Although Hippocrates could not determine the proportionals, his statement of the problem in this form was a great advance, for it was perceived that the problem of trisecting an angle was reducible to a similar form which, in the language of algebraic geometry, is to solve geometrically a cubic equation.
Borchers predicted that, at the high temperatures available with the electric furnace, every oxide would prove to be reducible by the action of carbon, and this prediction has in most instances been justified.
Logical Of course the first step was to approach the phenomena of human character and social existence with the expectation of finding them as reducible to general laws as the other phenomena of the universe, and with the hope of exploring these laws by the same instruments of observation and verification as had done such triumphant work in the case of the latter.
N-hedrons, is reducible to the problem: In how many ways can multiplets, i.e.
Within this essential co-ordination he distinguished three values: R-values of the environment as stimulus; C-values of the central nervous system; and E-values of human statements - the latter being characterized by that which at the time of its existence for the individual admits of being named, and including what we call sensations, &c., which depend indirectly on the environment and directly on the central nervous system, but are not, as the materialist supposes, in any way reducible to possessions of the brain or any other part of that system.
He maintained that the physical and the psychical are two orders which are parallel without interference; that the physical or objective order is merely phenomena, or groups of feelings, or " objects," while the psychical or subjective order is both a stream of feelings of which we are conscious in ourselves, and similar streams which we infer beyond ourselves, or, as he came to call them, " ejects "; that, if we accept the doctrine of evolution at all, we must carry these ejective streams of feelings through the whole organic world and beyond it to the inorganic world, as a " quasimental fact "; that at bottom both orders, the physical phenomena and the psychical streams, are reducible to feelings; and that therefore there is no reason against supposing that they are made out of the same " mind-stuff," which is the thing-in-itself.
In the petiole these strands may increase in number by branching, and thotigh usually reducible to the outline of the primitive horseshoe, more or less elaborated, they may in some of the complex polycylic dictyostelic types (Marattiaceae) be arranged in several concentric circles, thus imitating the arrangement of strands formed in the stem.
He starts from the Spinozistic thought that bodily facts and conscious facts, though not reducible one to the other, are different sides of one reality.
Demons, when they are regarded as spirits, may belong to either of the classes of spirits recognized by primitive animism; that is to say, they may be human, or non-human, separable souls, or discarnate spirits which have never inhabited a body; a sharp distinction is often drawn between these two classes, notably by the Melanesians, the West Africans and others; the Arab jinn, for example, are not reducible to modified human souls; at the same time these classes are frequently conceived as producing identical results, e.g.
The partition method of treating symmetrical algebra is one which has been singularly successful in indicating new paths of advance in the theory of invariants; the important theorem of expressibility is, directly we exclude unity from the partitions, a theorem concerning the expressibility of covariants, and involves the theory of the reducible forms and of the syzygies.
The simplest form to which the quartic is in general reducible is +6mxix2+x2, involving one parameter m; then Ox = 2m (xi +x2) +2 (1-3m2) x2 ix2; i = 2 (t +3m2); j= '6m (1 - m) 2; t= (1 - 9m 2) (xi - x2) (x21 + x2) x i x 2.
When 0=4 it is clear that no form, whose partition contains a part 3, can be reduced; but every form, whose partition is composed of the parts 4 and 2, is by elementary algebra reducible by means of perpetuants of degree 2.
When 0 _ 5, the reducible forms are connected by syzygies which there is some difficulty in enumerating.
Stroh assumes that every reducible seminvariant can in this way be reduced.
As all intellectual phenomena have by experimentalists been reduced to sensation, so all emotion has been and is regarded as reducible to simple mental affection, the element of which all emotional manifestations are ultimately composed.
If a fourth integral is obtainable, the solution is reducible to quadrature, but this is not possible except in a limited series of cases, investigated by H.
Lastly, when d-galactonic acid is heated with pyridine, it is converted into talonic acid, which is reducible to talose, an isomeride bearing to galactose the same relation that mannose bears to glucose.
Zinc blende, however, being zinc sulphide, is not directly reducible by charcoal; but it is easy to convert it into oxide by roasting: the sulphur goes off as sulphur dioxide whilst the zinc remains in the (infusible) form of oxide, ZnO.
On the other hand, if on the plan of Sigwart categorical universals were reducible to hypotheticals, the same inference would be a pure hypothetical syllogism, thus: If anything is a man it is mortal.
4 In the case of non-identical truths, too, there is a priori proof drawn from the notion of the terms, " though it is not always in our power to arrive at this analysis," 5 so that the question arises, specially in connexion with the possibility of a calculus, whether the contingent is reducible to the necessary or identical at the ideal limit.
In recognizing, further, that the relation of an actual individual fact to its sufficient ground was not reducible to identity, he set a problem diversely treated by Kant and Herbart.
The clue to the discovery of transcendental conditions Kant finds in the existence of judgments, most manifest in mathematics and in the pure science of nature, which are certain, yet not trifling, necessary and yet not reducible to identities, synthetic therefore and a priori, and so accounted for neither by Locke nor by Leibnitz.
Even in Hamilton's earlier work it was shown that all such questions were reducible to the solution of linear equations in quaternions; and he proved that this, in turn, depended on the determination of a certain operator, which could be represented for purposes of calculation by a single symbol.
In the case of a beautiful object the resultant pleasure borrows its specific quality from the presence of determinations essentially objective in their nature, though not reducible to the categories of science.
(2) The real must be absolutely simple; for if it contain two determinations, A and B, then either these are reducible to one, which is the true quality, or they are not, when each is conditioned by the other and their position is no longer absolute.
These relations Herbart finds to be reducible to five, which do not admit of further simplification; and corresponding to them are as many moral ideas (Musterbegriffe), viz.: (I) Internal Freedom, the underlying relation being that of the individual's will to his judgment of it; (2) Perfection, the relation being that of his several volitions to each other in respect of intensity, variety and concentration; (3) Benevolence, the relation being that between his own will and the thought of another's; (4) Right, in case of actual conflict with another; and (5) Retribution or Equity, for intended good or evil done.