Development of the Drawing: from the Magic of Geometry to
the Use of Several Logics in Technical Design
Antonio Donnarumma
Dipartimento di Ingegneria Meccanica, Università degli Studi di Salerno
Via Ponte Don Melillo – 84084 Fisciano (SA) – Italy
e-mail: donnaru@diima.unisa.it
1.
INTRODUCTION
It isn’t a fortuitous event that a lot of great mathematicians of our time had geometry
as their central interest, above all projective geometry. We can mention, among them,
Federigo Enriques, Francesco Severi, Cremona, Fano, Burali Forti, Castelnuovo, Guido
Zappa. And it isn’t a fortuitous event that Russell had focused the second volume of his
mathematical trilogy on “Foundation of Geometry” which is, in substance, projective
geometry; a great part of “Principia” discusses projective and descriptive geometry too.
Well-deserving work of Zanichelli publisher was anastatic reprint of “Lezioni di
geometria proiettiva” by Enriques, and, by the same Author, the volume “Per la storia della
logica”. Philosophic content of projective geometry, a “not metric” geometry, whose features
do not derive from any measure, is huge and in the meantime fascinating.
As regards our concern, that is “descriptive geometry”, I quote Enriques’ words:
“Abbracciando insieme i vari problemi tecnici di una teoria scientifica, Monge creò la
geometria descrittiva (1795) nella quale seppe fondere armonicamente vari indirizzi della
matematica pura ed applicata (sublime caratteristica di un uomo di genio!) E quanto anche
nella teoria egli si sia levato in alto, viene attestato dal fatto che egli poté fare dell’Algebra
con la Geometria come Cartesio aveva fatto della Geometria con l’Algebra1”.
Projective geometry has come into being, so to say officially, in 1822, and thus it is
subsequent to descriptive geometry; but these are just conventional dates.
We could rather say that technical drawing, as not connected to reality, used, and to
date still uses, a not metric geometry, combined with metric information, such as dimension,
and then with tolerances and other indications.
Because of the coming of calculators and several modelling techniques, geometry was
done using algebra and analysis.
After all nowadays tolerancing problem is a problem concerning analytic geometry.
An ideal surface, when it is possible, can be defined only by equations that represent it.
But we can’t strip substance of appearance. ; projective geometry with its axioms,
group property, cross ratios, conics, seems to be still at the base of the world of
1
Taking in the many technical problems of a scientific theory, Monge founded descriptive
geometry (1795) in which he combined harmonically the different directions of pure and applied Mathematics
(sublime quality of a man of genius!). And how high he was as a theorizer, is testified by the fact that he was
able to do Algebra with Geometry like Cartesio did Geometry with Algebra.
representation, and it is still liable of useful examinations in several fields of knowledge, such
as for example, besides Logic, image processing and so on.
Nowadays, the drawing has the larger sense of design; in this case too many themes
related to design, such as for example probability as belief, bi- and multivalued logic, and
others, show a deep logic-philosophic content. So we can deduce that the study of our roots,
besides enriching our culture, can make us discovering again (and using) potentialities still
alive, and can perhaps persuade us that modern methods are related to our past in a manner
narrower and more “timeless” than you could think.
2.
2. 1
FOUNDATION OF THECNICAL DRAWING
Foundation of Modern Geometry
An analysis on “Foundation of Technical Drawing” have necessarily to be based on
Descriptive Geometry, Projective Geometry and on metric geometry, that are narrowly related
to each other. The attribute “technical”, which accompanies the noun, saves us from starting
from a distance. Our references are above all Bertrand Russell (logicist) and Federigo
Enriquez, but we have always considered David Hilbert’s works, particularly “Grundlagen
der Geometrie”. So we can come to the point at once.
Russell states in “Principia mathematica”, that foundations of geometry have been
subjected, at the end of nineteenth century, to a threefold scrutiny:
1. The work of not Euclideans, which showed that various axioms, long known to be
sufficient for certain results, were also necessary, i. e. that results inconsistent with the
usual results but consistent with each other, followed from the denial of those axioms. For
example: possibility of drawing countless parallels or none to a given straight line, not
congruent with the Euclid’s fifth postulate, is on the contrary congruent with hyperbolic or
elliptical geometry;
2. Revision of analytical geometry due to Dedekind’s and Cantor’s results on the nature of
the continuity, destroying the prejudice of an “atomic” geometry and making it possible to
treat problems connected with quantities incommensurable among them, first of all those
connected with Pythagorean theorem;
3. Lastly, a great change introduced by the Italian work, which made Russell state:
“In Geometry, von Staudt’s quadrilateral construction and Pieri’s work on Projective
Geometry have shown how to give points, lines, and planes an order independent of
metrical considerations and of quantity while descriptive Geometry proves that a very
large part of Geometry demands only the possibility of serial arrangement. Moreover the
whole philosophy of space and time depends upon the view we take of order. Thus a
discussion of order, which is lacking in the current philosophies, has become essential to
any understanding of the foundations of mathematics”.
