Imágenes de páginas

The additions to the Zoological Society's Gardens during the is also recorded upon the plates. Photographs have been taken past week include a Sykes's Monkey (Cercopithecus albigularis )

of the spectra of spots and faculæ. The calcium lines at Hand K

often appear bright upon them, and are always stronger than from East Africa, presented by Mr. G. N. Wylie ; a Beatrix

the hydrogen lines. But no new facts appear to have been Antelope (Oryx beatrix ), an Indian Gazelle (Gazella bennetti) discovered in this direction of work. from Arabia, presented by Lieut.-Colonel Talbot ; a Goshawk

ON THE VARIATION OF LATITUDE.-Dr. S. C. Chandler (Astur palumbarius), European, presented by Captain Noble ; has published a series of papers on the variation of latitude, in a Common Quail (Coturnix communis), European, presented the Astronomical Journal from No. 248 to No. 251. The general by W. K. Purnell; a Hybrid Goose (between Anser cinercus result of a wide discussion indicates a revolution of the earth's and A. brachyrhynchus), captured in Holland, presented by axis of inertia about that of rotation from west to east, with a Mr. F. E. Blaauw, C.M.Z.S. ; a Gould's Monitor (Varanus

radius of 30 feet measured at the earth's surface, in a period of

427 days. 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.

NON-EUCLIDIAN GEOMETRY.1 H. E. Young ; two Yaks (Poëphagus grunniens 8 9 ) from Tibet, three Gigantic Salamanders (Megalobatrachus maximus) | EVERY conclusion supposes premisses ; these premisses

themselves are either self-evident and have no need of from Japan, deposited ; an Azara's Agouti (Dasyprocta azara), demonstration, or can only be established by assuming other a Pucheran's Hawk (Asturina pucherani), a Sulphury Tyrant propositions ; and as we cannot continue this process to infinity, (Pilangus sulphuratus), two Short-winged Tyrants (Machetornis every deductive science, and especially geometry, must rest on

a certain number of axioms which cannot be demonstrated. rixosa) a Brown Milvago (Milvago chimango), an Orange-billed

All treatises on geometry therefore commence with the enunciaCoot (Fulica leucoptera), a Cayenne Lapwing (Vanellus cayen. tion of these axioms. But a distinction must be made between nensis), six Rosy-billed Ducks (Metopiana peposaca 38 39 ) them : some-such as this for example, "Two quantities that are from South America, purchased ; an American Bison (Bison equal to a third quantity are equal to one another"--are not americanus 8 ) from North America, received in exchange ; a

geometrical propositions, but are analytical ones. I regard Gayal (Bibos frontalis !), born in the Gardens.

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


(1) Only one straight line can be drawn between two points. THE SOLAR DISTURBANCE OF 1891, JUNE 17.- In the October

(2) A straight line is the shortest distance between two points. number of the Observatory Mr.H. H. Turner publishes an article parallel to a given straight line.

(3) Only one straight line can be drawn through a point on the luminous outburst on the sun observed by M. Trouvelot on June 17, and recorded in these columns on July 9. The disturbance generally dispensed with, it would be possible to deduce it from

Although the demonstration of the second of these axioms is was of such an unusual character that M. Trouvelot hazarded the

the other two, and from those, of which the number is more suggestion that it was possibly accompanied by perturbations of

considerable, that we admit explicitly without stating them, as the magnetic elements. Mr. Whipple was good enough to look

I shall explain in the sequel. over the Kew curves to see if they showed any such variations,

Efforts have also for a long time been made without success and a negative result was obtained. Mr. Turner, however,

to demonstrate the third axiom, known under the name of the after an examination of the Greenwich records has succeeded in finding “a very minute, though unmistakable, disturbance at

The amount of trouble that has been

postulatum d'Euclide. almost precisely the time noted by Trouvelot. . : : The Finally, at the commencement of the century, and almost

taken in that chimerical hope is truly beyond imagination. disturbance is smaller than many others on the same day, simultaneously, Lowatchewski and Bolyai, two men of science, although the day itself was very quiet : but it differs from others in its abruptness, which is clearly shown in all three

a Russian and Hungarian respectively, established, in an irre

futable manner, that such a demonstration was impossible ; they curves. The change in declination is only about 1', and in H.F. 0.0005 of the whole H.F.". Diagrams illustrating these fuc

have very nearly rid us of the inventors of geometries without tuations accompanied Mr. Turner's paper. It seemed strange

postulates : since their time the Academy of Sciences only that the Kew and the Greenwich records should differ in their

receives annually one or two new demonstrations. indications, so a iurther enquiry was sent to Mr. Whipple, who

