Definizioni.23 | Postulati.5 | Nozioni comuni.5
https://www.maths.tcd.ie/~dwilkins/Euclid/Proclus_Taylor1792/ Proclus's Commentary on the First Book of Euclid's Elements (translated by Thomas Taylor, 1792)
A Plane Angle, is the inclination of two Lines to each other in a Plane, which meet together, but are not in the same direction.
Some of the ancient philosophers, placing an angle in the predicament of
relation, have said, that it is the mutual inclination of lines or planes to
each other. But others, including this in quality, as well as rectitude and
obliquity, say, that it is a certain passion of a superficies or a solid.
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91 Mr. Simson, in his note on this definition, supposes it to be the addition of
some less skilful editor; on which account, and because it is quite useless (in
his opinion) he distinguishes it from the rest by inverted double commas. But it
is surely strange that the definition of angle in general should be accounted
useless, and the work of an un-skilful geometrician. Such an assertion may,
indeed, be very suitable to a professor of experimental philosophy, who
considers the useful as inseparable from practice; but is by no means becoming a
restorer of the liberal geometry of the ancients. Besides, Mr. Simson seems
continually to forget that Euclid was of the Platonic sect; and consequently was
a philosopher as well as a mathematician. I only add, that the commentary on the
present definition is, in my opinion, remarkable subtle and accurate, and well
deserves the profound attention of the greatest geometricians.
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And
others, referring it to quantity, confess that it is a superficies or a solid.
For
the angle which subsists in superficies is divided by a line; but that which
is in solids, by a superficies. But (say they) that which is divided by these,
is no other than magnitude, and this is not linear, since a line is divided
by a point; and therefore it follows that it must be either a superficies or
a solid. But if it is magnitude, and all finite magnitudes of the same kind
have a mutual proportion; all angles of the same kind, i. e. which subsist in
superficies, will have a mutual proportion. And hence, the cornicular will be
proportionable to a right-lined angle. But things which have a mutual
proportion, may, by multiplication, exceed each other; and therefore it may be
possible for the cornicular to exceed a right-lined angle, which it is
well-known, is impossible, since it is shewn to be less than every right-lined
angle.
But if it is quality alone, like heat and cold, how is it divisible into equal
parts? For equality, inequality, and divisibility, are not less resident in
angles
than in magnitudes; but they are, in like manner, essential. But if the things
in which these are essentially inherent, are quantities, and not qualities, it
is manifest that angles also are not qualities. Since the more and the less are
the proper passions of quality92, but not equal and unequal. On this hypothesis,
therefore, angles ought not to be called unequal, and this greater, but the
other less; but they ought to be denominated dissimilars, and one more an angle,
but the other less. But that these appelations are foreign from the essence of
mathematical concerns, is obvious to every one: for every angle receives the
same definition, nor is this more an angle, but that less. Thirdly, if an angle
is inclination, and belongs to the category of relation, it must follow, that
from the existence of one inclination, there will also be one angle, and not
more than one. For if it is nothing else than the relation of lines or planes,
how is it possible there can be one relation of lines or planes, but many
angles? If, therefore, we conceive a cone cut by a triangle from the vertex to
the base, we shall behold one inclination of the triangular lines in the
semicone to the vertex; but two distinct angles: one of
which is plane, I mean that of the triangle; but the other subsists in the mixt
superficies of the cone, and both are comprehended by the two triangular lines.
The relation, therefore, of these, do not make the angle. Again, it is necessary
to call an angle either quality or quantity, or relation; for figures, indeed,
are qualities, but their mutual proportions belong to relation. It is necessary,
therefore, that an angle should be reduced under one of these three genera. Such
doubts, then, arising concerning an angle, and Euclid
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92 For a philosophical discussion of the nature of quality and quantity, consult
the Commentaries of Ammonius, and Simplicius on Aristotle's Categories, Plotinus
on the genera of beings, and Mr. Harris's Philosophical Arrangements.
