The Systematic Model

Systematic botany emerged as a fully developed science in the first half of the 18th century, as the culmination of a gradual development of concepts and methods over the preceding two hundred years. The stimulus of Renaissance humanism and naturalism in art, the growing popularity of natural history as a leisure pastime, and increasing exploration of the various habitats of European flora as well as the findings of European explorers in Asia, Africa, and America brought rapid increases in the numbers of known plants. A six-to-tenfold increase can be recorded between 1550 and 1623, when 6,000 species were listed in a leading textbook. By 1690, the number had more than tripled again, rising to 19,000.^[1]

From around 1600 problems of botanical nomenclature and classification became acute. Writers on botany recognized that creation of a standardized, uniform, and accurate system of naming, describing, and grouping plants was fundamental to the construction of a science of botany. Until such a system existed, the rapidly expanding knowledge of plants would yield more embarrassment than glory. By the time that Linnæus began his scientific career in the 1730s, many of the materials and methods of systematic botany were already in existence. It was only through Linnæus' work, however, that an authoritative system of nearly universal appeal emerged.^[2] In Linnæus' Systema naturae (1735–67, thirteen editions), Critica botanica (1737), Philosophia botanica (1751), Species plantarum (1753), and

[1] A.G. Morton, History of botanical science (London and New York: Academic Press, 1981), 115–286. On numbers of plant species, see also Gunnar Broberg, chap. 2 in this volume.

[2] W.T. Stearn, "Introduction" In Carl Linnæus, Species plantarum. A facsimile of the first edition 1753 , 2 vols. (London: The Ray Society, 1957), 1 , 1–176, esp. 3.

other writings, botanists were at last in possession of a comprehensive, rationalized language upon which to found their science.^[3]

The result was the relatively sudden appearance, around the middle of the 18th century, of a dogmatic confidence that the correct approach to botany had been found. Although particular points of Linnæus' classificatory schemes were questioned or modified, his methods of nomenclature and description had been generally adopted by botanists by the time of his death in 1778. Linnæus' work not only consolidated earlier developments in systematic botany, it also defined its future tasks. The search for new species and genera, their naming and systematic placement, became the main aim of botany, to the relative neglect of plant anatomy and physiology.

Linnæus' botany was a project of rationalization, an effort to create a set of concepts and procedures that would bring uniformity, consistency, and coherence to the representation of a clearly defined domain of natural objects. "Minerals grow, plants grow and live, animals grow, live, and have feeling," wrote Linnæus in the Systema naturae , thus marking off in epigrammatic style the three kingdoms to be organized.^[4] The essential tasks of the organizing system were to describe, to name, and to classify every object within its kingdom by uniform and consistent principles. In the Systema naturae Linnæus wrote that "the first step in wisdom is to know the things themselves; this notion consists in having a true idea of the objects; objects are distinguished and known by classifying them methodically and giving them names. Therefore, classification and name-giving will be the foundation of our science."^[5]

[3] Frans A. Stafleu, Taxonomic literature: A selective guide to botanical publications with dates, commentaries, and types (Utrecht: International Bureau for Plant Taxonomy and Nomenclature, 1967), 275–90; William T. Stearn, "Linnæan classification, nomenclature, and method," in Wilfrid Blunt, ed., The compleat naturalist: A life of Linnæus (New York: Viking Press, 1971), 242–49; Gunnar Eriksson, "Linnæus the botanist," in Tore Frängsmyr, ed., Linnæus: The man and his work (Berkeley: University of California Press, 1983), 63–109.

[4] Carolus Linnæus, Systema naturae 1735. Facsimile of the first edition , ed. and transl. M.S.J. Engel-Ledeboer and H. Engel (Nieuwkoop, Holland: B. de Graaf, 1964), 19 (quoted); Julius Sachs, History of botany 1530–1860 (Oxford: Clarendon Press, 1890), 7–9; Stearn, "Introduction," 3.

