Table of Temperatures corresponding to the Pressures of Steam in Atmospheres.

Pressures in Atmo. Temperatures in deg. of Pressures in inches of

ART. XXX.-Considerations on the Divisibility of Magnitude; by ALEXANDER MAC WHORTER, New Haven.

GEOMETRY is the science of magnitude; and magnitude, has been defined to be, that which has one or more of the three dimensions, length, breadth, and thickness. The conception denoted by the term magnitude, to which this science has reference, is an abstract general conception, notion, or idea, and like all other abstract notions is formed, first, by making a concrete thing or a real existence, an object of thought; next, by distinguishing in mental analysis between different characteristics of this real existence; and finally, by giving to the several characteristics thus separated and set apart in thought, names, so far as the case in hand may require. In the present instance, the concrete or real existence made the object of thought is a body, as that which occupies space. The characteristics distinguished by mental analysis and SECOND SERIES, Vol. VI, No. 18.-Nov., 1848.

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set apart under distinct names, are the three dimensions of body, length, breadth, and thickness; while to these characteristics, taken either separately or in combination, the name of magnitude has been assigned.

Thus the conception denoted by the term magnitude has for its object, like every other abstract general conception, some only of the characteristics of a concrete or existing reality, which object though a mere object of thought, corresponds with what is real, but not with all that is real in the concrete reality. That which is real in every existing body and to which the object of the abstract general conception corresponds, is the dimensions of that body, to which of course belongs an existence distinct and separate both from the conception itself and from the mind which forms the conception.

Space as fitted to be occupied by body, may be considered as extension in three directions, and thus as having three dimensions, and if the term magnitude be used to denote these three characteristics of space, taken either separately or in combination, the conception denoted by the term magnitude has also, in this instance, for its object only some of the characteristics of a concrete or existing reality, and the reality to which this object corresponds, is the dimensions of space, to which of course belongs an existence which is necessary, and which at the same time is distinct and separate both from the conception itself and from the mind which forms the conception. The distinction here made between the objects of mental conception and real existences, or existences which are such independent of our thinking, though quite obvious, is of the first importance, and may be made more apparent by considering the wide difference between those objects of thought, which are constituted what they are by the nature and action of the mind, and things, which are what they are in virtue of a constitution of their own; known by our minds, to a certain extent, but still, existing independent of our thinking.

It is also obvious that when a general term, such as magnitude, is used to denote an abstract general conception, the object denoted by this term, in these circumstances, is subject solely to the laws which control and determine the nature and relations of mere thoughts, that is, to the laws of thinking; while if the same general term be used to denote either an object, or some property of an object, which has a real existence independent of our thinking, then the object denoted by this term, under these circumstances is subject to the laws which control and determine the nature and relations of real existences, that is, to the laws of nature. It is because philosophers have failed to carry out this distinction consistently, and thus have overlooked and denied their own knowledge, that so much imposing logic has been at war with common sense. Failure on this very point constitutes the grand and vital error of Hegel, with whom "thoughts are the only concrete reali

ties, and logic, as being a true description of their processes, is at the same time a true description of the Laws of the Universe." In the confusion between thoughts and things is found the central fallacy of his Idealistic Pantheism, in which "the process of the evolution of ideas in the human mind is, at the same time, the process of all existence," in a word, in which "the dialectic process is the Method, the dialectic process the Deity, the dialectic process everything;" a system no less remarkable for the unsoundness of its premises, than for the logical rigor and scientific beauty with which it marches forward to its fatal results.

With the preceding distinction in view, the way is opened for a decision of the much vexed question respecting the divisibility of magnitude. In every age, from that of the earliest skeptic to the present moment, the contradictions on this subject, which the reasoning of geometry seems fairly to involve, have furnished weapons by which the certainty of human knowledge has been assailed; leaving room for a doubt, which the defenders of truth have appeared more willing to avoid than directly to meet; very many of them seeming to think, that the solution of the problem included something beyond the limits of the human understanding.

The difficulty referred to is this. It is well known to the geometrician that according to the principles of mathematical reasoning, every line representing magnitude is divisible beyond any assignable limit; that is, let any magnitude or line representing magnitude, be given, a magnitude can always be found, less than the given magnitude. For, let any magnitude whatever, be supposed, and, for the sake of argument, let it be considered as the least conceivable. This magnitude is some assignable distance. It can therefore become the radius of a circle. But* "if a point be taken in the diameter of a circle which is not the centre, of all the straight lines which can be drawn from it to the circumference, the greatest is that in which the centre is, and the other part of the diameter is the least."

