7 Geometrical Constructions based upon the Axioms I

Suppose we have given a geometry of space, in which all of the axioms I–V are valid. For the sake of simplicity, we shall consider in this chapter a a plane geometry which is contained in this geometry of space and shall investigate the question as to what elementary geometrical constructions may be carried out in such a geometry.

Upon the basis of the axioms of group I, the following constructions are always possible.

Problem 1. To join two points with a straight line and to ﬁnd the intersection of two straight lines, the lines not being parallel.

By the assistance of the axioms (IV) of congruence, it is possible to lay oﬀ segments and angles; that is to say, in the given geometry we may solve the following problems:

Problem 3. To lay oﬀ from a given point upon a given straight line a given segment.

Problem 4. To lay oﬀ on a given straight line a given angle; or what is the same thing, to construct a straight line which shall cut a given straight line at a given angle.

It is impossible to make any new constructions by the addition of the axioms of groups II and V. Consequently, when we take into consideration merely the axioms of groups I–V, all of those constructions and only those are possible, which may be reduced to the problems 1–4 given above.

We see at once that this construction can be made in diﬀerent ways by means of the problems 1–4.

In order to carry out the construction in problem 1, we need to make use of only a straight edge. An instrument which enables us to make the construction in problem 3, we will call a transferer of segments. We shall now show that problems 2, 4, and 5 can be reduced to the constructions in problems 1 and 3 and, consequently, all of the problems 1–5 can be completely constructed by means of a straight-edge and a transferer of segments. We arrive, then, at the following result:

Proof. In order to reduce problem 2 to the solution of problems 1 and 3, we join the given point $P$ with any point $A$ of the given straight line and produce $P A$ to $C$ , making $A C = P A$ . Then, join $C$ with any other point $B$ of the given straight line and produce $C B$ to $Q$ , making, $B Q = C B$ . The straight line $P Q$ is the desired parallel.

pict
Figure 50

We can solve problem 5 in the following manner. Let $A$ be an arbitrary point of the given straight line. Then upon this straight line, lay oﬀ in both directions from $A$ the two equal segments $A B$ and $A C$ . Determine, upon any two straight lines passing through the point $A$ , the points $E$ and $D$ so that the segments $A D$ and $A E$ will equal $A B$ and $A C$ .

pict
Figure 51

Suppose the straight lines $B D$ and $C E$ intersect in $F$ and the straight lines $B E$ and $C D$ intersect in $H$ . $F H$ is then the desired perpendicular. In fact, the angles $B D C$ and $B E C$ , being inscribed in a semicircle having the diameter $B C$ , are both right angles, and, hence, according to the theorem relating to the point of intersection of the altitudes of a triangle, the straight lines $F H$ and $B C$ are perpendicular to each other.

pict
Figure 52

Moreover, we can easily solve problem 4 simply by the drawing of straight lines and the laying oﬀ of segments. We will employ the following method which requires only the drawing of parallel lines and the erection of perpendiculars. Let $β$ be the angle to be laid oﬀ and $A$ its vertex. Draw through $A$ a straight line $l$ parallel to the given straight line, upon which we are to lay oﬀ the given angle $β$ . From an arbitrary point $B$ of one side of the angle $β$ , let fall a perpendicular upon the other side of this angle and also one upon $l$ . Denote the feet of these perpendiculars by $D$ and $C$ respectively. The construction of these perpendiculars is accomplished by means of problems 2 and 5. Then, let fall from $A$ a perpendicular upon $C D$ , and let its foot be denoted by $E$ . According to the demonstration given in Section Section 14, the angle $C A E$ equals $β$ . Consequently, the construction in 4 is made to depend upon that of 1 and 3 and with this our proposition is demonstrated. □

34. Analytical Representation of the Co-ordinates of Points which can be so Constructed

Besides the elementary geometrical problems considered in Section 33, there exists a long series of other problems whose solution is possible by the drawing of straight lines and the laying oﬀ of segments. In order to get a general survey of the scope of the problems which may be solved in this manner, let us take as the basis of our consideration a system of axes in rectangular co-ordinates and suppose that the co-ordinates of the points are, as usual, represented by real numbers or by functions of certain arbitrary parameters. In order to answer the question in respect to all the points capable of such a construction, we employ the following considerations.

