Quaternion

“ The question will then arise with respect to the comparatively limited field which is the peculiar property of Hamilton, How important are the advantages to be gained by the use of the quaternion? This question, unlike the preceding, is one into which a personal equation will necessarily enter. Everyone will naturally prefer the methods with which he is most familiar; but I think that it may be safely affirmed that in the majority of cases in this field the advantage derived from the use of the quaternion is either doubtful or very trifling. ”

“ Quaternions (as a mathematical method) is an extension, or improvement, of Cartesian geometry, in which the artifices of co-ordinate axes, &c., are got rid of, all directions in space being treated on precisely the same terms. It is therefore, except in some of its degraded forms, possessed of the perfect isotropy of Euclidian space. From the purely geometrical point of view, a quaternion may be regarded as the quotient of two directed lines in space—or, what comes to the same thing, as the factor, or operator, which changes one directed line into another. ”

“ In a system of vector analysis, in which the principle of development is not thus predetermined, it seems to me contrary to good method that the more simple and elementary notions should be defined by means of those which are less so.
The quaternion affords a convenient notation for rotations. The notation where is a quaternion and the operand is to be written in the parenthesis, produces on all possible vectors just such changes as a (finite) rotation of a solid body. Rotations may also be represented, in a manner which seems to leave nothing to be desired, by linear vector functions. ”

“ C. J. Joly's projective geometrical applications starting from the interpretation of the quaternion as a point-symbol; [8] these applications may be said to require no addition to the quaternion algebra; (b) W. K. Clifford's bi-quaternions and G. Combebiac's tri-quaternions, which require the addition of quasi-scalars, independent of one another and of true scalars, and analogous to true scalars. As an algebraic method quaternions have from the beginning received much attention from mathematicians. An attempt has recently been made under the name of multenions to systematize this algebra. ”

“ And that likewise the quaternion generates, [3] he says, the perfect number, as in the case of intelligibles (the monad) does the decade, they teach thus. If any, beginning to number, says one, and adds two, then in like manner three, these (together) will be six, and to these (add) moreover four, the entire (sum) , in like manner, will be ten. For one, two, three, four, become ten, the perfect number. Thus, he says, the quaternion in every respect imitated the intelligible monad, which was able to generate a perfect number. ”

“ Quāter′nion, a set or group of four: a word of four syllables: (pl.) in mathematics, a calculus of peculiar power and generality invented by Sir W. R. Hamilton of Dublin, as a geometry, primarily concerning itself with the operations by which one directed quantity or vector is changed into another.—v.t. to divide into quaternions: (Milt.) to divide into companies.—ns. Quāter′nionist, a student of quaternions; Quātern′ity, the state of being four, a group of four.—Quaternary number, ten; so called by the Pythagoreans because equal to 1 + 2 + 3 + 4. ”

“ Hamilton was not the only worker at the theory of sets. The year after the first publication of the quaternion method, there appeared a work of great originality, by Grassmann [6] in which results closely analogous to some of those of Hamilton were given. In particular, two species of multiplication (“inner” and “outer ”) of directed lines in one plane were given. The results of these two kinds of multiplication correspond respectively to the numerical and the directed parts of Hamilton’s quaternion product. ”

“ For one, two, three, four, become ten—as has been previously proved—if the proper denomination be preserved, according to substance, for each of the numbers. This is the sacred quaternion, according to Pythagoras, having in itself roots of an endless nature, that is, all other numbers; for eleven, and twelve, and the rest, derive the principle of generation from the ten. Of this decade—the perfect number—there are called four parts—number, monad, power, cube—whose connections and mixtures take place for the generation of increase, according to nature completing the productive number. ”

“ The properties of linear vector functions were further studied in 'Scalar invariants of two linear vector functions' (Trans. R.I.A. 1896, xxx. 709) and 'Quaternion invariants of linear vector functions' (Proc. R.I.A. 1896, iv. 1) , while the extension of the quaternion calculus to space of more than three dimensions was discussed in 'The associative algebra applicable to hyperspace' (Proc. R.I.A. 1897, v. 75) ; the algebras considered are those that are associative and distributive, and whose units satisfy equations of the same type as the units of quaternions. ”

“ But there has been little progress in recent years, except that made by Sylvester in the solution of quaternion equations, nor has the application of quaternions to physics been as extended as was predicted. The change in notation made in France by Hoüel and by Laisant has been considered in England as a wrong step, but the true cause for the lack of progress is perhaps more deep-seated. There is indeed great doubt as to whether the quaternionic product can claim a necessary and fundamental place in a system of vector analysis. ”

“ The second charge evidently relates to the notations and (see Nature, vol. xlvii, p. 592) . It is perfectly true that I have used a scalar and a vector connected with the linear vector operator, which, if combined, would form a quaternion. I have not. thus combined them. Perhaps Prof. Knott will say that since I use both of them it matters little whether I combine them or not. ”

“ To show what I mean, I have only to compare Hamilton's treatment of the quaternion and of the linear vector operator with respect to notations. Since quaternions have been identified with matrices, while the linear vector operator evidently belongs to that class of multiple quantities, it seems unreasonable to refuse to the one those notations which we grant to the other. Thus, if the quaternionist has why should not the vector analyst have where represents a linear vector operator? ”

“ We believe that this tri-quaternion solution of the very interesting problem proposed by Combebiac is the best one. But the first thing that strikes one is that it seems unduly complicated. A point and a plane fix a line or axis, viz. that of the perpendicular from point to plane, and therefore a calculus of points and planes is ipso facto a calculus of lines also. To fix a weighted point and a weighted plane in Euclidean space we require 8 scalars, and not the 12 scalars of a tri-quaternion. We should expect some species of bi quaternion to suffice. ”

