If the functions be subjected to linear transformation the Jacobian is multiplied by a constant factor or modulus, which is the determinant of the coefficients of the transformation. The Jacobian is thus a covariant of the functions from which it is derived. See INVARIANTS AND COVARIANTS.

If the Jacobian vanishes the functions are not independent, and conversely.

28. The Jacobian of the partial differential coefficients of a function, taken with respect to its several variables, is called the Hessian of the function. The Hessian is a symmetrical determinant.

Thus, the given function being the ternary quadric,

w= ax2+by2+ cz2 + 2 fyz + 2gzx + 2hxy,

a determinant, which in this instance is also known as the discriminant, for the reason that its vanishing is the condition that w be resolvable into linear factors.

Like the Jacobian the Hessian is a covariant. Both were originally called functional determinants.

29. The theory of determinants has two notable extensions which will be explained, though little more than the definition of the functions suggested can be undertaken in the present article.

Just as n elements may be arranged as a square array with two sets of parallel lines, so may n' elements be arranged in the form of a cube so as to lie in three sets of parallel planes. These three sets of planes may be called strata, planes and sections; the last two corresponding respectively to the rows and columns of the square array lying in any given stratum. An element of such an array may be represented by anks, where n is the number of its stratum, κ of its plane, and s of its section. Then | an,K,8 | represents a function known as a cubic determinant. Its principal term is a111a222... annn, and the others are obtained as follows: Permute the third suffixes and in each of the n! products obtained permute the second suffixes, thus obtaining (n!) products. Then apply to each product a sign factor which is the product of the sign factor appropriate to the permutation of the third suffixes by that appropriate to the permutation of the second. The aggregate of terms thus obtained is the cubic determinant.

The sign of the cubic-determinant is changed by the interchange of two planes or two sec

letting 2 +μn=p+o, the given array represents a determinant of order (n+1). If now n increase indefinitely, λ, on the one hand and p, on the other also becoming indefinitely large, while maintaining the above relation to n, the determinant may approach a definite limiting value, it may become indeterminate, or it may become infinite. Determinants of infinite order, as here defined, present certain analogies to infinite series.

History and Bibliography. The first definite reference to determinants is to be found in a manuscript of Leibnitz, left unpublished at the time of his death. His later development of the idea is embodied in a communication to De l'Hospital (1693) in which he explains the formation of the eliminant of a system of linear equations identical in form with that discussed in 17. In his notation he used double indices, representing the element ako by ks or by Kg. The next approach to the subject was made quite independently by Cramer (1750), who expounded a rule for writing out the values of the unknowns in a system of linear equations, in which he recognized the two classes of permutations as given in 2. This rule found instant favor and the development of the theory of determinants has since been almost continuous.

The first attempt at a formal exposition of the theory was made by Vandermonde (1771). His treatment of the functions as independent of their connection with the process of elimination and his recognition of several of their more fundamental properties entitles him to be ranked as the real founder of the theory. Almost simultaneously Laplace gave the method, partly anticipated by Vandermonde, of expanding determinants in terms of complementary The functions were called by him "resultants," a term now used in the restricted sense explained in 18.

minors.

New applications of the functions were at once found by Lagrange (1773), principally in the domains of geometry and the theory of numbers. Their application was further extended by Gauss (1801), who used the word determinant to designate the discriminant of the ternary quadric (see 28). He also noted the significance of reciprocal determinants (see 23), and was on the verge of discovering the multiplication theorem. This theorem (see 21 and 22) was first given by Binet (1812), but its satisfactory proof is due to Cauchy.

Cauchy was the first to use the word determinant in its present sense. His memoir in the

DETERMINATE PROBLEM.

Jour. de l'école polytechnique (1812) rounded the general theory of determinants into something like its present form. Jacobi early adopted the new calculus and greatly extended its field of usefulness. In his hands it became, not merely an indispensable system of notation, but a powerful instrument of research as well. His justly famous memoirs in 'Crelle's Journal' (1841) made the subject readily accessible and, since their publication, determinants have been freely used by mathematical writers without apology or explanation.

