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we obtain a general insight into the train of reasoning pursued by the author, and of the logical arrangement of his work, and we are thereby better prepared to read and understand it; for in every case it is from the body of the work only that we can derive true knowledge ; and all that either an index or table of contents can avail us is to point out to us the best way of pursuing our studies, and to show us the connexion of what we have acquired with what we have yet to attain.

It should be here observed, that the application of arrangements strictly artificial, to the animal kingdom, has long been given up, for reasons, too, which clearly prove the justice of the objections we have urged against the principle of such systems. The very complexity of animal organization, while it invites more constant and reiterated investigation, enables us, by a comparison of its component parts, to arrive at more conclusive inferences respecting their functions and their relative importance; and each new examination confirms the truth of what we have already stated, as to the constancy, within certain limits, with which forms of structure are associated with certain functions and habits: hence, the primary divisions of every classification of animals remain nearly constant; and it is only the subordinate ones of orders and genera

which fluctuate. Thus, for example, when an animal was found which had all the essential characters of the genus Felis, but did not possess retractility of its claws, naturalists, who regarded this as a character pertaining to animals intended to spring on their prey, might agree as to the nesssity for forming a new genus, and would discover a new, and, hitherto unsuspected, relation between genera of the same order, Carnivora; but we are so convinced of the harmony existing in the plan of creation, as to feel certain that we shall never meet with an animal which will combine the characters of a carnivorous and of a ruminating one, or having the complicated digestive organs of the latter, with canine teeth and claws. Again, among the species composing the sub-order of plantigrade carnivora, there occurs one, the Kinkajou, which has a prehensile tail, and a tongue capable of considerable protrusion, like that of the Manis, thus presenting an affinity between a carnivorous species and genera of the widely different orders of Quadrumana and Edentata. If we descend to the invertebrated division, we find the limits of even classes fluctuating: thus, there are many species of which naturalists are not yet agreed, whether to refer them to Articulata or Mollusca; but all these doubtful cases bear too small a proportion to the more definite ones, to cause any uncertainty as to the permanence of such general divisions as have for a long time been adopted in the animal kingdom.

The case, however, is widely different as regards the vegetable kingdom, the simplicity of the organization, and our uncertainty as to the functions exercised by the component parts, an uncertainty chiefly arising from the difficulty of making observations, prevent our yet deciding on even the primary divisions, and still more on fixing their limits with anything like distinctness; the numerous species might therefore seem still to justify the use of an artificial arrangement; and that of Linnæus being unquestionably the best ever contrived, and possessing merits of the highest order, it will probably, for some time to come, be employed even by many who acknowledge the imperfect principle of all artificial systems.

It is necessary, however, to put beginners, who have not considered the subject, on their guard, lest they should, as is but too commonly the case, mistake a facility in applying this system, for the acquisition of real knowledge of the science of Botany: we shall therefore proceed to show that the Linnæan system can only lead to an acquaintance with the names of plants, and not to that of their mutual affinities, which, as we have stated, is the only true foundation of real knowledge.

Most of our readers are aware that, besides the two primary divisions of species into flowering and flowerless plants, the former are arranged under two classes, distinguished by peculiarity of structure; which, from their generality and constancy, must be important; though, in the present imperfect state of our knowledge of the subject, we are even unable to indicate at what period the elementary organs, which are apparently the same in all plants, undergo that change in their functions that leads to such remarkable results. All that we know is, that, generally, certain species which proceed from seeds with only one, or with alternate cotyledons, have the structure of their stem entirely different from that which is perceptible in the stems of plants proceeding from seeds with two, or with verticillate, cotyledons: that the leaves also of monocotyledonous plants differ from the reticulated texture of those of the other class, in having their veins parallel, and that their flowers are characterized by a ternary instead of the quinary division of their component parts, which is usually met with in those of the dicotyledonous class.

It is impossible to say whether we shall ever be able to explain the origin of the numerous and singular exceptions to these general characters of the two classes; but it is clear that it is by endeavouring to do so that we shall make the greatest progress in ascertaining the laws of vegetable physiology, and our investigations will be mainly assisted by every improvement in correct classification of known species, by which they may be grouped according to their most numerous natural affinities: it is also impossible for any one to make a progress in the knowledge of that physiology, unless he is habituated to keep these affinities constantly before his mind during his researches.