“ If quaternions, of which I have just spoken, had not been so promptly utilized by the English physicists, many persons would doubtless see in them only a useless fancy, and yet, in teaching us to liken what appearances separate, they would have already rendered us more apt to penetrate the secrets of nature. ”

“ Now, these equations will hold if interpreted in the quatemionic sense, as is, indeed, a necessary consequence of their holding in the dyadic sense, although the converse would not be true. My critic was thus led, in consequence of the inadvertence mentioned, to suppose that I had departed from my ordinary usage and my express definitions, and had intended the products in these integrals to be taken in the quaternionic sense. This is the sole ground for the last charge. ”

“ Sir W. R. Hamilton's method of Quaternions is a new instrument of research; and, whether or not as valuable as some think, undoubtedly adds a large region to the world of known mathematical truth. And then, more important still, there are the achievements of Cayley and Sylvester in the development of the higher algebra. ”

“ Hamilton displayed wonderful fertility in their development. His Lectures on Quaternions, delivered in Dublin, were printed in 1852. His Elements of Quaternions appeared in 1866. Quaternions were greatly admired in England from the start, but on the Continent they received less attenttion. P. G. Tait's Elementary Treatise helped powerfully to spread a knowledge of them in England. Cayley, Clifford, and Tait advanced the subject somewhat by original contributions. ”

“ The cult of the quarters is intimately associated with the concept of the universe. With the four cardinal points a number of others are sometimes taken including the zenith, the nadir, and the middle. The sacred numbers 4, 5, 6, and 7 may thus conceivably be derived from the points of space, but it would be very unsafe to assume that they are necessarily so derived. The general concept of a universe divided into quarters, fifths, or sixths is a powerful conventionalizing factor in mythology, religion, and art. ”

“ FitzGerald's verse was evidently also influenced by distich 1866 of the Mantik ut-tair.
Heaven and hell are reflections, the one of thy goodness, and the other of thy wrath.
87
LXVIII.
We are no other than a moving row
Of Magic Shadow-shapes that come and go
Round with the Sun-illumined Lantern held
In Midnight by the Master of the Show;
This quatrain is translated from O. 108.
This vault of heaven beneath which we stand bewildered,
We know to be a sort of magic-lantern: [73]
Know thou that the sun is the flame and the universe is the lamp,
We are like figures that revolve in it. ”

“ The fragment of golden quartz lies on its clean white slab with a brilliant light upon it.
Mysterious little golden rock! What secrets are there, down beyond the vanishing point in the realm of the infinitely small? Our human longings go to Alan and Glora.
But sometimes we are swept by the greater viewpoint. Awed by the mysteries of nature, we realize how very small and unimportant we are in the vast scheme of things. We envisage the infinite reaches of astronomical space overhead. Realms of largeness unfathomable. And at our feet, everywhere, a myriad entrances into the infinitely small. ”

“ This Quaternary, among a host of other things, typifies the four lower planes, elements, principles, aspects, etc., of the Universe, with their Hierarchies of Angels, Archangels, Rulers, etc., each synthesized by a Lord who is supreme in his own domain. Seeing, however, that the outermost physical plane is so vast that it transcends the power of conception of even the greatest intellect, it is useless for us to speculate on the interplay of cosmic forces and the mysterious interaction of Spheres of Being that transcend all normal human consciousness. ”

“ The quarters in the spiritual world shall now be spoken of. There are quarters there in like manner as in the natural world, but like that world itself, they are spiritual; while the quarters in the natural world, like that world itself, are natural; the difference between them therefore is so great that they have nothing in common. In each world there are four quarters, which are called east, west, south, and north. In the natural world, these four quarters are constant, determined by the sun on the meridian ”

“ Also on the earth, as is well known, there is a directing of all things towards a common center; but there is this difference between this directing in the world and that in heaven, that in heaven the front parts are turned to the common center, but in the world the lower parts of the body. In the world this directing is called centripetal force, also gravitation. The interiors of angels are actually turned forwards; and since interiors manifest themselves in the face it is the face that determines the quarters. [2] ”

“ The happiness and misery that are predestined for us,
Do not impute them to the heavens, for, in the way of Wisdom,
Those heavens are a thousandfold more helpless than thou art.
Ref.: O. 41, C. 62, L. 80, B. 76, S.P. 95, P. 45.—W. 96, N. 95, V. 79.
LXXIII.
With Earth's first Clay They did the Last Man knead,
And there of the Last Harvest sow'd the Seed:
And the first Morning of Creation wrote
What the Last Dawn of Reckoning shall read.
In this quatrain we trace the influence of O. 31 (quoted in the parallel to quatrain No. 71, ante) and of O. 95.
90
Oh, heart! ”

“ Quetzalcoatl is coming with a new body, like a star, from the shadows of death.
“Yes, even as you sit upon the earth this moment, with the round of your body touching the round of the earth, say: Earth! Earth! you are alive as the globes of my body are alive. Breathe the kiss of the inner earth upon me, even as I sit upon you.
“And so, it is said. The earth is stirring beneath you, the sky is rushing its wings above. Go home to your homes, in front of the waters that will fall and cut you off forever from your yesterdays. ”

“ Seek only deceitful Speciosity, godlike Reality will be forever far from you. The Quack shall be legitimate inevitable King of you; no earthly machinery able to exclude the Quack. Ye shall be born thralls of the Quack, and suffer under him, till you hearts are near broken, and no French Revolution or Manchester Insurrection, or partial or universal volcanic combustions and explosions; never so many, can do more than 'change the figure of your Quack;' the essence of him remaining, for a time and times.—"How long, O Prophet?" ”