of the fundamental propositions of elementary geometry, we easily verify
their necessary want of meaning. They are all founded on a vicious
manner of conceiving the principle of _homogeneity_, the true general
idea of which was explained in the second chapter of the preceding book.
These demonstrations suppose that this principle does not allow us to
admit the coexistence in the same equation of numbers obtained by
different concrete comparisons, which is evidently false, and contrary
to the constant practice of geometers. Thus it is easy to recognize
that, by employing the law of homogeneity in this arbitrary and
illegitimate acceptation, we could succeed in "demonstrating," with
quite as much apparent rigour, propositions whose absurdity is manifest
at the first glance. In examining attentively, for example, the
procedure by the aid of which it has been attempted to prove
analytically that the sum of the three angles of any rectilinear
triangle is constantly equal to two right angles, we see that it is
founded on this preliminary principle that, if two triangles have two of
their angles respectively equal, the third angle of the one will
necessarily be equal to the third angle of the other. This first point
being granted, the proposed relation is immediately deduced from it in a
very exact and simple manner. Now the analytical consideration by which
this previous proposition has been attempted to be established, is of
such a nature that, if it could be correct, we could rigorously deduce
from it, in reproducing it conversely, this palpable absurdity, that two
sides of a triangle are sufficient, without any angle, for the entire
determination of the third side. We may make analogous remarks on all
the demonstrations of this sort, the sophisms of which will be thus
verified in a perfectly apparent manner.
The more reason that we have here to consider geometry as being at the
present day essentially analytical, the more necessary was it to guard
against this abusive exaggeration of mathematical analysis, according to
which all geometrical observation would be dispensed with, in
establishing upon pure algebraical abstractions the very foundations of
this natural science.
_Attempted Demonstrations of Axioms, &c._ Another indication that
geometers have too much overlooked the character of a natural science
which is necessarily inherent in geometry, appears from their vain
attempts, so long made, to _demonstrate_ rigorously, not by the aid of
the calculus, but by means of certain constructions, several fundamental
propositions of elementary geometry. Whatever may be effected, it will
evidently be impossible to avoid sometimes recurring to simple and
direct observation in geometry as a means of establishing various
results. While, in this science, the phenomena which are considered are,
by virtue of their extreme simplicity, much more closely connected with
one another than those relating to any other physical science, some must
still be found which cannot be deduced, and which, on the contrary,
serve as starting points. It may be admitted that the greatest logical
perfection of the science is to reduce these to the smallest number
possible, but it would be absurd to pretend to make them completely
disappear. I avow, moreover, that I find fewer real inconveniences in
extending, a little beyond what would be strictly necessary, the number
of these geometrical notions thus established by direct observation,
provided they are sufficiently simple, than in making them the subjects
of complicated and indirect demonstrations, even when these
demonstrations may be logically irreproachable.
The true dogmatic destination of the geometry of the ancients, reduced
to its least possible indispensable developments, having thus been
characterized as exactly as possible, it is proper to consider summarily
each of the principal parts of which it must be composed. I think that I
may here limit myself to considering the first and the most extensive of
these parts, that which has for its object the study of _the right
line_; the two other sections, namely, the _quadrature of polygons_ and
the _cubature of polyhedrons_, from their limited extent, not being
capable of giving rise to any philosophical consideration of any
importance, distinct from those indicated in the preceding chapter with
respect to the measure of areas and of volumes in general.
GEOMETRY OF THE RIGHT LINE.
The final question which we always have in view in the study of the
right line, properly consists in determining, by means of one another,
the different elements of any right-lined figure whatever; which enables
us always to know indirectly the length and position of a right line, in
whatever circumstances it may be placed. This fundamental problem is
susceptible of two general solutions, the nature of which is quite
distinct, the one _graphical_, the other _algebraic_. The first, though
very imperfect, is that which must be first considered, because it is
spontaneously derived from the direct study of the subject; the second,
much more perfect in the most important respects, cannot be studied till
afterwards, because it is founded upon the previous knowledge of the
other.
GRAPHICAL SOLUTIONS.
The graphical solution consists in constructing at will the proposed
figure, either with the same dimensions, or, more usually, with
dimensions changed in any ratio whatever. The first mode need merely be
mentioned as being the most simple and the one which would first occur
to the mind, for it is evidently, by its nature, almost entirely
incapable of application. The second is, on the contrary, susceptible of
being most extensively and most usefully applied. We still make an
important and continual use of it at the present day, not only to
represent with exactness the forms of bodies and their relative
positions, but even for the actual determination of geometrical
magnitudes, when we do not need great precision. The ancients, in
consequence of the imperfection of their geometrical knowledge, employed
this procedure in a much more extensive manner, since it was for a long
time the only one which they could apply, even in the most important
precise determinations. It was thus, for example, that Aristarchus of
Samos estimated the relative distance from the sun and from the moon to
the earth, by making measurements on a triangle constructed as exactly
as possible, so as to be similar to the right-angled triangle formed by
the three bodies at the instant when the moon is in quadrature, and when
an observation of the angle at the earth would consequently be
sufficient to define the triangle. Archimedes himself, although he was
the first to introduce calculated determinations into geometry, several
times employed similar means. The formation of trigonometry did not
cause this method to be entirely abandoned, although it greatly
diminished its use; the Greeks and the Arabians continued to employ it
for a great number of researches, in which we now regard the use of the
calculus as indispensable.
This exact reproduction of any figure whatever on a different scale
cannot present any great theoretical difficulty when all the parts of
the proposed figure lie in the same plane. But if we suppose, as most
frequently happens, that they are situated in different planes, we see,
then, a new order of geometrical considerations arise. The artificial
figure, which is constantly plane, not being capable, in that case, of
being a perfectly faithful image of the real figure, it is necessary
previously to fix with precision the mode of representation, which gives
rise to different systems of _Projection_.
It then remains to be determined according to what laws the geometrical
phenomena correspond in the two figures. This consideration generates a
new series of geometrical investigations, the final object of which is
properly to discover how we can replace constructions in relief by plane
constructions. The ancients had to resolve several elementary questions
of this kind for various cases in which we now employ spherical
trigonometry, principally for different problems relating to the
celestial sphere. Such was the object of their _analemmas_, and of the
other plane figures which for a long time supplied the place of the
calculus. We see by this that the ancients really knew the elements of
what we now name _Descriptive Geometry_, although they did not conceive
it in a distinct and general manner.
I think it proper briefly to indicate in this place the true
philosophical character of this "Descriptive Geometry;" although, being
essentially a science of application, it ought not to be included within
the proper domain of this work.
DESCRIPTIVE GEOMETRY.