Let a system of deﬁnite points be given. Combine the co-ordinates of these points into a domain

R

. This domain contains, then, certain real numbers and certain arbitrary parameters

p

. Consider, now, the totality of points capable of construction by the drawing of straight lines and the laying oﬀ of deﬁnite segments, making use of the system of points in question. We will call the domain formed from the co-ordinates of these points

Ω (R)

, which will then contain real numbers and functions of the arbitrary parameters

p

The discussion in Section 14 shows that the drawing of straight lines and of parallels amounts, analytically, to the addition, subtraction, multiplication, and division of segments. Furthermore, the well known formula given in Section 9 for a rotation shows that the laying oﬀ of segments upon a straight line does not necessitate any other analytical operation than the extraction of the square root of the sum of the squares of two segments whose bases have been previously constructed. Conversely, in consequence of the pythagorean theorem, we can always construct, by the aid of a right triangle, the square root of the sum of the squares of two segments by the mere laying oﬀ of segments.

From these considerations, it follows that the domain

Ω (R)

contains all of those and only those real numbers and functions of the parameters

p

, which arise from the numbers and parameters in

R

by means of a ﬁnite number of applications of the ﬁve operations; viz., the four elementary operations of arithmetic and, in addition, the ﬁfth operation of extracting the square root of the sum of two squares. We may express this result as follows:

From this proposition, we can at once show that not every problem which can be solved by the use of a compass can also be solved by the aid of a transferer of segments and a straight-edge. For the purpose of showing this, let us consider again that geometry which was constructed in Section 9 by the help of the domain

Ω

of algebraic numbers. In this geometry, there exist only such segments as can be constructed by means of a straight-edge and a transferer of segments, namely, the segments determined by the numbers of the domain

Ω

Now, if

ω

is a number of the domain

Ω

, we easily see from the deﬁnition of

Ω

that every algebraic number conjugate to

ω

must also lie in

Ω

. Since the numbers of the domain

Ω

are evidently all real, it follows that it can contain only such real algebraic numbers as have their conjugates also real.

Let us now consider the following problem; viz., to construct a right triangle having the hypotenuse 1 and one side

| \sqrt{2} | - 1

. The algebraic number

\sqrt{2 | \sqrt{2} | - 2}

, which expresses the numerical value of the other side, does not occur in the domain

Ω

, since the conjugate number

\sqrt{- 2 | \sqrt{2} | - 2}

is imaginary. This problem is, therefore, not capable of solution in the geometry in question and, hence, cannot be constructed by means of a straight-edge and a transferer of segments, although the solution by means of a compass is possible.

35. The Representation of Algebraic Numbers and of Integral Rational Functions as Sums of Squares

The question of the possibility of geometrical constructions by the aid of a straight-edge and a transferer of segments necessitates, for its complete treatment, particular theorems of an arithmetical and algebraic character, which, it appears to me, are themselves of interest. Since the time of Fermat, it has been known that every positive integral rational number can be represented as the sum of four squares. This theorem of Fermat permits the following remarkable generalization:

Definition. Let

k

be an arbitrary number ﬁeld and let

m

be its degree. We will denote by

k^{'}

k^{''}

, …,

k^{(m - 1)}

the

m - 1

number ﬁelds conjugate to

k

. If, among the

m

ﬁelds

k

k^{'}, k^{''}, \dots, k^{(m - 1)}

there is one or more formed entirely of real numbers, then we call these ﬁelds real. Suppose that the ﬁelds

k

k^{'}

, …,

k^{(s - 1)}

are such. A number

α

of the ﬁeld

k

is called in this case totally positive in

k

, whenever the

s

numbers conjugate to

α

, contained respectively in

k

k^{'}

k^{''}

, …,

k^{(s - 1)}

are all positive. However, if in each of the

m

ﬁelds

k

k^{'}

k^{''}

, …,

k^{(m - 1)}

there are also imaginary numbers present, we call every number

α

k

totally positive.