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calling it inclination, but Apollonius the collection of a superficies, or a
solid in one point, under a refracted line or superficies (for he seems to
define every angle universally), we shall affirm, agreeable to the sentiments of
our preceptor Syrianus, that an angle is of itself none of the aforesaid; but is
constituted from the concurrence of them all. And that, on this ccount, a doubt
arises among those who regard one category alone. But this is not peculiar to an
angle, but is likewise the property of a triangle. For this, too, participates
of quantity, and is called equal or unequal; because it has to quantity the
proportion of matter. But quality also, is present with this, in consequence of
its figure (since triangles are called as well similar as equal); but it
possesses this from one category, and that from another. Hence, an
angle is perfectly indigent of quantity, the subject of magnitude. But it is
also indigent of quality, by which it possesses, as it were, its proper form and
figure. Lastly, it is indigent of the relation of lines terminating, or of
superficies comprehending its form. So that an angle consists from all these,
yet is not any one of them in particular. And it is indeed divisible, and
capable of receiving equality and inequality, according to the quantity which it
contains. But it is not compelled to admit the proportion of magnitudes of the
same kind, since it has also a peculiar quantity, by which angles are also
incapable of a comparison with each other. Nor can one inclination perfect one
angle: since the quantity also, which is placed between the inclined lines,
completes its essence. If then we regard these distinctions, we shall dissolve
all absurdities, and discover that the property of an angle is not the
collection of a superficies or solid, according to Apollonius (since these also
complete its essence,) but that it is nothing else than a superficies itself,
collected into one point, and comprehended by inclined lines, or by one line
inclined to itself: and that a solid angle is the collection of superficies
mutually inclined to each other. Hence, we shall find that a formed quantum,
constituted in a certain relation, supplies its perfect definition. And thus
much we have thought requisite to assert concerning the substance of angles,
previously contemplating the common essence of every triangle, before we divide
it into species. But since there are three opinions of an angle, Eudemus the
Peripatetic, who composed a book concerning an angle, affirms that it is
quality. For, considering the origin of an angle, he says that it is nothing
else than the fraction of lines: because, if rectitude is quality, fraction also
will be quality. And hence, since its generation is in quality, an angle will be
entirely quality. But Euclid, and those who call it inclination, place it in the
category of relation. But they call it quantity, who say that it is the first
interval under a point, that is immediately subsisting after a point. In the
number of which is Plutarch, who constrains Apollonius also into the same
opinion. For it is requisite (says he) there should be some first interval,
under
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the inclination of containing lines or superficies. But since the interval,
which is under a point, is continuous, it is not possible that a first interval
can be assumed; since every interval is divisible in infinitum. Besides, if we
any how distinguish a first interval, and through it draw a right line, a
triangle is produced, and not one angle. But Carpus Antiochenus says, that an
angle is quantity, and is the distance of its comprehending lines, or
superficies;
and that this is distant by one interval, and yet an angle is not on that
account a line: since it is not true that every thing which is distant by only
one interval, is a line. But this surely is the most absurd of all, that there
should be any magnitude except a line, which is distant only by one interval.
