- Igal
- Bouillet
- Guthrie
- MacKenna
Igal
13 Pues bien, se ha dicho acertadamente que la cuantidad continua se distingue de la discreta porque en aquella el límite es común, y en ésta, propio. Una división ulterior dentro del número se basa en que unos son impares y otros pares. Las divisiones ulteriores de cada una de estas dos clases, si las hay, o hay que dejárselas ya a los que se ocupan del número o son divisiones aplicables a los números aritméticos, no ya a los que están en las cosas sensibles. Pero si la mente distingue entre los números y los números que están en las cosas sensibles, nada impide concebir en estos las mismas divisiones.
—Pero ¿cómo dividir la cuantidad continua, si la división en línea, superficie y sólido, atendiendo a que la primera se extiende en una dimensión, la segunda en dos y el tercero en tres, no parecería una división de un género en sus especies, sino un recuento de dimensiones?. Porque como en los números considerados de ese modo, bajo el aspecto de anterioridad y posterioridad, no hay un género que les sea común, tampoco habrá un género que sea común a la primera, segunda y tercera dimensión.
—Pero tal vez bajo el aspecto de cuantidad la misma nota está en todos por igual, y no son unos más cuantos y otros menos, aun cuando unos se extiendan en más dimensiones y otros en menos. Pues también en los números, en cuanto todos son números, hay una nota común a todos. Porque tal vez no sea verdad que el uno genere el dos, y el dos el tres, sino que todos los genera el principio. Pero si no son generados, sino que son, sólo que nosostros los pensamos como generados, demos que el menor sea anterior y el mayor posterior, pero en cuanto todos son números, caen bajo un mismo género. Pues bien, también a las magnitudes hay que transferir lo que se aplica a los números: las divisiones en línea, superficie y sólido, que Aristóteles llama «cuerpo», basándonos en que, aunque magnitudes, difieren específicamente.
Si hay que dividir cada una de estas magnitudes —la línea en recta, redonda y helicoidal, la superficie en polígono y círculo y el sólido en figuras sólidas: esfera y poliedros—, y si hay que subdividir éstas, por ejemplo los polígonos en triángulos y cuadriláteros, como hacen los geómetras, y volver a subdividir éstos, es un punto que debe ser examinado.
Bouillet
XIII. Nous avons déjà expliqué que la quantité discrète est bien distinguée de la quantité continue par sa définition propre et par la définition commune [de la quantité] (53). Noue ajouterons que les nombres sont distingués les uns des autres par le pair et l’impair. S’il y a en outre quelques différences parmi les nombres pairs ou les impairs, il faut rapporter ces différences aux objets dans lesquels se trouvent les nombres, ou bien aux nombres qui sont composés d’unités (54) et non plus à ceux qui sont dans les choses sensibles. Si la raison sépare des choses sensibles les nombres qui sont en elles, rien n’empêche alors d’attribuer à ces nombres les mêmes différences [qu’aux nombres composés d’unités].
Quant à la quantité continue, quelles distinctions admet-elle? Il y a la ligne, la surface, le solide : car on peut distinguer l’étendue à une dimension, l’étendue à deux dimensions, l’étendue à trois dimensions [et compter ainsi les éléments numériques de la grandeur continue] au lieu d’établir des espèces (55). — Dans les nombres considérés ainsi comme antérieurs ou postérieurs les uns aux autres, on ne trouve rien de commun qui constitue un genre. De même dans la première, la seconde et la troisième augmentation [dans la ligne, la surface et le solide], il n’y a rien de commun; mais en tant qu’on y trouve la quantité, on y trouve aussi l’égalité [et l’inégalité] : quoiqu’il n’y ait pas une étendue qui soit un quantitatif plus qu’une autre (56), cependant l’une a des dimensions plus grandes que l’autre. C’est donc seulement en tant qu’ils sont tous nombres que les nombres peuvent avoir quelque chose de commun. Peut-être en effet n’est-ce pas la monade qui engendre la dyade, ni la dyade qui engendre la triade, mais est-ce le même principe qui engendre tous les nombres. Si les nombres ne sont pas engendrés, mais existent par eux-mêmes, nous les concevons du moins dans notre pensée comme engendrés : nous nous représentons le nombre moindre comme antérieur, le plus fort comme postérieur. Mais les nombres, en tant que nombres, se ramènent tous à l’unité. — On peut appliquer aux grandeurs le mode de division adopté pour les nombres et distinguer ainsi la ligne, la surface et le solide ou corps, parce que ce sont là des grandeurs qui forment des espèces différentes. Si l’on veut diviser aussi chacune de ces espèces, on divisera les lignes en droites, courbes et spirales ; les surfaces, en planes et curvilignes ; les solides en corps ronds et polyèdres ; on considérera ensuite dans ces figures le triangle, le quadrilatère, etc., comme font les géomètres.
