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It may be observed, that the two parallelograms exhibited in fig. 2 partially lie on one another, and that the triangle whose base is BC is a common part of them, but that the triangle whose base is DE is entirely without both the parallelograms. After having proved the triangle ABE equal to the triangle DCF, if we take from these equals (fig. 2.) the triangle whose base is DE, and to each of the remainders add the triangle whose base is BC, then the parallelogram ABCD is equal to the parallelogram EBCF. In fig. 3, the equality of the parallelograms ABCD, EBCF, is shewn by adding the figure EBCD to each of the triangles ABE, dcf.
In this proposition, the word equal assumes a new meaning, and is no longer restricted to mean coincidence in all the parts of two figures.
Prop. XXXVIII. In this proposition, it is to be understood that the bases of the two triangles are in the same straight line. If in the diagram the point E coincide with C, and D with A, then the angle of one triangle is supplemental to the other. Hence the following property :-If two triangles have two sides of the one respectively equal to two sides of the other, and the contained angles supplemental, the two triangles are equal.
A distinction ought to be made between equal triangles and equivalent triangles, the former including those whose sides and angles mutually coincide, the latter those whose areas only are equivalent.
Prop. xxxix. If the vertices of all the equal triangles which can be described upon the same base, or upon the equal bases as in Prop. 40, be joined, the line thus formed will be a straight line, and is called the locus of the vertices of equal triangles upon the same base, or upon equal bases.
A locus in plane Geometry is a straight line or a plane curve, every point of which and none else satisfies a certain condition. With the exception of the straight line and the circle, the two most simple loci; all other loci, perhaps including also the Conic Sections, may be more readily and effectually investigated algebraically by means of their rectangular or polar equations.
Prop. XLI. The converse of this proposition is not proved by Euclid; viz. If a parallelogram is double of a triangle, and they have the same base, or equal bases upon the same straight line, and towards the same parts, they shall be between the same parallels. Also, it may easily be shewn that if two equal triangles are between the same parallels; they are either upon the same base, or upon equal bases.
Prop. XLIV. A parallelogram described on a straight line is said to be applied to that line.
Prop. XLV. The problem is solved only for a rectilineal figure of four sides. If the given rectilineal figure have more than four sides, it may be divided into triangles by drawing straight lines from any angle of the figure to the opposite angles, and then a parallelogram equal to the third triangle can be applied to LM, and having an angle equal to E: and so on for all the triangles of which the rectilineal figure is composed.
Prop. XLVI. The square being considered as an equilateral rectangle, its area or surface may be expressed numerically if the number of lineal units in a side of the square be given, as is shewn in the note on Prop. 1., Book II.
The student will not fail to remark the analogy which exists between the area of a square and the product of two equal numbers; and between the side of a square and the square root of a number. There is, however,
this distinction to be observed; it is always possible to find the product of two equal numbers, (or to find the square of a number, as it is usually called,) and to describe a square on a given line; but conversely, though the side of a given square is known from the figure itself, the exact number of units in the side of a square of given area, can only be found exactly, in such cases where the given number is a square number. For example, if the area of a square contain 9 square units, then the square root of 9 or 3, indicates the number of lineal units in the side of that square. Again, if the area of a square contain 12 square units, the side of the square is greater than 3, but less than 4 lineal units, and there is no number which will exactly express the side of that square: an approximation to the true length, however, may be obtained to any assigned degree of accuracy.
Prop. XLVII. In a right-angled triangle, the side opposite to the right angle is called the hypotenuse, and the other two sides, the base and perpendicular, according to their position.
In the diagram the three squares are described on the outer sides of the triangle ABC. The Proposition may also be demonstrated (1) when the three squares are described upon the inner sides of the triangle: (2) when one square is described on the outer side and the other two squares on the inner sides of the triangle: (3) when one square is described on the inner side and the other two squares on the outer sides of the triangle.
As one instance of the third case. If the square BE on the hypotenuse be described on the inner side of BC and the squares BG, HC on the outer sides of AB, AC; the point D falls on the side FG (Euclid's fig.) of the square BG, and KH produced meets CE in E. Let LA meet BC in M. Join DA; then the square GB and the oblong LB are each double of the triangle DAB, (Euĉ. 1. 41.); and similarly by joining EA, the square HC and oblong LC are each double of the triangle EAC. Whence it follows that the squares on the sides AB, AC are together equal to the square on the hypotenuse BC.
By this proposition may be found a square equal to the sum of any given 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 obvicus, 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 squares 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 effected; and thirdly, the twelve axioms, which express the principles by which a comparison is made
ween the ideas of the things defined.
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 knowledge of the demonstrations. And further, when he has become sufficiently familiar with the method of geometrical reasoning, he may dispense with the aid of letters altogether, and acquire the power of expressing 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 solution of Problems and the demonstration of Theorems.
QUESTIONS ON BOOK I.
1. What is the name of the Science of which Euclid gives the Elements? What is meant by Solid Geometry? Is there any distinction between Plane Geometry, and the Geometry of Planes?
2. Define the term 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, to a given straight line?
7. Define a triangle; shew how many kinds of triangles there are according 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 Geometry? What other method has been suggested?
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 definitions 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?
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. 1. 4, 5, 18, 19.
Give an example.
23. When is a problem said to be indeterminate ?
24. When is one proposition said to be the converse 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 word 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? State the 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 1. 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 1. 2, if DB a side of the equilateral triangle 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
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. 1. 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. 1. 2, should be equilateral? Could we, at this stage of the subject, describe an isosceles triangle on a given base?
37. State how Euc. 1. 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 you cut off from a straight line unlimited in both directions, a length equal to a given straight line?
39. In the proof of Euclid 1. 4, how much depends upon Definition, how much upon Axiom?
40. Draw the figure for the third case of Euc. 1. 7, and state why it needs no demonstration.
41. In the construction Euclid 1. 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. 1. 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. 1. 12, must the given line necessarily be "of unlimited length"?
46. Shew that (fig. Euc. 1. 11) every point without the perpendicular drawn from the middle point of every straight line DE, is at unequal distances from the extremities D, 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. 1. 16. 49. Is it essential to the truth of Euc. 1. 21, that the two straight lines be drawn from the extremities of the base?
50. In the diagram, Euc. 1. 21, by how much does the greater angle BDC exceed the less BAC?
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, √2, √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 DE 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." Enumerate 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?