THE DEFINITIONS OF GEOMETRY
When we consider the nature of geometry it is evident that more
attention must be paid to accuracy of definitions than is the case in
most of the other sciences. The essence of all geometry worthy of
serious study is not the knowledge of some fact, but the proof of that
fact; and this proof is always based upon preceding proofs, assumptions
(axioms or postulates), or definitions. If we are to prove that one line
is perpendicular to another, it is essential that we have an exact
definition of "perpendicular," else we shall not know when we have
reached the conclusion of the proof.
The essential features of a definition are that the term defined shall
be described in terms that are simpler than, or at least better known
than, the thing itself; that this shall be done in such a way as to
limit the term to the thing defined; and that the description shall not
be redundant. It would not be a good definition to say that a right
angle is one fourth of a perigon and one half of a straight angle,
because the concept "perigon" is not so simple, and the term "perigon"
is not so well known, as the term and the concept "right angle," and
because the definition is redundant, containing more than is necessary.
It is evident that satisfactory definitions are not always possible; for
since the number of terms is limited, there must be at least one that is
at least as simple as any other, and this cannot be described in terms
simpler than itself. Such, for example, is the term "angle." We can
easily explain the meaning of this word, and we can make the concept
clear, but this must be done by a certain amount of circumlocution and
explanation, not by a concise and perfect definition. Unless a beginner
in geometry knows what an angle is before he reads the definition in a
textbook, he will not know from the definition. This fact of the
impossibility of defining some of the fundamental concepts will be
evident when we come to consider certain attempts that have been made in
this direction.
It should also be understood in this connection that a definition makes
no assertion as to the existence of the thing defined. If we say that a
tangent to a circle is an unlimited straight line that touches the
circle in one point, and only one, we do not assert that it is possible
to have such a line; that is a matter for proof. Not in all cases,
however, can this proof be given, as in the existence of the simplest
concepts. We cannot, for example, prove that a point or a straight line
exists after we have defined these concepts. We therefore tacitly or
explicitly assume (postulate) the existence of these fundamentals of
geometry. On the other hand, we can prove that a tangent exists, and
this may properly be considered a legitimate proposition or corollary of
elementary geometry. In relation to geometric proof it is necessary to
bear in mind, therefore, that we are permitted to define any term we
please; for example, "a seven-edged polyhedron" or Leibnitz's "ten-faced
regular polyhedron," neither of which exists; but, strictly speaking, we
have no right to make use of a definition in a proof until we have shown
or postulated that the thing defined has an existence. This is one of
the strong features of Euclid's textbook. Not being able to prove that a
point, a straight line, and a circle exists, he practically postulates
these facts; but he uses no other definition in a proof without showing
that the thing defined exists, and this is his reason for mingling his
problems with his theorems. At the present time we confessedly sacrifice
his logic in this respect for the reason that we teach geometry to
pupils who are too young to appreciate that logic.
It was pointed out by Aristotle, long before Euclid, that it is not a
satisfactory procedure to define a thing by means of terms that are
strictly not prior to it, as when we attempt to define something by
means of its opposite. Thus to define a curve as "a line, no part of
which is straight," would be a bad definition unless "straight" had
already been explicitly defined; and to define "bad" as "not good" is
unsatisfactory for the reason that "bad" and "good" are concepts that
are evolved simultaneously. But all this is only a detail under the
general principle that a definition must employ terms that are better
understood than the one defined.
It should be understood that some definitions are much more important
than others, considered from the point of view of the logic of geometry.
Those that enter into geometric proofs are basal; those that form part
of the conversational language of geometry are not. Euclid gave
twenty-three definitions in Book I, and did not make use of even all of
these terms. Other terms, those not employed in his proofs, he assumed
to be known, just as he assumed a knowledge of any other words in his
language. Such procedure would not be satisfactory under modern
conditions, but it is of great importance that the teacher should
recognize that certain definitions are basal, while others are merely
informational.
It is now proposed to consider the basal definitions of geometry, first,
that the teacher may know what ones are to be emphasized and learned;
and second, that he may know that the idea that the standard definitions
can easily be improved is incorrect. It is hoped that the result will be
the bringing into prominence of the basal concepts, and the discouraging
of attempts to change in unimportant respects the definitions in the
textbook used by the pupil.
In order to have a systematic basis for work, the definitions of two
books of Euclid will first be considered.[53]
1. POINT. _A point is that which has no part._ This was incorrectly
translated by Capella in the fifth century, "Punctum est cuius pars
nihil est" (a point is that of which a part is nothing), which is as
much as to say that the point itself is nothing. It generally appears,
however, as in the Campanus edition,[54] "Punctus est cuius pars non
est," which is substantially Euclid's wording. Aristotle tells of the
definitions of point, line, and surface that prevailed in his time,
saying that they all defined the prior by means of the posterior.[55]
Thus a point was defined as "an extremity of a line," a line as "the
extremity of a surface," and a surface as "the extremity of a
solid,"--definitions still in use and not without their value. For it
must not be assumed that scientific priority is necessarily priority in
fact; a child knows of "solid" before he knows of "point," so that it
may be a very good way to explain, if not to define, by beginning with
solid, passing thence to surface, thence to line, and thence to point.
