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58. State the converse of the second case of Euc. 1. 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 propositions 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 knowledge 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 1. 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 First Book of Euclid.
67. Can you suggest anything to justify the assumption in the twelfth axiom upon which the proof of Euc. 1. 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. 1. 29.
70. From Euc. 1. 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. 1. 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 proposition. Also shew that a quadrilateral figure, is a paral
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. Show how to divide one of the parallelograms in Euc. 1. 35, by straight lines so that the parts when properly arranged 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. 1. 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. 1. 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. 1. 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?
91. 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 1. 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 simultaneously by any finite numbers?
96. By means of Euc. 1. 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. 1. 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 11. 2, be assumed, enunciate the form in which Euc. 1. 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 1. which are included in more general ones which follow.
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 reasoning from consequences to principles.
The synthetic method is pursued by Euclid in his Elements of Geometry. He commences with certain assumed principles, and proceeds 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 problem, the quæsita, or what is required to be done, is supposed to have been effected, and the consequences are traced by a series of geometrical 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 problem, 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 quæsita, the solution of the problem is effected.
But if, in the analysis, we arrive at a consequence which contradicts 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
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 investigating the demonstration of theorems.
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 conditions 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 restricted. 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 conditions, 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 indeterminate when it admits of more than one definite solution. This latter circumstance arises from the data not absolutely fixing, but merely restricting the quæsita, leaving certain points or lines not fixed in one position only. The number of given conditions may be insufficient for a single determinate 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 difference 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 triangle, the sum of the squares on the base and on the perpendicular, is equal to the square on the hypotenuse, it follows that the difference 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 D, then if a perpendicular be drawn from the point of bisection, it is manifest that a circle
described with any point in the perpendicular as a center, and a radius equal to its distance from one of the given points, will pass through the other point, and the perpendicular will be the locus of all the circles which can be described passing through the two given points.
Again, if a third point be taken, but not in the same straight line with the other two, and this point be joined with the first point A; then the perpendicular drawn from the bisection E of this line will be the locus of the centers of all circles which pass through the first and third points A and C. But the perpendicular at the bisection of the first and second points A and B is the locus of the centers of circles which pass through these two points. Hence the intersection F of these two perpendiculars, will be the center of a circle which passes through the three points and is called the intersection of the two loci. Sometimes this method of solving geometrical problems may be pursued with advantage, by constructing the locus of every two points separately, which are given in the conditions of the problem. In the Geometrical Exercises which follow, only those local problems are given where the locus is either a straight line or a circle.
Whenever the quæsitum is a point, the problem on being rendered indeterminate, becomes a locus, whether the deficient datum be of the essential or of the accidental kind. When the quæsitum is a straight line or a circle, (which were the only two loci admitted into the ancient Elementary Geometry) the problem may admit of an accidentally indeterminate case; but will not invariably or even very frequently do so. This will be the case, when the line or circle shall be so far arbitrary in its position, as depends upon the deficiency of a single condition to fix it perfectly—that is, (for instance) one point in the line, or two points in the circle, may be determined from the given conditions, but the remaining one is indeterminate from the accidental relations among the data of the problem.
Determinate Problems become indeterminate by the merging of some one datum in the results of the remaining ones. This may arise in three different ways; first, from the coincidence of two points; secondly, from that of two straight lines; and thirdly, from that of two circles. These, further, are the only three ways in which this accidental coincidence of data can produce this indeterminateness; that is, in other words, convert the problem into a Porism.
In the original Greek of Euclid's Elements, the corollaries to the propositions are called porisms (Tоpioμaтa); but this scarcely explains the nature of porisms, as it is manifest that they are different from simple deductions from the demonstrations of propositions. Some analogy, however, we may suppose them to have to the porisms or corollaries in the Elements. Pappus (Coll. Math. Lib. VII. pref.) informs us that Euclid wrote three books on Porisms. He defines " a porism to be something between a problem and a theorem, or that in which something is proposed to be investigated." Dr. Simson, to whom is due the merit of having restored the porisms of Euclid, gives the following definition of that class of propositions: "Porisma est propositio in qua proponitur demonstrare rem aliquam, vel plures datas esse, cui, vel quibus, ut et cuilibet ex rebus innumeris, non quidem, datis, sed quæ ad ea quæ data sunt eandem habent relationem, convenire osten