3.
RELATIONS
We cannot examine the three conditions presented above without introducing some
basic concepts.
Such concepts are seemingly simple; but, indeed, they involve a lot of philosophical,
logical as well as mathematical problems.
After these preliminary remarks, we report some considerations on relations.
Russell describes a relation between two terms as a “a concept which occurs in a
proposition in which there are two terms not occurring as concept, and in which the
interchange of two terms gives a different propositions”. This definition would be completed,
in our opinion, by the adverb “usually”, if we are going to introduce among the relations also
those between a term and itself, that is identity, on which Russell focuses his attention later.
The mark of diversity occurring between propositions due to terms exchange is
imposed by necessity of avoiding relations of the type “a and b are two ”, equals to the
relation “b and a are two ”. A proposition of the type just mentioned prevents from
determining the direction of the relation, which is of great importance in order to introduce
the concept of order.
Thus, marking with a R b the proposition “a is related to b”, we can say that the
former relation is a relation which is directed from a to b. It involves and is involved by the
relation a R-1 b, which goes from b to a; R-1 marks the opposite relation. The first term of the
direct relation is called by Russell referent, the second relatum.
The concept of direction is assumed as primitive.
3. 1.
Symmetry and transitivity
If R is the equal of R-1 then the relation is symmetric. In the example “Carlo is Luigi’s
friend”, the relation R is expressed by the predicate “is […] friend”; the relation R-1 is the
equal of R if also the relation “Luigi is Carlo’s friend” is true. The value of truth of the two
propositions is not deduced at once by the context (we should have information about the
relation between Carlo and Luigi!); it’s of different kind the mathematical relation expressed
for example from a<b, which has as its opposite the relation b> a. The two relations can be
read: “a goes b” and “b follows a”.
The transitive property is evident if a R b and b R c, then a R c.
A relation is asymmetric if a R b, but not b R-1 a. In this case it exists only one
direction, as it is in the case of Descriptive Geometry, in which the direction is determined by
the radium going from the centre of projection (proper or improper) to drawing’s plane; but
the image point is not in univocal relation with the projected point.
3. 2.
Relation of order
We have clarified the meaning of concepts “relation” and “order” and therefore of
“relation of order”. Russell talks at length and with rather complex words up on the term
“series” present in the propositions mentioned. In a naïve and very “restrictive” way, we
could give to the term “series” the meaning of progression (or succession,or ordered set)
produced by an asymmetric and transitive relation, according to which, given only two terms
a and b, you can say if a third term is between them or not. The mathematical series are
always open; the “closed series” philosophical meaning and are, simplifying, progressions
with the first term arbitrary. This justifies serial order of the descriptive straight line, in this
different from the projective straight line, which needs of four points in order to define its
arrangement.
Both Enriques and Russell consider the relations of order at the roots of coming to
axioms of Projective Geometry.
As a matter of fact, if the straight line can be understood as caused by the movement
of one of its points in the two directions, it can be conceived of as a closed line caused by the
movement of the point A of the straight line a, going back to A on the other side, passing
trough the point to infinity. As regards intuition, we can conceive the straight line a as the
limit of a circumference, tangent in A to a and with a radius growing to infinity.
The correspondence becomes perfect if you think of the movement of a radius r
(understood as a right line and not semidiameter) of a circumference around its centre, and of
that of the point A intersection of the same radius r with a: the position of r parallel to a
defines the improper point of a; but what seems to be more interesting, at least for those
people who are not very familiar with geometry, is the concept of the points of the straight
line on a circular (natural) arrangement, which has two directions.
From this new idea we can have an extension of the concept of the segment and of
complementary of the segment itself. Be for example the segment A B (of finished length); its
complementary is the segment A B of infinite length. To eliminate every ambiguity it is
therefore necessary to introduce a point (for example C) internal to AB, and a point D
external. To have a better understanding of the process think, in coherence with what
mentioned above, of a circumference (even if of infinite radius). Then the segment ACB
corresponds to the arch (finished) AB, while the complementary (considering the whole
straight line ADB corresponds to the infinitive arch A D B, covered by a point proceeding in
direction opposite ACB.
4.
4. 1
REFORM OF CONTEMPORARY LOGIC
Revolt against Idealism
As regards the point of view of the logic, Enriques argues that we have to consider and
compare some movement of thought which, although of different origins, interact and meet
with each other in the same reforming concept.
