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In Latin, obtusus means blunt; acutus, sharp.
More will be said about these definitions afterwards (note, Art. 65). If it were not for the advantage of having all the definitions in one collection for easy reference, it would be better to take these definitions themselves and some others afterwards.
The base of a triangle, as the name would suggest, is usually the lowest side. Still, whenever two sides of a triangle have been named, the remaining side is often called the base, whichever it may be. In an isosceles triangle, we usually call those which are equal “the sides.'
Notice the word 'three' in Def. 29. The side opposite the right angle of a right-angled triangle is called the hypotenuse.'
EXAMINATION IX. 1. Familiarly explain equilateral,' 'isosceles,' scalene,' having re. gard to their derivations. 2. Is an equilateral triangle isosceles ? 3. Given that AC is to be called the base of some triangle ABC, name the sides. 4. If AC, BC be called the sides of the triangle ABC, what is AB to be called ? 5. In the triangle ABC, supposing that AB and AC are equal, name the sides and base.
13, KINDS OF QUADRILATERALS.
Of four-sided figures30. A square is that which has all its sides equal and all its angles right angles.
31. An oblong is that which has all its angles right angles but not all its sides equal.
32. A rhombus is that which has all its sides equal, but its angles are not right angles.
33. A rhomboid is that which has its opposite sides equal to one another, but all its sides are not equal nor its angles right angles.
34. All other four-sided figures besides these are called trapeziums.
With respect to Definition 30, if a figure has four equal sides and one angle a right angle, we can prove that all the other angles must be right angles ; so that this definition says too much, and, for that reason, is not a good definition. It would be correct to state it thus,—“A square is a four-sided figure which has all its sides equal and one angle a right angle.” The learner knows well enough what a square is like, and will probably think Euclid's description is better than this; but he should remember that a description and a definition are not necessarily the same thing at all.
Figures which have all their angles right are sometimes called rect. angles. This term generally takes the place of oblong.'
Definition 34 is not much observed in modern books. Euclid employs it once in Book I.
EXAMINATION X. 1. State in what respect Euclid's definition of a square is redundant. 2. Give two classes of figures included under the name 'rectangle.' 3. What name is more common than 'oblong' for that figure? 4. In what respect does an oblong differ from a square ? 5. In what respect does a rhombus differ from a square? 6. How does a rhomboid differ from a rhombus ?
14. PARALLELS AND PARALLELOGRAMS. 35. Parallel straight lines are such as are in the same plane, and which being produced ever so far both ways do not meet.
(A.) A parallelogram is a four-sided figure whose opposite sides are parallel.
(B.) A diagonal of a parallelogram is a straight line joining two opposite angles. It is sometimes also called a diameter, and the word is used similarly for other quadrilaterals.
Definitions A and B are added to those of Euclid, as the words they define are in common use in his Elements.
We shall be able to show afterwards that squares, oblongs, rhombuses, and rhomboids are different classes of parallelograms; but we are not to assume this to be so, until we have given the proofs.
A trapezium having two sides parallel is called a "trapezoid.' The word trapezium itself is sometimes used with this meaning.
Obs. The next three Examinations are intended to be taken in conjunction with a revision of the definitions as given in the larger type. They will refer chiefly to the diagrams at the end of the book.
EXAMINATION XI. (The questions beginning at 5 are on Diagram I.) 1. What sort of point is it which possesses surface ? 2. Plato said that a straight line is such that, if the eye be placed in a continuation of it, one extremity hides all the rest. Could this be employed as a test of a geometrical line ? Give a reason. 3. Is there any kind of boundary which has no magnitude ? 4. The definition we have given as 7 is due to Hero the Elder. Euclid himself says: “A plane surface is that which lies evenly with the straight lines in it." State which other definition is uniform in manner with this. 5. Mention two angles adjacent to the angle AOB; also two adjacent to POQ and to ROU. 6. Name two angles which have 0 A as a common bounding line ; and similarly for OR, OD, OH. 7. Write in four different ways the angle which is between OP, OB. 8. Write the angle between OP, OQ in four different ways, using different sets of letters each time. 9. Write the three angles which are separate parts of the angle POR. 10. Write two others, each of which is a part of the same angle POR. 11. If POR be a right angle, of what kind is POQ, and also POS? 12. Write down the terms of the figure POVX. 13. If POT and QOV are both diameters, name four different semicircles of the larger circle. 14. Count the number of small segments cut off along the edge of the same circle.
