THE AXIOMS AND POSTULATES
The interest as well as the value of geometry lies chiefly in the fact
that from a small number of assumptions it is possible to deduce an
unlimited number of conclusions. With the truth of these assumptions we
are not so much concerned as with the reasoning by which we draw the
conclusions, although it is manifestly desirable that the assumptions
should not be false, and that they should be as few as possible.
It would be natural, and in some respects desirable, to call these
foundations of geometry by the name "assumptions," since they are simply
statements that are assumed to be true. The real foundation principles
cannot be proved; they are the means by which we prove other statements.
But as with most names of men or things, they have received certain
titles that are time-honored, and that it is not worth the while to
attempt to change. In English we call them axioms and postulates, and
there is no more reason for attempting to change these terms than there
is for attempting to change the names of geometry[42] and of
algebra.[43]
Since these terms are likely to continue, it is necessary to
distinguish between them more carefully than is often done, and to
consider what assumptions we are justified in including under each. In
the first place, these names do not go back to Euclid, as is ordinarily
supposed, although the ideas and the statements are his. "Postulate" is
a Latin form of the Greek [Greek: aitema] (_aitema_), and appears only
in late translations. Euclid stated in substance, "Let the following be
assumed." "Axiom" ([Greek: axioma], _axioma_) dates perhaps only from
Proclus (fifth century A.D.), Euclid using the words "common notions"
([Greek: koinai ennoiai], _koinai ennoiai_) for "axioms," as Aristotle
before him had used "common things," "common principles," and "common
opinions."
The distinction between axiom and postulate was not clearly made by
ancient writers. Probably what was in Euclid's mind was the Aristotelian
distinction that an axiom was a principle common to all sciences,
self-evident but incapable of proof, while the postulates were the
assumptions necessary for building up the particular science under
consideration, in this case geometry.[44]
We thus come to the modern distinction between axiom and postulate, and
say that a general statement admitted to be true without proof is an
axiom, while a postulate in geometry is a geometric statement admitted
to be true, without proof. For example, when we say "If equals are added
to equals, the sums are equal," we state an assumption that is taken
also as true in arithmetic, in algebra, and in elementary mathematics in
general. This is therefore an axiom. At one time such a statement was
defined as "a self-evident truth," but this has in recent years been
abandoned, since what is evident to one person is not necessarily
evident to another, and since all such statements are mere matters of
assumption in any case. On the other hand, when we say, "A circle may be
described with any given point as a center and any given line as a
radius," we state a special assumption of geometry, and this assumption
is therefore a geometric postulate. Some few writers have sought to
distinguish between axiom and postulate by saying that the former was an
assumed theorem and the latter an assumed problem, but there is no
standard authority for such a distinction, and indeed the difference
between a theorem and a problem is very slight. If we say, "A circle may
be passed through three points not in the same straight line," we state
a theorem; but if we say, "Required to pass a circle through three
points," we state a problem. The mental process of handling the two
propositions is, however, practically the same in spite of the minor
detail of wording. So with the statement, "A straight line may be
produced to any required length." This is stated in the form of a
theorem, but it might equally well be stated thus: "To produce a
straight line to any required length." It is unreasonable to call this
an axiom in one case and a postulate in the other. However stated, it is
a geometric postulate and should be so classed.
What, now, are the axioms and postulates that we are justified in
assuming, and what determines their number and character? It seems
reasonable to agree that they should be as few as possible, and that for
educational purposes they should be so clear as to be intelligible to
beginners. But here we encounter two conflicting ideas. To get the
"irreducible minimum" of assumptions is to get a set of statements quite
unintelligible to students beginning geometry or any other branch of
elementary mathematics. Such an effort is laudable when the results are
intended for advanced students in the university, but it is merely
suggestive to teachers rather than usable with pupils when it touches
upon the primary steps of any science. In recent years several such
attempts have been made. In particular, Professor Hilbert has given a
system[45] of congruence postulates, but they are rather for the
scientist than for the student of elementary geometry.
