causes of movements of the earth's crust; for if the fluid or viscous layer is chiefly due to internal heat and the relaxation of pressure near the surface, it may exist much nearer to our feet than could otherwise be admitted.

One of the gravest difficulties that the theory that added weight produces subsidence by acting on a fluid layer has had to contend with has been the great depth at which this fluid layer has had to be placed. It has always seemed to me next to inipossible that liquid lava could well up from any such depths as tose assigned to the viscous layer, or that a solid crust of so great a thickness should be sensitive to, as it is now shown to be, and rise and fall under, barometric changes. In acknowledging Mr. Fisher's letter and thanking him, I feel I am ungrateful in questioning that part of his work which interposes barriers which would break up the continuity of the viscous layer; I allude to his theory of the roots of mountains." There does seem to me to be little fact in support of so startling a proposition, and I think the existence of volcanic vents, scattered through and in the midst of some of the highest chains, renders its acceptance difficult.

Mr. Murray restates his theory of the formation of coral atolls and reefs in the clearest manner, but I do not see that he explains any fact left unexplained by Darwin, or exposes any flaw in Darwin's reasoning. These masses of coral may have been continuously forming throughout even successive geological

Sheet 21. Geological Survey of Ireland, Antrim Coast, facing north-east. periods, and their thickness is perhaps not exceptionally remarkable relative to that of slowly deposited oceanic sediments. There is no evidence that atolls are mere incrustations of volcanic craters, and it seems to me difficult to imagine so great a number of craters at the same level so completely masked. There are volcanic isles in abundance outside coral areas, but none I think, or few, of the form of a coral atoll. After all, Mr. Murray only shows that a second explanation is possible, though I still prefer

the first.

I regret, being from home, that I am unable to answer Mr. Stanley. I may have alluded to the sinking of Greenland myself, and if I did not it was because the illustration was too familiar and self-evident. The sinking on the Greenland coast is not, I have understood, universal.

I still think it would render a service to science if readers of

NATURE residing on sea-coasts would furnish authentic examples of elevation or subsidence or of waste. The magnificent Antrim coast, which I have recently visited, furnishes examples of subsidence among most unyielding rocks. The cliffs on the mainland are capped with basalt and dip inland, yet the basalt reappears in the Skerries out to sea with the same dip and at a much lower level. The same correspondence in stratification is seen between the mainland and Rathlin, but also with a great difference in elevation. The dip inland in all cases on this coast

should bring up much older rocks out to sea, unless we are prepared to admit a fault running parallel to the coast, and following its sinuosities, and at right angles to the general lines of faulting.

The way in which all the strata forming the cliffs along the Antrim coast dip inland is very remarkable. The accompanying tracing from the Geological Survey Map is of a particularly indented coast-line, and the arrows show that the dip is everywhere away from the sea, irrespective of any general strike. In fact the general strike must often be the reverse of that shown on the coast for the same strata crop out at much higher levels on the hills farther inland. I recollect that most cliffs that I have examined, particularly in Hampshire, dip away from the sea. It would appear that the removal of weight along a cliff line causes a local elevation, which gives a cant inward, whilst sub idence takes place under sediment farther out to sea. This seems to explain the observed facts connected with marine denudation; but I must take a future opportunity of entering more thoroughly into this part of the question. Glasgow, September 12

J. STARKIE GARDNER

"Zoology at the Fisheries Exhibition " LETTERS have been published in NATURE of August 9 and 16 (PP. 334 and 366) by Mr. Bryce-Wright of Regent Street and Prof. Honeyman of Canada, calling in question the accuracy of statements made in an article in NATURE (vol. xxviii. p. 289) which were condemnatory of exhibits for which these two gentlemen are respectively responsible. It is natural that they should seek to remove the unfavourable impression which the statements in question were intended to convey: they seem, however, to have been unacquainted with the complete character of the information upon which the statements were based. Mr. Bryce-Wright states that it is not the fact that some of the corals exhibited in Lady Brassey's case belong to him. Nevertheless it is the fact that when the jury of Class V. asked Mr. Bryce-Wright to point out the corals entered in the offcial catalogue under his name, No. "8136," he informed them that the corals so entered were in the same case with Lady Brassey's corals, and formed part of that collection. It is also the fact that in the opinion of experts the names attached by Mr. Bryce-Wright to many of these corals are incorrect; and as to his assertion that these specimens have been compared with those in the British Museum and with those obtained during the Challenger Expedition, it is a fact that neither the one series nor the other has been accessible for such purposes for some considerable time, and I have reason to believe that no qualified zoologist has made a comparison of the corals exhibited by Lady Brassey and Mr. Bryce-Wright with any collection at all.