The demonstration of this theorem presents serious diﬃculty. It depends essentially upon the theory of relatively quadratic number ﬁelds, which I have recently developed in several papers.²⁵ We will here call attention only to that proposition in this theory which gives the condition that a ternary diophantine equation of the form

can be solved when the coeﬃcients

α

β

γ

are given numbers in

k

and

ξ

η

ζ

are the required numbers in

k

. The demonstration of Theorem 42 is accomplished by the repeated application of the proposition just mentioned.

From Theorem 42 follow a series of propositions concerning the representation of such rational functions of a variable, with rational coeﬃcients, as never have negative values. I will mention only the following theorem, which will be of service in the following sections.

Proof. . We will denote the degree of the function $f (x)$ by $m$ , which, in any case, must evidently be even. When $m = 0$ , that is to say, when $f (x)$ is a rational number, the validity of Theorem 43 follows immediately from Fermat’s theorem concerning the representation of a positive number as the sum of four squares. We will assume that the proposition is already established for functions of degree $2$ , $4$ , $6$ , …, $m - 2$ , and show, in the following manner, its validity for the case of a function of the $m^{t h}$ degree.

Let us, ﬁrst of all, consider brieﬂy the case where $f (x)$ breaks up into the product of two or more integral functions of $x$ with rational coeﬃcients. Suppose $p (x)$ to be one of those functions contained in $f (x)$ , which itself cannot be further decomposed into a product of integral functions having rational coeﬃcients. It then follows at once from the “deﬁnite” character which we have given to the function $f (x)$ , that the factor $p (x)$ must either appear in $f (x)$ to an even degree or $p (x)$ must be itself “deﬁnite”; that is to say, must be a function which never has negative values for any real values of $x$ . In the ﬁrst case, the quotient $\frac{f (x)}{{p (x)}^{2}}$ and, in the second case, both $p (x)$ and $\frac{f (x)}{p (x)}$ , are “deﬁnite,” and these functions have an even degree $< m$ . Hence, according to our hypothesis, in the ﬁrst case, $\frac{f (x)}{{p (x)}^{2}}$ and, in the last case, $p (x)$ and $\frac{f (x)}{p (x)}$ may be represented as the quotient of the sum of squares of the character mentioned in theorem 43. Consequently, in both of these cases, the function $f (x)$ admits of the required representation.

Let us now consider the case where $f (x)$ cannot be broken up into the product of two integral functions having rational coeﬃcients. The equation $f (𝜃) = 0$ deﬁnes, then, a ﬁeld of algebraic numbers $k (𝜃)$ of the $m^{t h}$ degree, which, together with all their conjugate ﬁelds, are imaginary. Since, according to the deﬁnition given just before the statement of Theorem 42, each number given in $k (𝜃)$ , and hence also $- 1$ is totally positive in $k (𝜃)$ , it follows from Theorem 42 that the number $- 1$ can be represented as a sum of the squares of four deﬁnite numbers in $k (𝜃)$ . Let, for example

- 1 = α^{2} + β^{2} + γ^{2} + δ^{2},

where

α

β

γ

δ

are integral or fractional numbers in

k (𝜃)

. Let us put

\begin{matrix} α & = & a_{1} 𝜃^{m - 1} + a_{2} 𝜃^{m - 2} + \dots + a_{m} & = & ϕ (𝜃), \\ β & = & b_{1} 𝜃^{m - 1} + b_{2} 𝜃^{m - 2} + \dots + b_{m} & = & ψ (𝜃), \\ γ & = & c_{1} 𝜃^{m - 1} + c_{2} 𝜃^{m - 2} + \dots + c_{m} & = & χ (𝜃), \\ 𝜃 & = & d_{1} 𝜃^{m - 1} + d_{2} 𝜃^{m - 2} + \dots + d_{m} & = & ρ (𝜃); \end{matrix}

where $a_{1}$ , $a_{2}$ , …, $a_{m}$ , …, $d_{1}$ , $d_{2}$ , …, $d_{m}$ are the rational numerical coeﬃcients and $ϕ (𝜃)$ , $ψ (𝜃)$ , $χ (𝜃)$ , $ρ (𝜃)$ the integral rational functions in question, having the degree $(m - 1)$ in $𝜃$ .

