^^Book 1. Proposition 4.  1° criterio uguaglianza trilati SAS.

If two triangles have

• SS)  2 sides equal to 2 sides respectively
• A)    the angles contained by the equal straight lines equal,

then

they also have the base equal to the base, the triangle equals the triangle, and the remaining angles equal the remaining angles respectively, namely those opposite the equal sides.

Proposizione contestata dal punto di vista assiomatico

poiche' si ritiene che non sia dimostrabile in base ai soli assiomi. Vedi Elementi come testo assiomatico.

Rem terminologia

Euclide. Definizione8  angolo piano.

Euclide. Nozioni comuni.

Hardy commentary

Classificazione delle definizioni di angolo

1. Tlie angle is the difference of direction between two straight lines.

   (With this group may be compared Euclid's definition of an angle as an inclination.)

2. The angle is the quantity or amount (or the measure) of the rotation necessary to bring one of its sides from its own position to that of the other side without its moving out of the plane containing both.
3. The angle is the portion of a plane included between two straight lines in the plane which meet in a point (or two rays issuing from the point).

It is remarkable however that nearly all of the text-books which give definitions different from those in group 2 add to them something pointing to a connexion between an angle and rotation : a striking indication that the essential nature of an angle is closely connected with rotation, and that a good definition must take account of that connexion.

The definitions in the first group must be admitted to be tautologous, or circular, inasmuch as they really presuppose some conception of an angle.
Direction-(as between two given points) may no doubt be regarded as a primary notion; and it may be defined as "the immediate relation of two points which the ray enables us to realise" (Schotten). But "a direction is no intensive magnitude, and therefore two directions cannot have any quantitative difference" (Burklen). Nor is direction susceptible of differences such as those between qualities, e.g. colours. Direction is a singular entity: there cannot be different sorts or degrees of direction. If we speak of " a different direction," we use the word equivocally ; what we mean is simply "another" direction. The fact is that these definitions of an angle as a difference of direction unconsciously appeal to something outside the notion of direction altogether, to some conception equivalent to that of the angle itself.
cit.  Hardy 179

The phraseology of the propositions, e.g. 1. 4 and 1. 8, in which Euclid employs the "method of superposition", leaves no room for doubt that he regarded one figure as actually moved and placed upon the other. 

...

    seeing how much of the Elements depends on 1. 4, directly or indirectly, the method can hardly be regarded as being, in Euclid, of only subordinate importance; on the contrary, it is fundamental.

...

Killing (Einfuhrung in die Grundlagen der Geometrie, 11, pp. 4, 5) contrasts the attitude of the Greek geometers with that of the philosophers, who, he says, appear to have agreed in banishing motion from geometry altogether. In support of this he refers to the view frequently expressed by Aristotle that mathematics has to do with immovable objects (aKivrjTa), and that only where astronomy is admitted as part of mathematical science is motion mentioned as a subject for mathematics.

cit.  Hardy 225

Mr Bertrand Russell observes (Encyclopaedia Britannica, Suppl. Vol. 4, 1902, Art. " Geometry, non-Euclidean ") that

• the apparent use of motion here is deceptive; what in geometry is called a motion is merely the transference of our attention from one figure to another.

Actual superposition, which is nominally employed by Euclid, is not required; all that is required is the transference of our attention from the original figure to a new one defined by the position of some of its elements and by certain properties which it shares with the original figure.
cit.  Hardy 227

3 New systems of Congruence-Postulates.

(1) Pasch takes as primary the notion of

• congruence or equality between any figures which are made up of a finite number of points only.

The definitions of congruent segments and of congruent angles have to be deduced in the way shown on pp. 68—9 of the Article referred to, after which Eucl. 1. 4 follows immediately, and Eucl. 1. 26 (1) and 1. 8 by a method recalling that in Eucl. 1. 7, 8.
 

(2) Veronese takes as primary the notion of

• congruence between segments (rectilineal).

The transition to congruent angles, and thence to triangles is made by means of the following postulate:
"Let AB, AC and A'B', A'C' be two pairs of straight lines intersecting at A, A', and let there be determined upon them the congruent segments AB, A'B' and the congruent segments AC, A'C; then, if BC, B'C' are congruent, the two pairs of straight lines are congruent."
 

(3) Hilbert takes as primary the notion of

• congruence between both segments and angles.

cit.  Hardy 228

Proclus remarks that everything is divided by the same things as those by which it is bounded.

coincide ⇔ equal ?

Most editors seem to have failed to observe that at the very beginning of the proof a much more serious assumption is made without any explanation whatever, namely that, if A be placed on D, and AB on DE, the point B will coincide with E, because AB is equal to DE. That is, the converse of Common Notion 4 is assumed for straight lines. Proclus merely observes, with regard to the converse of this Common Notion, that it is only true in the case of things '' of the same form " (ofxoei&rj), which he explains as meaning straight lines, arcs of one and the same circle, and angles " contained by lines similar and similarly situated" (p. 241, 3—8).
cit.  Hardy 250