The question was still not settled ; soon a great step was replied as follows:-"I have again referred to the curves of June entitled Ueber die Hypothesen welche der Geometrie zum

made by the publication of the celebrated memoir of Riemann, 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 of recent works, of which I will make mention subsequently,

Grunde liegen. This small treatise has inspired the majority -accepting Sabine's interpretation of a magnetic disturbance

and among which must be mentioned those of Beltrami and (see Phil. Trans, vol. cliii., p. 274), and so avoiding loose ex

Helmholtz. pressions. According to the Observatory, October 1891, Father Sidgreaves is quite of our opinion as to the case in

The Geometry of Lowatchewski.- If it were possible to deduce point." The evidence in favour of a magnetic disturbance evidently happen that in denying the postulate and admitting

the postulatum d' Euclide from the other axioms, it would simultaneously with Trouvelot's observation is thus not very

the axioms, we should be led to contradictory results ; it strong.

would then be impossible to base a coherent geometry on such PHOTOGRAPHY OF SOLAR PROMINENCES.-In a communica- premisses. tion to the Paris Academy on February 8, M. Deslandres But this is precisely what Lowatchewski has done. He described some new results obtained by him in the photography supposes in the first place thatof solar prominences. The object of the research was to photo- "Several straight lines can be drawn through a point parallel graph the spectra of prominences further into the ultra-violet to a given straight line.' than had previously been done. In July of last year, M. And he moreover retains all the other axioms of Euclid. Deslandres, following Prof. Hale, succeeded in photographing From these hypotheses he deduces a series of theorems among the spectra to 1 380. He has now been able to obtain negatives which it is impossible to detect any contradiction, and be upon which the spectrum extends from 1 410 to 1 350. In order constructs a geometry the faultless logic of which is not inferior to obtain this result, a siderostat with a mirror 8 inches in to that of the Euclidian geometry. diameter has been employed to project the sun's image, a The theorems are, certainly, very different from those to Rowland grating has been used to produce the spectra, and the which we are accustomed, and they disconcert us a little at first. lenses of the observing telescope have been made of quartz. Thus, the sum of the angles of a triangle is always less than The photographs show eight bright lines of the ultra-violet two right angles ; and the difference between this sum and two hydrogen series, and it is believed that observations made from right angles is proportional 10 the surface of the triangle. an elevated station would lead to the detection of the remaining two. The line a little more refrangible than hydrogen a (1 388), Sciences, No. 23, by M. H. Poincaré.

Translation of an article that appeared in the Revue Générale des

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It is impossible to construct a figure similar to a given figure, a surface any figure. Imagine this figure, traced on a flexible but of different dimensions.

and inextensible cloth, to be laid on this surface, in such a way If a circle be divided into n equal parts, and tangents that when the cloth is moved and changes its shape, the various be drawn to the points of division, these n tangents will meet lines of this figure can change form without altering their length. and form a polygon, provided that the radius of the circle In general this flexible and inextensible figure canno: leave its be small enough; but if this radius is sufficiently large, they place without quitting the surface ; but there are certain parwill not meet. It is useless to multiply these examples; the ticular surfaces for which a similar movement would be possible ; propositions of Lowatchewski have no longer any connection these are the surfaces with constant curvature. with those of Euclid, but they are not less logically connected If we resume the comparison that we previously made, and together.

imagine beings without thickness living on one of these surfaces, The Geometry of Riemann.-Let us imagine a world peopled they will regard the movement of a figure all of whose lines only with beings deprived of thickness ; and let us suppose that preserve a constant length as possible. A like movement, on these animals, "infinitely flat,” are all in one plane, and are not the other hand, would appear absurd to animals without thick. able to get out of it. Let us admit, further, that this world is ness living on a surface whose curvature was variable. removed sufficiently from others to be free from their influence. These surfaces of constant curvature are of two kinds :As we are making these assumptions, we may as well endow Some are of positive curvature, and can be so deformed as to these beings both with reasoning powers and the capacity of be laid on a sphere. The geometry of these surfaces becomes founding a geometry. In this case they would certainly attribute then spherical geometry, which is that of Riemann. to space only two dimensions.

