nately takes place at the bright limb, and the observer's success will therefore depend upon the very uncertain contingency of his having the point of reappearance at the dark limb just under his eye. This may indeed be secured by a wire in the field with an equatoreal mounting; but it will demand very pertinacious watching, as the actual times, and the consequent amount of duration, will vary from those here given, in proportion to the distance of the station from Greenwich. 30th, 55 Leonis, 6 mag., will be hidden from 10h. 13m. to 10h. 31m.

ARITHMETICAL RECREATIONS.

THE following paper, communicated by M. Plateau to the Belgian Academy, is translated from Cosmos, and will not fail to interest a large circle of our readers by its exhibition of amusing and unexpected properties of numbers:

M. Plateau states that a long while ago he heard the following recreative problem proposed:-Given as a multiplicand the curious number 12,345,679, it is required to find a multiplier which shall furnish a product entirely composed of a repetition of any one number, chosen at pleasure. If the figure required to be repeated in the product be 1, the multiplier required will be 9. If the repeated figure is 2, 3, 4, etc., the multiplier must be the product of 9 multiplied by such figure. Thus, to have a product consisting of 2 repeated in succession, the multiplier must be 9 x 2 = 18; and to obtain a repetition of threes, it must be 9 x 327, etc.

Mr. Plateau supplies the following general rule applied to this class of proposition, and he believes it to be his own discovery :

Any unequal number being given as a multiplicand, provided it does not terminate in 5, it is always possible to find another number as a multiplier which shall give, with the former, a product composed entirely of the iteration of any figure that may be named.

To do this, divide unity by the number given as a multiplicand, which will give a periodic decimal fraction, in which the period will commence immediately after the point, it being understood that the ciphers preceding the significant figures belong to the period.

If the number given be not divisible by 3, the period will be divisible by 9; make this division, and the quotient will be the factor sought, taken singly, or multiplied by 2, 3, 4, according as the product is required to be composed of ones, twos, threes, fours, etc. If the number given be divisible by

3, and not by 9, the period may not be divisible by 9, but it may be by 3, and in this case divide the whole of the three periods by 9, and the quotient will be the factor sought.

Lastly, if the number given be divisible neither by 9 nor by 3, divide by 9 the whole of 9 periods, and the quotient will be the factor sought.

If, for example, N be the number given-subject to the condition of its being uneven, and not terminated by 5—and we convert into a periodic fraction, and if we represent the period by P, we shall have

1

N

Р
999....

from whence we

PN.

If the number N is not divisible by 9, the period P will be In this case let us call Q the quotient of P x 9, and we shall have 111 .... 2 N, and we shall find the factor

So.

which is sought.

=

As for the case in which N is a multiple of 3 or of 9, and in which P may consequently be divisible by 9, it is plain that we have only to take, instead of P, the reunion of three or nine periods, so that the whole may be divisible by 9. For example: suppose the number given be 7, we find = 0,142,857, 142,857 ......; the factor sought will be 142,857 15,873; and, in fact, 15,873 multiplied by 7, gives 111,111; and twice 15,873, or 31,746, multiplied by 7, gives 222,222, etc.

9

If the number given be 3, we have 0,3333 ... and the period is divisible by 3, but not by 9, we therefore divide by the reunion of three periods-that is to say, 333, and the quotient 37 will be the factor required.

If 27 be the number given, the period will be 037, which is neither divisible by 9 nor 3, and consequently we must take the reunion of nine of these periods to divide by 9, which gives the number 411,522, 633,744, 855,967, 0,781,893.

We see by this last example that for a small number given as multiplicand, we may require a very high multiplier; but, on the other hand, a high multiplicand may require a small multiplier, as, if the great number just recited had been that given as a multiplicand, we should have found 27 as the multiplier.

THE GENUS FISSIDENS; OR, THE FLAT

FORK-MOSSES.

BY M. G. CAMPBELL.

THE Fissidentes, like their toothed allies which we considered last month, are perennial plants, and grow on the ground, or on rocks, but rarely on the bark of trees. Their foliage, however, bears no resemblance to that of the Dicranums, being, instead of setaceous and secund, flattened, the upper part expanded into a vertical scalpelliform lamina, and inserted in two opposite rows alternately on the opposite sides of the stem, or rachis; for this arrangement gives a frond-like appearance to the

stem.

Indeed, the peculiar and highly curious structure and disposition of the leaves of this genus very naturally separate it from all other British mosses with which we have any acquaintance, while the insertion of the fruit seems to make it a connecting link between the Acrocarpous and Pleurocarpous sections; the capsule in some species being terminal, in others lateral, arising from a short fertile ramulus, to which arrangement Wilson gives the term Cladocarpous.

The singular structure of the leaves, besides their being vertical, consists in an expansion of tissue from the nerve, and a prolongation of one of the wings of the true leaf, making the leaf, on the stem side, double from above the middle to the base, the expanding wing clasping the stem. This is more conspicuous in the upper leaves, the length of the wing gradually diminishing, and the lower being almost destitute of this strange appendage.