The date of birth of projective geometry is considered to be 1822, year during the
which appears the first systematic work on this subject: Poncelet’s “Traité des proprietés
projectives des figures”; however most concepts reported in “Traité” were well-known or had
been discussed for a long period.
We can obviously refer to Euclid and to Eleatic School, but to not stray from the point
we can start from XVII century with Desargues and Pascal, going through Newton thus
reaching Poncelet. However Monge had already criticised Poncelet’s principle of continuity.
But, in our opinion, attention has to be focused on the principle of duality; it reaches a very
high degree in the considerations above transformations and correspondences (Moebius) and
above the use of the coordinates introduced by Pluecker in order to represent arbitrary
geometric elements.
Passing over issues related to Euclid’s fifth postulate (Lobacewski) and to
multidimensional spaces (Riemann, Helmotz), it seems to be of great interest, from a
philosophical point of view, the acknowledgement of a geometrical possibility which
coincides with our perception of space. This deals metaphysical rationalism of the XVIII
century a blow. Maybe we can say that idealism of every kinds is negated, from Platonic to
Gentile’s idealism passing by Kant.
Reality cannot be determined a priori, as choosing among possible geometries requires
simply an experimental verification.
5.
5. 1.
PROJECTIVE GEOMETRY ACCORDING TO ENRIQUES
Intuition as basis of geometry
It is remarkable for the coherence with the Author’s logical philosophical ideas the
incipit of Enriques’ renowned lectures on Projective Geometry:
“Dall’ordine delle cose esterne, nella rappresentazione data alla mente dai sensi,
scaturisce il concetto di spazio. La geometria studia questo concetto già formato nella mente
del geometra, senza porsi il problema (psicologico ma non matematico) della sua genesi.
Sono dunque oggetto di studio, nella geometria, i rapporti intercedenti fra gli elementi (punti,
linee, superficie, rette, piani, etc. ) che costituiscono il concetto complesso di spazio: a tali
rapporti si dà il nome di proprietà spaziali e geometriche2”.
Enrique considers intuition as basis of Geometry, although he is a purist rather than an
intuitionist like Heyting.
Such an act involves in depth the definition of knowledge origin: intuitionism,
positivism, a priori and a posteriori knowledge, and so on.
So disputes between Enriques on one side and Croce and Gentile, supporters of
Idealism on the other side, are not fortuitous.
Further on Enriques is more specific about his opinion: “la scelta degli elementi
fondamentali, he notes, non è a priori; gli elementi fondamentali sono scelti in quanto i più
semplici rispetto alla nostra intuizione psicologica, cioè sono quelli la cui nozione è formata
nella nostra mente come contenuto del concetto di spazio. Tali sono, per esempio il punto, la
retta, lo spazio. 3”
From the paragraph above we infer that:
1. it does not exist an “a priori world” (transcendent? Platonic?) distinct from the present
one by which we are swallowed;
2. it is not excluded the existence of other elements, beyond the point, the straight line,
the space.
In fact, the expression “per esempio” introduced in the former proposition, which is
also written in the short form “p. e. ”, seems to be indicative of a certain caution, a sort of fear
in hypothesizing primitives elements different from the point, the straight line, and the space.
I quote, f. i., the pointless Geometry of Whitehead and, nowadays,of Gerla.
The analysis of relations among the elements or among the entities through these
defined (for example, the straight line as intersection of planes), takes place in two ways:
 through (psychological) intuition
 through (logical) deduction of new properties from those that have been given.
2
From the order of the extreme things, in the representation given to the mind from sense, it
comes out the concept of space. Geometry studies this concept already formed in geometer’s mind, without
considering the problem (psychological but not mathematical) of its genesis. Therefore, in geometry subject of
study are relations which occur among elements (points, lines, surface, straight lines, planes and others)
constituting the complex concept of space: such relations are called special and geometrical properties.
3
The chose of fundamental elements is not an a priori chose; fundamental elements are
selected as they are the simplest in our psychological perception, are, namely, those whose notion is formed in
our mind as implied in the concept of space. Such are, for example the point, the straight line, the space.
We have in first case the “Postulates”, in second case the “Theorems”.
We would not report the above very famous propositions, if it were not important in
our opinion to highlight once again the fact that, according to Enriques, postulates have to be
chosen among those of greater intuitive evidence but they are not determined a priori.
Consequently postulates are independent among them and therefore every “right” reasoning
requires the least possible number of postulates.
We already feel a trend towards simplification which is a constant of different
domains of sciences. We would like, besides, to note semantic difference between
“postulates”, understood as evident and indemonstrable truths, and “axioms” which have a
meaning in a modern manner “hypothetical”.
