It will be noticed that this problem with its solvitur ambulando contains implicitly the very difficulty concerning the method of limits which has caused many to look with doubt on results obtained by this method-imagining that we cannot learn the exact truth by a method which causes us to approach it as near as we please. It is as true on the one hand that we may take any number of the stages considered in the problem without bringing Achilles on a strict level with the tortoise, as that if such a race were actually run the man would at the end of a definite time overtake the animal. And in like manner it is as true that in problems depending on the method of limits we do not obtain exact relations while applying the method, as that the result arrived at by the method is strictly exact. And that difficulty which in the case of the Achilles and Tortoise Problem some of the ancients chose to regard as insuperable, is the very one that troubles many modern students of mathematics, when they are told of those seemingly contradictory relations which appertain to the theory and method of limits.

I do not know that I can better introduce the doctrine of limits than by taking the above problem as an illustrative case, and showing how the method of limits applied to that problem leads to precisely the same result as the simpler method applicable in this case (and in many others which admit of being solved by the method of limits).

Suppose we take the problem, first, as one to be solved by simple algebraical considerations. We must first assign a definite velocity to Achilles and the tortoise. Suppose that the Swift-footed runs at the rate of a mile in 4 m. 24 s., or 400 yards per minute (our best professional runners covering a mile in 4 m. 20 s., and Achilles having much more than a mile to run). Then the problem would be thus dealt with by the algebraist :

Let x=time in minutes occupied by Achilles in overtaking the tortoise. Then the space covered by Achilles in yards will be 400 x; and the space traversed by the tortoise will be 4x. And since Achilles has in this time gained 10,000 yards on the tortoise we have as our equation for determining

in which number of minutes Achilles will overtake the tortoise. Now the method of limits would be applied somewhat on this wise. Achilles traverses the 10,000 yards in 25 minutes, and is then 100 yards behind the tortoise. He traverses the 100 yards in one 4th part of a minute, and is then 1 yard behind the tortoise. He traverses the yard in one 400th part of a minute, and is then one 100th part of a yard behind. He traverses the one 100th part of a yard in one 40,000th part of a minute, and is then the 10,000th part of a yard behind and so on continually, each stage occupying him one 100th part of the time occupied by the preceding. Hence he will overtake the tortoise in

+ &c. + &c.] ]m

minutes (A).

4 400 40000 4000000

by whatever line of reasoning it could be shown that he would turn first to one bundle, by a line of reasoning precisely similar it may be shown that he would turn first to the other. But he cannot turn first to both. Therefore, he will turn to neither." Another of these problems was thus worded:-" Epimenides the Cretan says that the Cretans are liars. Now Epimenides is himself a Cretan; therefore Epimenides is a liar. Therefore the Cretans are not liars. Therefore Epimenides is not a liar. Therefore the Cretans are liars. Therefore Epimenides is a liar. Therefore, &c., ad infinitum." Others stated the problem in a more simple form, thus: "When a man says I lie, does he lie or does he not lie? If he lies he speaks the truth, if he speaks the truth he lies." We are told that one philosopher, after vainly endeavouring to clear up this important question, flung himself, in despair, into the sea. Philosophy sustained no great loss, it may be conceived.

25 minutes

the same result which we obtained by the direct method.

Now the point to be noticed here is that although the method of limits does not here actually bring us to the exact value while we are still applying the method, it shows us the way to that value and not to an approximate value.

In order the more clearly to recognise the nature of the approximation which is actually involved in the method of limits we may take a much simpler case. Suppose I have a line of two inches before me as A B in Fig. 1. Then I may suppose this line directly measured and its length to be thus ascertained.

But

may also conceive that one-half of it, A C, is cut off, and then the half C D of the remainder, the half D E of what then remains, and so on ad infinitum. In this way I should never get to the end B, but I should get as near as I pleased to this point. The point B would obviously be the end to which this process of section would tend; and if I added together the length of all these pieces I should see that the length to which the sum continually approached was as by direct measurement a length of two inches. This method of determining the length would be in itself approximate; but the deduced length of two inches would be exact.

Now in both these instances we have gone out of our way (though not without a purpose) to apply the method of limits to matters which can be much better solved by a simpler method. But when the student learns that there are numerous problems-or rather an infinity of problems-which can only be solved by the method of limits, he can see the importance of establishing the exactness of the results obtained by the method. The method is approximative, but the results obtained by it are exact.