Others are of negative curvature. M. Beltrami has shown that But let us suppose, however, that these imaginary animals, all the geometry of these surfaces is none other than that of still devoid of thickness, have the form of a portion of a spherical | Lowatchewski. The two-dimensional geometries of Riemann figure, and not of a plane one, and are all on one and the same and Lowatchewski are thus found to be re-attached to Euclidian sphere without being able to leave it. What geometry would they geometry. construct? It is clear at once that they would only attribute to Interpretation of Non-Euclidian Geometries.—Thus the obspace two dimensions : that which will play for them the part of jection disappears as regards geometries of two dimensions. the straight line will be the shortest distance between two It would be easy to extend M. Beltrami's reasoning to points on the sphere--that is to say, an arc of a great circle ; in geometries of three dimensions. The minds which space of a word, their geometry would be spherical geometry.

four dimensions does not repel will see here no difficulty ; but What they will call space will be this sphere which they they are few. I prefer, then, to proceed otherwise. cannot leave, and on which occur all the phenomena of which Let us consider a particular plane that we will call fundathey can have any knowledge. Their space then will be without mental, and construct a kind of dictionary, making a double limits, since on a sphere one can always go forward, without series of words, written in the two columns, correspond each to ever coming to an end, and nevertheless it will be finite-one each, in the same way that the words of two languages, having will never find the limit, but one can make the circuit of it. the same signification correspond in ordinary dictionaries :

In fact, the geometry of Riemann is spherical geometry extended to three dimensions. To construct it, the German


Portion of space situated above the fundamathematician had to throw overboard not only the postulates

mental plane. of Euclid, but even the first axiom : Only one straight line can


Sphere cutting orthogonally the fundabe drawn between two points.

mental plane. On a sphere only one great circle in general can be drawn Right line... Circle cutting orthogonally the fundamental through two given points (which, as we have just seen, would

plane. 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


Angle. be made to pass through them.

Logarithm of the anharmonic ratio of In the same way, in the geometry of Riemann, only one Distance between

these two points and the intersections straight line in general can be drawn between two points; but two points

of the fundamental plane with a circle there are exceptional cases where an infinite number of straight

passing through these two points and lines can be drawn between them.

cutting it orthogonally.

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

Let us take, then, the theorems of Lowatchewski, and translate Thus, the sum of the angles of a triangle is—

them by means of this dictionary, as we should translate a Equal to two right angles in Euclid's geometry.

German text with the aid of a German-French dictionary. We Less than two right angles in that of Lowatchewski. shall obtain then the theorems of ordinary geometry. Greater than two right angles in that of Riemann.

For example, this theorem of Lowatchewski-"The sum of The number of parallels that can be drawn to a given straight the angles of a triangle is less than two right angles”—is transline through a given point is equal

lated thus : “If a curvilinear triangle has for its sides the arcs To one in the geometry of Euclid.

of a circle which if prolonged would cut orthogonally the To zero in that of Riemann.

fundamental plane, the sum of the angles of this curvilinear To infinity in that of Lowatchewski.

triangle will be less than two right angles.” Thus, however Let us add that the space of Riemann is finite although without far one pushes the results of the hypotheses of Lowatchewski, limit, in the sense already given to these two words.

one will never be led to a contradiction. Indeed, if two of Surfaces of Constant Curvature.-There was, however, one Lowatchewski's theorems were contradictory, the translations of possible objection. The theorems of Lowatchewski and of these two theorems, made with the help of our dictionary, would Riemann present no contradiction, but, however numerous the also be contradictory ; but these translations are theorems of consequences which these two geometers have drawn from their ordinary geometry, and everyone agrees that ordinary geometry hypotheses, they were compelled to stop before they had ex- is free from contradictions. Whence comes this certainty, hausted all of them, for the number would be infinite : who can and is it justified ? This is a question that I cannot treat here, say, therefore, that, if they had carried their deductions further, but which is very interesting, and, as I believe, soluble. The they would not finally have found such contradictions ?

objection that I have formulated above no longer then exists. This difficulty does not exist for the geometry of Riemann, But this is not all. The geometry of Lowatchewski, susprovided that it is limited to two dimensions ; for, in fact, the ceptible of a concrete interpretation, ceases to be a frivolous geometry of Riemann for two dimensions does not differ, as we logical exercise, and is capable of application : I have not the have seen, from spherical geometry, which is only a branch of time to mention here either these applications or the use that ordinary geometry, and consequently outside all discussion. M. Klein and myself had made of them for the integration of