That is, a point being taken in the radius which is neither at the centre nor at the circumference of the circle, a magnitude can be cut off between this point and the circumference, (or this point and the centre,) less than radius. But, by supposition, radius was the least conceivable magnitude, a magnitude therefore has been found less than the least conceivable. Now this process can be repeated as often as a new radius is taken, so that, on the principles of geometry, magnitude is divisible beyond any assignable limit, in other words, magnitude is indefinitely, or as it is commonly called, infinitely divisible.

On the other hand, it is urged by the natural philosopher as well as by common sense, that there is a world which exists independent of our conceptions, governed by its own laws, which

are not the laws of logic but the laws of nature; and that in this world are found a series of facts as certain, and which involve consequences as sure, as are the conclusions deduced by the geometrician from his definitions, postulates and axioms. The natural philosopher considers the existence of body and space, of distance and motion, to be among such facts; he knows by observation and experiment, that bodies moving in opposite directions through space sometimes pass by each other, that a body moving in the same direction with another body will sometimes overtake it, and he believes, that the opposite faces of approaching solids sometimes come in contact.

He also reasons, that if a body is in motion through space toward another body at any given distance from it, and if these two finally come in contact, then antecedent to such contact there must be a least intervening distance not occupied, or filled up, by the approaching body. In other words, that when the bodies are in contact no distance intervenes, and when not in contact some distance intervenes; distance, in the case supposed, being the interval between the opposite faces of each solid unoccupied by the presence of either body; while, of all the distances which can be supposed to intervene, the least is that next antecedent to contact. The same consequence follows in respect to bodies moving in opposite directions, and also in respect to bodies which overtake each other. If Achilles overtake the tortoise, the greyhound the hare, or the minute-hand the hour-hand of a clock, then there must, in the nature of things, be a least intervening distance next antecedent to each event. Should the distance thus limited be denoted by the term magnitude, then this will be the least magnitude which, in the nature of things, can be conceived to be, that is, can be conceived as having a real as opposed to an ideal existence. And if there be a least distance or length, then for the same reason, there must be a least breadth and depth, in other words, a least space; and if there be a least space, then for anything we know to the contrary respecting the nature of matter, a body may be conceived as occupying this least space, and this will be the least body, the least impenetrable solid which, in the nature of things, can be conceived to be, that is, be conceived as having a real as opposed to an ideal existence.

It follows from this, first, that every body which has a real existence, if it be a definite body such as a tree, the globe or a star, must be made up of indivisible units, or impenetrable and extended particles having a form; and next, that the magnitude of these particles can in no case be less than the least space; in other words, that the ultimate particles of a body can in no case be actually reduced so as to become less than the magnitude, which, by reasoning from the nature of actual extension, has been shown to be the least conceivable, that is, the least which can be conceived to have a real as opposed to an ideal existence.

There is an apparent objection to the existence of a least distance, or least magnitude, turning upon an ingenious fallacy respecting motion, which may as well be resolved at this point.

It appears from experiment that bodies move with unequal velocity, that is, traverse equal spaces in unequal times; time being measured by some common standard. Now, let two bodies traverse a given distance, greater than the least, with unequal velocity, that is, in unequal times; on the hypothesis of continuous motion, other things being equal, since their velocities differ, they must traverse some one least space or least distance, of which the whole distance is the sum, in unequal times; time being measured by some standard common to both. But to traverse the least distance is to make the least motion, or the least change of place, which, in the nature of things, can be; consequently, the bodies supposed make a least motion or a least change of place, which in the nature of things, can be, in unequal times. And if the bodies supposed make a least motion in unequal times, then bodies of unequal velocity in making some one least motion or least change of place, while their velocity remains unequal, require unequal times as necessary to the existence of this least motion or least change of place, in each particular case. And if bodies of unequal velocity require unequal times as NECESSARY TO THE EXISTENCE of some one least motion or least change of place, then this least motion cannot be supposed to occur in any time less than the one necessary to its existence in the particular case, without supposing a contradiction; that is, without supposing that a body moving with a given velocity requires a certain time to make some one least motion; this time being necessary in the nature of things to the existence of the motion; and then to suppose, that the same body, moving with the same velocity, can make the same least motion in a time less than the one necessary in the nature of things to the existence of that motion, which is absurd. Yet this is the fallacy at the basis of the various and celebrated sophisms used by the ancients against the possibility of motion. It generally comes in the shape of an interrogation, thus:

On the hypothesis of continuous motion, other things being equal, suppose two bodies moving with unequal velocity, and suppose the one moving with the greater velocity to have traversed a least distance by making the least motion or the least change of place. What has the body moving with the less velocity, and consequently requiring more time as necessay to the existence of its least motion, done;-how far has it moved, during the time the first body moved? Now to ask this question, is simply to ask, what has a body done, how far has it moved, when one of the conditions necessary to the very existence of its least motion, viz. the requisite time is, by supposition, NON-EXISTENT. In other words, it is to ask, if a contradiction be true, what consequences