# Guide Theory of determinants. Up to 1841

An Elementary Treatise on the Calculus. With Illustra- tions from Geometry, Mechanics, and Physics. Gib- son, M. Differential Calculus for Beginners. Edwards, M. Integral Calculus for Beginners. With an Introduction to the Study of Differential Equations. By Joseph Edwards, M. Calculus Made Easy. Being a very-simplest Introduction to those beautiful Methods of Reckoning which are generally called by the terrifying names of the Differential Calculus and the Integral Calculus.

New Edition, with many Examples. Osgood, Ph. Practical Integration for the use of Engineers, etc. Percival, M. Differential Calculus. With Applications and numerous Exam- ples. Lambert, M. Differential and Integral Calculus. With Applica- tions. By Sir A. Greenhill, F. A Treatise on the Integral Calculus and its Ap- plications. Todhunter, F. DICK, G. DUiLP, H. Voss, A. HAHN, J.

## Thomas Muir: "History of determinants"

BERG, F. D'Ovinio, E. LE pp. IGEL, B. Ii Bibliographical Note. Isk, E. DI , , E. LE p p. THE number of writings to be considered under this heading amounts to about one hundred and eighty , and the number of writers to about one hundred and twenty , these being almost three times the corresponding numbers for the twenty-year period immediately preceding. A striking new feature is the large proportion of text-books,-more than a third-there being over sixty 60 of them dealing with determinants alone and several others in which the subject forms an important part.

It is also noteworthy that for the first time text-books and other expositions of a really elementary character make their appearance, Germany becoming especially productive from the time that the subject began to be introduced into her higher schools. As a further evidence of this increase of interest attention may be called to the fact that, besides the usual number of works in English, French, German and Italian, we have now not only as before three or four publications in Russian, but also contributions in eight of the less widely spread languages of Europe,-Czech, Polish, Dutch, etc.

Annales de Math. In France and elsewhere determinants were still viewed as having originated with Cramer. English Encyclopaedia, v. Papers, iv. The list includes about two dozen, but some of them, like determinant, quantic, invariant, include quite a number of subordinates. It recalls the'' Glossary' given by Sylvester at the end of his memoir on Syzygetic Relations Runkle's Math. Monthly, iii. Traduit de l'allemand par J. Second edition, see pp.

A chapter of elementary Algebra. Messenger of Math. Isander's is a carefully written introduction based on Cauchy's memoir of Oliver's is a mere fragment. Houel's Baltzer is nothing more than a close translation of the original. Ferrers' is a short chapter simply and clearly written. Todhunter's plan and style are quite similar to Ferrers', and his three chapters are therefore still more helpful. Tait's also resembles Ferrers', and is of the same extent. Gould's Astron. His mode of writing the final expansion of , In specifying the number of terms of each type in the expansion, he gives for the number of terms in a zero-axial determinant of the 3rd, 4th, 5th, Magazine, 4 , xxi.

Papers, v. For all three it is necessary that the elements of the determinant be specified by their row and column numbers. He denotes 11, 22, He omits to note, however, that he also intends a sign of summation, Z say, to be prefixed mentally in each case: for example, - should be - 1il, as it stands for the six-termed aggregate - - We may remark that in the notation of his paper of the year before Hist.

Further, there should be noted the difference between this and the same author's development of the year Hist. London, cli. Papers, i. There are, however, several points of contact with Determinants, just as in the case of previous papers of the same kind by Hermite , Bazin , Heger The main subject of the memoir was continued in three papers -published under the conjoint title "Arithmetical Notes" in the Proceed.

London Math. The writer's purpose, however, is quite different, his subjects being substitutions, matrices not so called , and groups. The nearest approach to the field of determinants is in the fifth chapter pp. See Liouville's Journal, 2' ser. Early in the chapter on the 'Combinatorial Product' chap. They are even found to be needed for the statement of theorems in the 'Ausdehnungslehre' itself, for example, the theorem on the inner product of two magnitudes each of the mntl 'Stufe ' and consisting of m simple factors. In this matter, too, there is a new procedure which is of considerable interest 'when brought into comparison with the old.

This equation, Grassmann says, takes the place of the given equations, and in order to find the value of any one of the x's, xl say, we apply the factor 2a2 LE Annali di Mat. It must be noted, however, that what Painvin establishes is something more important, namely, the identity of the two determinants ' elementa element,' that is to say, the identity of their matrices.

