effected by a shutter, with an opening sliding in front of the slit; a diagram of its action and form is given.

The arrangement of the spectroscope, heliostat, &c., for obtaining the sun's light is described. The image of the sun was brought to a focus between the poles of the lamp by an extra lens interposed between the lamp and the heliostat.

The use of the shutter enables us to compare either two or more spectra upon a single plate, or the solar spectrum may be compared with two metallic spectra, being made to occupy the position between the two.

III. On the Lines coincident in different Spectra

The bearing of the former papers on the lengths of the lines of the elements is briefly recapitulated.

The examination of the various spectra of metals and alloys indicated the great impurity of most of the metals used, and suggested the possibility of the coincidences observed by Thalen and others being explained in the light of former work.

It is observed that coincidences are particularly numerous in the spectra of iron titanium, and calcium, and that nearly every other solar metallic spectrum has one or more lines coincident with lines of the last element. These coincident lines are, as a rule, very variable in length and intensity in various specimens of the metals in which they occur, and are sometimes altogether absent.

One of the longest calcium lines, that at wave-length 4226'3, is also seen in the strontium spectrum as a line of medium length, and 4607'5, a very long line in strontium, appears in calcium as a short line. Another very long strontium line, 42153, is asserted by Thalén to be seen in calcium; but the author has never seen it till lately, and then only in a specimen of calcium known

to contain strontium.

We have here, then, a case of coincident lines, in which the one that is long and bright in one spectrum is short and faint in the other, and a case of a line said to be coincident in two spectra being, though always visible in one, sometimes absent in the other of them, and only appearing in it when the two substances were mixed. The hypothesis of impurity at once explains the whole case, even without the third line, which renders the fact of mixture certain.

The longest lines of calcium occur in iron, cobalt, nickel, barium, strontium, &c., and the longest lines of iron occur in calcium, strontium, barium, and other metals.

Other cases are adduced, and the following general statements are hazarded, with a premise that further inquiry may modify

them.

I. If the coincident lines of the metals be considered, those cases are rare in which the lines are of the first order of length in all the spectra to which they are common: those cases are much more frequent in which they are long in one spectrum and shorter in the others.

2. As a rule, in the instances of those lines of iron, cobalt, nickel, chromium, and manganese which are coincident with lines of calcium, the calcium lines are long, while the lines as they appear in the spectra of the other metals are shorter than the longest lines of those metals. Hence we are justified in assuming that short lines of iron, cobalt, nickel, chromium, and manganese, coincident with long and strong lines of calcium, are really due to traces of the latter metal occurring in the former as an impurity.

3. In cases of coincidences of lines found between various spectra the line may be fairly assumed to belong to that one in which it is longest and brightest.

A description of some photographs of spectra is then given, a photograph of the coincident lines of calcium and strontium being amongst them, and proving that strontium occurs in the sun; and the section concludes with a brief description of the method employed in making the new map, showing lengths and thicknesses, and enumerating coincident lines. This is done thus papers are pasted on to photographs of the solar spectrum on glass; the lengths of the lines of the metallic spectrum under examination (e.g. that of iron) are marked on this paper in prolongation of the solar lines to which they correspond. They are then copied upon a map, and another piece of paper being fixed down, another spectrum is proceeded with in the same

way.

:

IV. The Preliminary Inquiry into the Existence of Elements in the Sun not previously traced

The previous researches having shown that the former test for the presence or absence of a metal in the sun, namely, the pre

sence or absence of its brightest or strongest lines in the average solar spectrum, was not conclusive, a preliminary search for other metals was determined on; and as a guide, Mr. R. J. Friswell was requested to prepare two lists, showing broadly the chief chemical characteristics of the elements traced and not traced in the sun.

The tables showed that in the main those metals which had been traced formed stable compounds with oxygen.

The author therefore determined to search for the metals which formed strong oxides, but which had not yet been traced.