All questions of geometry of three dimensions necessarily give rise,
when we consider their graphical solution, to a common difficulty which
is peculiar to them; that of substituting for the different
constructions in relief, which are necessary to resolve them directly,
and which it is almost always impossible to execute, simple equivalent
plane constructions, by means of which we finally obtain the same
results. Without this indispensable transformation, every solution of
this kind would be evidently incomplete and really inapplicable in
practice, although theoretically the constructions in space are usually
preferable as being more direct. It was in order to furnish general
means for always effecting such a transformation that _Descriptive
Geometry_ was created, and formed into a distinct and homogeneous
system, by the illustrious MONGE. He invented, in the first place, a
uniform method of representing bodies by figures traced on a single
plane, by the aid of _projections_ on two different planes, usually
perpendicular to each other, and one of which is supposed to turn about
their common intersection so as to coincide with the other produced; in
this system, or in any other equivalent to it, it is sufficient to
regard points and lines as being determined by their projections, and
surfaces by the projections of their generating lines. This being
established, Monge--analyzing with profound sagacity the various partial
labours of this kind which had before been executed by a number of
incongruous procedures, and considering also, in a general and direct
manner, in what any questions of that nature must consist--found that
they could always be reduced to a very small number of invariable
abstract problems, capable of being resolved separately, once for all,
by uniform operations, relating essentially some to the contacts and
others to the intersections of surfaces. Simple and entirely general
methods for the graphical solution of these two orders of problems
having been formed, all the geometrical questions which may arise in any
of the various arts of construction--stone-cutting, carpentry,
perspective, dialling, fortification, &c.--can henceforth be treated as
simple particular cases of a single theory, the invariable application
of which will always necessarily lead to an exact solution, which may be
facilitated in practice by profiting by the peculiar circumstances of
each case.
* * * * *
This important creation deserves in a remarkable degree to fix the
attention of those philosophers who consider all that the human species
has yet effected as a first step, and thus far the only really complete
one, towards that general renovation of human labours, which must
imprint upon all our arts a character of precision and of rationality,
so necessary to their future progress. Such a revolution must, in fact,
inevitably commence with that class of industrial labours, which is
essentially connected with that science which is the most simple, the
most perfect, and the most ancient. It cannot fail to extend hereafter,
though with less facility, to all other practical operations. Indeed
Monge himself, who conceived the true philosophy of the arts better than
any one else, endeavoured to sketch out a corresponding system for the
mechanical arts.
Essential as the conception of descriptive geometry really is, it is
very important not to deceive ourselves with respect to its true
destination, as did those who, in the excitement of its first discovery,
saw in it a means of enlarging the general and abstract domain of
rational geometry. The result has in no way answered to these mistaken
hopes. And, indeed, is it not evident that descriptive geometry has no
special value except as a science of application, and as forming the
true special theory of the geometrical arts? Considered in its abstract
relations, it could not introduce any truly distinct order of
geometrical speculations. We must not forget that, in order that a
geometrical question should fall within the peculiar domain of
descriptive geometry, it must necessarily have been previously resolved
by speculative geometry, the solutions of which then, as we have seen,
always need to be prepared for practice in such a way as to supply the
place of constructions in relief by plane constructions; a substitution
which really constitutes the only characteristic function of descriptive
geometry.
It is proper, however, to remark here, that, with regard to intellectual
education, the study of descriptive geometry possesses an important
philosophical peculiarity, quite independent of its high industrial
utility. This is the advantage which it so pre-eminently offers--in
habituating the mind to consider very complicated geometrical
combinations in space, and to follow with precision their continual
correspondence with the figures which are actually traced--of thus
exercising to the utmost, in the most certain and precise manner, that
important faculty of the human mind which is properly called
"imagination," and which consists, in its elementary and positive
acceptation, in representing to ourselves, clearly and easily, a vast
and variable collection of ideal objects, as if they were really before
us.
Finally, to complete the indication of the general nature of descriptive
geometry by determining its logical character, we have to observe that,
while it belongs to the geometry of the ancients by the character of its
solutions, on the other hand it approaches the geometry of the moderns
by the nature of the questions which compose it. These questions are in
fact eminently remarkable for that generality which, as we saw in the
preceding chapter, constitutes the true fundamental character of modern
geometry; for the methods used are always conceived as applicable to any
figures whatever, the peculiarity of each having only a purely secondary
influence. The solutions of descriptive geometry are then graphical,
like most of those of the ancients, and at the same time general, like
those of the moderns.
* * * * *
After this important digression, we will pursue the philosophical
examination of _special_ geometry, always considered as reduced to its
least possible development, as an indispensable introduction to
_general_ geometry. We have now sufficiently considered the _graphical_
solution of the fundamental problem relating to the right line--that
is, the determination of the different elements of any right-lined
figure by means of one another--and have now to examine in a special
manner the _algebraic_ solution.
ALGEBRAIC SOLUTIONS.
This kind of solution, the evident superiority of which need not here be
dwelt upon, belongs necessarily, by the very nature of the question, to
the system of the ancient geometry, although the logical method which is
employed causes it to be generally, but very improperly, separated from
it. We have thus an opportunity of verifying, in a very important
respect, what was established generally in the preceding chapter, that
it is not by the employment of the calculus that the modern geometry is
essentially to be distinguished from the ancient. The ancients are in
fact the true inventors of the present trigonometry, spherical as well
as rectilinear; it being only much less perfect in their hands, on
account of the extreme inferiority of their algebraical knowledge. It
is, then, really in this chapter, and not, as it might at first be
thought, in those which we shall afterwards devote to the philosophical
examination of _general_ geometry, that it is proper to consider the
character of this important preliminary theory, which is usually, though
improperly, included in what is called _analytical geometry_, but which
is really only a complement of _elementary geometry_ properly so called.
Since all right-lined figures can be decomposed into triangles, it is
evidently sufficient to know how to determine the different elements of
a triangle by means of one another, which reduces _polygonometry_ to
simple _trigonometry_.
TRIGONOMETRY.
The difficulty in resolving algebraically such a question as the above,
consists essentially in forming, between the angles and the sides of a
triangle, three distinct equations; which, when once obtained, will
evidently reduce all trigonometrical problems to mere questions of
analysis.
_How to introduce Angles._ In considering the establishment of these
equations in the most general manner, we immediately meet with a
fundamental distinction with respect to the manner of introducing the
angles into the calculation, according as they are made to enter
_directly_, by themselves or by the circular arcs which are proportional
to them; or _indirectly_, by the chords of these arcs, which are hence
called their _trigonometrical lines_. Of these two systems of
trigonometry the second was of necessity the only one originally
adopted, as being the only practicable one, since the condition of
geometry made it easy enough to find exact relations between the sides
of the triangles and the trigonometrical lines which represent the
angles, while it would have been absolutely impossible at that epoch to
establish equations between the sides and the angles themselves.
_Advantages of introducing Trigonometrical Lines._ At the present day,
since the solution can be obtained by either system indifferently, that
motive for preference no longer exists; but geometers have none the less
persisted in following from choice the system primitively admitted from
necessity; for, the same reason which enabled these trigonometrical
equations to be obtained with much more facility, must, in like manner,
as it is still more easy to conceive _à priori_, render these equations
much more simple, since they then exist only between right lines,
instead of being established between right lines and arcs of circles.
Such a consideration has so much the more importance, as the question
relates to formulas which are eminently elementary, and destined to be
continually employed in all parts of mathematical science, as well as in
all its various applications.
It may be objected, however, that when an angle is given, it is, in
reality, always given by itself, and not by its trigonometrical lines;
and that when it is unknown, it is its angular value which is properly
to be determined, and not that of any of its trigonometrical lines. It
seems, according to this, that such lines are only useless
intermediaries between the sides and the angles, which have to be
finally eliminated, and the introduction of which does not appear
capable of simplifying the proposed research. It is indeed important to
explain, with more generality and precision than is customary, the great
real utility of this manner of proceeding.
