ?

45. In Euc. I. 12, must the given line necessarily be "of unlimited
length" ?

46. Shew that (fig. Euc. i. 11) every point without the perpendi-
cular drawn from the middle point of every straight line DEy is at unequal
distances from the extremities Z), E of that line.

47. From what proposition may it be inferred that a straight line is
the shortest distance between two points ?

48. Enunciate the propositions you employ in the proof of Euc. i. 16.

49. Is it essential to the truth of Euc. i. 21, that the two straight
lines be drawn from the extremities of the base ?

50. In the diagram, Euc. i. 21, by how much does the greater angle
BDC exceed the less BAG ?

51. To form a triangle with three straight lines, any two of them
must be greater than the third : is a similar limitation necessary with
respect to the three angles ?

52. Is it possible to form a triangle with three lines whose lengths are
1, 2, 3 units : or one with three lines whose lengths are 1, V'2, V 3 ?

53. Is it possible to construct a triangle whose angles shall be as the
numbers 1,2,3? Prove or disprove your answer.

54. What is the reason of the limitation in the construction of Euc.
1. 24. viz. ** that BE is that side which is not greater than the other ?"

55. Quote the first proposition in which the equality of two areas
which cannot be superposed on each other is considered.

56. Is the following proposition universally true ? ♦* If two plane
triangles have three elements of the one respectively equal to three
elements of the other, the triangles are equal in every respect." Enu-
merate all the cases in which this equality is proved in the First Book.
What case is omitted ?

57. What parts of a triangle must be given in order that the triangle
may be described ?



62

58. State the converse of the second case of Euc. i. 26? Under
what limitations is it true ? Prove the proposition so limited ?

59. Shew that the angle contained between the perpendiculars drawn
to two given straight lines which meet each other, is equal to the angle
contained by the lines themselves.

60. Are two triangles necessarily equal in all respects, where a side and
two angles of the one are equal to a side and two angles of the other,
each to each ?

61. Illustrate fully the difference between analytical and synthetical
proofs. What propositioijs in Euclid are demonstrated analytically ?

62. Can it be properly predicated of any two straight lines that they
never meet if indefinitely produced either way, antecedently to our know-
ledge of some other property of such lines, which makes the property
first predicated of them a necessary conclusion from it ?

63. Enunciate Euclid's definition and axiom relating to parallel
straight lines ; and state in what Props, of Book t. they are used.

64. What proposition is the converse to the twelfth axiom of the
First Book ? What other two propositions are complementary to these ?

65. If lines being produced ever so far do not meet; can they be
otherwise than parallel ? If so, under what circumstances ?

66. Define adjacent angles, opposite angles, vertical angles, and alternate
angles ; and give examples from the Eirst Book of Euclid.

67. Can you suggest anything to justify the assumption in the
twelfth axiom upon which the proof of Euc. i. 29, depends ?

68. What objections have been urged against the definition and the
doctrine of parallel straight lines as laid down by Euclid ? Where does
the difficulty originate ? What other assumptions have been suggested
and for what reasons ?

69. Assuming as an axiom that two straight lines which cut one
another cannot both be parallel to the same straight line ; deduce Euclid's
twelfth axiom as a corollary of Euc. i. 29.

70. From Euc. i. 27, shew that the distance between two parallel
straight lines is constant ?

71. If two straight lines be not parallel, shew that all straight lines
falling on them, make alternate angles, which differ by the same angle.

72. Taking as the definition of parallel straight lines that they are
equally inclined to the same straight line towards the same parts ; prove
that " being produced ever so far both ways they do not meet?" Prove
also Euclid's axiom 12, by means of the same definition.

73. What is meant by exterior and interior angles ? Point out examples.

74. Can the three angles of a triangle be proved equal to two right
angles without producing a side of the triangle ?

75. Shew how the corners of a triangular piece of paper may be
turned down, so as to exhibit to the eye that the three angles of a
triangle are equal to two right angles.

76. Explain the meaning of the term corollary. Enunciate the two
corollaries appended to Euc. i. 32, and give another proof of the first.
What other corollaries may be deduced from this proposition ?

77. Shew that the two lines which bisect the exterior and interior
angles of a triangle, as well as those which bisect any two interior
angles of a parallelogram, contain a right angle.

