The Teaching of Geometry Part 9

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 quant.i.ties.

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 quant.i.ties. 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 cla.s.s by the use of numbers.

8. _Quant.i.ties that are equal to the same quant.i.ty or to equal quant.i.ties are equal to each other._ In this axiom the word "quant.i.ty"

is used, in the common manner of the present time, to include number and all geometric magnitudes (length, area, volume).

9. _A quant.i.ty may be subst.i.tuted for its equal in an equation or in an inequality._ This axiom is tacitly a.s.sumed 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 a.s.sumptions 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 subst.i.tute _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 subst.i.tuting, 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 quant.i.ties 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 a.s.sert 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 s.p.a.ce. 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 a.s.sumes 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 a.s.serts 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 a.s.sumption of the _invariability of figures_, which would have to be a.s.serted as an antecedent postulate. Euclid preferred to a.s.sert 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 _h.o.m.ogeneity_ of s.p.a.ce.[48]

It is better educational policy, however, to a.s.sert this fact more definitely, and to state the additional a.s.sumption that figures may be moved about in s.p.a.ce 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 a.s.sumptions.[49] Every proof adduced contains some a.s.sumption that practically conceals the postulate itself. Thus the great English mathematician John Wallis (1616-1703) gave a proof based upon the a.s.sumption 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 a.s.sumption.

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 subst.i.tute is that of the Italian Ingrami (1904), "Two parallel straight lines intercept, on every transversal which pa.s.ses 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 a.s.sume 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 a.s.sumptions 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 compa.s.ses to measure the distance. The straightedge and the compa.s.ses are the only two drawing instruments recognized in elementary geometry.[51] While this involves more than Euclid's postulate, it is a better working a.s.sumption 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 compa.s.ses.

5. _Any figure may be moved from one place to another without altering the size or shape._ This is the postulate of the h.o.m.ogeneity of s.p.a.ce, and a.s.serts that s.p.a.ce 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 a.s.sumption 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 a.s.sume it, just as we a.s.sume 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 a.s.sumptions, 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 quant.i.ty 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.