From (1), we have

1 + {ϕ (𝜃)}^{2} + {ψ (𝜃)}^{2} + {χ (𝜃)}^{2} + {ρ (𝜃)}^{2} = 0

Because of the irreducibility of the equation

f (x) = 0

, the expression

F (x) = 1 + {ϕ (𝜃)}^{2} + {ψ (𝜃)}^{2} + {χ (𝜃)}^{2} + {ρ (𝜃)}^{2}

represents, necessarily, an integral rational function of

x

which is divisible by

f (x)

F (x)

is, then, a “deﬁnite” function of the degree

(2 m - 2)

or lower. Hence, the quotient

\frac{F (x)}{f (x)}

is a “deﬁnite” function of the degree

(m - 2)

or lower in

x

, having rational coeﬃcients. Consequently, by the hypothesis we have made,

\frac{F (x)}{f (x)}

can be represented as the quotient of two sums of squares of the kind mentioned in Theorem 43 and, since

F (x)

is itself such a sum of squares, it follows that,

f (x)

must also be a quotient of two sums of squares of the required kind. The validity of Theorem 43 is accordingly established. □

It would be perhaps diﬃcult to formulate and to demonstrate the corresponding proposition for integral functions of two or more variables. However, I will here merely remark that I have demonstrated in an entirely diﬀerent manner the possibility of representing any “deﬁnite” integral rational function of two variables as the quotient of sums of squares of integral functions, upon the hypothesis that the functions represented may have as coeﬃcients not only rational but any real numbers.²⁶

36. Criterion for the Possibility of a Geometrical Construction by means of A Straight-Edge and a Transferer of Segments

Suppose we have given a problem in geometrical construction which can be aﬀected by means of a compass. We shall attempt to ﬁnd a criterion which will enable us to decide, from the analytical nature of the problem and its solutions, whether or not the construction can be carried out by means of only a straight-edge and a transferer of segments. Our investigation will lead us to the following proposition.

Proof. . We shall demonstrate this proposition merely for the case where the co-ordinates of the given points are rational functions, having rational coeﬃcients, of a single parameter $p$ .

It is at once evident that the proposition gives a necessary condition. In order to show that it is also suﬃcient, let us assume that it is fulﬁlled and then, among the $n$ square roots, consider that one which, in the calculation of the co-ordinates of the desired points, is ﬁrst to be extracted. The expression under this radical is a rational function $f_{1} (p)$ , having rational coeﬃcients, of the parameter $p$ . This rational function cannot have a negative value for any real value of the parameter $p$ ; for, otherwise the problem must have imaginary solutions for certain values of $p$ , which is contrary to the given hypothesis. Hence, from Theorem 43, we conclude that $f_{1} (p)$ can be represented as a quotient of the sums of squares of integral rational functions.

Moreover, the formulæ

\begin{array}{rcl} \sqrt{a^{2} + b^{2} + c^{2}} & = & \sqrt{{(\sqrt{a^{2} + b^{2}})}^{2} + c^{2}} \\ \sqrt{a^{2} + b^{2} + c^{2} + d^{2}} & = & \sqrt{{(\sqrt{a^{2} + b^{2} + c^{2}})}^{2} + d^{2}} \end{array}

\cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot

show that, in general, the extraction of the square root of a sum of any number of squares may always be reduced to the repeated extraction of the square root of the sum of two squares.