IT

Our Whist Column.

BY" FIVE OF CLUBS."

LATE SIGNALS.

is well remarked by both Cavendish and Clay that if a player fails to signal at the first opportunity, his partner need not regard a signal given later as having the same authoritative character which an original signal possesses. An original signal means more than a trump lead. It means, or should mean (only some players are too ready to signal), that the signaller is not only very strong in trumps, but has such strength in other suits that (1) he can answer for the absolute safety of a trump lead, and (2) can give good promise of a great game. A signal after the first chance for signalling has passed, means much the same as a trump lead; and, whatever rule to the contrary may be set up, a trump lead does not involve the return of trumps by partner as necessary or even always proper. Very often a trump lead is tentative, and in not a few cases where it is so, the return of trumps would be bad play. So, a late signal means little more, usually, than that a lead of trumps seems likely to be advantageous.

In

But it occasionally happens that a late signal points to the one sole way of making the game, and should be answered at once. fact, after the middle of a hand, a signal-if possible, which is not often-acquires a very pointed meaning. Take a case such as occurred to the editor a few evenings ago. We will call the editor B, his partner A, and players to right and left of B, Y and Z, as in our games. Seven rounds remained to be played, and one round of trumps (diamonds) had been already taken out, in such sort as to leave the best, 3rd, and 5th best trumps with B, the 2nd and

4th with Y and three trumps between Z and A, their positions unknown but one certainly with A. One trump had been forced from Y, the original trump leader. The best and third best hearts lay with Y and three small Hearts were with A, command in Clubs being with Z. B, who has not had a lead, holds, besides his three trumps, Queen, Knave, Three, and Two of Spades. As it chanced, every trick was wanted to make the game. At this juncture A led Spade King, Spades having been as yet unplayed, but (from the play) being Z's suit. Here B's course to a won game (with A's concurrence) is plain and obvious, while it is equally clear that any other course must lead to the loss of one trick at least by A, B. The one sole way of making the game is by signalling. Therefore, B dropped the Three and Two of Spades in that order to the King and Ace; A responded to the signal by a trump lead; and every trick went to A, B. On the contrary, if B had not signalled, or 4 had failed to respond, A would have led a small Heart which I would have covered with the third best, and B would have been forced to ruff; for if he passed the trick, Y would have simply repeated the force. Then B could have done nothing with his command in Spades but force the enemy, uncertain whether he were forcing Z or Y; if Z, then the lead of a winning Club would again force B, and three tricks in all would be made by Y, Z; if luckily Y, then but one trick would be made by Y Z, but still their game would be saved.

Some players seem to think that if they look out for the signal in the first suit led they have done all that is required of them; but the above case and others which might easily be cited show that even towards the end of a hand the signal may be played with effect; and that therefore it should be looked for to the last. In other words, it is always worth while to attend, to the very last, to the play of the small cards.

Volume IV., comprising the numbers published from July to December, 1883, is now ready, price 7s. 6d.; including parcels postage, 88.

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CHELTENHAM (Assembly Rooms), Feb. 5, 8, 12, 15 (1, 2, 4, 6). At 3 o'clock, Feb. 5 and 12 (3, 5).

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I

BY RICHARD A. PROCTOR.

HAVE just read the Edinburgh Review on Mr. Spencer's System of Philosophy, or rather on the first volume (First Principles) of the series relating to the system to which Mr. Spencer has given the appropriate name Synthetic Philosophy. I must confess to being saddened by the reading. The author of the review is one of those whose name is "writ large" in every line he pens. No one who has ever read a page from his pen can fail to recognise thereafter whatsoever he may write. A masterly advocate in a good cause, which is well though not always best in science, he here shows his power in special pleading for what I cannot but regard as a bad cause. I do not mean the cause of that philosophy for which he seems to contend, but his obvious wish to cast absurdities on (not to show absurdities in) the philosophy of one whose aims he detests. From the first lines of the review, in which he jeers at the letterpress and binding-"First Principles " inside, the "System of Philosophy" outside, and the "Synthetic Philosophy" on the back (where however "First Principles" appears in letter five times as large)the review is unfair. The kind of ridicule cast on Mr. Spencer is such as a lover of the old Aristotelian philosophy might as effectively, nay much more effectively, have cast on Bacon's Novum Organum,-more effectively because in many matters Bacon was not abreast of the science of his day, and many of his suggestions were open to ridicule even in his own time, faults which cannot be urged against the teachings of the greater Bacon of our era. To begin with, the author of the review touches on what would scarcely affect our estimate of a system of philosophy, -the language and even the grammar in which it is presented. He permits himself to say, "In quoting from Mr. Spencer, we must occasionally alter the grammar of a quotation to make it fit our own writing without abrupt changes." This, by the way, might not seem very severe criticism if the reviewer's writing were judged by the sentence immediately following, which begins thus, "No writer that we know of requires his definitions so carefully