The letter of Prof. Honeyman in reference to the naming and state of preservation of the Collection in the Canadian Department, for which he is responsible, is misleading. The discreditable state of that collection, to which a passing allusion only was made in NATURE, has been remedied in one or two instances since the visit of the jury of Class V. Should there be any doubt as to the justice of the opinion expressed in the article in NATURE, I would simply ask Prof. Honeyman whether he would have any objection to allowing the matter to be decided by reference to the report of the jury of Class V., of which he was a member. I should be surprised (and so I think would he) were the report of that jury, when published, found to be at variance with the opinion expressed in the article in NATURE. Prof. Honeyman's statement that the specimen of Cryptochiton Stelleri is properly exhibited in a convenient glass jar and labelled inside and out, is calculated to mislead. When first exhibited it was not labelled with any name; subsequently it was labelled with the name of a genus of Holothurians, "Psolus." After the visit of the jury of Class V., probably as the re ult of informa tion imparted by some of the eminent zoologists who served on that jury, it was labelled with its proper name. Without citing details, I shall simply state that there are (or were when the article in NATURE was written) far more serious blunders in the identification of specimens and worse instances of bad preserva

tion in the Canadian collection of Invertebrata than those to which special allusion has been made.

THE WRITER OF THE ARTICLE

A Complete Solar Rainbow

MR. D. MORRIS, in his account of this rainbow (p. 436) appears to have fallen into a mistake in stating that its inner dia

rampart of the camp, with remains of burnt wood and late Celtic pottery. I immediately saw that several of the flakes had been struck from the same block of flint, and after a short examination I managed to replace two as illustrated, one-half real size, in Fig. 1. The front of the two conjoined flakes is shown in the lefthand bottom figure, the side at B, the top at C, and the line of junction at D D. Behind EE are two cones of percussion, one belonging to each flake, and at F is the depression into which the cone of the missing frontal flake at one time fitted. The fractured part of the flint is deep chocolate brown, and lustrous, and the bark of the flint is dull ochreous; the flakes are undoubtedly artificial, and as old as the rampart of the camp, not less than two thousand years. This example, with other relics, will be placed in the Guildhall Museum.

Greater interest attaches to the replacing of Paleolithic flakes, as these are enormously older than Neolithic, and the chances are so very much against lighting on a perfectly undisturbed Paleolithic position.

K

FIG. 2.

M

At Fig. 2 is illustrated (one-half actual size) two Paleolithic flakes from the "Palæolithic floor" at Stoke Newington Com

mon, found and replaced by me. The front of the conjoined flakes is shown at G and the side at H. I found the lower flake two days before, and some distance from where I found the upper one; but as I have a method of placing newly found sharp flakes on a table, arranged temporarily in accordance with their colour and markings, I speedily saw that the upper flake would fit on to the lower one. Each flake has a cone of percussion, as shown at JK, and the upper flake has a well-marked

depression at L, corresponding with the missing flake, which, if it had been found, would have fitted on to the front of the two conjoined examples. Both flakes are sharp and slightly stained with the ochreous river sand which overlaid them. Both (especially the upper one) show unmistakable signs of having been used as scrapers, the upper curved edge (and that edge only) being worn away by use. The worn upper edge of the superimposed flake at M M is distinctly shown in the illustration. A small intermediate piece belonging to the position at N I did not find. Both are naturally mottled in a peculiar manner, and the pattern and colour of the mottling exactly agree.