A new impulse was given through the study of linear transformations, in which Cayley and Sylvester led the way. The more recent developments relate mainly to special forms of determinants. Skew determinants and Pfaffians were developed by Cayley, in connection with the orthogonal transformation. He was the first writer to use the vertical bars enclosing the determinant array. Continuants were investigated by Sylvester, who also named and made much use of Jacobians and Hessians. Alternants, first recognized by Cauchy and studied by Jacobi, have been further discussed by Trudi, Nägelbach, and Garbieri. Important results relating to compound determinants were published by Schweins (1825), but these have been rediscovered and greatly extended by Sylvester, Reiss, Kronecker, and Picquet. Determinants of infinite order have been used by G. W. Hill in a memoir on the linear theory appearing in the Acta Math., Vol. VIII. Their properties have been investigated by Poincare, von Koch, and Cazzaniga. In recent years the arithmetical properties of determinants and the theory of elementary divisors have been discussed by H. J. S. Smith, Kronecker, Frobenius and Weierstrass.

Bibliography.- Muir's "Theory of Determinants in the Historical Order of its Development' (London 1890) is the standard work on the history of the subject down to 1841. The same author has also compiled a complete bibliography of determinates down to 1900 (Quar. Jour. of Math.). Spottiswoode (1851) prepared the first regular treatise. Other wellknown texts are those of Baltzer (Leipzig 1857); Günther (Erlangen 1875); Dostor (Paris 1877); Scott (Cambridge 1880); Muir (London 1882); Hanus (Boston 1886); Weld (New York 1893); Scott and Mathews (Cambridge 1904).

LAENAS GIFFORD WELD,

Author of Theory of Determinants'; 'Determinants.

DETERMINATE PROBLEM, a problem in geometry which admits of a limited number of solutions, an indeterminate problem admitting of an indefinite number of solutions.

DETERMINATIVE MINERALOGY. That division of the science of mineralogy (q.v.) which treats of the identification of minerals. While the occasional new or undescribed species is necessarily identified by a determination of the percentages of its various ingredients (see CHEMICAL ANALYSIS), the symmetry and geometrical constants of its crystals (see CRYSTALLOGRAPHY), and its optical and other physical properties (see PHYSICAL CRYSTALLOGRAPHY); specimens of previously described species are accurately and rapidly determined by much simpler methods, varying from sight

23

recognition to systematic testing with the aid of tables or "schemes." Sight recognition is the result of experience in handling and testing and has no fixed order of procedure. Except for this, it is similar to the later mentioned schemes by external signs, inasmuch as it relies chiefly on lustre, color, crystals, cleavage, associates, etc., and supplements these by tests such as the color of the mark (streak) made by it upon unglazed porcelain, or its hardness as tested by a knife point. Testing at random is nearly worthless and in all cases of doubt it is better to follow the guidance of some systematic scheme in which a few simple physical or chemical tests made in definite order successively place the specimens in smaller and smaller groups until the group is so small that the differences between its members are readily recognized and it becomes possible_conclusively to eliminate all but one species. There are many schemes which may be said to be chiefly of three types. (1) Schemes in which the classifying characters are the so-called "external signs" such as lustre, color, crystalline form and simple physical tests such as hardness, color of powder (streak) or specific gravity. These schemes are invaluable in the classroom, enormously lessening the time spent in looking up characters and they foster acute observation of appearance and require only simple apparatus, such as a pocket lens, streak plate and file or knife. The best known and most followed scheme of this type was prepared in 1866 by Prof. Albin Weisbach of Freiberg, and while for the great group of minerals with non-metallic lustre and white or gray streak it is not very conclusive, the minerals of metallic lustre or with colored powders (streaks) are often very rapidly determined. For instance by the Frazer-Brown modification of Weisbach's tables only three species are at