If now we combine this conclusion with the preceding results, it follows that the expression $\sqrt{f_{1} (p)}$ can certainly be constructed by means of a straight-edge and a transferer of segments. Among the $n$ square roots, consider now the second one to be extracted in the process of calculating the co-ordinates of the required points. The expression under this radical is a rational function $f_{2} (p, \sqrt{f_{1}})$ of the parameter $p$ and the square root ﬁrst considered. This function $f_{2}$ can never be negative for any real arbitrary value of the parameter $p$ and for either sign of $\sqrt{f_{1}}$ ; for, otherwise among the $2^{n}$ solutions of our problem, there would exist for certain values of $p$ also imaginary solutions, which is contrary to our hypothesis. It follows, therefore, that $f_{2}$ must satisfy a quadratic equation of the form

f_{2}^{2} - ϕ_{2} (p) f_{2} + ψ_{1} (p) = 0,

where

ϕ_{1} (p)

and

ψ_{1} (p)

are, necessarily, such rational functions of

p

as have rational coeﬃcients and for real values of

p

never become negative. From this equation, we have

f_{2} = \frac{f_{2}^{2} + ψ_{1} (p)}{ϕ_{1} (p)} .

Now, according to Theorem 43, the functions $ϕ_{1} (p)$ and $ψ_{1} (p)$ must again be the quotient of the sums of squares of rational functions, and, on the other hand, the expression $f_{2}$ may be, from the above considerations, constructed by means of a straight-edge and a transferer of segments. The expression found for $f_{2}$ shows, therefore, that $f_{2}$ is a quotient of the sum of squares of functions which may be constructed in the same way. Hence, the expression $\sqrt{f_{2}}$ can also be constructed by means of a straight-edge and a transferer of segments.

Just as with the expression $f_{2}$ , any other rational function $ϕ_{2} (p, \sqrt{f_{1}})$ of $p$ and $\sqrt{f_{1}}$ may be shown to be the quotient of two sums of squares of functions which may be constructed, provided this rational function $ϕ_{2}$ possesses the property that for real values of the parameter $p$ and for either sign of $\sqrt{f_{1}}$ , it never becomes negative.

This remark permits us to extend the above method of reasoning in the following manner.

Let $f_{3} (p, \sqrt{f_{1}}, \sqrt{f_{2}})$ be such an expression as depends in a rational manner upon the three arguments $p, \sqrt{f_{1}}, \sqrt{f_{2}}$ and of which, in the analytical calculation of the co-ordinates of the desired points, the square root is the third to be extracted. As before, it follows that $f_{3}$ can never have negative values for real values of $p$ and for either sign of $\sqrt{f_{1}}$ and $\sqrt{f_{2}}$ . This condition of aﬀairs shows again that $f_{3}$ must satisfy a quadratic equation of the form

f_{3}^{2} - ϕ_{2} (p, \sqrt{f_{1}}) f_{3} - ψ_{2} (p, \sqrt{f_{1}}) = 0,

where

ϕ_{2}

and

ψ_{2}

are such rational functions of

p

and

\sqrt{f_{1}}

as never become negative for any real value of

p

and either sign of

\sqrt{f_{1}}

. But, according to the preceding remark, the functions

ϕ_{2}

and

ψ_{2}

are the quotients of two sums of squares of functions which may be constructed and, hence, it follows that the expression

f_{3} = \frac{f_{3}^{2} + ψ_{2} (p, \sqrt{f_{1}})}{ϕ_{2} (p, \sqrt{f_{1}})}

is likewise possible of construction by aid of a straight-edge and a transferer of segments. □

The continuation of this method of reasoning leads to the demonstration of Theorem 44 for the case of a single parameter

p

The truth of Theorem 44 for the general case depends upon whether or not Theorem 43 can be generalized in a similar manner to cover the case of two or more variables.

As an example of the application of Theorem 44, we may consider the regular polygons which may be constructed by means of a compass. In this case, the arbitrary parameter,

p

does not occur, and the expressions to be constructed all represent algebraic numbers. We easily see that the criterion of Theorem 44 is fulﬁlled, and, consequently, it follows that the above-mentioned regular polygons can be constructed by the drawing of straight lines and the laying oﬀ of segments. We might deduce this result also directly from the theory of the division of the circle (Kreisteilung).