WORTHINGTON G. SMITH

NOTES ON THE POST-GLACIAL GEOLOGY OF THE COUNTRY AROUND SOUTHPORT

SINCE the writer carried out the geological survey of

the western coast of Lancashire in 1868 he has constantly been asked, "Is there any geology to be studied at Southport? Is not the country a sandy expanse fringing peat-mosses of ceaseless monotony?" The meeting of the British Association this week at Southport renders this a fitting time to reply to these questions; for, strange as it may appear, in these apparently unpromising surroundings exists a record of the complete sequence of events from the commencement of the Glacial episode down to the present time. The sand dunes, rising to 50 and even 80 feet in height, that form so prominent a feature between Liverpool and Southport, rest upon a wedge-shaped mass of sand blown from the coast by westerly winds over the thick peat-mosses that intervene between the coast and the rising ground about Ormskirk; the surface of the Glacial beds, with the overlying deposits, dip steadily towards the sea, and fragments of peat are frequently trawled up by the fishermen.

Beneath the sand dunes on the sea coast the peat is seen cropping out, and at the base of the peat occur the roots of forest trees embedded in clay beneath, while trunks of trees lie scattered in many directions, but generally with their heads lying to the north-east, as if they had been blown over by a gale from the south-west. The bases of the trunks are left standing in the places where they grew; all appear to have been broken off at a uniform level, and it is most probable that through the drainage being obstructed water surrounded the trees, which gradually became rotten at the point of contact of the air and the water, and thus the way was prepared for the effects of storms and hurricanes. Sections of these beds near High Town, at the mouth of the Alt, will be found of great interest. Sections also occur on the coast at Dunkirk, near Crossens. At the Palace Hotel, Kirkdale, a boring was put down in 1867, that proved the sand to be 78 feet in thickness, resting on 18 inches of peat, which occurs at about 90 feet beneath high-water mark. When the land stood this amount above its present level, the coast would range in a straight north and south line from St. Bees Head to the mouth of the Clywd at Rhyl, but there is no reason to suppose that this amount represents the subsequent submergence since the era of the peat in Lancashire and North Wales. It is far more probable that when the trees flourished, found at the bottom of the peat fringing these coasts, this coast nearly coincided with the present twenty-fathom line, which passes from Anglesea round the Isle of Man; in that island the same sequence of post glacial deposits is found, and the Irish elk alike occurs in the grey slags beneath peat.

At the mouth of the Ribble very interesting sections occur at Freckleton and Dow Brook; the latter is crossed by a Roman road, and has upon it a "Roman bath," only ten feet above the present high-water mark, proving the elevation of this coast has not been great since Roman times. The same fact is brought out by the interesting find of Roman coins near Rossal landmark, near Fleetwood, which were found in a salt-marsh clay lying on the peat beds, at about eight feet below the

surface, or at about high-water mark, the coins having been apparently lost by the Romans scrambling over the soft slippery mud. This discovery proves the thick peat beds to be of older date than the Romans; this is also borne out by the very remarkable sections along the north coast of Wirral, especially near Leasowe, which have afforded the fine collection of antiquities preserved in the Liverpool Free Museum; the silty beds over the peat yield Roman coins of Nero, Antoninus Pius, and Marcus Aurelius, while in the peat beds beneath occur flint implements of the Neolithic type. When the peat beds of Western Lancashire are followed into the valleys of the large rivers that traverse the country, they are found to pass insensibly into a peaty seam occurring at the base of the alluvium of the lowest plain of these rivers. This is well seen in the valley of the Ribble at Preston; it is more than a mile in width, and 180 feet in depth; it is excavated entirely in the Glacial deposits, down to the rocky floor, which lies somewhat below high-water mark, and nearer the sea slopes down considerably beneath it. On the slopes of the valley lie terraces of old alluvium, marking successive stages in the process of denudation, commenced since the deposition of the Upper Boulder Clay, as the bottom of the valley is the ordinary alluvial plain, made of silt, resting on a peaty bed, with trunks of trees lying on rough river gravel, the latter marking a period of great fluviatile denudation, when the land was at least as high, if not higher, above the sea as it is at present. To this era belong the marine beds lying beneath the peat I have called the Presall shingle, occurring east of Fleetwood, and the Shirley Hill sands near Southport, which mark the position of old sea-beaches and old sand dunes respectively.

