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Plane affine geometry is developed from four axioms, up to the construction of a field (in general non-commutative), introduction of coordinates and derivation of equations for lines. The development is intended to be intelligible and rewarding to better high school students.

AMONG the various solutions to quad ratic and cubic equations, some possess an intuitive appeal arising from direct identification with geometric analogs. One analog, discussed here, shows that the quadratic solution known as "completing the square" actually conforms to the pic torial situation of fitting rectangles to gether in the shape of a square figure. A second analog likewise deals with the cubic equation and demonstrates the cor respondence between graphical and analy tical versions of a solution that might be appropriately called "completing the cube." Geometric analogs usually lack the ele gance that abstract symbolism can give to analytic solutions. Examples include the matrix formulation of vectors in theoreti cal mechanics and the complex-number representation of phasors in electrical engineering. Historically, however, pic torial solutions were the first to come within reach of man's intuitive compre hension. One historian speaks of a graphical solution to the cubic equation as early as the eleventh century, which he credits to the celebrated poet Omar Khayyam (Lamb 1936). Analytic solu tions that appeared in the sixteenth cen tury were due to the work of Scipio Ferro and Niccol? Fontana, also known as Tartaglia (Newman 1956, pp. 118-19). The most elegant of all analytic solutions to the cubic equation, due to Niels Abel, appeared only a little over one hundred years ago (Encyclopaedia Britannica 1960, p. 33). The present article begins with a geometric construction for obtaining one solution of a quadratic equation when both roots are real. Progressive generali zations then demonstrate constructions for the second real root, for both real roots simultaneously, and finally for the case of two complex roots. Geometric construc tions for the cubic equation demonstrate (a) the removal of the quadratic term and (b) a means for associating one solu tion with the edge-length of a "completed cube" under certain conditions. These constructions nevertheless serve as a de tailed guide to an analytic solution, free of all restrictive conditions.

Let us sum up the history of geometry from its beginnings in peg-and-cord constructions for circles and squares.

Based on the concept of quaternionic extension of general linear group, we consider the case of group extensions using generalised hypercomplex systems such as the Clifford algebra, and the Grassman algebra.

Our earlier studies [l, 2, 31 on the generalised Clifford algebra (G.C.il.) formulated by K. Yamazaki [4] 1 e d us to a surprising connection between the generalised Clifford algebra and the unitary groups which describe the internal symmetry of elementary particles. We shall now show that it is possible to obtain the matrices of the Duffin-Kemmer-Petiau [5] (D.K.P.) algebra which enter the space-time description of particles having spin zero or one through a wave equation, known in literature as the D.K.P. equation. Such a derivation of D.K.P. algebra from the generalised Clifford algebra leads us automatically to a method of constructing the elements of the algebra of the orthogonal groups also.

A geometric program concerns minimizing a function subject to constraint functions, all functions being of posynomial form. In this paper the posynomial functions are condensed to monomial form by the use of the inequality reducing a weighted arithmetic mean to a weighted geometric mean. The geometric mean is a monomial and by a logarithmic transformation it becomes a linear function. This observation shows that the condensed program is equivalent to a linear program. Moreover by suitable choice of the weights it is found that the minimum of the condensed program is the same as the minimum of the original programs. This fact together with the duality theorem of linear programming proves that the maximum of the dual geometric program is equal to the minimum of the primal geometric program. With this result as a basis a new approach to the duality properties of geometric programs is carried through. In particular it is shown that a “duality gap” cannot occur in geometric programming.

A new method of approach to Eddington’s wave-tensor calculus is given. The basis of the method is the concept of scale invariance. This leads to the idea of a scale basis for the algebra employed, and it is shown that this is only possible for the even Clifford algebras. Thus the origin of the E -number algebra is clearly shown, and the requirement of scale invariance then defines a four-dimensional space in which E -numbers are to be represented. The theory is carried far enough to provide for the simplest physical application— the finestructure of the hydrogen spectrum—and the connexion with Eddington’s statistical theory is also discussed.

The Maxwell equations in their general form, corresponding to the presence of electric and magnetic charge and currents, are shown to be invariant with respect to a previously unknown group, the “group of external transformations” of electromagnetic quantities. The introduction of this group is based essentially on the use of an algebraic method of writing the Maxwell equations in terms of a real-spinor basis algebra (one of the real forms of Clifford algebra). The algebraic notation is also used to determine the transformation properties of electromagnetic quantities with respect to spatial and temporal reflections.

About the middle of last century, there were invented several forms of multiple algebra, some of which are extensively used today in mathematical physics. The two types discussed here are quaternions, a quadruple algebra, and vector analysis, a triple algebra. The notations and forms of analysis used in the two systems are contrasted, using for each their representation of scalar and vector products and of Gauss's, Stokes's and Green's theorems. Maxwell made use of the vector part of quaternions in his Treatise on Electricity, and it was largely due to this that Gibbs came to invent his form of vector analysis. By 1890, about fifty years after the invention of quaternions, it became apparent that Gibbs's vector analysis was being used in place of quaternions, and this gave rise to a heated controversy in the pages of Nature. In the seventy years since this controversy, it is apparent that Gibbs's work and point of view have been completely vindicated.

is a manic irreducible polynomial in R[,$ with a,,+, # 0. If in particular v(e) = 5 1, then the corresponding algebra is called an r-Clifford algebra and denoted by C,(Q, V). In [3], it was shown that any r-Clifford algebra C,(Q, V) is isomorphic, as a vector space, to the special r-Clifford algebra C,(O, V) which may be called an r-exterior algebra. Furthermore, the dimension of r-Clifford algebras over a finite field k with 4 elements was considered, and shown that if dim, V > 2 and Y > & + 1)/2, then C,(Q, V) is of infinite dimension. It was conjectured that C,(Q, V) is infinite dimensional whenever dim, V > 2 and Y > 1 for an arbitrary field k. The purpose of this paper is to give a partial answer to this question.

The inherent U4 symmetry over spinor spaces of the regular decomposition indicated previously is discussed from the viewpoint of a single spinor space. It is noted that this symmetry does not yet include isotopic spin. Isotopic space is introduced by considering that the space‐time vectors and its vector Clifford algebra are imbedded in the algebra of a six‐dimensional Euclidean space. Three of the latter's dimensions are identified with ordinary space, and three are identified with isotopic space. The ``time vector'' corresponds to the pseudoscalar element in isospace, and the imaginary unit is expressed as a linear combination of the bivectors in isospace. The relation among complex conjugation, time inversion, and space inversion is thereby clarified. Some comments on the application of these ideas to particle symmetry discussions are made. In particular one obtains as the first extension of the SU2 symmetry of isotopic spin, a U3 symmetry for particles.

The physical quantities with which the science of mechanics is concerned can be loosely classified as scalars which can be completely specified by single real numbers, and vectors which require both magnitude and direction for their complete specification. Typical scalar quantities are mass, density, time, whereas vectors are typified by velocity, force, moment and acceleration. In this chapter an algebra is developed which can be applied to these vector quantities and which leads to a more concise and more comprehensible method of expressing some of the basic equations of mechanics. Once vector algebra has been developed a vector quantity can more properly be defined as a quantity with magnitude and direction which obeys the laws of vector algebra; not all quantities specified by a magnitude and a direction are vectors (see exercise 5, page 438).