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will follow, it being the purpose of the inquirer to use the conclusions drawn from such a premise, in order to overthrow some other conclusion resting on independent evidence. So far then as this reasoning is concerned, the circumstance that existing bodies have different velocities, furnishes nothing to disprove the existence of a least distance or a least magnitude.

In fact, the proposition that every finite body is definitely divisible, in other words, is so limited by its finite nature as to be reducible only to a certain extent, is a proposition which would never have been questioned had philosophers paid more attention to the nature of the objects of thought which the premises of geometry involve; since an attentive consideration of these premises will show, that the conclusions of the geometrician respecting the infinite divisibility of magnitude do in no way conflict with the decisions of natural philosophy and common sense, but, on the contrary, are simply the enunciation of what is true in respect to the relations in which the objects of certain abstract general notions stand to each other, as deduced solely from what belongs of necessity to their nature as mere objects of thought.

The distance just described has been shown to be the least magnitude which, in the nature of things, can be conceived to be. Now things have been very beautifully classified into things existing as mere thinks, that is, as mere objects of thought; and things which are real existences, whether we think of them or not. Thus, in the phrase "the least magnitude which in the nature of things can be conceived to be," the term magnitude may denote either the object of an abstract conception, or a characteristic of a real existence, viz. a dimension of space; that is, it may denote either a mere object of thought or a real existence; and the phrase "nature of things" is equally ambiguous, and may denote either the nature of mere objects of thought or the nature of real existences. Now if the term magnitude be taken as denoting a real as opposed to an ideal magnitude, that is, as denoting a dimension of space, to which belongs an existence distinct and apart from the mind's conception of it, and the phrase "nature of things" as denoting the nature of real existences, then manifestly there is a least magnitude which in the nature of things can be conceived to be, or can be conceived to have a real as opposed to an ideal existence, viz. that distance next antecedent to the contact of bodies.

If, on the other hand, the term magnitude be taken as denoting the object of an abstract conception, that is, be taken as denoting a mere object of thought; and the phrase "nature of things" as denoting the nature of thinks, that is, the nature of mere objects of thought, then it is equally true that magnitude must be conceived as infinitely divisible, or more properly, as divisible beyond any assignable limit.

In this case three questions arise; first, what is the meaning of the term divisible when applied to mere objects of thought? secondly, what is the nature of the conceptions denoted by the terms mathematical line and mathematical point? and thirdly, what is it to divide a mathematical line at a mathematical point?

Respecting the first inquiry, it is evident, that while to divide means primarily to dispart, or disjoin, when material existences are spoken of, to divide when used in reference to things which are mere objects of thought, must mean simply to distinguish between them as mere objects of thought; so that, in this case, infinitely divisible means merely infinitely distinguishable, or distinguishable beyond any assignable limit.

In the next place a line, by definition, denotes the object of the abstract conception of length or distance, considered as a mere object of thought, apart from the abstract conceptions of breadth and depth; while a point is defined by Legendre to be, “a limit terminating a line."

In this definition of a point the abstract notion of length, represented by a line, is considered as ceasing or ending, and this notion of limit to length being a notion of the cessation and consequent negation of length, it is a conception which, in its own nature, excludes from itself the conception thus denied. It is in this view that Euclid defines a point to be "that which hath no parts or which hath no magnitude;" a definition imperfect because not convertible, everything "having no parts or no magnitude" not being a point.

But as in things a given distance is in actual space, that is, sustains the relation to actual space denoted by the terms place, situation, or position, so in mere objects of thought the coëxisting conceptions of abstract length, breadth, and thickness, are viewed as holding such relations to each other, as to authorize the application of the term place, or position, borrowed from real existences and applied to these conceptions in order to denote a particular relation between them. Now as mere objects of thoughts are not in space, the only relation which they can sustain to each other at all resembling the relations subsisting between actual existences in respect to place, is THE RELATION OF LIMITATION, the term limitation when applied to mere objects of thought denoting that the given conception is just what it is, AND NOT ANOTHER conception.

Thus, in answer to the second inquiry, it appears that a mathematical line is merely the object of the abstract conception of length or distance; and in respect to a mathematical point it also appears, first, that it is not a thing "in re" and in this sense a real existence, but on the contrary, is a mere object of thought formed by the mind, analytically, in accordance with the laws of thinking; and next, that this object of thought is limit as applied to the dimensions of body or of space when these are viewed as objects of abstract conception.

Now since to divide means to distinguish, when mere objects of thought are spoken of; in answer to the third inquiry it appears, that to divide a magnitude represented by a line, at a mathematical point, means,-to distinguish one magnitude viewed as a mere object of thought from another magnitude viewed as a mere object of thought, by the notion of limitation, as the differentia ;and then, to represent this distinction by the division of an actual and visible line. Thus, let any two magnitudes be conceived, and let them be made distinct objects of thought; each object is distinguished from the other by being conceived to be just what it is, as an object of thought, and, at the same time, being conceived to be NOT THE OTHER object. Now this mode of distinguishing between mere objects of thought can be repeated just as often as any two conceptions of magnitude can be formed, each of which is not the other, that is, can be repeated indefinitely ; and this distinction between objects of thought can be represented by the division of actual lines;-this difference between these objects being simply numerical as mere objects of thought, having no real or substantial existence. So that in this sense, magnitudes, as mere objects of thought, are distinguishable, and in this sense only is magnitude divisible beyond any assignable limit. The fallacy involved in the mathematical argument against the possibility of a least conceivable magnitude consists in assuming, without a shadow of warrant, that the least conceivable magnitude may be divided at a mathematical point. For the phrase, "least conceivable distance" denotes an actual distance, a concrete or existing reality. The term mathematical point, on the contrary, denotes merely the object of an abstract general conception, notion, or idea. This object is also a mere object of thought, having no counterpart or corresponding object, among real existences. Now to suppose that a mere object of thought, which has no corresponding object "in re," can divide a concrete or existing reality, and thus diminish it by partition, is to suppose, what is impossible in the known nature of things; it is to suppose, a real existence divided by a mere object of thought, the properties of a body or of space divided by an abstract idea; which is absurd. Moreover, even as a mere object of thought, a point, by definition, has no extension or no magnitude, being simply the notion of a limit, by which one extension is conceived of as not another. Can then extension be divided by that which has no extension, and magnitude be diminished by that which has no magnitude? Plainly, to divide means to distinguish, when mere objects of thought are spoken of; and it is in this sense, and in this sense only, that magnitude, which, when the subject of geometrical reasoning, is considered merely as an object of thought, is indefinitely or as it is usually, but improperly termed, infinitely divisible.