The result up to the present time has been that strentum, cadmium, lead, cerium, and uranium would seem with consider able probability to exist in the solar reversing layer. Should the presence of cerium and uranium be subsequently confirmed, the whole of the iron group of metals will thus have been found in the sun.

Certain metals forming unstable oxides, such as gold, silver, mercury, &c., were sought for and not found. The same was the case when chlorine, bromine, iodine, &c., were sought by means of their lines produced in tubes by the jar-spark. These elements are distinguishable as a group by forming compounds with hydrogen.

It is observed that certain elementary and compound gases effect their principal absorption in the most refrangible part of the spectrum when they are rare, and that as they become dense the absorption approaches the less refrangible end; that the spectra of compounds are banded or columnar, the bands or columns lying at the red end of the spectrum; that the absorp tion spectra of chlorine, iodine, bromine, &c., are columnar, and that these are broken up by the spark just as the band spectra of compounds are broken up and that it is probable that no com pounds exist in the sun. The following facts, gathered from the work already accomplished by Rutherford and Secchi ate stated :

There are three classes of stars :

1. Those like Sirius, the brightest (and therefore hottest?) star in the northern sky, their spectra showing only hydrogen lines very thick, and metallic lines exceedingly thin.

2. A class of stars with a spectrum differing only in degree from those of the class of Sirius, and to this our sun belongs. 3. A class of stars with columnar or banded spectra indicat ing the formation of compounds,

The question is asked whether all the above facts cannot be grouped together in a working hypothesis, which assumes that in the reversing layers of the sun and stars various degrees of "celestial dissociation are at work which prevents the coming together of the atoms which, at the temperature of the earth, and at all artificial temperatures yet attained here, form the metals, the metalloids, and compounds.

In other words, the metalloids are regarded as quasi compound bodies when in the state in which we know them; and it is sup posed that in the sun the temperature is too great to permit them to exist in that state in the reversing layer, though they may be found at the outer portions of the chromosphere or in the

corona.

It is suggested that if this hypothesis should gain strength from subsequent work, stony meteorites will represent the third class of metalloidal or compound stars, and iron meteorites the other, or metallic stars.

The paper concludes as follows:

"An interesting physical speculation connected with this working hypothesis is the effect on the period of duration of a star's heat which would be brought about by assuming that the original atoms of which a star is composed are possessed of the increased potential energy of combination which this hypothesis endows them with. From the earliest phase of a star's life the dissipation of energy would, as it were, bring into play a new supply of heat, and so prolong the star's life.

66

May it not also be, if chemists take up this question, which has arisen from the spectroscopic evidence of what I have before termed the plasticity of the molecules of the metalloids taken as a whole, that much of the power of variation which is at present accorded to metals may be traced home to the metalloids? need only refer to the fact that, so far as I can learn, all sc called changes of atomicity take place when metalloids are 10volved, and not when the metals alone are in question.

"As instances of these, I may refer to the triatomic combina tions formed with chlorine, oxygen, sulphur, &c. in the case of tetrad or hexad metals. May not this be explained by the plas ticity of the metalloids in question?

May we not from these ideas be justified in defining a metal, provisionally, as a substance the absorption spectrum of which is generally the same as the radiation spectrum, while the metalloids are substances the absorption spectrum of which, generally, is not the same?

"In other words, in passing from a hot to a comparatively cold state, the plasticity of these latter comes into play, and we get a new molecular arrangement. Hence are we not justified in asking whether the change from oxygen to ozone is but a type of what takes place in all metalloids?"

Abstract of paper "On the Quantitative Analysis of certain Alloys by means of the Spectroscope," by J. Norman Lockyer, F.R.S., and William Chandler Roberts, Chemist of the Mint.

The authors, after referring to experiments which showed clearly that the spectroscope might be employed to detect minute differences in the composition of certain alloys, proceed to give an account of the researches which they had instituted with a view to ascertain the degree of accuracy of which the method is capable.

un

The image of an electric-spark passing between the known alloy and a fixed electrode being thrown by means of a lens on the slit of the spectroscope, the phenomena observed were found to vary with the composition of the alloys; and further, by arranging them together with known check-pieces on a suitable stand, and bringing them in turn under the fixed electrode, the composition of the unknown alloys was determined by comparison with the known check-pieces.

