reverted to the Wernerian view, but with some important modifications; these he explained in his "crenitic hypothesis."

Dr. Hunt's earlier papers (1846-49) were wholly on chemistry and mineralogy, and to these subjects he always gave much attention. Some of his latest writings are purely chemical, dealing mainly with the more speculative aspects of that science. Perhaps in these questions, as is certainly the case with many of his theoretical views on geology, Dr. Hunt failed to carry conviction to the minds of his fellow-workers; and it may well be doubted if some of his views on these matters will ultimately add to his scientific reputation. But it would be unjust on this account to ignore the mass of solid work which he accomplished, and the suggestive hints which are scattered throughout his writings.

Dr. Hunt was a man of wide reading and general culture; he possessed a marvellous memory, and great conversational powers. In his company one might for hours forget that science was his special study, so well informed was he in history, literature, and philosophy. His conversation on such subjects possessed an additional interest from his personal acquaintance with many American authors. He was thus an excellent travelling companion, and the writer will not soon forget with what thrilling effect he recited Macaulay's "Horatius," within sight of Cortona and its Etruscan walls.

NOTES.

W. TOPLEY.

THE date of the Bakerian Lecture to be delivered before the Royal Society has been altered to March 10. Prof. James Thomson has chosen as his subject "The Trade Winds."

THE general arrangements for the Edinburgh meeting of the British Association have now been completed. The first general meeting will be held on Wednesday, August 3, at 8 p.m., when Dr. William Huggins, F. R.S., will resign the chair, and Sir Archibald Geikie, For. Sec. R. S., Director-General of the Geological Survey of the United Kingdom, President-Elect, will assume the Presidency, and deliver an address. On Thursday evening, August 4, at 8 p.m., there will be a soirée; on Friday evening, August 5, at 8.30 p.m., a discourse will be delivered by Prof. A. Milnes Marshall, F. R. S.; on Monday evening, August 8, at 8.30 p.m., a discourse on magnetic induction will be delivered by Prof. J. A. Ewing, F.R.S.; on Tuesday evening, August 9, at 8 p.m., there will be another soirée; and on Wednesday, August 10, the concluding general meeting will be held at 2.30 p.m. The different Sections will assemble for the reading and discussion of Reports and other communications on Thursday, August 4, and on the following Friday, Saturday, Monday, and Tuesday. The delegates of Corresponding Societies will meet on Thursday, August 4, and Tuesday, August 9, at 3.30 p.m. Excursions to places of interest in the neighbourhood of Edinburgh will be made on the afternoon of Saturday, August 6, and on Thursday, August II.

IT is proposed that Englishmen shall celebrate the fourth centenary of the discovery of the New World, and do honour to the memory of Columbus, by establishing in Jamaica a marine biological station on the lines of the marine laboratories at Naples and Plymouth. The institution would be called "the Columbus Marine Biological Station." An excellent letter on the subject by Lady Blake appeared in the Times on Wednesday. The scheme has been laid before Prof. Huxley, Prof. Ray Lankester, Prof. Flower, Dr. Günther, Dr. Ball, Sir John Lubbock, Mr. Scott, Mr. Sclater, and numerous other scientific men, all of whom warmly approve of it. For the establishment of the laboratory on a sound basis an outlay of £15,000 will be

required. The following have consented to receive subscriptions:-Prof. Ray Lankester, Oxford; Dr. Günther, British Museum (Natural History), Cromwell Road; Dr. Ball, Science and Art Museum, Dublin; the Duchess of St. Albans, Bestwood Lodge, Arnold, Notts. ; and Messrs. Coutts and Co., bankers, 59 Strand. The Hon. Walter Rothschild, 148 Piccadilly, has undertaken the duties of honorary secretary.

