Any quantity, commensurable or incommensurable, can be expressed uniquely as a simple continued fraction, terminating in the case of a commensurable quantity, non-terminating in the case of an incommensurable quantity.
When we say that an asteroid's period iscommensurable with that of Jupiter, we mean that a certain whole number of the former is equal to another whole number of the latter.
The mathematical continuum of one dimension admitted of a scale whose divisions, infinite in number, corresponded to the different values, commensurable or not, of one same magnitude.
Evidently there is not in the first class a number less than all the others, for, however near the square of a number may be to 2, we can always find a commensurable number whose square is still closer to 2.
As an abbreviation, let me call a mathematical continuum of the first order every aggregate of terms formed according to the same law as the scale of commensurable numbers.
Let us see now what Dedekind's definition is: The commensurable numbers can in an infinity of ways be partitioned into two classes, such that any number of the first class is greater than any number of the second class.
Suppose, for example, we put in the first class all commensurable numbers whose squares are greater than 2 and in the second all whose squares are less than 2.
Stated briefly, dimension as commensurable quality makes the existence of the fourth dimension a transcendental problem, but as mere direction, an empirical problem.
It is now understood that conceivably only in the subjective world, and in theory and the imagination, do we deal with identically similar units, and with absolutely commensurable quantities.
Days, months, and years are not exactly commensurable with each other.
In a commensurable manner; so as to be commensurable.
B 600 (permutando, and assuming a value for A and B, so as to make them commensurable with the respectiy sums).
In arithmetic he was the first to expound the theory of means and of proportion as applied to commensurable quantities.
The main proposition, that two magnitudes balance at distances reciprocally proportional to the magnitudes, is proved first for commensurable and then for incommensurable magnitudes.
In geometry Eudoxus discovered the great theory of proportion, applicable to incommensurable as well as commensurable magnitudes, which is expounded in Euclid, Book V.
Taking into consideration that the law of Gay-Lussac holds good, not only for elements, but also for compounds, it should be expressed as follows: Substances interact with one another in commensurable volumes of their vapours.
This assumption is involved in the very notion of Maximum Happiness; as the attempt to make 'as great as possible' a sum of elements not quantitatively commensurable would be a mathematical absurdity.
The difference between necessary and contingent truths is indeed the same as that between commensurable and incommensurable numbers.
For the reduction of commensurable numbers to a common measure is analogous to the demonstration of necessary truths, or their reduction to such as are identical.
Of course, with a bright class a teacher may well afford to take it as it is given in the textbook, but the important thing is that the commensurable case should be proved and the incommensurable one recognized.
It is a little easier to start with the hexagon, however, for we are already nearer the circle, and the side and perimeter are both commensurable with the radius.
This is usually proved first for the commensurable case and then for the incommensurable one.
In the modern treatment by limits the proof is divided into two parts: first, for commensurable bases; and second, for incommensurable ones.
It should be said, however, that it is scientifically correct, that it covers the case of incommensurable magnitudes as well as that of commensurable ones, and that it is the Greek forerunner of the modern theories of irrational numbers.
The one reduces the primitive animistic world to the lower end of its scale, the other construes it in terms of a purposive utility commensurable with that of human action.
Conversation, as we know, denotes an interchange of commensurable meanings.
In order to make his data commensurable with the phenomena of nature, he discovers or defines bodily conditions for the subjective content which he analyzes.
Suppose now a particle or planet close to the commensurable point inside it.
Free from the rivalry of crowded furnishings, men and women take on a quite singular quality of dignity and importance.
What we are speaking of are the deep impressions, which cannot properly be made commensurable at all, which may spring up directly out of an inward experience, an apprehension of nature, the world and history, in the depths of the spirit.
The clear-cut, luminous, conception of the world which expresses everything in terms of commensurable concepts is thoroughly Aristotelian.
In the real world it is reasonable to suppose we deal at most with practically similar units and practically commensurable quantities.
Gold and milk must be, then, commensurable quantities, i.
Now a quantitative ratio is between commensurable quantities.
On the other hand, since commodities acquire only in price the form of exchange value with respect to one another, he makes them commensurable through money.
He gets out of the difficulty by making commensurable through money what is in itself incommensurable, so far as it is necessary for practical purposes.
The breadth of this zone is such as to contain several portions in which the periods of asteroids would be commensurable with that of Jupiter.
Portions of the ring in which the periods of asteroids would be commensurable with that of Jupiter.
The interval, therefore, occupies precisely the space in which the periods would be commensurable with those of the four members of the system immediately exterior.
First, the error of supposing that the three dimensions of length, breadth, and height, are really commensurable with one another.