The first definition of point of which Proclus could learn is attributed
by him to the Pythagoreans, namely, "a monad having position," the early
form of our present popular definition of a point as "position without
magnitude." Plato defined it as "the beginning of a line," thus
presupposing the definition of "line"; and, strangely enough, he
anticipated by two thousand years Cavalieri, the Italian geometer, by
speaking of points as "indivisible lines." To Aristotle, who protested
against Plato's definitions, is due the definition of a point as
"something indivisible but having position."
Euclid's definition is essentially that of Aristotle, and is followed by
most modern textbook writers, except as to its omission of the reference
to position. It has been criticized as being negative, "which has _no_
part"; but it is generally admitted that a negative definition is
admissible in the case of the most elementary concepts. For example,
"blind" must be defined in terms of a negation.
At present not much attention is given to the definition of "point,"
since the term is not used as the basis of a proof, but every effort is
made to have the concept clear. It is the custom to start from a small
solid, conceive it to decrease in size, and think of the point as the
limit to which it is approaching, using these terms in their usual sense
without further explanation.
2. LINE. _A line is breadthless length._ This is usually modified in
modern textbooks by saying that "a line is that which has length without
breadth or thickness," a statement that is better understood by
beginners. Euclid's definition is thought to be due to Plato, and is
only one of many definitions that have been suggested. The Pythagoreans
having spoken of the point as a monad naturally were led to speak of the
line as dyadic, or related to two. Proclus speaks of another definition,
"magnitude in one dimension," and he gives an excellent illustration of
line as "the edge of a shadow," thus making it real but not material.
Aristotle speaks of a line as a magnitude "divisible in one way only,"
as contrasted with a surface which is divisible in two ways, and with a
solid which is divisible in three ways. Proclus also gives another
definition as the "flux of a point," which is sometimes rendered as the
path of a moving point. Aristotle had suggested the idea when he wrote,
"They say that a line by its motion produces a surface, and a point by
its motion a line."
Euclid did not deem it necessary to attempt a classification of lines,
contenting himself with defining only a straight line and a circle, and
these are really the only lines needed in elementary geometry. His
commentators, however, made the attempt. For example. Heron (first
century A.D.) probably followed his definition of line by this
classification:
{ Straight
Lines { { Circular circumferences
{ Not straight { Spiral shaped
{ Curved (generally)
Proclus relates that both Plato and Aristotle divided lines into
"straight," "circular," and "a mixture of the two," a statement which
is not quite exact, but which shows the origin of a classification not
infrequently found in recent textbooks. Geminus (_ca._ 50 B.C.) is said
by Proclus to have given two classifications, of which one will suffice
for our purposes:
{ Composite (broken line forming an angle)
{
Lines { { Forming a figure, or determinate. (Circle,
{ { ellipse, cissoid.)
{ Incomposite { Not forming a figure, or indeterminate and
{ extending without a limit. (Straight
{ line, parabola, hyperbola, conchoid.)
Of course his view of the cissoid, the curve represented by the equation
_y_^2(_a_ + _x_) = (_a_ - _x_)^3, is not the modern view.
3. _The extremities of a line are points._ This is not a definition in
the sense of its two predecessors. A modern writer would put it as a
note under the definition of line. Euclid did not wish to define a point
as the extremity of a line, for Aristotle had asserted that this was not
scientific; so he defined point and line, and then added this statement
to show the relation of one to the other. Aristotle had improved upon
this by stating that the "division" of a line, as well as an extremity,
is a point, as is also the intersection of two lines. These statements,
if they had been made by Euclid, would have avoided the objection made
by Proclus, that some lines have no extremities, as, for example, a
circle, and also a straight line extending infinitely in both
directions.
4. STRAIGHT LINE. _A straight line is that which lies evenly with
respect to the points on itself._ This is the least satisfactory of all
of the definitions of Euclid, and emphasizes the fact that the straight
line is the most difficult to define of the elementary concepts of
geometry. What is meant by "lies evenly"? Who would know what a
straight line is, from this definition, if he did not know in advance?
The ancients suggested many definitions of straight line, and it is well
to consider a few in order to appreciate the difficulties involved.