The sug

ON Saturday last a meeting was held in the Combination Room of St. John's College, Cambridge, to discuss a proposal for the provision of a national monument to the late Prof. Adams. The Rev. Dr. Taylor, the Master of the College, presided; and among those present were Dr Peile (Master of Christ's, and Vice-Chancellor), Dr. Ferris (Master of Caius), Dr. Porter (Master of Peterhouse), Mr. Aldis Wright (ViceMaster of Trinity), Dr. Forsyth, Prof. Hughes, Dr. Hobson, Prof. Thomson, Dr. Glaisher, Dr. Frost, Dr. Sandys, Prof. Mayor, and Sir George G. Stokes, M.P. The Master said that Prof. Adams had memorials in Cambridge in the Adams Prize, and his portraits at that College and at Pembroke. His own work was his monument in the annals of science. They wished to commemorate his name and personality in the eyes of the world in that central sanctuary where, age after age, they commemorated their national types of various kinds of supreme excellence which were the glory of the world. The first suggestion of that came to him from Archdeacon Farrar. gestion had been mentioned at a College meeting and by it adopted, and they were met that day to carry it out. He thought the better method would be to form a large and influential committee, containing the most prominent names in mathematics and science, which would enable them to show there was a general feeling in favour of it. Then he thought the request might be made to the Dean and Chapter, on behalf of the Committee, by the Chancellor, the Duke of Devonshire, and in a letter which he had received from the Duke he stated that he should be very glad to give any assistance in his power to carry out the wishes of the Committee. Among those who had agreed to join the Committee were the Astronomer-Royal, the Master of Trinity, Dr. Salmon (Provost of Trinity College, Dublin), the Master of Corpus, Mr. Justice Romer, Prof. Jebb, Mr. Courtney, Lord Rayleigh, Prof. Newton, the Gresham Professor of Astronomy, Prof. Cayley, and Sir Donald Smith (Chancellor of Montreal University), who asked to be allowed to subscribe £100. The following motion, proposed by the Master, seconded by Sir G. G. Stokes, and supported by Dr. Glaisher and Prof. Liveing, was carried unanimously: "That the late Prof. John Couch Adams, by his discovery of the planet Neptune, and other masterly work, published or unpublished, is entitled to be named with the great astronomers of the world; and that this meeting pledges itself (so far as in lies) to promote and carry out the scheme for placing a memorial to the late Professor in Westminster Abbey." The following resolutions were also carried :—“That the memorial consist of a bust, with tablet and inscription." "That a Committee be formed (with power to add to their number) to carry out the scheme; that the Master of Pembroke College and Prof. Liveing be the Treasurers, and the Master of Peterhouse, Dr. D. MacAlister, and Dr. Glaisher the Secretaries, and that such and such persons be the Executive Committee." "That any surplus from subscriptions after payment of the necessary expenses to be used in the first instance to defray the cost of presenting copies of the collected papers of Prof. Adams to learned Societies and libraries at home and abroad, and that the remainder (which, if of sufficient amount, shall be constituted a permanent memorial fund) be offered to the Master and Fellows of St. John's College to form an Exhibition or Scholarship fund for the encouragement of the study of mathematics or physics by the undergraduate students of the

was published in the Kew Bulletin for October and November 1891. A translation of the botanical section of the reports made by the French members of the Commission is given in the February number of the Bulletin for the purpose of supplementing Dr. Brown Lester's notes.

APPENDIX II., 1892, of the Kew Bulletin contains a list of the new garden plants of the year 1891. The list includes, besides the plants brought into cultivation for the first time in 1891, the most noteworthy of those which have been reintroduced after being lost from cultivation. Other plants in the list have been in gardens for several years, but either were not described or their names had not been authenticated till recently.