The explanations are consequently fuller, the illustrations are more numerous, the less simple theorems are broken up for gradual absorption, and to deductions that are self-evident are given formal enunciations. The scheme of arrangement resembles Baltzer's, there being a first part devoted to the theory and a second part to the so-called applications. Whereas, however, Trudi apportions to the parts and pages respectively, the like apportionment in Houel's Baltzer is 63 and ; further, on account of the unequal treatment of geometry pages of Trudi's are algebraical as against of Baltzer's Save this marked fullness and clearness of explanation there is not much fresh to be noted so far as general determinants are concerned.

A number of additional expressions of more or less importance are by definition given a special sense for the sake oi definiteness or convenience; for example p. In the. When, however, we get beyond the section on skew determinants p. Napoli, i. The results being burdened with exceptions are a little disappointing. It may also be pointed out that the elements in the latter form are three-line minors of the former. Part IV. British Assoc Cambridge , pp. See Crelle's Journ. Liege , 2 xii. Deutsch bearbeitet von Dr. Wilhelm Fiedler. Traduit de l'anglais par M. Augmente de notes par M.

Salmon himself does likewise, but increases the number of lessons to six, marking for the first time the importance of "the reduction and calculation of determinants," by assigning thereto a separate lesson. Bazin, on the other hand, makes considerable curtailments. In none of the three does anything really fresh appear in regard to the genera] theory. Napoli, The third edition of the 'Elementi,' a superior text-book, contains in two chapters vii. In the 'Complementi' there are an equally helpful lesson on elimination xxxi.

This will be fully understood from its application to one of the simplest cases, namely, the case of the two arrays, a1 a2 a3 h1 h2 h3 bl b2 b3, 1 k2 k3. I -h2 -Jo2. In his lectures on Higher Alg. In his lectures on Analyt. V Praze. In Pokorny's book the section pp. Dolp's school-program is of slighter interest.

Taking I a1,a For example, n, p, q being 4, 2, 3, the result is a. Wien , Math. Two pages pp. London, xv. Stated in the form of a rule it is: Compute every two-line minor consisting of four adjacent elements in the given determinant A, and form with the results in order a square array B: compute every similar two-line minor of B, performing however this time the additional operation of division by the particular element of A which entered into the composition of the four elements of the minor in question, and form with the quotients in order a new array C: thereafter let successive arrays be computed in the same way as C until an array is reached with only one element: this element is the value of the original determinant.

For example, the given determinant being -2 -1 -1 -4 -1 -2 -1 -6 -1 -1 2 4 2 1 -3 -8 the series of derived arrays is 3 -1 2 8 -2 -1 -5 8 The rule is stated to be dependent on Jacobi's theorem regarding a minor of the adjugate, and is formally proved for determinants of the 3r and 4th orders. Care is also taken to show how the difficulty caused by the appearance of a zero as one of the divisors can be obviated, and how the labour of computation may be minimized when a number of related determinants have to be evaluated, as happens in solving a set of simultaneous linear equations.

We may note for ourselves in passing that by applying the rule to a determinant with general elements we may obtain not only M. Thus, the given determinant being a. In this way we see that the determinants of the intermediate arrays are not equivalents of the given determinant: indeed, if the given determinant be of the nt' degree in the elements, the second is of the degree n-1 2, the third of the degree n-2 3, the fourth of the degree n-3 4, and so on by a rise and a subsequent fall until the degree 1.

Edinburgh, vi. Tait draws attention. The only property of the six that has an appearance of freshness is one which resembles Catalan's of the year Hist. Giornale di Mat. Sardi's has more appearance of being the result of a suggestion from the equatement of the two denominators in Spottiswoode's proof of Theil, Theorie der complexen Zahlensysteme.

They recall vividly Cauchy's memoir of. The 'alternating units' j1, i, The penultimate result is the basis of the proposition that a general determinant of the nth order can by means of alternating numbers be resolved into n linearfactors,-a proposition from which, as Cauchy originally pointed out, all the fundamental properties of determinants are readily deducible. I cA21,. In a historical note p. Hoffmann's Math. Worterbuch, vi. As its title 'Ueber erweiterte Determinanten' implies, it concerns theorems of the kind known at a later date as 'extensionals' and exemplified in the writings of Desnanot , Schweins , Reiss himself , and others.