Concerning the other known problems of construction in the elementary geometry, we will here only mention that the problem of Malfatti may be constructed by means of a straight-edge and a transferer of segments. This is, however, not the case with the contact problems of Appolonius.

37. Conclusion

The preceding work treats essentially of the problems of the euclidean geometry only; that is to say, it is a discussion of the questions which present themselves when we admit the validity of the axiom of parallels. It is none the less important to discuss the principles and the fundamental theorems when we disregard the axiom of parallels. We have thus excluded from our study the important question as to whether it is possible to construct a geometry in a logical manner, without introducing the notion of the plane and the straight line, by means of only points as elements, making use of the idea of groups of transformations, or employing the idea of distance. This last question has recently been the subject of considerable study, due to the fundamental and proliﬁc works of Sophus Lie. However, for the complete elucidation of this question, it would be well to divide into several parts the axiom of Lie, that space is a numerical multiplicity. First of all, it would seem to me desirable to discuss thoroughly the hypothesis of Lie, that functions which produce transformations are not only continuous, but may also be diﬀerentiated. As to myself, it does not seem to me probable that the geometrical axioms included in the condition for the possibility of diﬀerentiation are all necessary.

In the treatment of all questions of this character, I believe the methods and the principles employed in the preceding work will be of value. As an example, let me call attention to an investigation undertaken at my suggestion by Mr. Dehn, and which has already appeared.²⁷ In this article, he has discussed the known theorems of Legendre concerning the sum of the angles of a triangle, in the demonstration of which that geometer made use of the idea of continuity.

The investigation of Mr. Dehn rests upon the axioms of connection, of order, and of congruence; that is to say, upon the axioms of groups I, II, IV. However, the axiom of parallels and the axiom of Archimedes are excluded. Moreover, the axioms of order are stated in a more general manner than in the present work, and in substance as follows: Among four points

A, B, C, D

of a straight line, there are always two, for example

A, C

, which are separated from the other two and conversely. Five points

A, B, C, D, E

upon a straight line may always be so arranged that

A, C

shall be separated from

B, E

and from

B, D

. Consequently,

A, D

are always separated from

B, E

and from

C, E

, etc. The (elliptic) geometry of Riemann, which we have not considered in the present work, is in this way not necessarily excluded.

Upon the basis of the axioms of connection, order, and congruence, that is to say, the axioms I, II, IV, we may introduce, in the well known manner, the elements called ideal,—-ideal points, ideal straight lines, and ideal planes. Having done this, Mr. Dehn demonstrates the following theorem.

This euclidean geometry, superimposed upon the non-euclidean plane, may be called a pseudo-geometry and the new kind of congruence a pseudo-congruence.

By aid of the preceding theorem, we may now introduce an algebra of segments relating to the plane and depending upon the developments made in Section 14, pp. §–§. This algebra of segments permits the demonstration of the following important theorem:

The case where the sum of the angles is equal to two right angles gives the well known theorem of Legendre. However, in his demonstration, Legendre makes use of continuity.

Mr. Dehn then discusses the connection between the three diﬀerent hypotheses relative to the sum of the angles and the three hypotheses relative to parallels.

In order to demonstrate part (a) of this theorem, Mr. Dehn constructs a geometry where we may draw through a point an inﬁnity of lines parallel to a given straight line and where, moreover, all of the theorems of Riemann’s (elliptic) geometry are valid. This geometry may be called non-legendrian, for it is in contradiction with that theorem of Legendre by virtue of which the sum of the angles a triangle is never greater than two right angles. From the existence of this non-legendrian geometry, it follows at once that it is impossible to demonstrate the theorem of Legendre just mentioned without employing the axiom of Archimedes, and in fact, Legendre made use of continuity in his demonstration of this theorem.

