Chapter 3
The Theory of Proportion¹⁶

13. Complex Number-Systems

At the beginning of this chapter, we shall present briefly certain preliminary ideas concerning complex number systems which will later be of service to us in our discussion.

The real numbers form, in their totality, a system of things having the following properties:

1.

From the number $a$ and the number $b$, there is obtained by “addition” a definite number $c$, which we express by writing

$$a+b=c\text{ or }c=a+b.$$

2.

There exists a definite number, which we call $0$, such that, for every number $a$, we have

$$a+0=a\text{ and }0+a=a.$$

3.

If $a$ and $b$ are two given numbers, there exists one and only one number $x$, and also one and only one number $y$, such that we have respectively

$$a+x=b,y+a=b.$$

4.

From the number $a$ and the number $b$, there may be obtained in another way, namely, by “multiplication,” a definite number $c$, which we express by writing

$$ab=c\text{ or }c=ab.$$

5.

There exists a definite number, called $1$, such that, for every number $a$, we have

$$a\cdot 1=a\text{ and }1\cdot a=a.$$

6.

If $a$ and $b$ are any arbitrarily given numbers, where $a$ is different from $0$, then there exists one and only one number $x$ and also one and only one number $y$ such that we have respectively

$$ax=b,ya=b.$$

If $a$, $b$, $c$ are arbitrary numbers, the following laws of operation always hold:

7.

8.

$a+b=b+a$

9.

10.

11.

12.

$ab=ba.$

13.

If $a$, $b$ are any two distinct numbers, one of these, say $a$, is always greater than the other. The other number is said to be the smaller of the two. We express this relation by writing

$$a>b\text{ and }b<a.$$

14.

If $a>b$ and $b>c$, then is also $a>c$.

15.

If $a>b$, then is also $a+c>b+c$ and $c+a>c+b$.

16.

If $a>b$ and $c>0$, then is also $ac>bc$ and $ca>cb$.

17.

If $a$, $b$ are any two arbitrary numbers, such that $a>0$ and $b>0$, it is always possible to add $a$ to itself a sufficient number of times so that the resulting sum shall have the property that

$$a+a+a+\cdots +a>b.$$

A system of things possessing only a portion of the above properties (1–17) is called a complex number system, or simply a number system. A number system is called archimedean, or non-archimedean, according as it does, or does not, satisfy condition (17).

Not every one of the properties (1–17) given above is independent of the others. The problem arises to investigate the logical dependence of these properties. Because of their great importance in geometry, we shall, in Sections 29, 30, pp. §–§, answer two definite questions of this character. We will here merely call attention to the fact that, in any case, the last of these conditions (17) is not a consequence of the remaining properties, since, for example, the complex number system , considered in Section 12, possesses all of the properties (1–16), but does not fulfil the law stated in (17).

14. Demonstration of Pascal’s Theorem

In this and the following chapter, we shall take as the basis of our discussion all of the plane axioms with the exception of the axiom of Archimedes; that is to say, the axioms I, 1–2 and II–IV. In the present chapter, we propose, by aid of these axioms, to establish Euclid’s theory of proportion; that is, we shall establish it for the plane and that independently of the axiom of Archimedes.

For this purpose, we shall first demonstrate a proposition which is a special case of the well known theorem of Pascal usually considered in the theory of conic sections, and which we shall hereafter, for the sake of brevity, refer to simply as Pascal’s theorem. This theorem may be stated as follows:

In order to demonstrate this theorem, we shall first introduce the following notation. In a right triangle, the base $a$ is uniquely determined by the hypotenuse $c$ and the base angle $\alpha$ included by $a$ and $c$. We will express this fact briefly by writing

Hence, the symbol $\alpha c$ always represents a definite segment, providing $c$ is any given segment whatever and $\alpha$ is any given acute angle.

Furthermore, if $c$ is any arbitrary segment and $\alpha$, $\beta$ are any two acute angles whatever, then the two segments $\alpha \beta c$ and $\beta \alpha c$ are always congruent; that is, we have

In order to prove this statement, we take the segment $c=AB$, and with $A$ as a vertex lay off upon the one side of this segment the angle $\alpha$ and upon the other the angle $\beta$. Then, from the point $B$, let fall upon the opposite sides of the $\alpha$ and $\beta$ the perpendiculars $BC$ and $BD$, respectively. Finally, join $C$ with $D$ and let fall from $A$ the perpendicular $AE$ upon $CD$.