The shape of the electrode ultimately adopted was stated; the pieces were held in their places by suitable metallic clips. Special attention was then directed to the adjustment of the length of the spark, which was found to materially influence the phenomena. The method adopted consisted in placing the variable electrode in the field of a fixed microscope having a 3- or 4-inch objective, and adjusting the summit of this electrode to coincide with the spider lines of the eye-piece.

After a series of experiments on alloys of zinc and cadmium of various compositions, the results of which were shown on a curve, more extended trials were made with the gold-copper alloy employed in coinage, which was peculiarly suited to these researches in consequence of the known method of assay having been brought to so high a state of perfection (the composition being determined with accuracy to the root part of the original assay-piece of about 7 grains), and from the fact that reliance can be placed on its homogeneity. The paper is accompanied by a series of four curves, which show the results of experiments, and in which the coordinates are given by the ordinary method of assay, and by the spectroscopic readings.

The chief practical advantage which appeared to flow from this inquiry was that, if it were possible to replace the parting assay by the spectroscopical method, a great saving of time in ascertaining the value of gold bullion would be effected.

Institution of Civil Engineers, Dec. 9.-T. Hawksley, president, in the chair.-"On the Geological Conditions affecting the Constructing of a Tunnel between England and France," by Mr. Joseph Prestwich, F.R.S. The author reviewed the geological conditions of all the strata between Harwich and Hastings on one side of the Channel, and between Ostend and St. Valery on the other side, with a view to serve as data for any future projects of tunnelling, and to show in what directions inquiries should be made. The points considered were the lithological characters, dimensions, range and probable depth of the several formations. The London clay, at the mouth of the Thames, was from 200 feet to 400 feet thick, while under Calais it was only 10 feet, at Dunkirk it exceeded 264 feet, and at Ostend it was 448 feet thick. He considered that a trough of London clay from 300 feet to 400 feet, or more, in thickness extended from the coast of Essex to the coast of France, and, judging from the experience gained in the Tower Subway, and the known impermeability and homogeneity of this formation, he saw no difficulty, from a merely geological point of view, in the construction of a tunnel, but for the extreme distance -the nearest suitable points being 80 miles apart. The lower Tertiary strata were too unimportant and too permeable for tunnel work. The chalk in this area was from 400 feet to 1,000 feet thick; the upper beds were soft and permeable, but the lower beds were so argillaceous and compact as to be com. paratively impermeable. In fact, in the Hainaut coal fields they effectually shut out the water of the water-bearing tertiary strata from the underlying coal measures. Still, the author did