In fact, a correct proposition requires not so much the truth of propositions, often
unreachable, as the coherence of preliminary remarks with deduction.
From mathematical point of view the main distinction is that one existing among
 Graphic Properties (eg. more straight lines pass through a point, more planes through
a line or through a point, and so on)
 Metrical properties (concerning the concept of distance (or length of a segment), of
angular quantity, and so on.
The two categories of properties derive from graphical and metrical intuition, in turn
linked, for reasons deduced from physiologic psychology, respectively with visual sensations
(and this is quite clear) and with the tactile sensations and of movement, because of the
necessity of putting a rigid tool (for example a rule divided into millimeters) on top of the
segment to be measured. The two sensations are then combined for association.
In fact the men can evaluate, even if in an approximately way, the shape and size of a
figure.
The concept of geometry linked in this way with psychology and human physiology is
typical of Enriques. It provokes admiration and perplexity and it let us venture to wonder:
“would there be a geometry in a world of blind and maimed persons?” Most of us would
answer I think so. Then we should make an hypothesis about a transcendent world or in any
case different from that sensible, and accept Idealism, or at least some kind of it, which is
perfectly legitimate, but in contrast with that one who is not idealist.
Let the question be without answer.
However we have to mention the contribution given to debate by psychologists and
famous pedagogists. One name is sufficient: Piaget.
Let’s go back to Projective Geometry; it considers only the properties that keep
themselves unaltered during operations of projection and section, and that for this reason are
defined projective. They are only graphic but there are relations also with metric geometry.
Perhaps metric-projective geometry is more useful on a strictly practical level; the pure
projective geometry saves however its great fascination.
6.
6. 1.
CHARACTER OF PROJECTIVE GEOMETRY
Fundamentals of main propositions
Intuition, to which the author always refers, justifies the following propositions valid
for proper and improper forms, and written in short as we eliminate the locution “which it
belongs to” or “ which they belong to”:
1. Two points determine a straight line
2. Two planes determine a straight line
3. Three points not belonging to a straight line determine a plane
1.
2.
3.
Three planes not belonging to straight line determine point
A point and a straight line do not belonging to each other determine a plane
A plane a straight line that do not belong to each other determine a point.
6. 2.
Law of Duality: movement
From above it comes out the most important feature of projective geometry: principle
of duality. In order to point out other features we remember shortly that we call forms of first
kind the dotted line, a bundle of straight lines and of a bundle of planes; forms of second kind
the dotted plane, the lined plane, the star of straight lines and the star of planes; forms of third
kind the dotted space and the space of planes. It’s of great importance the concept of
movement that Enriques considers the base of the generation of the three forms: those
belonging to the first kind are generated by the movement of an element of theirs
(respectively: point, straight line (rotatory), plane (rotatory); those belonging to the second
kind have in themselves forms of the first kind, by the movement of which those belonging to
the second kind are generated; finally forms of the third kind are generated by the movement
of the forms belonging to the second kind. To identify the three forms it’s enough to have
respectively one, two, three coordinates.
The fundamental forms mentioned before are enriched by improper forms (improper
dotted straight line, improper bundle of planes, improper dotted and lined, improper bundle of
straight lines, improper star of straight lines and planes).
This “dynamic” vision of the generation of the forms of first, second, third categories
looks congruent with the concept of “series” (according to Russell. ) or of succession
(according to Enriques): in fact n points P1, P2, . . ., Pn on a segment can be understood as the n
positions reached by the point P generator in instants 1, 2, …, n.
The movement appears also in definition of equality between two plane figures, which
are axiomatically equal if superimposable.
In metric geometry the movement take a sensible aspect and determining for the
measure of figure.
6. 3.
Postulates of projective Geometry. Generation of forms. Belonging.
Associating the propositions 1 and 2, 3 and 4, 5 and 6 in the paragraph 6. 1,
respectively in the propositions I, II, III below, we obtain the postulates of projective
geometry
I. In a form of third kind two essential elements determine a form of the first kind,
(contained in the one belonging to the third kind) to which they belong.
II.
In a form of third kind three essential elements not belonging to a form of the
first kind determine a form of second kind (contained in the one belonging to the third
kind) to which they belong.
III.
In a form of third kind an essential element and a form of the first kind which
do not belong to each other determine a form of the second kind to which they belong.
As homage to the principles of movement (in a graphical sense) of order and series (in
Russell’s sense) we can write the following postulates.
IV.
The elements of a form of the first kind can be thought in a natural circular
arrangement which has two directions, the one opposite the other.
V.
If two forms of the first kind are perspective and one element moves on the
other and describes a segment, also the corresponding element moves on the other
tracing a segment.