In the matter of notation Reiss is no longer peculiar. His symbolism for a determinant is now professedly the same as Sylvester's, and his mode of specifying a sum of products of pairs of determinants is not far removed from that of Schweins. A closer imitation of the latter,-whom, by the way, he never mentions, -would not have been a disadvantage: and instead of Sylvester he might more appropriately have mentioned Desnanot for the additional reason just given. Thus, the expansion of a five-line determinant in terms of the minors formed from the 2nd and 4th rows and the minors formed from the 1st, 3rd and 5th rows is indicated by s i 2 4 1.

In the way of new contributions the first point to be noted is that, whereas preceding writers had dealt with the 'extension' of only special cases of Laplace's expansion-theorem, Reiss gave the full generalization of the theorem. For example, while Desnanot showed that the identities ab2c3d4, M More important, however, is the fact that having set himself to inquire what the left-hand member of Laplace's expansion-theorem would become if each first factor in the right-hand member were.

Lastly, he gives what he calls an application of Sylvester's and Jacobi's theorems to functions of the form. Ca,,, where the c's are determinants, the A's are coefficients independent of the c's, the e's are signs, and the function as a whole is, like the individual determinants, homogeneous in the elements. Such a 'doppelt-homogene Function,' if it vanishes identically, he finds has two interesting properties, namely, 1 it may be 'extended' and still vanish, 2 the identity will continue to hold if the determinants be replaced by their coefficients in another determinant to which they all belong as minors.

It is also carefully noted that the same properties belong to the difference of two such functions if they be identical. It is much to be regretted that these theorems of Reiss' received no attention from his contemporaries. His work is not even mentioned by any of the German text-books giving bibliographical references, for example, Baltzer's editions of , , and Gunther's of , Had it been otherwise, the generalisations known as the Law of Complementary Minors and the Law of Extensible Minors would have been formulated much earlier than they were. The second and third Sections of the memoir pp.

Professedly its main aim is logical exactitude. In pursuance thereof all definitions, conventions, axioms, propositions and corollaries are carefully formulated, labelled and numbered: every step in a train of reasoning is kept scrupulously separate from its antecedent and consequent, and in order to guard against possible contamination of the reasoning illustrative examples are relegated to the footnotes.

Clearness is also sought to be promoted by the use of more than the ordinary variety of printers' type, and precision by the introduction of new terms and symbols. One of the latter, namely, rUs which is used for the element in the r, s th place, is naturally of frequent occurrence. The consequence is that the pages, though broad-margined and well-printed, present an unwonted and rather bizarre appearance. Leaving out twenty-two propositions chap. Of the latter twenty-three chap.

The most important chapters are iii. Chapters v. Chapter v. For this purpose it must be premised that an ' oblong ". The propositions,-which should be viewed in connection with Kronecker's of ,-are pp. Proof of part of the second may be given by way of sample. For, he says, by reason of data 1 and 2 the first three equations are consistent: and similarly data 1 and 3 may be used: and. The entire set is thus consistent, and the initial array consequently evanescent. Taking then the determinant a b c d ef ghi, he in the first place divides the 2, 2 tl element by itself and multiplies each of the elements of its complementary minor by the same: secondly, he divides the 3, 3 t1 element and multiplies the elements of its complementary minor by i: thirdly, he multiplies the 2"d and 3rd rows by b and c respectively: and fourthly, he multiplies the 21d and 3"r columns by d and g respectively.

We should then have reached a square array, whose 3-by-5 minor arrays we could treat in the same fashion without additional data, and so prove that if each of the 9 three-line minors be evanescent, of which the non-evanescent alb2 is a common minor, then all the other three-line minors would vanish. Thus, denoting by pqrst any term abqc,de, of the determinant alb2c3d4e5 we have ,b2ce ! Ib-H 7 iI- 1 tltJ2U3tv4s 5 1 Memorias y documentos Madrid , viii.

The third is of no moment. Grelle's Journ. The theorem may consequently be viewed as a generalisation of Salmon's of the year It may be noted, however, in passing that Hesse's notation rather obscures the true character of his result, which is nothing more nor less than the simplest case of Jacobi's theorem regarding a minor of the adjugate.

For dispensing with differentiation by using A,. A and that therefore Hesse's result is aja Fir Studirende an Mittelschulen und technischen Anstalten. The edition in Russian, published the same year, I have not seen. Zelewski's so-called 'Beitrag ' is a very simple but very formally arranged introductory sketch. Appendix to Folkierski's Zasady ruchunku vozniczkowego i calkowego Tom i.