IN their Irish Education Bill the Government propose that a large proportion of the funds at their disposal for the improvement of national education in Ireland shall be spent for the benefit of the teachers, who as a class have hitherto been too much neglected. The rest of the amount will be devoted to a capitation grant, and to the freeing of all schools in which the fees do not exceed six shillings a year per child. Attendance at elementary schools, if the Bill becomes law, will be compulsory in Irish towns, but in rural disiricts it will be open to the people to accept or reject compulsion as they may think fit. THE National Association for the Promotion of Technical and Secondary Education has issued an appeal to the electors of the London County Council on the subject of technical instruction. As everyone interested in technical education knows, London has devoted to the relief of the rates the whole of its share of the grant obtained from the proceeds of the beer and spirits duties. This has been done in direct opposition to the wish of Parliament; and the Association has no difficulty in showing that the grant will be continued only if it is used for the purposes to which the House of Commons intended it to be applied. It may be hoped that the appeal will be widely read, and that voters will perceive that it deals with a matter by which their interests must sooner or later be vitally affected.

THE principal article of interest to meteorologists in the American Meteorological Journal for January is by A. L. Rotch, on the mountain meteorological stations of the United States. At the present time the only stations in operation throughout the year are the Lick Observatory, in California, and the Blue Hill Observatory, in Massachusetts.

That on

Mount Washington (6280 feet above the sea) was established in 1870, and partially closed in 1887; during the three following years it was opened during the summer months only. At no other station in the world was such severe weather experienced, as the highest wind velocity often occurred with the lowest temperature. During a storm in February 1876, when the temperature fell to 50°, a wind velocity of 184 miles an hour was recorded. In foggy weather the frost formed upon the anemometer cups in such quantity as to break off the arms. The observations at this station have been much lessened in value, owing to their not being published in detail, and to the want of a low-level station for comparison. The Blue Hill Observatory is only 640 feet above the sea, and was opened in 1885. The hourly values for five years have been printed in the Harvard College Observatory. For several years hourly observations of clouds have been made, with a view to benefit weather predictions. The Observatory on Pike's Peak, Colorado (14,134 feet), was built in 1873, and for fifteen years was maintained by the Signal Service. It was closed in 1888, and the observations have been published in the Annals of the Harvard College Observatory. The average annual temperature was 19°, and the extremes 64° and 39°. Pike's Peak is remarkable for its electrical storms. When the air is moist, and

generally when snow is falling, sparks emanate from the fingers of the outstretched hands; but the station was only once struck by lightning. The Lick Observatory is on Mount Hamilton, 4300 feet above the Pacific Ocean, which is plainly visible from the summit. Fragmentary observations have been made at various other stations, the most important of which were those by Prof. Langley, on Mount Whitney, California, in 1881, which have served to change the theory of the nature of the heat received from the sun, and to show that the sun is much hotter than had been supposed. The article is accompanied by photographic illustrations of several of the stations.

ELECTRICITY is being applied to a novel use in the U.S. Navy. Four electric fans have been placed by the Crocker Wheeler Company in the turrets of the powerful iron vessel Miantonomah, the intention being that they shall blow away the smoke from the guns.

AN interesting compound of carbon with the metal barium, possessing the composition C,Ba, is described by M. Maquenne in the current number of the Comptes rendus. It may be considered, perhaps, as an acetylide of barium—that is, a compound formed by the replacement of the hydrogen of acetylene, CH2, by metallic barium. For immediately it is brought in contact with water pure acetylene gas is evolved with great rapidity. M. Maquenne has obtained the new substance by the direct action of metallic barium, employed in the form of an amalgam consisting of one part barium and four parts mercury, upon powdered retort-charcoal. Upon distilling such a mixture in a current of hydrogen, when the mercury had been expelled and the temperature attained redness, an energetic reaction was found to occur between the barium and the carbon, with production of the new carbide or acetylide. The hydrogen took no part in the reaction, and M. Maquenne has subsequently found that it may be replaced by nitrogen; the latter, however, being less advantageous, inasmuch as the carbide produced is then admixed with more or less cyanide. The new substance, as obtained when hydrogen is employed to furnish the atmosphere, consists of a grey, friable mass, which remains quite unaitered when heated to bright redness. The moment, however, it is thrown into cold water it is decomposed, with a rapid effer. vescence of a gas which possesses the odour of acetylene, burns in the air with a luminous flame, precipitates a red substance resembling acetylide of copper from an ammoniacal solution of cuprous chloride, and, in short, possesses all the properties of is remarkably pure. acetylene. M. Maquenne adds that the acetylene thus obtained The reaction with water may be expressed by the equation—