_Division of Trigonometry into two Parts._ It consists in the fact that
the introduction of these auxiliary magnitudes divides the entire
question of trigonometry into two others essentially distinct, one of
which has for its object to pass from the angles to their
trigonometrical lines, or the converse, and the other of which proposes
to determine the sides of the triangles by the trigonometrical lines of
their angles, or the converse. Now the first of these two fundamental
questions is evidently susceptible, by its nature, of being entirely
treated and reduced to numerical tables once for all, in considering all
possible angles, since it depends only upon those angles, and not at all
upon the particular triangles in which they may enter in each case;
while the solution of the second question must necessarily be renewed,
at least in its arithmetical relations, for each new triangle which it
is necessary to resolve. This is the reason why the first portion of the
complete work, which would be precisely the most laborious, is no longer
taken into the account, being always done in advance; while, if such a
decomposition had not been performed, we would evidently have found
ourselves under the obligation of recommencing the entire calculation in
each particular case. Such is the essential property of the present
trigonometrical system, which in fact would really present no actual
advantage, if it was necessary to calculate continually the
trigonometrical line of each angle to be considered, or the converse;
the intermediate agency introduced would then be more troublesome than
convenient.
In order to clearly comprehend the true nature of this conception, it
will be useful to compare it with a still more important one, designed
to produce an analogous effect either in its algebraic, or, still more,
in its arithmetical relations--the admirable theory of _logarithms_. In
examining in a philosophical manner the influence of this theory, we see
in fact that its general result is to decompose all imaginable
arithmetical operations into two distinct parts. The first and most
complicated of these is capable of being executed in advance once for
all (since it depends only upon the numbers to be considered, and not at
all upon the infinitely different combinations into which they can
enter), and consists in considering all numbers as assignable powers of
a constant number. The second part of the calculation, which must of
necessity be recommenced for each new formula which is to have its
value determined, is thenceforth reduced to executing upon these
exponents correlative operations which are infinitely more simple. I
confine myself here to merely indicating this resemblance, which any one
can carry out for himself.
We must besides observe, as a property (secondary at the present day,
but all-important at its origin) of the trigonometrical system adopted,
the very remarkable circumstance that the determination of angles by
their trigonometrical lines, or the converse, admits of an arithmetical
solution (the only one which is directly indispensable for the special
destination of trigonometry) without the previous resolution of the
corresponding algebraic question. It is doubtless to such a peculiarity
that the ancients owed the possibility of knowing trigonometry. The
investigation conceived in this way was so much the more easy, inasmuch
as tables of chords (which the ancients naturally took as the
trigonometrical lines) had been previously constructed for quite a
different object, in the course of the labours of Archimedes on the
rectification of the circle, from which resulted the actual
determination of a certain series of chords; so that when Hipparchus
subsequently invented trigonometry, he could confine himself to
completing that operation by suitable intercalations; which shows
clearly the connexion of ideas in that matter.
_The Increase of such Trigonometrical Lines._ To complete this
philosophical sketch of trigonometry, it is proper now to observe that
the extension of the same considerations which lead us to replace angles
or arcs of circles by straight lines, with the view of simplifying our
equations, must also lead us to employ concurrently several
trigonometrical lines, instead of confining ourselves to one only (as
did the ancients), so as to perfect this system by choosing that one
which will be algebraically the most convenient on each occasion. In
this point of view, it is clear that the number of these lines is in
itself no ways limited; provided that they are determined by the arc,
and that they determine it, whatever may be the law according to which
they are derived from it, they are suitable to be substituted for it in
the equations. The Arabians, and subsequently the moderns, in confining
themselves to the most simple constructions, have carried to four or
five the number of _direct_ trigonometrical lines, which might be
extended much farther.
But instead of recurring to geometrical formations, which would finally
become very complicated, we conceive with the utmost facility as many
new trigonometrical lines as the analytical transformations may require,
by means of a remarkable artifice, which is not usually apprehended in a
sufficiently general manner. It consists in not directly multiplying the
trigonometrical lines appropriate to each arc considered, but in
introducing new ones, by considering this arc as indirectly determined
by all lines relating to an arc which is a very simple function of the
first. It is thus, for example, that, in order to calculate an angle
with more facility, we will determine, instead of its sine, the sine of
its half, or of its double, &c. Such a creation of _indirect_
trigonometrical lines is evidently much more fruitful than all the
direct geometrical methods for obtaining new ones. We may accordingly
say that the number of trigonometrical lines actually employed at the
present day by geometers is in reality unlimited, since at every
instant, so to say, the transformations of analysis may lead us to
augment it by the method which I have just indicated. Special names,
however, have been given to those only of these _indirect_ lines which
refer to the complement of the primitive arc, the others not occurring
sufficiently often to render such denominations necessary; a
circumstance which has caused a common misconception of the true extent
of the system of trigonometry.
_Study of their Mutual Relations._ This multiplicity of trigonometrical
lines evidently gives rise to a third fundamental question in
trigonometry, the study of the relations which exist between these
different lines; since, without such a knowledge, we could not make use,
for our analytical necessities, of this variety of auxiliary magnitudes,
which, however, have no other destination. It is clear, besides, from
the consideration just indicated, that this essential part of
trigonometry, although simply preparatory, is, by its nature,
susceptible of an indefinite extension when we view it in its entire
generality, while the two others are circumscribed within rigorously
defined limits.
It is needless to add that these three principal parts of trigonometry
have to be studied in precisely the inverse order from that in which we
have seen them necessarily derived from the general nature of the
subject; for the third is evidently independent of the two others, and
the second, of that which was first presented--the resolution of
triangles, properly so called--which must for that reason be treated in
the last place; which rendered so much the more important the
consideration of their natural succession and logical relations to one
another.
It is useless to consider here separately _spherical trigonometry_,
which cannot give rise to any special philosophical consideration;
since, essential as it is by the importance and the multiplicity of its
uses, it can be treated at the present day only as a simple application
of rectilinear trigonometry, which furnishes directly its fundamental
equations, by substituting for the spherical triangle the corresponding
trihedral angle.
This summary exposition of the philosophy of trigonometry has been here
given in order to render apparent, by an important example, that
rigorous dependence and those successive ramifications which are
presented by what are apparently the most simple questions of elementary
geometry.
* * * * *
Having thus examined the peculiar character of _special_ geometry
reduced to its only dogmatic destination, that of furnishing to general
geometry an indispensable preliminary basis, we have now to give all our
attention to the true science of geometry, considered as a whole, in the
most rational manner. For that purpose, it is necessary to carefully
examine the great original idea of Descartes, upon which it is entirely
founded. This will be the object of the following chapter.
CHAPTER III.
MODERN OR ANALYTICAL GEOMETRY.
_General_ (or _Analytical_) geometry being entirely founded upon the
transformation of geometrical considerations into equivalent analytical
considerations, we must begin with examining directly and in a thorough
manner the beautiful conception by which Descartes has established in a
uniform manner the constant possibility of such a co-relation. Besides
its own extreme importance as a means of highly perfecting geometrical
science, or, rather, of establishing the whole of it on rational bases,
the philosophical study of this admirable conception must have so much
the greater interest in our eyes from its characterizing with perfect
clearness the general method to be employed in organizing the relations
of the abstract to the concrete in mathematics, by the analytical
representation of natural phenomena. There is no conception, in the
whole philosophy of mathematics which better deserves to fix all our
attention.