78. The opposite sides and angles of a parallelogram are equal to
one another, and the diameters bisect it. State and prove the converse
of this proposition. Also shew that a quadrilateral figure, is a paral-



I



QUESTIONS ON BOOK I. 63



lelogram, when its diagonals bisect each other : and when its diagonals
divide it into four triangles, which are equal, two and two, viz. those
which have the same vertical angles.

79. If two straight lines join the extremities of two parallel straight
lines, but not towards the same parts, when are the joining lines equal,
and when are they unequal ?

80. If either diameter of a four-sided figure divide it into two equal
triangles, is the figure necessarily a parallelogram ? Prove your answer.

81. Shew how to divide one of the parallelograms in Euc. i. 3o,
by straight lines so that the parts when properly Eirranged shall make
up the other parallelogram.

82. Distinguish between equal triangles and equivalent triangles, and
give examples from the First Book of Euclid.

83. What is meant by the locus of a point? Adduce instances of
loci from the first Book of Euclid.

84. How is it shewn that equal triangles upon the same base or
equal bases have equal altitudes, whether they are situated on the same
or opposite sides of the same straight line ?

85. In Euc. I. 37, 38, if the triangles are not towards the same parts,
shew that the straight line joining the vertices of the triangles is
bisected by the line containing the bases.

86. If the complements (fig. Euc. i. 43) be squares, determine their
relation to the whole parallelogram.

87. What is meant by a parallelogram being applied to a straight line ?

88 . Is the proof of Euc. i. 45, perfectly general ?

89. Define a square without including superfluous conditions, and
explain the mode of constructing a square upon a given straight line
in conformity with such a definition.

90. The sum of the angles of a square is equal to four right angles.
Is the converse true ? If not, why ?

9 1 . Conceiving a square to be a figure bounded by four equal straight
lines not necessarily in the same plane, what condition respecting the
angles is necessary to complete the definition ?

92. In Euclid i. 47, why is it necessary to prove that one side of
each square described upon each of the sides containing the right angle,
should be in the same straight line with the other side of the triangle ?

93. On what assumption is an analogy shewn to exist between the
product of two equal numbers and the surface of a square ?

94. Is the triangle whose sides are 3, 4, 5 right-angled, or not?

95. Can the side and diagonal of a square be represented simul-
taneously by any finite numbers ?

96. By means of Euc. i. 47, the square roots of the natural numbers,
1, 2, 3, 4, &c. may be represented by straight lines.

97. If the square on the hypotenuse in the fig. Euc. i. 47, be
described on the other side of it : shew from the diagram how the
squares on the two sides of the triangle may be made to cover exactly
the square on the hypotenuse.

98. If Euclid II. 2, be assumed, enunciate the form in which Euc. i. 47
may be expressed.

99. Classify all the properties of triangles and parallelograms ^ proved
in the First Book of Euclid.

100. Mention any propositions in Book i. which are included in more
general ones which follow.



64 Euclid's elements.

ON THE ANCIENT GEOMETRICAL ANALYSIS.

Synthesis, or the method of composition, is a mode of reasoning which
begins with something given, and ends with something required, either
to be done or to be proved. This may be termed a direct process, as it
leads from principles to consequences.

Analysis, or the method of resolution, is the reverse of synthesis,
and thus it may be considered an indirect process, a method of reason-
ing from consequences to principles.

The synthetic method is pursued by Euclid in his Elements of
Geometry. He commences with certain assumed principles, and pro-
ceeds to the solution of problems and the demonstration of theorems
by undeniable and successive inferences from them.

The Geometrical Analysis was a process employed by the ancient
Geometers, both for the discovery of the solution of problems and for
the investigation of the truth of theorems. In the analysis of a proh-
letn, the quaesita, or what is required to be done, is supposed to have
been effected, and the consequences are traced by a series of geometri-
cal constructions and reasonings, till at length they terminate in the
data of the problem, or in some previously demonstrated or admitted
truth, whence the direct solution of the problem is deduced.

In the Synthesis of a prnhlem, however, the last consequence of the
analysis is assumed as the first step of the process, and by proceeding
in a contrary order through the several steps of the analysis until the
process terminate in the quaesita, the solution of the problem is effected.

