For possible geographical reasons for this duality of type see Crete.
In God there is no distinction or difference, whereas in all substances or things there is duality, arising from the element of matter.
But the followers of Cyril of Alexandria, and with them those of Eutyches, saw in the Chalcedon decree of two natures only another form of the "Nestorian" duality of persons in Christ, and rose everywhere in opposition.
Under the general heading "Geometry" occur the subheadings "Foundations," with the topics principles of geometry, non-Euclidean geometries, hyperspace, methods of analytical geometry; "Elementary Geometry," with the topics planimetry, stereometry, trigonometry, descriptive geometry; "Geometry of Conics and Quadrics," with the implied topics; "Algebraic Curves and Surfaces of Degree higher than the Second," with the implied topics; "Transformations and General Methods for Algebraic Configurations," with the topics collineation, duality, transformations, correspondence, groups of points on algebraic curves and surfaces, genus of curves and surfaces, enumerative geometry, connexes, complexes, congruences, higher elements in space, algebraic configurations in hyperspace; "Infinitesimal Geometry: applications of Differential and Integral Calculus to Geometry," with the topics kinematic geometry, curvature, rectification and quadrature, special transcendental curves and surfaces; "Differential Geometry: applications of Differential Equations to Geometry," with the topics curves on surfaces, minimal surfaces, surfaces determined by differential properties, conformal and other representation of surfaces on others, deformation of surfaces, orthogonal and isothermic surfaces.
They held that the duality of natures implied a distinction between two modes of sonship in Christ - the natural or proper, and the adoptive.
In this he went beyond Cyril and the Alexandrine school generally, who, although they expressed the unity of the two natures in Christ so as almost to nullify their duality, yet took care verbally to guard themselves against the accusation of in any way circumscribing or modifying his real and true humanity.
There seems to be, however, not a unity but a duality in its plan of construction, for the two parts, North and South America, resemble each other not only in outline but, roughly speaking, in geological evolution also; and the resemblances thus discovered are the more remarkable when it is considered how extremely small is the probability that among all the possible combinations of ancient mountain systems, modern mountain systems and plains, two continents out of five should present so many points of correspondence.
Hefele himself, one of the most learned and acute of Cyril's partisans, is compelled to admit that Nestorius accurately held the duality of the two natures and the integrity of each, was equally explicitly opposed to Arianism and Apollinarianism, and was perfectly correct in his assertion that the Godhead can neither be born nor suffer; all that he can allege against him is that "the fear of the communicatio idiomatum pursued him like a spectre."
The retention of the old duality of dignities was the one reminiscence of the original separation.
The incessant change which experience brings before us, taken in conjunction with the thought of unity in productive force of nature, leads to the all-important conception of the duality, the polar opposition through which nature expresses itself in its varied products.
Indeed, in order to oppose this unity of being to the realistic duality, both materialists and idealists describe themselves as monists, and call realists dualists by way of disparagement.
According to him, we begin with an experience of ideas, in which object and idea are originally identical (V orstellungsobject); we divide this unitary experience into its subjective and objective factors; and especially in natural science we so far abstract the objects as to believe them at last to be independent things; but it is the office of psychology to warn us against this popular dualism, and to teach us that there is only a duality of psychical and physical, which are divisible, not separable, factors of one and the same content of our immediate experience; and experience is our whole knowledge.
Hence he explains, what is a duality for us is only a " quasi-duality " from a universal standpoint.
James Ward, in Naturalism and Agnosticism (1899), starts from the same phenomenalistic views of Mach and Kirchhoff about mechanics; he proceeds to the hypothesis of duality within experience, which we have traced in James Ward.
Under the second head, according to Ward, as according to Wundt, knowledge is experience; we must start with the duality of subject and object, or perpetual reality, phenomenon, in the unity of experience, and not believe, as realists do, that either subject or object is distinct from this unity; moreover, experience requires " conation," because it is to interesting objects that the subject attends; conation is required for all synthesis, associative and intellective; thinking is doing; presentation, feeling, conation are one inseparable whole; and the unity of the subject is due to activity and not to a substratum.
He infers the corollary that universal experience contains the same duality of subjective and objective factors without dualism.
What, then, could we know from this " duality in experience" ?
When we combine his view of Nature under the first head that whatever is inferred in the natural sciences is ideas, with his view of knowledge under the second head that knowledge is experience, and experience, individual or universal, is of duality of subject and object in the unity of experience, it follows that all we could know from the data would be one experience of the race, one subject consisting of individual subjects, and in Nature single objects in the unity of this universal experience; and beyond we should be able to form conceptions dependent on the perceptions of individual experience in the unity of universal experience: that is all.
