Horst Steinke

62

themes, […] of contemplation

*sub specie aeternitatis,*

and

*amor Dei intellectualis*

)».

(Id.,

*Vico e la tradizione “platonica”. “La filosofia dell’umanità e la storia universale delle*

*nazioni”*

, in «BCSV», XXII-XXIII, 1992-1993, pp. 65-102, p. 102).

105

G. Vico,

*Liber metaphysicus*

, Chapter VII, § IV, according to

*On the Most*

*Ancient Wisdom of the Italians*

, trans. by J. Taylor, with an introduction by R.

Miner, New Haven-London, Yale University Press, 2010, p. 121.

106

*Ibid.*

, chapter III, p. 53. The reference is to Section IV of

*De ratione*

,

found in

*On the Study Methods of Our Time*

, cit., p. 23, which reads: «We are able

to demonstrate geometrical propositions because we create them; were it pos-

sible for us to supply demonstrations of propositions of physics, we would be

capable of creating them

*ex nihilo*

as well».

107

*Liber metaphysicus*

, Chapter I, § I,

*On the Most Ancient Wisdom of the Ital-*

*ians*

, cit., p. 25. The originality of defining the

*point*

as an entitity that is “sim-

ple” in that it has no parts can be seen by remembering that it was not con-

sidered the only definition possible. Vico himself referred to the competing

view that «even the smallest particles […] are infinitely divisible”, and of «a

geometry which defines the point as a minimal particle divided endlessly»

(

*ibid.*

, pp. 61, 63).

108

An important means of devising such constructions are so-called “aux-

iliary constructions”, such as bisecting an angle, bisecting a line, drawing a line

at a right angle from a given point, and drawing a straight line perpendicular

to another line, as outlined in Book I, Propositions 9-12, of Euclid’s

*Elements*

.

See Euclid,

*The Thirteen Books of Euclid’s Elements*

, trans. by T. L. Heath, Cam-

bridge, Cambridge University Press, 1908, vol. 1, pp. 264-275. A specific ex-

ample of an important auxiliary construction, to prove the Triangle Sum Con-

jecture, can be found in M. Serra,

*Discovering Geometry: An Investigative Approach*

,

Emeryville, Key Curriculum Press, 3

rd

edition, 2003, p. 200. On the crucial

heuristic role of auxiliary constructions, see also J. Hintikka - U. Remes,

*The*

*Method of Analysis: Its Geometrical Origin and Its General Significance*

, Dordrecht-

Boston, D. Reidel, 1974, pp. 41-48.

109

§ 349 is discussed in some detail by Remaud (

*Vico lector de Espinosa*

, cit.,

pp. 198-201), and Otto (

*“Contextualidad” científica y “convertibilidad” filosófica*

, cit.,

p. 173), but without drawing a distinction between geometrical method and

demonstration.

110

See T. Crilly with D. Johnson,

*The Emergence of Topological Dimension The-*

*ory*

, in

*History of Topology*

, ed. by I. M. James, Amsterdam, Elsevier, 1999, pp. 1-

24, pp. 1, 20-22. Actually, there are several “dimension theories” which under-

lines the “creative” rather than “constructive” nature of the process of

*demon-*

*stration*

. Each “dimension theory” has been “composed”, to use a Vichian

term, in a different way, according to its own logic.