M. Beltrami, in considering in the same way the two-dimen- linear equations. sional geometry of Lowatchewski to be only a branch of This interpretation, moreover, is not unique, and one could ordinary geometry, has equally refuted the objection in this construct several dictionaries analogous to that given above, and case.

| by which we could by a simple "translation transform the This be has done this in the following manner :-Consider on , theorems of Lowatchewski into theorems of ordinary geometry,


Implicit Axioms. - Are then the axioms explicitly enunciated We thus assume iwo propositions : first, that such a rotain treatises the only foundations of geometry? One can be tion is possible, and then that it can be continued until the two assured to the contrary when one sees that, after having succes- lines are in one straight line. sively abandoned them, there still remain some propositions If the first point be admitted, and the second rejected, we common to theorems of Euclid, Lowatchewski, and Riemann. are led to a series of theorems still more curious than those of These propositions ought to rest on some premisses, as geometers Lowatchewski and Riemann, but equally free from contradiction. admit, although they do not state them. It is interesting to try I will quote only one of them, and that not the most singular : to liberate them from classical demonstrations.

A true straight line can be perpendicular to itself. Stuart Mill has made the assertion that every definition The Theorem of Lie.—The number of implicit axioms introcontains an axiom, since, in defining it, the existence of the duced in classical demonstrations is greater than it need be, and it object defined is implicitly affirmed. This is going too far : it would be interesting to reduce them to a minimum. We can is seldom that one gives a definition in mathematics without ask ourselves, in the first place, if this reduction is possible, if following it by the demonstration of the existence of the object the number of necessary axioms, and imaginable geometries is defined, and when it is omitted, it is generally because the not infinite. reader can easily supply it. It must not be forgotten that the M. Sophus Lie's theorem dominates all this discussion : it can word existence has not the same sense when it is the question of be thus stated :a mathematical creation as when we have to do with a material Let us suppose that the following premisses are admitted :object. A mathematical creation exists, provided that its (1) Space has n dimensions. definition involves no contradiction either in itself or with the (2) The movement of an invariable figure is possible. properties previously admitted.

(3) To determine the position of this figure in space, p conBut if Stuart Mill's remark cannot be applied to all definitions, ditions are necessary. it is none the less true for some of them.

The number of geometries compatible with these premisses will A plane is sometimes defined in the following manner :- be limited. The plane is a surface such that the straight line which joins I can even add that, if n be given, a higher limit to p can be any two points in it lies altogether in the surface.

assigned. This definition manifestly hides a new axiom : we could, it is if, then, the possibility of movement be admitted, only a true, alter it, and that would be better, but then it would be finite number (and that a restricted one) of geometries can be necessary to enunciate the axiom more explicitly.

invented. Other definitions give place to reflections no less important. The Geometries of Riemann.-However, this result seems

Such is, for example, that of the equality of two figures : two to be contradicted by Riemann, because this investigator configures are equal when they can be superposed ; to superpose structed an infinite number of different geometries, and the one them it is necessary to displace one until it coincides with the which generally bears his name is only a particular case. other ; but how must it be displaced ? If we ask, we should be Everything depends, he says, on the way in which we define answered that it ought to be done without changing its shape, the length of a curve. But there are an infinite number of ways and in the manner of an invariable solid. The “reasoning in a of defining this length, and each of these can become the starting circle” would then be evident.

point of a new geometry. In truth, this definition implies nothing. It would have no

That is perfectly true ; but most of these definitions are inmeaning for a being who lived in a world where there were only compatible with the movement of an invariable figure, which is Auids. If it seems clear to us, it is that we are accustomed to supposed possible in Lie's theorem. These geometries of the properties of natural solids that do not differ greatly from Riemann, so interesting on many grounds, can only then remain those of ideal solids whose dimensions are all invariable.

purely analytical, and do not lend themselves to demonstrations Meanwhile, however imperfect it may be, this definition analogous to those of Euclid. implies an axiom.

The Nature of Axioms.—Most mathematicians regard the The possibility of the movement of an invariable figure is not geometry of Lowatchewski only as a simple logical curiosity ; a truth evident by itself, or at least it is only one in the same some of them, however, have gone further. Since several way as the postulatum d'Euclide, and not as an analytical a geometries are possible, is it certain that ours is the true one ? priori judgment would be.