ANALYTICAL REPRESENTATION OF FIGURES.
In order to succeed in expressing all imaginable geometrical phenomena
by simple analytical relations, we must evidently, in the first place,
establish a general method for representing analytically the subjects
themselves in which these phenomena are found, that is, the lines or the
surfaces to be considered. The _subject_ being thus habitually
considered in a purely analytical point of view, we see how it is
thenceforth possible to conceive in the same manner the various
_accidents_ of which it is susceptible.
In order to organize the representation of geometrical figures by
analytical equations, we must previously surmount a fundamental
difficulty; that of reducing the general elements of the various
conceptions of geometry to simply numerical ideas; in a word, that of
substituting in geometry pure considerations of _quantity_ for all
considerations of _quality_.
_Reduction of Figure to Position._ For this purpose let us observe, in
the first place, that all geometrical ideas relate necessarily to these
three universal categories: the _magnitude_, the _figure_, and the
_position_ of the extensions to be considered. As to the first, there is
evidently no difficulty; it enters at once into the ideas of numbers.
With relation to the second, it must be remarked that it will always
admit of being reduced to the third. For the figure of a body evidently
results from the mutual position of the different points of which it is
composed, so that the idea of position necessarily comprehends that of
figure, and every circumstance of figure can be translated by a
circumstance of position. It is in this way, in fact, that the human
mind has proceeded in order to arrive at the analytical representation
of geometrical figures, their conception relating directly only to
positions. All the elementary difficulty is then properly reduced to
that of referring ideas of situation to ideas of magnitude. Such is the
direct destination of the preliminary conception upon which Descartes
has established the general system of analytical geometry.
His philosophical labour, in this relation, has consisted simply in the
entire generalization of an elementary operation, which we may regard as
natural to the human mind, since it is performed spontaneously, so to
say, in all minds, even the most uncultivated. Thus, when we have to
indicate the situation of an object without directly pointing it out,
the method which we always adopt, and evidently the only one which can
be employed, consists in referring that object to others which are
known, by assigning the magnitude of the various geometrical elements,
by which we conceive it connected with the known objects. These elements
constitute what Descartes, and after him all geometers, have called the
_co-ordinates_ of each point considered. They are necessarily two in
number, if it is known in advance in what plane the point is situated;
and three, if it may be found indifferently in any region of space. As
many different constructions as can be imagined for determining the
position of a point, whether on a plane or in space, so many distinct
systems of co-ordinates may be conceived; they are consequently
susceptible of being multiplied to infinity. But, whatever may be the
system adopted, we shall always have reduced the ideas of situation to
simple ideas of magnitude, so that we will consider the change in the
position of a point as produced by mere numerical variations in the
values of its co-ordinates.
_Determination of the Position of a Point._ Considering at first only
the least complicated case, that of _plane geometry_, it is in this way
that we usually determine the position of a point on a plane, by its
distances from two fixed right lines considered as known, which are
called _axes_, and which are commonly supposed to be perpendicular to
each other. This system is that most frequently adopted, because of its
simplicity; but geometers employ occasionally an infinity of others.
Thus the position of a point on a plane may be determined, 1°, by its
distances from two fixed points; or, 2°, by its distance from a single
fixed point, and the direction of that distance, estimated by the
greater or less angle which it makes with a fixed right line, which
constitutes the system of what are called _polar_ co-ordinates, the most
frequently used after the system first mentioned; or, 3°, by the angles
which the right lines drawn from the variable point to two fixed points
make with the right line which joins these last; or, 4°, by the
distances from that point to a fixed right line and a fixed point, &c.
In a word, there is no geometrical figure whatever from which it is not
possible to deduce a certain system of co-ordinates more or less
susceptible of being employed.
A general observation, which it is important to make in this connexion,
is, that every system of co-ordinates is equivalent to determining a
point, in plane geometry, by the intersection of two lines, each of
which is subjected to certain fixed conditions of determination; a
single one of these conditions remaining variable, sometimes the one,
sometimes the other, according to the system considered. We could not,
indeed, conceive any other means of constructing a point than to mark it
by the meeting of two lines. Thus, in the most common system, that of
_rectilinear co-ordinates_, properly so called, the point is determined
by the intersection of two right lines, each of which remains constantly
parallel to a fixed axis, at a greater or less distance from it; in the
_polar_ system, the position of the point is marked by the meeting of a
circle, of variable radius and fixed centre, with a movable right line
compelled to turn about this centre: in other systems, the required
point might be designated by the intersection of two circles, or of any
other two lines, &c. In a word, to assign the value of one of the
co-ordinates of a point in any system whatever, is always necessarily
equivalent to determining a certain line on which that point must be
situated. The geometers of antiquity had already made this essential
remark, which served as the base of their method of geometrical _loci_,
of which they made so happy a use to direct their researches in the
resolution of _determinate_ problems, in considering separately the
influence of each of the two conditions by which was defined each point
constituting the object, direct or indirect, of the proposed question.
It was the general systematization of this method which was the
immediate motive of the labours of Descartes, which led him to create
analytical geometry.
After having clearly established this preliminary conception--by means
of which ideas of position, and thence, implicitly, all elementary
geometrical conceptions are capable of being reduced to simple numerical
considerations--it is easy to form a direct conception, in its entire
generality, of the great original idea of Descartes, relative to the
analytical representation of geometrical figures: it is this which forms
the special object of this chapter. I will continue to consider at
first, for more facility, only geometry of two dimensions, which alone
was treated by Descartes; and will afterwards examine separately, under
the same point of view, the theory of surfaces and curves of double
curvature.
PLANE CURVES.
_Expression of Lines by Equations._ In accordance with the manner of
expressing analytically the position of a point on a plane, it can be
easily established that, by whatever property any line may be defined,
that definition always admits of being replaced by a corresponding
equation between the two variable co-ordinates of the point which
describes this line; an equation which will be thenceforth the
analytical representation of the proposed line, every phenomenon of
which will be translated by a certain algebraic modification of its
equation. Thus, if we suppose that a point moves on a plane without its
course being in any manner determined, we shall evidently have to regard
its co-ordinates, to whatever system they may belong, as two variables
entirely independent of one another. But if, on the contrary, this point
is compelled to describe a certain line, we shall necessarily be
compelled to conceive that its co-ordinates, in all the positions which
it can take, retain a certain permanent and precise relation to each
other, which is consequently susceptible of being expressed by a
suitable equation; which will become the very clear and very rigorous
analytical definition of the line under consideration, since it will
express an algebraical property belonging exclusively to the
co-ordinates of all the points of this line. It is clear, indeed, that
when a point is not subjected to any condition, its situation is not
determined except in giving at once its two co-ordinates, independently
of each other; while, when the point must continue upon a defined line,
a single co-ordinate is sufficient for completely fixing its position.
The second co-ordinate is then a determinate _function_ of the first;
or, in other words, there must exist between them a certain _equation_,
of a nature corresponding to that of the line on which the point is
compelled to remain. In a word, each of the co-ordinates of a point
requiring it to be situated on a certain line, we conceive reciprocally
that the condition, on the part of a point, of having to belong to a
line defined in any manner whatever, is equivalent to assigning the
value of one of the two co-ordinates; which is found in that case to be
entirely dependent on the other. The analytical relation which expresses
this dependence may be more or less difficult to discover, but it must
evidently be always conceived to exist, even in the cases in which our
present means may be insufficient to make it known. It is by this simple
consideration that we may demonstrate, in an entirely general
manner--independently of the particular verifications on which this
fundamental conception is ordinarily established for each special
definition of a line--the necessity of the analytical representation of
lines by equations.
