Implicit Axioms. -Are then the axioms explicitly enunciated in treatises the only foundations of geometry? One can be assured to the contrary when one sees that, after having successively abandoned them, there still remain some propositions common to theorems of Euclid, Lowatchewski, and Riemann. These propositions ought to rest on some premisses, as geometers admit, although they do not state them. It is interesting to try to liberate them from classical demonstrations.

Stuart Mill has made the assertion that every definition contains an axiom, since, in defining it, the existence of the object defined is implicitly affirmed. This is going too far: it is seldom that one gives a definition in mathematics without following it by the demonstration of the existence of the object defined, and when it is omitted, it is generally because the reader can easily supply it. It must not be forgotten that the word existence has not the same sense when it is the question of a mathematical creation as when we have to do with a material object. A mathematical creation exists, provided that its definition involves no contradiction either in itself or with the properties previously admitted.

But if Stuart Mill's remark cannot be applied to all definitions, it is none the less true for some of them.

A plane is sometimes defined in the following manner :The plane is a surface such that the straight line which joins any two points in it lies altogether in the surface.

This definition manifestly hides a new axiom: we could, it is true, alter it, and that would be better, but then it would be necessary to enunciate the axiom more explicitly.

Other definitions give place to reflections no less important. Such is, for example, that of the equality of two figures: two figures are equal when they can be superposed; to superpose them it is necessary to displace one until it coincides with the other; but how must it be displaced? If we ask, we should be answered that it ought to be done without changing its shape, and in the manner of an invariable solid. The "reasoning in a circle" would then be evident.

In truth, this definition implies nothing. It would have no meaning for a being who lived in a world where there were only fluids. If it seems clear to us, it is that we are accustomed to the properties of natural solids that do not differ greatly from those of ideal solids whose dimensions are all invariable.

Meanwhile, however imperfect it may be, this definition implies an axiom.

The possibility of the movement of an invariable figure is not a truth evident by itself, or at least it is only one in the same way as the postulatum d'Euclide, and not as an analytical a priori judgment would be.

Moreover, in studying the definitions and the demonstrations of geometry, we see that one is obliged to admit, without demonstrating it, not only the possibility of this movement, but even some of its properties.

This results, first of all, from the definition of the straight line. Many defective definitions have been given, but the true one is that which is understood in all the demonstrations where the straight line is in question :

"It may happen that the movement of a constant figure is such that all points of a line belonging to this figure remain immovable while all the points situated outside this line are displaced. Such a line will be called a straight line."

We have in this enunciation purposely separated the definition from the axiom that it implies.

Several proofs, such as those relating to the equality of triangles which depend on the possibility of letting fall a perpendicular from a point on a line, assume propositions that are not enunciated, since we must admit that it is possible to carry a figure from one place to another in a certain manner.

The Fourth Geometry.-Among these implicit axioms, there is one which seems to me worth mentioning, not only because it has given rise to a recent discussion, but because in abandoning it, one can construct a fourth geometry, as coherent as those of Euclid, Lowatchewski, and Riemann.

To demonstrate that we can always raise from a point, A, a perpendicular to a straight line, AB, a straight line, AC, is considered movable round the point A, and in the first instance coinciding with the fixed line AB; and it is made to turn round the point A until it lies in the prolongation of AB.

We thus assume two propositions: first, that such a rotation is possible, and then that it can be continued until the two lines are in one straight line.

If the first point be admitted, and the second rejected, we are led to a series of theorems still more curious than those of Lowatchewski and Riemann, but equally free from contradiction. I will quote only one of them, and that not the most singular : A true straight line can be perpendicular to itself.

The Theorem of Lie.-The number of implicit axioms introduced in classical demonstrations is greater than it need be, and it would be interesting to reduce them to a minimum. We can ask ourselves, in the first place, if this reduction is possible, if the number of necessary axioms, and imaginable geometries is not infinite.

M. Sophus Lie's theorem dominates all this discussion: it can be thus stated:

Let us suppose that the following premisses are admitted :(1) Space has n dimensions.

(2) The movement of an invariable figure is possible. (3) To determine the position of this figure in space, conditions are necessary.

The number of geometries compatible with these premisses will be limited.