Guthrie
DISCRETE QUANTITY QUITE DISTINCT FROM CONTINUOUS QUANTITY.
13. We have already explained that discrete quantity is clearly distinguished from continuous quantity, both by its own definition, and the general definition (for quantity). We may add that numbers are distinguished from each other by being even and odd. If besides there be other differences amidst the even and odd numbers, these differences will have to be referred to the objects in which are the numbers, or to the numbers composed of unities, and not any more to those which exist in sense-beings. If reason separate sense-things from the numbers they contain, nothing hinders us then from attributing to these numbers the same differences (as to the numbers composed of unities).
ELEMENTS OF CONTINUOUS QUANTITY.
What distinctions are admitted by continuous quantity? There is the line, the surface, and the solid; for extension may exist in one, two or three dimensions (and thus count the numerical elements of continuous size) instead of establishing species. In numbers thus considered as anterior or posterior to each other, there is nothing in common, which would constitute a genus. Likewise in the first, second and third increases (of a line, surface, and solid) there is nothing in common; but as far as quantity is found, there is also equality (and inequality), although there be no extension which is quantitative more than any other. However, one may have dimensions greater than another. It is therefore only in so far as they are all numbers, that numbers can have anything in common. Perhaps, indeed, it is not the monad that begets the pair, nor the pair that begets the triad, but it may be the same principle which begets all the numbers. If numbers be not derivative, but exist by themselves, we may, at least within our own thought, consider them as begotten (or, derivative). We conceive of the smaller number as the anterior, the greater as posterior. But numbers, as such, may all be reduced to unity.
STUDY OF GEOMETRICAL FIGURES.
The method of classification adopted for numbers may be applied to sizes, and thus distinguish the line, the surface, and the solid or body, because those are sizes which form different species. If besides each of these species were to be divided, lines might be subdivided into straight, curved and spiral; surfaces into straight and curved; solids into round or polyhedral bodies. Further, as geometers do, may come the triangle, the quadrilateral, and others.
MacKenna
13. It has been remarked that the continuous is effectually distinguished from the discrete by their possessing the one a common, the other a separate, limit.
The same principle gives rise to the numerical distinction between odd and even; and it holds good that if there are differentiae found in both contraries, they are either to be abandoned to the objects numbered, or else to be considered as differentiae of the abstract numbers, and not of the numbers manifested in the sensible objects. If the numbers are logically separable from the objects, that is no reason why we should not think of them as sharing the same differentiae.
But how are we to differentiate the continuous, comprising as it does line, surface and solid? The line may be rated as of one dimension, the surface as of two dimensions, the solid as of three, if we are only making a calculation and do not suppose that we are dividing the continuous into its species; for it is an invariable rule that numbers, thus grouped as prior and posterior, cannot be brought into a common genus; there is no common basis in first, second and third dimensions. Yet there is a sense in which they would appear to be equal – namely, as pure measures of Quantity: of higher and lower dimensions, they are not however more or less quantitative.
Numbers have similarly a common property in their being numbers all; and the truth may well be, not that One creates two, and two creates three, but that all have a common source.
Suppose, however, that they are not derived from any source whatever, but merely exist; we at any rate conceive them as being derived, and so may be assumed to regard the smaller as taking priority over the greater: yet, even so, by the mere fact of their being numbers they are reducible to a single type.
What applies to numbers is equally true of magnitudes; though here we have to distinguish between line, surface and solid – the last also referred to as “body” – in the ground that, while all are magnitudes, they differ specifically.
It remains to enquire whether these species are themselves to be divided: the line into straight, circular, spiral; the surface into rectilinear and circular figures; the solid into the various solid figures – sphere and polyhedra: whether these last should be subdivided, as by the geometers, into those contained by triangular and quadrilateral planes: and whether a further division of the latter should be performed.