Experience, doubtless, teaches us that the sum of the angles of Moreover, in studying the definitions and the demonstrations

a triangle is equal to two right angles ; but this is only because of geometry, we see that one is obliged to admit, without de.

we operate on too small triangles ; the difference, according to monstrating it, not only the possibility of this movement, but Lowatchewski, is proportional to the surface of the triangle ; even some of its properties.

will it not become sensible if we work with larger triangles, or This results, first of all, from the definition of the straight line. if our means of measurement grow more accurate ? Euclidian Many desective definitions have been given, but the true one is geometry would only then be a provisional geometry. that which is understood in all the demonstrations where the To discuss this question, we ought in the first instance to straight line is in question :

inquire into the nature of geometrical axioms. “ It may happen that the movement of a constant figure is

Are they synthetical conclusions a priori, as Kant used to such that all points of a line belonging to this figure remain immovable while all the points situated outside this line are They would appeal to us then with such force, that we could displaced. Such a line will be called a straight line.” not conceive the contrary proposition, nor construct on it a

We have in this enunciation purposely separated the definition theoretical edifice. There could not be a non-Euclidian from the axiom that it implies.

geometry. Several proofs, such as those relating to the equality of To convince oneself of it, let us take a true synthetical a triangles which depend on the possibility of letting fall a perpen- priori conclusion ; for example, the following :dicular from a point on a line, assume propositions that are If an infinite series of positive whole numbers be taken, all not enunciated, since we must admit that it is possible to carry different from each other, there will always be one number that a figure from one place to another in a certain manner.

is smaller than all the others. The Fourth Geometry. -Among these implicit axioms, there Or this other, which is equivalent to it :is one which seems to me worth mentioning, not only because it If a theorem be true for the number 1, and if it has been shown has given rise to a recent discussion, but because in abandoning to be true for n + 1, provided that it is true for n, then it will it, one can construct a fourth geometry, as coherent as those of

be true for all positive whole numbers. Euclid, Lowatchewski, and Riemann.

Let us next try to free ourselves from this conclusion, and, To demonstrate that we can always raise from a point, A, a denying these propositions, to invent a false arithmetic analogous perpendicular to a straight line, AB, a straight line, AC, is con- to the non-Euclidian geometry. We find that we cannot ; we sidered movable round the point A, and in the first instance shall be even tempted in the first instance to regard these conclucoinciding with the fixed 'line AB; and it is made to turn sions as the results of analysis. round the point A until it lies in the prolongation of AB.

Moreover, let us resume our idea of the indefinitely thin

animals : surely we can scarcely admit that these beings, if they "See MM. Renouvier, Léchalas, Calinon, Revue Philosophique, June have minds like ours, would adopt Euclidian geometry, which 1889 : Critique Philosophique, September 30 and November 30, 1889; Revue

would be contrary to all their experience. Philosophique, 1890, p. 158. See especially the discussion on the “postulate of perpendicularity.'

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


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experimental truths ? But we do not experiment on straight Desiring to be brief, I have affirmed more than I have proved : lines or ideal circles ; only material objects can be dealt with. the reader must pardon me for this. So much has been written On what would depend, then, the experiments which would on this subject, so many different opinions have been put serve to found a geometry? The answer is easy.

forward, that the discussion of them would fill volume. We have seen above that one argues constantly as if

W. J. L. geometrical figures behaved like solids. That which geometry would borrow from experience is therefore the properties of these bodies.

SOCIETIES AND ACADEMIES. But a difficulty exists, and it cannot be overcome. If

LONDON. geometry were an experimental science, it would not be an exact science-it would be liable to a continual revision.

Royal Society, February 11.—“The Role played by Sugar

What do I say? It would from to-day be convicted of error, since we

in the Animal Economy: Preliminary Note on the Behaviour of know that a rigorously invariable solid does not exist.

