It seems apparent that anyone with an *earnest* interest in the prehistory,
history, and reception of non-Archimedean mathematics should take a close
look at the controversial and juicy history of HORN ANGLES.
This concept -- loosely defined as the configuration formed by two curves
starting at a point, called the vertex, in a common direction - is also found
in the literature with colorful names such as
o angle of contingence
o curvilinear angle
o angle of contact
o cornicular angle
o horn-shape angle
o horn-like angle
o horned angle,
and its historical profile may be roughly sketched as follows:
(1) Greek Prelude
(2) Medieval Transition
(3) Renaissance Age
(4) Calculus Interlude
(5) Non-Archimedean Period
(6) Conformal Mapping Breakthrough
(7) Non-standard New Age
Let us give a general and quick and light survey of each one of these stages:
(1) GREEK PRELUDE
Sir Thomas Little Heath (1861--1940), in his masterpiece "A History of Greek
Mathematics" [1,a], argues about the bizarre sentence _On a difference of
opinion (_gnw/mhs_: v.l. _gnw/monos_, gnomon), or on the contact of a circle
and a sphere_, possibly a (corrupted) title of one of Democritus's treatises.
With regard to this, Heath remarks:
I think that the attempts to extract a sense out of
Cobet's reading _gnw/monos_ (on a difference of a
gnomon) have failed, and that _gnw/mhs_ (Diels) is
better. But 'On a difference of opinion' seems
scarcely determinative enough, if this was really an
alternative title to the book. We know that there
were controversies in ancient times about the nature
of the 'angle of contact' (the 'angle' formed, at
the point of contact, between an arc of a circle and
the tangent to it, which angle was called by the
special name *hornlike*, _keratoeidh/s_), and the
'angle' complementary to it (the 'angle of a semi-
circle'). The question was whether the 'hornlike
angle' was a magnitude comparable with the recti-
lineal angle, i.e. whether by being multiplied a
sufficient number of times it could be made to
exceed a given rectilineal angle. [1,a]
As we know, Euclid, in his "Elements", handled almost exclusively rectilinear
angles. A single reference, however, to angles with curved sides indeed is
discussed in Book III, proposition 16 [III.16]:
The straight line drawn at right angles to the
diameter of a circle from its extremity will fall
outside the circle, and into the space between the
straight line and the circumference another straight
line cannot be interposed; further the angle of the
semicircle is greater, and the remaining angle less,
than any acute rectilineal angle. [3,a]
This is a very special proposition, being the only reference also to the
ANGLE OF A SEMICIRCLE. Euclid also introduced the concept of the ANGLE OF A
SEGMENT [3,b] in Book III, definition 7 [III.7], and discussed about it in
III.31. Let us conclude Heath's previous disquisition on Democritus's curious
title:
But we know from a passage of Aristotle that before
his time 'angles of segments' came into geometrical
text-books as elements in figures which couldbe used
in the proofs of propositions; thus eg. the equality
of two the two angles of a segment(assumed as known)
was to prove the theorem of Eucl. I.5. Euclid aban-
doned the use of all such angles in proofs, and the
references to them abovementioned are only survivals
The controversies doubtless arose long before his
[ Euclid's ] time, and such a question as the nature
of the contact of a circle with its tangent would
probably have a fascination for Democritus, who, as
we shall see, broached other questions involving
infinitesimals. As, therefore, the questions of the
nature of the contact of a circle with its tangent
and of the character of the 'hornlike' angle are
obviously connected, I prefer to read _gwni/hs_ ('of
an angle') instead of _gnw/mhs_; this would give the
perfectly comprehensible title, 'On a difference in
an angle, or on the contact of a circle and a sphere'
[1,b]
In many passages of his writings, Proclus mentions HORN ANGLES, and uses this
very term (_keratoeidh/s gwnia_):
[T]here are mixed lines, such as spirals; mixed
angles, such as the semicircular and the horned
angle; mixed figures, such as sections of plane
figures and arches; and mixed solids, such as cones,
cylinders, and the like. [2,a]
It is not the aim of these postings to define all these 'odd' and old terms.
For an ancient classification of 'angles' may I suggest you to read [3,c], in
which Heath shows a clear and useful diagram of a full discussion of angles
given by Proclus, probably borrowed from Geminus, the Greek philosopher with
a Latin name, who held a Stoic view of the universe, and defended mathematics
from Epicurean's attacks.
The idea that "Euclid wanted to _exclude_ non-Archimedean magnitudes", as has
been remarked seems far-fetched to me. Euclid did not actually 'play' with
HORN ANGLES, but this does not necessarily imply that he actually wanted to
_exclude_ non-Archimedean objects, let alone that he wished to kill them off.