From these facts it appears that the excavation of the Western Lancashire river valleys was entirely carried out since the Glacial episode, that they had reached their present depth when Neolithic man inhabited the northwest of England, and that since that era much land has been destroyed, now covered by the Irish Sea, but since Roman times there has been but little change.

C. E. DE RANCE

THE BRITISH ASSOCIATION

THE HE Southport meeting promises to be one of the most successful since the Association met in Liverpool twelve years ago. According to the latest statistics it is expected that in attendance it may even rival the York meeting, when over 2500 people gathered to celebrate the jubilee of the Association. From the information we have already published it will have been seen that Southport has shown the greatest zeal in preparing to give a generous reception to the representatives of British science; and if only the weather be propitious, there can be little doubt that the meeting will be a success. Both the papers to be read and the reports to be presented are expected this year to suggest some specially interesting subjects for discussion.

Last night Sir C. W. Siemens resigned the presidential chair to Prof. Cayley, who then delivered the opening address.

INAUGURAL ADDRESS BY ARTHUR CAYLEY, M.A., D.C.L.,

LL.D., F.R.S., SADLERIAN Professor of PURE MATHEMATICS IN THE UNIVERSITY OF CAMBRIDGE, President. SINCE our last meeting we have been deprived of three of our most distinguished members. The loss by the death of Prof. Henry John Stephen Smith is a very grievous one to those who knew and admired and loved him, to his University, and to mathematical science, which he cultivated with such ardour and success. I need hardly recall that the branch of mathematics to which he had specially devoted himself was that most interesting and difficult one, the Theory of Numbers. The immense range of this subject, connected with and ramifying into so many others, is nowhere so well seen as in the series of re

ports on the progress thereof, brought up unfortunately only to the year 1865, contributed by him to the Reports of the Association; but it will still better appear when to these are united (as will be done in the collected works in course of publication by the Clarendon Press) his other mathematical writings, many of them containing his own further developments of theories referred to in the reports. There have been recently or are being published many such collected editions-Abel, Cauchy, Clifford, Gauss, Green, Jacobi, Lagrange, Maxwell, Riemann, Steiner. Among these the works of Henry Smith will occupy a worthy position.

More recently, General Sir Edward Sabine, K.C. B., for twenty-one years general secretary of the Association, and a trustee, president of the meeting at Belfast in the year 1852, and for many years treasurer and afterwards president of the Royal Society, has been taken from us at an age exceeding the ordinary age of man. Born October, 1788, he entered the Royal Artillery in 1803, and commanded batteries at the siege of Fort Erie in 1814; made magnetic and other observations in Ross and Parry's North Polar exploration in 1818-19, and in a series of other voyages. He contributed to the Association reports on Magnetic Forces in 1836-7-8, and about forty papers to the Philosophical Transactions; originated the system of Magnetic Observatories, and otherwise signally promoted the science of Terrestrial Magnetism.

There is yet a very great loss: another late president and trustee of the Association, one who has done for it so much, and has so often attended the meetings, whose presence among us at this meeting we might have hoped for-the president of the Royal Society, William Spottiswoode. It is unnecessary to say anything of his various merits: the place of his burial, the crowd of sorrowing friends who were present in the Abbey, bear witness to the esteem in which he was held.

I take the opportunity of mentioning the completion of a work promoted by the Association: the determination by Mr. James Glaisher of the least factors of the missing three out of the first nine million numbers: the volume containing the sixth million is now published.

I wish to speak to you to-night upon Mathematics. I am quite aware of the difficulty arising from the abstract nature of my subject; and if, as I fear, many or some of you, recalling the Presidential Addresses at former meetings-for instance, the résumé and survey which we had at York of the progress, during the half century of the lifetime of the Association, of a whole circle of sciences-Biology, Paleontology, Geology, Astronomy, Chemistry-so much more familiar to you, and in which there was so much to tell of the fairy-tales of science; or at Southampton, the discourse of my friend who has in such kind terms introduced me to you, on the wondrous practical applications of science to electric lighting, telegraphy, the St. Gothard Tunnel, and the Suez Canal, gun-cotton, and a host of other purposes, and with the grand concluding speculation on the conservation of solar energy: if, I say, recalling these or any earlier addresses, you should wish that you were now about to have, from a different president, a discourse on a different subject, I can very well sympathise with you in the feeling.