And thus much concerning the nature of an angle. But with respect to the
division of angles, some consist in superficies, but others in solids. And of
those which are in superficies, some are in simple ones, but others in such as
are mixt. For an angle may be produced in a cylindric, conic, spherical, and
plane superficies. But of those which consist in simple superficies, some
are constituted in the spherical; but others in the plane. For the zodiac itself
forms angles, dividing the equinoctal in two parts, at the vertex of the cutting
superficies. And angles of this kind subsist in a spherical superficies. But of
those which are in planes, some are comprehended by simple lines, others by mixt
ones; and others, again, by both. for in the shield-like figure93, an angle is
comprehended by the axis, and the line of the shield: but one of these lines is
mixt, and the other simple. But if a circle cuts a shield, the angle will be
comprehended by the circumference, and the ellipsis. And when cissoids, or lines
similar to an ivy leaf, closing in one point like the leaves of ivy (from whence
they derive their appellation) make an angle, such an angle is comprehended by
mixt lines. Also, when the hippopeda, or line similar to the foot of a mare,
which is one of the spirals, inclining to another line, forms an angle, it is
comprehended by mixt lines. Lastly, the angles contained by
a circumference and a right line, are comprehended by simple lines. But of these
again, some are contained by such as are similar in species, but others by such
as are dissimilar. For two circumferences, mutually cutting, or touching each
other, produce angles: and these triple, for they are either on both sides
convex, when the convexities of the circumference are external:
or on both sides concave, when both the concavities are external; which they
call sistroides; or mixt from convex and concave lines, as the lines called
lunulas. But besides this, angles are contained in a twofold manner, by a right
line, and a circumference: for they are either contained by a right line and a
concave circumference, as the semicircular angle; or by a right line and a
convex circumference, as the cornicular angle. But all those which are
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93This is, the ellipsis.
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comprehended by two right lines, are called rectilinear angles, which have
likewise a triple di erence94. The geometrician, therefore, in the present
hypothesis, defines all those angles which are constituted in plane superficies,
and gives them the common name of a plane angle. And the genus of these he
denominates inclination: but the place, the plane itself, for angles have
position: but their origin such, that it is requisite there should be two lines
at least, and not three as in a solid. And that these should touch each other,
and by touching, must not lie in a right line, as an angle is the inclination
and comprehension of lines: but is not distance only, according to one interval.
But if we examine this definition, in the first place it appears that it does
not admit, an angle can be perfected by one line; though a cissoid, which is but
one, perfects an angle. And, in like manner, the hippopeda. For we call the
whole a cissoid, and not its portions (least any one should say, that the
conjunction of these forms an angle) and the whole a spiral, but not its parts.
Each, therefore, since it is one, forms an angle to itself, and not to another.
But after this, he is faulty, in defining an angle to be inclination. For how,
on this hypothesis, will there be two angles, from one inclination? How can we
call angles equal and unequal? And whatever else is usually objected against
this opinion. Thirdly, and lastly, that part of the definition, which says,
and not placed in a right line, is superfluous in certain angles, as in
those which are formed from orbicular lines. For without the assistance of this
part, the definition is perfect; since the inclination of one of the lines to
the other, forms the angle. And it is not possible that orbicular angles
should be placed in a right line. And thus much we have thought proper to say
concerning the definition of Euclid; partly, indeed, interpreting, and partly
doubting its truth.
But when the Lines containing the Angle, are right,
the Angle is called Rectilinear.
An angle is the symbol and image of the connection and compression, which subsists in the divine genera, and of that order which collects divisibles into one, partibles into an impartible nature, and the many into conciliating community. For it is the bond of a multitude of lines and superficies, the collector of magnitude into the impartibility of points, and the comprehender of every figure which is composed by its confining nature. On which account, the oracles95 call the angular junctions of figures, knots, so far as they bring
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94That is, they are either right, acute or obtuse.
95This oracle is not mentioned by any of the collectors of the Zoroastrian
oracles.
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with them an image of connecting union, and divine conjunctions, by which
discrete natures mutually cohere with each other. The angles, therefore,
susisting in superficies, express the more immaterial, simple, and perfect
unions which superficies contain: but those which are in solids, represent the
unions, which proceed even to inferiors, and supply a community to things
disjunct, and a construction of the same nature, to things which on every side
receive a perfect partition. But of the angles in superficies, some shadow forth
primary and unmixt unions; but others, such as omprehend in themselves, an
infinity of progressions. And some, indeed, are the sources of union to
intellectual forms; but others, to sensible reasons; and others, again, are
copulative of those forms which obtain between these, a middle situation. Hence
the angles which are made from circumferences, imitate those causes which
envelop intellectual variety in coercive union; for circumferences, hastening to
coalesce with each other, are images of intellect, and intellectual forms.