[5] Linnæus, Systema naturae , 19, 22–4. In practice Linnæus sometimes allowed his sense of affinities to overcome the rigid application of his criteria. See Stearn, "Linnæan classification," 244. On consistency as a criterion of systems, see Michel Adanson, Familles des plantes , 2 vols. (Paris: Vincent, 1763), 1 , xli. Daudin contrasts Linnæus' truly systematic approach, in which the number of categories is universally and necessarily limited to five (class, order, genus, species, variety), with "a simple 'synopsis,' in which the number of successive dichotomies vary at will from one section to another," Henri Daudin, De Linné à Jussieu: Méthodes de classification et idée de série en botanique et en zoologie (Paris: Librairie Felix Alcan, 1926), 38.

In the Critica botanica and Philosophia botanica , Linnæus laid down in aphoristic form detailed rules for describing, naming, and classifying plants. His binomial nomenclature for designating genus and species placed the plant within a logically integrated system. Binomials had been used in a haphazard way before Linnæus to define more or less isolated groups, but in the Linnæan system they became instruments of a rationalizing project to impose unity, consistency, and logical order on the whole field of botany.

Also central to Linnæus' botanical work was his sexual system of classification, the fundamental features of which were abstraction, numeration, and artificiality. Abstraction, because Linnæus proceeded by setting aside as irrelevant all but a few select qualities of the plant. The essence of plants, Linnæus said, consists in sexual reproduction or fructification. The essence of fructification consists in the flower and fruit, and the essence of the flower consists in the stamen and pistil. The essence of the stamen consists in anthers, the essence of the pistil consists in the stigma. Thus did Linnæus justify the narrowing of classificatory critieria to a part of a part of the plant. As W.T. Stearn and James L. Larson have remarked, Linnæus' thinking was informed by Aristotelian logic, and Linnæus drew an analogy between logical and natural forms. Yet it is founded also on a close study of particulars. Linnæus went well beyond his predecessors in distinguishing and naming the organs of fructification.^[6]

Having distilled the essence of the plant by abstraction, Linnæus classified it by numeration. The sexual system, in Stearn's words, is

[6] Linnæus, Systema naturae , 22–4; J.L. Larson, Reason and experience: The representation of natural order in the work of Carl von Linné (Berkeley: University of California Press, 1971), 2–3, 149; Stearn, "Introduction," 2–3. See also V.H. Heywood, "Linnæus—the conflict between science and scholasticism," in John Weinstock, ed., Contemporary perspectives on Linnæus (New York: University Press of America, 1985), 1–15.

"a basically simple but ingenious arithmetical system, whereby the genera are grouped into twenty-four classes according to the number of stamens (together with their relative lengths, their distinctness or fusion, their occurrence in the same flower as the pistil or their separation in unisexual flowers, or their apparent absence), while division into orders within each class is determined by number of pistils"^[7] (fig. 3.1). Such a system gives a privileged place to the numerical or spatial qualities of its objects, and to the visual sense. In the Systema naturae Linnæus refers to his science as one of "describing and picturing," and defines a naturalist as one who distinguishes and names the parts of natural bodies by sight.^[8]

Abstraction and numeration gave the sexual system a clarity and simplicity that translated into ease of use and certainty of identification. These advantages came at a price, however, namely the divergence that opened up between the classifications so constituted and those based on a less sharply defined but more intuitively satisfying grouping of the vegetable kingdom that resulted from simultaneous consideration of multiple characteristics. Julius von Sachs argued that the major aims of systematic botany from the late 16th century to Linnæus were to arrive at a natural grouping of plants and to identify the groups by a few easily recognizable marks. Linnæus was, in Sachs' view, the first to recognize that these two aims were incompatible. Linnæus sketched the elements of a natural system—an outline that became the basis of later attempts by others—but devoted his major effort to a classification, based on the sexual system, that was avowedly artificial. Even his use of the parts of the flower, justified by the flower's functional importance, contained an irony, since just those aspects of the parts used as criteria—number and connection—are functionally irrelevant.^[9]

Paying the price of artificiality proved a good investment for systematic botany. In the Systema naturae Linnæus compared tables of

[7] Stearn, "Introduction," 22–6.

[8] Linnæus, Systema naturae , 19.