not consider even the lower chalk suited for tunnel work, owing to its liabilty to fissures, imperfect impermeability, and exposure in the Channel. The gault was homogeneous and impermeable, but near Folkstone it was only 130 ft. thick reduced to 40 ft. at Wissant, so that a tunnel would hardly be feasible. The Lower Greensands, 260 ft thick at Sandgate, thinned off to 50 ft. or 60 ft. at Wissant, and were all far too permeable for any tunnel work. Again, the Wealden strata, 1,200 ft. thick in Kent, were reduced to a few unimportant rubbly beds in the Boulonnais. To the Portland beds the same objections existed as to the Lower Greensands, both were water-bearing strata. The Kimmeridge clay was 360 ft. thick near Boulogne, and no doubt passed under the Channel, but in Kent it was covered by so great a thickness of Wealden strata as to be almost inaccessible; at the same time it contained subordinate water-bearing beds. Still, the author was of opinion that, in case of the not improbable denudation of the Portland beds, it might be questionable to carry a tunnel in by the Kimmeridge clay on the French coast, and out by the Wealden beds on the English coast. The oolitic series presented conditions still less favourable, and the lower beds had been found to be water-bearing in a deep artesian well recently sunk near Boulogne. The experimental deepboring now in progress near Battle would throw much light on this part of the question. The author then passed on to the consideration of the Paleozoic series, to which his attention was more particularly directed while making investigations, as a member of the Royal Coal Commission, on the probable range of the coal measures under the south-east of England. He showed that these rocks, which consisted of hard Silurian slates, Devonian and carboniferous limestone and coal measures, together 12,000 ft. to 15,000 t. thick, passed under the chalk in the North of France, outcropped in the Boulonnais, were again lost under newer formations near to the coast, and did not reappear until the neighbourhood of Frome and Wells was reached. But, although not exposed on the surface, they had been encountered at a depth of 1,032 ft. at Calais, 985 ft. at Ostend, 1,026 ft. at Harwich, and 1,114 ft. in London. They thus seemed to form a subterranean table land of old rocks, covered immediately by the chalk and Tertiary strata. It was only as the southern flank of this old ridge that the Jurassic and Wealden series set in, and beneath these the Palaeozoic rocks rapidly descended to great depths. Near Boulogne these strata were already 1,000 ft. thick; and at Hythe the author estimated their thickness might be that or more. Supposing the strike of the coal measures and the other Paleozoic rocks to be prolonged from their exposed area in the Boulonnais across the Channel, they would pass under the Cretaceous strata somewhere in the neighbourhood of Folkestone, at a depth estimated by the author at about 300 ft., and near Dover at about 600 ft., or nearly at the depth at which they had been found under the chalk at Guines, near Calais, where they were 665 ft. deep. These Paleozoic strata were tilted at high angles, and on the original elevated area they were covered by horizontal Cretaceous strata, the basement beds of which had filled up the interstices of the older rocks as though with a liquid grouting. The overlying mass of gault and lower chalk also formed a barrier to the passage of water so effectual, that the coal measures were worked without difficulty under the very permeable Tertiary and upper chalk of the North of France; and in the neighbourhood of Mons, notwith. standing a thickness of from 500 ft. to 900 ft. of strata charged with water, the lower chalk shut the water out so effectually that the coal measures were worked in perfect safety, and were found to be perfectly dry under 1,200 ft. of these strata combined. No part of the Straits exceeded 186 ft. in depth. The author, therefore, considered that it would be perfectly practicable, so far as safety from the influx of the sea water was concerned, to drive a tunnel through the Paleozoic rocks under the Channel between Blanc Nez and Dover, and he stated that galleries had actually been carried in coal, under less favourable circumstances, for two miles under the sea near Whitehaven. But while in the case of the London clay the distance seemed almost an insurmountable bar, here again the depth offered a formidable difficulty. As a collateral object to be attained, the author pointed to the great problem of the range of the coal measures from the neighbourhood of Calais in the direction of East Kent, which a tunnel in the Paleozoic strata would help to solve. These were, according to the author, the main conditions which bore on the construction of a submarine tunnel between England and France. He was satisfied that on geological grounds alone, it was in one case perfectly practicable, and in one or two others it was possibly so; but there were other considerations besides those of a geolo

gical nature, and whether or not they admitted of so favourable a solution was questionable. In any case, the author would suggest that, the one favourable solution admitted, it might be desirable, in a question involving so many and such great interests, not to accept an adverse verdict without giving all those considerations the attention and deliberation which the importance of the subject deserved. Granting the possibility of the work in a geological point of view, there were great and for midable engineering difficulties; but the vast progress made in engineering science during the last half century, led the author to imagine that they would not prove insurmountable, if the necessity for such a work were to arise, and the cost were not a bar.

