## The Quarterly Journal of Pure and Applied Mathematics, Volume 22J.W. Parker, 1887 - Mathematics |

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### Common terms and phrases

A₁ acnodes ALGEBRA angle asymptote B₁ c-equation C₁ c₂ cercles cercles doubles coefficients COLENSO'S complexe linéaire complexe polaire complexe quadratique conic constants coordonnées corresponding courbe bicirculaire crunodes cubic curve cusp D₁ denote determined deux cercles differential equation droite e₁ elliptic functions équations evanescent evanescent quadrilaterals evanescent triangles faisceau formulæ function ƒ µ G₁ given Hence infinite infinity integral intersection l'équation locus modular equation multiplication obtain ordre p-discriminant P₁ peut plane quadric quadrilaterals quartic quaternion quatre rapport relation self-conjugate singular solutions singularity solution species Species II sphere spherical harmonic surface symbols symmetric function système tac-locus tangents theorem u₁ vanish velocity velocity potential XXII Y₁ αμ µ³

### Popular passages

Page 271 - ... the original units may be replaced by linear functions of these units, so as to give rise, for the units finally adopted, to a multiplication table of the most simple form; and it is very remarkable, how frequently in these simplified forms we have nilpotent or idempotent symbols (i...

Page 271 - ... and consequently how simple are the forms of the multiplication tables which define the several systems respectively. I have spoken of this multiple algebra before referring to various geometrical theories of earlier date, because I consider it as the general analytical basis, and the true basis, of these theories. I do not realise to myself directly the notions of the addition or multiplication of two lines, areas, rotations, forces, or other geometrical, kinematical, or mechanical entities;...

Page 271 - I, &c., with coefficients which are ordinary analytical magnitudes, real or imaginary, viz. the coefficients are in general of the form x + iy, where | i is the before-mentioned imaginary or >f^) of ordinary analysis.

Page 271 - ... so that there is a determinate and unique product of three or more letters; or, what is the same thing, the laws of combination of the units...

Page 280 - ... or forwards, but can be measured in the same plane above the line,, or (as appears elsewhere) at right angles to the line either in the plane, or in the plane at right angles thereto. Considered as a history of algebra, this work is strongly objected to by...

Page 282 - ... merits were fully recognized by De Morgan and acknowledgments of indebtedness were frankly made by Hamilton. Throughout Warren's work the term quantity, like Cauchy's geometric quantity, indicates a line given in length and direction. Some of his definitions are as follows : The sum of two quantities is the diagonal of the parallelogram whose sides are the two quantities. The first of four quantities is said to have to the second the same ratio which the third has to the fourth ; when the first...

Page 271 - ... system of multiple algebra or linear associative algebra, developed in the valuable memoir by the late Benjamin Peirce, Linear Associative Algebra (1870, reprinted 1881 in the American Journal of Mathematics, vol. IV., with notes and addenda by his son, CS Peirce) ; we here consider symbols A, B, &c.

Page 270 - X greater than 3, of the equation = z\ has given rise to investigations of very great interest and difficulty. Outside of ordinary mathematics, we have some theories which must be referred to : algebraical, geometrical, logical. It is, as in many other cases, difficult to draw the line ; we do in ordinary mathematics use symbols not denoting quantities, which we nevertheless combine in the way of addition and multiplication, a + b, and ab, and which may be such as not to obey the commutative law...

Page 281 - C'est un nouvel adjectif joint au substantif ordinaire i, et non un nouveau substantif. Mais que veut dire ce signe ? Il n'indique ni une addition, ni une soustraction, ni une suppression, ni une opposition par rapport aux signes + et — . Une quantité accompagnée de \/— l n'est ni additive, ni soustractive, ni égale à zéro. La Es qualité marquée par \/ — i n'est opposée ni à cell« qu'indiqué -|-, ni à celle qui est désignée par — . Cju 'est-elle donc?

Page 271 - ... and which may be such as not to obey the commutative law ab = ba : in particular, this is or may be so in regard to symbols of operation; and it could hardly be said that any development whatever of the theory of such symbols of operation did not belong to ordinary algebra. But I do separate from ordinary mathematics the system of multiple algebra or linear associative algebra, developed in the valuable memoir by the .late Benjamin Peirce, Linear Associative Algebra (1870, reprinted 1881 in the...