Our humanity cannot annul the incommensurable sorts of good that may be pursued in the world, though it cannot itself pursue them.
It would appear also that he considers the rotations of the Moon and Sun to beincommensurable with each other, both of them being members included in the Circle of the Different.
In the Platonic city, both arithmetic and geometry will be taught, so far as to guard the youth against absurd blunders about measurement, and against confusion of incommensurable lines and spaces with commensurable.
He might also fancy that even the attempt to meddle with the problem betokened that confusion of the incommensurable with the commensurable, which he denounces in this very treatise (vii.
Do not let us be deceived by this appearance: it signifies only that language is incommensurable with thought, that speech admits of endless multiplication in approximations incapable of exhausting their object.
These bordered on one of the water-highways of civilisation; a great fleet passed annually in their view, and of the shipwrecks of the world they were the scene and cause of a proportion wholly incommensurable to their size.
For he has not told everything; this incommensurable region, which had hitherto remained unworked, is far from being exhausted.
No; we are far from having made the tour of this miraculous and incommensurable kingdom through which this admirable master leads us, and I should never be done were I to attempt to exhaust all the spectacles which he offers us.
It easily follows that the continued fraction is incommensurable if a/b, c/d, etc.
She was outside--an adventure--a perplexing incommensurable with all these things.
Now he was enormously exercised in the search for a reconciliation between these, he felt, incommensurable factors.
This includes the incommensurable case, but this case may be omitted.
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.
It is also apparent that even if the incommensurable cases are generally omitted, the notion of limit is needed at this time, and that it must briefly be reviewed before proceeding further.
It seems to reduce the propositions to be proved in plane geometry to about one hundred fifteen, and it recommends the omission of the incommensurable case.
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.
They were also the first to study seriously the incommensurable number, and with this study the treatment of proportion from the standpoint of rational numbers lost its scientific position with respect to geometry.
The second definition is intended to exclude zero and infinite magnitudes, and to show that incommensurable magnitudes are included.
Somewhat related to the preceding paragraph is the fact that the edges of the five regular solids are incommensurable with the radius of the circumscribed sphere.
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.
We may start with a square, for example, but this is not so good as the hexagon because its side is incommensurable with the radius, and its perimeter is not as near the circumference.
The notion of the rational number seeming to them to present no difficulty, they have chiefly striven to define the incommensurable number.
If we afterwards intercalate new steps according to the law of formation of incommensurable numbers, we shall obtain what we will call a continuum of the second order.
We have made hitherto only the first stride; we have explained the origin of continua of the first order; but it is necessary to see why even they are not sufficient and why the incommensurable numbers had to be invented.
That would not yet be sufficient, because we should get in this way only certain incommensurable numbers and not all those numbers.
So for the incommensurable number, the vague idea of continuity, which we owe to intuition, has resolved itself into a complicated system of inequalities bearing on whole numbers.
The incommensurable of Pythagoras and the paradoxes of Zeno present the "no thoroughfares" of ancient mathematical thought.
And these "more important" consequences are important because there appears to lie in them the possibility of a type of personal character divergent from the present type and from any present point of view incommensurable with it.
Enjoyment, which some people call criticism, is something aesthetic, spontaneous, and irresponsible; the aesthetic perfection of anything is incommensurable with that of anything else.
Their several fruits become incommensurable in beauty and in value, like the poetry of different languages, and more disparate the more each is perfected after its kind.
Furthermore, incommensurable numbers are defined as the limits arrived at as the result of certain procedures with rational numbers.
Defn: Any number, proper or improper fraction, or incommensurable ratio.
Yet the state of consciousness overflows the intellect; it is indeed incommensurable with the intellect, being itself indivisible and new.
The free act is incommensurable with the idea, and its "rationality" must be defined by this very incommensurability, which admits the discovery of as much intelligibility within it as we will.
Before we name this other spectator, let us pause here a moment in order to recall our own impression, as previously described, of the discordant and incommensurableelements in the nature of Æschylean tragedy.
No,' andincommensurable variations in saying 'Yes.
The new event can wake but partial echoes in his soul or none at all; it can neither be received into, nor can it create a complete relation, and so it passes incommensurable from limbo into forgetfulness.
The essential quality of great artists is incommensurable with biography; they seem to be unconsciously engaged in a perpetual evasion of the event.
Thus, if you are seeking to prove that the diagonal is incommensurable with the side, you may put as a question, Is not the side incommensurable with the diagonal?
Through them it is that we detect the enormous but incomprehensible unity which underlies this incommensurable multiplicity.
If incommensurable suns swung high overhead, he at least was the centre of his own little world, and not the most astounding facts of science could alter or remove that egocentric view.
And qualitatively as well as quantitatively the effect may be absolutely incommensurable with the cause.
The causes which operate in these incommensurable cycles are connected with one another only if we take the whole universe into account.
Sidenote: Alexander was to all the theorists an incommensurable quantity.
The main proposition, that two magnitudes balance at distances reciprocally proportional to the magnitudes, is proved first for commensurable and then for incommensurable magnitudes.
This was, doubtless, first discovered with reference to the diagonal of a square which is incommensurable with the side, being in the ratio to it of [root]2 to 1.
In geometry Eudoxus discovered the great theory of proportion, applicable to incommensurable as well as commensurable magnitudes, which is expounded in Euclid, Book V.
Lastly, the Pythagoreans discovered the existence of incommensurable lines, or of irrationals.
