that is, the distinction between the mathematics and the objects of Natural Philosophy.

10. The objects of Natural Philosophy are the facts or data of the science. The knowledge of these is only to be obtained by observation. Jupiter placed at a certain distance from the sun, and moving in a certain direction, and with a certain velocity, is an object. His satellites, with their positions and their motions, are also so many objects. Any piece of matter, including those attributes which it is the part of Natural Philosophy to take cognizance of, such as weight, and magnitude, and movement, and situation, is an object of this science. Altogether they form what may be called the individual and existent realities of the science. And Lord Bacon has done well in having demonstrated that for the knowledge of these we must give ourselves up exclusively to the informations of experience; that is, to obtain a knowledge of the visible properties of material things we must look at them, or of their tangible properties we must handle them, or of their weights or motions or distances we must measure them.

11. Thus far, then, do the applications of the Baconian Philosophy go, and no farther. After that the facts or objects of the science have in this way been ascertained, we perceive certain mathematical relations between the objects from which we can derive truths and properties innumerable. But it is not experience now which lights us on from one truth or property to another. The objects

meaning, as comprehensive of the duties owing to God in heaven, as well as to our fellow-men upon earth.

or data of the science are ascertained by the evidence of observation; but the mathematics of the science proceed on an evidence of their own, and land us in sound and stable mathematical conclusions, whether the data at the outset of the reasoning be real or hypothetical. The moral proprieties founded on equity between man and man would remain like so many fixtures in ethical science, though the whole species were swept away, and no man could be found to exemplify our conclusions. The mathematical properties founded on an equality between line and line would in like manner abide as eternal truths in geometry, although matter were swept away from the universe, and there remained no bodies whose position or whose distances had to be reasoned on. It has been already said that we do not need to extend the domain of observation in order to have a clear and a right notion of the moral proprieties; and it may now be said that we do not need to extend the domain of observation in order to have a clear and a right notion of the mathematical properties. If straight lines be drawn between the centres of the earth and the sun and Jupiter, they would constitute a triangle, the investigation of whose properties might elicit much important truth on the relations of these three bodies. But all that is purely mathematical in the truth would remain, although it were not exemplified, or although these three bodies had no existence. Nay, the triangle might serve as the exemplar of an infinity of triangles, which required only a corresponding infinity of objects, in order that the general and abstract truth might become the symbol or

representative of an endless host of applicable and actually existent truths. For the objects of both sciences you must have inductive or observational evidence; but by a moral light in the one science, and a mathematical light in the other, we arrive at the ethics of the first science, at the mathematics of the second, without the aid of the inductive philosophy.

12. It is interesting to note if aught may have fallen from Lord Bacon himself upon this subject. In his English treatise on "the advancement of learning," he says, "that in mathematics I can report no deficience." So that this great author of the experimental method by which to arrive at a true philosophy of facts, had no improvement to propose on the methods of mathematical investigation. And in his more extended Latin treatise on the same subject, entitled, "De augmentis scientiarum," where he takes so comprehensive a view of all the possible objects of human knowledge, he says, speaking of geometry and arithmetic, “Quæ duo artes, magno certe cum acumine, et industria, inquisitæ et tractatæ sunt: veruntamen et Euclidis laboribus in geometricis nihil additum est a sequentibus quod intervallo tot seculorum dignum sit;" or "which two arts have certainly been investigated and handled with much acuteness and industry; notwithstanding which, however, nothing has been added to the labours of Euclid in geometry by those who have followed him, that is worthy of so long a series of ages."

13. The proper discrimination then to be made in natural philosophy, is between the facts or data of the science, and the relations that by means of

mathematics might be educed from these data. The former are ascertained by observation-after which no further aid is required from observation, while we prosecute that reasoning which often brings the most weighty and important discoveries in its train. It is well to consider how much can be achieved by mathematics in this process, and how distinct its part is from that of wide and distant observation; insomuch that by the light which it strikes out in the little chamber of one's own thoughts, we are enabled to proceed from one doctrine and discovery to another. From three distant points in the firmament, a triangle may be formed to which the very mathematics are applicable that we employ upon a triangle constructed upon paper by our own fingers. Whether they be the positions and the distances that lie within the compass of a diagram, or the positions and distances that obtain in wide immensity, it is one and the same geometry which, from a few simple and ascertained data, guides the inquirer to the various and important relations of both. After that observation hath done its office, and made over to mathematics the materials which it hath gathered--this latter science can guide the way to discoveries and applications innumerable; and without one look more upon the heavens, with nought but the student's concentrated regard on the lines and the symbols that lie in little room upon his table, might the whole mystery and mechanism of the heavens be unravelled.

14. Let those things, then, be rightly distinguished which are distinct from one another. They were not the objects of the science which gave the

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observer his mathematics. These objects were only addressed to his previous and independent mathematics; and he, in virtue of his mathematics, was enabled rightly to estimate many important relations which subsisted between the objects. Nay, it is conceivable that the objects might have remained for ever obscure and unknown to him. He, in this case, would have wanted an application which he now has for his mathematics; but the mathematics themselves would have been still as much within his reach or his power of acquisition as before. His mathematical nature, if we may so speak, would have been entire notwithstanding; and he have had as clear a sense of the mathematical relations, and as prompt and powerful a faculty of prosecuting these to their results. Things might have been so constituted, as that every star in the firmament should have been beyond the discernment of our naked eye; or what is still more conceivable, the lucky invention might never have been made by which the wonders of a remoter heavens have been laid open to our view. But still they were neither the informations of the eye nor of the telescope which furnished man with his geometry; they only furnished him with data for his geometry. And thus, while the objects of astronomy are brought to him by a light from afarthere enters, as a constituent part of the science, the mathematics of astronomy, immediately seen by him in the light of his own spirit, and to master the lessons of which he needs not so much as one excursion of thought beyond the precincts of his own little home.