Plato spoke of it as "that of which the middle covers the ends," meaning
that if looked at endways, the middle would make it impossible to see
the remote end. This is often modified to read that "a straight line
when looked at endways appears as a point,"--an idea that involves the
postulate that our line of sight is straight. Archimedes made the
statement that "of all the lines which have the same extremities, the
straight line is the least," and this has been modified by later writers
into the statement that "a straight line is the shortest distance
between two points." This is open to two objections as a definition: (1)
a line is not distance, but distance is the _length_ of a line,--it is
measured on a line; (2) it is merely stating a property of a straight
line to say that "a straight line is the shortest path between two
points,"--a proper postulate but not a good definition. Equally
objectionable is one of the definitions suggested by both Heron and
Proclus, that "a straight line is a line that is stretched to its
uttermost"; for even then it is reasonable to think of it as a catenary,
although Proclus doubtless had in mind the Archimedes statement. He also
stated that "a straight line is a line such that if any part of it is in
a plane, the whole of it is in the plane,"--a definition that runs in a
circle, since plane is defined by means of straight line. Proclus also
defines it as "a uniform line, capable of sliding along itself," but
this is also true of a circle.
Of the various definitions two of the best go back to Heron, about the
beginning of our era. Proclus gives one of them in this form, "That line
which, when its ends remain fixed, itself remains fixed." Heron proposed
to add, "when it is, as it were, turned round in the same plane." This
has been modified into "that which does not change its position when it
is turned about its extremities as poles," and appears in substantially
this form in the works of Leibnitz and Gauss. The definition of a
straight line as "such a line as, with another straight line, does not
inclose space," is only a modification of this one. The other definition
of Heron states that in a straight line "all its parts fit on all in all
ways," and this in its modern form is perhaps the most satisfactory of
all. In this modern form it may be stated, "A line such that any part,
placed with its ends on any other part, must lie wholly in the line, is
called a straight line," in which the force of the word "must" should be
noted. This whole historical discussion goes to show how futile it is to
attempt to define a straight line. What is needed is that we should
explain what is meant by a straight line, that we should illustrate it,
and that pupils should then read the definition understandingly.
5. SURFACE. _A surface is that which has length and breadth._ This is
substantially the common definition of our modern textbooks. As with
line, so with surface, the definition is not entirely satisfactory, and
the chief consideration is that the meaning of the term should be made
clear by explanations and illustrations. The shadow cast on a table top
is a good illustration, since all idea of thickness is wanting. It adds
to the understanding of the concept to introduce Aristotle's statement
that a surface is generated by a moving line, modified by saying that
it _may_ be so generated, since the line might slide along its own
trace, or, as is commonly said in mathematics, along itself.
6. _The extremities of a surface are lines._ This is open to the same
explanation and objection as definition 3, and is not usually given in
modern textbooks. Proclus calls attention to the fact that the statement
is hardly true for a complete spherical surface.
7. PLANE. _A plane surface is a surface which lies evenly with the
straight lines on itself._ Euclid here follows his definition of
straight line, with a result that is equally unsatisfactory. For
teaching purposes the translation from the Greek is not clear to a
beginner, since "lies evenly" is a term not simpler than the one
defined. As with the definition of a straight line, so with that of a
plane, numerous efforts at improvement have been made. Proclus,
following a hint of Heron's, defines it as "the surface which is
stretched to the utmost," and also, this time influenced by Archimedes's
assumption concerning a straight line, as "the least surface among all
those which have the same extremities." Heron gave one of the best
definitions, "A surface all the parts of which have the property of
fitting on [each other]." The definition that has met with the widest
acceptance, however, is a modification of one due to Proclus, "A surface
such that a straight line fits on all parts of it." Proclus elsewhere
says, "[A plane surface is] such that the straight line fits on it all
ways," and Heron gives it in this form, "[A plane surface is] such that,
if a straight line pass through two points on it, the line coincides
with it at every spot, all ways." In modern form this appears as
follows: "A surface such that a straight line joining any two of its
points lies wholly in the surface is called a plane," and for teaching
purposes we have no better definition. It is often known as Simson's
definition, having been given by Robert Simson in 1756.
The French mathematician, Fourier, proposed to define a plane as formed
by the aggregate of all the straight lines which, passing through one
point on a straight line in space, are perpendicular to that line. This
is clear, but it is not so usable for beginners as Simson's definition.
It appears as a theorem in many recent geometries. The German
mathematician, Crelle, defined a plane as a surface containing all the
straight lines (throughout their whole length) passing through a fixed
point and also intersecting a straight line in space, but of course this
intersected straight line must not pass through the fixed point.
Crelle's definition is occasionally seen in modern textbooks, but it is
not so clear to the pupil as Simson's. Of the various ultrascientific
definitions of a plane that have been suggested of late it is hardly of
use to speak in a book concerned primarily with practical teaching. No
one of them is adapted to the needs and the comprehension of the
beginner, and it seems that we are not likely to improve upon the
so-called Simson form.
8. PLANE ANGLE. _A plane angle is the inclination to each other of two
lines in a plane which meet each other and do not lie in a straight
line._ This definition, it will be noticed, includes curvilinear angles,
and the expression "and do not lie in a straight line" states that the
lines must not be continuous one with the other, that is, that zero and
straight angles are excluded. Since Euclid does not use the curvilinear
angle, and it is only the rectilinear angle with which we are concerned,
we will pass to the next definition and consider this one in connection
therewith.