Royal Astronomical Society, Dec. 14.-Prof. Cayley, president, in the chair.-Prof. Pritchard gave a verbal account of the Physical Observatory about to be established at Oxford. He said that the University authorities had been induced to grant a site for a physical observatory in the noble park of sixty acres, which they had recently thrown open to the public. He had been anxious that such a site should not be disgraced by an unsightly building such as observatories usually were. He found himself fortunately situated in having amongst his old pupils Mr. Barry, the well-known architect, who had furnished them with a design which he showed to the meeting, and had devised, amongst other things, a dome with a fine broad shutter, which he trusted would be really ornamental as well as useful. There would be a central tower of three rooms, one above the other; the basement room would be used for storage; above would be the professor's room; and in the floor above that would be mounted the noble reflector which had been presented to the University by Dr. De La Rue. In a side wing there would be a transit instrument to be used for educational purposes, and another telescope which he hoped would be well worked by members of the University. Mr. Barry informed the society that their new rooms at Burlington House would probably be ready by the middle of April.— Capt. Noble mentioned to the society that in the new volume of the Nautical Almanac for 1877 tables of Uranus were given, but it was no credit to England that we should have been kept waiting for them until they were presented to us from across the Atlantic by the labour of Prof. Simon Newcomb.

Entomological Society, Dec. 1.-H. T. Stainton, F.L.S., vice-president, in the chair.-Mr. Bond exhibited a hybrid specimen between Clostera curtula and C. reclusa partaking of the characters of both parents.-Mr. Jenner Weir exhibited specimens of a minute Hymenopterous insect (a species of Psen), which he had observed in large numbers (probably 150) in June last on a pear leaf at Lewes. They had congregated together on the surface of the leaf like a swarm of bees, though it was not apparent what motive brought them together.-Mr. Dunning read extracts from a letter from New Zealand stating that the red clover had been introduced into that colony, but that they had no humble bees to fertilise the plant. Also that certain Lepidopterous insects had been accidentally imported into the islands, but that the corresponding Ichneumon flies were wanted to keep down their numbers. It was suggested that the nests of humble bees might be imported, when the bees were in a dormant condition, keeping them in that state (by means of ice) during the voyage. Mr. Baly communicated a paper on the Phytophagous Coleoptera of Japan, being a continuation of a former paper on the same subject.-Mr. Bates communicated a supplementary paper on the Longicorn Beetles recently brought from Chontales, Nicaragua, by Mr. Thomas Belt.-Mr. W. H. Miskin, of Queensland, communicated criticisms on Mr. Masters' Catalogue of the described species of Diurnal Lepidoptera of Australia.A fourth portion of the catalogue of British Insects, now being published by the society, was on the table. It contained the Hymenoptera (Oxyura), by Rev. T. A. Marshall, M. A.

PARIS

Academy of Sciences, Dec. 8.-M. de Quatrefages, president, in the chair.-The president announced the death of M. Cl. Gay, member of the Botanical Section; and the Perpetual Secretary also announced the death of the well-known mineralo. gist, C. F. Naumann, Corresponding Member of the Mineralogical Section.-The following papers were read :-An answer to M. Pasteur's paper on the origin of beer yeast, by M. A. Trécul. The author contradicted M. Pasteur's statement that the development of Pencillium glaucum fom putrid yeast was an admitted fct. On the contrary, it had been observed to develop itself