once

Submetallic or non-metallic,
Red streak,

Hardness about 4 (that of fluorite). These are easily distinguished by the shade of red of the streak and by the specific gravity. Zincite, orange red, Cuprite, brownish-red, sp. gr. 6, Hauerite, brownish-red, sp.gr. 4; (2) Schemes in which the so-called "blowpipe tests" such as fusibility or qualitative determination of principal constituents, or solubility in acids, are the classifying tests, while the external signs and physical tests are subordinate. These, because they are largely independent of the condition in which the mineral occurs, use only minute amounts of material, and give positive proof of the presence of the component elements, may fairly be said to be of the most general applicability. They require in addition to the simple apparatus used in the previously mentioned schemes, a blowpipe and its accompanying lamp, forceps, wire, reagents, etc. Undoubtedly the most used schemes of this type are translations and modifications of von Kobell's Tables for Determination of Minerals by Simple Chemical Tests,' first published in 1833 "to facilitate determination of minerals so that by simple testing with the blowpipe and in the dry way one is led quickly to a small group which includes the mineral being tested." As with the Weisbach tables, there is first a division by lustre, but thereafter the route followed is different, the next division being by

fusibility and the further subdivisions in some cases being the detection of particular elements as arsenic, sulphur, antimony, iron, etc., and in others some detail of fusion such as "magnetic after heating" or of solubility such as "effervescence. For instance, a specimen of green granular Chrysolite (Mg, Fe)2SiO, would be placed by the Brush-Penfield modification of the Von Kobell tables successively in the following divisions and sub-divisions:

1. Non-metallic lustre. One of several hundred. 2. Infusible. True of 150 of the non-metallic species. 3. Dissolves in HCl leaving a gelatinous residue on evaporation. Therefore one of 16 species,

4. Gives reactions for Magnesium. Therefore one of 6 species.

5. Contains no water. Therefore one of 2 species. 6. Gives reactions for iron. Therefore chrysolite. This confirmed by color, hardness, structure,

etc.

(3) Schemes in which characters determined by the polarizing microscope are used as the classifying tests, these including single or double refraction, index of refraction, birefringence, pleochroism, shape, cleavage, color, etc. Such schemes are extensively used for the determination of the minerals in fine-grained rocks and are coming into more general use for identification not only of those minerals which can be readily made thin enough to transmit light but for the so-called opaque minerals. The testing requires specially prepared sections as thin as paper, or finely crushed powder or, in some cases, polished sections, and the tests while usually easily made require for their proper understanding a considerable knowledge of crystal structure and optics. They, however, supplement admirably the blowpipe and physical tests.

Bibliography. Works in English in addition to tables in most textbooks of mineralogy. External Signs: Eakle, A. S., 'Mineral Tables'; Frazer-Brown, Tables for the Determination of Minerals' (6th ed.); KrausHunt, Tables for the Determination of Minerals (1911); Moses, A. J., Guide to Sight Recognition of 120 Minerals. Blowpipe Schemes: Brush-Penfield, 'Manual of Determinative Mineralogy); Lewis, J. V., 'Determinative Mineralogy) (2d ed.). Optical Schemes: Johannsen, A., 'Determination of Rock Forming Minerals'; Luquer, L. McI., 'Minerals in Rock Sections' (4th ed.); Moses, A. J., 'Utilizing Polarizing Microscope in Determination of Minerals'; Murdoch, J., Microscopical Determination of the Opaque Minerals.'

Also

ALFRED J. MOSES, Professor of Mineralogy, Columbia University. DETERMINING TENDENCY. called attitude, task (Aufgabe) and is partly synonymous with purpose in its more empirical use. Determining tendency was a term introduced by Ach and taken over into English to designate one of the wider influences which control the course of associations and so of thought and action. One ordinarily thinks of the course of thought as controlled by the closeness of the connection between the elements of which it is made up, and of these connections in turn as due to the number of times the two experiences have occurred together in the past, the recency of their appearance together, etc. Determining tendency is looked upon as a factor of wider influence, which will make some one or some one class of the pos

sible associates more likely to appear than others which are in themselves equally strong or even stronger. Thus if one is given a series of words and asked to speak the first other word that comes to mind each word may call up a number of associates. "Water," e.g., might suggest "drink"; "H.O," "ice," etc. If, however, one has been asked to give the chemical composition of the substances named, "H2O» alone comes to mind; if the request was to name the use of the objects, drinking or washing or other words of that class alone will The request is the determining ten

appear.