But if, in the analysis, we arrive at a consequence which contra-
dicts any truth demonstrated in the Elements, or which is inconsistent
with the data of the problem, the problem must be impossible : and
further, if in certain relations of the given magnitudes the construction
be possible, while in other relations it is impossible, the discovery
of these relations will become a necessary part of the solution of the
problem.

In the analysis of a theorem, the question to be determined, is,
whether by the application of the geometrical truths proved in the
Elements, the predicate is consistent with the hypothesis. This point
is ascertained by assuming the predicate to be true, and by deducing
the successive consequences of this assumption combined with proved
geometrical truths, till they terminate in the hypothesis of the theorem
or some demonstrated truth. The theorem will be proved synthetically
by retracing, in order, the steps of the investigation pursued in the
analysis, till they terminate in the predicate, which was^ assumed
in the analysis. This process will constitute the demonstration of the
theorem.

If the assumption of the truth of the predicate in the analysis lead
to some consequence which is inconsistent with any demonstrated
truth, the false conclusion thus arrived at, indicates the falsehood of
the predicate ; and by reversing the process of the analysis, it may
be demonstrated, that the theorem cannot be true.

It may here be remarked, that the geometrical analysis is more
extensively useful in discovering the solution of problems than for in-
vestigating the demonstration of theorems.



ANCIENT GEOMETRICAL ANALYSIS. 65

From the nature of the subject, it must be at once obvious, that no
general rules can be prescribed, which will be found applicable in all
cases, and infallibly lead to the solution of every problem. The con-
ditions of problems must suggest what constructions may be possible ;
and the consequences which follow from these constructions and the
assumed solution, will shew the possibility or impossibility of arriving
at some known property consistent with the data of the problem.

Though the data of a problem may be given in magnitude and
position, certain ambiguities will arise, if they are not properly re-
stricted. Two points may be considered as situated on the same side,
or one on each side of a given line ; and there may be two lines drawn
from a given point making equal angles with a line given in position;
and to avoid ambiguity, it must be stated on which side of the line
the angle is to be formed.

A problem is said to be determinate when, with the prescribed con-
ditions, it admits of one definite solution ; the same construction which
may be made on the other side of any given line, not being considered
a different solution : and a problem is said to be indetenninate when it
admits of more than one definite solution. This latter circumstance
arises from the data not ahsolutely fixing, but merely restricting the
quaesita, leaving certain points or lines not fixed in one position only.
The number of given conditions may be insufficient for a single deter-
minate solution ; or relations may subsist among some of the given
conditions from which one or more of the remaining given conditions
may be deduced.

if the base of a right-angled triangle be given, and also the differ-
ence of the squares on the hypotenuse and perpendicular, the triangle
is indeterminate. For though apparently here are three things given,
the right angle, the base, and the difference of the squares on the
hypotenuse and perpendicular, it is obvious that these three apparent
conditions are in fact reducible to two : for since in a right-angled tri-
angle, the sum of the squares on the base and on the perpendicular,
is equal to the square on the hypotenuse, it follows that the differ-
ence of the squares on the hypotenuse and perpendicular, is equal to
the square on the base of the triangle, and therefore the base is known
from the difference of the squares on the hypotenuse and perpendicular
being known. The conditions therefore are insufficient to determine
a right-angled triangle ; an indefinite number of triangles may be
found with the prescribed conditions, whose vertices will lie in the line
which is perpendicular to the base.

If a problem relate to the determination of a single point, and the
data be sufficient to determine the position of that point, the problem
is determinate : but if one or more of the conditions be omitted, the
data which remain may be sufficient for the determination of more
than one point, each of which satisfies the conditions of the problem ;
in that case, the problem is indeterminate : and in general, such points
are found to be situated in some line, and hence such line is called the
locus of the point which satisfies the conditions of the problem.

If any two given points A and B (fig. Euc. IV. 5.) be joined by
a^ straight line AB, and this line be bisected in J), then if a perpen-
dicular be drawn from the point of bisection, it is manifest that a circle



66 ANCIENT GEOMETRICAL ANALYSIS.



described ■with ani/ point in the perpendicular as a center, and a radius
equal to its distance from one of the given points, will pass through



Online LibraryEuclidEuclid's Elements of geometry, the first six books, chiefly from the text of Dr. Simson, with explanatory notes; a series of questions on each book ... Designed for the use of the junior classes in public and private schools → online text (page 7 of 38)