This volume contains a brief exposition of the geometrical calculus and some of its applications to elementary differential geometry. The analytical geometry of Descartes (1637), operates on numbers which have an indirect relation with the geometrical elements which they represent. Leibniz in 1679 recognized the advantages of a geometrical calculus operating directly on geometrical elements, but the operation suggested by Leibniz does not possess the ordinary properties of algebraic operations. The idea, however was fruitful, and in 1797 Caspar Wessel gave an analytical representation of direction which contains Argand's (1806) geometrical interpretation of complex numbers and several of the operations introduced by Hamilton (1843-1854) in his method of quaternions. Later the barycentric calculus (1827-1842) of Möbius and the method of equipollences (1832-1854) of Bellavitis brought forward two independent methods of geometric calculus which their authors applied to various questions of geometry and mechanics. In 1843 Hamilton published his first essay on quaternions; the complete development of this theory in 1854 gives a complete geometrical calculus which finds at present its most extensive applications in mathematical physics. The works of Hamilton were preceded by the Ausdehnungslehre (1844), of H Grassmann which, in the power and simplicity of its operations, surpasses all other known forms of geometrical calculus. The method of exposition adopted by Grassmann is exceedingly abstract and this fact has stood stubbornly in the way of the general adoption of the Ausdehnungslehre to such an extent that we use today the barycentric calculus, the theory of equipollences, quaternions, or the Cartesian geometry, for the resolution of geometric questions which are capable of much more simple resolution by the methods of Grassmann. These classic objections to Grassmann's exposition have been met recently by Peano who has given concrete geometric interpretations to the forms and operations of the Ausdehnungslehre. There is a splendid account of the importance of this discipline in geometry, mechanics and physics to be found in the historical memoir of Schlegel. M Burali Forti gives the, elements of Grassmann's calculus as reconstructed by Peano. The latter took the idea of a tetrahedron as his starting point and defined the product of two and three points; he then defined the products of these elements by numbers and finally gave definitions of the sums of these products. The theory of forms of the first order gives the barycentric calculus and that of vectors; the geometric forms of the second order represent straight lines, orientations, and systems of forces applied to rigid bodies; the forms of the third order represent planes and the plane at infinity. Among the operations, the progressive and regressive products give the geometric operations of projecting and cutting; the inner product gives the orthogonal projections and the elements which we designate in mechanics by the terms moment, work, et cetera. In ordinary differential geometry simple properties most frequently yield themselves only after very complicated calculations. This complication is due in general to the use of coordinates; with these coordinates algebraic transformations are made on numbers in order to obtain certain formulae, namely, invariants, which are susceptible of geometric interpretations. On the other hand the geometrical calculus makes no use whatever of coordinates; it operates directly on the geometric elements; each formula which it produces is an invariant, capable of a simple geometric interpretation and leading directly to the graphic representation of the elements considered. Burali-Forti's work, though by no means a pioneer in the application of Grassmann's theories to differential geometry (note for example the memoirs of the younger Grassmann in the theory of curves and surfaces), shows the elegant power and simplicity of the geometrical calculus in elementary differential geometry and points the student to a vast field of transformations and researches in higher geometry.

A vector quantity may be combined with another similar quantity according to the parallelogram law which is familiar in the composition of velocities and forces. That may be taken as a definition of a vector quantity. The simplest example of a vector quantity is a displacement in space and this may be represented by a directed line. Suppose we take it so and define a vector to be a finite real directed line in Euclidean space. It then becomes a question of fundamental mathematical enquiry, what simplest set of axioms will give the law of Vector Addition. This is the object of Rudolf Shimmack's memoir, which therefore appeals to those mathematicians who are interested in such enquiries. Most of us are quite content to take the law of vector addition for granted as a direct result of physical experience; and push on to the developments of vector analysis, such as we find presented to us in the books named above.

But why will vector analysts always be rushing to the front with new notations? I have carefully read the modern books and articles on vector analysis by Gibbs, Heaviside, Föppl, Macfarlane, Henrici, Gans, Bücherer, Jahnke, Grassmann, and now these last, and - leaving out of account Gibbs's Dyadic - I find that they are all, where they treat of the same things, remarkably like one another, although they differ superficially in notation. In one thing they do agree - namely, a total disregard for the real essence of the quaternion vector analysis, from which to a large extent they have, sometimes without acknowledgment, and perhaps without knowledge, drawn their inspiration. In the essential feature which marks these systems off from Hamilton's method, they are all more or less copies of O'Brien's vector algebra, invented about the time during which Hamilton was rapidly developing his powerful calculus. ...