In view of these efforts it is well to go back to Euclid and see what
this great teacher of university men[46] had to suggest. The following
are the five "common notions" that Euclid deemed sufficient for the
purposes of elementary geometry.
1. _Things equal to the same thing are also equal to each other._ This
axiom has persisted in all elementary textbooks. Of course it is a
simple matter to attempt criticism,--to say that -2 is the square root
of 4, and +2 is also the square root of 4, whence -2 = +2; but it is
evident that the argument is not sound, and that it does not invalidate
the axiom. Proclus tells us that Apollonius attempted to prove the axiom
by saying, "Let _a_ equal _b_, and _b_ equal _c_. I say that _a_ equals
_c_. For, since _a_ equals _b_, _a_ occupies the same space as _b_.
Therefore _a_ occupies the same space as _c_. Therefore _a_ equals
_c_." The proof is of no value, however, save as a curiosity.
2. _And if to equals equals are added, the wholes are equal._
3. _If equals are subtracted from equals, the remainders are equal._
Axioms 2 and 3 are older than Euclid's time, and are the only ones given
by him relating to the solution of the equation. Certain other axioms
were added by later writers, as, "Things which are double of the same
thing are equal to one another," and "Things which are halves of the
same thing are equal to one another." These two illustrate the ancient
use of _duplatio_ (doubling) and _mediatio_ (halving), the primitive
forms of multiplication and division. Euclid would not admit the
multiplication axiom, since to him this meant merely repeated addition.
The partition (halving) axiom he did not need, and if needed, he would
have inferred its truth. There are also the axioms, "If equals are added
to unequals, the wholes are unequal," and "If equals are subtracted from
unequals, the remainders are unequal," neither of which Euclid would
have used because he did not define "unequals." The modern arrangement
of axioms, covering addition, subtraction, multiplication, division,
powers, and roots, sometimes of unequals as well as equals, comes from
the development of algebra. They are not all needed for geometry, but in
so far as they show the relation of arithmetic, algebra, and geometry,
they serve a useful purpose. There are also other axioms concerning
unequals that are of advantage to beginners, even though unnecessary
from the standpoint of strict logic.
4. _Things that coincide with one another are equal to one another._
This is no longer included in the list of axioms. It is rather a
definition of "equal," or of "congruent," to take the modern term. If
not a definition, it is certainly a postulate rather than an axiom,
being purely geometric in character. It is probable that Euclid included
it to show that superposition is to be considered a legitimate form of
proof, but why it was not placed among the postulates is not easily
seen. At any rate it is unfortunately worded, and modern writers
generally insert the postulate of motion instead,--that a figure may be
moved about in space without altering its size or shape. The German
philosopher, Schopenhauer (1844), criticized Euclid's axiom as follows:
"Coincidence is either mere tautology or something entirely empirical,
which belongs not to pure intuition but to external sensuous experience.
It presupposes, in fact, the mobility of figures."
5. _The whole is greater than the part._ To this Clavius (1574) added,
"The whole is equal to the sum of its parts," which may be taken to be a
definition of "whole," but which is helpful to beginners, even if not
logically necessary. Some writers doubt the genuineness of this axiom.
Having considered the axioms of Euclid, we shall now consider the axioms
that are needed in the study of elementary geometry. The following are
suggested, not from the standpoint of pure logic, but from that of the
needs of the teacher and pupil.
1. _If equals are added to equals, the sums are equal._ Instead of this
axiom, the one numbered 8 below is often given first. For convenience in
memorizing, however, it is better to give the axioms in the following
order: (1) addition, (2) subtraction, (3) multiplication, (4) division,
(5) powers and roots,--all of equal quantities.
2. _If equals are subtracted from equals, the remainders are equal._
3. _If equals are multiplied by equals, the products are equal._
4. _If equals are divided by equals, the quotients are equal._
5. _Like powers or like positive roots of equals are equal._ Formerly
students of geometry knew nothing of algebra, and in particular nothing
of negative quantities. Now, however, in American schools a pupil
usually studies algebra a year before he studies demonstrative geometry.