I can even add that, if n be given, a higher limit to can be assigned.

if, then, the possibility of movement be admitted, only a finite number (and that a restricted one) of geometries can be invented.

The Geometries of Riemann.-However, this result seems to be contradicted by Riemann, because this investigator constructed an infinite number of different geometries, and the one which generally bears his name is only a particular case.

Everything depends, he says, on the way in which we define the length of a curve. But there are an infinite number of ways of defining this length, and each of these can become the starting point of a new geometry.

That is perfectly true; but most of these definitions are incompatible with the movement of an invariable figure, which is supposed possible in Lie's theorem. These geometries of Riemann, so interesting on many grounds, can only then remain purely analytical, and do not lend themselves to demonstrations analogous to those of Euclid.

The Nature of Axioms.-Most mathematicians regard the geometry of Lowatchewski only as a simple logical curiosity; some of them, however, have gone further. Since several geometries are possible, is it certain that ours is the true one? Experience, doubtless, teaches us that the sum of the angles of a triangle is equal to two right angles; but this is only because we operate on too small triangles; the difference, according to Lowatchewski, is proportional to the surface of the triangle; will it not become sensible if we work with larger triangles, or if our means of measurement grow more accurate? Euclidian geometry would only then be a provisional geometry.

To discuss this question, we ought in the first instance to inquire into the nature of geometrical axioms. Are they synthetical conclusions a priori, as Kant used to say?

They would appeal to us then with such force, that we could not conceive the contrary proposition, nor construct on it a theoretical edifice. There could not be a non-Euclidian geometry.

To convince oneself of it, let us take a true synthetical a priori conclusion; for example, the following:

If an infinite series of positive whole numbers be taken, all different from each other, there will always be one number that is smaller than all the others.

Or this other, which is equivalent to it :

If a theorem be true for the number 1, and if it has been shown to be true for n + 1, provided that it is true for n, then it will be true for all positive whole numbers.

Let us next try to free ourselves from this conclusion, and, denying these propositions, to invent a false arithmetic analogous to the non-Euclidian geometry. We find that we cannot; we shall be even tempted in the first instance to regard these conclusions as the results of analysis.

Moreover, let us resume our idea of the indefinitely thin animals surely we can scarcely admit that these beings, if they have minds like ours, would adopt Euclidian geometry, which would be contrary to all their experience.

Ought we, then, to conclude that the axioms of geometry are

experimental truths? But we do not experiment on straight lines or ideal circles; only material objects can be dealt with. On what would depend, then, the experiments which would serve to found a geometry? The answer is easy.

We have seen above that one argues constantly as if geometrical figures behaved like solids. That which geometry would borrow from experience is therefore the properties of these bodies.

But a difficulty exists, and it cannot be overcome. If geometry were an experimental science, it would not be an exact science-it would be liable to a continual revision. What do I say? It would from to-day be convicted of error, since we know that a rigorously invariable solid does not exist.

Geometrical axioms, therefore, are neither synthetic a priori conclusions nor experimental facts.

They are conventions: our choice, amongst all possible conventions, is guided by experimental facts; but it remains free, and is only limited by the necessity of avoiding all contradiction. It is thus that the postulates can remain rigorously true, even when the experimental laws which have determined their adoption are only approximate.

In other words, axioms of geometry (I do not speak of those of arithmetic) are only definitions in disguise,

This being so, what ought one to think of this question: Is the Euclidian geometry true?

The question is nonsense.

One might just as well ask whether the metric system is true and the old measures false; whether Cartesian co-ordinates are true and polar co-ordinates false; whether one geometry cannot be more true than another-it can only be more convenient.

Now, Euclidian geometry is, and will remain, the most convenient :

(1) Because it is the simplest; and it is not so simply on account of our habits of thought, or any kind of direct intuition which we may have of Euclidian space; it is the most simple in itself in the same way as a polynomial of the first order is simpler than one of the second.

(2) Because it agrees sufficiently well with the properties of natural solids, those bodies which come nearer to our members and our eye, and with which we make our instruments of

It is useless to add that everyone would regard the latter solution as the more advantageous.

Euclidian geometry, then, has nothing to fear from new experiments.