[9] Linnæus, Systema naturae , 22–4; Stearn, "Introduction," 24–6; Sachs, History of botany , 7–9, 82–3. Linnæus considered his genera to be less artificial or conventional than his classes or orders. Cf. Daudin, De Linné à Jussieu , 34–48.

classification to maps. They were like maps in abstracting from a complex reality, in representing degrees of proximity and distance of a sort, and, above all, in their utility. The sexual system's clarity, simplicity, and ease of use opened up botany to the contributions of amateurs. The very step that constituted systematic botany as a science made it more, not less, accessible to popular participation. In this case, at least, it appears that the simplification effected by a successful effort of rationalization facilitated a real, although subordinate, role for the nonexpert.^[10]

In all of this the formal qualities of Linnæus' science are evident. Linnæus' formalism is expressed in his insistence on explicit rules, especially in the Critica botanica and Philosophia botanica , and their consistent application in the systematic works. It is found in the analogy of logical and natural forms that is implicit in the very idea of a system of nature and explicit in the parallel Linnæus drew between the hierarchies of logical and systematic categories. It is embodied in the a priori habit of mind that led Linnæus to embrace an artificial system placing clarity of concept over empirical intuition, and that may have led him to the sexual system in the first place. It is, finally, most obvious in his presentation, which is methodical, concise, impersonal, and—to some—arid. The formality of the Linnæan system was facilitated and reinforced by its atemporality. Time did not appear in the Linnæan scheme because within it species were constant, while each natural group had as its basis a common type, which, like a Platonic form, was beyond the reach of temporal change.^[11]

Linnæus' system was universal by intention and also in effect. In ways not yet systematically studied, botanists in the 18th century

[10] Linnæus, Systema naturae , 19; Stearn, "Introduction," 3; Blunt, The compleat naturalist , 183–92; and Stearn, "Linnæan classification," 244

[11] Linnæus, Systema naturae , 18, 22; Sachs, History of botany , 7–9; Stearn, "Introduction," 2–3, 17, 24–26; Larson, Reason and experience , 149–50. The epistemological issues raised by the analogy of logical and natural categories, and by the problematic status of systematics in relation to mathematical physics, were taken up by Kant in the Critique of judgment . See Ernst Cassirer, "Das Problem der Klassifikation und der Systematik der Naturformen" in Cassirer, Das Erkenntnis problem in der Philosophie und Wissenschaft der neueren Zeit von Hegels Tod bis zur Gegenwart (1832–1932), 4 (Stuttgart: W. Kohlhammer, 1957), 127–44.

formed an international community, corresponding, exchanging specimens and visits, and struggling toward a common scientific language. Publishing in Latin, and developing his system through an extensive network of international correspondents, Linnæus achieved widespread acceptance. By his death in 1778, the binomial system of nomenclature was universal and the sexual system in general use.^[12]

Related to the universalism of Linnæus' system was its ease of generalization. It was readily—almost promiscuously—transferable from one domain of objects to another. Nowhere is this better illustrated than in Linnæus himself, whom Sachs rightly compared to a "classifying, coordinating, and subordinating machine." Besides plants, he provided systematic arrangements for animals, minerals, and diseases. In the Bibliotheca botanica (1736) he classified botanists, in the process nicely revealing the relative value and place he assigned to the various components of the science (fig. 3.2).^[13]

The universality of these qualities of the systematic model may be seen in counterpoint in the work of one of Linnæus' major critics, the French botanist Michel Adanson. Confronted for the first time by a tropical flora during his six-year service in Senegal with the Compagnie des Indes, Adanson brought back to France a conviction of the inadequacy of the Linnean and every other system. In place of "systems," Linnæan or otherwise, which based classification on one or a few parts of the plant defined at the outset, Adanson proposed the "natural method." The botanist was first to consider "the ensemble of all the parts of plants," including roots, twigs, and fruits as well as flowers, insofar as these could be studied in their number, figure, situation, relative proportion, and symmetry. Affinities were to be determined by subsets of common features arrived at empirically and open to revision by new experience. In his Familles des plantes (1763), Adanson elaborated the principles of the natural method and applied it to the formation of fifty-eight families of plants.^[14]

[12] Stearn, "Introduction," 2–3.

[13] Sachs, History of botany , 89–91; Sten Lindroth, "The two faces of Linnæus," in T. Frängsmyr, ed., Linnæus , 1–62 (22–4); Linnæus, Bibliotheca botanica (Amsterdam, 1736).