It is therefore better, in speaking of roots, to limit them to positive
numbers, since the two square roots of 4 (+2 and -2), for example, are
not equal. If the pupil had studied complex numbers before he began
geometry, it would have been advisable to limit the roots still further
to real roots, since the four fourth roots of 1 (+1, -1, +[sqrt](-1),
-[sqrt](-1)), for example, are not equal save in absolute value. It is
well, however, to eliminate these fine distinctions as far as possible,
since their presence only clouds the vision of the beginner.
It should also be noted that these five axioms might be combined in one,
namely, _If equals are operated on by equals in the same way, the
results are equal_. In Axiom 1 this operation is addition, in Axiom 2 it
is subtraction, and so on. Indeed, in order to reduce the number of
axioms two are already combined in Axiom 5. But there is a good reason
for not combining the first four with the fifth, and there is also a
good reason for combining two in Axiom 5. The reason is that these are
the axioms continually used in equations, and to combine them all in one
would be to encourage laxness of thought on the part of the pupil. He
would always say "by Axiom 1" whatever he did to an equation, and the
teacher would not be certain whether the pupil was thinking definitely
of dividing equals by equals, or had a hazy idea that he was
manipulating an equation in some other way that led to an answer. On the
other hand, Axiom 5 is not used as often as the preceding four, and the
interchange of integral and fractional exponents is relatively common,
so that the joining of these two axioms in one for the purpose of
reducing the total number is justifiable.
6. _If unequals are operated on by positive equals in the same way, the
results are unequal in the same order._ This includes in a single
statement the six operations mentioned in the preceding axioms; that is,
if _a_ > _b_ and if _x_ = _y_, then _a_ + _x_ > _b_ + _y_,
_a_ - _x_ > _b_ - _y_, _ax_ > _by_, etc. The reason for thus combining
six axioms in one in the case of inequalities is apparent. They are
rarely used in geometry, and if a teacher is in doubt as to the pupil's
knowledge, he can easily inquire in the few cases that arise, whereas it
would consume a great deal of time to do this for the many equations
that are met. The axiom is stated in such a way as to exclude
multiplying or dividing by negative numbers, this case not being needed.
7. _If unequals are added to unequals in the same order, the sums are
unequal in the same order; if unequals are subtracted from equals, the
remainders are unequal in the reverse order._ These are the only cases
in which unequals are necessarily combined with unequals, or operate
upon equals in geometry, and the axiom is easily explained to the class
by the use of numbers.
8. _Quantities that are equal to the same quantity or to equal
quantities are equal to each other._ In this axiom the word "quantity"
is used, in the common manner of the present time, to include number and
all geometric magnitudes (length, area, volume).
9. _A quantity may be substituted for its equal in an equation or in an
inequality._ This axiom is tacitly assumed by all writers, and is very
useful in the proofs of geometry. It is really the basis of several
other axioms, and if we were seeking the "irreducible minimum," it would
replace them. Since, however, we are seeking only a reasonably abridged
list of convenient assumptions that beginners will understand and use,
this axiom has much to commend it. If we consider the equations
(1) _a_ = _x_ and (2) _b_ = _x_, we see that for _x_ in equation (1) we
may substitute _b_ from equation (2) and have _a_ = _b_; in other words,
that Axiom 8 is included in Axiom 9. Furthermore, if (1) _a_ = _b_ and
(2) _x_ = _y_, then since _a_ + _x_ is the same as _a_ + _x_, we may, by
substituting, say that _a_ + _x_ = _a_ + _x_ = _b_ + _x_ = _b_ + _y_. In
other words, Axiom 1 is included in Axiom 9. Thus an axiom that includes
others has a legitimate place, because a beginner would be too much
confused by seeing its entire scope, and because he will make frequent
use of it in his mathematical work.
10. _If the first of three quantities is greater than the second, and
the second is greater than the third, then the first is greater than the
third._ This axiom is needed several times in geometry. The case in
which _a_ > _b_ and _b_ = _c_, therefore _a_ > _c_, is provided for in
Axiom 9.
