3 Systematics and the Geometrical Spirit

### Varia

If the geometrical spirit is found in systematics, the systematic spirit is also found in geometry, more precisely in analytic geometry. The field that joined algebraic analysis and geometry for the benefit of each was the subject of intense activity in the first half of the 18th century. The study of curves was taken up first primarily as a means to determine the roots of equations. Increasingly curves came to be of interest in themselves, however; and by midcentury they had become a candidate for systematization. Leonhard Euler's Introductio in analysin infinitorum (1748) and Gabriel Cramer's Introduction à l'analyse des lignes courbes algébriques (1750) were largely successful, though different, efforts to meet the need for systematic order acknowledged by both.[50]

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Cramer's treatise, described by Speziali as "a true encyclopedia of algebraic curves," was well received in the latter 18th century and remained a classic text well into the 19th. Although Cramer conceded that the ancients had formulated useful particular propositions about curves, he gave modern mathematics high marks for its method, its art of deducing from a single universal principle a great number of truths, submitting them to general rules, and connecting them so as to stimulate new discoveries. Algebra, the "universal key of mathematics," provided the mathematician with "an ingenious means of reducing problems to the simplest and easiest calculation that the question proposed can admit."[51] And the key to algebra was the curve. A proper distribution of curves into orders, classes, genera, and species would serve mathematicians as a "well-arranged arsenal," enabling them to choose without hesitation the arms that might serve in the resolution of a proposed problem. Cramer credited Newton, whose Enumeratio linearum tertii ordinis had first appeared in 1704, with opening the way to such a classification.[52]

Cramer defines the order of algebraic curves "according to the degrees of their equations. . . . Thus one may form, for each order of lines, a general equation that represents all the possible lines of this order." To do so, Cramer makes use of J.P. de Gua de Malves' analytic triangle, a modified version of Newton's parallelogram arrangement of the terms of algebraic equations. The analytic triangle gives the different terms of a general equation of a given degree on a horizontal line, starting with a degree zero (a ) at the bottom, then degree one (by, cx ), degree two (dy[2] , exy, fx[2] ) and so on.[53]

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The subdivision of the first five orders of curves into their classes and genera is accomplished via the number, nature, and position of the infinite branches of each type of curve. The second order contains three classes: the ellipse, of which the circle is one species; the hyperbola; and the parabola. In essentials, Cramer follows Newton's Enumeratio in his division of third-order curves into four classes and fourteen genera. His fourth order has nine classes, based on the number and the hyperbolic or parabolic character of the infinite branches of its curves. For example, the eighth class contains the curves that have six hyperbolic branches. This class contains three genera: those curves that have only two, nonparallel asymptotes; those that have three asymptotes, of which two are parallel; and those that have three nonparallel asymptotes. The fifth order has eleven classes, defined by procedures similar to those for the fourth order (fig. 3.5).[54]

Cramer's classification, although comprehensive for its subject, was confined to plane curves. In the 1770s several mathematicians, including Euler, Joseph Louis Lagrange, and Gaspard Monge, began to take analytic geometry into three dimensions. From the outset of his epoch-making work in this field, Monge had as one of his concerns the groupings of its objects. In his Feuilles d'analyse (1795 and 1801) he discussed some twenty families of surfaces defined by their mode of generation.[55]

The Feuilles d'analyse first appeared as notes to Monge's course at the Ecole polytechnique, where as professor he was responsible for solid analytic geometry. Finding himself without a satisfactory

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textbook, he was compelled to put his results in systematic order. Similar pedagogical challenges created by the new Ecoles helped motivate other efforts at systematization and rationalization during the Revolution and the First Empire. In these efforts, the full systematic model in its Linnæan form, with logically nested categories—especially the canonical five, class, order, genus, species, variety—and binomial nomenclature, was frequently replaced by other arrangements more or less improvised for the matter at hand. Natural history categories might or might not be used, rationalized nomenclature might or might not be a goal, and other qualities of the full model such as numeration or formalism might be present in different degrees. What these undertakings did consistently have in common with the full systematic model were its emphasis on rationalization and method, its aim for comprehensiveness, and its promise of utility.

The work of Etienne Barruel, an examiner in physics at the Ecole polytechnique, is a case in point. As its title advertises, Barruel's textbook, Physics reduced to systematic tables (1799), offered students a complete summary of current physics in tabular form. Apart from brief prefatory remarks, the entire volume consists of thirty-eight tables, most in large fold-out format.[56]

Barruel defined physics "properly speaking" as the science that considers the properties of natural bodies, in contrast to natural history, which studies their varieties, organization, and so on, and chemistry, which considers their combination. "In the methodical order that I have adopted," Barruel wrote, "a science that has for its object the properties of bodies, cannot be subject to any other division than that of these same properties." Accordingly the first table, labeled tableau général de la physique , lists twenty-one fundamental physical properties, divided into two major groups. The first group comprises properties "that affect bodies in a constant manner" (extension, impenetrability, mobility, inertia, gravity). The second is constituted by properties "that affect bodies in a variable manner" (porosity, sonority, affinity, caloricity, electricity, elasticity, solidity,

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liquidity, gaseousness, capillarity, hygrometricity, meteoricity, crystallizability, light, magnetism, galvanism). Barruel subdivides the first group into "constant and essential" properties such as extension and impenetrability, and "constant and non-essential" properties, made up of gravity alone. He subdivides the second major group into "variable properties that belong to all bodies in general," such as porosity and affinity, and "variable properties that belong only to certain bodies," for example hygrometricity and light.[57]