[14] Michael Adanson, Familles des plantes, 1 , esp. clv. On Adanson, see Morton, History of botanical science , 301–14; Frans A. Stafleu, "Adanson and the 'Familles des plantes,'" in Adanson: The bicentennial of Michel Adanson's 'Familles des plantes' , 2 vols. (Pittsburgh: The Hunt Botanical Library, Carnegie Institute of Technology, 1963), 1 , 123–264, and Linnæus and the Linnoeans: The spreading of their ideas in systematic botany, 1735–1789 (Utrecht: Oosthoek, 1971), 310–39. An instructive comparison of "system" and "method" is given by Michel Foucault, The order of things: An archeology of the human sciences (New York: Vintage Press, 1973; translation of Les mots et les choses , 1966), 128–45.

Adanson gave the first full theoretical statement and justification of the method by which a natural arrangement of plants could be constructed. Despite his thoroughness, however, Adanson was not successful in his aim to displace the Linnæan system. That achievement fell to his younger contemporary and associate Antoine-Laurent de Jussieu, who benefited not only from the teachings of his uncle Bernard and of Adanson, but also from detailed study of the many new plants yielded by voyages of exploration, and from his adoption of the popular Linnean binomials. It was largely through A.-L. de Jussieu's Genera plantarum (1789) and other writings that the natural method gained ascendency in systematic botany in the early 19th century.^[15]

Although the differences between natural and artificial systems loomed large to their protagonists, on most points the two approaches shared the general features of the systematic model. Adanson's project was one of rationalization that stressed comprehensiveness, internal consistency, and disciplined use of language. Adanson insisted that his method was comprehensive—or, as he put it, "universal"—in that he strove to include all plants, tropical as well as temperate and glacial. Consistency, too, was to be sought in steady adherence to the more empirical rules of the natural method. Even Adanson's rejection of binomial nomenclature resulted from his commitment to principles of the priority and stability of names, in opposition to the precedence given by Linnæus to their significance.^[16]

[15] Linnæus' attempt at a natural arrangement—the Methodi naturalis fragmenta —first appeared as part of his Classes plantarum (1738). See Morton, History of botanical science , 301–14; Stafleu, Linnæus , 310–39; and Stearn, "Linnæan classification," 244. Stafleu notes common features in the views of Linnæus and advocates of the natural method. See "Adanson and the 'Familles des plantes,'" esp. 155, 166, 167, 229, 236.

[16] Adanson, Familles des plantes, 1 , clii-cliii, cxci-cxcii. On Adanson and nomenclature, see Stafleu, Linnoeus , 311; and "Adanson and the 'Familles des plantes,'" 187.

Adanson, like Linnæus, abstracted from the totality of the plant's features criteria for defining degrees of identity and difference. The two botanists differed not on the goal of abstraction but on the tactics used to obtain it. Where Linnæus' procedure was a priori and essentialist, at least at the level of classes and orders, Adanson's was inductive or "experimental," first looking at all features, then narrowing to a subset. The two naturalists also converged in their emphasis on numerically or spatially definable traits and the associated precedence of the visual sense. Here Adanson was, if anything, more explicit than Linnæus, remarking that botany "distinguishes plants only by their relations of quantity, whether numerical or discrete, or continuous, which gives us the extent of their surface or their size, their figure, their solidity."^[17]

On one point—simplicity, or ease of mastery and use—Linnæus' system had a seeming advantage. Adanson conceded as much and also that artificial systems had increased the popularity of botany and the numbers of botanical publications. But he complained that the result was merely superficiality, the spoiling of the most penetrating minds, and the production of quantities of catalogues on Linnæan principles. Most important, Adanson regarded artificial systems as incapable of bringing a permanent end to the confusion and uncertainty of botany. Certainty, stability—in short, a true system based on true principles—would come to botany only through the slower, more empirical, but ultimately more reliable procedures of the natural method.^[18]

In its insistence on explicit rules, its positing of a hierarchy of systematic categories implicitly analogous to logical ones, and its methodical, precise, and impersonal form of presentation, Adanson's natural method can be described as formalistic. The categories of Adanson's method, like those of Linnæus' system, are untouched by

[17] Morton, History of botanical science , 303, 306–8; Adanson, Familles des plantes, 1 , cc–cci (quote).