These postulates are enough in order to develop further the projective geometry using
exclusively Logic, with renunciation of the intuition.
Enriques, called by Carruccio “punto di arrivo della sistemazione assiomatica della
Geometria proiettiva4” is able to state the subsequent law of duality (through space): “To each
theorem deduced from propositions I, II, III, IV, V, corresponds a correlative theorem (or
dual) which is enounced substituting to the word < point > of the first proposition the word
<plane>, and reciprocally to the word <plane> the word <point> and remaining unchanged
the word <straight line>. ”
To the five postulates we can add the postulate relative to continuity:
VI.
If an orderly segment AB of a form belonging to the first kind is divided so
that: “every element of the segment AB belongs to one of the two parts; the extreme A
belongs to the first part and B to the second; any element of the first part always
precedes the second, then it exists an element C (which can belong to the first or to the
second part), such that every element from AB going before C belongs to the first part
and every element following C belong to the second one in the determined succession.
”
7.
EUCLIDEAN AND NOT EUCLIDEAN SPACES
Russell states that only in metric properties, save the exception relative to the axiom of
the straight line, Euclidean spaces differ from those one not Euclidean. Properties of
Projective Geometry, taken without using imaginary, are properties common to all spaces.
Russell defines projective geometry as completely a priori, and therefore subjective. As
concerns this aspect, the “subjectivism”, it seems to be congruent with “psychological”
definition of Enriques. The absolute independence of projective geometry from measure
makes projective coordinates distinct from metric ones. They are, in conclusion, just numbers
arbitrarily arranged, but systematically, in order to distinguish different points, in the same
way as street numbering which is apt to distinguish different buildings. Every metric, when it
interferes, is just the quantitive superstructure of the qualitative substratum; it happens, for
example, in treatment of cross ratios, which requires, without doubt, measures of segments
and angles, but do not alter the qualitative aspect of the cross ratio.
7. 1
Similarity
As seen above, every geometric reasoning is circular.
Therefore if two points determine a straight line, in turn two straight lines determine a
point and so on as regards different forms, now properties of generating a certain straight line
are characteristic of all couples of points which belong to that straight line, as well as
properties of determining a given point belong to all couples of straight lines of the bundle of
straight lines to which the point (that is the centre of the bundle) belongs. These elements
(points and straight lines generators and therefore also generated) are qualitatively similar.
The couples of points or generator straight lines are qualitatively equivalent.
The principle of duality is, according to Russell, the mathematical form of a
philosophical circle, being consequence of relativity of space. This principle makes all
definitions of a point contradictory.
4
Arrival point of the axiomatic systematisation of Projective Geometry
Fig. 1
8.
Fig. 2
ANHARMONIC RATIO AND QUADRILATERAL CONSTRUCTION
According to Russell essential entities of projective geometry are the disharmonic
ratio (or anharmonic or cross ratio) and the quadrilateral construction.
Fig. 3
In order to define properties of the cross ratio, it is enough to consider a group of four
straight lines having a common point: if we trace another couple of straight lines we have the
disharmonic ratio (a, b, c, d) = (a’, b’, c’, d’), and so on for other straight lines. Qualitative
property relative to the cross ratios derives from the comment just mentioned. We can have
the following qualitative definition (Fig. 1 and 2):
Two groups of four points each are in the same disharmonic ratio when:
1. Every group of four (points) lies on the same straight line;
2. Corresponding points of different groups lies two by two on four straight lines passing
trough a point or when the two groups are in this relation as regards a third group.
Reciprocally two groups of four straight lines are in anharmonic ratio when:
1. Every group of four (lines) pass through a single point;
2. Corresponding lines of corresponding groups pass two by two through four points on a
straight line, or when two groups are in this ratio as regards a third group.
Two groups of lines having the same cross ratio are projectively equivalent and
substitute the quantitative equivalence in metric geometry.
As regards the complete (that is comprehensive of the two diagonals) quadrilateral (or
quadrangle, Fig. 3), let’s remember that four points ABCD on a straight line constitute a
harmonic succession if it is (AB/BC) / (AD/DC) = -1, from which descends also that
AB/BC=AD/CD. On the other end, given three points on a straight line, for example ABD,
the fourth is found immediately by tracing a straight line for D and by arranging two points on
it (for example O,P). By tracing the joining straight lines from B and A to O and P we get the
quadrangle QRPO( T = intersection AO,DQ; R=intersection BP,DQ; C = int. OR,r). Trough
successive projective transformations we have the series or (succession).CBAD.
