Popular Science Monthly/Volume 77/November 1910/Mathematical Definitions in Text-Books and Dictionaries
| MATHEMATICAL DEFINITIONS IN TEXT-BOOKS AND DICTIONARIES |
By Dr. JOS. V. COLLINS
STATE NORMAL SCHOOL, STEVENS POINT, WIS.
THE word definition is defined as "fixing the bounds of," or "determining the precise signification of." It may be distinguished from the word description by saying that the latter merely makes its object known by words or signs, very often by some non-essential quality, as a lady by her dress. Young, in his "Teaching of Mathematics," says a definition is simply an agreement making clear the precise meaning of the word defined. As mathematics is an exact science, its definitions are important and play a significant part in the development of the subject. Formerly the tendency was to give a large number of definitions at the beginning of a study, but latterly only essential ones are furnished, and others are introduced as needed.
The distinction between definitions, axioms and postulates is often not clear, though it would appear that there should be definite boundaries between them. Doubtless so far as their etymological meanings go the words postulate and axiom could be used interchangeably. A few late geometries class axioms and postulates together and call them all postulates. German texts usually avoid the use of these terms altogether. To the writer the distinction between axiom and postulate in Euclid is valuable and should be retained. Fortunately, most American authors follow Euclid in regarding the postulates as the fundamental propositions of constructions, one the straight-line postulate, and the other the circle postulate. Similarly, some writers do not distinguish clearly between axioms and definitions. For instance, it is usually given as an axiom that quantities that can be made to coincide are equal. This, on the face of things, simply defines the meaning of the term equal. Again, some writers following the lead of the popular French geometer, Legendre, define a straight line as the shortest distance between two points, whereas Euclid gives this property as an axiom. This test for a straight line implies measurement, and hence the idea of measurement of lines would have to be developed before a straight line could be defined. Evidently Euclid's view of the matter is much preferable to Legendre's.
The definitions of the fundamental concepts by different authors should amount to the same, however differently they are expressed. But it turns out that definitions apparently meaning the same thing are really very different. Thus some authors have defined parallel lines as lines that are everywhere the same distance apart (Webster); others that they are lines in the same plane that will not meet however far produced; and others still that they are lines having the same direction. In one sense these definitions mean the same thing, but in the development of geometry they give rise to very different series of propositions.
Perhaps the most far-reaching of differences in definitions are found in those for parallel lines. Wentworth, following Chauvenet, says parallel lines are those having the same direction throughout their whole extent. This definition is very objectionable for two reasons; first, because the meaning of the word direction is ambiguous, the word being used to signify either one way or the exactly opposite, or in the sense of the angle a line makes with a standard line; second, because the idea of direction in the sense intended is difficult to explain. The Century Dictionary gives this definition: "The direction of point A from point B is or is not the same as that of another point C from point D, according as a straight line drawn from B to A and continued to infinity would or would not cut the celestial sphere at the same point as the straight line from D to C continued to infinity." Chauvenet and Wentworth thought they had found a way to simplify the definition of parallelism. It is clear from the preceding that what they did was to slur over a very complex concept. As a matter of fact the use of the word direction in trying to define parallels was not new. Thus Dr. Johnson defines parallels "as lines extended in the same direction, and preserving always the same distance." The definition used by Euclid, viz., lines in the same plane that will not meet, however far produced, is practically the best, and has the merit of preparing the student for the non-Euclidean geometry.
Not only have parallel lines been defined differently by different authors, but other important terms have met the same fate. The concept angle has been presented in three or four ways: (1) As the figure formed by two lines meeting, which is essentially a description, not a definition. (2) As the difference in direction of two lines. (3) As the inclination of one line to another. (4) As the amount of divergence of two lines that meet. The objection to the first is that it does not call attention to an angle as a magnitude, but rather as a shape. A recent author gives this definition and then asks on the next line whether increasing the lengths of the lines would increase the size of the angle? Of course it would if the pupil judged by the definition given. The use of the term direction to define angle is as objectionable as for parallels. The third definition, Euclid's, is better than the others, but not as clear as it might be on account of the meaning of the word inclination. Thus we are led to the fourth definition, which is objectionable chiefly on the ground that, following the usual custom in English, a Latin word is introduced to possess a technical signification, which has little or no meaning for most young persons. The word divergence should be explained etymologically in the definition as turning apart.