[18] Adanson, Familles des plantes, 1 , xli–xlii, clii–cliv, cxci–cxcv. Adanson sounded an Enlightenment theme when he appealed to nature as a source of truth in opposition to "the old prejudice in favor of systems and the ideas on which they are based." Ibid., clvii.

time or change.^[19] Where Adanson's formalism departed from that of Linnæus was in his clear break with essentialism and with Linnæus' a priori habits of mind. Like Linnæus, Adanson intended his method to be universally adopted. But the success of the Linnæan system, Adanson's rejection of binomials, and his lack of an institutional teaching position doomed the Familles des plantes to an oblivion from which it had to be rescued by Henri Baillon in the following century. Only through its adoption and extension by A.-L. de Jussieu did Adanson's method achieve a delayed acceptance and play a role in the formation of modern systematics.^[20]

A collaborator of Diderot and d'Alembert, described by Stafleu as plus encyclopédiste que les encyclopédistes , Adanson conceived a vast work that would extend the natural method to other parts of natural history, physics, chemistry, ethnology, philology, and related subjects. Although never brought to fruition, the project expresses Adanson's conviction of the compatibility of his method with the systematic model.^[21]

In one respect, at least, Adanson's natural method had an advantage over artificial systems that added a significant quality to the systematic model. Having identified the general features of a family, the Adansonian botanist could predict that other members of the group as yet unknown would have numbers of features in common. No such predictive quality could be relied upon for artificial systems. Besides its value for systematics, predictiveness enhanced the utility of the method, particularly at a time when voyages of exploration were bringing new plants to Europe in ever greater numbers. For example, plants of medicinal value might be found among the newly discovered specimens belonging to families already known to include medically useful plants.^[22]

[19] Morton, History of botanical science , 309–11.

[20] Stafleu, Linnæus , 311, and "Adanson and the 'Familles des plantes,'" 126–7, 187.

[21] Stafleu, "Adanson and the 'Familles des plantes,'" 136, 197; Jean-Paul Nicolas, "Adanson, the Man," in Adanson: The bicentennial, 1 , 1–121, esp. 35–6, 65–78.

[22] Adanson, Familles des plantes, 1 , lxxiii–lxxx, cxcv–cxcvi; Morton, History of botanical science , 306; Stafleu, Linnoeus , 330; and "Adanson and the 'Familles des plantes,'" 161, 196. When, in the first decades of the 19th century, Paris pharmacists turned to the chemical analysis of medicinal plants, they incorporated the botanists' prediction into their research program. See John E. Lesch, Science and medicine in France: The emergence of experimental physiology 1790–1855 (Cambridge, Mass.: Harvard University Press, 1984), 125–44.

The Model and the Geometrical Spirit

The systematic model was one expression of the geometrical spirit of the late Enlightenment. In its most general sense the geometrical spirit may be identified with analysis, a term and concept placed by Ernst Cassirer at the center of Enlightenment thought. In its first meaning analysis is indeed mathematical—or "geometrical" in 18th-century usage—referring especially to algebra. Enlightenment thinkers, however, allowed the term a wider formulation that gave it nearly unlimited applicability. In this formulation, "analysis" refers to a double movement of analysis and synthesis by which the phenomena of a field are reduced to their elements, and then restructured into a true whole that can be known by reason because, in Cassirer's words, reason "can reproduce it in its totality and in the ordered sequence of its individual elements." As Cassirer remarks further, for the Enlightenment "to 'know' a manifold of experience is to place its component parts in such a relationship to one another that, starting from a given point, we can run through them according to a constant and general rule." For Michel Foucault, too, analysis represents a universal method of classical thought that includes but is not confined to its mathematical expression in algebra. Systematics, in this view, is made possible by analysis in the form of a system of signs, and is a nonmathematical expression of the quest for a mathesis or universal science of measurement and order.^[23] The kinship of the systematic model with mathematics is therefore not a question of derivation or a direct modeling of systematics on one or another field of mathematics, but of the sharing of a generalized method of analysis and of the qualities that make such a method possible.