C,Ba + 2H2O = C2H2 + Ba(OH)2. Barium acetylide would appear to be analogous to the conpounds obtained by M. Berthelot by heating the metals of the alkalies in a current of acetylene, and also to the acetylide of calcium prepared by Wöhler. The direct formation of this substance from barium and carbon, together with its reaction with water, afford another mode of synthesizing acetylene, which M. Maquenne considers to be of interest from the point of view of the formation of the natural hydrocarbons. He considers it probable that other metals possess this same property of forming acetylides under the influence of high temperatures. If, therefore, as M. Berthelot has attempted to show, it is a fact that acetylene forms the primary material, or starting-point, for the formation of other hydrocarbons, it is quite possible that such compounds of metals with carbon, upon coming in contact with water under conditions of more or less pressure, may give rise to the production of the immense stores of natural hydrocarbons, such as those which exist in the petroleum wells of Russia and the New World.

THE additions to the Zoological Society's Gardens during the past week include a Sykes's Monkey (Cercopithecus albigularis ? ) from East Africa, presented by Mr. G. N. Wylie; a Beatrix Antelope (Oryx beatrix ? ), an Indian Gazelle (Gazella bennetti) from Arabia, presented by Lieut. -Colonel Talbot; a Goshawk (Astur palumbarius), European, presented by Captain Noble ; a Common Quail (Coturnix communis), European, presented by W. K. Purnell; a Hybrid Goose (between Anser cinereus and A. brachyrhynchus), captured in Holland, presented by Mr. F. E. Blaauw, C. M.Z.S.; a Gould's Monitor (Varanus gouldi), a Stump-tailed Lizard (Trachydosaurus rugosus) from New South Wales, presented by Mr. T. Hellberg; a Chub (Leuciscus cephalus), British fresh waters, presented by Mr. H. E. Young; two Yaks (Poëphagus grunniens 8 ) from

is also recorded upon the plates. Photographs have been taken of the spectra of spots and faculæ. The calcium lines at H and K often appear bright upon them, and are always stronger than the hydrogen lines. But no new facts appear to have been discovered in this direction of work.

ON THE VARIATION OF LATITUDE.-Dr. S. C. Chandler has published a series of papers on the variation of latitude, in the Astronomical Journal from No. 248 to No. 251. The general result of a wide discussion indicates a revolution of the earth's

axis of inertia about that of rotation from west to east, with a radius of 30 feet measured at the earth's surface, in a period of 427 days.

NON-EUCLIDIAN GEOMETRY.1

from Japan, deposited; an Azara's Agouti (Dasyprocta azaræ), a Pucheran's Hawk (Asturina pucherani), a Sulphury Tyrant (Pitangus sulphuratus), two Short-winged Tyrants (Machetornis rixosa) a Brown Milvago (Milvago chimango), an Orange-billed Coot (Fulica leucoptera), a Cayenne Lapwing (Vanellus cayennensis), six Rosy-billed Ducks (Metopiana peposaca 38 39) from South America, purchased; an American Bison (Bison americanus) from North America, received in exchange; a Gayal (Bibos frontalis ), born in the Gardens.

OUR ASTRONOMICAL COLUMN.

