## Lectures on Quaternions |

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algebra already analogous angle arbitrary Articles axes axis binary products biradial calculus called cardinal analysis circle coefficients common compare art conceived conception conjugate connexion considered construction coplanar corresponding denote differential direction ellipsoid equa equal equation equivalent example expression Faciend factor Factor x factum foregoing formula fourth proportional function Geocentric Vector geometrical given Heliocentric identity imaginary inscribed interpretation inversion Lecture length line of curvature minus mode multiplication namely negative numbers neral nions notation operation opposite ordinal relation ordinary algebra Pages perpendicular plane polygon position present Profactor proposed Provector Provectum quadrant quadratic equation quadrilateral quater quaternion quotient rectangular regarded represented respecting result REVECTOR right line rotation round scalar sides sphere spherical polygons spherical triangle spherical trigonometry square step subtraction successive Sun's Geocentric supposed surface symbol synthesis telescope tensor theorem theory tion Transvector triplets Vectum Vehend versor write

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Page 33 - Ba, as being in all cases constructed by the line BC. drawn to the middle point C of the line Aa: which would again agree with many modern systems. Thus Wallis seems to have possessed, in 1685, at least in germ (for I do not pretend that he fully and consciously possessed them), some elements of the modern methods of Addition and Subtraction of directed lines. But on the equally essential point of Multiplication of directed lines in one plane, it does not appear that Wallis, any more than Buee (see...

Page 30 - I thus perceived to exist, in that very general theory, deterred me from pursuing it farther at the time above referred to. |36| There was, however, a motive which induced me then to attach a special importance to the consideration of triplets, as distinguished from those more general sets, of which some account has been given. This was the desire to connect, in some new and useful (or at least interesting) way, calculation with geometry, through some undiscovered extension, to space of three dimensions,...

Page 13 - Perhaps I ought to apologize for having thus ventured here to reproduce (although only historically . . . ) a view so little supported by scientific authority. I am very willing to believe that (though not unused to calculation) I may have habitually attended too little to the symbolical character of Algebra, as a Language, or organized system of signs: and too much (in proportion) to what I have been accustomed to consider its scientific character, as a Doctrine analogous to Geometry, through the...

Page 1 - Reason," which appeared to justify the expectation that it should be possible to construct a priori a science of time as well as a science of space. The principal passage is as follows: " Time and space are two sources of knowledge from which various a priori synthetical cognitions can be derived. Of this pure mathematics gives a splendid example in the case of our cognitions of space and its various relations. As...

Page 13 - Graves. |19| After remarking that it was he who had proposed those names, of orders and ranks of logarithms, that early Essay of my own, of which a very abridged (although perhaps tedious) account has thus been given, continued and concluded as follows:— But because Mr. GRAVES employed, in his reasoning, the usual principles respecting Imaginary Quantities, and was content to prove the symbolical necessity without shewing...

Page 1 - It early appeared to me that these ends might be attained by our consenting to regard algebra as being no mere art, nor language, nor primarily a science of quantity ; but rather as the science of order in progression. It was, however, a part of this conception, that the progression here spoken of was understood to be continuous and unidimensional ; extending indefinitely forward and backward, but not in any lateral direction. And although the successive states of such a progression might, no doubt,...

Page 1 - ... regard algebra as being no mere art, nor language, nor primarily a science of quantity ; but rather as the science of order in progression. It was, however, a part of this conception, that the progression here spoken of was understood to be continuous and unidimensional ; extending indefinitely forward and backward, but not in any lateral direction. And although the successive states of such a progression might, no doubt, be represented by points upon a line, yet I thought that their simple successiveness...

Page 32 - ABu} is equal to the Double of AC. So that, whereas in case of Negative Roots, we are to say, The Point B cannot be found, so as is supposed in AC Forward, but Backward from A it may in the same Line: We must here say, in case of a Negative Square, the Point B cannot be found so as was supposed, in the Line AC; but Above that Line it may in the same Plain. This I have the more largely insisted...

Page 61 - Nothwithstanding these, and perhaps some other coincidences of view, Prof. Grassmann's system and mine appear to be perfectly distinct and independent of each other, in their conceptions, methods, and results. At least, that the profound and philosophical author of the Ausdehnungslehre was not, at the time of its publication, in possession of the theory of the quaternions, which had in the preceding year (1843) been applied by me as a sort of organ or calculus for spherical trigonometry , seems clear...

Page 540 - C + sin B sin C cos a. Similarly cos B = - cos C cos A + sin C sin A cos b, and cos C = — cos A cos B + sin A sin B cos c.