from perfectly healthy yeast.-On the vitreous substre found included in Santorin lava, by M. F. Fouque-1 the determination of the ratio of two specific heats by the e pression of a limited volume of gas, by M. E. H. Amgat— the distribution of the neolithic populations in the depar of the Oise, by M. R. Guérin. On the habits of the Ph (continued), by M. Max. Cornu.-A further notice on the nection of storms and sunspots as observed at Paris and Fea was received from M. Poey.—Preliminary note on the elem existing in the sun, by Mr. Norman Lockyer. M. Bea then criticised the paper. He held that the phenomen specific heat, &c., indicated that the elements, so-called "e. on a very different basis from the compounds, and th phenomena they presented in this respect could not be e plained if they were not regarded as actually simple bodies Dumas thought that, as he had himself maintained les the Academy, elements ought only to be regarded as ments in relation to human experience and not as lute elements, a fact which he considered Lavoisier to b established. He considered that modern experiments tended confirm this opinion.-Note on the identity of Cauchy's form for the determination of the conditions of convergence of L grange's series with those given by Lagrange himself, by M. L Ménabrea. On the November meteors, by M. Wolf.-Nott Faye's periodic comet and on the discovery and observations twenty nebulae made at the Marseilles observatory, by M Stephan.-On the movement of an elastic wire one end of wh has a vibratory motion, by M. E. Mercadier.-Observation: the action of certain poisons on sea fish, by MM. A. Ralu and F. Papillon.-On the embryo cell of the egg of osseous. by M. Balbiani.-On the age of the dental follicle in the mifere, by MM. E. Magitot and Ch. Legros.-On the use electrical cauterisation in surgical operations, by MM. Ch. Le and Onimus.-On the Ostracious marl of Fresnes-les-Rug(Seine), by M. Stan. Meunier.-Note on a meteor observed Versailles on Dec. 3, by M. Martin de Brettes.-New analy of the water of St. Thiebaut's fountain at Nancy, by M. P. Ab -Studies on certain combustibles from the basin of Donet: 5. Toula, Russia, by MM. Scheurer-Kestner and Meunier-Doll

BOOKS RECEIVED

2 vols.: Ic

ENGLISH.-Guide to Latin Prose: R. M. Millington Relfe)Animals: Wolf (Macmillan & Co.)-Problems of Life and Mind: Ge Henry Lewes (Trübner & Co).-Theory of Attraction. hunter (Macmillan & Co.).-The Borderland of Science: R. A. Pret (Smith, Elder & Co)-Memoir of Mary Somerville: Martha S ville (John Murray).-Manual of Comparative Anatomy and Physiol J. M. Bradley (Simpkin and Marshall).-The River Amazon: H WE John Murray)-The Chase: Somerville (W. Tegg).-Virgil's Ecles Translated: Millington (Longmans).-Quantitative Chemical Analysis" edition: Fresenius (Churchill)-Nautical Almanac, 1877 (John MurrayThe Simplicity of Life: Dr. Ralph Richardson (H. K. Lewis)-Introduct to Quaternions: Kelland and Tait (Macmillan)-Free-thinking and Fam Speaking: Leslie Stephens (Longmans).-United States Geological Surve 6th Annual Report: F. W. Haydyn (Trübner & Co.) -Harvest of the Se Bertram (John Murray).-Mountain, Meadow, and Mere: G. C. Das (H. S. King & Co.).-Legal Handbook for Architects: Jenkins and Raym H. S. King & Co.).-The Conservation of Energy: Balfour Stewart (H King & Co.).-Telegraphic Journal, vol. i. (Gillman).-Primer of Geol A Geikie (Macmillan & Co.).-Darwinism and Design: G. St Claire (Ho & Stoughton).-From January to December (Longmans).-Pheasants Coverts and Aviaries: W. B. Tegetmeier (Horace Cox).

A

THURSDAY, DECEMBER 25, 1873

QUATERNIONS

MATHEMATICIAN is one who endeavours to secure the greatest possible consistency in his thoughts and statements, by guiding the process of his reasoning into those well-worn tracks by which we pass from one relation among quantities to an equivalent relation. He who has kept his mind always in those paths which have never led him or anyone else to an inconsistent result, and has traversed them so often that the act of passage has become rather automatic than voluntary, is, and knows himself to be, an accomplished mathematician. The very important part played by calculation in modern mathematics and physics has led to the development of the popular idea of a mathematician as a calculator, far more expert, indeed, than any banker's clerk, but of course immeasurably inferior, both in resources and in accuracy, to what the "analytical engine" will be, if the late Mr. Babbage's design should ever be carried into

execution.

But though much of the routine work of a mathematitician is calculation, his proper work-that which constitutes him a mathematician-is the invention of methods. He is always inventing methods, some of them of no great value except for some purpose of his own; others, which shorten the labour of calculation, are eagerly adopted by all calculators. But the methods on which the mathematician is content to hang his reputation are generally those which he fancies will save him and all who come after him the labour of thinking about what has cost himself so much thought.