Since the two angles $ACB$ and $ADB$ are right angles, the four points $A$, $B$, $C$, $D$ are situated upon a circle. Consequently, the angles $ACD$ and $ABD$, being inscribed in the same segment of the circle, are congruent. But the angles $ACD$ and $CAE$, taken together, make a right angle, and the same is true of the two angles $ABD$ and $BAD$. Hence, the two angles $CAE$ and $BAD$ are also congruent; that is to say,

From these considerations, we have immediately the following congruences of segments:

Returning now to the figure in connection with Pascal’s theorem, denote the intersection of the two given straight lines by $O$ and the segments $OA$, $OB$, $OC$, $OA\prime$, $OB\prime$, $OC\prime$, $CB\prime$, $BC\prime$, $CA\prime$, $AC\prime$, $BA\prime$, $AB\prime$ by $a$, $b$, $c$, $a\prime$, $b\prime$, $c\prime$, $l$, $l^{\ast }$, $m$, $m^{\ast }$, $n$, $n^{\ast }$, respectively.

Let fall from the point $O$ a perpendicular upon each of the segments $l$, $m$, $n$. The perpendicular to $l$ will form with the straight lines $OA$ and $OA\prime$ acute angles, which we shall denote by $\lambda \prime$ and $\lambda$, respectively. Likewise, the perpendiculars to $m$ and $n$ form with these same lines $OA$ and $OA\prime$ acute angles, which we shall denote by $\mu \prime$, $\mu$ and $\nu \prime$, $\nu$, respectively. If we now express, as indicated above, each of these perpendiculars in terms of the hypotenuse and base angle, we have the three following congruences of segments:

Multiplying both members of congruence (3) by the symbol $\lambda \prime \mu$, and remembering that, as we have already seen, the symbols in question are commutative, we have

If now we consider the perpendicular let fall from $O$ upon $n$ and draw perpendiculars to this same line from the points $A$ and $B\prime$, then congruence (6) shows that the feet of the last two perpendiculars must coincide; that is to say, the straight line $n^{\ast }=AB\prime$ makes a right angle with the perpendicular to $n$ and, consequently, is parallel to $n$. This establishes the truth of Pascal’s theorem.

Having given any straight line whatever, together with an arbitrary angle and a point lying outside of the given line, we can, by constructing the given angle and drawing a parallel line, find a straight line passing through the given point and cutting the given straight line at the given angle. By means of this construction, we can demonstrate Pascal’s theorem in the following very simple manner, for which, however, I am indebted to another source.

Through the point $B$, draw a straight line cutting $OA\prime$ in the point $D\prime$ and making with it the angle $OCA\prime$, so that the congruence
(1$\ast$)      $\angle OCA\prime \equiv \angle OD\prime B$
is fulfilled. Now, according to a well known property of circles, $CBD\prime A\prime$ is an inscribed quadrilateral, and, consequently, by aid of the theorem concerning the congruence of angles inscribed in the same segment of a circle, we have the congruence
(2$\ast$)      $\angle OBA\prime \equiv \angle OD\prime C$.
Since, by hypothesis, $CA\prime$ and $AC\prime$ are parallel, we have
(3$\ast$)      $\angle OCA\prime \equiv \angle OAC\prime$,
and from (1$\ast$) and (3$\ast$) we obtain the congruence

An Algebra of Segments, Based Upon Pascal’s Theorem

Pascal’s theorem, which was demonstrated in the last section, puts us in a position to introduce into geometry a method of calculating with segments, in which all of the rules for calculating with real numbers remain valid without any modification.

Instead of the word “congruent” and the sign $\equiv$, we make use, in the algebra of segments, of the word “equal” and the sign $=$.

If $A,B,C$ are three points of a straight line and if $B$ lies between $A$ and $C$, then we say that $c=AC$ is the sum of the two segments $a=AB$ and $b=BC$. We indicate this by writing

From the linear axioms of congruence (axioms IV, 1–3), we easily see that, for the above definition of addition of segments, the associative law

In order to define geometrically the product of two segments $a$ and $b$, we shall make use of the following construction. Select any convenient segment, which, having been selected, shall remain constant throughout the discussion, and denote the same by $1$. Upon the one side of a right angle, lay off from the vertex $O$ the segment $1$ and also the segment $b$. Then, from $O$ lay off upon the other side of the right angle the segment $a$. Join the extremities of the segments $1$ and $a$ by a straight line, and from the extremity of $b$ draw a line parallel to this straight line. This parallel will cut off from the other side of the right angle a segment $c$. We call this segment $c$ the product of the segments $a$ and $b$, and indicate this relation by writing

We shall now demonstrate that, for this definition of the multiplication of segments, the commutative law

If now we apply the commutative law which we have just demonstrated, we obtain the above formula, which expresses the associative law for the multiplication of two segments.