Some writers introduce the simple geometrical concepts without attempting to give any definition of them, these concepts being thought of as fundamental, and not needing or capable of definition, just as certain truths are taken as axiomatic. Thus Hilbert, in his "Foundations of Geometry," so regards the concepts of point, straight line, plane and angle. Among his axioms he includes one which says a straight line is "determined" when two of its points are given, and another which says a plane is determined when three of its points are given, the (virtual) definition for a plane thus corresponding to that for a straight line. Nearly all American authors say a plane is a surface such that if any two of its points be joined by a straight line, the line will lie wholly in the surface. To be consistent they should define a straight line as Euclid does, viz., as a line that lies evenly between any two of its points. Thus a plane surface is tested by laying a straight edge on it, and a straight line is tested by sighting between its points to see if they are in the line of sight.
There are four principal definitions for straight line, all different in character. Three of these have already been referred to. One of them, Legendre's (that a straight line is the shortest distance between two points), the Century Dictionary criticizes. The fourth definition, that of the English Society for the Improvement of Geometrical Teaching, viz., A straight line is such a line that any part of it, however turned, will coincide with any other part, if its extremities lie in that other part, has merit as a practical description, but pedagogically it is not very satisfactory on account of the difficulty students have in understanding it, and because the definition is not used after it is made. Theoretically the Hilbert plan of going at the matter is far superior to the association definition.
Some terms in geometry are used ambiguously, notably circle, straight line and equals. By circle is meant either an area or a circumference, the latter being the usual meaning of circle in higher mathematics. By straight line is meant either an indefinite line, or a line segment or sect. By equal is meant either equal in area or volume, i. e., numerically equal, or equal in all respects, often called congruent. These ambiguities lead to much confusion in the minds of learners. It is probable that the words straight for indefinite straight line, ray for half indefinite straight line, sect for line segment, and congruent for "equal in all respects," will soon be generally adopted.
A difference in the definitions of area and volume as given by various authors and by the dictionaries is of considerable interest. Wells says the area of a surface is its ratio to a standard unit of surface. Wentworth, on the other hand, says it is the ratio to a standard unit times the unit of measure. Certainly these definitions do not give the word the same meaning. Which is right? The writer is inclined to think the Wentworth definition correct, for the reason that if one is asked for the area of a field, he does not say, e. g., 11, but 11 acres. There is, of course, the same distinction made in the definitions of volume, and the same distinction could be made in defining contents, weight, length, etc.
Perhaps the most striking thing in connection with mathematical definitions is the weakness of their statement in our dictionaries. These definitions are often stated in synonyms when the real thing could just as well be given; they are stated obscurely when it is just as easy to give clear definitions; and they are stated in Latin terms, the language of the schools two or three centuries ago, though the vast body of users of the dictionary do* not have the least idea of the meaning of the Latin roots. Thus in defining number, Webster's International says "it is a unit or an aggregate of units; a numerable aggregate or collection of individuals; an assemblage made up of distinct things expressible by figures; that which admits of being counted." Compare these definitions with this: a number is one or more units, or ones. They all have this meaning. Only one of the Webster definitions is simple, and it could be simplified still further to advantage by saying that a number is that which can be counted. Notice that number has virtually been defined by a synonym in this definition, since counting would have to be defined. However, counting is a familiar act to every one.
The Old International, latest edition, defines a perpendicular as a line that makes right angles with another, and then a right angle as one that is formed by a perpendicular! The Standard Dictionary does the same thing. If a pupil in a geometry class were to do this, the teacher, metaphorically at least, would box his ears. There is no occasion for defining one of these terms by the other, and then the latter by the former. It would be all right to define either by the other, providing an essential definition were given for the other. In this case the geometries say right angles (or a perpendicular) are formed when one line meets another so as to make the adjacent angles formed equal. The dictionary should do the same. The Century Dictionary says a perpendicular is the shortest distance from a point to a line, and then defines right angle as one formed by a perpendicular. This definition for perpendicular is open to the same objection as the definition of a straight line, which says it is the shortest distance between two points. But the Century evidently does not fall into the silly course of Webster and the Standard. Worcester says a right angle is one of 90°, and defines one degree as one three-hundred-and-sixtieth of a circumference.