9. RECTILINEAR ANGLE. _When the lines containing the angle are straight,
the angle is called rectilinear._ This definition, taken with the
preceding one, has always been a subject of criticism. In the first
place it expressly excludes the straight angle, and, indeed, the angles
of Euclid are always less than 180 deg., contrary to our modern concept. In
the second place it defines angle by means of the word "inclination,"
which is itself as difficult to define as angle. To remedy these defects
many substitutes have been proposed. Apollonius defined angle as "a
contracting of a surface or a solid at one point under a broken line or
surface." Another of the Greeks defined it as "a quantity, namely, a
distance between the lines or surfaces containing it." Schotten[56] says
that the definitions of angle generally fall into three groups:
_a._ An angle is the difference of direction between two lines that
meet. This is no better than Euclid's, since "difference of direction"
is as difficult to define as "inclination."
_b._ An angle is the amount of turning necessary to bring one side to
the position of the other side.
_c._ An angle is the portion of the plane included between its sides.
Of these, _b_ is given by way of explanation in most modern textbooks.
Indeed, we cannot do better than simply to define an angle as the
opening between two lines which meet, and then explain what is meant by
size, through the bringing in of the idea of rotation. This is a simple
presentation, it is easily understood, and it is sufficiently accurate
for the real purpose in mind, namely, the grasping of the concept. We
should frankly acknowledge that the concept of angle is such a simple
one that a satisfactory definition is impossible, and we should
therefore confine our attention to having the concept understood.
10. _When a straight line set up on a straight line makes the adjacent
angles equal to one another, each of the equal angles is right, and the
straight line standing on the other is called a perpendicular to that on
which it stands._ We at present separate these definitions and simplify
the language.
11. _An obtuse angle is an angle greater than a right angle._
12. _An acute angle is an angle less than a right angle._
The question sometimes asked as to whether an angle of 200 deg. is obtuse,
and whether a negative angle, say -90 deg., is acute, is answered by saying
that Euclid did not conceive of angles equal to or greater than 180 deg. and
had no notion of negative quantities. Generally to-day we define an
obtuse angle as "greater than one and less than two right angles." An
acute angle is defined as "an angle less than a right angle," and is
considered as positive under the general understanding that all
geometric magnitudes are positive unless the contrary is stated.
13. _A boundary is that which is an extremity of anything._ The
definition is not exactly satisfactory, for a circle is the boundary of
the space inclosed, but we hardly consider it as the extremity of that
space. Euclid wishes the definition before No. 14.
14. _A figure is that which is contained by any boundary or boundaries._
The definition is not satisfactory, since it excludes the unlimited
straight line, the angle, an assemblage of points, and other
combinations of lines and points which we should now consider as
figures.
15. _A circle is a plane figure contained by one line such that all the
straight lines falling upon it from one point among those lying within
the figure are equal to one another._
16. _And the point is called the center of the circle._
Some commentators add after "one line," definition 15, the words "which
is called the circumference," but these are not in the oldest
manuscripts. The Greek idea of a circle was usually that of part of a
plane which is bounded by a line called in modern times the
circumference, although Aristotle used "circle" as synonymous with "the
bounding line." With the growth of modern mathematics, however, and
particularly as a result of the development of analytic geometry, the
word "circle" has come to mean the bounding line, as it did with
Aristotle, a century before Euclid's time. This has grown out of the
equations of the various curves, _x_^2 + _y_^2 = _r_^2 representing the
circle-_line_, _a_^2_y_^2 + _b_^2_x_^2 = _a_^2_b_^2 representing the
ellipse-_line_, and so on. It is natural, therefore, that circle,
ellipse, parabola, and hyperbola should all be looked upon as lines.
Since this is the modern use of "circle" in English, it has naturally
found its way into elementary geometry, in order that students should
not have to form an entirely different idea of circle on beginning
analytic geometry. The general body of American teachers, therefore, at
present favors using "circle" to mean the bounding line and
"circumference" to mean the length of that line. This requires
redefining "area of a circle," and this is done by saying that it is the
area of the plane space inclosed. The matter is not of greatest
consequence, but teachers will probably prefer to join in the modern
American usage of the term.
17. DIAMETER. _A diameter of the circle is any straight line drawn
through the center and terminated in both directions by the
circumference of the circle, and such a straight line also bisects the
circle._ The word "diameter" is from two Greek words meaning a "through
measurer," and it was also used by Euclid for the diagonal of a square,
and more generally for the diagonal of any parallelogram. The word
"diagonal" is a later term and means the "through angle." It will be
noticed that Euclid adds to the usual definition the statement that a
diameter bisects the circle. He does this apparently to justify his
definition (18), of a semicircle (a half circle).
Thales is said to have been the first to prove that a diameter bisects
the circle, this being one of three or four propositions definitely
attributed to him, and it is sometimes given as a proposition to be
proved. As a proposition, however, it is unsatisfactory, since the proof
of what is so evident usually instills more doubt than certainty in the
minds of beginners.