It would not be a wild speculation at this point that ancient mathematicians
may have tried to axiomatize non-Archimedean quantities, but they may have
given up their (re-)search so soon as they realized of _paradoxes_ involved.
Apropos of this issue, not long ago, Gordon Fisher wrote on the [HM] list:
Possibly, they [ the Ancients ] even got as far as
recognizing, though I expect not proving, that any
system which incorporated their analogs of our stan-
dard real numbers with actual infinitesimals would
have to be more complicated than some analog of a
complete linearly ordered field, which must satisfy
the Archimedean postulate. At least they appeared to
have found a likable ground for staying away from
infinitesimals: the Archimedean postulate,which per-
haps sounded to them like a reasonable assumption,
perhaps more reasonable than the parallel postulate.
-------------
What did the "real" nature of HORN ANGLES mean for the ancients? ...
There was much discussion as to which of the Aristotelian categories a HORN
ANGLE might/should belong: namely, QUALITY, QUANTITY, or RELATION.
i. Might the HORN ANGLE fit in the fourth Aristotelian sort of QUALITY? ...
As we may recall, Aristotle asserts that:
The fourth sort of quality is figure and the shape
that belongs to a thing,and besides these, straight-
ness,curvature and any other qualities of this type;
each of these defines a thing as being such and such
Because it is triangular or quadrangular a thing is
said to have a specific character, or again because
it is straight or curved; in fact a thing's shape in
every case gives rise to a qualification of it.
Although this wording might sound queer to our modern ears, this category was
the leit motiv of those scholars that defended the thesis that the HORN ANGLE
was a QUALITY.
ii. Should the HORN ANGLE be put in the 'basket' of MAGNITUDES?
And to this respect, I would like to cite Proclus' own words again:
[I]f it [= an angle] is a magnitude, and all finite
homogeneous magnitudes have a ratio to one another,
then all homogeneous angles, at least those in
planes, will have a ratio to one another, so that a
horned angle will have a ratio to a rectilinear. But
all quantities that have a ratio to one another can
exceed one another by being multiplied; a horned an-
gle, then, may exceed a rectilinear, which is impos-
sible, for it has been proved that a horned angle is
less than any rectilinear angle. [2,b]
This is an explicit assumption of what we call nowadays the Archimedean pos-
tulate, which does indeed prevent us from introducing actual infinitesimals
which have ratios to standard real numbers, or Eudoxian magnitudes.
iii. If we trust the sources, Proclus was a pupil of Syrianus, and succeeded
him as head of the Platonic Academy in Athens. According to Syrianus, anyone
who called an angle an *inclination* must classify it as a RELATION.
The notion of HORN ANGLE seems to have been loosely handled in the first
stages of its controversial and long history, but not so badly treated as one
may think at first sight -- apparently the pros and cons were pondered upon
in ancient Greece, not a minor point after all.
(2) MEDIEVAL TRANSITION
I am not quite certain how strong this belief is rooted in Western minds, but
I happen to think that it was not until the XIIIth century, especially the
second half of it, that Aristotle's *scientific* seeds began to spread widely
in Western Europe.
Aristotle, in his "Physica" [III. 207b], not only dealt with continuity but
also considered infinite magnitudes and infinitesimals:
Every assigned magnitude is surpassed in the direc-
tion of smallness, while in the other direction
there is no infinite magnitude.
Number, on the other hand, is a plurality of 'ones'
and a certain quantity of them. Hence, number must
stop at the indivisible ... But in the direction of
largeness it is always possible to think of a larger
number.
With magnitudes the contrary holds. What is continu-
ous is divided ad infinitum,but there is no infinite
in the direction of increase. For the size which it
can potentially be, it can also actually be.
Such notions as continuity, the infinite, and the infinitesimal, became hot
topics amongst some scholastic philosophers of the XIIIth century. The well-
known historian of mathematics, Carl B Boyer, has suggested that these issues
were studied in the light of Peripatetic philosophy,
rather than in terms of mathematical postulational
thoughts, but the resulting speculations were of
service in sustaining an interest insuch conceptions
until, at a later date, they became a part of mathe-
matics. [4,a]
The first traces of the term ANGLE OF CONTINGENCE (_angulus contingencie_)
seem to be found in "De triangulis", a major geometrical work of Jordanus de
Nemore [Nemorarius], who, along with Fibonacci, was a leading mathematician
of the first half of the XIIIth century.