The Linnæan system is strictly an artificial one, and was intended to be so by its great author: it is founded on the selection of modifications of one organ only as the basis of the divisions; we might therefore anticipate, by the analogy of what we have shown to be the consequence of such a principle in the classification of the animal kingdom, that the most glaring violations of natural affinities would be the result. It is found by observation that the precise number of any of the parts of the reproductive organs of plants is liable to constant derangement, from the laws which govern their developement; that is to say, that of several plants having decidedly the closest connexion in every peculiarity of structure, qualities, and even form, one will have only two stamens, while another has four, and a third, perhaps, five or ten. All these would be separated and placed in different classes in the Linnæan system; and since the primary distinction of structure of the seed, and all its attendant consequences, formed no part of the plan on which that

system was built, the student who employs it, and who is not aware of its true principle, is habituated to associate together in his mind monocotyledonous and dicotyledonous plants, provided they have an equal number of stamens or styles.

It would be foreign to the object of this essay, to enter here into any account of the principles on which the normal number of the parts of fructification varies in plants; but we will show, by one or two examples, the inconsistencies resulting from the artificial arrangement in question. The establishment of the very first class is a fallacy of the most fatal tendency; the existence of one stamen only, though constant in certain species, is the result of that peculiar law of non-developement of some of the parts arranged round a central axis, when, from their position, they are crowded together: the botanist who is imbued with the spirit of natural affinities, regards those species thus distinguished only as regular and constant deviations from the type of the family to which their other characters unite them; a deviation he can explain by the principle just alluded to. The Linnæan student, who always commences his studies with mastering the system, on account of the aid it affords to him in his pursuits, becomes habituated to the idea, that the character of having only one stamen is just as important as that of having three, six, five, or ten: when he sees a specimen of the genus Maranta, he does not, as he ought to do, view it as one of a family, having a ternary arrangement of its flowers, but that two of its anthers ever remain undeveloped, or abortive, as botanists term this state of non-appearance of any organ which exists in the regular or ideal type of the order. That this is the true explanation of the deficient number of stamens, is shown by the existence of the petaloid filaments in the plants in question. Of all the natural orders of plants, of those which are distinguished by the greatest number and constancy of characters in the species composing it, those of palms and grasses are pre-eminent for their distinctness; now of the first of these two we find the genera in the Linnæan system distributed under four different classes, and eight different orders or subdivisions of those classes. The same unnatural separation of genera takes place with respect to the grasses; and in both cases it is caused by deviations either in the number of stamens, or by the equally explicable anomaly of the male and female flowers being distinct on the same or on different individuals.

But this is not the whole of the incongruity resulting from this application of one character only to the distribution of species of the vegetable kingdom, not only are those separated which are really closely allied, but others are brought into juxta-position which have no relation whatever to each other; in the instances to which we have been alluding, genera of grasses and palms are placed together with dicotyledonous ones of the most opposite qualities and characters.

It must not be supposed that these are singular instances collected to justify a partial statement; there is not a class or order throughout the system which does not exhibit nearly equally striking absurdities, as for example, the separating a few genera of the well-defined order of Leguminosæ from the rest, because the union of the filaments is more perfect, or because it does not exist at all, as in the mimosa.

We should be less urgent in insisting on the errors of artificial systems, when used, as they unfortunately always are, for any other purpose than that of a mere index to names, and for which they were only intended, if it were not clear that the true principles of classification are still misunderstood, as is proved by the endeavours made, even in the present day, to improve the Linnæan system, so as to adapt it to our present state of knowledge. These corrections invariably consist in removing species and genera from the place assigned to them by the spirit of that system, in order to unite them to others with which they are naturally allied. If this plan were followed out, the principle of the system must be abandoned, as it has been long ago, with respect to the animal kingdom, for reasons before mentioned. By this operation, however, the Linnæan system loses its peculiar merit of simplicity and ready application, without attaining that of one grounded on natural affinities.

Let us suppose a beginner resolved to study botany in the best possible manner, by examining the plants of his own country, and by collecting the specimens himself; and that he has been persuaded to provide himself with an English Flora arranged according to the Linnæan system in its most approved form. We will suppose that on his first expedition he collects species of the following genera of plants *, with none of which he is at all acquainted as a botanist.