(Diagram I. is referred to.) 1. If HK, KX, XH are all equal, what sort of triangle is HKX ? 2. If PQ is equal to OP, what sort of triangle is POQ? 3. Give the names, in letters, of eight isosceles triangles in the circle ABCDH. 4. TE is not equal to the radius of either circle, nor is TG; to what sort of triangles do O ET and UGT belong? 5. Distinguish between the angle HOK and triangle HOK. 6. Are the triangles HOK and OKH different? 7. Are the angles HOK and OKH different ? 8. How many different triangles, and how many different angles, may be denoted by the following :-COD, COR, ODC, ORC, DRC, RCD, RCO ? 9. Suppose OEU to be a right angle, then name the triangles EOU, EOT according to the nature of their angles. 10. Which side of the triangle EO U is th, hypotenuse? 11. If, also, each of the angles EFO, FOE is acute, say why FOE must be an acute-angled triangle. 12. OH is hypotenuse of the triangle OHL, and OK of the triangle OKL; name a right angle in each of those triangles.
EXAMINATION XIII. (For Questions 1—6 see Diagram II., and for 7—9 see Diagram III.)
1. AD and PV are parallel, and so are PV and BC; will AD meet BC if both are produced ever so far both ways ? Say why. 2. Show that, if PV and BC are each parallel to QU, they cannot meet one another. 3. AB, YR, XS, DC are all parallel ; also AD, PV, QU, BC; also PY, QX, BD, RV, SU. Say what kind of figure each of these is APEY, PQXY, PHDY, ABSX, BSUD—remembering that all the sides of the little triangles are equal. 4. If A were a right angle, what would APEY become then ? 5. Write the name of angle A in ten different ways, using a different set of letters each time. In how many more ways can you write the same angle? 6. Write down the geometrical terms of PQFHXY. 7. All the angles at E are equal to one another; show that they must each be a right angle. 8. If a straight line joining AC passed through E, G, how many right-angled triangles would it form ?-how many hypotenuses ? 9. Which figure may be described as ' lozenge-shaped'? 10. From derivations previously given, infer that of diagonal.'
15. INSTRUMENTS FOR THE CONSTRUCTION OF THE FIGURES.
We shall require to draw many figures in order to help our reasoning; and for this purpose we might use various instruments, some of which the learner will most likely have seen. Euclid, however, only requires us to know something about the use of two, namely—some instrument having a straight edge, with which straight lines may be ruled, and when necessary lengthened ; and a pair of compasses, or some other means of making a circle. At this stage, Euclid also proceeds to state exactly what he requires that we shall be able to do with these instruments. His demands are called Postulates,' which is from a Latin word postulo, I demand.
POSTULATES. Let it be granted
1. That a straight line may be drawn from any one point to any other point.
2. That a terminated straight line may be produced to any length in a straight line.
3. And that a circle may be described from any centre at any distance from that centre.
In Postulate 2, produced' means lengthened out.
After reading the above 'postulates,' or ' demands,' the learner may, perhaps, think that they do not ask for much. This is certainly true enough; for, in fact, just as little is asked for as can be made to answer the purpose. It is Euclid's general rule, in regard to the use of an instrument, to apply neither less nor more than he expressly demands or postulates. This point is dwelt upon in the questions which will now follow.
EXAMINATION XIV. 1. What is a postulate'? 2. What instrument will suffice to perform what is required by Postulate 1 ? 3. What instrument will suffice to perform what is required by Postulate 2 ? 4. What instrument also for Postulate 3? 5. Some straight edges' have marks upon them denoting length. Is there any authority in the postulates for using these ? 6. Distances may be compared and also transferred by means of compasses. Is this permitted by any postulate? 7. Give the meaning of described,' in Postulate 3. 8. In the same postulate, what other word should strictly