For the demonstration of case (b), Mr. Dehn constructs a geometry where the axiom of parallels does not hold, but where, nevertheless, all of the theorems of the euclidean geometry are valid. Then, we have the sum of the angles of a triangle equal to two right angles. There exist also similar triangles, and the extremities of the perpendiculars having the same length and their bases upon a straight line all lie upon the same straight line, etc. The existence of this geometry shows that, if we disregard the axiom of Archimedes, the axiom of parallels cannot be replaced by any of the propositions which we usually regard as equivalent to it.

This new geometry may be called a semi-euclidean geometry. As in the case of the non-legendrian geometry, it is clear that the semi-euclidean geometry is at the same time a non-archimedean geometry.

This theorem shows that, with respect to the axiom of Archimedes, the two non-euclidean hypotheses concerning parallels act very diﬀerently.

However, as I have already remarked, the present work is rather a critical investigation of the principles of the euclidean geometry. In this investigation, we have taken as a guide the following fundamental principle; viz., to make the discussion of each question of such a character as to examine at the same time whether or not it is possible to answer this question by following out a previously determined method and by employing certain limited means. This fundamental rule seems to me to contain a general law and to conform to the nature of things. In fact, whenever in our mathematical investigations we encounter a problem or suspect the existence of a theorem, our reason is satisﬁed only when we possess a complete solution of the problem or a rigorous demonstration of the theorem, or, indeed, when we see clearly the reason of the impossibility of the success and, consequently, the necessity of failure.

Thus, in the modern mathematics, the question of the impossibility of solution of certain problems plays an important role, and the attempts made to answer such questions have often been the occasion of discovering new and fruitful ﬁelds for research. We recall in this connection the demonstration by Abel of the impossibility of solving an equation of the ﬁfth degree by means of radicals, as also the discovery of the impossibility of demonstrating the axiom of parallels, and, ﬁnally, the theorems of Hermite and Lindeman concerning the impossibility of constructing by algebraic means the numbers

e

and

π

This fundamental principle, which we ought to bear in mind when we come to discuss the principles underlying the impossibility of demonstrations, is intimately connected with the condition for the “purity” of methods in demonstration, which in recent times has been considered of the highest importance by many mathematicians. The foundation of this condition is nothing else than a subjective conception of the fundamental principle given above. In fact, the preceding geometrical study attempts, in general, to explain what are the axioms, hypotheses, or means, necessary to the demonstration of a truth of elementary geometry, and it only remains now for us to judge from the point of view in which we place ourselves as to what are the methods of demonstration which we should prefer.



	THOUGH A GIVEN POINT, WE MAY DRAW


THE SUM OF	NO PARALLELS	ONE PARALLEL	AN INFINITY OF PARALLELS




THE ANGLES	TO A	TO A	TO A STRAIGHT LINE




OF A TRIANGLE IS	STRAIGHT LINE	STRAIGHT LINE

$> 2$ right	Riemann’s	This case is	Non-legendrian geometry




angles	(elliptic) geometry	impossible

$< 2$ right	This case is	Euclidean	Semi-euclidean geometry




angles	impossible	(parabolic) geometry

$= 2$ right	This case is	This case is	Geometry of Lobatschewski




angles	impossible	impossible	(hyperbolic)

Chapter 7
Geometrical Constructions based upon the Axioms I–V

33. Geometrical Constructions by means of A Straight-Edge and a Transferer of Segments

34. Analytical Representation of the Co-ordinates of Points which can be so Constructed

35. The Representation of Algebraic Numbers and of Integral Rational Functions as Sums of Squares

36. Criterion for the Possibility of a Geometrical Construction by means of A Straight-Edge and a Transferer of Segments

37. Conclusion

Chapter 7Geometrical Constructions based upon the Axioms I–V

33. Geometrical Constructions by means of A Straight-Edge and a Transferer of Segments

34. Analytical Representation of the Co-ordinates of Points which can be so Constructed

35. The Representation of Algebraic Numbers and of Integral Rational Functions as Sums of Squares

36. Criterion for the Possibility of a Geometrical Construction by means of A Straight-Edge and a Transferer of Segments

37. Conclusion

Chapter 7
Geometrical Constructions based upon the Axioms I–V