« ΠροηγούμενηΣυνέχεια »
may have other qualities besides the one or more which we use in defining its name; but any definition which gives more of those qualities than are necessary for its own purposes, is not a strictly proper definition.
We may usefully notice two sorts of improper definitions.
A redundant definition of a class of things is a definition which states more qualities than are sufficient to mark out that class from all other things.
A definition of a class of things is deficient when it does not state such qualities as are sufficient to distinguish that class from all other things.
Almost all the definitions which will be given hereafter were written by Euclid nearly 2,200 years ago. They are very good, taken as a whole; but, like all other human work, they have their faults also, here and there. Some of these we will try to find out, and notice as we go on,
The word 'definition' is derived from the Latin definio, I limit.
The learner may here be reminded that Euclid's definitions will be in larger print. He should study them, as they come before him, until he can repeat them from memory fluently. In order to give him, for the present at least, the opportunity of doing a little at a time and doing it well, these early articles will each embrace a very narrow, although of course increasing, range.
1. A point is that which has no parts, or which has no magnitude.
All such points as a dot on a blackboard, or a full stop on paper, have magnitude or size, however small they may be; they have both length and breadth, and even thickness or depth; and can therefore be divided into parts. Yet in these cases the magnitude is generally an accidental quality, which we do not want for its own sake, but rather that we may see the point, and have something to guide the eye in fixing the position. The position or place, which is marked by the point, is what we usually wish to consider. In Geometry it is always so; hence some writers prefer to say on this subject," A point is that which has position but no parts."
A point, as imagined by Euclid, is called a geometrical point; such a dot as we must make in order to represent it, is called a physical point. The word 'magnitude' is from the Latin magnitudo, size.
1. Give the derivation and your own explanation of the word 'definition." 2. Say when a definition becomes redundant; also when it is deficient. 3. If you have given a correct definition of a square without mentioning more than one corner, what sort of definition will this become if the
statement in it about the one corner be changed into a similar one about all the corners? 4. When making a dot on a blackboard with chalk, you leave something on the board which can be felt with the finger. Will such a dot have parts? 5. Would you expect a dot of ink dried on the paper to have parts, even if you could not feel it with your finger? 6. A dot on paper is not truly a geometrical point; then of what use is it to make such a dot and call it a point? 7. Distinguish between a 'geometrical' or 'mathematical point' and a 'physical' one. 8. Give one or two other English words for 'magnitude,' and also the Latin. 9. If the word 'or' in Definition 1 be changed into 'and,' show that the definition would become redundant.
2. A line is length without breadth.
3. The extremities of a line are points.
4. A straight line is that which lies evenly between its extreme points.
A physical line has breadth, however little, and even some thickness or depth; we could not see its position otherwise.
The word 'even' is usually applied to surfaces; an even surface" meaning a perfectly flat one; that is, one in which there are no 'ups and downs.' So an 'even line' is one in which there are no ins and outs' to the right or left in going along it.
To express a straight line, we put letters to it; thus, A or BC. We usually require two letters placed at the two ends.
1. Explain the difference between a mathematical' and a 'physical ' line. 2. Are the extremities the only points in a line? 3. What will the place of meeting or crossing of two mathematical lines form? 4. Imagine yourself starting at one end of a line to go along it to the other, and suppose that, in walking the whole distance, you never change your direction, what sort of line will you consider it to be? 5. Suppose, on the other hand, while always looking along the line just in front of you, you find yourself now and then looking away from the end you are going to, would you then consider the line to lie evenly between its extreme points? 6. If A be placed at one point, and B at another, what would the straight line joining them be called?
5. A superficies, or surface, is that which has only length and breadth.
6. The extremities of a superficies are lines.
7. A plane superficies is that in which, any two points being taken, the straight line between lies wholly in that superficies.
The words 'superficies' and 'surface' are two words both derived from the same Latin words-super, above, and facies, a face. Their meaning is very much the same; but surface' is the word in common use for outward faces of things; and 'superficies' is chiefly used for imaginary surfaces in the sciences, especially Geometry. A 'physical' superficies will have some thickness, however little.
The word 'plane' is from the Latin planus, even or level; and a plane superficies may be familiarly described as a perfectly flat surface, either really existing, or only imagined to exist, anywhere we choose in space. It seldom gets its full name, but is usually called a 'plane' simply.