We have already seen the pomposity and verbosity displayed by Webster in the definition of number. These characteristics can be duplicated in the definitions of numerous other common words. Thus Webster states that the area of a surface is its "superficial contents": The Century defines area as "the superficies of an enclosed or defined surface space"! Webster defines a proportion as "the relation or adaptation of one portion to another or to the whole, as respects magnitude, quantity or degree." This definition (not intended, of course, as a mathematical one) is hazy enough to suit a mystic. He says equals means "exactly agreeing with respect to quantity," which is not bad aside from the fact that every word is Latin with the exception of the two prepositions. Webster defines ratio as the relation which one magnitude or quantity has to another of the same kind, and in a note distinguishes two kinds of ratio, arithmetical and geometrical. Now nine thousand nine hundred and ninety-nine times out of ten thousand, by the ratio of two numbers, is meant the geometric ratio or the quotient of one by the other, nearly always the first divided by the second. So far as the dictionary definitions go the reader would be likely to think one definition as important as the other.
The foregoing definitions from Webster are those in the Old International now in wide use over the country. It will be found interesting to compare them with those of the New International recently published which contains numerous new features. Looking for the word number we find instead of the four definitions given above, the following two: "The or a total, aggregate, or amount of units (whether of things, persons, or abstract units); arithmetical aggregate; as odd or even number." Now bad as the preceding definitions were, probably every one will say these are inferior in their crude awkwardness. Ideas, for instance, would hardly be included in the parenthesis list, and yet they can be counted when they exist. What the last phrase about odd and even means in its setting does not appear.
Fortunately the bad definition of number just referred to seems to be a very poor example by which to judge the new dictionary. The definitions of ratio and proportions, for instance, unlike in the old dictionary, are above reproach, with a single exception. Under ratio it is said that it is sometimes called the "rule of three." This is evidently a continuation of the old confusion of ratio and proportion. Ratio has only two terms, while rule of three has three, with a fourth implied. Under "proportion" one definition is, the rule of three, which is correct.
Under the word area are given the same old definitions which have been handed down from Dr. Johnson's time, hazy in meaning and oozing with Latin roots. Under the word volume, on the other hand, strangely enough, is found a simple and correct definition. Thus, "volume is the space occupied as measured by cubic units, i. e., cu. ft., cu. in., etc.
The new dictionary, unlike the old, defines a right angle, not by a synonym, but by saying it is an angle included between two radii subtended by a quarter circle. Under angle, "right angle" is not given. Thus the new dictionary can not be criticized as the old was. But it does seem a pity that the dictionaries can not give the simple, plain essential definition found in almost all geometry text-books.
Under "straight line," the first definition, a line having an invariable direction, is credited to Newcomb, thus retaining the old weak idea of direction, used in defining parallel lines. Next, Euclid's definition is given, and then the Hilbert axiom as a definition. Legendre's definition is criticized. Of course the reader of the dictionary will not learn the meaning of a straight line from the Newcomb definition, but will learn the meaning "of the same direction" from his knowledge of a straight line. It would have been far wiser to have told what Euclid meant by his definition.
In defining angle the same discredited definition of the difference in direction of two lines appears again. Curiously enough the generalized, or trigonometrical, definition of an angle is found not under the word angle at all, but from a cross reference to "Mathematical angle." It is only at this place that the essential quality of an angle as a magnitude is given.
The word congruent, for some unknown reason is not given the meaning applied to it in foreign and recent American text-books on elementary geometry.
Under "Parallel lines" we find "lying evenly everywhere in the same direction, however far extended; in all parts equally distant." This is said to be the Euclid idea of the term! Then there is given the modern geometry conception of a point and line at infinity in which parallel lines and parallel planes meet.
Under "Parallel Postulate" is presented a good idea of the difference between Euclidean, Lobachevskian and Elliptical space. Such features as this go far to show that the dictionary is up-to-date in dealing with important ideas of mathematics which the general public has not had a chance to know about heretofore. While this explanation of the parallel postulate deals with one of the most abstruse matters in modern mathematics it is still reasonably intelligible to the ordinary reader. But there have been introduced into the New International the definitions of numerous highly technical mathematical terms whose meaning is quite beyond the ken of all except a very limited number of technical school graduates. Thus, under Dirichlet's theorem is found a triple integral involving perhaps a dozen elements. Similar technical matter will be found under numerous headings.
Evidently in the introduction of such matter the New International has broken away from the long established custom followed by dictionaries and popular cyclopedias, of inserting only what will be fairly intelligible to any well-informed person. This rule probably holds still in other fields of knowledge in the dictionary, but it certainly does not hold in the fields of pure and applied mathematics.
