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general and indisputable conceptions which constitute this lofty science, to be on their guard against the baseless notions of a vital principle, vital forces, and entities of that character. Hitherto, all advance in positivity in biology has been obtained at the expense of its dignity, which has always been implicated with an imaginary origin of life, of sensibility, etc.: but when physiologists have learned from their study of gravitation and other primary laws how to confine themselves to true science, their subject will rise to the highest elevation that positivity admits of, that rational prevision of events which is, as I have so often said, the end of true science :-an end to be aimed at in biology, as it is perfectly fulfilled in astronomy.

Here, too, must biologists learn the character of sound scientific hypothesis. This method is eminently wanted in so complex a study as physiology; but it has been as yet. used with very little effect. The way is, undoubtedly, to determine the organ from the function, or the function

from the organ. It is permissible to form the most plausible hypothesis as to the unknown function of a given organ, or the concealed organ of a manifest function. If the supposition be in harmony with existing knowledge, if it be held provisionally, and if it be capable of a positive verification, it may contribute to the progress of discovery, and is simply a use of a right of the human mind, exercised as in astronomy. The only eminent example known to me of sound hypothesis in biology is that of M. Broussais, in proposing the mucous membrane of the alimentary canal as the seat of so-called essential fevers. Whether he was mistaken or not, is not the question. His hypothesis being open to unquestionable confirmation or subversion, it gave a great impulse to the study of pathology in a positive manner and it will stand in the history of the human mind, as the first example of the spontaneous introduction of a sound hypothetical method into the positive study of living beings: a method derived from the region of astronomy.

It remains to consider the relation of biology to mathematics.

The encroachments of the pure geometers upon the domain of biology have been at

To mathe

matics.

tended with the same mischief, but in an aggravated form, that we have witnessed in the case of other sciences. This mischief has led physiologists to repudiate mathematics altogether, and open an impassable gulf between themselves and the geometers. This is a mistake; inasmuch as their science cannot be severed from that which is the basis of the whole of natural philosophy; and it is only through the admission of this that they can maintain the originality and independence of their scientific labours. The rational study of nature proceeds on the ground that all phenomena are subject to invariable laws, which it is the business of philosophical speculation to discover. It is needless to prove that on any other supposition, science could not exist, and our collections of facts could yield no result. In the phenomena of living bodies, as in all others, every action proceeds according to precise, that is, mathematical laws, which we should ascertain if we could study each phenomenon by itself. The phenomena of the inorganic world are, for the most part, simple enough to be calculable: those of the organic world are too complex for our management: but this has nothing to do with any difference in their nature. And this is the view which both geometers and biologists should bear in mind.

If in astronomy our calculations are baffled when we pass beyond two or three essential conditions, it is evident how impracticable they must be amidst the inextricable complications of physiology. And again, this complexity prevents our ever effecting a mathematical disclosure of the elementary laws of the science. This excludes all idea of this method of philosophizing in biology; for these laws are no otherwise accessible than by the immediate analysis of their numerical effects. Now, whichever way vital phenomena are looked at, they present such endless and incessant variations in their numbers, that geometers are baffled as completely as if those degrees were entirely arbitrary. Even numerical chemistry is inapplicable to bodies whose molecular composition varies incessantly; and this is precisely the distinguishing character of living organisms. However hurtful may have been the incursions of the geometers, direct and indirect, into a domain

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BIOLOGY AND MATHEMATICS.

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which it is not for them to cultivate, the physiologists are not the less wrong in turning away from mathematics altogether. It is not only that without mathematics they could not receive their due preliminary training in the intervening sciences: it is further necessary for them to have geometrical and mechanical knowledge, to understand the structure and the play of the complex apparatus of the living, and especially the animal organism. Animal mechanics, statical and dynamical, must be unintelligible to those who are ignorant of the general laws of rational mechanics. The laws of equilibrium and motion are, as we saw when treating of them, absolutely universal in their action, depending wholly on the energy, and not at all on the nature of the forces considered: and the only difficulty is in their numerical application in cases of complexity. Thus, discarding all idea of a numerical application in biology, we perceive that the general theorems of statics and dynamics must be steadily verified in the mechanism of living bodies, on the rational study of which they cast an indispensable light. The highest orders of animals act, in repose and motion, like any other mechanical apparatus of a similar complexity, with the one difference of the mover, which has no power to alter the laws of motion and equilibrium. The participation of rational mechanics in positive biology is thus evident. Mechanics cannot dispense with geometry; and besides, we see how anatomical and physiological speculations involve considerations of form and position, and require a familiar knowledge of the principal geometrical laws which may cast light upon those complex relations.