11. _The whole is greater than any of its parts and is equal to the sum
of all its parts._ The latter part of this axiom is really only the
definition of "whole," and it would be legitimate to state a definition
accordingly and refer to it where the word is employed. Where, however,
we wish to speak of a polygon, for example, and wish to say that the
area is equal to the combined areas of the triangles composing it, it is
more satisfactory to have this axiom to which to refer. It will be
noticed that two related axioms are here combined in one, for a reason
similar to the one stated under Axiom 5.
In the case of the postulates we are met by a problem similar to the one
confronting us in connection with the axioms,--the problem of the
"irreducible minimum" as related to the question of teaching. Manifestly
Euclid used postulates that he did not state, and proved some statements
that he might have postulated.[47]
The postulates given by Euclid under the name [Greek:
aitemata](_aitemata_) were requests made by the teacher to his pupil
that certain things be conceded. They were five in number, as follows:
1. _Let the following be conceded: to draw a straight line from any
point to any point._
Strictly speaking, Euclid might have been required to postulate that
points and straight lines exist, but he evidently considered this
statement sufficient. Aristotle had, however, already called attention
to the fact that a mere definition was sufficient only to show what a
concept is, and that this must be followed by a proof that the thing
exists. We might, for example, define _x_ as a line that bisects an
angle without meeting the vertex, but this would not show that an _x_
exists, and indeed it does not exist. Euclid evidently intended the
postulate to assert that this line joining two points is unique, which
is only another way of saying that two points determine a straight line,
and really includes the idea that two straight lines cannot inclose
space. For purposes of instruction, the postulate would be clearer if it
read, _One straight line, and only one, can be drawn through two given
points_.
2. _To produce a finite straight line continuously in a straight line._
In this postulate Euclid practically assumes that a straight line can be
produced only in a straight line; in other words, that two different
straight lines cannot have a common segment. Several attempts have been
made to prove this fact, but without any marked success.
3. _To describe a circle with any center and radius._
4. _That all right angles are equal to one another._
While this postulate asserts the essential truth that a right
angle is a _determinate magnitude_ so that it really serves as
an invariable standard by which other (acute and obtuse) angles
may be measured, much more than this is implied, as will easily
be seen from the following consideration. If the statement is
to be _proved_, it can only be proved by the method of applying
one pair of right angles to another and so arguing their
equality. But this method would not be valid unless on the
assumption of the _invariability of figures_, which would have
to be asserted as an antecedent postulate. Euclid preferred to
assert as a postulate, directly, the fact that all right angles
are equal; and hence his postulate must be taken as equivalent
to the principle of _invariability of figures_, or its
equivalent, the _homogeneity_ of space.[48]
It is better educational policy, however, to assert this fact more
definitely, and to state the additional assumption that figures may be
moved about in space without deformation. The fourth of Euclid's
postulates is often given as an axiom, following the idea of the Greek
philosopher Geminus (who flourished in the first century B.C.), but this
is because Euclid's distinction between axiom and postulate is not
always understood. Proclus (410-485 A.D.) endeavored to prove the
postulate, and a later and more scientific effort was made by the
Italian geometrician Saccheri (1667-1733). It is very commonly taken as
a postulate that all straight angles are equal, this being more evident
to the senses, and the equality of right angles is deduced as a
corollary. This method of procedure has the sanction of many of our best
modern scholars.
5. _That, if a straight line falling on two straight lines make the
interior angle on the same side less than two right angles, the two
straight lines, if produced indefinitely, meet on that side on which are
the angles less than the two right angles._
This famous postulate, long since abandoned in teaching the beginner in
geometry, is a remarkable evidence of the clear vision of Euclid. For
two thousand years mathematicians sought to prove it, only to
demonstrate the wisdom of its author in placing it among the
assumptions.[49] Every proof adduced contains some assumption that
practically conceals the postulate itself. Thus the great English
mathematician John Wallis (1616-1703) gave a proof based upon the
assumption that "given a figure, another figure is possible which is
similar to the given one, and of any size whatever." Legendre
(1752-1833) did substantially the same at one time, and offered several
other proofs, each depending upon some equally unprovable assumption.