Finally ABCD can be transformed in CBAD only in a projective way(from
R,ABCDSPOD;fromA,SPODRQTD;from O, RQTDCBAD).The procedure is
indefinitely iterative. By combining A, B, C, D, respectively with the numbers 0, 1, 2 , the
differences AB, AC, AD are in harmonic progression. By taking BCD as a new triad
corresponding to ABD we find an harmonic point with B as to C, D, to which we assign the
number 3, and so on. In such a way we have certainty that there isn’t a point or a number
twice reiterated, as well as the bijective mapping between points and numbers (co-ordinates),
which represent them.
It’s immediate the extension of projective co-ordinates in the plane and in the space.
9.
AXIOMS OF PROJECTIVE GEOMETRY ACCORDING TO RUSSELL
Without lingering anymore on these topics, we report the three axioms of projective
geometry, as they were put by Russell, that is different from the classical ones exposed above:
1. Different parts of the space are qualitatively similar, and are distinguished just for the
immediate fact that they are positioned one outside the other.
2. The space is continuous and divisible to infinity; the extension 0 is called point.
3. Two any points determine an only figure called straight line; three points, as a general
rule, determine an only figure: the plane; four points determine a figure with three
dimensions, and, as it can seem different, the same can be say as regards any number
of points.
The Russel’s “unexpected” observation is that “the process just described, before or
later come to the end with a number of points determining the whole space;if it wasn’t in such
a manner, any number of relation of a point as regards a set of given points could not
determine the relation with new points, and geometry would become impossible”.
Axioms just mentioned are, so to say, deep in very interesting philosophical
considerations; we cannot however insist on it.
The position of Russell and Enriques seems more “revolutionary” than Hilbert’s one,
who is rather linked with the first even if he renewed classical geometry with an original
axiomatic formulation. His theory lies on very five groups of axioms, and precisely those of
“connection, order, congruency, of parallels, of continuity”.
10.
AXIOMS OF METRIC GEOMETRY
Axioms of projective geometry are, according to Russell, “a priori deductions” (it
seems to be a contradiction in terms), due to the opportunity of experimenting the
“externalism” that is a multiplicity of different but connected things.
However projective geometry does not succeed in discriminating between Euclidean
and not Euclidean spaces. In order to reach this aim a measurement is necessary. The three
axioms of projective geometry are:
1. Axiom of free movement, which allows the measurement of figures;
2. Axioms of dimensions;
3. Axiom of distance (corresponding to that of the straight line in projective geometry).
Measurement is necessary also for the so called physic geometry (for example graphic
statics).
11.
CONICS
Handling of conic, unified, acquires a character of great generality and elegance in
projective geometry. It’s interesting the way with which Enriques faces the problem of conics;
it would be better to say “the ways”, as those presented by Enriques are numerous and all
interesting
1. Definition based on polarity:
“The set of points and straight lines autoconigates is called fundamental conic of
polarity”
The conic, considered as the set of its point, is called “conic locus”.
As to the law of duality:
“The conic considered simply as the envelope of its tangents is called “conic
envelope”
From a historical point of view, it is Poncelet that in 1824 enunciated the law of
duality in the plane as consequence of the theory of polars of conics.
2. An intuitive idea, corresponding to the psychological vision of Enriquez, is provided
by the following situation:
Let us imagine that our eyes follow the genesis of a conic, starting from a point A of it.
The point of the straight line correspond to the straight lines for A; to the moving of a
straight line for A, such as it describes the bundle A starting from the position of tangent,
corresponds the moving of a point, which starting from A describes all the line going back to
A. The conic therefore seems to be a closed line, as regards to which the separation among
regions of external and internal points has the usual intuitive meaning, as the varying tangent
always leaves a conic from a band, without invade internal points.
It is important to note the coherence of this concept with that of the straight line
understood as a closed line, in our opinion, in the mean time psychological and hypothetical
deductive.
It exists however a discrepancy between this “projective” and the metric one, as in the
last named geometry “open” conics remain open.
3. Definition based on focal properties.
It is to be noted beyond the general elegant handling of the three conics, the
arrangement according to which the different aspects are presented: first of all the projective
one, then the psychological one, the geometric elementary one. Such order corresponds to the
importance that the Author gives to the different aspects.
11. 1. Outstanding theorems
Among the other properties, beyond the diameters of conics, which imply the use of
the imaginary, we mention the theorem of Staudt, which states, in the two dual forms (Fig. 4):
Given a conic and a triangle ABC inscribed in it (such that its vertexes are on the
conic), every straight line conjugate to a side BC, intersect the other two sides in conjugate
points. On the contrary a straight line intersect two sides AB, AC of the triangle in two
conjugate pints, it is conjugate to the third side, that is passes for its pole.