Let me be pardoned for stating a little paradox in conclusion:

The beings which had minds like ours, and who had the same senses as we have, but who had not received any previous education, might receive conventionally from an exterior world choices of impressions such that they would be led to construct a geometry different from that of Euclid, and to localize the phenomena of this exterior world in a non-Euclidian space, or even in a space of four dimensions.

For us, whose education has been formed by our real world, if we were suddenly transported in this new one, we should not have any difficulty in referring the phenomena to our Euclidian

Desiring to be brief, I have affirmed more than I have proved : the reader must pardon me for this. So much has been written on this subject, so many different opinions have been put forward, that the discussion of them would fill a volume.

W. J. L.

SOCIETIES AND ACADEMIES.

LONDON.

Royal Society, February 11.-"The Role played by Sugar in the Animal Economy: Preliminary Note on the Behaviour of Sugar in Blood." By Vaughan Harley, M.D.

whole amount of added sugar can seldom be recovered from blood This communication was to show that the causes why the are threefold. Firstly, the imperfections in the as yet known methods of analysis. Secondly, the different ways in which the albumens of the blood behave themselves while coagulating; some coagulating in the form of firm clots, which retain the saccharine matter in their interstices, rendering it impossible to extract all the sugar from them by washing; others separating as loose, comparative facility. While, thirdly, as bacteria were distinctly flocculent curds, from which the sugar can be regained with ascertained to have nothing to do in the matter, and yet the loss of the sugar added to the blood is in every instance distinctly progressive-according to the period of time the sugar is left in contact with the blood before the analysis is begun-Dr. Vaughan Harley considered himself justified in saying that there must exist in the normal blood itself a sugar-transforming agent. This he described as an enzyme; but refrained from going into any further particulars regarding it until his researches upon the subject are more advanced.

He gave tables of the results of his experiments, and compared Seegen; showing that while the percentages of the sugar them with those recently published by Schenk, Rohmann, and regained by the first observer ranged from 20 to 55 per cent., and

those recovered by the two last experimenters fluctuated between So and 96 per cent., in his three different series of experiments, where different methods of analysis were employed, the percentages of the added sugar regained ranged respectively between 85 and 100; 92'9 and 99'3; and 94'7 and 99'9 per cent.

Mathematical Society, February 11.-Prof. Greenhill, F.R.S., President, in the chair.-The following communications were made:-On the logical foundations of applied mathematical sciences, by Mr. Dixon. He maintained the importance of distinguishing in all sciences between what is dependent en verbal conventions and what is not. He thus distinguished between that part of the meaning of a term which is laid down as its definition, and the part which remains to be discovered as a consequence of the definition. So also sciences might be divided into purely symbolic sciences, which being based on definitions alone conveyed no real information ; subjective sciences, which deal with concepts and objective sciences, which deal with actual things. He then stated the conditions under which a set of assertions might be arbitrarily laid down as the definition of a term; and applied these condi

tions to show that Newton's three laws of motion could be

regarded as a definition of the term force, that if this was done there could no longer be any discussion as to whether or not force alone is sufficient to account for the movements of matter. The anomaly that we are apparently able to determine directions absolutely, though we can determine positions only relatively, was explained, and a formal proof of all the elementary theorems of mechanics, including the principle of virtual work, might be deduced.-Note on the inadmissibility of the usual reasoning by which it appears that the limiting value of the ratio of two infinite functions is the same as the ratio of their first derived, with instances in which the result obtained by it is erroneous, by Mr. Culverwell.-On Saint Venant's theory of the torsion of prisms, by Mr. A. B. Bass F.R. S.