But, be this as it may, I think it is more respectful to you that I should speak to you upon and do my best to interest you in the subject which has occupied me, and in which I am myself most interested. And in another point of view, I think it is right that the Address of a President should be on his own subject,

and that different subjects should be thus brought in turn before the meetings. So much the worse, it may be, for a particular principles must be sacrificed for the development of the race. meeting; but the meeting is the individual, which on evolution

Mathematics connect themselves on the one side with common life and the physical sciences; on the other side with philosophy, in regard to our notions of space and time; and in the questions which have arisen as to the universality and necessity of the truths of mathematics, and the foundation of our knowledge of them. I would remark here that the connection (if it exists) of arithmetic and algebra with the notion of time is far less obvious than that of geometry with the notion of space.

As to the former side, I am not making before you a defence of mathematics, but if I were I should desire to do it-in such manner as in the "Republic" Socrates was required to defend justice, quite irrespectively of the worldly advantages which may accompany a life of virtue and justice, and to show that, independently of all these, justice was a thing desirable in itself and for its own sake-not by speaking to you of the utility of mathematics in any of the questions of common life or of physi

cal science. Still less would I speak of this utility before, I trust, a friendly audience, interested or willing to appreciate an interest in mathematics in itself and for its own sake. I would, on the contrary, rather consider the obligations of mathematics to these different subjects as the sources of mathematical theories now as remote from them, and in as different a region of thought -for instance, geometry from the measurement of land, or the Theory of Numbers from arithmetic-as a river at its mouth is from its mountain source.

On the other side the general opinion has been and is that it is indeed by experience that we arrive at the truths of mathematics, but that experience is not their proper foundation: the mind itself contributes something. This is involved in the Platonic theory of reminiscence; looking at two things, trees or stones or anything else, which seem to us more or less equal, we arrive at the idea of equality: but we must have had this idea of equality before the time when first seeing the two things we were led to regard them as coming up more or less perfectly to this idea of equality; and the like as regards our idea of the beautiful, and in other cases.

The same view is expressed in the answer of Leibnitz, the nisi intellectus ipse, to the scholastic dictum, nihil in intellectu quod non prius in sensu : there is nothing in the intellect which was not first in sensation, except (said Leibnitz) the intellect itself. And so again in the "Critick of Pure Reason," Kant's view is that, while there is no doubt but that all our cognition begins with experience, we are nevertheless in possession of cognitions a priori, independent, not of this or that experience, but absolutely so of all experience, and in particular that the axioms of mathematics furnish an example of such cognitions a priori. Kant holds further that space is no empirical conception which has been derived from external experiences, but that in order that sensations may be referred to something external, the representation of space must already lie at the foundation; and that the external experience is itself first only pos-ible by this representation of space. And in like manner time is no empirical conception which can be deduced from an experience, but it is a necessary representation lying at the foundation of all intuitions.

And so in regard to mathematics, Sir W. R. Hamilton, in an introductory lecture on astronomy (1836), observes: "These purely mathematical sciences of algebra and geometry are sciences of the pure reason, deriving no weight and no assistance from experiment, and isolated or at least isolable from all outward and accidental phenomena. The idea of order, with its subordinate ideas of number and figure, we must not indeed call innate ideas, if that phrase be defined to imply that all men must possess them with equal clearness and fulness: they are, however, ideas which seem to be so far born with us that the possession of them in any conceivable degree is only the development of our original powers, the unfolding of our proper humanity.'

The general question of the ideas of space and time, the axioms and definitions of geometry, the axioms relating to number, and the nature of mathematical reasoning, are fully and ably discussed in Whewell's "Philosophy of the Inductive Sciences" (1840), which may be regarded as containing an exposition of the whole theory.