Fig. 4
The dual states: given a conic and a triangle a b c circumscribed, so that its sides are
tangent of it, every point conjugate to a vertex bc of the trilateral project the other to vertexes
according to two conjugates straight lines.
On the contrary, if one point projects two vertexes ab, ac of the trilateral according to
two conjugates straight lines, it is coniugate to the third vertex, namely id belongs to its pole.
It is also essential the theorem of Steiner on projective generation of conics which
states:
The place of the intersections of homologous straight lines of two projective sheaves,
neither perspective nor concentric, is a conic.
The dual: the envelope of straight lines joining points homologous of two dotted
projective, neither perspective nor coincident, is a conic.
On the basis of this theorem, 5 points (not more of three aligned) are enough to
generate a conic: we have to assume two points as the centres of projective sheaves passing
through the other three. It is therefore possible to trace the couples of homologous straight
lines which with their intersections provide other points of the conic.
It is useful to mention the famous theorems of Pascal and Brianchon (Fig. 5)
Theorem of Pascal:
If a simple hexagon is inscribed in a conic, the three couples of opposite sides meet in
three points on a straight line (Pascal’s straight line). The hexagon is called “Pascal’s
hexagon”.
Fig. 5
Theorem of Brianchon:
If an hexalateral is circumscribed to a conic, the straight lines joining the three
couples of opposite vertexes pass through a point (Brianchon’s point). The hexalateral is
called “hexalateral of Brianchon”.
12.
DESCRIPTIVE GEOMETRY
The aim of descriptive geometry is the representation of real objects. It has to happen
in such a way that:
 It corresponds to our natural vision,
 Its reconstruction is univocal;
We include in Descriptive Geometry axonometry, perspective, dimensioned
projections (contour lines), and the decisive method of the double projection or of Monge.
The first two methods, which we can unify under the names of perspective, respectively conic
or axonometric, are not method of representation, as the reconstruction of the object from an
only view is impossible. Obviously it is possible to reconstruct univocally on the base of two
prospectical representations, but the method would be very laborious. Therefore the only
representation method is the Monge’s one and, in topography, those of dimensioned
projections.
We can leave out axioms of descriptive geometry which are very theoretical. However
we want to stress the fact that, even if modern electronic and optical tools reduced quite a bit
its practical importance, the conceptual importance of descriptive geometry, above all of that
subsequent to the introduction of projective geometry is still huge. It allows to deduct
properties of figures in the space through plane representation, using projective
correspondences, in particular homologies.
12. 1. “A posteriori” psychological reconstruction
The reconstruction of a figure by only a perspective (always ambiguous) is, in our
opinion, of the kind “a posteriori psychological”, that is founded on previous experiences as
it recalls to the mind similar objects, already seen and anyway existing in the world.
In any case, the methods of representation follow the present procedure: introduction
of elements of reference and conventions, methods of representation of the point, of the
straight line and of the plane, resolution of problems of belonging, parallelism and
perpendicularity (that is typically projective problems), resolution of metric problems, such as
the measurement of the sizes, therefore it is often used the operation of overturn.
It is useful at this point to report a fundamental distinction between projective and
descriptive geometry: according to Russell, who also, from a different point of view, does not
consider the two geometries completely distinct: also the Descriptive Geometry has an ordinal
character whose base is represented by points. Two points determine a class, formed by the
only points inside the two extreme points. There is contrast about this point with Peano, who
thinks that the points necessary to define the class must be at least three. We can speak in this
case of transitive asymmetric relations contrarily to what happens for projective geometry.
The descriptive geometry, as we have already hinted, is also called “geometry of position”,
translation of “Geometrie der Lage”, as the methods of representation identify univocally the
points represented, independently from every system of coordinates and therefore from every
measure. The technical design is therefore the descriptive geometry complete with norms and
dimensions. With reference to logical philosophical mathematical and normative implications,
which we have only hinted, we can say that, even in its most elementary meaning, technical
drawing has and has had character of science.
13.
ANALYTICAL GEOMETRY
Graphical problems either can be resolved directly by graphic way, with the methods
of descriptive geometry, or after transforming the graphic problem into an analytic one we
can resolve it and then we transform it again into graphic problem. Which method must we
prefer? We think that nowadays (1970), in drawing, it’s better to proceed, by choosing
depending on the case, the proper method, either the synthetic or the analytic one. We believe
however that many problems require the analytic use: for example, the problem of tolerances,
in which deviation of real surface from the ideal one, can be resolved if the ideal surface can
be described analytically. We can say the same for problems of parallelism, perpendicularity,
belonging, distance.
We remember, besides, the physic geometry, in particular the graphic statistics which
considers the representation as a powerful help for the resolution of problem concerning it.