dency. Determining tendencies are frequently not definite tasks, but are furnished by the knowledge of the circumstances or situation. One tips one's hat to a male friend when he is with a lady, but only speaks when he is alone. One plays one card in a given situation when one suit is trumps; another under the same circumstances when the trump is different, One strikes one note for a given position when the selection is written in one key and another when the key is different. The awareness of the companion, the knowledge of the trump or of the key constitute in each case the determining tendency.

The influence of the wider setting has recently been given wide vogue, chiefly through the work of the Würzburg school. It was noticed as a factor in the control of perception by Pillsbury in 1897, who spoke of it as "attitude" or "setting," by Watt in the control of association in 1904, who called it Aufgabe, was emphasized under the present name by Ach in 1905 as a factor in the control of action and thought, and by Pillsbury again as mood or attitude as one of the conditions of attention in 1906.

The importance of emphasizing the wider controls as opposed to the immediate links which alone were considered by the associationist school is obvious. Were the strength of the mechanical connections alone to be taken into account, we would have no explanation at all of the adaptability and flexibility of human thought and conduct. If one were limited to a single response to a given stimulus or to a single thought in succession to any other thought, act and thinking would alike follow unvarying courses. One's existence would be a continuous treadmill. While we still think of all the successions as made possible by earlier associations, selection of the proper one in accordance with the situation or context increases the adequacy of the thought and act.

We must look to the nervous interactions for the real explanation of these influences. The determining tendency acts only as it is accompanied by or induces activities in the neurones. In the nervous system an association has as its analogue a connection between neurone groups induced by simultaneous action at some earlier times. Any neurone group is connected with several other groups in such a way that it may excite any one of them when it is itself excited. The determining tendency decides which of these shall be aroused. When a task is set or a wider context is effective, these preliminary stimuli partially arouse larger or smaller groups of neurones. These larger groups will have elements in common with the groups that will be aroused by the other direct

stimulus. When this finds one associated group partially aroused, it will completely arouse that one, while the others associated with it will not respond sufficiently to influence consciousness. Possibly, too, the other associates may be actually inhibited, prevented from responding at all. More likely there is both a positive and a negative effect. One group of neurones connected with the stimulus is prepared to respond by partial excitation and the others are blocked or inhibited. While we speak of determining tendency as an influence of task or setting or purpose, these terms are merely descriptive of the antecedents of the act or thought, of the circumstances under which determining tendencies act. The actual

cause of the selection is to be found in the preparation for action of a group of cerebral elements by an earlier stimulus.

WALTER B. PILLSBURY.

DETERMINISM is the doctrine that everything that exists, as well as every event that happens, is absolutely fixed and necessary in the sense that it has been made what it is by the action of some ground or cause. From this point of view there is accordingly no element of chance or contingency in the world: nothing which is could possibly have been otherwise. While this doctrine conceives of all of the parts of the world as determined, it derives its main interest and importance from its application to the sphere of mind and the thoughts, feelings, and actions of human beings. At the present day we have become accustomed to the conception of physical nature as a uniform series of events that happen in a fixed and necessary way in accordance with uniform laws. The crucial question is whether the same mode of conception is to be extended to mind and its operations, or whether the view can be justified that mind occupies a unique place in nature and is characterized by real freedom. "Most writers on the emotions and on human conduct," Spinoza tells us, "seem to be treating rather of matters outside of nature than of phenomena which follow nature's general laws. They appear to conceive man to be situated in nature as a kingdom within a kingdom, for they believe that he disturbs rather than follows nature's order, that he has absolute control over his actions and is determined solely by himself." As against such a view, Spinoza maintains the doctrine of determinism: "that nature's laws and ordinances whereby all things come to pass and are changed from one form to another are everywhere and always the same." Hence "it is impossible that man should not be a part of nature," and subject, like everything else, to nature's universal laws.