attending to, and the subsequent use of his defined terms so carefully watching." Only a few sentences before, the hypercritical reviewer (for I do not think any writer requires his writing so carefully attended to as for the moment I attend to his) speaks of something "which we should be sure to be told that we have misunderstood, or overlooked qualifying statements somewhere else," and I should very much like to hear his parsing of the words "which we should be sure to be told that we have overlooked qualifying statements," &c.,-[which it seems very bad grammar to me, or my last eight words good grammar, which they are not, nor which are the last four or these nine.]

If I smile at such trifles for a moment it is because the reviewer tries to give them prominence in pretending to deal with the Spencerian philosophy. He is perhaps wise in his generation. It is easier in this case to criticise style than matter; for such a subject as Mr. Spencer deals with in his "First Principles" is one in which the use of abstract language easily cavilled at is almost unavoidable.

When the Edinburgh reviewer attacks or rather tackles the subject matter of Mr. Spencer's volume, he still keeps clear of essentials to lay chief stress on misinterpreted words or manufactured absurdities. Among the latter the most remarkable perhaps is that, because Mr. Spencer speaks of the Unknowable First Cause as only known to us through the known effects of persistent force, therefore Mr. Spencer practically presents the Unknowable First Cause as identical with Persistent Force: whereas if there is a point on which Mr. Spencer lays special stress in dealing with THAT of which he speaks as the "First Cause, in every sense perfect, complete, total, including within itself all power and transcending all law," it is that It may be, nay must be, utterly unlike that by which we know of Its existence.

After this it is a trifle that the reviewer misrepresents Mr. Spencer in detail,-as for instance at p. 47, where he quotes professedly "a specimen of Mr. Spencer's most careful and precise style unreduced," yet omits several important words, and actually gives a concrete illustration of what he says he understands Mr. Spencer to mean, an illustration corresponding only with the garbled extract and naturally leading to an absurd abstract proposition.* The reviewer takes Mr. Spencer to task for regarding the laws of Newton as results of experience, and points out that Newton himself established them by the sufficient cause" argument. This is altogether new to me. In the "Principia" Newton presents these axioms as the results of experience and describes some of the experiments which establish them. In fact Newton uses the word "axiom " in its proper sense as meaning a fact or law established by experience and known to be worthy (aos) of acceptance.

66

*The concrete illustration really corresponding to Mr. Spencer's abstract general statement would have been this,-If the indifferent mental idea conveyed to the mind by the aspect of a tiger leads always to the idea of physical injury and thus to the movements necessary for escape from the tiger's ferocity, it is a matter of no moment whether one idea or the other is identical with the reality or utterly unlike it. Of this it may be remarked, indeed, that it is obvious; but the very circumstance that a proposition can be laid down in abstract terms implies that any concrete example must be obvious. All the examples of the action of gravity_are obvious, but it required a Newton to deduce from them the Law of Gravitation.

"Sartor Resartus" "clotted nonsense"; and so it was no doubt to that angry writer: the Edinburgh reviewer, who seems to attach weight to what Carlyle wrote, would probably consider that criticism hasty and ill-judged, we may even say false and spiteful. The criticism has so far become a thing of the past that now, less than fortyfive years after it was uttered, an admirer of Carlyle forgets that it was said of that writer and not by him. Perhaps in less time still our reviewer's angry abuse of Mr. Herbert Spencer may similarly have passed into oblivion.

I

THE EVOLUTION OF FLOWERS.

BY GRANT ALLEN.