The definite proof that the postulate cannot be demonstrated is due to
the Italian Beltrami (1868).
Of the alternative forms of the postulate, that of Proclus is generally
considered the best suited to beginners. As stated by Playfair (1795),
this is, "Through a given point only one parallel can be drawn to a
given straight line"; and as stated by Proclus, "If a straight line
intersect one of two parallels, it will intersect the other also."
Playfair's form is now the common "postulate of parallels," and is the
one that seems destined to endure.
Posidonius and Geminus, both Stoics of the first century B.C., gave as
their alternative, "There exist straight lines everywhere equidistant
from one another." One of Legendre's alternatives is, "There exists a
triangle in which the sum of the three angles is equal to two right
angles." One of the latest attempts to suggest a substitute is that of
the Italian Ingrami (1904), "Two parallel straight lines intercept, on
every transversal which passes through the mid-point of a segment
included between them, another segment the mid-point of which is the
mid-point of the first."
Of course it is entirely possible to assume that through a point more
than one line can be drawn parallel to a given straight line, in which
case another type of geometry can be built up, equally rigorous with
Euclid's. This was done at the close of the first quarter of the
nineteenth century by Lobachevsky (1793-1856) and Bolyai (1802-1860),
resulting in the first of several "non-Euclidean" geometries.[50]
Taking the problem to be that of stating a reasonably small number of
geometric assumptions that may form a basis to supplement the general
axioms, that shall cover the most important matters to which the student
must refer, and that shall be so simple as easily to be understood by a
beginner, the following are recommended:
1. _One straight line, and only one, can be drawn through two given
points._ This should also be stated for convenience in the form, _Two
points determine a straight line_. From it may also be drawn this
corollary, _Two straight lines can intersect in only one point_, since
two points would determine a straight line. Such obvious restatements of
or corollaries to a postulate are to be commended, since a beginner is
often discouraged by having to prove what is so obvious that a
demonstration fails to commend itself to his mind.
2. _A straight line may be produced to any required length._ This, like
Postulate 1, requires the use of a straightedge for drawing the physical
figure. The required length is attained by using the compasses to
measure the distance. The straightedge and the compasses are the only
two drawing instruments recognized in elementary geometry.[51] While
this involves more than Euclid's postulate, it is a better working
assumption for beginners.
3. _A straight line is the shortest path between two points._ This is
easily proved by the method of Euclid[52] for the case where the paths
are broken lines, but it is needed as a postulate for the case of curve
paths. It is a better statement than the common one that a straight line
is the shortest _distance_ between two points; for distance is measured
on a line, but it is not itself a line. Furthermore, there are
scientific objections to using the word "distance" any more than is
necessary.
4. _A circle may be described with any given point as a center and any
given line as a radius._ This involves the use of the second of the two
geometric instruments, the compasses.
5. _Any figure may be moved from one place to another without altering
the size or shape._ This is the postulate of the homogeneity of space,
and asserts that space is such that we may move a figure as we please
without deformation of any kind. It is the basis of all cases of
superposition.
6. _All straight angles are equal._ It is possible to prove this, and
therefore, from the standpoint of strict logic, it is unnecessary as a
postulate. On the other hand, it is poor educational policy for a
beginner to attempt to prove a thing that is so obvious. The attempt
leads to a loss of interest in the subject, the proposition being (to
state a paradox) hard because it is so easy. It is, of course, possible
to postulate that straight angles are equal, and to draw the conclusion
that their halves (right angles) are equal; or to proceed in the
opposite direction, and postulate that all right angles are equal, and
draw the conclusion that their doubles (straight angles) are equal. Of
the two the former has the advantage, since it is probably more obvious
that all straight angles are equal. It is well to state the following
definite corollaries to this postulate: (1) _All right angles are
equal_; (2) _From a point in a line only one perpendicular can be drawn
to the line_, since two perpendiculars would make the whole (right
angle) equal to its part; (3) _Equal angles have equal complements,
equal supplements, and equal conjugates_; (4) _The greater of two
angles has the less complement, the less supplement, and the less
conjugate._ All of these four might appear as propositions, but, as
already stated, they are so obvious as to be more harmful than useful to
beginners when given in such form.