18. SEMICIRCLE. _A semicircle is the figure contained by the diameter
and the circumference cut off by it. And the center of the semicircle is
the same as that of the circle._ Proclus remarked that the semicircle is
the only plane figure that has its center on its perimeter. Some writers
object to defining a circle as a line and then speaking of the area of a
circle, showing minds that have at least one characteristic of that of
Proclus. The modern definition of semicircle is "half of a circle," that
is, an arc of 180 deg., although the term is commonly used to mean both the
arc and the segment.
19. RECTILINEAR FIGURES. _Rectilinear figures are those which are
contained by straight lines, trilateral figures being those contained by
three, quadrilateral those contained by four, and multilateral those
contained by more than four, straight lines._
20. _Of trilateral figures, an equilateral triangle is that which has
its three sides equal, an isosceles triangle that which has two of its
sides alone equal, and a scalene triangle that which has its three sides
unequal._
21. _Further, of trilateral figures, a right-angled triangle is that
which has a right angle, an obtuse-angled triangle that which has an
obtuse angle, and an acute-angled triangle that which has its three
angles acute._
These three definitions may properly be considered together.
"Rectilinear" is from the Latin translation of the Greek _euthygrammos_,
and means "right-lined," or "straight-lined." Euclid's idea of such a
figure is that of the space inclosed, while the modern idea is tending
to become that of the inclosing lines. In elementary geometry, however,
the Euclidean idea is still held. "Trilateral" is from the Latin
translation of the Greek _tripleuros_ (three-sided). In elementary
geometry the word "triangle" is more commonly used, although
"quadrilateral" is more common than "quadrangle." The use of these two
different forms is eccentric and is merely a matter of fashion. Thus we
speak of a pentagon but not of a tetragon or a trigon, although both
words are correct in form. The word "multilateral" (many-sided) is a
translation of the Greek _polypleuros_. Fashion has changed this to
"polygonal" (many-angled), the word "multilateral" rarely being seen.
Of the triangles, "equilateral" means "equal-sided"; "isosceles" is from
the Greek _isoskeles_, meaning "with equal legs," and "scalene" from
_skalenos_, possibly from _skazo_ (to limp), or from _skolios_
(crooked). Euclid's limitation of isosceles to a triangle with two, and
only two, equal sides would not now be accepted. We are at present more
given to generalizing than he was, and when we have proved a proposition
relating to the isosceles triangle, we wish to say that we have thereby
proved it for the equilateral triangle. We therefore say that an
isosceles triangle has two sides equal, leaving it possible that all
three sides should be equal. The expression "equal legs" is now being
discarded on the score of inelegance. In place of "right-angled
triangle" modern writers speak of "right triangle," and so for the
obtuse and acute triangles. The terms are briefer and are as readily
understood. It may add a little interest to the subject to know that
Plutarch tells us that the ancients thought that "the power of the
triangle is expressive of the nature of Pluto, Bacchus, and Mars." He
also states that the Pythagoreans called "the equilateral triangle the
head-born Minerva and Tritogeneia (born of Triton) because it may be
equally divided by the perpendicular lines drawn from each of its
angles."
22. _Of quadrilateral figures a square is that which is both equilateral
and right-angled; an oblong that which is right-angled but not
equilateral; a rhombus that which is equilateral and not right-angled;
and a rhomboid that which has its opposite sides and angles equal to one
another, but is neither equilateral nor right-angled. And let all
quadrilaterals other than these be called trapezia._ In this definition
Euclid also specializes in a manner not now generally approved. Thus we
are more apt to-day to omit the oblong and rhomboid as unnecessary, and
to define "rhombus" in such a manner as to include a square. We use
"parallelogram" to cover "rhomboid," "rhombus," "oblong," and "square."
For "oblong" we use "rectangle," letting it include square. Euclid's
definition of "square" illustrates his freedom in stating more
attributes than are necessary, in order to make sure that the concept is
clear; for he might have said that it "is that which is equilateral and
has one right angle." We may profit by his method, sacrificing logic to
educational necessity. Euclid does not use "oblong," "rhombus,"
"rhomboid," and "trapezium" (_plural_, "trapezia") in his proofs, so
that he might well have omitted the definitions, as we often do.
23. PARALLELS. _Parallel straight lines are straight lines which, being
in the same plane and being produced indefinitely in both directions, do
not meet one another in either direction._ This definition of parallels,
simplified in its language, is the one commonly used to-day. Other
definitions have been suggested, but none has been so generally used.
Proclus states that Posidonius gave the definition based upon the lines
always being at the same distance apart. Geminus has the same idea in
his definition. There are, as Schotten has pointed out, three general
types of definitions of parallels, namely:
_a._ They have no point in common. This may be expressed by saying that
(1) they do not intersect, (2) they meet at infinity.
_b._ They are equidistant from one another.
_c._ They have the same direction.