THE SOLAR DISTURBANCE OF 1891, JUNE 17.—In the October number of the Observatory Mr. H. H. Turner publishes an article on the luminous outburst on the sun observed by M. Trouvelot on June 17, and recorded in these columns on July 9. The disturbance was of such an unusual character that M. Trouvelot hazarded the suggestion that it was possibly accompanied by perturbations of the magnetic elements. Mr. Whipple was good enough to look over the Kew curves to see if they showed any such variations, and a negative result was obtained. Mr. Turner, however, after an examination of the Greenwich records has succeeded in finding "a very minute, though unmistakable, disturbance at almost precisely the time noted by Trouve lot. . . . disturbance is smaller than many others on the same day, although the day itself was very quiet: but it differs from others in its abruptness, which is clearly shown in all three curves. The change in declination is only about 1', and in H. F. 0'0005 of the whole H.F." Diagrams illustrating these fluctuations accompanied Mr. Turner's paper. It seemed strange

The

that the Kew and the Greenwich records should differ in their indications, so a iurther enquiry was sent to Mr. Whipple, who replied as follows:-"I have again referred to the curves of June 17, 1891, and fail to find any trace of what can by any means be termed to be a magnetic disturbance at the time in question -accepting Sabine's interpretation of a magnetic disturbance (see Phil. Trans, vol. cliii., p. 274), and so avoiding loose expressions. According to the Observatory, October 1891, Father Sidgreaves is quite of our opinion as to the case in point." The evidence in favour of a magnetic disturbance simultaneously with Trouvelot's observation is thus not very strong.

PHOTOGRAPHY OF SOLAR PROMINENCES.-In a communication to the Paris Academy on February 8, M. Deslandres described some new results obtained by him in the photography of solar prominences. The object of the research was to photograph the spectra of prominences further into the ultra-violet than had previously been done. In July of last year, M. Deslandres, following Prof. Hale, succeeded in photographing the spectra to A 380. He has now been able to obtain negatives upon which the spectrum extends from A 410 to λ 350. In order to obtain this result, a siderostat with a mirror 8 inches in diameter has been employed to project the sun's image, a Rowland grating has been used to produce the spectra, and the lenses of the observing telescope have been made of quartz. The photographs show eight bright lines of the ultra-violet hydrogen series, and it is believed that observations made from an elevated station would lead to the detection of the remaining two. The line a little more refrangible than hydrogen a (A 388),

themselves are either self-evident and have no need of demonstration, or can only be established by assuming other propositions; and as we cannot continue this process to infinity, every deductive science, and especially geometry, must rest on a certain number of axioms which cannot be demonstrated. All treatises on geometry therefore commence with the enunciation of these axioms. But a distinction must be made between them some-such as this for example, "Two quantities that are equal to a third quantity are equal to one another"—are not geometrical propositions, but are analytical ones. I regard them as analytical a priori judgments, and as such I will not discuss them. But I must insist on other axioms which are special to geometry. Text-books for the most part state them very explicitly:

(1) Only one straight line can be drawn between two points. (2) A straight line is the shortest distance between two points. (3) Only one straight line can be drawn through a point parallel to a given straight line.

Although the demonstration of the second of these axioms is generally dispensed with, it would be possible to deduce it from the other two, and from those, of which the number is more considerable, that we admit explicitly without stating them, as I shall explain in the sequel.

Efforts have also for a long time been made without success to demonstrate the third axiom, known under the name of the

postulatum d'Euclide. The amount of trouble that has been taken in that chimerical hope is truly beyond imagination. simultaneously, Lowatchewski and Bolyai, two men of science, Finally, at the commencement of the century, and almost a Russian and Hungarian respectively, established, in an irrefutable manner, that such a demonstration was impossible; they have very nearly rid us of the inventors of geometries without postulates since their time the Academy of Sciences only receives annually one or two new demonstrations.

The question was still not settled; soon a great step was made by the publication of the celebrated memoir of Riemann, entitled "Ueber die Hypothesen welche der Geometrie zum Grunde liegen." This small treatise has inspired the majority of recent works, of which I will make mention subsequently, and among which must be mentioned those of Beltrami and Helmholtz.