I. THE STARTING-POINT.

PROPOSE in this set of papers that we should follow out together, so far as is possible, the various steps in the evolution of a single great group of plants, illustrated for the most part either by native English wild flowers, or by such common garden favourites as are within the easy reach and familiar knowledge of almost everybody. Starting from the simplest known form, which we may conceive to represent very nearly the peculiarities of the primitive ancestor, we must trace the gradual changes by which the various successively higher forms have been developed ; and at each stage we must try to discover what was the advantage gained by the plant through the different new arrangements, and in consequence of what special agency these arrangements became finally stereotyped in the persons of its descendants. In this way, we shall obtain a more clear and connected view of the methods of evolution in the vegetable world than we could ever obtain by the study of mere casual isolated instances, and we shall be able more fully to understand the underlying meaning and reasons for the classifications long since half blindly (though very wisely) adopted by the earlier pre-Darwinian botanists. We shall see that the classes they mapped out are really genealogical divisions, and that all the members of each family or genus are really bound together by genuine ties of blood in their common descent from a single central and typical ancestor.

The great group of plants to which I propose to apply our present scrutiny is one that may be roughly described for unbotanical readers as that of the Lilies. Botanists will know more clearly what is meant if I say that our subject is to be the Monocotyledons, especially those with conspicuous petals or perianths, comprising the main central body of the class, from the Alismas up to the Orchids. This group may fairly enough be described throughout for popular purposes under the general name of Lilies, both because most of the flowers are moderately lily-like in form and texture, and because the true lilies occupy a central place in the class as a whole, presenting the peculiarities of the entire body in a comparatively simple and recognisable form.

What, then, is the simplest and most primitive existing type of lily, or, to speak more correctly, of Monocotyledon? I believe, if we take relative simplicity in the arrangement of parts as our guide, we shall come to the conclusion that no lily-like plant is more primitive or antiquated in type than our own common English water-plantains. Let us begin, therefore, by looking briefly at the nature and structure of this familiar and pretty little British pondhaunter; and then let us inquire what are the marks which it still bears on its very face of its own archaic and

ancient characteristics.

Everybody must often have seen and noticed the waterplantain, with its tall sparse whitish flowers rising in large, loose masses high above the stagnant surface of still pools or flooded ditches. It is a pretty, glossy-leaved plant, with long-stalked bright green blades, and a spreading panicle of starry little blossoms, which look white in the mass as you see them growing, but turn out to be delicately pink or rose-colour when you gather them for close inspection. In fact, if ever you have seen a lush and succulent water-weed, with a perfect pyramid of straggling white bloom clustered in its centre, overtopping the calm levels of a shallow English pond, you may be pretty sure that that was a water-plantain. Its botanical name (which I shall always add here for the benefit of those readers who already take an interest in structural botany) is Alisma plantago.

Now, what are the reasons which induce us to begin our review of the lily tribe with this little inconspicuous English wild-flower? Well, let us premise first of all that evolution runs habitually from the simpler to the more complex; from the like to the unlike; from the less consolidated to the more consolidated. Suppose we find two flowers, one of which has five distinct petals, all alike, and the other of which, resembling it in every other way, has two of those petals specially modified into a peculiar form, we rightly conclude that the former is the more primitive and original of the two. Not necessarily that the second is directly derived from the first; about that we can only judge by means of very minute and circumstantial evidence; but that, at least, the first stands nearer in type than the second to the common ancestor from which both are presumably descended. Again, if we find one flower with five separate petals, and another just like it, only with the five petals united into a single tubular corolla, we once more rightly conclude that the former is more primitive and original than the latter. Distinctness of parts is almost always a mark of the early unconsolidated stage; coalescence of parts is almost always a mark of the later consolidated stage. For example, most simpler crustaceans have the body divided into several nearly equal and similar joints or segments; but in the crabs and lobsters, the princes among crustaceans, seven such segments have become united together to form the large head-piece with its single solid shell or carapace. In such a case, everybody can see at once that the union of

parts is an obvious sign of higher and more complete development.

If we look at the flower of the water-plantain, we shall similarly see that it presents many such symptoms of an early, uncompounded, simple type. In the technical language of botany, there is very little cohesion or adhesion among its parts: it shows us in the easiest and most separate form the ground-plan upon which all the lilies, high or low, are ultimately constructed. Only, while in the higher lilies we have to pick out the various component elements of the flower with some difficulty from their entangled and combined condition, in the waterplantain we get them all distinct and individualised, so that there need be no hesitation at all in recognising their nature and meaning.