But it is maintained by John Stuart Mill that the truths of mathematics, in particular those of gec metry, rest on experience; and, as regards geometry, the same view is on very different grounds maintained by the mathematician Riemann.

It is not so easy as at first sight it appears to make out how far the views taken by Mill in his "System of Logic Ratiocinative and Inductive" (ninth edition, 1879) are absolutely contradictory to those which have been spoken of; they profess to be so; there are most definite assertions (supported by argument),

for instance, p. 263:—"It remains to inquire what is the ground

of our belief in axioms, what is the evidence on which they rest. I answer, they are experimental truths, generalisations from experience. The proposition 'Two straight lines cannot inclose a spice,' or, in other words, two straight lines which have once me: cannot meet again, is an induction from the evidence of our senses." But I cannot help considering a previous argument (p. 259) as very materially modifying this absolute contradiction. After inquiring "Why are mathematics by almost all philosoph rs.... considered to be independent of the evidence of experience and observation, and characterised as systems of necessary truth?" Mill proceeds (I quote the whole passage) as follows:-"The answer I conceive to be that this character of necessity ascribed to the truths of mathematics, and even (with some reservations to be hereafter made) the peculiar certainty

ascribed to them, is a delusion, in order to sustain which it is necessary to suppose that those truths relate to and express the properties of purely imaginary objects. It is acknowledged that the conclusions of geometry are derived partly at least from the so-called definitions, and that these definitions are assumed to be correct representations, as far as they go, of the objects with which geometry is conversant. Now we have pointed out that from a definition as such no proposition, unless it be one concerning the meaning of a word, can ever follow, and that what apparently follows from a definition follows in reality from an implied assumption that there exists a real thing conformable thereto This assumption in the case of the definitions of geometry is not strictly true: there exist no real things exactly conformable to the definitions. There exist no real points without magnitude, no lines without breadth, nor perfectly straight, no circles with all their radii exactly equal, nor squares with all their angles perfectly right. It will be said that the assumption does not extend to the actual but only to the possible existence of such things. I answer that according to every test we have of possibility they are not even possible. Their existence, so far as we can form any judgment, would seem to be inconsistent with the physical constitution of our planet at least, if not of the universal [sic]. To get rid of this difficulty, and at the same time to save the credit of the supposed system of necessary truths, it is customary to say that the points, lines, circles, and squares which are the subjects of geometry, exist in our concep tions merely, and are parts of our minds: which minds, by working on their own materials, construct an a priori science, the evidence of which is purely mental and has nothing to do with outward experience. By howsoever high authority this doctrine has been sanctioned, it appears to me psychologically incorrect. The points, lines, and squares which any one has in his mind, are (as I apprehend) simply copies of the points, lines, and squares which he has known in his experience. Our idea of a point I apprehend to be simply our idea of the minimum visibile, the small portion of surface which we can see. We can reason about a line as if it had no breadth, because we have a power which we can exercise over the operations of our minds: the power, when a perception is present to our senses or a conception to our intellects, of attending to a part only of that perception or conception instead of the whole. But we cannot conceive a line without breadth we can form no mental picture of such a line: all the lines which we have in our mind are lines rossessing breadth. If any one doubt this, we may refer him to his own experience. I much question if any one who fancies that he can conceive of a mathematical line thinks so from the evidence of his own consciousness. I suspect it is rather because he suppo es that unless such a perception be possible, mathematics could not exist as a science: a supposition which there will be no difficulty in showing to be groundless."