One interesting addition p. Times, xxiii. Times, xv. Annalen, iii. Teorija opred'litelej. It opens with a historical introduction of ten pages, and then follow three chapters pp. The order followed is Bezout's of Hist. It has no advantage save that the r"t permutation from the end is obtainable from the r11t permutation from the beginning by reversing the order of the integers, a fact to which Bourget does not refer. A new definition of 'conjugate' permutations is introduced which entails the equivalence of conjugate and reverse, and which therefore, unlike the old, does not permit of the interesting idea of self-conjugateness.

The fact, too, that the total number of inverted-pairs in a permutation and its now so-called conjugate is i-n n-1 becomes self-evident when we change the word 'conjugate' into 'reverse. Zeitschrift f. As his own contribution he shows that the two numbers are at least of the same kind as regards influence on the sign. In effect he says that if one of the interchanges requisite for the transformation of A into B be the interchange of a,, and a4,, and this be performed, the double set of indices will be changed from x y z w Further, the like being the case for every such interchange it follows that the number of inverted-pairs in the first double set x y z w In het Hollandsch overgebracht door Dr.

In his A nalytische Geometrie des Raumes he had already devoted a lecture to the subject. As might have been expected, his new exposition is in its way excellent, twenty pages being devoted to linear equations and alternating functions, and the rest to sixteen carefully formulated propositions regarding determinants.

It is more suitable, however, for teachers than for gymnasium pupils. Hattendorff, on the other hand, aims at keeping the wants of the young student constantly in view, and therefore assumes less at the outset, advances more slowly, illustrates more fully by examples, and does not neglect to give numerous exercises for practice. Annalen, iv. The first theorem may be formulated as follows: If two arrays C and D, each consisting of n-1 rows and n columns, be so related to one another that in every case the product of the rth and every succeeding row of C by the rt" row of D vanishes, then the principal minors of the last k rows of C are proportional to the principal minors of the first n-k rows of D, the series of minors in the latter case being arranged in reverse order.

Taking for shortness' sake the case where n is 5 and k is 2, the arrays being al a X b, b C5 Z1 Z Z d1 d By the multiplication-theorem the product of these 2am Ian. The proof turns on postulating a set of linear homogeneous equations in ut, u2,.. It would therefore be an improvement to append to the enunciation of the theorem the words " and the members of one of the two series being signed. At the outset pp. The subjects to which they are applied and the manner of application recall Trudi No proof is given. Casopis pro pestovni math.

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Torino, vii. The theorem concerns one of that class of determinants whose every element is a constant function of the corresponding elements of two given general determinants, the exact connection in this case being that the new element is the sum of constant multiples of the parent elements. Proceeding further with the coefficients of the latter type he substitutes for each determinant in them a sum of binary products, one factor being a minor from A and the other a minor from B.

The remaining portions of the two papers are occupied with deductions, some of which fall under Orthogonants and one under iRecurrents. Casopis pro p'Cstovt'nimath. Times, xxiv. Times, xvii. The theorem should be compared with Dodgson's of GGiornale di Mat. The result is thus closely like that reached by Siacci earlier in the same year.

The remainder of the paper pp. It is strange, for example, to find in the short opening chapter valuable space given to two unnecessary though interesting theorems regarding derangements while skew determinants and other important special forms are nowhere mentioned. The theorems in question are 1. The total number of derangements in the n! Save for a very special persymmetric determinant there is nothing else worth noting. It being understood that the given array consists of in rows and n columns, the said increment 1 6 '.

This agrees with the result obtained in a different way by Sylvester in where n - m-1, and where, therefore, the product of the given arrays. M e'langes Math. Liege , 2 xiii. Messenniger of Math. Crelle's Journ. Of course, it is impossible that more than three connecting sets of equations can be independent. A second property is also enunciated and proved, namely, that if in the case of the one determinant all the r-line minors vanish for a particular value of 0 while all the r — -line minors do not vanish, the same holds in the case of the other determinant;-in later phraseology, the 'rank' or 'characteristic' of the two determinants is the same.

Parte seconda.

