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aAftB afty ALEXANDER MACFARLANE analysis axes circle circular ratios circular sector circular versor components cos B sin cosAR cosh coshu coshw cosine deduce defined denote the ratio differentiation directed sine dyad ellipsoid elliptic equal equation equilateral equivalent excircle excircular expression exsphere find the product functions fundamental theorem geometrical Hence hyperbolic hyperbolic sector hyperbolic trigonometry hyperboloid i j k imaginary initial line length logarithm magnitude meaning meant mutually rectangular negative notation OA OA OA OP OA obtained ordinary algebra perpendicular to OA positive principal axis principle product versor Professor Gibbs quaternion radius vector ratio of twice right angles rotation rule scalar sin2 sinh A sinh sinhu sinhw sinw spherical trigonometry symbol tangent tion triangle twice the area vector quantity versor yy soo
Page 49 - I am convinced, however, that the introduction of the ideas, as distinguished from the operations and methods of Quaternions, will be of great use to us in the study of all parts of our subject, and especially in electrodynamics, where we have to deal with a number of physical quantities, the relations of which to each other can be expressed far more simply by a few expressions of Hamilton's, than by the ordinary equations.
Page 80 - How much more deeply rooted in the nature of things are the functions Sa/3 and Va/3 than any which depend on the definition of a quaternion, will appear in a strong light if we try to extend our formulae to space of four or more dimensions. It will not be claimed that the notions of quaternions will apply to such a space, except indeed in such a limited and artificial manner as to rob them of their value as a system of geometrical algebra. But vectors exist in such a space, and there must be a vector...
Page 124 - A single example of reasoning, in which symbols are employed in obedience to laws founded upon their interpretation, but without any sustained reference to that interpretation, the chain of demonstration conducting us through intermediate steps which are not interpretable, to a final result which is interpretable, seems not only to establish the validity of the particular application, but to make known to us the general law manifested therein.
Page 83 - B to have a common point of application 0 (fig. 10), their resultant or sum is the diagonal of the parallelogram of which A and B are the sides. The principle of the parallelogram of forces is thus one of the fundamental principles of the algebra of physics. Subtraction. — 'By subtracting one quantity of a vector from another quantity is meant finding the quantity which added to the former produces the latter. Let A (lig.
Page 124 - ... trigonometry, furnishes an illustration of what has been said. I apprehend that there is no mode of explaining that application which does not covertly assume the very principle in question. But that principle, though not, as I conceive, warranted by formal reasoning based upon other grounds, seems to deserve a place among those axiomatic truths, which constitute, in some sense, the foundation of the possibility of general knowledge, and which may properly be regarded as expressions of the mind's...
Page 77 - ... numerical quantities combined (as in algebra) with the notion of positive or negative quality are incomparably more amenable to analytical transformation than the simple numerical quantities of arithmetic. I do not know of anything which can be urged in favor of the quaternionic product of two vectors as a fundamental notion In vector analysis, which does not appear trivial or artificial in comparison with the above considerations. The same Is true of the quaternionic quotient and of the quaternion...
Page 123 - VOL. xi.r. 3 (33) ate steps which are not interpretable to a final result which Is interpretable, seems not only to establish the validity of the particular application, but to make known to us the general law manifested therein.
Page 124 - Whatever algebraical forms are equivalent, when the symbols are general in form but specific in value, will be equivalent likewise when the symbols are general in value as well as in form.
Page 78 - This objection is not valid against the method of quaternions as the algebra of versors or directed quotients, that, is, geometric ratios; but it is valid against it as claiming to be the algebra of vectors or physical magnitudes. The primary definition of the quaternion is the quotient, not the product, of two directed lines. "From the purely geometrical point of view, a quaternion may be regarded as the quotient of two directed lines In space, or what conies to the same thing as the factor or operator...
Page 95 - Application of Algebraical Symbols to Geometry," says, " If we combine more symbols than three, we find no geometrical interpretation for the result. In fact, it may be looked on as an impossible geometrical operation; just as \/ — 1 is an impossible arithmetical one.