On the contrary, rectilineal angles, are the sumbols of those unions which
preside over sensibles, and a ord a conjunction of the reasons subsisting in
these: but mixt angles represent the preservers of the communion, as well of
sensible, as of intellectual forms, according to one immoveable union. It is
requisite, therefore, by regarding these paradigms, or exemplars, to render the
causes of each. For among the Pythagoreans we shall find various angles
dedicated to various gods. Thus, Philolaus, consecrates to some a triangular,
but to others a quadrangular angle; and to others, again, different angles.
Likewise, he permits the same to many gods, and many to the same god, according
to the di erent powers which they contain. And with a view to this, and to the
demiurgic triangle, which is the primary cause of all the ornament of the
elements, it appears to me, that Theodorus Asinǽus
the philosopher, constitutes some of the gods, according to sides; but others,
according to angles. The first, indeed, supplying progression and power; but the
second, the conjunction of the universe, and the collection of progressive
natures again into one. But these, indeed, direct us to the knowledge of the
things which are. And we must not wonder that lines are here said to contain an
angle. For the one and impartible nature which is found in these, is
adventitious: but in the gods themselves, and in true beings, the whole, and
impartible good, precedes things many, and divided.
When a Right Line standing on a Right Line, makes the successive Angles on each side equal to one another, each of the equal Angles, is a Right Angle; and the instituting Right Line, is called a Perpendicular to that upon which it stands.
96
An Obtuse Angle is that which is greater than a
Right Angle.
But an Acute Angle, is that which is less than a Right Angle.
These are the triple species of angles, which Socrates speaks of in the
Republic, and which are received by geometricians from hypothesis; a right-line
constituting these angles, according to a division into species; I mean, the
right, the obtuse, and the acute. The first of these being defined by equality,
identity and similitude; but the others being composed through the nature of the
greater and the lesser; and lastly, through inequality and diversity, and
through the more and the less, indeterminately assumed. But many geometricians,
are unable to render a reason of this division, and use the assertion, that
there are three angles, as an hypothesis96. So that, when we interrogate
them concerning its cause, they answer, this is not to be required of them as
geometricians. However, the Pythagoreans, referring the solution of this triple
distribution to principles, are not wanting in rendering the causes of this difference of right-lined angles. For, since one of the principles subsists
according to bound, and is the cause of limitation, identity, and equality,
and lastly, of the whole of a better co-ordination: but the other is of an
infinite nature, and confers on its progeny, a progression to infinity,
increase,
and decrease, inequality, and diversity of every kind, and entirely presides
over the worse series; hence, with great propriety, since the principles of a
right-lined angle are constituted by these, the reason proceeding from bound,
produces a right angle, one, with respect to the equality of every right angle, endued with similitude, always finite and determinate, ever abiding the
same, and neither receiving increment nor decrease. But the reason proceeding from infinity, since it is the second in order, and of a dyadic nature,
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96This, indeed, must always be the case with those geometricians, who are not at
the same time, philosophers; a conjunction no less valuable than rare. Hence,
from their ignorance of principles and intellectual concerns, when any contemplative enquiry is
proposed,
they immediately ask, in what its utility consists; considering every thing as
superfluous,
which does not contribute to the solution of some practical problem.
