Thus the illusory nature of the traditional definition of mathematics is again illustrated.
The conditions of peace were naturally humiliating for Valdemar,' though, ultimately, he contrived to render illusory many of the inordinate privileges he was obliged to concede.
They borrowed his money and his armies, but fed him with vain promises and illusory treaties.
For a long time this hope proved illusory, and in the case of Egyptian archaeology the results have proved disappointing even up to the very present.
He had long passed the traditional years of Peter's pontificate, had reigned longer than any previous wearer of the tiara, and had seen some brilliant days - days of illusory glory.
The management of finance was scarcely satisfactory, for though Giolitti, who had succeeded Magliani and Perazzi at the treasury, suppressed the formers illusory pension fund, he lacked the fibre necessary to deal with the enormous deficit of nearly 10,000,000 in 1888-1889, the existence of which both i Perazzi and he had recognized.
By abolishing the illusory pensions fund, by applying and amending the Bank Laws, effecting economies, and increasing taxation upon corn, incomes from consolidated stock, salt and matches, he averted national bankruptcy, and placed Italian finance upon a sounder basis than at any time since the fall of the Right.
P. 444), he is ready to amend nervous into psychical shocks, which is no doubt what he ought to have meant but could not say without ruining the illusory bridge between the psychical and the physiological which is suggested in the phrase nervous shock."
Comparisons of light grasp derived from small, fresh, carefully silvered surfaces are sometimes given which lead to illusory results, and from such experiments Foucault claimed superiority for the silvered speculum over the object-glass.
A composition committee was actually formed, but it proved illusory from the first, the patriotism of neither of the factions being equal to the puniest act of selfdenial.
Progress is illusory; there is no satisfactory goal to which moral development inevitably tends; religion in which some take refuge when distressed by the inexplicable contradictions of moral conduct itself " contains and rests upon an element of make believe " (p. 489).
Further protests were made by many of the powers when the illusory nature of these concessions became known, and it was not until May 1906 that diplomatic relations with Servia were resumed by Great Britain.
Disdaining the illusory dream of the imperial crown, still cherished by his legal advisers, he pushed forward towards that fluctuating eastern frontier, the line of least resistance, which would have yielded to him had it not been for the unfortunate interruption of the Hundred Years War.
Moreover, Schopenhauer's subjective idealism, and his view of time as something illusory, hindered him from viewing this process as a sequence of events in time.
But this concession was illusory, and as the statute prevented Jews from engaging in finance - the only occupation which had been open to them - it was a prelude to their expulsion in 1290.
The rent stipulated for must not be illusory, and must fairly represent the value of the subjects leased, and the term of the lease must not be excessive (as to rent generally, see Rent).
And though Spencer's general position - that it is absurd to suppose that organisms after being modified by their life should give birth to offspring showing no traces of such modifications - seems the more philosophic, yet it does not dispose of the facts which go to show that most of the evidence for the direct transmission of adaptations is illusory, and that beings are organised to minimize the effects of life on the reproductive tissues, so that the transmission of the effects of use and disuse, if it occurs, must be both difficult and rare - far more so than is convenient for Spencer's psychology.
+Zn(n -I) a n_ 1= Zn(n+3) -2 (without which the transformation would be illusory); and the conclusion is that a 1, a 2, ...
If our belief in necessary connexion in the physical world is in reality an illusion, it follows that the opposition between freedom and necessity will be illusory also.
Contrasted with this belief in necessity the supposition we have of freedom is illusory, and, if extended so as to involve a belief that men's actions do not proceed from character or habitual disposition, immoral.
The phenomena which deceived us may continue to persist, but they no longer persist as illusory: the appearance which deceived us is seen in its true nature, even though it should still retain those characteristic marks or signs of reality which hitherto we regarded as significant of a nature which we now no longer believe it to possess.
Expert knowledge and judicial insight must decide the point; but, so far as the present writer can judge, it is illusory to imagine that Duns points us beyond the medieval assumptions.