Determinism follows directly when substance and causality are employed as the ultimate principles of explanation. From the point of view of substance, the parts are viewed as determined by the nature of the whole. This whole, whether called God, or "substance," or "matter," or "energy," is conceived as the one ultimate reality, of which particular phenomena are "accidents" or "modifications." The nature of the whole being given as a fixed datum or prior term, the nature of the parts, as its modes or modifications, are therefore also fixed and determined. More frequently, however, the causal category is employed to support deterministic

conclusions, either alone or in conjunction with that of substance. No part of the universe, it is argued, stands by itself: everything gets its constitution and nature through its connection with and dependence upon something other than itself. Universal causal relationship, then, implies universal dependence, necessary determination. If the principle of cause and effect is valid throughout the whole field of reality in the mental realm as well as in the physical determinism is the logical and inevitable consequence. The adherents of determinism have nevertheless maintained these principles with varying degrees of strictness so far as their applications to mental life are concerned. Many of those professing determinism have given to the causal principle in its application to mind a meaning which they believe guarantees a rational view of freedom. Some of these positions will be referred to below under the various headings with reference to which the subject is to be further discussed. These headings, it will be noticed, indicate different aspects and applications of the problem as already outlined, and not separate or exclusive doctrines.

(a) Metaphysical Determinism arrives at its conclusions from a consideration of the general nature of the world and the universal form in which events happen. The determinism of the will is simply one instance among others of the universal order and uniformity of all events. In strict monistic systems it is difficult to avoid the conclusion that the nature of the one ultimate reality is that which determines everything. This is true, as we have seen, whether that one substance is regarded as God, or as matter or energy. The opponents of determinism accordingly usually fall back upon pluralism, and emphasize the uniqueness and self-centred reality of human personality. But this position is confronted, on the other hand, with the task of accounting for the unity which experience shows actually to exist. How, if there is a plurality of completely independent individuals, is it possible to think of the whole as forming a single world a universe? It is clear that this world must be something more than a manifold, and that human beings least of all are mere points which exclude relation to others. This antinomy is of course just one aspect of the fundamental metaphysical problem of the one and the many, and cannot receive a full discussion here. It is however obvious from what has already been said that a solution can be found only by adopting a metaphysical system that is at the same time both monistic and pluralistic that holds fast to the unity of things and also maintains the reality and provides for the possibility of self-determination on the part of finite individuals. Such a result is not possible for materialism, or any mechanical system of metaphysics. If, however, the view can be defended that the universe exists in its ultimate form as life and consciousness, freedom can still be maintained. For life and consciousness are principles of inclusion, not of exclusion, and at the same time they leave within their unity free place and scope for individual differences. From the metaphysical point of view, the solution of the apparent contradiction between freedom and determinism depends, therefore, upon the possibility of finding logical grounds for maintain

ing a teleological and spiritual view of the world.

(b) Theological Determinism.—The problem here is in principle identical with that of metaphysics, though stated in different terms and generated by a special interest. The theological doctrine is variously known as predestination, foreordination, election, etc. (qq.v.). The whole course of history, both in its physical and moral aspects, is assumed to take place in accordance with the eternal decrees or purposes of God. The crucial problem is regarding man's ability to secure his own salvation. Theological determinists maintain that God has "elected some to everlasting life," and that such election, as an act of free grace, is the necessary condition of salvation. Thus

Jonathan Edwards, in summing up his defense of Calvinism against the Arminians, claims: "If we put these things together, it will follow that God's assistance or influence must be determining and decisive, or must be attended with a moral necessity of the event; and so that God gives virtue, holiness, and conversion to sinners, by an influence which determines the effect in such a manner that the effect will infallibly follow by a moral necessity, which is what Calvinists mean by efficacious and irresistible grace." ("On the Freedom of the Will, Ch. XIV).