The most obvious of these are abstraction and numeration. Just as the mathematical sciences confined their treatment of the physical

[23] Ernst Cassirer, The philosophy of the Enlightenment , transl. Fritz C.A. Koelln and James P. Pettegrove (Boston: Beacon Press, 1955; original German edition 1932), 13–6, 23–4; and Foucault, The order of things , 46–77.

world to its numerable or measurable qualities of extension and motion, so did Linnæan and Adansonian systematics reduce the plant to the number, form, connection, and spatial arrangement of its characters. The formal qualities of Linnæus' and Adanson's approaches—their insistence on explicit rules, their methodical, impersonal, and economical presentation, and, in the case of Linnæus, a priori thinking subordinating the empirical to the conceptual clarity of the artificial—all have clear analogues in mathematics, as do the certainty and simplicity each system offered its users.

Adanson grasped the connection clearly. Insisting that botany was a science not merely of names but also of facts, he added that "we even believe that we find in it an immediate relationship with geometry." This relationship consisted not only in the exclusive use of characters subject to number and measure but also in the botanist's ability to pose questions analogous to "the most sublime geometry" in difficulty and instructiveness. "Find the most sensible point that establishes the line of separation or of definition between the family of the Scabiosa and that of the honeysuckle," or "Find a known genus of plants (natural or artificial, it does not matter) which occupies an accurate middle point between the family of dogbane and that of borage." Properly constructed, such questions would yield conclusions on the possibilities "that would be as evident and as well-demonstrated as the truths of the best geometry." In this way, too, the botanist would be able to estimate how many families or genera were lacking between two distant families or genera whose intermediaries were unknown, "presumptions which, if they would not have all the precision of mathematics, nevertheless would yield large views, and would furnish new means of extending our knowledge in botany." While admitting that absolute perfection in a botanical system—which he equated with "the necessary exactitude, which characterizes mathematical perfection"—was not possible, he nevertheless credited the families determined by the natural method with bringing to botany all the certitude, stability, comprehensiveness, concision, ease of use, and utility of which it was capable.^[24]

[24] Adanson, Familles des plantes, 1, cxci–cxcii, cc–cci. Underlying Adanson's presumption of the existence of unknown intermediaries between known families and genera was his commitment to the idea of a progression that connects "in a continuous series the families that resemble one another the most, and in each family, the genera that have the most general relationships." Cf. Adanson, Familles des plantes, 1, clxxxviii; Stafleu, Linnoeus , 328; and "Adanson and the 'Familles des plantes,'" 194. On the idea of series in botany and zoology, see Daudin, De Linné à Jussieu , esp. 79–187.

Another who saw systematics as an extension of the mathematical way of thinking to the ordering of empirical objects was Johann Heinrich Lambert. Much of his life work was an effort to reshape science and philosophy in the image of mathematics and to assure for those fields of knowledge the exactness and certitude of their exemplar. Lambert treated the problem of order and its measure not in relation to natural history or any domain of particulars, but abstractly as a problem of knowledge in general.^[25] Mathematics, he pointed out, had been most successfully applied where the objects of knowledge could be construed as homogeneous entities, because only such entities could be added, subtracted, and related to one another as more and less. In its formation of generic names and the abstract ideas derived from them, language often lost sight of the homogeneities on which the names and ideas were originally based. The result was conceptual confusion and failure in communication. Where homogeneities could be defined, however, "mathematics shows us that. . .they present ideas that are simple, very knowable, and exempt from logomachy. And this is what is necessary for a clear and well-arranged system."^[26]

According to Lambert, the degree of order of a given arrangement may often be calculated as a proper fraction. In the simplest case, a linear succession of objects, each of which is assigned a rank or value, absolute order is rank order. The degree of disorder that results from displacement of one object from this absolute order is taken as the product of the number of places the object is displaced and the value

[25] Johann Heinrich Lambert, "Essai de taxéometrie, ou sur la mesure de l'ordre," Akademie der Wissenschaften, Berlin, Nouveau mémoires , 1770, 327–42 and 1773, 347–68. See Christoph J. Scriba, s.v. "Lambert, Johann Heinrich" Dictionary of scientific biography (DSB) , 7, 595–600; Colloque international et interdisciplinaire Jean-Henri Lambert (Paris: Editions Ophyrs, 1979); and Gereon Wolters, Basis und Deduktion: Studien zur Entstehung und Bedeutung der Theorie der axiomatischen Methode bei J.H. Lambert (1728–1777) (Berlin: Walter de Gruyter, 1980).