I think it may be at once conceded that the truths of geometry are truths precisely because they relate to and express the properties of what Mill calls "purely imaginary objects"; that these objects do not exist in Mill's sense, that they do not exist in nature, may also be granted; that they are "not even possible," if this means not possible in an existing nature, may also be granted. That we cannot "conceive" them depends on the meaning which we attach to the word conceive. I would myself say that the purely imaginary objects are the only realities, the ŎVTws Ŏvra, in regard to which the corresponding physical objects are as the shadows in the cave; ard it is only by means that we are able to deny the existence of a corresponding physical object; if there is no conception of straightness, then it is meaningless to deny the existence of a perfectly straight line.

them

But at any rate the objects of geometrical truth are the socalled imaginary objects of Mill, and the truths of geometry are only true, and a fortiori are only necessarily true, in regard to these so-called imaginary objects; and these objects, points, lines, circles, &c., in the mathematical sense of the terms, have a likeness to and are represented more or less imperfectly, and from a geometer's point of view no matter how imperfectly, by corresponding physical points, lines, circles, &c. I shall have to return to geometry, a d will then speak of Riemann, but I will first refer to another passage of the "Logic."

Speaking of the truths of arithmetic Mill says (p. 297) that even here there is one hypothetical element: "In all propositions concerning numbers a condition is implied without which none of them would be true, and that condition is an assumption which may be false. The condition is that II: that all the numbers are numbers of the same or of equal units." Here at least the assumption may be absolutely true; one shilling

=one

shilling in purchasing power, although they may not be absolutely of the same weight and fineness: but it is hardly necessary; one coin + one coin two coins, even if the one be a shilling and the other a half-crown. In fact, whatever difficulty be raisable as to geometry, it seems to me that no similar difficulty applies to arithmetic; mathematician or not, we have each of us, in its most abstract form, the idea of a number; we can each of us appreciate the truth of a proposition in regard to numbers; and we cannot but see that a truth in regard to numbers is something different in kind from an experimental truth generalised from experience. Compare, for instance, the proposition that the sun, having already risen so many times, will rise to-morrow, and the next day, and the day after that, and so on; and the proposition that even and odd numbers succeed each other alternately ad infinitum: the latter at least seems to have the characters of universality and necessity. Or, again, suppose a proposition observed to hold good for a long series of numbers, one thousand numbers, two thousand numbers, as the case may be: this is not only no proof, but it is absolutely no evidence, that the proposition is a true proposition, holding good for all numbers whatever; there are in the Theory of Numbers very remarkable instances of propositions observed to hold good for very long series of numbers and which are nevertheless untrue.

I pass in review certain mathematical theories.

In arithmetic and algebra, or say in analy is, the numbers or magnitudes which we represent by symbols are in the first instance ordinary (that is, positive) numbers or magnitudes. We have also in analysis and in analytical geometry negative magnitudes; there has been in regard to these plenty of philosophical discussion, and I might refer to Kant's paper, "Ueber die negativen Grössen in die Weltweisheit" (1763), but the notion of a negative magnitude has become quite a familiar one, and has extended itself into common phraseology. I may remark that it is used in a very refined manner in bookkeeping by double entry.

But it is far otherwi-e with the notion which is really the fundamental one (and I cannot too strongly emphasise the assertion) underlying and pervading the whole of modern analysis and geometry, that of imaginary magnitude in analysis and of imaginary space (or space as a locus in quo of imaginary points and figures) in geometry: I use in each case the word imaginary as including real. This has not been, so far as I am aware, a subject of philosophical discussion or inquiry. As regards the older metaphysical writers, this would be quite accounted for by saying that they knew nothing, and were not bound to know anything, about it; but at present, and considering the prominent position which the notion occupies-say even that the conclusion were that the notion belongs to mere technical mathematics, or has reference to nonentities in regard to which no science is possible, still it seems to me that (as a subject of philosophical discussion) the notion ought not to be thus ignored; it should at least be shown that there is a right to ignore it.

Although in logical order I should perhaps now speak of the notion just referred to, it will be convenient to speak first of some other quasi-geometrical notions; those of more-than-threedimensional space, and of non-Euclidian two- and threedimensional space, and also of the generalised notion of distance. It is in connection with these that Riemann considered that our notion of space is founded on experience, or rather that it is only by experience that we know that our space is Euclidian space.