The tableau général characterizes each property in one or two lines. For example, hygrometricity is the property by virtue of which liquids capable of wetting bodies enter their pores. Light is the property by virtue of which bodies excite a vivid impression in the organ of sight. At least one additional table is devoted to each property, elaborating on such points as its manner and circumstances of appearance, the laws to which it is subject, its relation to the senses, or instruments that may make use of it. Light, to give a single example, is the subject of seven tables. The initial division is made according to whether light reaches the eye directly, after reflection, or after refraction. Among the later subdivisions under refraction is a table that considers light in relation to instruments including telescopes and microscopes, subdivided in turn into their kinds.[58]

Except for its initial division of physics according to the properties of bodies, Barruel's text does not have the form of a classification of objects of the same general type (plants, animals, minerals, chemicals, algebraic curves, etc.) according to a determined set of categories. Instead it offers an arrangement of the different aspects of knowledge about what Barruel took to be single things (here properties, such as light) in a table formed by successive dichotomizations. No rule limits the number of dichotomies, and the kinds of criteria by which they are made may vary substantially within the same table. Barruel

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does not attempt a rationalized system of names, and indeed he derides reliance on "a simple nomenclature" for its aridity. Barruel's was nevertheless a project of systematization and rationalization, and he remarked that "it is to be desired that the elements of all sciences were accompanied by similar tables."[59]

One who heeded Barruel's call was Gaspard Prony. In 1800, when he published his Mécanique philosophique , Prony could draw upon five years of teaching experience at Ecole polytechnique. The title deliberately imitated Antoine Fourcroy's Philosophie chimique (1792), for, like Fourcroy, Prony intended to present a synoptic view of his science systematically organized on the basis of an analysis of its elements. Every even-numbered page contained formulas, definitions, and brief discussion. Each facing odd-numbered page was divided into four columns, the first defining the letters in the formulas, the second listing items defined in the text, the third and fourth stating theorems and problems. As far as possible, Prony eliminated demonstrations and "intermediate calculations" in favor of a concise presentation of results.[60]

Prony's procedure embodied a double movement of analysis and synthesis. He began with a complex mass of material—his lectures, current knowledge of mechanics in all its detail—and analyzed it into its matériaux primitifs , its simplest, most fundamental propositions or elements. Out of these he then constructed the complex structure of the science of mechanics, but in such a way that its components were grouped so that their relations were transparent. In this effort of

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systematization, in his implicit commitment to a sensationalist theory of knowledge, and in his remarks on science as a well-made language, Prony revealed his links with the encyclopedic tradition and the philosophy of Condillac as well as the kinship of his work with other systematic endeavors.[61]

Mechanics of a more immediately practical sort was the subject of Jean-Nicolas-Pierre Hachette's Traité élémentaire des machines (1811). A colleague of Barruel and Prony and student of Monge, Hachette taught a course on machines for engineers at the Ecole polytechnique. The Traité was based on these lectures and on the ideas of Monge, who treated the theory of machines as a branch of descriptive geometry.[62]

Hachette limited his treatment to machines that transform motion of one type into motion of another type. The types could be exhaustively enumerated: continuous circular, alternating circular, continuous rectilinear, and alternating rectilinear. These four types of motion make six when taken two at a time; and to these six combinations may be added the four that result when each movement reproduces itself. Hachette concluded, therefore, that there are ten different series of elementary machines.[63] Part of his table of elementary machines is reproduced here (fig. 3.6). Each machine is represented by a picture in a small box. The table presents the ten series of machines in numbered horizontal ranks; capital letters placed above the vertical columns allow a brief designation of each box. So, for example, the box 3a D designates a machine (the windmill) of the third series, which changes continuous rectilinear movement (wind) into continuous circular movement (the mill). Two series, the second and eighth, are empty, for no known machines changed continuous rectilinear into alternating rectilinear motion, or alternating rectilinear motion into itself.[64]

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The inclusion of two empty series in the table reveals in a graphic way the a priori and arithmetically determined character of Hachette's systematization. He identifies an exhaustive set of possibilities, subdivides them, and distributes existing machines into the preexisting categories. Once again, mathematics intersects the systematic model in criteria of classification.

Prompted in part by the same pedagogical needs that moved Monge, Prony, and Hachette, physiologists and anatomists analyzed and reanalyzed the human body into its functional and structural elements, classified these elements, and provided them with rationalized nomenclature. Xavier Bichat, who taught surgery at the Hôtel-Dieu in Paris, ordered the vital properties into which he analyzed the phenomena peculiar to living things into classes, genera, species, and varieties, and associated the divisions with the elementary tissues yielded by his anatomical analysis. Bichat published his work in a series of textbooks. So, too, did the professors of medicine, François Chaussier and Charles Louis Dumas, who were determined to provide their students with a rationalized nomenclature and classification of the parts of the human body.[65]

Physicians and pharmacists disturbed by radical criticism of traditional materia medica and drug therapy turned to the systematic model as the key to rationalization of knowledge of medicines. Pharmacists had an especially strong motive for reform, since the legitimacy of the body of esoteric knowledge on which their claim to a professional status comparable to that of physicians and surgeons was being threatened. From the 1780s on, systematic arrangements of the materia medica proliferated, especially in France, where differing

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versions were published by Antoine Fourcroy, Xavier Bichat, C.J.A. Schwilgué, Jean Alibert, and Jean Baptiste Barbier.[66]

3 Systematics and the Geometrical Spirit