The author above named enumerates and describes in his valuable Bryologia Britannica eight species of Fissidens, most of which may be found in fruit during this month. They derive their generic name from the Latin fissus, split, and dens, a tooth.

Among the most common is Fissidens taxifolius, or the yewleaved flat fork-moss. It inhabits moist, shady banks, chiefly in a clayey soil. The inflorescence is monoicous; the gemmiform barren flower sitting at the base of the fertile stem. The stems are branched, and fasciculate, i.e., with short lateral branches of unequal height; the taller in this instance are about half an inch long; they are clustered several together from a common rooting base, where both barren and fertile flowers originate. The leaves are crowded, lanceolate, apiculate, very minutely crenulate, of a light green, not twisted

when dry, and with a pellucid nerve extending almost to the apex, the dorsal wing broad to the base, but not decurrent. As in the rest of the genus, they are equitant, i.e., inserted in two ranks, the leaves of each rank partly sheathing each other at the base of the leaf.

The seta arises from near the root to about, or rather more

than half an inch in height, is flexuose, curved at the top, and its perichetial leaves are pointed, convolute, ovate, and sheathing; the capsule is oblong, or obovate, inclined or horizontal, of thick texture, constricted below the mouth when dry, reddish brown; the lid convex at the base, with a beak nearly as long as the capsule; the calyptra whitish, rather longer than the lid, and dimidiate, or splitting along one side. The oblong vaginula is distinct, and not insensibly passing into the perichetial ramulus as in truly pleurocarpous mosses.

FISSIDENS TAXIFOLIUS.

The illustration given is, of course, considerably magnified. Another example of the cladocarps, and looking at first sight much more like a pleurocarp, is Fissidens adiantoides, or the marsh flat fork-moss, the smaller forms of which somewhat resemble F. taxifolius, from which however it is clearly distinguished by the position of the fruit stalk, which is lateral at a considerable distance from the base of the main stem, which latter is elongated and branched, with crowded, ovate, lanceolate leaves, denticulate at the apex, minutely serrulate below, and nerved to the point. In moist situations the stem reaches more than two inches in length, with dark green leaves and sub-fasciculate branches, more loosely cæspitose; in dryer situations it is shorter and more densely tufted. The leaves are crisped and incurved when dry. The fruit stalks rise from about the middle of the stem to from half an inch to an inch in length. The capsule is more or less inclined, of thick texture, constricted below the mouth, and somewhat turbinate when dry. The lid with a long beak, and the calyptra still longer. Its favourite habitats are shady banks, wet pastures and bogs, and on wet rocks near waterfalls.

We have found it in great abundance on a rock just above Longfords Lake, under the shade of trees in Gatcombe Wood, Gloucestershire.

As an example of the terminal-fruited section, we may instance the common flat fork-moss, Fissidens bryoides, to be met with almost everywhere, even in the sandy deserts of Africa, where its exquisite beauty and wondrous structure spoke so eloquently to the heart of the renowned traveller, Mungo Park, at a time when, worn with fatigue, plundered by banditti, and surrounded by all the horrors of the desert, his sinking courage had well-nigh failed, and he threw himself down to rest his weary limbs and ponder on his destitute condition. He says himself, after describing his state-" At this moment, painful as my reflections were, the extraordinary beauty of a small moss irresistibly caught my eye; and though the whole plant was not larger than the tip of one of my fingers, I could not contemplate the delicate conformation of the roots, leaves, etc., without admiration. Can that Being, I thought, who planted, watered, and brought to perfection, in this obscure part of the world, a thing of so small importance, look with unconcern upon the situation and sufferings of creatures formed after his own image? Surely not! Reflections such as these would not allow me to despair. I started up, and disregarding both hunger and fatigue, travelled forwards, assured that relief was at hand, and I was not disappointed."

Such is Mungo Park's own account of the incident, and from original specimens of the moss given by his brother-inlaw, Mr. Dickson, to our celebrated botanist Hooker, the latter was able to ascertain the species as Fissidens bryoides, which we will now proceed briefly to describe. The general structure and arrangement of its leaves sufficiently characterize the family group to which it belongs, but it is smaller than either of those already noticed, the stems rarely averaging more than from two lines to half an inch long. The barren flowers are numerous, gemmiform and axillary, never terminal, which the fertile flower is, producing a symmetrical capsule on a red seta, and having a red peristome with a persistent rudimentary annulus and a rostellate lid. The leaves are spreading, widely lanceolate, apiculate, and with a thick cartilaginous border. It is met with in fructification at various seasons, but particularly in the winter months, December, January, and February, and its exquisite beauty cannot fail to strike with admiration any one who will bestow upon it an attentive examination. All the family, in common with the Dicranums, possess sixteen bifid teeth. For a description of the latter, see the INTELLECTUAL OBSERVER for November.

The remaining members of the group are F. exilis, slender

VOL. IV.-NO. V.

BB