What, then, is the relation of God to the one universal experience, the experience of the race, which was under the second head the unity in duality of all knowledge ?
Like Aphrodite and Adonis in Syria, Baal and Astarte at Sidon, and Isis and Osiris in Egypt, the Great Mother and Attis formed a duality which symbolized the relations between Mother Earth and her fruitage.
" As the Unbegotten, God is an absolutely simple being; an act of generation would involve a contradiction of His essence by introducing duality into the Godhead."
They went back to the later period of Plato's thought, the period when Plato endeavoured to combine his doctrine of Ideas with the Pythagorean number-theory, and identified the Good with the One, the source of the duality of the Infinite and the Measured (rd .bretpov and 71-pas) with the resultant scale of realities from the One down to the objects of the material world.
Like Hamilton, Mansel maintained the purely formal character of logic, the duality of consciousness as testifying to both self and the external world, and the limitation of knowledge to the finite and "conditioned."
Or again, within the latter in his admission of a duality of thought and " the given " in knowledge, which within knowledge was apparently irreducible, concurrently with hints as to the possibility, upon a wider view, of the sublation of their disparateness at least hypothetically and speculatively.
Schleiermacher's formula obviously ascribes a function in knowledge to thought as such, and describes in a suggestive manner a duality of the intellectual and organic functions, resting on a parallelism of thought and being whose collapse into identity it is beyond human capacity to grasp. It is rather, however, a statement of a way in which the relations of the terms of the problem may be conceived than a system of necessity.
" single uterus," - in allusion to the fusion of at least the basal portions of this organ, and in contradistinction to their duality in the Didelphia, or Marsupialia), Cuvier's name for the group which includes all the orders of mammals (See MAMMALIA) except the Marsupialia and Monotremata; other titles for this group being Placentalia and Eutheria.
The results of her study of German philosophy were seen in philosophical essays; in lectures on "Doubt and Belief," "The Duality of Character," &c., delivered in1860-1861in her home in Boston, and later in Washington; and in addresses before the Boston Radical Club and the Concord school of philosophy.
Hence the Nestorians, who insisted upon the duality of the natures to such a degree as to lose sight of the unity of the person, and who rejected the term Theotokos, repudiated the decrees both of Ephesus and of Chalcedon, and upon the promulgation of the decrees of Chalcedon formally separated from the church.
Cybele was usually worshipped in connexion with Attis, as Aphrodite with Adonis, the two being a duality interpreted by the philosophers as symbolic of Mother Earth and her vegetation.
Subject-object, knowledge, or, more widely, self-consciousness with its implicates - this unity in duality is the ultimate aspect which reality presents.
Thus, whereas the Ionians, confounding the unity and the plurality of the universe, had neglected plurality, and the Pythagoreans, contenting themselves with the reduction of the variety of nature to a duality or a series of dualities, had neglected unity, Parmenides, taking a hint from Xenophanes, made the antagonistic doctrines supply one another's deficiencies; for, as Xenophanes in his theological system had recognized at once the unity of God and the plurality of things, so Parmenides in his system of nature recognized at once the rational unity of the Ent and the phenomenal plurality of the Nonent.
In this memoir by Gergonne, the theory of duality is very clearly and explicitly stated; for instance, we find " da p s la geometrie plane, a chaque theoreme ii en repond necessairement un autre qui s'en deduit en echangeant simplement entre eux les deux mots points et droites; tandis que dans la geometrie de l'espace ce sont les mots points et plans qu'il faut echanger entre eux pour passer d'un theoreme a son correlatif "; and the plan is introduced of printing correlative theorems, opposite to each other, in two columns.
The theorem of duality as regards plane figures may be thus stated: two figures may correspond to each other in such manner that to each point and line in either figure there correspond in the other figure a line and point respectively.
The theorem of duality is considered and developed, but chiefly in regard to its metrical applications, by Michel Chasles in the Memoire de geometrie sur deux principes generaux de la science, la dualite et l'homographie, which forms a sequel to the Apercu historique l'origine t.
It is obvious that we cannot by consideration of the equation u = o in point-co-ordinates obtain the remaining three of Pliicker's equations; they might be obtained in a precisely analogous manner by means of the equation v= o in line-co-ordinates,but they follow at once from the principle of duality, viz.