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In his discussion of Vieta, Klein asserts that Vieta's concept "species"
is the modern concept of number. There is some difficulty in
understanding what Klein means. Robert Schmidt in "The Early theory of
equations" (Golden Hind Press, Annapolis, 1986), provides a useful
glossary of some of Vieta's terms. The entry on Species may be of
interest here:

Species (Species). The counterparts of magnitudes in Vieta's algebraic
logistic, or calculus. An equivalent Greek term is used by Diophantus
in his Arithmetic.
Vieta would probably have defined logic as the art of comparing things
correctly. Different logics would compare things under different
aspects. One way of comparing geometrical entities is from a
quantitative point of view. Another might be from a figural point of
view. Vieta introduced still another logic that was interested in the
formal aspects of magnitudes, which may roughly be defined as the role
they play in a problem. For example, instead of considering a rectangle
to be a figure or an area, it might be regarded as something two
dimensional - a kind of function of its sides where the sides are
present to one another in such a way that they create something of a
higher order or genus. Vieta calls this a plane or a factate; he calls
the sides latitudes & longitudes. Alternatively, we might regard this
rectangle as a homogeneum. This is to look at the rectangle as a kind
of function of its substantial parts, which are present to one another
in a conjunction of affirmation or denial Thus, the parts are of the
same genus as the whole. For example, the rectangle may be understood
as the sum of two squares.
Vieta's introduction of species was meant to direct attention to the
ways in which magnitudes show themselves in algebraic comparison. This
formal aspect is not something intrinsic to magnitudes like figure &
size are. In fact, in the course of one and the same algebraic argument
it is possible for a magnitude to change its role (see transformation).
But Vieta thinks that this is the proper way of regarding things when
practicing the algebraic art.
A logic may be served by a logistic, which for Vieta is the art of
making vicarious comparisons of things. A logistic employs easily
manipulable counterparts or proxies to exhibit the intended comparison
of the things under examination - magnitudes, for instance. The
numerical logistic of Vieta's day compared things under the aspect of
quantity, and it employed written numbers to exhibit this comparison.
Vieta's new logic also was served by a suitable logistic, and he called
the elements of this logistic species.
Since there does not seem to be any one realm to which these functions
intrinsically belong, and since even for magnitudes it is necessary to
set up conventions defining how they are factates or homogenea, just
about any written symbols could serve as the elements for this logistic.
But it would be nice if we could find counterpart elements whose
intrinsic relationships were not at variance with the very ideas behind
factate or homogeneum functions. Ideally, they would even express these
functions in some relatively uncontrived way.
At least two considerations would rule out written numbers. First of
all, all numbers are homogeneous, while magnitudes are not. Again every
numerical homogeneum can ultimately be reexpressed as a sum of units, as
a consequence of the fact that the unit elements of numbers are all
equal. In the case of magnitudes, however, the homogeneum that is a
conjunction by denial is not in general reducible to one that is a
conjunction by affirmation, at least not in terms of the unequal but
proportional roots that constitute the equation in recognition, which
are the elements from which these function must be made. Denial is an
equally primitive operation.
Vieta chose letters. From his formal point of view, letters can be
elements in a factate or homogeneum function just as uncontrivedly as
magnitudes themselves, because in different types of configurations
letters can have a different import as sounds. A certain configuration
of letters generates a syllable, which might be interpreted as a
factate. A certain configuration of syllables (or occasionally
individual letters) produces a word, which might be interpreted as a
homogeneum. What Vieta does is to take the principle behind syllable
and word formation and refine it to the point where there is a perfect
correspondence between the collections of letters, and the different
types of factate and homogeneum functions that come up during the
algebraic comparison of magnitudes.
The manipulable elements of a logistic have an ambivalent status as
symbols. On the one hand, they are the tangible counterparts to things
not readily compared directly. But, in so far as they are written on
paper, they tend to be taken as signs pointing to a universal, much in
the manner of any other written word. One is naturally led to wonder
whether Vieta thought that the species signified some pure and
non-sensible entity, considering the philosophical importance of this
word. It is hard to tell.
But I don't think that the introduction of species posed a new
ontological problem for Vieta. In the numerical logistic of the time,
written numbers were not so much symbols meant to designate pure or
ideals numbers, as easily manipulable examples of multitudes. The marks
on paper were substitutes for the actual multitudes much as the beads on
an abacus might serve as counterparts for a number of real things. The
multitudes were implicit in the numerical figures themselves, the
isolated figure 5, say, standing for a group of five single strokes of
the pen. This is particularly clear in the case of Roman 'numerals',
where the first three are simply one more stroke of the pen. The Arabic
figures were introduced into the West merely as a better way of writing
numbers, and preserved this sense of 'standing for' multitudes.
So the written collections of letters are species in the same sense
that the numerical figures are numbers. At the very least we might say
that they are each the privileged instances of the mathematical entities
they exemplify.