DUBLIN

Royal Society, January 20.-Prof. W. N. Hartley, F. R. S., in the chair.-Reports on the zoological collections made by Prof. Haddon in Torres Straits, 1888-89: the Hydrocorallinæ, by S. J. Hickson. The specimens described are a female stock of Stylaster gracilis, Distichopora violacea, and Millepora Murrayi. Some of the smaller colonies of Distichopora are bright orange in colour, others vandyke brown, and the larger ones are deep purple with pale yellowish tips. The author

believes that the differences in colour mean a difference in age and sexual condition. The smallest colonies are not sexually mature, the brownish have male ampullæ, and the oldest stocks are violet in colour, and are apparently female. In any case, the colour of Distichopora can no longer be regarded as the principal character for specific definition.-Sir Howard Grubb, F.R.S., exhibited and described a 4-inch equatorially-mounted refracting telescope of novel construction, in which the right ascension and declination circles were situated at the eye end of the telescope itself, instead of, as usual, on the polar and declination axes; thereby rendering the working of the instrument most convenient for the observer. The circles at the eye end of the telescope are connected by gearing to the polar and declination axes; but Sir H. Grubb described the method by which the "loss of time," and other errors, consequent on gearing, were almost totally eliminated, and the readings rendered quite sufficiently accurate for all ordinary purposes to which such a telescope would be put.-The first part of a memoir on the fossil fishes of the Coal-measures of the British Islands, by Mr. James W. Davis, was communicated by the Honorary Secretaries of the Society.

PARIS.

Academy of Sciences, February 15.-M. d'Abbadie in the chair. On a new method of organic analysis, by M. Berthelot. The method consists in heating the compound in oxygen under a pressure of 25 atmospheres in a calorimetric bomb. Combustion is total and instantaneous, and therefore differs from that which appertains to the use of copper oxide.-On the employment of compressed oxygen in the calorimetric bomb, by the same author.-Action of alkaline metals on boric acid: critical study of the processes used in the preparation of amorphous boron, by M. Henri Moissan. The general result of the investigation is that when an alkaline metal acts on boric acid the reaction that occurs is accompanied with considerable heat, and, on account of the elevation of temperature, the greatest part of the boron set free combines with the excess of alkali, and with parts of the metal vessel used for the experiment. When this is afterwards washed out with water and hydrochloric acid, a mixture of boron, boride of sodium, boride of iron, boron hydride, nitride of boron, and hydrated boric | acid is obtained after desiccation. This mixture is said to have the same composition as the substance which has hitherto been regarded as amorphous boron. M. Moissan will describe a method of preparing amorphous boron in a future paper.Experimental researches on the transmissibility of cancer, by M. Simon Duplay.-The temporary star in Auriga, by M. G. Rayet. On February 10 and 11 the star appeared to M. Rayet to be orange-yellow or pale yellow. Its spectrum was examined by means of the 14-inch equatorial fitted with a spectroscope. It appeared to be continuous, the red and violet portions being comparatively bright. Four bright lines were seen in the green, and their wave-lengths were determined as 518, 501, 493, and 487.-Extension of Lagrange's equations to the case of sliding friction, by M. Paul Appell.-On the distribution of prime numbers, by M. Phragmen.-On the measure of high temperatures; reply to M. H. Becquerel, by M. H. Le Chatelier.-Remarks on the surface tension of liquid metals; a reply to a note by M. Pellat, by M. GouyVariation, with temperature, of the dielectric constant of liquids, by M. D. Negreano. Experiments on benzine, toluene, and xylene, between 5° and 45° C., indicate that the dielectric constant diminishes with increase of temperature.-On the influence exercised on electro-magnetic resonance by an unsymmetrical arrangement of the long circuit along which the waves are propagated, by MM. Blondlot and M. Dufour. The experiments show that the wave-length, measured by means of a resonator, is independent of the dissymmetry of the two wires which transmit the electro-magnetic undulations.-The propagation of electric waves studied by a telephonic method, by M. R. Colson. -Magnetic perturbation of February 13 and 14, by M. Moureaux. The disturbance was first indicated on the magnetograph of the Parc Saint-Maur Observatory at 5h. 42m. on the morning of the 13th inst. The declination and horizontal force curves suffered a simultaneous rise, while the vertical component decreased. The most important phase of the perturbation occurred between II p.m. and 2 a.m.; and about 5 p.m. of the 14th the elements had returned to their normal value. The disturbance in declination amounted to 1° 25', and the horizontal and vertical components varied respectively more