We remember further that is merit of the geometry extending the concept of space to
multidimensional or abstract spaces. But it is better to avoid excessive extensions, based on
more or less remote links.
14.
THE PROGRAM OF ERLANGEN
We cannot omit the group properties of collineation or projectivity in the forms
belonging to the second kind:
In that case projectivities in the forms of second type form group: to use the same
word used by Klein, in the famous program of Erlangen, the projective geometry is concerned
with the properties of geometrical forms resulting invariant for the group of collineations;
A subgroup of projective geometry is formed by affine geometry, which is concerned
with collineations with improper centre ; another subgroup is formed by similar geometry,
in which the collineations are with improper axis. The elementary geometry form a
subgroup of the similar Geometry in relation to the equivalence(or congruence, according to
Hilbert).
The importance, also present in our times, of definition of Klein, lies in the fact that it
gives the opportunity of distinguishing, as regards a determined group, what is accidental
from what is objective. The program has even didactic, psychological purpose.
Geometrization of physics, absolute differential calculus and relativity are based on
such definition.
15.
DOES IT EXIST ABSOLUTE INVARIANT?
Let’s put in interrogative form the existence of the absolute invariant which on one
hand seems to merit its name while on the other hand the same name has something esoteric.
Simplifying, we can consider the plane and not the space; in the plan we can consider a
particular involution on the improper straight line, which associates every direction with the
related perpendicular. Introduced a system of homogeneous coordinates {0, x1, x2, x3}, and
set x = x1/x3; y = x2/x3 the unified points, called cyclic points, have the coordinates (1, i, 0),
(1, -i, 0). The set of cyclic points is said absolute circle.
In the space, introduced the coordinates x = x1/x4, y = x2/x4, z = x3/x4, the absolute is
given by equations
x12 + x22 + x 32 = 0
x42 = 0
The straight lines that projected the cyclic points are the isotropic straight lines of the
bundle of centre P; in the space we speak of “isotropic cone” of vertex P.
We would not speak of absolute in the plane and in the space if it did not exist another
“magic” aspect of the projective geometry, expressed by the fact that every property of metric
character can be expressed through projective relations with the absolute of the plane and the
space. The absolute invariant show the subordination of elementary Geometry to Projective
Geometry.
16.
CONCLUSIONS: RETURN TO ENRIQUES
Enriques says that “la storia della scienza si identifica con la scienza stessa
considerata nel suo divenire, nel pensiero di coloro che l’hanno costruita e nei suoi rapporti
con i diversi aspetti della cultura e della vita umana”. We have drawn inspiration from this
concept in this work, sure of the persisting utility of Projective Geometry, beyond its
fascination.
The end of Projective Geometry as pure mathematic discipline, has caused its
becoming object of engineering studies; I will mention, among all, the studies carried out in
Spain (Prieto and Alberga, Sondesa Freire, G. Marti and C. Calleja) and, at least for some
aspects, the computational spheric geometry (Lin Lin Chen, T. C. Woo)) used in problems of
tolerancing e and automated machining. .
Beyond “direct derivations”, we have the unifying attempt of Hilbert, founded on logic
formalism, which would have avoided any contradiction in mathematical theories. The dream
was frustrated by Gödel, with his “metamathematical theorems”, but the methodological,
fundamental abstract rigor, imposed by Hilbert himself, has led to the discovery of
connections among theories apparently very far.
The crisis of mathematics, in particular of geometry, which loses its character of
“absolute science” and replaces evident postulates with axioms that are only preliminary
remarks, encourages the study of multivalued Logics especially with Lukasievicz, and at
moment with Zadeh; the same probability assumes with de Finetti, the meaning of belief.
Concepts typical of Topology, such as those of completeness, compactness, filter, are
found with analogous meaning in pure Logic. The Topology itself is seen as an extension of
Geometry.
A more general case is that of distance. It is a purely “metric” concept, to which we
associate the idea of distance between two points, of length of a segment, but that we use also
to measure the distances among abstract entities such as probabilistic distributions, the
membershipfunctions, by the way in problems of planning or of integration design production.
The generalized inverse generated for Statistics problems, are used also in problems
for image reconstruction; the C-Calculus is used as a filter in the Pattern recognition and as
refiner of decision.
We have not lingered over Projective and Descriptive Geometry, which we consider
“the foundation of technical drawing”, so that it is not perhaps hazardous to think of an
axiomatic theory of Technical Drawing, even in its elementary form, which could be based on
axioms of Descriptive Geometry and on that of information.
We are sure that “Projective Geometry” with its generalizations and its invariants and
methodological refinement, is a very large category of the thought. Most of present days
principles of science can be included in it.
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