This, in brief, is the original ground-plan of the blossom in the common ancestor from which the great lily group has ultimately descended. Its parts were all arranged in whorls of three members each. It must have had (as we know by comparison of all existing forms) first of all a protective calyx whorl of three outer green sepals, enclosing and shielding the unopened bud from all attacks of cold weather or greedy insects. Inside this must have come a second or corolla whorl of three brightly-coloured and delicate petals, intended for the attraction of its insect fertilizers. Within the petals, again, were the pollen-bearing stamens, arranged in alternate rows of three each; and of these rows there may have been one, two, three, or more; though the fact that most existing monocotyledons have six stamens apiece, or else exhibit traces of having originally had six, would seem to show that two rows were most probably the contingent possessed by the prime ancestor. Last of all, in the very centre, came the carpels or young seed-vessels, of which there were also three, six, nine, or more, according to circumstances.

another. In the third place, the water-plantains have only one seed in each carpel; and we also know by analogies elsewhere that primitive flowers always have only one seed in each carpel, but that more advanced types, while lessening the number of carpels, increase the number of seeds in each.

I know this first exposition has necessarily been a little dull, because we have here to dwell chiefly on fundamental points of structure, which are always dry, and to say very little about points of function and the practical use of parts, which are always comparatively interesting; but that could hardly be helped in an introductory sketch, where it is needful, above all things, that we should have a clear conception of the raw form from which we take our first departure. In future papers, I trust we shall be able to make the final development of the various lily-like plants from this simple original a little more graphic and a little less dull. Meanwhile, I hope my readers will try to master the first principles laid down in this opening part; as a firm grasp of the architectural plan of the water-plantain will greatly assist in following out the subsequent course of evolution on which we are about to embark.

One word more, as the preachers say, and I have done. It is a very significant fact that the water plantain and its congeners are all, without exception, aquatic plants of the marshes, ponds, and ditches. Now, it frequently happens that fresh-water animals and plants preserve for us very antique and otherwise extinct types-creatures of a sort which have become extinct in the fiercer competition of the great continents and the great oceans, but which have lingered on in the less-occupied reaches of inland rivers, lakes, or pools. It has been ingeniously noted that meres or ponds may be regarded in this respect as the aquatic analogues of oceanic islands, where so many very archaic forms have been preserved for us, far from the wild struggle for life which rages so incessantly in the wider stretches of land or water. Indeed, it may be said, roughly speaking, that almost all very early or primæval types of plants or animals yet existing belong to one or other of three peculiar habitats-islands, freshwater lakes or streams, and caves. And the one point these three habitats have in common is just this-freedom from competition save by the members of a very small and local fauna or flora.

tacle, which forms the boss or axis of the whole flower.

It is to these carpels that we must most especially direct our attention at the outset, because they are, so to speak, the very patents of nobility of the Alisma family, the grand evidence that the water-plantain and its congeners do really form the most primitive existing members of the great lily group. In the first place, all the other lilies without exception (save only the Alisma family and a few closely related small orders) have the carpels more or less combined into a single compound ovary, the walls of the different carpels having coalesced, for a reason which we shall have hereafter to consider. In the second place, the number of carpels in the water-plantains is exceptionally large; and we know by the analogy of the buttercups, which are the simplest members of the other great group of flowering plants (the Dicotyledons), that primitive flowers always have a great many distinct carpels, and that with the advance towards higher types, the carpels tend to become reduced in number as well as to cohere with one

GHOSTS AND GOBLINS.

BY R. A. PROCTOR.

HERE are few subjects more perplexing, on a close the men the super

natural (as distinguished from the religious). Whether we analyse particular superstitions and endeavour to understand what is actually believed respecting them, or whether, taking a wider view, we consider the origin of the widespread belief in supernatural agencies, we find ourselves beset with difficulties; and these are only preliminary to the great difficulty of all-that of determining how far it is reasonable or likely that any of the common ideas about the supernatural have any basis of fact whatever.

But the first difficulty to be encountered resides in oneself. I who write have my superstitions. If I simply had them and believed in them, there would be little diffi culty. But I do not believe in them. I know that they exist, because on certain occasions I have felt them in operation. Every reader of these lines must have had similar experiences-vague terrors coming we know not whence, and refusing to be exorcised by reason; the feeling -not momentary though transient-that a sight or sound