than and of their normal value.- Observations of atmospheric electricity, made by means of a captive balloon, by M. E. Semmola.-On the determination of the state of dissolved salts from a study of contraction, by M. Georges Charpy.-On some properties of bismuthic acid, by M. G. André.-On a carbide of barium, by M. Maquenne. (See Notes.)-Transformation of aromatic amines into chlorinated hydrocarbons, by MM. Prud'homme and C. Rabaut.-On the principles which accompany chlorophyll in leaves, by M. A. Etard.-Improvement of the culture of industrial and fodder potatoes in France : results of the season 1891, by M. Aimé Girard.-Contributions to the study of unplastered wines, by M. H. Quantin. —On the assimilation of carbohydrates, by M. Hanriot.-On the presence of numerous diatoms in the Cretaceous of the Paris basin, by M. Cayeux.-On the existence of zeolites in the calcareous Jurassic rocks of Ariège, and on the dissemination of these minerals in the Pyrenees, by M. A. Lacroix.

AMSTERDAM.

Royal Academy of Sciences, January 30.-Prof. van de Sande Bakhuyzen in the chair.—Prof. Pekelharing spoke of the composition of the fibrin ferment. When oxalated blood-plasma is diluted with water and treated with acetic acid till moderate acid reaction, the precipitate consists chiefly of a substance which is soluble in alkali, in an excess of acid, and in neutral saltsolutions, and from which, by the action of pepsin-hydrochloric acid, is split off a nuclein-a substance that thus must be considered as a nucleo-albumin. This nucleo-albumin acquires, combined with lime, all the properties of fibrin ferment. It is very probable that this nucleo-albumin issues from the corpuscles of the blood.-Prof. Max Weber gave some results of his investigations of the fresh-water fauna of the islands of Sumatra, Java, Flores, Celebes, and Saleyer. Among the Crustacea, the Entomostraca are not essentially different from European forms. Isopods are only represented by marine species: Ichthyoxenus, Tachæa, Rocinela, and Bopyridæ. Amphipods are extremely rare, and only Orchestia was found. Nearly 70 species of Decapods were collected, out of which 33 are living also in brackish and sea water. It could be proved that immigration out of the sea into the rivers had taken place. An account was given of the life-history of Ichthyoxenus Tellinghausii.

Cirques.-Prof. T. G. Bonney, F.R.S.

Bedford College and the Gresham University.-Dr.
W. J. Russell, F.R.S., and Lucy J. Russell
The Implications of Science.-Edward T. Dixon
The Value of Useless Studies.-Prof. Geo, Fras.
Fitzgerald, F.R.S.

The Nickel Heat Engine.-W. B. Croft
The University of London. By W. T. Thiselton-
Dyer, C.M.G., F.R.S..

A Preliminary Statement of an Investigation of the
Dates of some of the Greek Temples as derived
from their Orientation. By F. C. Penrose .
Volcanic Action in the British Isles .
The Centenary of Murchison

H. W. Bates, the Naturalist of the Amazons. By
A. R. W.
Thomas Archer Hirst

391

391

391

392

392

392

395

398

398

398

Dr. Thomas Sterry Hunt. By W. Topley, F.R.S. Notes

399

400

401

THURSDAY, MARCH 3, 1892.

DEEP-SEA DEPOSITS.

Report on the Scientific Results of the Voyage of H.M.S. “Challenger" during the Years 1873-76, under the command of Captain George S. Nares, R.N., F.R.S., and the late Captain Frank Tourle Thomson, R.N. Prepared under the Superintendence of the late Sir C. Wyville Thomson, F.R.S., and now of John Murray, LL.D., Ph.D., &c., one of the Naturalists of the Expedition. Report on Deep-Sea Deposits, based on the Specimens collected during the Voyage. By John Murray, LL.D., Ph.D., and the Rev. A. F. Renard, LL.D., Ph.D., Professor of Geology and Mineralogy in the University of Ghent. Pp. xxix. and 496; with 43 Charts, 22 Diagrams, and 29 Lithographic Plates. (London: Published by Order of Her Majesty's Government, 1891.)

GEOLOGISTS have had to wait long for this very

important work, but now that it lies before them, we believe that the general verdict will be that it was worth while to wait even sixteen years for a monograph so excellent in design and so complete in execution. It must not be forgotten, too, that much of the information contained in this volume has been already given to the scientific world-first in Mr. Murray's Preliminary Report on the subject, published in the Proceedings of the Royal Society; and secondly in a series of papers written by him in conjunction with Prof. Renard, and published in the Proceedings of the Royal Society of Edinburgh.