_Expression of Equations by Lines._ Taking up again the same reflections
in the inverse direction, we could show as easily the geometrical
necessity of the representation of every equation of two variables, in a
determinate system of co-ordinates, by a certain line; of which such a
relation would be, in the absence of any other known property, a very
characteristic definition, the scientific destination of which will be
to fix the attention directly upon the general course of the solutions
of the equation, which will thus be noted in the most striking and the
most simple manner. This picturing of equations is one of the most
important fundamental advantages of analytical geometry, which has
thereby reacted in the highest degree upon the general perfecting of
analysis itself; not only by assigning to purely abstract researches a
clearly determined object and an inexhaustible career, but, in a still
more direct relation, by furnishing a new philosophical medium for
analytical meditation which could not be replaced by any other. In fact,
the purely algebraic discussion of an equation undoubtedly makes known
its solutions in the most precise manner, but in considering them only
one by one, so that in this way no general view of them could be
obtained, except as the final result of a long and laborious series of
numerical comparisons. On the other hand, the geometrical _locus_ of the
equation, being only designed to represent distinctly and with perfect
clearness the summing up of all these comparisons, permits it to be
directly considered, without paying any attention to the details which
have furnished it. It can thereby suggest to our mind general analytical
views, which we should have arrived at with much difficulty in any other
manner, for want of a means of clearly characterizing their object. It
is evident, for example, that the simple inspection of the logarithmic
curve, or of the curve _y_ = sin. _x_, makes us perceive much more
distinctly the general manner of the variations of logarithms with
respect to their numbers, or of sines with respect to their arcs, than
could the most attentive study of a table of logarithms or of natural
sines. It is well known that this method has become entirely elementary
at the present day, and that it is employed whenever it is desired to
get a clear idea of the general character of the law which reigns in a
series of precise observations of any kind whatever.
_Any Change in the Line causes a Change in the Equation._ Returning to
the representation of lines by equations, which is our principal object,
we see that this representation is, by its nature, so faithful, that the
line could not experience any modification, however slight it might be,
without causing a corresponding change in the equation. This perfect
exactitude even gives rise oftentimes to special difficulties; for
since, in our system of analytical geometry, the mere displacements of
lines affect the equations, as well as their real variations in
magnitude or form, we should be liable to confound them with one another
in our analytical expressions, if geometers had not discovered an
ingenious method designed expressly to always distinguish them. This
method is founded on this principle, that although it is impossible to
change analytically at will the position of a line with respect to the
axes of the co-ordinates, we can change in any manner whatever the
situation of the axes themselves, which evidently amounts to the same;
then, by the aid of the very simple general formula by which this
transformation of the axes is produced, it becomes easy to discover
whether two different equations are the analytical expressions of only
the same line differently situated, or refer to truly distinct
geometrical loci; since, in the former case, one of them will pass into
the other by suitably changing the axes or the other constants of the
system of co-ordinates employed. It must, moreover, be remarked on this
subject, that general inconveniences of this nature seem to be
absolutely inevitable in analytical geometry; for, since the ideas of
position are, as we have seen, the only geometrical ideas immediately
reducible to numerical considerations, and the conceptions of figure
cannot be thus reduced, except by seeing in them relations of situation,
it is impossible for analysis to escape confounding, at first, the
phenomena of figure with simple phenomena of position, which alone are
directly expressed by the equations.
_Every Definition of a Line is an Equation._ In order to complete the
philosophical explanation of the fundamental conception which serves as
the base of analytical geometry, I think that I should here indicate a
new general consideration, which seems to me particularly well adapted
for putting in the clearest point of view this necessary representation
of lines by equations with two variables. It consists in this, that not
only, as we have shown, must every defined line necessarily give rise to
a certain equation between the two co-ordinates of any one of its
points, but, still farther, every definition of a line may be regarded
as being already of itself an equation of that line in a suitable system
of co-ordinates.
It is easy to establish this principle, first making a preliminary
logical distinction with respect to different kinds of definitions. The
rigorously indispensable condition of every definition is that of
distinguishing the object defined from all others, by assigning to it a
property which belongs to it exclusively. But this end may be generally
attained in two very different ways; either by a definition which is
simply _characteristic_, that is, indicative of a property which,
although truly exclusive, does not make known the mode of generation of
the object; or by a definition which is really _explanatory_, that is,
which characterizes the object by a property which expresses one of its
modes of generation. For example, in considering the circle as the line,
which, under the same contour, contains the greatest area, we have
evidently a definition of the first kind; while in choosing the property
of its having all its points equally distant from a fixed point, we have
a definition of the second kind. It is, besides, evident, as a general
principle, that even when any object whatever is known at first only by
a _characteristic_ definition, we ought, nevertheless, to regard it as
susceptible of _explanatory_ definitions, which the farther study of the
object would necessarily lead us to discover.
This being premised, it is clear that the general observation above
made, which represents every definition of a line as being necessarily
an equation of that line in a certain system of co-ordinates, cannot
apply to definitions which are simply _characteristic_; it is to be
understood only of definitions which are truly _explanatory_. But, in
considering only this class, the principle is easy to prove. In fact, it
is evidently impossible to define the generation of a line without
specifying a certain relation between the two simple motions of
translation or of rotation, into which the motion of the point which
describes it will be decomposed at each instant. Now if we form the most
general conception of what constitutes _a system of co-ordinates_, and
admit all possible systems, it is clear that such a relation will be
nothing else but the _equation_ of the proposed line, in a system of
co-ordinates of a nature corresponding to that of the mode of generation
considered. Thus, for example, the common definition of the _circle_ may
evidently be regarded as being immediately the _polar equation_ of this
curve, taking the centre of the circle for the pole. In the same way,
the elementary definition of the _ellipse_ or of the _hyperbola_--as
being the curve generated by a point which moves in such a manner that
the sum or the difference of its distances from two fixed points remains
constant--gives at once, for either the one or the other curve, the
equation _y_ + _x_ = _c_, taking for the system of co-ordinates that in
which the position of a point would be determined by its distances from
two fixed points, and choosing for these poles the two given foci. In
like manner, the common definition of any _cycloid_ would furnish
directly, for that curve, the equation _y_ = _mx_; adopting as the
co-ordinates of each point the arc which it marks upon a circle of
invariable radius, measuring from the point of contact of that circle
with a fixed line, and the rectilinear distance from that point of
contact to a certain origin taken on that right line. We can make
analogous and equally easy verifications with respect to the customary
definitions of spirals, of epicycloids, &c. We shall constantly find
that there exists a certain system of co-ordinates, in which we
immediately obtain a very simple equation of the proposed line, by
merely writing algebraically the condition imposed by the mode of
generation considered.