The definitions of ratio and proportion as given by lexicographers in times past strike the present-day reader as curious, and thereby hangs a tale. The old writers following Euclid looked upon ratio and proportion as expressing the relation of quantities, such as lines, and were not ready to admit that ratio could be always a number, since two lines taken at random in general are incommensurable. The old Euclid definition of a proportion (given at the beginning of his Book V.) avoided entirely the question as to whether the ratio of two numbers would always give rise to a number.[1] Whether Euclid's ratios are or are not always numbers, it certainly is true that Euclid cuts irrationals out of his theory of proportion. The modern tendency is to teach that ratios and quantities in algebra are numbers. Certainly in elementary mathematics it is highly desirable for pedagogical reasons that the ratio of two quantities be defined specifically as the quotient of the first divided by the second, a proportion as an equality of ratios, and a quantity in algebra as a number.
Oddly enough, the old writers did not distinguish between ratio and proportion, using the two interchangeably. To understand how this probably came about, it must be observed that proportion is applied to two or more quantities that are thought of as changing, or assuming different values. Thus if two bushels of wheat cost $2.10, five bushels will cost $5.25. Here there are only two kinds of quantity, bushels and dollars. With this view of proportion in mind, we can see what Dr. Johnson meant when he said proportion is the comparative relation of one thing to another; ratio; settled relation of comparative quantity. Ash (1775), Fenning (1761) and Webster (1806) give practically the same definition. Bailey (1736), whose work shows he was something of a mathematician, gives the definitions for ratio and proportion now in use, except that he adds at the end that a ratio is a proportion. Webster in his first large dictionary (that of 1828) gives for the definition of proportion "the comparative relation of one thing for another; the identity of two ratios."
It is interesting to find some of the old lexicographers falling into errors of pupils at the present time. Thus Ash defines an angle as the point or corner where two lines meet. Bailey says an angle is the space comprehended between the meeting of two lines, but he hastens to add "which is either greater or less, as those lines incline towards one another, or stand farther distant asunder." He says perpendicular means level, or a plumb line. Webster, in his 1828 dictionary, says an angle is the space comprised between two straight lines that meet in a point; but he adds also that an angle is the quantity by which two lines diverge from each other.
Though the term per cent, in arithmetic is a simple one, the definition of it in practically all the arithmetics and dictionaries wraps it in a fog. A very considerable part of the trouble pupils in school have with percentage is due to obscurity in this definition. This seems like a bold statement to make, but the reader can judge for himself whether it is probably true when he hears the case stated. In simple English the word per cent, means hundredths. Thus, six per cent, means six hundredths. Instead of giving this definition the books say per cent, means "by the hundred" or "in the hundred." If this is so then 6 per cent, means six by the hundred, which means nothing. Dr. Johnson, who was no mathematician, said that per cent, meant "in the hundred," and all the lexicographers seem to have followed his lead, merely varying his preposition. Webster (1828) says: "In commerce per cent, denotes a certain rate by the hundred. Ten per cent, means ten in the hundred." Webster's International states that per cent, means "by the hundred," "in the hundred." Of course there is a way of giving concrete meaning to these words. Thus 2 per cent, in commission can mean that the agent gets $2 in every $100 worth of goods he sells, or he gets $2 by every $100 worth of goods he buys. One sees the obscurity in these phrases, however, very clearly, by using another denominator. Thus, what does 2 by the 8 mean?
In the preface to his dictionary Dr. Johnson says: "Every other author may aspire to praise; the lexicographer can only hope to escape reproach, and even this negative recompense has been yet granted to a very few." Evidently from this lexicographers had a harder time in those early days than they have now; for nowadays it would seem as though the dictionaries were to be regarded as sacred writings, to criticize which would be sacrilege. Had the dictionary makers been criticized more, most likely they would have improved the quality of their work. The writer has heard it said that the dictionaries are as weak in defining other similar technical terms as in their definitions of mathematical terms, and that they are far behind the progressive cyclopedia makers in the quality of the matter they print. It would seem judging from the preceding mathematical definitions as though the greatest opening for a progressive publisher would lie along the way of bringing out a new unabridged dictionary adapted for the use of the mass of twentieth century Americans.
- ↑ See Encyclopedia Britannica, article "Geometry."