Thus by reasoning in one mode it appears that there must be a least conceivable magnitude; that is, a least magnitude which,

in the nature of things, can be conceived to have a real as opposed to an ideal existence. Here the term magnitude denotes existing dimensions of body or of space, and the reasoning is conducted wholly in reference to what is true respecting real existences. By reasoning in another mode it appears, that there cannot be a least conceivable magnitude, but that magnitude must be conceived as divisible beyond any assignable limit. Here the term magnitude denotes an abstract conception, whose object is a mere object of thought, and the reasoning is conducted wholly in reference to what the mind can do in forming mere objects of thought, analytically, and, when they are thus formed, in distinguishing between them.

The solution of this apparent contradiction, which has hitherto perplexed every inquiry respecting the divisibility of magnitude, turns upon the difference between things existing as mere thinks, that is, as mere objects of thought, and things which are real existences, whether we think of them or not. It is a distinction, which a German would immediately recognize as made between thought in itself—(Der Gedanke an sich), and thing in itself— (Das Ding an sich); and claims our assent not merely as true, but as the starting point of all truth, in respect to anything which can with propriety be called mental science. In the preceding discussion this principle has been applied to the solution of a single difficulty; what light a similar analysis may cast upon kindred problems may be left, at least for the present, to the reflections of the geometrician.

ART. XXXI.-Researches on Salts; by C. GERHARDT. Translated from the Journal de Pharmacie et de Chimie, t. xii: third series.

THE following experiments, form the outlines of a more extended research into the laws which govern the formation and the composition of salts. They are intended particularly to determine with precision the phenomena of the double decomposition of salts. I hope, through them, to arrive at a rigorous appreciation of what is to be understood by a neutral salt, an acid and a basic salt.

The part most neglected in the history of the salts, is undoubtedly that which relates to the basic or subsalts. Since the introduction of polybasic acids, it is no longer possible to give a precise definition of them; for we do not so readily perceive the difference which can exist between a salt said to be neutral, containing two or three equivalents of oxyd, and a salt called basic containing the same number. In most cases, the reactions do not indicate, in this respect, any difference.

SECOND SERIES, Vol. VI, No. 18.-Nov., 1848.

44

It is generally thought that the number of subsalts formed by the same acid and the same base, is much greater than that of the corresponding persalts or acid salts. So that five or six subnitrates of lead are admitted, five or six subsulphates of copper, three or four subacetates of copper, etc., and the most irregular relations are considered as occurring in the composition of these bodies.

My experiments do not give the same results. Whenever I have been able to obtain a crystallized subsalt, or it has been possible for me so to control circumstances as to avoid a mixture, the composition of the product was exceedingly simple. I have thus been able to satisfy myself that there does not exist but a single subsalt, or at most but two, for the same acid and the same base. There is one fact whose importance is not perceived at first sight, but which must nevertheless be borne in mind to prevent errors; it is the influence of masses in the double decomposition of salts. It is not a matter of indifference, when one solution is to be precipitated by another, whether we pour the first liquid into the second or the second into the first. It is stated, for example, in all the books, that potash if added to a solution of sulphate of copper precipitates the hydrate from it. The truth of the matter is this; if potash is added drop by drop to the salt of copper in such a way as to keep this latter in excess, we obtain a green sub-sulphate of constant composition; by operating in an inverse manner, we produce the blue hydrate perfectly pure. Without the precaution of always maintaining in excess the liquid into which we pour the other, we obtain nothing but mixtures. The observance of this simple rule cannot be too strongly insisted upon : very often the products of a reaction seem to have a complicated composition, when in reality they are only mixtures.

Another not less striking example of this influence of masses, is one I have noticed in the case of phosphate of soda and nitrate of lead. If we pour neutral nitrate of lead into ordinary phosphate of soda kept in excess, we shall have a flocculent precipitate of tribasic phosphate of lead anhydrous at 212° F. [PO3, 3Pb O or PO (Pb3)]; if, on the contrary, we pour the phosphate into the nitrate in excess, we shall obtain a crystalline precipitate of a new salt to which I give the name of nitro-phosphate of lead. This salt crystallizes without alteration from nitric acid, in the form of small hexagonal tables, derived from a symmetrical oblique prism, and resembling the crystals of cane sugar; it does not lose weight at 212° F.

This salt constitutes a phosphate in which half the phosphorus is replaced by nitrogen, or a subnitrate in which half the nitrogen is replaced by phosphorus:

2

Bibasic phosphate of soda, PO (Na2 H)=P2 O5, 2Na2 O, H2 O
Bibasic subnitrate of lead, N2 O (Pb2 H)=No 05, 2Pb2 O,HO
Bibasic nitrophosphate S N
SN2 Os, 2Pb2 O, HO
P2 05 2Pb2 O, H2 O

of lead,

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O1 (Pb2 H)=

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