Besides its direct importance as a means of rendering perfectly apparent
the necessary representation of every line by an equation, the preceding
consideration seems to me to possess a true scientific utility, in
characterizing with precision the principal general difficulty which
occurs in the actual establishment of these equations, and in
consequently furnishing an interesting indication with respect to the
course to be pursued in inquiries of this kind, which, by their nature,
could not admit of complete and invariable rules. In fact, since any
definition whatever of a line, at least among those which indicate a
mode of generation, furnishes directly the equation of that line in a
certain system of co-ordinates, or, rather, of itself constitutes that
equation, it follows that the difficulty which we often experience in
discovering the equation of a curve, by means of certain of its
characteristic properties, a difficulty which is sometimes very great,
must proceed essentially only from the commonly imposed condition of
expressing this curve analytically by the aid of a designated system of
co-ordinates, instead of admitting indifferently all possible systems.
These different systems cannot be regarded in analytical geometry as
being all equally suitable; for various reasons, the most important of
which will be hereafter discussed, geometers think that curves should
almost always be referred, as far as is possible, to _rectilinear
co-ordinates_, properly so called. Now we see, from what precedes, that
in many cases these particular co-ordinates will not be those with
reference to which the equation of the curve will be found to be
directly established by the proposed definition. The principal
difficulty presented by the formation of the equation of a line really
consists, then, in general, in a certain transformation of co-ordinates.
It is undoubtedly true that this consideration does not subject the
establishment of these equations to a truly complete general method, the
success of which is always certain; which, from the very nature of the
subject, is evidently chimerical: but such a view may throw much useful
light upon the course which it is proper to adopt, in order to arrive at
the end proposed. Thus, after having in the first place formed the
preparatory equation, which is spontaneously derived from the definition
which we are considering, it will be necessary, in order to obtain the
equation belonging to the system of co-ordinates which must be finally
admitted, to endeavour to express in a function of these last
co-ordinates those which naturally correspond to the given mode of
generation. It is upon this last labour that it is evidently impossible
to give invariable and precise precepts. We can only say that we shall
have so many more resources in this matter as we shall know more of true
analytical geometry, that is, as we shall know the algebraical
expression of a greater number of different algebraical phenomena.
CHOICE OF CO-ORDINATES.
In order to complete the philosophical exposition of the conception
which serves as the base of analytical geometry, I have yet to notice
the considerations relating to the choice of the system of co-ordinates
which is in general the most suitable. They will give the rational
explanation of the preference unanimously accorded to the ordinary
rectilinear system; a preference which has hitherto been rather the
effect of an empirical sentiment of the superiority of this system, than
the exact result of a direct and thorough analysis.
_Two different Points of View._ In order to decide clearly between all
the different systems of co-ordinates, it is indispensable to
distinguish with care the two general points of view, the converse of
one another, which belong to analytical geometry; namely, the relation
of algebra to geometry, founded upon the representation of lines by
equations; and, reciprocally, the relation of geometry to algebra,
founded on the representation of equations by lines.
It is evident that in every investigation of general geometry these two
fundamental points of view are of necessity always found combined,
since we have always to pass alternately, and at insensible intervals,
so to say, from geometrical to analytical considerations, and from
analytical to geometrical considerations. But the necessity of here
temporarily separating them is none the less real; for the answer to the
question of method which we are examining is, in fact, as we shall see
presently, very far from being the same in both these relations, so that
without this distinction we could not form any clear idea of it.
1. _Representation of Lines by Equations._ Under _the first point of
view_--the representation of lines by equations--the only reason which
could lead us to prefer one system of co-ordinates to another would be
the greater simplicity of the equation of each line, and greater
facility in arriving at it. Now it is easy to see that there does not
exist, and could not be expected to exist, any system of co-ordinates
deserving in that respect a constant preference over all others. In
fact, we have above remarked that for each geometrical definition
proposed we can conceive a system of co-ordinates in which the equation
of the line is obtained at once, and is necessarily found to be also
very simple; and this system, moreover, inevitably varies with the
nature of the characteristic property under consideration. The
rectilinear system could not, therefore, be constantly the most
advantageous for this object, although it may often be very favourable;
there is probably no system which, in certain particular cases, should
not be preferred to it, as well as to every other.
2. _Representation of Equations by Lines._ It is by no means so,
however, under the _second point of view_. We can, indeed, easily
establish, as a general principle, that the ordinary rectilinear system
must necessarily be better adapted than any other to the representation
of equations by the corresponding geometrical _loci_; that is to say,
that this representation is constantly more simple and more faithful in
it than in any other.
Let us consider, for this object, that, since every system of
co-ordinates consists in determining a point by the intersection of two
lines, the system adapted to furnish the most suitable geometrical
_loci_ must be that in which these two lines are the simplest possible;
a consideration which confines our choice to the _rectilinear_ system.
In truth, there is evidently an infinite number of systems which deserve
that name, that is to say, which employ only right lines to determine
points, besides the ordinary system which assigns the distances from two
fixed lines as co-ordinates; such, for example, would be that in which
the co-ordinates of each point should be the two angles which the right
lines, which go from that point to two fixed points, make with the right
line, which joins these last points: so that this first consideration is
not rigorously sufficient to explain the preference unanimously given to
the common system. But in examining in a more thorough manner the nature
of every system of co-ordinates, we also perceive that each of the two
lines, whose meeting determines the point considered, must necessarily
offer at every instant, among its different conditions of determination,
a single variable condition, which gives rise to the corresponding
co-ordinate, all the rest being fixed, and constituting the _axes_ of
the system, taking this term in its most extended mathematical
acceptation. The variation is indispensable, in order that we may be
able to consider all possible positions; and the fixity is no less so,
in order that there may exist means of comparison. Thus, in all
_rectilinear_ systems, each of the two right lines will be subjected to
a fixed condition, and the ordinate will result from the variable
condition.
_Superiority of rectilinear Co-ordinates._ From these considerations it
is evident, as a general principle, that the most favourable system for
the construction of geometrical _loci_ will necessarily be that in which
the variable condition of each right line shall be the simplest
possible; the fixed condition being left free to be made complex, if
necessary to attain that object. Now, of all possible manners of
determining two movable right lines, the easiest to follow geometrically
is certainly that in which, the direction of each right line remaining
invariable, it only approaches or recedes, more or less, to or from a
constant axis. It would be, for example, evidently more difficult to
figure to one's self clearly the changes of place of a point which is
determined by the intersection of two right lines, which each turn
around a fixed point, making a greater or smaller angle with a certain
axis, as in the system of co-ordinates previously noticed. Such is the
true general explanation of the fundamental property possessed by the
common rectilinear system, of being better adapted than any other to the
geometrical representation of equations, inasmuch as it is that one in
which it is the easiest to conceive the change of place of a point
resulting from the change in the value of its co-ordinates. In order to
feel clearly all the force of this consideration, it would be sufficient
to carefully compare this system with the polar system, in which this
geometrical image, so simple and so easy to follow, of two right lines
moving parallel, each one of them, to its corresponding axis, is
replaced by the complicated picture of an infinite series of concentric
circles, cut by a right line compelled to turn about a fixed point. It
is, moreover, easy to conceive in advance what must be the extreme
importance to analytical geometry of a property so profoundly
elementary, which, for that reason, must be recurring at every instant,
and take a progressively increasing value in all labours of this kind.