Personally I regard these systems as inferior to Hamilton's not only in their analytical conception but also in their practical power as a calculus. This inferiority is a feature of all the non-associative vector algebras which have been devised, including that which is now offered to us with all the authority of the two Italian mathematicians, Burali-Forti and Marcolongo.

Burali-Forti and Marcolongo claim that their system is the minimum system suitable for a large field of geometrical, physical, and mechanical problems. Instead of Gibbs's a.b and a × b for the "scalar and vector products" they use a × b and a∧b, introducing an inverted v as the symbol of the vector product. Of course there is absolutely nothing in this shuffling with symbols. The real virtue of a vector analysis cannot depend upon whether we use a dot or a cross or a wedge or a stroke interposed between the two vectors to represent what is generally called the vector product. The objection I have to all these notations is that they are essentially artificial and not in accordance with the usual methods of analysis. For example, Burali-Forti and Marcolongo follow the usual custom and write ang(ab) and sin(ab) for the angle between a and b, and the sine of that angle respectively, and yet maintain that the "vector product" is best symbolised by a separating symbol placed between the two constituent vectors. A most important factor of this "vector-product" is this very sin(ab). The "vector-product" is in fact a function of a and b, and in no true analytical sense a product of them. Thus we cannot pass from the equation a∧b = c to an equation of the form a = c ÷b. Why will so many vector analysts refuse to have anything to do with the complete analytical product of vectors? Analytically the vector quantity differs from the ordinary algebraic quantity in not satisfying the commutative law in multiplication. The products ab and ba are not the same. A geometrical meaning can be assigned to them; and the functions ab + ba and ab - ba are found to possess certain important properties, which make them of great use in geometrical and physical investigations. This is practically Hamilton's mode of approach; and his Sab and Vab, the Scalar and the Vector respectively of the product ab, are immediately deduced in all their significance without further definitions. It is the necessity for the introduction of definition after definition which declares the artificiality of the modern systems of vector analysis. Burali-Forti and Marcolongo have some novel ways of doing this. Having defined their vector product after the usual fashion they define the " internal product" a × b as the real number by which we must multiply any vector r normal to b so as to obtain the vector (a∧r)∧b. Then by introducing a second vector s also perpendicular to b they deduce the meaning of a × b. What advantage has this circuitous definition with its adventitious vectors which in no way affect the value of a × b - what advantage has this over the simple statement, let a × b = ab cos q? By the non-quaternionic approach, some definition must be given - why not choose the simplest?
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Outside these superficial changes there is absolutely nothing in their omografie to distinguish it from Hamilton's linear vector function. In spite of their claim in the preface no quaternionist could ever admit that this part of their calculus has greater potentiality than Hamilton's. It cannot have - it is exactly the same thing. It is of great interest, however, to see how Burali-Forti and Marcolongo work out the "derivates" of vector functions - not that there is anything new in their conceptions, but they have developed very skilfully a notation which Hamilton himself noted as a possible though not an analytically convenient mode of representation.
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I have intentionally made this review a comparison of Burali-Forti and Marcolongo's system with Hamilton's; for only in that way can we see what advantage, if any, the new system has. But the Italian analysts take what they regard as a much higher ground. In their preface to the Omografie Vettoriale they pride themselves on having presented the homographs or linear vector functions as absolute wholes and not as "tachigraphs" of the coordinates. But this is exactly what Hamilton did with the f, several years before Cayley came forward with the matrix notation. Hamilton's linear vector function is not an array as treated by him. "Further," they say, "it is well to consider linear transformations of vectors in vectorial homographs (of one function of these, namely, the Gradient, we make great use), which cannot be replaced, not even indirectly, by quaternions, when these are employed as absolute wholes, such as they have been given by Hamilton, and not as 'tachigraphs' of Cartesian coordinates." This means that Burali-Forti and Marcolongo graciously permit Hamilton to use the quaternion as a quantity involving four independent units, but on no account is he to be allowed to consider the properties of parts of the quaternion in which one, two, or three only of these units are explicitly contained. The vector part of Hamilton's quaternion is not necessarily a "tachigraph" of Cartesian coordinates. It can be made to represent fully a quantity which in ordinary analysis is represented by the coordinates $x, y, z$; but that by no means exhausts the possibilities of Hamilton's vector. In truth Burali-Forti and Marcolongo, like many other vector analysts, appropriate Hamilton's word "vector," give it a modified and restricted meaning, and then say that Hamilton has no right to use the vector at all. As I think I have shown in the above comparison, the gradient and other linear transformations of vectors in vectorial homographs are part and parcel of the quaternion vector analysis as developed by Hamilton, Tait, McAulay and Joly, and were known to quaternion workers years before Burali-Forti and Marcolongo, by imposing arbitrary limitations upon Hamilton's calculus, ruled it as out of order so that their so-called minimum system might have undisputed sway.