It is well known that Euclid's twelfth axiom, even in Playfair's form of it, has been considered as needing demonstration; and that Lobatschewsky constructed a perfectly consistent theory wherein this axiom was assumed not to hold good, or say a system of non-Euclidian plane geometry. There is a like system of non-Euclidian solid geometry. My own view is that Euclid's twelfth axiom in Playfair's form of it does not need demonstration, but is part of our notion of space, of the physical space of our experience-the space, that is, which we become acquainted with by experience, but which is the representation lying at the foundation of all external experience. Riemann's view before referred to may I think be said to be that, having in intellectu a more general notion of space (in fact a notion of non-Euclidian space), we learn by experience that space (the physical space of our experience) is, if not exactly, at least to the highest degree of approximation, Euclidian space.

But, suppose the physical space of our experience to be thus only approximately Euclidian space, what is the consequence

which follows? Not that the propositions of geometry are only approximately true, but that they remain absolutely true in regard to that Euclidian space which has been so long regarded as being the physical space of our experience.

It is interesting to consider two different ways in which, without any modification at all of our notion of space, we can arrive at a system of non-Euclidian (plane or two-dimensional) geometry; and the doing so will, I think, throw some light on the whole question.

First, imagine the earth a perfectly smooth sphere; understand by a plane the surface of the earth, and by a line the apparently straight line (in fact an arc of great circle) drawn on the surface; what experience would in the first instance teach would be Euclidian geometry; there would be intersecting lines which produced a few miles or so would seem to go on diverging, ard apparently parallel lines which would exhibit no tendency to approach each other; and the inhabitants might very well conceive that they had by experience established the axiom that two straight lines cannot inclose a space, and the axiom as to parallel lines. A more extended experience and more accurate measurements would teach them that the axioms were each of them false; and that any two lines if produced far enough each way would meet in two points: they would in fact arrive at a spherical geometry, accurately representing the properties of the twodimensional space of their experience. But their original Euclidian geometry would not the less be a true system; only it would apply to an ideal space, not the space of their experience.

Secondly, consider an ordinary, indefinitely extended plare; and let us modify only the notion of distance. We measure distance, say, by a yard measure or a foot rule, anything which is short enough to make the fractions of it of no consequer ce (in mathematical language by an infinitesimal element of length); imagine, then, the length of this rule constantly changing (as it might do by an alteration of temperature), but under the condition that its actual length shall depend only on its situation on the plane and on its direction: viz., if for a given situation and direction it has a certain length, then whenever it comes back to the same situation and direction it must have the same length. The distance along a given straight or curved line between any two points could then be measured in the ordinary manner with this rule, and would have a perfectly determinate value; it could be measured over and over again, and would always be the same; but of course it would be the distance, not in the ordinary acceptation of the term, but in quite a different acceptation. Or in a somewhat different way: if the rate of progress from a given point in a given direction be conceived as depending only on the configuration of the ground, and the distance along a given path between any two points thereof be measured by the time required for traversing it, then in this way also the distance would have a perfectly determinate value; but it would be a distance, not in the ordinary acceptation of the term, but in quite a different acceptation. And corresponding to the new notion of distance, we should have a new, nonEuclidian system of plane geometry; all theorems involving the notion of distance would be altered.

We may proceed further. Suppose that as the rule moves away from a fixed central point of the plane it becomes shorter and shorter; if this shortening takes place with sufficient rapidity, it may very well be that a distance which in the ordinary sense of the word is finite will in the new sense be infinite; no number of repetitions of the length of the evershortening rule will be sufficient to cover it. There will be surrounding the central point a certain finite area such that (in the new acceptation of the term distance) each point of the boundary thereof will be at an infinite distance from the central point; the points outside this area you cannot by any means arrive at with your rule; they will form a terra incognita, or rather an unknowable land: in mathematical language, an imaginary or impossible space: and the plane space of the theory will be that within the finite area that is, it will be finite instead of infinite.

We thus with a proper law of shortening arrive at a system of non-Euclidian geometry which is essentially that of Lobatschewsky. But in so obtaining it we put out of sight its relation to spherical geometry: the three geometries (spherical, Euclidian, and Lobatschewsky's) should be regarded as members of a system: viz., they are the geometries of a plane (two-dimensional) space of constant positive curvature, zero-curvature, and constant negative curvature respectively; or, again, they are the plane geometries corresponding to three different notions of distance;