Of these, the first is Euclid's, the idea of the point at infinity being
suggested by Kepler (1604). The second part of this definition is, of
course, unusable for beginners. Dr. (now Sir Thomas) Heath says, "It
seems best, therefore, to leave to higher geometry the conception of
infinitely distant points on a line and of two straight lines meeting at
infinity, like imaginary points of intersection, and, for the purposes
of elementary geometry, to rely on the plain distinction between
'parallel' and 'cutting,' which average human intelligence can readily
grasp."
The direction definition seems to have originated with Leibnitz. It is
open to the serious objection that "direction" is not easy of
definition, and that it is used very loosely. If two people on different
meridians travel due north, do they travel in the same direction? on
parallel lines? The definition is as objectionable as that of angle as
the "difference of direction" of two intersecting lines.
From these definitions of the first book of Euclid we see (1) what a
small number Euclid considered as basal; (2) what a change has taken
place in the generalization of concepts; (3) how the language has
varied. Nevertheless we are not to be commended if we adhere to Euclid's
small number, because geometry is now taught to pupils whose vocabulary
is limited. It is necessary to define more terms, and to scatter the
definitions through the work for use as they are needed, instead of
massing them at the beginning, as in a dictionary. The most important
lesson to be learned from Euclid's definitions is that only the basal
ones, relatively few in number, need to be learned, and these because
they are used as the foundations upon which proofs are built. It should
also be noticed that Euclid explains nothing in these definitions; they
are hard statements of fact, massed at the beginning of his treatise.
Not always as statements, and not at all in their arrangement, are they
suited to the needs of our boys and girls at present.
Having considered Euclid's definitions of Book I, it is proper to turn
to some of those terms that have been added from time to time to his
list, and are now usually incorporated in American textbooks. It will be
seen that most of these were assumed by Euclid to be known by his
mature readers. They need to be defined for young people, but most of
them are not basal, that is, they are not used in the proofs of
propositions. Some of these terms, such as magnitudes, curve line,
broken line, curvilinear figure, bisector, adjacent angles, reflex
angles, oblique angles and lines, and vertical angles, need merely a
word of explanation so that they may be used intelligently. If they were
numerous enough to make it worth the while, they could be classified in
our textbooks as of minor importance, but such a course would cause more
trouble than it is worth.
Other terms have come into use in modern times that are not common
expressions with which students are familiar. Such a term is "straight
angle," a concept not used by Euclid, but one that adds so materially to
the interest and value of geometry as now to be generally recognized.
There is also the word "perigon," meaning the whole angular space about
a point. This was excluded by the Greeks because their idea of angle
required it to be less than a straight angle. The word means "around
angle," and is the best one that has been coined for the purpose. "Flat
angle" and "whole angle" are among the names suggested for these two
modern concepts. The terms "complement," "supplement," and "conjugate,"
meaning the difference between a given angle and a right angle, straight
angle, and perigon respectively, have also entered our vocabulary and
need defining.
There are also certain terms expressing relationship which Euclid does
not define, and which have been so changed in recent times as to require
careful definition at present. Chief among these are the words "equal,"
"congruent," and "equivalent." Euclid used the single word "equal" for
all three concepts, although some of his recent editors have changed it
to "identically equal" in the case of congruence. In modern speech we
use the word "equal" commonly to mean "like-valued," "having the same
measure," as when we say the circumference of a circle "equals" a
straight line whose length is 2[pi]_r_, although it could not coincide
with it. Of late, therefore, in Europe and America, and wherever
European influence reaches, the word "congruent" is coming into use to
mean "identically equal" in the sense of superposable. We therefore
speak of congruent triangles and congruent parallelograms as being those
that are superposable.
It is a little unfortunate that "equal" has come to be so loosely used
in ordinary conversation that we cannot keep it to mean "congruent"; but
our language will not permit it, and we are forced to use the newer
word. Whenever it can be used without misunderstanding, however, it
should be retained, as in the case of "equal straight lines," "equal
angles," and "equal arcs of the same circle." The mathematical and
educational world will never consent to use "congruent straight lines,"
or "congruent angles," for the reason that the terms are unnecessarily
long, no misunderstanding being possible when "equal" is used.
The word "equivalent" was introduced by Legendre at the close of the
eighteenth century to indicate equality of length, or of area, or of
volume. Euclid had said, "Parallelograms which are on the same base and
in the same parallels are equal to one another," while Legendre and his
followers would modify the wording somewhat and introduce "equivalent"
for "equal." This usage has been retained. Congruent polygons are
therefore necessarily equivalent, but equivalent polygons are not in
general congruent. Congruent polygons have mutually equal sides and
mutually equal angles, while equivalent polygons have no equality save
that of area.
In general, as already stated, these and other terms should be defined
just before they are used instead of at the beginning of geometry. The
reason for this, from the educational standpoint and considering the
present position of geometry in the curriculum, is apparent.
We shall now consider the definitions of Euclid's Book III, which is
usually taken as Book II in America.