The Geometry of Lowatchewski.-If it were possible to deduce the postulatum d'Euclide from the other axioms, it would evidently happen that in denying the postulate and admitting the axioms, we should be led to contradictory results; it would then be impossible to base a coherent geometry on such premisses.

But this is precisely what Lowatchewski has done. He supposes in the first place that—

"Several straight lines can be drawn through a point parallel to a given straight line."

And he moreover retains all the other axioms of Euclid. From these hypotheses he deduces a series of theorems among which it is impossible to detect any contradiction, and he constructs a geometry the faultless logic of which is not inferior to that of the Euclidian geometry.

The theorems are, certainly, very different from those to which we are accustomed, and they disconcert us a little at first.

Thus, the sum of the angles of a triangle is always less than two right angles; and the difference between this sum and two right angles is proportional to the surface of the triangle.

Translation of an article that appeared in the Revue Générale des Sciences, No. 23, by M. H. Poincaré.

It is impossible to construct a figure similar to a given figure, but of different dimensions.

If a circle be divided into n equal parts, and tangents be drawn to the points of division, these n tangents will meet and form a polygon, provided that the radius of the circle be small enough; but if this radius is sufficiently large, they will not meet. It is useless to multiply these examples; the propositions of Lowatchewski have no longer any connection with those of Euclid, but they are not less logically connected together.

The Geometry of Riemann.-Let us imagine a world peopled only with beings deprived of thickness; and let us suppose that these animals, "infinitely flat," are all in one plane, and are not able to get out of it. Let us admit, further, that this world is removed sufficiently from others to be free from their influence. As we are making these assumptions, we may as well endow these beings both with reasoning powers and the capacity of founding a geometry. In this case they would certainly attribute to space only two dimensions.

But let us suppose, however, that these imaginary animals, all still devoid of thickness, have the form of a portion of a spherical figure, and not of a plane one, and are all on one and the same sphere without being able to leave it. What geometry would they construct? It is clear at once that they would only attribute to space two dimensions: that which will play for them the part of the straight line will be the shortest distance between two points on the sphere-that is to say, an arc of a great circle; in a word, their geometry would be spherical geometry.

What they will call space will be this sphere which they cannot leave, and on which occur all the phenomena of which they can have any knowledge. Their space then will be without limits, since on a sphere one can always go forward, without ever coming to an end, and nevertheless it will be finite-one will never find the limit, but one can make the circuit of it.

In fact, the geometry of Riemann is spherical geometry extended to three dimensions. To construct it, the German mathematician had to throw overboard not only the postulates of Euclid, but even the first axiom: Only one straight line can be drawn between two points.

On a sphere only one great circle in general can be drawn through two given points (which, as we have just seen, would play the part of the straight line to our imaginary beings); but to this there is an exception; for, if the two given points are diametrically opposed, an infinite number of great circles can be made to pass through them.

In the same way, in the geometry of Riemann, only one straight line in general can be drawn between two points; but there are exceptional cases where an infinite number of straight lines can be drawn between them.

There is a kind of opposition between the geometry of Riemann and that of Lowatchewski.

Thus, the sum of the angles of a triangle is

Equal to two right angles in Euclid's geometry.

Less than two right angles in that of Lowatchewski. Greater than two right angles in that of Riemann. The number of parallels that can be drawn to a given straight line through a given point is equal

To one in the geometry of Euclid.

To zero in that of Riemann.

To infinity in that of Lowatchewski.

Let us add that the space of Riemann is finite although without limit, in the sense already given to these two words.

Surfaces of Constant Curvature.-There was, however, one possible objection. The theorems of Lowatchewski and of Riemann present no contradiction, but, however numerous the consequences which these two geometers have drawn from their hypotheses, they were compelled to stop before they had exhausted all of them, for the number would be infinite: who can say, therefore, that, if they had carried their deductions further, they would not finally have found such contradictions?