(c) Psychological Determinism.-Psychology has to describe and explain the conscious processes which constitute mind; and, like all the sciences, is necessarily deterministic in standpoint. From the psychological point of view every mental process without exception finds its explanation in its relation, either to some other mental content, or (as some would maintain) to accompanying neural changes in the body. An uncaused or undetermined mental event, as, e.g., an act of will which is not connected causally with other states of body or mind, is a contradiction from this point of view. Nor is the difficulty to be avoided by supposing that the volition is caused by a free act on the part of the ego, that the self intervenes to produce changes in the natural order of events at this point or that. For if the self is thus distinguished from its states, it is no longer a member of the mental series, and cannot be invoked as a cause of any particular event. To appeal to any such special agent is to abandon the scientific problem, which is to discover the causal relation of the phenomena that constitute the mental life. As Kant in the "Critique of Pure Reason' clearly shows, there is no possibility of freedom in a phenomenal series of mental events; since, as occurrences in time, these are subject without exception to the law of cause and effect. The oft-repeated attempts of defenders of free-will to overthrow determinism while still assuming that the psychological point of view is final are as contradictory and fruitless as the analogous attempts to vindicate teleology in the physical order by supposing that at times the natural order is interrupted by a miracle. If determinism is to be avoided, there must be an advance to a new standpoint and principle of explanation. This was perceived by Kant who, in his doctrine of man as belonging to a noumenal order of reality, passes beyond the psychological and causal point of view to the idea of selfdetermination in accordance with ends. It is

true that Kant did not regard this new category as one that is valid for knowledge, but describes it as a "postulate" of practical or moral reason. The result of his analysis, however, was to establish the fact that while the psychological or phenomenal description of mind must employ everywhere the category of cause and effect, and can therefore only issue in determinism, yet from another point of view it is equally necessary, in order to do justice to the facts of moral experience, to introduce the idea of freedom, or determination in accordance with ends.

(d) Historical Determinism may be a deduction from theological determinism, by simply regarding all the events of history as determined by the will of God, who makes even the "wrath of man to praise him." But what is usually known by this name is the theory according to which historical events are to be explained by referring them to natural powers and circumstances as their causes. The demand that history shall be made a science involves such an explanation of historical events by means of causes, and carries with it the assumption of determinism. To explain historically is accordingly regarded by some writers as requiring exhibition of the various causes -geographical location, climatic conditions, racial and economic rivalries, etc.- which are to be taken as determining the course of events in the period with which they are dealing. Since, however, historical events are determined by the volitions of men, the sphere of determination must be psychological, and the remarks made in the preceding paragraph will also have application here.

(e) Ethical Determinism.-It is because of the supposed ethical implications of Determinism that the doctrine has been most vigorously opposed. If all human actions take place, like natural occurrences, in a strictly determined way, then it seems an immediate corollary that the sense of obligation is unmeaning and the feeling of responsibility an illusion. Consequences of this character, which seem to the opponents of the doctrine to follow from it directly, are taken by them as involving the refutation of the theory of determinism. The advocates of freedom are often content with this "practical" disproof of the opposing doctrine, admitting that from the theoretical point of view it is unanswerable. Kant, e.g., bases the proof of freedom upon the demands of the moral consciousness. Without the "postulate" of freedom the fact of moral obligation would be illusory; the "ought" implies and presupposes the "can." On the other hand, the determinists as a rule are not willing to admit that it is necessary to assume freedom in the sense of indeterminism in order to give significance to moral distinctions. In the first place they point out that determinism is by no means identical with fatalism: that the causes of an action are no compelling powers outside the individual, but his own motives and character. Moreover, as H. Sidgwick has remarked: "The determinist allows that, in a sense, the 'ought' implies can,' that a man is only morally bound to do what is in his power' and that only acts from which a man could have abstained' are proper subjects of punishment or moral condemnation. But he explains 'can' and in his power' to imply only the presence