In regard to Method, the necessity of recurring to a perfect model of reasoning, the more earnestly in proportion to the complexity of the science concerned, is applicable in regard to Mathematics, as to Astronomy; only with still greater urgency. In mathematics we find the primitive source of rationality; and to mathematics must the biologists resort for means to carry on their researches. If biologists have hitherto not done this, but contented themselves with what is called logic, apart from all determinate reasoning, much of the fault is chargeable upon the indifference of geometers about duly organizing the whole of

mathematical knowledge. The imperfect and inadequate character of the elementary treatises on mathematics that have hitherto been given to the world quite accounts for the neglect of the fundamental logical properties of mathematical science by even intelligent minds. It accounts also for the exaggerations of some philosophers, who maintain that, far from preparing the intellectual organ for the rational interpretation of nature, a mathematical education rather tends to develop a spirit of sophistical argumentation and illusory speculation. Such an abuse, however, cannot affect the real value of mathematics as a means of positive education; but rather exhibits the necessity of a philosophical renovation of the whole system of mathematical instruction. Whatever advantage can be attributed to logic in directing and strengthening the action of the understanding is found in a higher degree in mathematical study, with the immense added advantage of a determinate subject, distinctly circumscribed, admitting of the utmost precision, and free from the danger which is inherent in all abstract logic,-of leading to useless and puerile rules, or to vain ontological speculations. The positive method, being everywhere identical, is as much at home in the art of reasoning as anywhere else: and this is why no science, whether biology or any other, can offer any kind of reasoning, of which mathematics does not supply a simpler and purer counterpart. Thus, we are enabled to eliminate the only remaining portion of the old philosophy which could even appear to offer any real utility; the logical part, the value of which is irrevocably absorbed by mathematical science. Hither, then, must biologists come, to study the logical art so as to apply it to the advancement of their difficult researches. In this school must they learn familiarly the real characters and conditions of scientific evidence, in order to transfer it afterwards to the province of their own theories. The study of it here, in the most simple and perfect cases, is the only sound preparation for its recognition in the most complex.

The study is equally necessary for the formation of intellectual habits; for obtaining an aptitude in forming and sustaining positive abstractions, without which the comparative method cannot be used in either anatomy or phy

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USE OF SCIENTIFIC FICTIONS.

37

siology. The abstraction which is to be the standard of comparison must be first clearly formed, and then steadily maintained in its integrity, or the analysis becomes abortive and this is so completely in the spirit of mathematical combinations, that practice in them is the best preparation for it. A student who cannot accomplish the process in the more simple case may be assured that he is not qualified for the higher order of biological researches, and must be satisfied with the humbler office of collecting materials for the use of minds of another order. Hence arises another use of mathematical training;-that of testing and classifying minds, as well as preparing and guiding them. Probably as much good would be done by excluding the students who only encumber the science by aimless and desultory inquiries, as by fitly instituting those who can better fulfil its conditions.

Use of scien

tific fictions.

There seems no sufficient reason why the use of scientific fictions, so common in the hands of geometers, should not be introduced into biology, if systematically employed, and adopted with sufficient sobriety. In mathematical studies, great advantages have arisen from imagining a series of hypothetical cases, the consideration of which, though artificial, may aid the clearing up of the real subject, or its fundamental elaboration. This art is usually confounded with that of hypotheses; but it is entirely different; inasmuch as in the latter case the solution alone is imaginary; whereas in the former, the problem itself is radically ideal. Its use can never be in biology comparable to what it is in mathe

matics: but it seems to me that the abstract character of the higher conceptions of comparative biology renders them susceptible of such treatment. The process would be to intercalate, among different known organisms, certain purely fictitious organisms, so imagined as to facilitate their comparison, by rendering the biological series more homogeneous and continuous: and it might be that several might hereafter meet with more or less of a realization among organisms hitherto unexplored. It may be possible, in the present state of our knowledge of living bodies, to conceive of a new organism capable of fulfilling certain given conditions of existence. However that may be, the