The postulate of parallels may properly appear in connection with that
topic in Book I, and it is accordingly treated in Chapter XIV.
There is also another assumption that some writers are now trying to
formulate in a simple fashion. We take, for example, a line segment
_AB_, and describe circles with _A_ and _B_ respectively as centers, and
with a radius _AB_. We say that the circles will intersect as at _C_ and
_D_. But how do we know that they intersect? We assume it, just as we
assume that an indefinite straight line drawn from a point inclosed by a
circle will, if produced far enough, cut the circle twice. Of course a
pupil would not think of this if his attention was not called to it, and
the harm outweighs the good in doing this with one who is beginning the
study of geometry.
With axioms and with postulates, therefore, the conclusion is the same:
from the standpoint of scientific geometry there is an irreducible
minimum of assumptions, but from the standpoint of practical teaching
this list should give place to a working set of axioms and postulates
that meet the needs of the beginner.
=Bibliography.= Smith, Teaching of Elementary Mathematics, New
York, 1900; Young, The Teaching of Mathematics, New York, 1901;
Moore, On the Foundations of Mathematics, _Bulletin of the
American Mathematical Society_, 1903, p. 402; Betz, Intuition
and Logic in Geometry, _The Mathematics Teacher_, Vol. II, p.
3; Hilbert, The Foundations of Geometry, Chicago, 1902; Veblen,
A System of Axioms for Geometry, _Transactions of the American
Mathematical Society_, 1904, p. 343.
FOOTNOTES:
[42] From the Greek [Greek: ge], _ge_ (earth), + [Greek: metrein],
_metrein_ (to measure), although the science has not had to do directly
with the measure of the earth for over two thousand years.
[43] From the Arabic _al_ (the) + _jabr_ (restoration), referring to
taking a quantity from one side of an equation and then restoring the
balance by taking it from the other side (see page 37).
[44] One of the clearest discussions of the subject is in W. B.
Frankland, "The First Book of Euclid's 'Elements,'" p. 26, Cambridge,
1905.
[45] "Grundlagen der Geometrie," Leipzig, 1899. See Heath's "Euclid,"
Vol. I, p. 229, for an English version; also D. E. Smith, "Teaching of
Elementary Mathematics," p. 266, New York, 1900.
[46] We need frequently to recall the fact that Euclid's "Elements" was
intended for advanced students who went to Alexandria as young men now
go to college, and that the book was used only in university instruction
in the Middle Ages and indeed until recent times.
[47] For example, he moves figures without deformation, but states no
postulate on the subject; and he proves that one side of a triangle is
less than the sum of the other two sides, when he might have postulated
that a straight line is the shortest path between two points. Indeed,
his followers were laughed at for proving a fact so obvious as this one
concerning the triangle.
[48] T. L. Heath, "Euclid," Vol. I, p. 200.
[49] For a resume of the best known attempts to prove this postulate,
see Heath, "Euclid," Vol. I, p. 202; W. B. Frankland, "Theories of
Parallelism," Cambridge, 1910.
[50] For the early history of this movement see Engel and Staeckel, "Die
Theorie der Parallellinien von Euklid bis auf Gauss," Leipzig, 1895;
Bonola, Sulla teoria delle parallele e sulle geometrie non-euclidee, in
his "Questioni riguardanti la geometria elementare," 1900;
Karagiannides, "Die nichteuklidische Geometrie vom Alterthum bis zur
Gegenwart," Berlin, 1893.
[51] This limitation upon elementary geometry was placed by Plato (died
347 B.C.), as already stated.
[52] Book I, Proposition 20.