_Perpendicularity of the Axes._ In pursuing farther the consideration
which demonstrates the superiority of the ordinary system of
co-ordinates over any other as to the representation of equations, we
may also take notice of the utility for this object of the common usage
of taking the two axes perpendicular to each other, whenever possible,
rather than with any other inclination. As regards the representation of
lines by equations, this secondary circumstance is no more universally
proper than we have seen the general nature of the system to be; since,
according to the particular occasion, any other inclination of the axes
may deserve our preference in that respect. But, in the inverse point of
view, it is easy to see that rectangular axes constantly permit us to
represent equations in a more simple and even more faithful manner; for,
with oblique axes, space being divided by them into regions which no
longer have a perfect identity, it follows that, if the geometrical
_locus_ of the equation extends into all these regions at once, there
will be presented, by reason merely of this inequality of the angles,
differences of figure which do not correspond to any analytical
diversity, and will necessarily alter the rigorous exactness of the
representation, by being confounded with the proper results of the
algebraic comparisons. For example, an equation like: _x^m_ + _y^m_ =
_c_, which, by its perfect symmetry, should evidently give a curve
composed of four identical quarters, will be represented, on the
contrary, if we take axes not rectangular, by a geometric _locus_, the
four parts of which will be unequal. It is plain that the only means of
avoiding all inconveniences of this kind is to suppose the angle of the
two axes to be a right angle.
The preceding discussion clearly shows that, although the ordinary
system of rectilinear co-ordinates has no constant superiority over all
others in one of the two fundamental points of view which are
continually combined in analytical geometry, yet as, on the other hand,
it is not constantly inferior, its necessary and absolute greater
aptitude for the representation of equations must cause it to generally
receive the preference; although it may evidently happen, in some
particular cases, that the necessity of simplifying equations and of
obtaining them more easily may determine geometers to adopt a less
perfect system. The rectilinear system is, therefore, the one by means
of which are ordinarily constructed the most essential theories of
general geometry, intended to express analytically the most important
geometrical phenomena. When it is thought necessary to choose some
other, the polar system is almost always the one which is fixed upon,
this system being of a nature sufficiently opposite to that of the
rectilinear system to cause the equations, which are too complicated
with respect to the latter, to become, in general, sufficiently simple
with respect to the other. Polar co-ordinates, moreover, have often the
advantage of admitting of a more direct and natural concrete
signification; as is the case in mechanics, for the geometrical
questions to which the theory of circular movement gives rise, and in
almost all the cases of celestial geometry.
* * * * *
In order to simplify the exposition, we have thus far considered the
fundamental conception of analytical geometry only with respect to
_plane curves_, the general study of which was the only object of the
great philosophical renovation produced by Descartes. To complete this
important explanation, we have now to show summarily how this elementary
idea was extended by Clairaut, about a century afterwards, to the
general study of _surfaces_ and _curves of double curvature_. The
considerations which have been already given will permit me to limit
myself on this subject to the rapid examination of what is strictly
peculiar to this new case.
SURFACES.
_Determination of a Point in Space._ The complete analytical
determination of a point in space evidently requires the values of three
co-ordinates to be assigned; as, for example, in the system which is
generally adopted, and which corresponds to the _rectilinear_ system of
plane geometry, distances from the point to three fixed planes, usually
perpendicular to one another; which presents the point as the
intersection of three planes whose direction is invariable. We might
also employ the distances from the movable point to three fixed points,
which would determine it by the intersection of three spheres with a
common centre. In like manner, the position of a point would be defined
by giving its distance from a fixed point, and the direction of that
distance, by means of the two angles which this right line makes with
two invariable axes; this is the _polar_ system of geometry of three
dimensions; the point is then constructed by the intersection of a
sphere having a fixed centre, with two right cones with circular bases,
whose axes and common summit do not change. In a word, there is
evidently, in this case at least, the same infinite variety among the
various possible systems of co-ordinates which we have already observed
in geometry of two dimensions. In general, we have to conceive a point
as being always determined by the intersection of any three surfaces
whatever, as it was in the former case by that of two lines: each of
these three surfaces has, in like manner, all its conditions of
determination constant, excepting one, which gives rise to the
corresponding co-ordinates, whose peculiar geometrical influence is thus
to constrain the point to be situated upon that surface.
This being premised, it is clear that if the three co-ordinates of a
point are entirely independent of one another, that point can take
successively all possible positions in space. But if the point is
compelled to remain upon a certain surface defined in any manner
whatever, then two co-ordinates are evidently sufficient for determining
its situation at each instant, since the proposed surface will take the
place of the condition imposed by the third co-ordinate. We must then,
in this case, under the analytical point of view, necessarily conceive
this last co-ordinate as a determinate function of the two others, these
latter remaining perfectly independent of each other. Thus there will be
a certain equation between the three variable co-ordinates, which will
be permanent, and which will be the only one, in order to correspond to
the precise degree of indetermination in the position of the point.
_Expression of Surfaces by Equations._ This equation, more or less easy
to be discovered, but always possible, will be the analytical definition
of the proposed surface, since it must be verified for all the points of
that surface, and for them alone. If the surface undergoes any change
whatever, even a simple change of place, the equation must undergo a
more or less serious corresponding modification. In a word, all
geometrical phenomena relating to surfaces will admit of being
translated by certain equivalent analytical conditions appropriate to
equations of three variables; and in the establishment and
interpretation of this general and necessary harmony will essentially
consist the science of analytical geometry of three dimensions.
_Expression of Equations by Surfaces._ Considering next this fundamental
conception in the inverse point of view, we see in the same manner that
every equation of three variables may, in general, be represented
geometrically by a determinate surface, primitively defined by the very
characteristic property, that the co-ordinates of all its points always
retain the mutual relation enunciated in this equation. This geometrical
locus will evidently change, for the same equation, according to the
system of co-ordinates which may serve for the construction of this
representation. In adopting, for example, the rectilinear system, it is
clear that in the equation between the three variables, _x_, _y_, _z_,
every particular value attributed to _z_ will give an equation between
at _x_ and _y_, the geometrical locus of which will be a certain line
situated in a plane parallel to the plane of _x_ and _y_, and at a
distance from this last equal to the value of _z_; so that the complete
geometrical locus will present itself as composed of an infinite series
of lines superimposed in a series of parallel planes (excepting the
interruptions which may exist), and will consequently form a veritable
surface. It would be the same in considering any other system of
co-ordinates, although the geometrical construction of the equation
becomes more difficult to follow.
Such is the elementary conception, the complement of the original idea
of Descartes, on which is founded general geometry relative to surfaces.
It would be useless to take up here directly the other considerations
which have been above indicated, with respect to lines, and which any
one can easily extend to surfaces; whether to show that every definition
of a surface by any method of generation whatever is really a direct
equation of that surface in a certain system of co-ordinates, or to
determine among all the different systems of possible co-ordinates that
one which is generally the most convenient. I will only add, on this
last point, that the necessary superiority of the ordinary rectilinear
system, as to the representation of equations, is evidently still more
marked in analytical geometry of three dimensions than in that of two,
because of the incomparably greater geometrical complication which would
result from the choice of any other system. This can be verified in the
most striking manner by considering the polar system in particular,
which is the most employed after the ordinary rectilinear system, for
surfaces as well as for plane curves, and for the same reasons.
In order to complete the general exposition of the fundamental
conception relative to the analytical study of surfaces, a philosophical
examination should be made of a final improvement of the highest
importance, which Monge has introduced into the very elements of this
theory, for the classification of surfaces in natural families,
established according to the mode of generation, and expressed
algebraically by common differential equations, or by finite equations
containing arbitrary functions.