1. EQUAL CIRCLES. _Equal circles are those the diameters of which are
equal, or the radii of which are equal._
Manifestly this is a theorem, for it asserts that if the radii of two
circles are equal, the circles may be made to coincide. In some
textbooks a proof is given by superposition, and the proof is
legitimate, but Euclid usually avoided superposition if possible.
Nevertheless he might as well have proved this as that two triangles are
congruent if two sides and the included angle of the one are
respectively equal to the corresponding parts of the other, and he might
as well have postulated the latter as to have substantially postulated
this fact. For in reality this definition is a postulate, and it was so
considered by the great Italian mathematician Tartaglia (_ca._
1500-_ca._ 1557). The plan usually followed in America to-day is to
consider this as one of many unproved propositions, too evident, indeed,
for proof, accepted by intuition. The result is a loss in the logic of
Euclid, but the method is thought to be better adapted to the mind of
the youthful learner. It is interesting to note in this connection that
the Greeks had no word for "radius," and were therefore compelled to use
some such phrase as "the straight line from the center," or, briefly,
"the from the center," as if "from the center" were one word.
2. TANGENT. _A straight line is said to touch a circle which, meeting
the circle and being produced, does not cut the circle._
Teachers who prefer to use "circumference" instead of "circle" for the
line should notice how often such phrases as "cut the circle" and
"intersecting circle" are used,--phrases that signify nothing unless
"circle" is taken to mean the line. So Aristotle uses an expression
meaning that the locus of a certain point is a circle, and he speaks of
a circle as passing through "all the angles." Our word "touch" is from
the Latin _tangere_, from which comes "tangent," and also "tag," an old
touching game.
3. TANGENT CIRCLES. _Circles are said to touch one another which,
meeting one another, do not cut one another._
The definition has not been looked upon as entirely satisfactory, even
aside from its unfortunate phraseology. It is not certain, for instance,
whether Euclid meant that the circles could not cut at some other point
than that of tangency. Furthermore, no distinction is made between
external and internal contact, although both forms are used in the
propositions. Modern textbook makers find it convenient to define
tangent circles as those that are tangent to the same straight line at
the same point, and to define external and internal tangency by
reference to their position with respect to the line, although this may
be characterized as open to about the same objection as Euclid's.
4. DISTANCE. _In a circle straight lines are said to be equally distant
from the center, when the perpendiculars drawn to them from the center
are equal._
It is now customary to define "distance" from a point to a line as the
length of the perpendicular from the point to the line, and to do this
in Book I. In higher mathematics it is found that distance is not a
satisfactory term to use, but the objections to it have no particular
significance in elementary geometry.
5. GREATER DISTANCE. _And that straight line is said to be at a greater
distance on which the greater perpendicular falls._
Such a definition is not thought essential at the present time.
6. SEGMENT. _A segment of a circle is the figure contained by a straight
line and the circumference of a circle._
The word "segment" is from the Latin root _sect_, meaning "cut." So we
have "sector" (a cutter), "section" (a cut), "intersect," and so on. The
word is not limited to a circle; we have long spoken of a spherical
segment, and it is common to-day to speak of a line segment, to which
some would apply a new name "sect." There is little confusion in the
matter, however, for the context shows what kind of a segment is to be
understood, so that the word "sect" is rather pedantic than important.
It will be noticed that Euclid here uses "circumference" to mean "arc."
7. ANGLE OF A SEGMENT. _An angle of a segment is that contained by a
straight line and a circumference of a circle._
This term has entirely dropped out of geometry, and few teachers would
know what it meant if they should hear it used. Proclus called such
angles "mixed."
8. ANGLE IN A SEGMENT. _An angle in a segment is the angle which, when a
point is taken on the circumference of the segment and straight lines
are joined from it to the extremities of the straight line which is the
base of the segment, is contained by the straight lines so joined._
Such an involved definition would not be usable to-day. Moreover, the
words "circumference of the segment" would not be used.
9. _And when the straight lines containing the angle cut off a
circumference, the angle is said to stand upon that circumference._
10. SECTOR. _A sector of a circle is the figure which, when an angle is
constructed at the center of the circle, is contained by the straight
lines containing the angle and the circumference cut off by them._
There is no reason for such an extended definition, our modern
phraseology being both more exact (as seen in the above use of
"circumference" for "arc") and more intelligible. The Greek word for
"sector" is "knife" (_tomeus_), "sector" being the Latin translation. A
sector is supposed to resemble a shoemaker's knife, and hence the
significance of the term. Euclid followed this by a definition of
similar sectors, a term now generally abandoned as unnecessary.
It will be noticed that Euclid did not use or define the word "polygon."
