The Strengthening of Proof in Elementary Geometry
at the turn of the 20th Century

Near the end of the 19th century and at the beginning of the 20th century, starting with M. Pasch and especially following D. Hilbert (Grundlagen der Geometrie ,1899), an attempt was made to establish elementary geometry on a solid basis, no longer implicitly referring to the details of certain particular figures.
   Here we present the contribution to this debate of a relatively modest professor of preparatory classes for the French "Grandes Écoles ", apparently largely forgotten in our time: Louis Gérard, professor at Lyon and then at Paris, specialist in non-Euclidean geometry (Thesis, 1892) and very careful to present the theorems of elementary geometry in the most rigorous possible manner. Editor of the Bulletin de Sciences mathématiques et physiques , he published in it numerous notes on this subject, and is the author (with one exception) of each of the passages cited below. In particular, he was the indefatigable promoter of the orientation of the angle between two lines, a notion which permits us to give numerous geometrical propositions absolute generality, independent of the details of the illustrating figure.

Over the centuries, the work of Euclid has served, with greater or lesser fidelity to the initial text, as a model for the teaching of elementary geometry. Each textbook author has felt himself bound to respect the overall division into Books, and within each book the disposition and formulation of the different results or theorems, though a certain number of more or less original commentaries were appended as needed. In particular, relatively rarely did an author add complementary figures covering specific cases overlooked or ignored by Euclid.

One must never cease to emphasize that the principal defect of the old proofs is that they leave aside the relations of the situations, and that these relations are often more difficult to prove than the theorem which one has in view, to such an extent that it is simpler to find a new proof than to correct the old one. (B.M.E., 6, 1901, p. 132).

To establish the properties of a parallelogram, we assume implicitly that two opposing vertices are situated on opposite sides of the diagonal which joins the other two vertices; the students would, I believe, be quite embarassed if we asked them to give a precise proof. (B.M.E., 6, 1901, p. 132)

Thus, in Proposition 20 of Book III, Euclid proves that the magnitude of a central angle is twice that of the corresponding inscribed angle (for any acute inscribed angle), considering two cases:

Certain commentators (from Clavius in the 16th century until the end of the 19th century) add a third case of the figure (point D is coincident with point B), but no word is said of what might happen if the inscribed angle is obtuse. In other words, while in the Elements an angle can equally well be either acute or obtuse, when we arrive at Proposition III, 20, a totally implicit restriction is imposed, without apparently disturbing a soul over the course of centuries.

Passing from thence to the theorem of the inscribed quadrilateral (conditions under which four points may be on a single circle), the authors content themselves in calling on the notion of the arc of constant angle, or the equality of two angles subtending the same arc. It is remarked of course, in Euclid (III, 22), that the opposite angles of an inscribed quadrilateral are supplementary (not equal!), but on each occasion that the property of an inscribed quadrilateral intervenes in the proof of a new result, the proof is reasoned on the case of the figure presented, without concern for that which would need to be modified in another case, or even without concern as to whether the case in question exists at all.

In elementary geometry, one normally considers only the absolute value of angles, without taking account of their direction, and without distinguishing the original side from the terminal side. This results in a multitude of complications. For example, if A, B, C, and D are four points on a circle, the angles ACB and ADB are supplementary or equal depending on whether the points C and D are on opposite sides of AB or not. By the same token, two angles of which the sides are parallel or perpendicular are sometimes equal, sometimes perpendicular. Therefore, in the questions in which angles of this nature intervene, one should review all of the possible cases; almost always, these are ignored, and people content themselves with examining the case of the figure in front of them. (B.M.E., 1, 1895, p. 1).

The consideration of the ray, or, which comes to the same thing, of direction on a line, is necessary only if we need to take account of the segments on that line; in all other cases, it creates an embarassment, which disappears if one attributes a new quality, the direction, to the angle formed by two lines: that is, we distinguish one side of the angle from the other. And in this case we have only one line, instead of two, which forms a given angle with a given line; the distinction between symmetry with respect to a point and symmetry with respect to a line on a plane makes itself clear; the set of points from which one sees a line segment under a given angle is no longer composed of two arcs of circumference, but rather of a single entire circumference, a result more consistent with calculation, and as a result the narrow and outdated concept of a line segment capable of a given angle disappears. The authors [B. Niewenglowski and L. Gérard] give simultaneously the necessary and sufficient condition, independent of the particular form of the figure, for four points on a single circumference. (C. Michel, B.M.E., 5, 1900, p. 93).

It seems that it is in the Géométrie of J. Hadamard (1897) that the theorem on the inscribable quadrilateral is given in the now classic necessary and sufficient form:

If four points A, B, C, and D are on the a single circumference, the angles in the same direction formed by the lines AC and AD on the one hand, and BC and BD on the other, are equal (and reciprocally) (Hadamard, Géometrie plane, 1897, pp. 71-72).

This text lacks only the notation (AC, AD) (used by L. Gérard) and an enumeration of the properties of the oriented angle between two lines (Chasles relation, passage to the opposite, etc...) for this notion to be usable in the solution of problems in geometry.