Sugar in Blood.” By Vaugban Harley, M.D. Geometrical axioms, therefore, are neither synthetic a priori whole amount of added sugar can seldom be recovered from blood

This communication was to show that the causes why the conclusions nor experimental facts. They are conventions : our choice, amongst all possible con:

are threefold. Firstly, the imperfections in the as yet known ventions, is guided by experimental facts ; but it remains free, albumens of the blood behave themselves while coagulating; some

methods of analysis. Secondly, the different ways in which the and is only limited by the necessity of avoiding all contradiction. It is thus that the postulates can remain rigorously true, even

coagulating in the form of firm clots, which retain the saccharine when the experimental laws which have deterinined their

matter in their interstices, rendering it impossible to extract all adoption are only approximate.

the sugar from them by washing; others separating as loose,

flocculent curds, from which the sugar can be regained with In other words, axioms of geometry (I do not speak of those of comparative facility. While, thirdly, as bacteria were distinctly arithmetic, are only definitions in disguise, This being so, what ought one to think of this question : Is

ascertained to have nothing to do in the matter, and yet the loss the Euclidian geometry true ?

of the sugar added to the blood is in every instance distinctly proThe question is nonsense.

gressive-according to the period of time the sugar is lefi in One might just as well ask whether the metric system is true

contact with the blood before the analysis is begun--Dr. and the old measures false ; whether Cartesian co-ordinates Vaughan Harley considered himself justified in saying that there are true and polar co-ordinates false ; whether one geometry This he described as an enzyme; but refrained from going

must exist in the normal blood itself a sugar-transforming agent. cannot be more true than another-it can only be more convenient.

into any further particulars regarding it until his researches upon Now, Euclidian geometry is, and will remain, the most

the subject are more advanced. convenient :

He gave tables of the results of his experiments, and compared (1) Because it is the simplest ; and it is not so simply on them with those recently published by Schenk, Rohmann, and account of our habits of thought, or any kind of direct intuition regained by the first observer ranged from 20 to 55 per cent., and

Seegen; showing that while the percentages of the sugar which we may have of Euclidian space ; it is the most simple those recovered by the two last experimenters fluctuated between in itself in the same way as a polynomial of the first order is 80 and 96 per cent., in his three different series of experiments, simpler than one of the second.

where different methods of analysis were employed, the percent. (2) Because it agrees sufficiently well with the properties of natural solids, those bodies which come nearer to our mem

ages of the added sugar regained ranged respectively between 85 bers and our eye, and with which we make our instruments of

and 100; 92-9 and 99*3 ; and 947 and 99.9 per cent. measurement.

Mathematical Society, February 11.-Prof. Greenhill, Geometry and Astronomy.-The above question has also been

F.R.S., President, in the chair.—The following communicastated in another way. If the geometry of Lowatchewski is tions were made :-On the logical foundations of applied true, the parallax of a very distant star would be finite; if that mathematical sciences, by Mr. Dixon. He maintained the of Riemann be true, it would be negative. Here we have importance of distinguishing in all sciences between what is results which seem subject to experience, and it has been hoped dependent on verbal conventions and what is not. He thus that astronomical observations would have been able to decide distinguished between that part of the meaning of a term which between the three geometries.


is laid down as its definition, and the part which remains to be But what one calls a straight line in astronomy is simply the discovered as a consequence of the definition. So also sciences trajectory of a ray of light. If then, as is impossible, we had might be divided into purely symbolic sciences, which being discovered negative parallaxes, or shown that all parallaxes based on definitions alone conveyed no real information ; subare greater up to a certain limit, we should have the choice jective sciences, which deal with concepts and objective between two conclusions :

sciences, which deal with actual things. He then stated the We could renounce Euclidian geometry, or modify the laws of conditions under which a set of assertions might be arbitrarily optics, and admit that light is not propagated strictly in straight laid down as the definition of a term; and applied these condilines.

tions to show that Newton's three laws of motion could be It is useless to add that everyone would regard the latter regarded as a definition of the term force, that if this was solution as the more advantageous,

done there could no longer be any discussion as to whether or Euclidian geometry, then, has nothing to fear from new

not force alone is sufficient to account for the movements of experiments.

matter. The anomaly that we are apparently able to determine Let me be pardoned for stating a little paradox in con- directions absolutely, though we can determine positions only clusion :

relatively, was explained, and a formal proof of all the The beings which had minds like ours, and who had the elementary theorems of mechanics, including the principle of same senses as we have, but who had not received any previous virtual work, might be deduced.-Note on the inadmissibility education, might receive conventionally from an exterior world of the usual reasoning by which it appears that the limiting choices of impressions such that they would be led to construct

value of the ratio of two infinite functions is the same as the a geometry different from that of Euclid, and to localize the ratio of their first derived, with instances in which the result phenomena of this exterior world in a non-Euclidian space, or

obtained by it is erroneous, by Mr. Culverwell.-On Saint even in a space of four dimensions.