1. Veronica; 2. Circæa; 3. Salvia; 4. Epilobium; 5. Paris; 6. Antirrhinum; 7. Urtica; 8. Bryonia; 9. Humulus; 10. Tamus.

After comparing the number, &c., of stamens and styles with those given as the characters of the classes in his book, he will finally refer his specimens to the following classes and orders:-1, 2, 3, DIANDRIA monogynia; 4, 5, OCT ANDRIA monogynia and tetragynia; 6, DIDYNAMIA angiospermia; 7, 8, Monacia tetrandria and pentandria; 9, 10, DICCIA pentandria and hexandria. And he will learn to associate these genera together in his mind according to this arrangement. Now, these plants are in every case transposed from their natural position, which, according to their intimate relations, should be as follows :-1, 6, of the order Scrophularinæ ; 2, 3, Labiatæ ; 4, Onagrariæ ; 8, Cucurbitaceæ ; 7,9, Urticæ; all of which are dicotyledonous; while 5, 10, belong to the monocolyledonous order of Smilacev.

If, as is too frequently the case, our student is contented with having ascertained the names, and arranged his specimens according to his system, the worthlessness of such information as he thus gains, may be rendered obvious by the exposure of its fallaciousness as regards the acquirement of any correct knowledge of the properties of the plants in question. The harmless, and in some respects useful, nettle, is associated with the deleterious and valueless bryony; while the connexion of this last with the melon, cucumber, gourd, &c., is unsuspected by him. A plant of the wholesome and valuable order of Labiatæ is brought into connexion with the suspicious one of Scrophularinæ, which contains the foxglove, and is nearly allied to that which comprises the nightshade and Belladonna.

They all might be found in flower at one time and in one neighbourhood, but an anachronism in this respect, of course, does not affect our argument.

These objections cannot be answered by urging the difficulty of applying the arrangement according to natural affinities as an index; for even in this point of view the Linnæan system has no great advantages over the other: it merely leads to the determination of a class and order, under which a certain plant is arbitrarily placed, but that order may contain a great number of genera, and the precise one to which the plant belongs must be ascertained by the same investigations as are requisite to determine that point with regard to a group of genera arranged in a natural order. Now, taking into account those doubtful cases in which either system may mislead, and which furnish the true test of their relative merits, as mere indices, we are persuaded that little difference will be found in this respect by any unbiassed

person. But, conceding this as a point of secondary importance, what we have endeavoured to point out in this paper is, that though classifications may serve as indices to enable a learner to discover the name, they ought to aim at fulfilling the more important object of presenting the endless number of species separated into groups, each containing those only which agree

in
every

detail of their organization which we may presume to influence the habits, mode of life, form, &c., of the beings composing it. Some naturalists have endeavoured to explain these complicated relations by the analogy of spheres mutually intersecting, the centre of each sphere representing the type, real or assumed, which combines the characters of the group expressed by the sphere: this, and other fanciful conceptions propounded by different authors, tend to show the difficulty, if not impossibility, of ever perfecting such a system; but we may constantly strive to do so, and every step gained is an addition to the strength and solidity of the foundation on which to build a superstructure of accurate knowledge of organic creation.

ON THE PROBABLE ORIGIN OF THE ALGEBRAIC

SIGNS + AND

I have always found it conduce much to the right appreciation of the signification of the symbols employed for the purpose of algebraic condensation, to point out the history, as far as it can be done, of their introduction and subsequent modifications. Such a procedure gives the mind a more complete grasp over their full meaning, and the proper limitations to which their interpretation must be subjected, and removes from the feelings that disagreeable air of mysteriousness, which, more than any other cause whatever, tends to create a painful uncertainty of the legitimacy of any step which it may occur to the unpractised inquirer to take, in his early investigations.

It is well known to every one who has looked into the earlier writings on algebra, that the words expressive of the operation were written at full length, or merely with such contractions as were employed in writing those words in general literary composition. This is the case amongst the Italian algebraists, as attested both by their remaining MSS. and their printed works. It was always the case amongst the Hindoo, Persian,

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