97
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produces twofold angles about the right angle, distinguished by inequality,
according to the nature of the greater and the lesser, and possessing an infinite motion, according to the more and the less, since the one becomes more
or less obtuse; but the other more or less acute. Hence, in consequence of
this reason, they ascribe right angles to the pure and immaculate gods of
the divine ornaments, and divine powers which proceed into the universe,
as the authors of the invariable providence of inferiors; for rectitude, and
an inflexibility and immutability to subordinate natures accords with these
gods: but they affirm, that the obtuse and acute angles should be ascribed to
the gods, who afford progression, and motion, and a variety of powers. Since
obtuseness is the image of an expanded progression of forms; but acuteness
possesses a similitude to the cause dividing and moving the universe. But
likewise, among the things which are, rectitude is, indeed, similar to essence,
preserving the same bound of its being; but the obtuse and acute, shadow
forth the nature of accidents. For these receive the more and the less, and
are indefinitely changed without ceasing. Hence, with great propriety, they
exhort the soul to make her descent into generation, according to this invariable species of the right angle, by not verging to this part more than to that;
and by not affecting some things more, and others less. For the distribution
of a certain convenience and sympathy of nature, draws it down to material
error, and indefinite variety97. A perpendicular line is, therefore, the symbol
of inflexibility, purity, immaculate, and invariable power, and every thing of
this kind. But it is likewise the symbol of divine and intellectual measure:
since we measure the altitudes of figures by a perpendicular, and define other
rectilineal angles by their relation to a right angle, as by themselves they are
indefinite and indeterminate. For they are beheld subsisting in excess and
defect, each of which is, by itself, indefinite. Hence they say, that virtue
also stands according to rectitude; but that vice subsists according to the
infinity of the obtuse and acute, that it produces excesses and defects, and
that the more and the less exhibit its immoderation, and inordinate nature.
Of rectilineal angles, therefore, we must establish the right angle, as the
image of perfection, and invariable energy, of limitation, intellectual bound,
and the like; but the obtuse and acute, as shadowing forth infinite motion,
unceasing progression, division, partition and infinitity. As thus much for
the theological speculation of angles. But here we must take notice, that
the genus is to be added to the definitions of an obtuse and acute angle; for
each is right-lined, and the one is greater, but the other less than a right-angle.
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97Concerning the soul's descent into body, see lib. ix. Ennead iv. of Plotinus;
and for
the method by which she may again return to her pristine felicity, study the
first book of
Porphyry's Treatise on Abstinence.
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But it is not absolutely true, that every angle which is less than a
right one, is acute. For the cornicular is less than every right angle, because
less than an acute one, yet is not on this account an acute angle. Also, a
semi-circular is less than any right-angle, yet is not acute. And the cause of
this property is because they are mixt, and not rectilineal angles. Besides,
many curve-lined angles appear greater than right-lined angles, yet are not
on this account obtuse; because it is requisite that an obtuse should be a
right lined angle. Secondly, as it was the intention of Euclid, to define a
right-angle, he considers a right line standing upon another right-line, and
making the angles on each side equal. But he defines an obtuse and acute
angle, not from the inclination of a right line to either part, but from their
relation to a right-angle. For this is the measure of angles deviating from the
right, in the same manner as equality of things unequal. But lines inclined
to either part, are innumerable, and not one alone, like a perpendicular. But
after this, when he says, (the angles equal to one another) he exhibits to us a
specimen of the greatest geometrical diligence; since it is possible that angles
may be equal to others, without being right. But when they are equal to one
another, it is necessary they should be right. Besides, the word successive
appears to me not to be added superfluously, as some have improperly considered it; since it exhibits the reason of rectitude. For it is on this account
that each of the angles is right; because, when they are successive, they are
equal. And, indeed, the insisting right-line, on account of its inflexibility to
either part, is the cause of equality to both, and of rectitude to each. The
cause, therefore, of the rectitude of angles, is not absolutely mutual equality,
but position in a consequent order, together with equality. But, besides all
this, I think it here necessary to call to mind, the purpose of our author; I
mean, that he discourses in this place, concerning the angles consisting in
one plane. And hence, this definition is not of every perpendicular; but of
that which is in one and the same plane. For it is not his present design to
define a solid angle. As, therefore, he defines, in this place, a plane angle,
so
likewise a perpendicular of this kind. Because a solid perpendicular ought
not to make right angles to one right-line only; but to all which touch it, and
are contained in its subject plane: for this is its necessary peculiarity.