Now Quaternions, or the doctrine of Vectors, is a mathematical method, but it is a method of thinking, and not, at least for the present generation, a method of saving thought. It does not, like some more popular mathematical methods, encourage the hope that mathematicians may give their minds a holiday, by transferring all their work to their pens. It calls upon us at every step to form a mental image of the geometrical features represented by the symbols, so that in studying geometry by this method we have our minds engaged with geometrical ideas, and are not permitted to fancy ourselves geometers when we are only arithmeticians.

This demand for thought-for the continued construction of mental representations-is enough to account for the slow progress of the method among adult mathematicians. Two courses, however, are open to the cultivators of Quaternions: they may show how easily the principles of the method are acquired by those whose minds are still fresh, and in so doing they may prepare the way for the triumph of Quaternions in the next generation; or they may apply

the method to those problems which the science of the day presents to us, and show how easily it arrives at those solutions which have been already expressed in ordinary mathematical language, and how it brings within our reach other problems, which the ordinary methods have hitherto abstained from attacking.

Sir W. R. Hamilton, when treating of the elements of the subject, was apt to become so fascinated by the metaphysical aspects of the method, that the mind of his disciple became impressed with the profundity, rather VOL. IX.-No. 217

than the simplicity of his doctrines. Professors Kelland and Tait in the opening chapter (II.) of their recently published work have, we think, successfully avoided this element of discouragement. They tell us at once what a vector is, and how to add vectors, and they do this in a way which is quite as intelligible to those who are just beginning to learn geometry as to the most expert mathematician.

The subject, like all other subjects, becomes more intricate as the student advances in it; but at the same time his ideas are becoming clearer and more firmly established as he works out the numerous examples and exercises which are placed before him.

The technical terms of the method-Scalar, Vector, Tensor, Versor-are introduced in their proper places, and their meaning is sufficiently illustrated to the beginner by the examples which he is expected to work out. The pride of the accomplished mathematician, however (for whom this book is not written), might have been somewhat mollified if somewhere in the book a few pages had been devoted to explaining to him the differences between the Quaternion methods and those which he has spent his life in mastering, and of which he has now become the slave. He is apt to be startled by finding that when one vector is multiplied into another at right angles to it, the product is still a vector, but at right angles to both. His only idea of a vector had been that of a line, and he had expected that when one vector was multiplied into another the result would be something of a different kind from a line, such, for instance, as a surface. Now if it had been pointed out to him in the chapter on vector multiplication that a surface is a vector, he would be saved from a painful mental shock, for a mathematician is as sensitive about "dimensions" as an English schoolboy is about "quantities.”

The fact is, that even in the purely geometrical applications of the Quaternion method we meet with three different kinds of directed quantities: the vector proper, which represents transference from A to B ; the area or "aperture,” which is always understood to have a positive and a negative aspect, according to the direction in which it is swept out by the generating vector; and the versor, which represents turning round an axis.

The Quaternion ideas of these three quantities differ from the old ideas of the line, the surface, and the angle only by giving more prominence to the fact that each of them has a determinate direction as well as a determinate magnitude. When Euclid tells us to draw the line A B, he supposes it to be done by the motion of a point from A to B or from B to A. But when the line is once generated he makes no distinction between the results of these two operations, which, on Hamilton's system, are each the opposite of the other.

Surfaces also, according to Euclid, are generated by the motion of lines, so that the idea of motion is an old one, and we have only to take special note of the direction of

the motion in order to raise Euclid's idea to the level of Hamilton's.

With respect to angles, Euclid appears to treat them as if they arose from the fortuitous concourse of right lines;

"Introduction to Quaternions, with numerous Examples." By P. Kelland, F. R.S., formerly Fellow of Queen's College, Cambridge; and P. G. Tait, formerly Fellow of St. Peter's College, Cambridge; Professors in the Department of Mathematics in the University of Edinburgh. (Macmillan, 1873.

I