Venant's theory of the torsion of prisms, by Mr. A. B. For us, whose education has been formed by our real world, Bassel, F.R.S. if we were suddenly transported in this new one, we should not

DUBLIN have any difficulty in referring the phenomena to our Euclidian

Royal Society, January 20.-Prof. W. N. Hartley, F.R.S., space. Anyone who should dedicate his life to it could, perhaps,

in the chair. -Reports on the zoological collections made by

Prof. Haddon in Torres Straits, 1888-89: the Hydrocorallina, eventually imagine the fourth dimension,

by S. J. Hickson. The specimens described are a female stock I fear that in the last few lines I have not been very clear. I

of Stylaster gracilis, Distichopora violacea, and Millepora can only be so by introducing new developments ; but I have

Murrayi. Some of the smaller colonies of Distichopora are already been too long, and those whom these explanations might bright orange in colour, others vandyke brown, and the larger interest have read their Helmholtz.

ones are deep purple with pale yellowish tips. The author

believes that the differences in colour mean a difference in age than it and its of their normal value.- Observations of
and sexual condition. The smallest colonies are not sexually atmospheric electricity, made by means of a captive balloon, by
mature, the brownish have male ampullæ, and the oldest stocks M. E. Semmola.- On the determination of the state of dissolved
are violet in colour, and are apparently female. In any case, salts from a study of contraction, by M. Georges Charpy.-On
the colour of Distichopora can no longer be regarded as the some properties of bismuthic acid, by M. G. André.—On a
principal character for specific definition. —Sir Howard Grubb, carbide of barium, by M. Maquenne. (See Notes.)— Trans-
F.R.S., exhibited and described a 4-inch equatorially-mounted formation of aromatic amines into chlorinated hydrocarbons, by
refracting telescope of novel construction, in which the right | MM. Prud'homme and C. Rabaut. On the principles which
ascension and declination circles were situated at the eye end accompany chlorophyll in leaves, by M. A. Etard. - Improve-
of the telescope itself, instead of, as usual, on the polar and ment of the culture of industrial and sodder potatoes in France :
declination axes; thereby rendering the working of the results of the season 1891, by M. Aimé Girard.-Contribu-
instrument most convenient for the observer. The circles at the tions to the study of unplastered wines, by M. H. Quantin.
eye end of the telescope are connected by gearing to the polar - On the assimilation of carbohydrates, by M. Hanriot.-On
and declination axes ; but Sir H. Grubb described the method by the presence of numerous diatoms in the Cretaceous of the Paris
which the "Joss of time," and other errors, consequent on basin, by M. Cayeux.-On the existence of zeolites in the
gearing, were almost totally eliminated, and the readings rendered calcareous Jurassic rocks of Ariège, and on the dissemination of
quite sufficiently accurate for all ordinary purposes to which such these minerals in the Pyıenees, by M. A. Lacroix.
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.

Royal Academy of Sciences, January 30.-Prof. van de

Sande Bakhuyzen in the chair.-Prof. Pekelharing spoke of the Paris.

composition of the fibrin ferment. When oxalated blood-plasma

is diluted with water and treated with acetic acid till moderate Academy of Sciences, February 15.-M. d'Abbadie in the acid reaction, the precipitate consists chiefly of a substance which chair.-On a new method of organic analysis, by M. Berthelot. is soluble in alkali, in an excess of acid, and in neutral saltThe method consists in heating the compound in oxygen under solutions, and from which, by the action of pepsin-hydrochloric a pressure of 25 atmospheres in a calorimetric bomb. Com- acid, is split off a nuclein-a substance that thus must be conbustion is total and instantaneous, and therefore differs from sidered as a nucleo albumin. This nucleo-albumin acquires, comthat which appertains to the use of copper oxide.-On the bined with lime, all the properties of fibrin ferment. It is very employment of compressed oxygen in the calorimetric bomb, by probable that this nucleo-albumin issues from the corpuscles of the same author.-Action of alkaline metals on boric acid : the blood.-Prof. Max Weber gave some results of his investicritical study of the processes used in the preparation of gations of the fresh-water fauna of the islands of Sumatra, Java, amorphous boron, by M. Henri Moissan. The general result of Flores, Celebes, and Saleyer. Among the Crustacea, the the investigation is that when an alkaline metal acts on boric Entomostraca are not essentially different from European forms. acid the reaction that occurs is accompanied with considerable Isopods are only represented by marine species : Ichthyoxenus, heat, and, on account of the elevation of temperature, the Tachæa, Rocinela, and Bopyridæ. Amphipods are extremely greatest part of the boron set free combines with ihe excess of rare, and only Orchestia was found. Nearly 70 species of alkali, and with parts of the metal vessel used for the experi. Decapods were collected, out of which 33 are living also in ment. When this is afterwards washed out with water and brackish and sea water. It could be proved that immigration hydrochloric acid, a mixture of boron, boride of sodium, boride out of the sea into the rivers had taken place. An account was of iron, boron hydride, nitride of boron, and hydrated boric given of the life-history of Ichthyoxenus Tellinghausii. 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