Also Johannes Campanus of Novara (1205? -- 1296?), chaplain to Pope Urban IV,
was interested in the nature of HORN ANGLES. It has been said that Campanus
prepared a Latin translation of Euclid's "Elements" from texts of different
sources. Campanus noted that there was a clear incompatibility between III.16
and X.1 of Euclid's "Elements". Book X begins with the famous proposition, on
which the 'method of exhaustion' as used in Book XII depends:
Propositis duabus magnitudinibus inaequalibus, si a
majore plus quam dimidium subtrahitur, et a reliqua
plus quam dimidium, et hoc semper fit, magnitudo
relinquitur, quae minor erit proposita magnitudine
minore. [5,a]
that is to say,
if from any magnitude there be subtracted more than
its half (or its half simply), from the remainder
more than its half (or its half), and so on continu-
ally, there will at length remain a magnitude less
than any assigned magnitude of the same kind. [1,c]
Campanus asserted that the inconsistency between III.16 and X.1 takes place
if one accepts that both rectilinear angles and HORN ANGLES are magnitudes of
the same kind (*univoce anguli*):
Attendere autem oportet, quod huic propositioni
[X.1] videtur quintadecima tertii contradicere, pro-
ponens angulum contingentiae minorem fore quolibet
angulo a duabus lineis rectis contento. Posito enim
angulo quolibet rectilineo, si ab ipso majus dimidio
dematur, itemque de residuo majus dimidio, necesse
videtur hoc toties posse fieri quosque angulus
rectilineus minor angulo contingentiae relinquatur,
cuius oppositum quintadecima tertii syllogizat. Sed
hi non sunt univoce anguli, non enim ejusdem sunt
generis simpliciter curvum et rectum. Et vero nec
angulum contingentiae toties contingit sumi, ut
qualemcumque rectilineum excedat, quod necessarium
est (ut ex praehabita demonstratione patet) ad hoc
ut consequens ex antecedente sequatur. Planum ergo
est etiam quemlibet angulum rectilineum inifinitis
angulis contingentiae esse majorem. [Campanus]
Let us assume -- as Campanus did -- that the angle between two straight lines
is greater than the ANGLE OF CONTINGENCE. Thereby, according to X.1, if from
the larger angle, that is the rectilinear, there be subtracted more than its
half (or its half simply), and from the remainder more than its half (or its
half), and so on, ultimately there will remain a rectilinear angle less than
any assigned magnitude of the same kind. Then, according to X.1 there would
reach a rectilinear angle less than the ANGLE OF CONTINGENCE, which obviously
is not true. Campanus thought that this flaw lay in the wrong assumption that
the ANGLE OF CONTINGENCE and the rectilinear were magnitudes of the same kind
But earlier in his "Elementa mathematica", Campanus had noticed another flaw
if both horn-shape and rectilinear angles are regarded as magnitudes of the
same kind.
Heath noticed, as did Vivanti [6,a], that Campanus had inferred from III.16
that there was a flaw - "a curieux paradox" wrote Vivanti - in the principle
that
the transition from the less to the greater, or vice
versa, takes place through all intermediate quanti-
ties and therefore through the equal. [3,d]
Campanus also observed that
[i]f a diameter of a circle ... be moved about its
extremity until it takes the position of the tangent
to that circle, then, as long as it cuts the circle,
it makes an acute angle _less_ than the "angle of a
semicircle"; but the moment it ceases to cut, it
makes a right angle _greater_ than the same "angle
of a semicircle". The rectilineal angle is never,
during the transition, _equal_ to the "angle of a
semicircle". There is therefore an apparent incon-
sistency with X.I, and Campanus could only observe
(as he does on that proposition), in explanation of
the paradox, that "these are not angles in the same
sense (univoce), for the curved and the straight are
not things of the same kind without qualification
(simpliciter)" The argument assumes, of course, that
the right angle _is_ greater than the "angle of a
semicircle". [3,d]
In his "De quadratura circuli", Albert of Saxony (1316? -- 1390) expressed
analogous views as those of Campanus [7].
Thomas Bradwardine (ca. 1290 -- 1349), Doctor profundus, who was a prominent
English mathematician as well as philosopher and theologian cannot pass in
silence. In his "Geometria speculativa", he asserts:
Angulus contingentie est omni angulo rectilineo mi-
nor tamen est divisibilis in infinitum. Angulus semi
circuli est omni angulo rectilineo acuto major et
omni angulo recto vel obtuso minor et tamen est aug-
mentabilis in infinitum. [8]
Doctor profundus thought that the ANGLE OF CONTINGENCE could not possibly be
null:
... sed non per equale et sic in rectilineis est
reperiri majorem angulum semicirculi et minorem: non
tamen equalem.
Those interested readers in "An Examination of Bradwardine's Geometry" might
consult Molland's thoughtful article [9]. See also [10].
[end of part 1/3]
Julio Gonzalez Cabillon