It is a most fortunate circumstance that the naturalist on board the Challenger, who had charge of the collection, examination, and preservation of the samples of the deposits collected by the sounding-apparatus and the dredge, as well as of those obtained by means of the tow-nets and tangles, has been able during the long period that has elapsed since the return of the vessel to England, to devote his attention to their careful study and description. In the work of dealing with this vast mass of materials, as the preface informs us, Mr. Murray has had the cooperation of Mr. Frederick Pearcey, who accompanied the Expedition, and was afterwards assistant in the Challenger Office, and also of Mr. James Chumley. Especially fortunate has been the circumstance that Sir Wyville Thomson and Mr. Murray were, in 1878, able to secure the aid of the eminent Belgian petrographer, Prof. Renard, who is so great a master of those microscopic methods of research which have played no unimportant part in the development of geological science during recent years. In the exact determination of the minute fragments of minerals which occur in these deosits, Prof. Renard's knowledge of the optical and chemical methods of microscopic research has proved of especial value; and the assurance that, during several months in the years 1881 and 1882, the Belgian petrographer was able to devote himself to the work of investigating these deposits will invest the mineralogical determinations with an authority which they could not otherwise possess.

The introduction to the work consists of an excellent summary of the references contained in various authors,

beginning with Herodotus, Plato, and Skylax, to the supposed nature of the sea-bottom. The sagacious remarks on the subject by the Italian naturalists, who were the real founders of the science of geology in the fifteenth century, receive appreciative notice; and the earliest attempts to deal with the deposits of the deep seas, especially those of Soldani, Ehrenberg, Sir Joseph Hooker, Edward Forbes, and Prof. J. W. Bailey, have full recognition. The important memoir of Prof. W. C. Williamson on the mud of the Levant is noticed; but the authors seem to be scarcely aware how many of the later discoveries in this branch of science were foreshadowed in the remarkable monograph of the Manchester Professor. A general account of the results obtained by the chief expeditions fitted out for the study of the deep ocean and its deposits-expeditions which preceded and followed that of the Challenger-leads up to a division of marine deposits into "Terrigenous" and "Pelagic," a classification which, if not too rigidly applied, appears to be serviceable and even necessary.

The first chapter is devoted to the various methods of obtaining, examining, and describing deep-sea deposits, and here the general arrangements made on board the Challenger, which are familiar to most readers from the description given by Sir Wyville Thomson in his “Voyage of the Challenger," and the narrative volumes of the Report, receive very full and exhaustive treatment. The precise account of the apparatus, illustrated as it is by numerous woodcuts, cannot fail to be of great value to those engaged in fitting out similar expeditions. The study of the methods employed in the sifting, fractional decantation, and chemical examination of the several deposits is essential to the proper understanding of the results detailed in succeeding chapters of the work. The methods of analysis employed by Prof. Brazier at Aberdeen, and by MM. Renard, Sipöcz, Hornung, and Klement, in the laboratory of Prof. Ludwig, of Vienna, and in M. Renard's laboratory at Brussels, are given in full detail, and will prove of great service when the results described in the present volume come to be compared with those of future investigators.

The second chapter consists of a series of synoptical tables, occupying 114 pages, in which the nature and composition of every deep-sea deposit collected during the voyage of the Challenger is described. In each case the number of the station, the date, the latitude and longitude, the depth in fathoms, the temperature at the surface and the bottom are given; and these particulars are followed by (1) a general description of the material brought up; (2) the percentage of calcium carbonate; (3) a list of the chief Foraminifera present; (4) an enumeration of the other calcareous organisms; (5) the percentage of insoluble residue; (6) a list of the siliceous organisms; (7) of the minerals; (8) a description of the fine washings; the last column being devoted to additional observations. These synoptical tables are followed by a discussion of the variation of the deposits with change of conditions along the different lines of soundings and dredgings. This general summary of the results, which occupies 36 pages of the work, constitutes an admirable résumé of the information contained in the tables.

