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 spuare 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 the 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 hypotenuɛe 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, (Euc. 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 11.), 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 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 belys ike 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 so lution 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 approximately 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. 23. When is a problem said to be indeterminate? Give an example. 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 principal parts, in the entire process of establishing ■ 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." merate all the cases in which this equality is proved in the First Book. What case is omitted? Enu 57. What parts of a triangle must be given in order that the triangle may be described ? 58. State the converse of the second case of Euc. 1. 26 Ünder 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 of this proposition. Also shew that a quadrilateral figure, is a paral |