Unit 0: Preface (Vorrede) by Karl Buzengeiger (pp. III–XVI, March 1822)¶
Karl Buzengeiger — Ordinarius Professor of Mathematics at the Grand-Ducal University of Freiburg
I. Mathematics as Science (Wissenschaft) and as Art (Kunst)¶
Mathematics permits itself to be contemplated from two points of view that differ greatly from one another: on the one hand it appears as a science (Wissenschaft), and on the other hand as an art (Kunst). In the former aspect one perceives its method of connecting a sequence of truths and uniting them into a system, in such a way that each single truth can be recognized out of those that precede it in a simple and evident manner. In the latter aspect, the methods and the heuristic artifices (Kunstgriffe) must be set forth that admit of application in order to discover something sought.
To become acquainted with mathematics from the first side, there exists among ancient and modern works none better than Euclid’s Elements. The geometric analytical works of Euclid and Apollonius—of which, to be sure, we no longer possess all originals themselves, though we do possess the admirable restorations of Robert Simson—and above all the extant writings of Diophantus are the oldest monuments of mathematical acumen from which the beginner may learn mathematics from the other side. In order to solve geometric problems, the Greeks employed a method which they called the analytical, whose invention is commonly attributed to Plato.
It consists in general in conceiving the problem as already solved, and then investigating backwards onto what simpler questions it allows itself to be reduced, proceeding thus until one arrives at a question whose resolution is already known. From Euclid we still possess a very beautiful and systematically ordered collection of such propositions serving as the foundation for this art, under the title Data. Since, however, it is often very difficult so to combine the given pieces of such a question with one another that serviceable consequences follow therefrom, the study of the works of those great geometers is of prime importance for learning the methods and artifices that must be applied in order to overcome such difficulties.
Diophantus’s writings concern another branch of mathematics; they consist of a numerous collection of arithmetical questions of a quite peculiar nature, and in their resolutions he endeavored to expound the artifices and methods by whose aid questions of this kind may be resolved. Whoever studies these writings of Diophantus will admire the sagacity and wealth of resources which his genius offered him, and will take pleasure in observing that through this study—augmented by the advantages which the modern state of arithmetic and algebra contributes—he has rendered himself skilled at resolving more general and harder problems of this kind. Other Greek treatises that are not mere problem-collections, such as the two books On the Sphere and Cylinder of Archimedes and the Conics of Apollonius, even when their authors annex the Analysis to the problems occurring therein, nevertheless exhibit principally the aim of presenting individual parts of geometry as deductive science.
II. Modern Algebraic & Trigonometric Symmetry (Synopsis of pp. V–XIV)¶
[Across pages V–XIV of the Preface, Buzengeiger traces the historical transition from ancient geometric analysis to modern symbolic algebra, Viète’s specious logistic, Descartes’s coordinate geometry, and symmetric trigonometric analysis—observing how symmetric algebraic relations among the sides and angles of rectilinear figures reveal whole families of geometric theorems simultaneously, provided the analyst guards against opaque symbolic sprawl.]
III. Combinatorial Operations, the Inexhaustible Triangle & Tribute to Feuerbach¶
[…] There exist perhaps analytical operations not yet noticed and not yet duly heeded, together with functions that accomplish something analogous with respect to them to what logarithms accomplish for arithmetical operations; naturally nothing definite can yet be said to illuminate this obscure conception, for if one could state it, the thing itself would already be at hand. Yet attempts of this kind have already been made: factorials (Fakultäten) and combinatorial operations belong here. The former have manifestly thus far received too little attention from analysts. A vast multitude of integrals whose numerical evaluation cannot be carried out through the ordinary expansion into series can be accomplished very simply by means of factorials, and their relations furnish new resources for finding series for such integrals yielding any desired degree of approximation. Combinatorial operations, on the other hand, have hitherto been viewed from the right side by very few, and what is important and beautiful in them has been hidden and disfigured beneath an exceedingly repulsive and unserviceable wilderness of notation (Zeichenwust).
And thus one stands precisely at the point of having to confess that, while one cannot yet conceive how mathematics might experience another epoch equal in its effect to those that have gone before, this was doubtless always the case before a new epoch entered. Yet this much is certain: each succeeding advance must become harder, for in its progress this science resembles collections of natural specimens—at the outset such a collection can be rapidly augmented, whereas the larger it grows, the more slowly additions proceed. Nevertheless, this study remains in itself good, noble, and of the weightiest influence for the whole of life; hence let this deter no one from devoting himself to it. Moreover, the domain of this science remains forever inexhaustible,
and infinitely much may still be accomplished that is worthy of the gratitude of contemporaries and of posterity, even though it may no longer be so easy as it was a century ago to bring forth something of the first magnitude.
I believe I cannot substantiate this better than by appealing to the treatise that follows. The plane triangle is the simplest geometric figure, and from the very first origin of geometry down to the present day geometers have striven to investigate its properties without being able to exhaust this source, as this very treatise demonstrates, containing as it does a considerable series of the most remarkable and most beautiful propositions belonging hither. Since, moreover, the Author was for several years my auditor (mein mehrjähriger Zuhörer), I refrain from all further praise, contenting myself merely to remark that the manner in which he has treated his subject attests a thoroughly systematic mind (einen sehr systematischen Kopf), and adding besides that he might just as well have ventured upon far higher subjects.