This difficulty does not exist for the geometry of Riemann, provided that it is limited to two dimensions; for, in fact, the geometry of Riemann for two dimensions does not differ, as we have seen, from spherical geometry, which is only a branch of ordinary geometry, and consequently outside all discussion.

M. Beltrami, in considering in the same way the two-dimensional geometry of Lowatchewski to be only a branch of ordinary geometry, has equally refuted the objection in this

case.

This he has done this in the following manner :-Consider on

a surface any figure. Imagine this figure, traced on a flexible and inextensible cloth, to be laid on this surface, in such a way that when the cloth is moved and changes its shape, the various lines of this figure can change form without altering their length. In general this flexible and inextensible figure cannot leave its place without quitting the surface; but there are certain particular surfaces for which a similar movement would be possible; these are the surfaces with constant curvature.

If we resume the comparison that we previously made, and imagine beings without thickness living on one of these surfaces, they will regard the movement of a figure all of whose lines preserve a constant length as possible. A like movement, on the other hand, would appear absurd to animals without thickness living on a surface whose curvature was variable.

These surfaces of constant curvature are of two kinds :Some are of positive curvature, and can be so deformed as to be laid on a sphere. The geometry of these surfaces becomes then spherical geometry, which is that of Riemann.

Others are of negative curvature. M. Beltrami has shown that the geometry of these surfaces is none other than that of Lowatchewski. The two-dimensional geometries of Riemann and Lowatchewski are thus found to be re-attached to Euclidian geometry.

Interpretation of Non-Euclidian Geometries.-Thus the objection disappears as regards geometries of two dimensions. It would be easy to extend M. Beltrami's reasoning to geometries of three dimensions. The minds which space of four dimensions does not repel will see here no difficulty; but they are few. I prefer, then, to proceed otherwise.

Let us consider a particular plane that we will call fundamental, and construct a kind of dictionary, making a double series of words, written in the two columns, correspond each to each, in the same way that the words of two languages, having the same signification correspond in ordinary dictionaries :Space...

Plane

Right line...

Sphere Circle Angle

Distance between two points

&c.,

Portion of space situated above the fundamental plane.

Sphere cutting orthogonally the funda-
mental plane.

Circle cutting orthogonally the fundamental
plane.
Sphere.
Circle.
Angle.

Logarithm of the anharmonic ratio of
these two points and the intersections
of the fundamental plane with a circle
passing through these two points and
cutting it orthogonally.
&c.

Let us take, then, the theorems of Lowatchewski, and translate them by means of this dictionary, as we should translate a German text with the aid of a German-French dictionary. We shall obtain then the theorems of ordinary geometry.

For example, this theorem of Lowatchewski-" The sum of the angles of a triangle is less than two right angles"-is translated thus: "If a curvilinear triangle has for its sides the arcs of a circle which if prolonged would cut orthogonally the fundamental plane, the sum of the angles of this curvilinear triangle will be less than two right angles.' Thus, however far one pushes the results of the hypotheses of Lowatchewski, one will never be led to a contradiction. Indeed, if two of Lowatchewski's theorems were contradictory, the translations of these two theorems, made with the help of our dictionary, would also be contradictory; but these translations are theorems of ordinary geometry, and everyone agrees that ordinary geometry is free from contradictions. Whence comes this certainty, and is it justified? This is a question that I cannot treat here, but which is very interesting, and, as I believe, soluble. The objection that I have formulated above no longer then exists. But this is not all. The geometry of Lowatchewski, susceptible of a concrete interpretation, ceases to be a frivolous logical exercise, and is capable of application: I have not the time to mention here either these applications or the use that M. Klein and myself had made of them for the integration of linear equations.

This interpretation, moreover, is not unique, and one could construct several dictionaries analogous to that given above, and by which we could by a simple "translation transform the theorems of Iowatchewski into theorems of ordinary geometry,