IX. PARALLEL LINES *

The next assumption which we shall set down is the justly famous assumption of Euclid about parallel lines. It has been stated in many different forms, of which the following is perhaps the simplest.

Assumption XIII. If A is any point and a any line not passing through A, there is not more than one line through A coplanar with a and not meeting a.

B

FIG. 48.

α

That there is at least one line through A, coplanar with a and not meeting it is easily seen by dropping a perpendicular AB from A to a, and observing that the perpendicular, b, to the line AB at the point A could not meet a without contradicting Theorem 38. The same result follows directly from Euclid, I, 27 or I, 28.

The assumption of parallels was stated by Euclid in his Postulate 5 as follows:

"If a straight line falling on two straight lines makes the interior angles 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."

B

FIG. 49.

E

C

This is a consequence of our
assumption, for let the rays AC and
BD be such that the sum of the
CAB and ABD, is less

Dangles
than two right angles.

Let AE be the ray such that the sum of 4.EAB

and ABD is two right angles.

Then by Euclid, I, 28, the line AE does not meet the line BD.

Hence the line AC does meet the line BD.

Since the sum of the

*From this point forward the essay is a mere outline, intended to suggest

how the rest of the subject may be developed.

angles CAB and 4ABD is less than two right angles and the sum of 4EAB and 4ABD is equal to two right angles, it follows that the ray AC falls within 4EAB. Hence the ray AC is on the same side of the line AE with the line BD. It is also on the same side of the line AB with the ray BD. Therefore the point of intersection of the line AC with the line BD is on the rays AC and BD.

For a further discussion of the theory of parallels the reader may consult Euclid's Elements, and the memoir in this collection by Professor Woods.

X. MENSURATION

Defining the sum of two segments and a multiple of a segment (or point-pair) and the terms equality and inequality of segments in the obvious way, it is easy to prove first that if A and B are any two points and n is any whole number, there is a point C on the line AB such that

n(A, B) = (A, C),

and second that there is a point D such that

n(A, D) = (A, B).

From this it follows that if m and n are any whole numbers there exists a point E such that

m(A, B)=n(A, E).

Thus, with an extension of our definition, we have that

Calling m/n the ratio of (A, E) to (A, B) this states that there is a point-pair having to (A, B) the same ratio as that of any two whole numbers. Two such segments are said to be com

mensurable.

It is not hard to show that there are segments which are not commensurable and there is thus propounded the problem of extending the notion of ratio to incommensurable segments.

Euclid's method of doing this is a purely geometrical one, and similar methods have been preferred by nearly all the great geometers, the latest notable example being the Algebra of Segments of Hilbert.

The method, however, which is more or less approximated to in elementary teaching, is that of defining the ratio of two incommensurable segments as an irrational number. The theory of irrational numbers is taken for granted from arithmetic and algebra.

The following proposition, known as the Postulate of Archimedes, is fundamental in this method.

Assumption XIV. If A, B, C are three points in the order {ABC and B1, B2, B3,... are points in the order {ABB1}, {AB1B2},... such that (A, B), is congruent to each of the pointpairs (B, B1), (B1, B2), then there are not more than a

finite number of the points B1, B2,. between A and C.

i B B1 B2

FIG. 50.

.

In other words, by laying off the segment AB a finite number of times in the way indicated a point is reached which is beyond C; that is to say, there exists a number n such that

(A, C)< n(A, B).

Another phrasing of this assumption would be: there exists no infinitely great interval (A, C).

A direct consequence of Assumption XIV is that if D is any point of the ray AB there exists a number n such that

for, if not, there would exist no number n such that

(A, B)< n(A, D).

This may be expressed by saying that there is no infinitely small interval.

Let A。 and A1 be any two points and let us denote by Am that point of the ray Ao4, which is such that the ratio

of the segment 404 to 4041 is If B is any point of

773

m
n

the ray such that (40, B) is incommensurable with (Ao, A1), the points [Am] fall into two classes, those on the segment

AoB, which we may call [4] and those on its prolongation which we may call [4]. The numbers, [x], associated with points in the first class, are all less than the numbers [y] associated with points in the second class. With the aid of

Assumption XIV it can be proved that B is the only point which is between every A, and every A.

By Dedekind's principle of definition of the irrational numbers there exists a unique irrational number, b, greater than every and less than every y. This number, b, we define to be the ratio of the segments AB and A041.

Since any segment whatever is congruent to one of the segments AA, which have 4o as one end, we have now established a scale of magnitudes for the comparison of segments and are in a position to develop a complete theory of proportion.

The theory of the measure (that is to say, length) of segments depends essentially on showing how to arrange segments in order of magnitude. In like manner, the theory of the measure, that is to say, of the area, of regions in the plane depends on showing how to arrange areas in an order of magnitude. For the purpose of elementary geometry we may confine attention to convex regions. A convex region A may be said to be less than a convex region B, if it is possible to decompose A, into a finite set of convex regions congruent to a nonoverlapping set of convex regions contained in B, and such

* See Monograph IV, Appendix I.

that B contains at least one convex region not in this set. Two convex regions A and B, may then be said to be equivalent if neither is less than the other. In order to give this definition value it must be proved that two congruent regions are equivalent. This amounts to proving the following proposition:

It is not possible to decompose two congruent convex regions R1, R2 into convex regions so that all the regions into which R1 is decomposed are congruent to a subset of the regions into which R2 is decomposed.

By associating with an arbitrary square the number 1, a number, called the area, can now be assigned to each region in such a way that two equivalent regions have the same area, and if one region is less than another the less region has the smaller area. The theory of volumes can be developed similarly.

It has been shown by Hilbert that a theory of the areas of polygonal regions can be developed independently of Assumption XIV, and by Dehn that a fully corresponding theory for polyhedral regions does not exist. On this subject the reader should consult the second edition of Hilbert's Grundlagen der Geometrie and the article by Amaldi, "Sulla teoria dell' equivalenza," in Questioni riguardanti la Geometria Elementare,† edited by F. Enriques.

*

XI. THREE-DIMENSIONAL SPACE

Definition. If A, B, C, D are four points not all in the same plane the set of all points on and interior to the four triangles ABC, BCD, CDA, ABD, is called a tetrahedron. The set of all points collinear with pairs of points of a tetrahedron is called a three-space.

By a discussion analogous to that made in IV it is possible to prove that if A'B'C'D' are any four points of a

* Leipzig, 1903.

† Bologna, 1900.

Cf. Transactions of the American Mathematical Society, Volume V,

page 360.