PAGE method of preparing amorphous boron in a future paper.Experimental researches on the transmissibility of cancer, by M.

The Science Museum and the Tate Gallery

385 Simon Duplay:-The temporary star in Auriga, by M. G. Chemical Technology. By W. A. T. Rayet. On February 10 and 11 the star appeared to M. An Agricultural Text-book. By W. T. Rayet to be orange-yellow or pale yellow. Its spectrum was Hylo-Idealism. By C. LI. M.. examined by means of the 14-inch equatorial fitted with a spectro- Physiography. By A. F.

390 scope. It appeared to be continuous, the red and violet Our Book Shelf:portions being comparatively bright. Four bright lines were Vasey : “Grasses of the South-West"

390 seen in the green, and their wave-lengths were determined as Kennedy : “ Sporting Sketches in South America 390 518, 501, 493, and 487.—Extension of Lagrange's equations to Letters to the Editor :the case of sliding friction, by M. Paul Appell.-On the dis- Cirques.-Prof. T. G. Bonney, F.R.S.

391 tribution of prime numbers, by M. Phragmen.-On the measure Bedford College and the Gresham University. ---Dr. of high temperatures ; reply to M. H. Becquerel, by M. H. W. J. Russell, F.R.S., and Lucy J. Russell 391 Le Chatelier.-Remarks on the surface tension of liquid The Implications of Science.-Edward T. Dixon

391 metals ; a reply to a note by M. Pellat, by M. Gouy.

The Value of Useless Studies.- Prof. Geo. Fras. Variation, with temperature, of the dielectric constant of Fitzgerald, F.R.S.

392 liquids, by M. D. Negreano. Experiments on benzine, toluene,

The Nickel Heat Engine.-W. B. Croft

392 and xylene, between 5° and 45° C., indicate that the dielectric con- The University of London. By W. T. Thiseltonstant diminishes with increase of temperature.-On the influence Dyer, C.M.G., F.R.S. .

392 exercised on electro-magnetic resonance by an unsymmetrical A Preliminary Statement of an Investigation of the arrangement of the long circuit along which the waves are pro- Dates of some of the Greek Temples as derived pagated, by MM. Blondlot and M. Dufour. The experiments from their Orientation. By F. C. Penrose . 395 show that the wave-length, measured by means of a resonator, Volcanic Action in the British Isles is independent of the dissymmetry of the two wires which The Centenary of Murchison transmit the electro-magnetic undulations. The propagation of H. W. Bates, the Naturalist of the Amazons. By electric waves studied by a telephonic method, by M. R. Colson. A. R. W.. - Magnetic perturbation of February 13 and 14, by M. Thomas Archer Hirst

399 Moureaux. The disturbance was first indicated on the magneto- Dr. Thomas Sterry Hunt. By W. Topley, F.R.S. 400 graph of the Parc Saint-Maur Observatory at 5h. 42m. on


401 the morning of the 13th inst. The declination and horizontal Our Astronomical Column:force curves suffered a simultaneous rise, while the vertical com- The Solar Disturbance of 1891, June 17

404 ponent decreased. The most important phase of the perturba- Photography of Solar Prominences

404 tion occurred between 11 p.m. and 2 a.m.; and about 5 p.m. On the Variation of Latitude

404 of the 14th the elements had returned to their normal value. Non-Euclidian Geometry. By Prof. H. Poincaré ; The disturbance in declination amounted to 1° 25', and the Translated by W. J. L. .

404 horizontal and vertical components varied respectively more Societies and Academies


386 388 389

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