CURVES OF DOUBLE CURVATURE.
Let us now consider the last elementary point of view of analytical
geometry of three dimensions; that relating to the algebraic
representation of curves considered in space, in the most general
manner. In continuing to follow the principle which has been constantly
employed, that of the degree of indetermination of the geometrical
locus, corresponding to the degree of independence of the variables, it
is evident, as a general principle, that when a point is required to be
situated upon some certain curve, a single co-ordinate is enough for
completely determining its position, by the intersection of this curve
with the surface which results from this co-ordinate. Thus, in this
case, the two other co-ordinates of the point must be conceived as
functions necessarily determinate and distinct from the first. It
follows that every line, considered in space, is then represented
analytically, no longer by a single equation, but by the system of two
equations between the three co-ordinates of any one of its points. It is
clear, indeed, from another point of view, that since each of these
equations, considered separately, expresses a certain surface, their
combination presents the proposed line as the intersection of two
determinate surfaces. Such is the most general manner of conceiving the
algebraic representation of a line in analytical geometry of three
dimensions. This conception is commonly considered in too restricted a
manner, when we confine ourselves to considering a line as determined by
the system of its two _projections_ upon two of the co-ordinate planes;
a system characterized, analytically, by this peculiarity, that each of
the two equations of the line then contains only two of the three
co-ordinates, instead of simultaneously including the three variables.
This consideration, which consists in regarding the line as the
intersection of two cylindrical surfaces parallel to two of the three
axes of the co-ordinates, besides the inconvenience of being confined to
the ordinary rectilinear system, has the fault, if we strictly confine
ourselves to it, of introducing useless difficulties into the analytical
representation of lines, since the combination of these two cylinders
would evidently not be always the most suitable for forming the
equations of a line. Thus, considering this fundamental notion in its
entire generality, it will be necessary in each case to choose, from
among the infinite number of couples of surfaces, the intersection of
which might produce the proposed curve, that one which will lend itself
the best to the establishment of equations, as being composed of the
best known surfaces. Thus, if the problem is to express analytically a
circle in space, it will evidently be preferable to consider it as the
intersection of a sphere and a plane, rather than as proceeding from any
other combination of surfaces which could equally produce it.
In truth, this manner of conceiving the representation of lines by
equations, in analytical geometry of three dimensions, produces, by its
nature, a necessary inconvenience, that of a certain analytical
confusion, consisting in this: that the same line may thus be expressed,
with the same system of co-ordinates, by an infinite number of different
couples of equations, on account of the infinite number of couples of
surfaces which can form it; a circumstance which may cause some
difficulties in recognizing this line under all the algebraical
disguises of which it admits. But there exists a very simple method for
causing this inconvenience to disappear; it consists in giving up the
facilities which result from this variety of geometrical constructions.
It suffices, in fact, whatever may be the analytical system primitively
established for a certain line, to be able to deduce from it the system
corresponding to a single couple of surfaces uniformly generated; as,
for example, to that of the two cylindrical surfaces which _project_ the
proposed line upon two of the co-ordinate planes; surfaces which will
evidently be always identical, in whatever manner the line may have been
obtained, and which will not vary except when that line itself shall
change. Now, in choosing this fixed system, which is actually the most
simple, we shall generally be able to deduce from the primitive
equations those which correspond to them in this special construction,
by transforming them, by two successive eliminations, into two
equations, each containing only two of the variable co-ordinates, and
thereby corresponding to the two surfaces of projection. Such is really
the principal destination of this sort of geometrical combination, which
thus offers to us an invariable and certain means of recognizing the
identity of lines in spite of the diversity of their equations, which is
sometimes very great.
IMPERFECTIONS OF ANALYTICAL GEOMETRY.
Having now considered the fundamental conception of analytical geometry
under its principal elementary aspects, it is proper, in order to make
the sketch complete, to notice here the general imperfections yet
presented by this conception with respect to both geometry and to
analysis.
_Relatively to geometry_, we must remark that the equations are as yet
adapted to represent only entire geometrical loci, and not at all
determinate portions of those loci. It would, however, be necessary, in
some circumstances, to be able to express analytically a part of a line
or of a surface, or even a _discontinuous_ line or surface, composed of
a series of sections belonging to distinct geometrical figures, such as
the contour of a polygon, or the surface of a polyhedron. Thermology,
especially, often gives rise to such considerations, to which our
present analytical geometry is necessarily inapplicable. The labours of
M. Fourier on discontinuous functions have, however, begun to fill up
this great gap, and have thereby introduced a new and essential
improvement into the fundamental conception of Descartes. But this
manner of representing heterogeneous or partial figures, being founded
on the employment of trigonometrical series proceeding according to the
sines of an infinite series of multiple arcs, or on the use of certain
definite integrals equivalent to those series, and the general integral
of which is unknown, presents as yet too much complication to admit of
being immediately introduced into the system of analytical geometry.
_Relatively to analysis_, we must begin by observing that our inability
to conceive a geometrical representation of equations containing four,
five, or more variables, analogous to those representations which all
equations of two or of three variables admit, must not be viewed as an
imperfection of our system of analytical geometry, for it evidently
belongs to the very nature of the subject. Analysis being necessarily
more general than geometry, since it relates to all possible phenomena,
it would be very unphilosophical to desire always to find among
geometrical phenomena alone a concrete representation of all the laws
which analysis can express.
There exists, however, another imperfection of less importance, which
must really be viewed as proceeding from the manner in which we conceive
analytical geometry. It consists in the evident incompleteness of our
present representation of equations of two or of three variables by
lines or surfaces, inasmuch as in the construction of the geometric
locus we pay regard only to the _real_ solutions of equations, without
at all noticing any _imaginary_ solutions. The general course of these
last should, however, by its nature, be quite as susceptible as that of
the others of a geometrical representation. It follows from this
omission that the graphic picture of the equation is constantly
imperfect, and sometimes even so much so that there is no geometric
representation at all when the equation admits of only imaginary
solutions. But, even in this last case, we evidently ought to be able to
distinguish between equations as different in themselves as these, for
example,
_x²_ + _y²_ + 1 = 0, _x⁶_ + _y⁴_ + 1 = 0, _y²_ + _e^x_ = 0.
We know, moreover, that this principal imperfection often brings with
it, in analytical geometry of two or of three dimensions, a number of
secondary inconveniences, arising from several analytical modifications
not corresponding to any geometrical phenomena.
* * * * *
Our philosophical exposition of the fundamental conception of analytical
geometry shows us clearly that this science consists essentially in
determining what is the general analytical expression of such or such a
geometrical phenomenon belonging to lines or to surfaces; and,
reciprocally, in discovering the geometrical interpretation of such or
such an analytical consideration. A detailed examination of the most
important general questions would show us how geometers have succeeded
in actually establishing this beautiful harmony, and in thus imprinting
on geometrical science, regarded as a whole, its present eminently
perfect character of rationality and of simplicity.
_Note._--The author devotes the two following chapters of his
course to the more detailed examination of Analytical Geometry of
two and of three dimensions; but his subsequent publication of a
separate work upon this branch of mathematics has been thought to
render unnecessary the reproduction of these two chapters in the
present volume.
THE END.
End of Project Gutenberg's The philosophy of mathematics, by Auguste Comte