Finally, the distributive law

Proportion and the Theorems of Similitude.

By aid of the preceding algebra of segments, we can establish Euclid’s theory of proportion in a manner free from objections and without making use of the axiom of Archimedes.

If $a,b,a\prime ,b\prime$ are any four segments whatever, the proportion

From the theorem just demonstrated, we can easily deduce the fundamental theorem in the theory of proportion. This theorem may be stated as follows:

Equations of Straight Lines and of Planes

To the system of segments already discussed, let us now add a second system. We will distinguish the segments of the new system from those of the former one by means of a special sign, and will call them “negative” segments in contradistinction to the “positive” segments already considered. If we introduce also the segment $O$, which is determined by a single point, and make other appropriate conventions, then all of the rules deduced in Section 13 for calculating with real numbers will hold equally well here for calculating with segments. We call special attention to the following particular propositions:

We have always $a\cdot 1=1\cdot a=a$.

If $a\cdot b=0$, then either $a=0$, or $b=0$.

If $a>b$ and $c>0$, then $ac>bc$.

In a plane $\alpha$, we now take two straight lines cutting each other in $O$ at right angles as the fixed axes of rectangular co-ordinates, and lay off from $O$ upon these two straight lines the arbitrary segments $x$ and $y$. We lay off these segments upon the one side or upon the other side of $O$, according as they are positive or negative. At the extremities of $x$ and $y$, erect perpendiculars and determine the point $P$ of their intersection. The segments $x$ and $y$ are called the co-ordinates of $P$. Every point of the plane $\alpha$ is uniquely determined by its co-ordinates $x$, $y$, which may be positive, negative, or zero.

Let $l$ be a straight line in the plane $\alpha$, such that it shall pass through $O$ and also through a point $C$ having the co-ordinates $a,b$. If $x,y$ are the co-ordinates

of any point on $l$, it follows at once from Theorem 22 that

From these considerations, we may easily conclude, independently of the axiom of Archimedes, that every straight line of a plane is represented by an equation which is linear in the co-ordinates $x$, $y$, and, conversely, every such linear equation represents a straight line when the co-ordinates are segments appertaining to the geometry in question. The corresponding results for the geometry of space may be easily deduced.

The remaining parts of geometry may now be developed by the usual methods of analytic geometry.

So far in this chapter, we have made absolutely no use of the axiom of Archimedes. If now we assume the validity of this axiom, we can arrange a definite correspondence between the points on any straight line in space and the real numbers. This may be accomplished in the following manner.

We first select on a straight line any two points, and assign to these points the numbers $0$ and $1$. Then, bisect the segment thus determined and denote the middle point by the number $\frac{1}{2}$. In the same way, we denote the middle of by $\frac{1}{4}$, etc. After applying this process $n$ times, we obtain a point which corresponds to $\frac{1}{2^n}$. Now, lay off $m$ times in both directions from the point $O$ the segment . We obtain in this manner a point corresponding to the numbers and . From the axiom of Archimedes, we now easily see that, upon the basis of this association, to each arbitrary point of a straight line there corresponds a single, definite, real number, and, indeed, such that this correspondence possesses the following property: If $A$, $B$, $C$ are any three points on a straight line and $\alpha$, $\beta$, $\gamma$ are the corresponding real numbers, and, if $B$ lies between $A$ and $C$, then one of the inequalities,

From the development given in Section 9, p. §, it is evident, that to every number belonging to the field of algebraic numbers $\Omega$, there must exist a corresponding point upon the straight line. Whether to every real number there corresponds a point cannot in general be established, but depends upon the geometry to which we have reference.

However, it is always possible to generalize the original system of points, straight lines, and planes by the addition of “ideal” or “irrational” elements, so that, upon any straight line of the corresponding geometry, a point corresponds without exception to every system of three real numbers. By the adoption of suitable conventions, it may also be seen that, in this generalized geometry, all of the axioms I–V are valid. This geometry, generalized by the addition of irrational elements, is nothing else than the ordinary analytic geometry of space.