Chapter iii. is devoted to the description of recent

marine formations and the different types of deep-sea deposits their composition, geographical and bathymetrical distribution. All marine deposits which are not "Littoral," or formed between high- and low-water marks, or "Shallow-water," a term which the authors limit to the interval between low-water mark and a depth of 100 fathoms, are classified in this work as deep-sea deposits. They include Coral Mud, Volcanic Mud, Green Mud, Red Mud, and Blue Mud (which are classed as Terrigenous Deposits, formed in deep and shallow water close to landmasses), and the Pteropod ooze, Globigerina ooze, Diatom ooze, Radiolarian ooze, and Red Clay (which are grouped as Pelagic Deposits, formed in deep water removed from land). In the case of each of these deposits the proportions and characters of the organic and inorganic materials are given, and the results of a large number of chemical analyses, some of which are now published for the first time, are discussed. Perhaps one of the most interesting of the many valuable discussions contained in this chapter is that which deals with the proportions of the ocean-floor covered by different kinds of deposits. A map (Chart I.) is devoted to an attempt to illustrate the nature of the ocean-floor over the whole of the globe, and we cannot resist the temptation of quoting the general estimate to which the authors have been led by their laborious and patient researches. These results are based not only on the collections made during the Challenger Expedition, but on many obtained before and since, which have all passed through the hands of the authors; they include the materials brought up in no less than 1600 soundings from the Atlantic, 300 from the Indian Ocean, and 400 from the Pacific, all from depths exceeding 1000 fathoms. It is evident, therefore, that the map and estimates, though admittedly only approximate, are based on a mass of data such as has never been brought together before.

The total area of the surface of the globe is estimated at 196,940,700 square miles, of which dry land occupies about 53,681,400 square miles, and the waters of the ocean 143,259,300 square miles. The approximate extent of the areas of the sea-floor occupied by each type of marine deposits is given as follows:

composed of the remains of animals on the bottom of the Antarctic Ocean.

Chapter iv., dealing with the materials of organic origin, is, we are informed in the preface, entirely from the pen of Mr. Murray. The Reports of the late Mr. H. B. Brady, of Prof. Haeckel, and of Count Castracane, on the Foraminifera, the Radiolarians, and the Diatomaceæ brought home by the Challenger Expedition, have already supplied naturalists with the means of drawing many important deductions; but Mr. Murray still finds much to say on the subject, which is not only new, but of very great interest. In the couple of pages devoted to the description of those curious and abundant organisms the Coccospheres and Rhabdospheres, which Mr. Murray here refers without doubt to the Calcareous Algr, we could have wished that he had been able to announce that he had succeeded in inducing some competent botanist to undertake the study of the material brought home. One of the most important discussions in this chapter is that on the disappearance of calcic carbonate in the deeper deposits. The estimate made by Mr. Murray of the mean percentage of calcic carbonate in the different deposits, as the result of a large number of chemical analyses, is as follows:

The facts cited by Mr. Murray, on the authority of Mr. John Rathay (p. 282), on the ease with which the remains of the Diatomaceæ are dissolved, are of especial importance to the geologist who is called upon to explain the origin of the silica now forming nodules and bands in beds of limestone, and which he is tempted to refer entirely to the larger organisms like Siliceous Sponges, because remains of these are sometimes preserved. All the observations made in the existing deep seas, however, point to the conclusion that the minute Diatoms and Radiolarians play a much more important part in separating the soluble silica from sea-water than do the Siliceous Sponges.

Chapter v., dealing with the mineral substances found in deep sea deposits, is full of interest. The mineral particles which are obviously derived from the solid crust of the globe are first dealt with, and in the account of the pumice, the basic volcanic glass, and the palagonite of the deep-sea deposits, Prof. Renard exhibits alike his wide mineralogical knowledge and his skill in dealing. with microscopical and often obscure materials. The coloured lithographic plates illustrating this part of the work, which have been drawn by Prof. Renard, and engraved in Vienna, are of wonderful beauty and fidelity. A list of mineral particles detected in deep-sea deposits is given, and includes all, or nearly all, the common rockforming minerals; but it is admitted that, with respect to the very minute particles in the finest washings, a considerable margin of doubt must always exist regarding their identification. We could wish that it were possible, in the space at our command, to give a summary of the