The example of the proof of the so-called property "of Simson's line".

A property of elementary geometry resulting from the property of the inscribed quadrilateral figures as an example of application or as and exercise in every textbook or course in geometry of the 19th century (and also of the following century...). Here is its statement:

If one considers a point M on a circle circumscribed around a triangle ABC, the orthogonal projections of M on the three sides of the triangle are colinear (following a line known as Simson's Line).

This property apppears to be due not to Simson, but to Wallace; nonetheless, despite efforts to promote the appellation "Wallace's Line" (see for example Intermédiaires des Mathématiciens, 1894 and 1895), both older and modern manuals speak entirely of Simson's line. Numerous properties of elementary geometry using Simson's line have been discovered over the course of time, forming if you will a veritable chapter in the geometry of the triangle. No matter the provenance, this property was popularized at the beginning of the 19th century. Below follows a classic proof, which appears to orignate with Duhamel (around 1840):

The Simson's line theorem (classical proof (Duhamel?))

 

Angle 1: AQR Angle 2: AMR Angle 3: CQP Angle 4: CMP
Angle 5: ABC Angle 6:AMC Angle 7: RMP

Proof:

All of our books, all of our professors in their courses think themselves obliged to reproduce, as a model of analysis or of synthesis, the classic proof (owed, I believe, to Duhamel) of the theorem concerning Simson's Line; this proof is far from simple, and, in the form in which it is presented, it proves nothing. (B.M.E., 4, 1899, p. 273).

Such a reasoning is only valid for the case of the specific figure, and it is also possible that this case cannot appear, if the figure is improperly drawn. In any case, to be rigorous, one should begin by finding all of the possible figures, which would seem inconvenient in the present case. (B.M.E., 4, 1899, p. 293).

Many readers have written me that the classic proof is doubtless incomplete, but that one feels that in the other cases of the figure, one could create an analogous proof, and that should suffice. This is an error: if one wishes to render the classic proof rigorous, the difficulty consists precisely of finding all of the possible cases of the figure. (B.M.E., 4, 1899, p. 273).

If one does in fact attempt to determine which are the possible configurations (of which only one serves as a support to the classic proof), one is brought to the consideration of the following figures. For each of them, the proof must be modified (supplementary angles in the place of equal angles, coincident or opposed points, etc...)

Configurations de Simson

In addition, the sums of angles, considered in item 6 of the classic proof, must sometimes be replaced by differences, or in any case be submitted to the classical restriction: the sum of two angles cannot, if we wish still to speak of angles, be more than two straight lines (Euclid's definition of an angle). Equally, the differences must not lead to negative angles, which implies that in a figure where we can see that angle B is less than angle A (and in which we intend to consider the difference A-B), it is necessary to prove precisely that in this case of the figure, we do indeed have B<A. And this is applicable of course to every aspect of each case of the figure.

L. Gérard gives two mathematical proofs of the Simson property, based on the theorem of the inscribable quadrilateral formulated in terms of oriented angles (and therefore independent of the case of the figure considered). The first is calculated on the classic proof, the second is even more simple:

A - Adaptation of the "classical" proof: (cf. the preceding figure)

B - A more simple proof:

This proof would appear more complicated than the ordinary proof, where we measure the angles by the arcs of circles; it is, however, incomparably more simple, for to arrive at a measurement of the angles, one must first distinguish the different possible cases of the figure and, I repeat, even in holding to the case of the figure that is before one's eyes, one must prove that this case is possible: otherwise one may easily prove anything one wishes. It is in this manner that one can establish, by a well-known reasoning, that an acute angle is equal to an obtuse angle. (B.M.E., 4, 1899, p. 293).

Other mathematical proofs, that one may consider more modern, are possible, using for example the properties of similarity, but fundamentally it is always the theorem of the inscribed quadrilateral that is at the origin of the proof.

C - A proof based on similarities:

 

Let M be a point on a plane. There exists a unique transformation ƒ, preserving similarities (in French: "une similitude"), with center M which will map AB to AC. The image of a point N on AB through this transformation is the point N', the intersection of AC and a circle passing through A, M and N.
   Let H be the orthagonal projection of M on NN'; the triangle MNH remains similar to itself when N moves, so H describes a line (d) arrived at from AB by a certain similitude. Two points on this line are in fact known, points Q (when N is coincident with A) and R (when N' is coincident with A). The line (d) is therefore the line QR.
   If M is on the circle circumscribed about ABC, C is the image of B by ƒ and point H is coincident with P, the orthagonal projection of M onto BC.
   Points P, Q and R are therefore colinear.

Let us leave the final word once more to L. Gérard:

When we compare two proofs, one general and the other depending on a single case of the figure, we are inclined to find the second more clear than the first; but after some thought, we perceive that the expressions, the details that appeared obscure or bizarre in the general proof are necessary to embrace this or that case to which we had not at first paid attention. (B.M.E., 4, 1899, p. 273).

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