Imagine we are watching a stone-mason at work upon the surface of a block of stone, trying to make it even. How does he test, when he wishes to do so, the evenness of the surface? He has a bar of wood with one of its edges forming a tolerably straight line. This straight edge he applies to the surface, and then looks at it from one side, trying to see between it and the surface. If he cannot see between them, the straight edge forms a straight line which lies, from one end of it to the other, wholly' in the surface. But it is not sufficient to try the straight edge thus in one position only. Afterwards the workman places it in another position, and tries to make it join two other points in the surface. If he can do so, he tries another position, and another, and another, and so on, till he is satisfied. This workman's test is evidently in strict accordance with our definition.
1. Give the derivation of the word 'plane': and compare its spelling with that of a geographical term of similar meaning. 2. Which of the words, superficies' and 'surface,' is most unlike the common origin of the two? Which is in more common use? 3. Give the usual abbreviated form of 'plane superficies.' 4. Describe a practical mode of testing evenness of surface, which illustrates Def. 7.
8. A plane angle is the inclination of two lines to one another in a plane, which meet together, but are not in the same direction.
9. A plane rectilineal angle is the inclination of two straight lines to one another, which meet together, but are not in the same straight line.
Notice three points in which this last definition differs from the one before it. First, the word 'rectilineal' is introduced, which means "contained by straight lines." Secondly, 'straight' is introduced, as is required by the previous term 'rectilineal." Thirdly, the words "in a plane" are left out; for they are expressed, in accordance with Def. 8, by the adjective 'plane' which precedes 'rectilineal.'
The word 'angle' is from the Latin angulus, a corner; 'rectilineal’ from rectus, right or straight, and linea, a line. In future, 'rectilineal' will nearly always be understood before 'angle.'
Definition 9 does not express the real nature of an angle as well as could be done; if, indeed, it can be said to express it at all. Instead of 'inclination to one another, it would be better to say 'from one another; and it would be better still to say 'divergence from.' To make a line incline more to another, is to make the angle less; while to make two lines diverge more, is to make the angle between them more also. We shall find another argument to urge against the same definition on coming to Definition 10.
It is important to notice the different ways in which angles may be expressed when we come to discuss them.
When two lines meet without cutting, as AB, AC, in the first of the next figures, the angle between them may be denoted by the single letter placed at the corner, as A; or, also, by the three letters which are on the two lines, but keeping 4 in the middle, as BAC or CAB.
When two straight lines intersect or cut one another, as BD, EC inter
sect at 4, we have four angles, each of which may be denoted by three letters, as just shown, Thus, the angle between AC and AD is CAD or DAC. We might also sometimes denote the angle by one letter placed within the corner, as BAE in the same figure by A.
When more than two lines meet at a point, we have various angles, each denoted in a similar manner, as BAC, CAE, &c.
When either of the straight lines containing an angle has more letters than one along it, we may take any one of them for our purpose. Thus, the angle O, in the last figure above, may be denoted by AOC, AOD, BOC, or BOD.
This arises from a fact which should be carefully remembered, namely, that the length of either line containing an angle has nothing to do with the magnitude of the angle. Thus, OC above has the same inclination to OA as OD has, and so on. We may state the fact otherwise by saying that, to lengthen one of two lines which meet, does not alter the divergence of the one from the other.
1. What does 'rectilineal' mean? Give its derivation, and that of 'angle.' 2. Give an instance of abbreviation of language in Def. 8. 3. State the three points in which Defs. 8 and 9 differ. 4. What word in Def. 9 will usually be left understood afterwards? 5. Explain the use we make of Def. 8 in stating Def. 9. 6. What little word might be changed in these definitions with advantage? 7. Give a definition of ' rectilineal angle,' using the words 'divergence' and 'from' instead of 'inclination' and 'to.' 8. Write out each of the angles formed by the straight lines BD and CE when they intersect in A, as in the second figure of Art. 5. 9. Write out six angles formed in the third figure of the same, each in two ways. 10. Write out the adjoining angle in as many different ways as you can, and say how many.
6. KINDS OF ANGLES.
10. When a straight line standing on another straight line makes the adjacent angles equal to each other, each of these angles is called a right angle; and the straight line which stands on the other is called a perpendicular to it.
11. An obtuse angle is that which is greater than a right angle.
12. An acute angle is that which is less than a right angle.