[26] Lambert, "Essai de taxéometrie," 327–9.

of the object displaced. Similar calculations of degrees of order may be carried out in the more complex case of systems of classification. Lambert gives the example of a well-arranged library in which the books are classified first according to the sciences, next according to their age, their format, their binding, and so on. If each book satisfies all conditions, the library will be absolutely well-arranged. Its order will then be unity. It cannot be greater, but can admit of fractions.^[27] Suppose that there are n books and that each book must satisfy three conditions, a,b,c . The product n(a + b + c) = 1. But suppose an arrangement in which all books satisfy a , while of the other two conditions m books satisfy b and c, p books satisfy only b , q books satisfy only c , and r books satisfy neither b nor c . Then the degree of order of this arrangement would be expressed as the fraction:

What such a calculation measures is the degree to which a given classificatory system conforms to a set of explicitly defined, consistent criteria. It does not touch the fit of the scheme with nature, and Lambert, who was no naturalist, did not discuss the specific problems of botany or any other particular field. Nevertheless, Lambert's conviction that systematics could be treated as an extension of mathematics, and his implicit commitment to the wide applicability of the systematic model, do indicate once more the confluence of that model and the late Enlightenment's ambition to establish a general science of order.

Considered in a wider perspective, mathematics and the systematic model may also be associated as instruments of a movement of rationalization that was pervasive in the late Enlightenment. This movement found literary expression in numerous encyclopedias and dictionaries, of which the Encyclopédie was only the most famous.^[28] It appeared in government in forms as diverse as attempts to monitor

[27] Ibid., 329–37.

[28] Stearn, "Introduction," 11–2.

and improve public health through statistics, formulation of the metric system, and the shaping of armies as tools of the absolutist state. It entered the empirical sciences as an effort to classify their objects and reform their nomenclatures. Often its actions, especially in the sciences, were driven by pedagogy, as professors sought to order their subjects for presentation to students. Everywhere rationalization harnessed the geometrical spirit that, in different ways, informed both mathematics and the systematic model.

The development of systematic botany and of the fields for which it came to serve as a model was conditioned by material and social factors that remain to be investigated in detail. By the mid-18th century botany had long since achieved its intellectual independence of medicine, although—as the case of Linnæus indicates—important institutional links were preserved. Exploration and empire now provided the richest opportunities and resources for botanists and zoologists. Like early Christians spreading their creed over Roman roads, 18th-century naturalists were moved by their own sort of zeal to the far-flung corners of European colonial and commercial empires, and brought or sent back the specimens that gave substance to the systematists' projects. Adanson's experience in Senegal is an excellent case in point. Mineralogy owed much to the increasing need for technical expertise in mining and metallurgy, and chemistry was stimulated by its ever-closer association with pharmacy and industry. Nosology, of course, was an integral part of medical theory. The stimulus afforded by the high popularity of natural history among the educated classes must not be overlooked. Linnæus' productive stay in Holland from 1735 to 1738 was largely supported by the patronage of a wealthy banker, George Clifford. The mineralogist, Jean Baptiste Louis Romé de l'Isle, too, long derived his sole financial support from wealthy patrons with amateur interests in natural history. Another mineralogist, René Just Haüy, was himself an amateur botanist. And the medical profession, historically sensitive to areas of science invested with prestige by the lay public, could not fail to be affected by the popular prestige accorded the natural history disciplines.^[29]

[29] Stearn, "Introduction," 8–10; Blunt, The compleat naturalist , 102–8, 116–8.

The need for rationalization was felt most acutely in fields untouched by the great synthesis of Newton and the continental mathematicians. The mathematicians had provided a new basis for sciences—including astronomy, optics, and mechanics—that had already acquired mathematical form. In other fields, for which the conceptual structure was less secure or coherent, the order of the day was the gradual establishment of foundations. For some of these, such as electricity and magnetism, the path led from discursive theorizing and experimental manipulation to mathematization.^[30] For others, including chemistry and medicine as well as natural history, the systematic model of order was decisive. Where electricity and magnetism had to deal with specialized physical phenomena, the latter fields had to contend with an ever increasing quantity of diverse specimens.^[31]