He uses "rectilinear figure" instead. Polygon may be defined to be a
bounding line, as a circle is now defined, or as the space inclosed by a
broken line, or as a figure formed by a broken line, thus including both
the limited plane and its boundary. It is not of any great consequence
geometrically which of these ideas is adopted, so that the usual
definition of a portion of a plane bounded by a broken line may be taken
as sufficient for elementary purposes. It is proper to call attention,
however, to the fact that we may have cross polygons of various types,
and that the line that "bounds" the polygon must be continuous, as the
definition states. That is, in the second of these figures the shaded
portion is not considered a polygon. Such special cases are not liable
to arise, but if questions relating to them are suggested, the teacher
should be prepared to answer them. If suggested to a class, a note of
this kind should come out only incidentally as a bit of interest, and
should not occupy much time nor be unduly emphasized.
[Illustration]
It may also be mentioned to a class at some convenient time that the old
idea of a polygon was that of a convex figure, and that the modern idea,
which is met in higher mathematics, leads to a modification of earlier
concepts. For example, here is a quadrilateral with one of its
diagonals, _BD_, _outside_ the figure. Furthermore, if we consider a
quadrilateral as a figure formed by four intersecting lines, _AC_, _CF_,
_BE_, and _EA_, it is apparent that this _general quadrilateral_ has six
vertices, _A_, _B_, _C_, _D_, _E_, _F_, and three diagonals, _AD_, _BF_,
and _CE_. Such broader ideas of geometry form the basis of what is
called modern elementary geometry.
[Illustration]
The other definitions of plane geometry need not be discussed, since all
that have any historical interest have been considered. On the whole it
may be said that our definitions to-day are not in general so carefully
considered as those of Euclid, who weighed each word with greatest
skill, but they are more teachable to beginners, and are, on the whole,
more satisfactory from the educational standpoint. The greatest lesson
to be learned from this discussion is that the number of basal
definitions to be learned for subsequent use is very small.
Since teachers are occasionally disturbed over the form in which
definitions are stated, it is well to say a few words upon this subject.
There are several standard types that may be used. (1) We may use the
dictionary form, putting the word defined first, thus: "_Right
triangle_. A triangle that has one of its angles a right angle." This is
scientifically correct, but it is not a complete sentence, and hence it
is not easily repeated when it has to be quoted as an authority. (2) We
may put the word defined at the end, thus: "A triangle that has one of
its angles a right angle is called a right triangle." This is more
satisfactory. (3) We may combine (1) and (2), thus: "_Right triangle_. A
triangle that has one of its angles a right angle is called a right
triangle." This is still better, for it has the catchword at the
beginning of the paragraph.
There is occasionally some mental agitation over the trivial things of a
definition, such as the use of the words "is called." It would not be a
very serious matter if they were omitted, but it is better to have them
there. The reason is that they mark the statement at once as a
definition. For example, suppose we say that "a triangle that has one of
its angles a right angle is a right triangle." We have also the fact
that "a triangle whose base is the diameter of a semicircle and whose
vertex lies on the semicircle is a right triangle." The style of
statement is the same, and we have nothing in the phraseology to show
that the first is a definition and the second a theorem. This may
happen with most of the definitions, and hence the most careful writers
have not consented to omit the distinctive words in question.
Apropos of the definitions of geometry, the great French philosopher and
mathematician, Pascal, set forth certain rules relating to this subject,
as also to the axioms employed, and these may properly sum up this
chapter.
1. Do not attempt to define terms so well known in themselves that there
are no simpler terms by which to express them.
2. Admit no obscure or equivocal terms without defining them.
3. Use in the definitions only terms that are perfectly understood or
are there explained.
4. Omit no necessary principles without general agreement, however clear
and evident they may be.
5. Set forth in the axioms only those things that are in themselves
perfectly evident.
6. Do not attempt to demonstrate anything that is so evident in itself
that there is nothing more simple by which to prove it.
7. Prove whatever is in the least obscure, using in the demonstration
only axioms that are perfectly evident in themselves, or propositions
already demonstrated or allowed.
8. In case of any uncertainty arising from a term employed, always
substitute mentally the definition for the term itself.
=Bibliography.= Heath, Euclid, as cited; Frankland, The First
Book of Euclid, as cited; Smith, Teaching of Elementary
Mathematics, p. 257, New York, 1900; Young, Teaching of
Mathematics, p. 189, New York, 1907; Veblen, On Definitions, in
the _Monist_, 1903, p. 303.
FOOTNOTES:
[53] Free use has been made of W. B. Frankland, "The First Book of
Euclid's 'Elements,'" Cambridge, 1905; T. L. Heath, "The Thirteen Books
of Euclid's 'Elements,'" Cambridge, 1908; H. Schotten, "Inhalt und
Methode des planimetrischen Unterrichts," Leipzig, 1893; M. Simon,
"Euclid und die sechs planimetrischen Buecher," Leipzig, 1901.
[54] For a facsimile of a thirteenth-century MS. containing this
definition, see the author's "Rara Arithmetica," Plate IV, Boston, 1909.
[55] Our slang expression "The cart before the horse" is suggestive of
this procedure.
[56] Loc. cit., Vol. II, p. 94.