### Bulletin of the American Mathematical Society

Traduzione del Valeriani Valeriano. Annalen, vii. A little examination will also show that for each case a different selection of data must be made, and that in fact the theorem as stated gives the aggregate of the data requisite for all the values of k. Manifestly there would be considerable advantage in recasting the theorem so as to have indicated the exact data requisite in each case; and this is what Gordan here does. The concluding condition of this does not appear in the original, but it is necessary, as we have already shown. No proof is given, but it can of course be readily effected in the same manner as before, namely, by annexing any s rows to P and any r rows to Q, and then multiplying together the two resulting square arrays.

This suggests, too, -that the solution of a set of simultaneous linear equations is involved in the theorem; for example, the satisfying arrays being a. It is the first instance-not a translationof the subject being dealt with in the language of Holland. Elementar behandelt. As regards the arrangement of matter the second is superior. V [Prispevek k theorii determinantu. Casopis pro pestovdn't math. Prag , Jahrg. The theorem referred to in the title of the second is to the effect that if the difference of any two rows be a multiple of the difference of any other two, the determinant vanishes; and the generalization derived from it is of still less moment.

It may perhaps be usefully compared with Dodgson's sixth proposition regarding the evanescence of arrays.

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Leipzig , xxv. The eighth note concerns Gundelfinger's paper of the same year , the chief object apparently being to register. Following on this are other properties of a null determinant established in his own way. He then proves Hermite's condensation-theorem Hist. His last result is that a determinant is not altered by changing the sign of every element whose place-indices have an odd sum, this change being equivalent to changing the signs of the even-numbered rows and thereafter the signs of the even-numbered columns.

Berlin , Jahrg. Tom ii. Teorya Wyznacznikow i ich przedniejsze zastowania This naturally suggests the statement which is likely to hold for all cases; and on the hypothesis that it is true for the case of p-termed elements he sets about trying to prove it true for the next higher case. Consideration is next given to another ultra-symbolic form and to certain rather fanciful applications pp. Parte Prima. Utile agli studiosi di matematica nei primi corsi universitari. Notwithstanding its greater fullness and general soundness, it cannot be said to show a marked advance in any particular.

The selection of matter is also somewhat arbitrary, some forms of special determinants being treated with considerable fullness and others not even mentioned. Under the head of general determinants very little falls to be noted. The procedure, however, is not so methodical as to suggest generalization.

Like Trudi, therefore, he ended by writing a lengthy book. The first distinguishing feature of it is a fluently written and interesting chapter i. This deservedly brings Rothe into notice, but in the desire for comprehensiveness does the same somewhat unnecessarily for Euler.

A less justifiable peculiarity is a chapter iv. A third noteworthy characteristic is the assigning of a whole chapter vi. The geometrical applications extend to only twelve pages. And, lastly, we may mention that, there being no footnotes, the bibliographical references are collected at the end of each chapter, where they form rather formidable-looking lists. So far as general determinants are concerned, there is nothing fresh to note. A l usage des etablissements d'instruction moyenne.

Nothing so good for beginners had as yet appeared in French. The third, as may be guessed, is still more elementary, having its origin in notelets contributed to the Revue de Vl'nstruction Publique en Belgique , Unterricht, vi. Under the last head the attempted proof of Cayley's property is based on a surprisingly questionable lemma. Atti del R. Istituto veneto, 5 i. Besides those to which we have in the foregoing directed attention he makes mention of his fifth, sixth, seventh and eighth 'riviste di giornali,' as bearing on determinants.

These appeared in the Atti for the years Vierte verbesserte Auflage. Third edition. The matter is as condensed as ever. In the third edition of Salmon the first six chapters are in: creased, but as in Baltzer the added matter had already appeared elsewhere. DI Casopis pro pestovdOfnt math. Eine literarisch-historische Studie. Prag , 6 viii. Mellberg's dissertation, it is true, devotes about seventy pages to an exposition of the theory and its applications, but these are of little moment compared with the fifty pages which precede.

His plan is to enable the reader to judge for himself as to the exact nature and amount of the contributions of the early writers by giving the actual words of the relevant passages in their works. Studnicka's memoir, although avowedly seeking to establish a special thesis, is constructed on the same model as Mellberg's.

His list of authors is the same, save for the inclusion of Rothe and the omission of Binet. For the sake of additional clearness he appends to the account of each one's work a concise statement of what was essentially new in it, and before entering on the subject,of Cauchy's memoir he gives a summing up of these epitomes. The result is a very readable and fair-minded 'study.

Their independent and practically simultaneous appearance,-Mellberg's dated the 1st of March and the other the 24th-show that it had begun to be felt that curt historical footnotes, which necessarily appear in the order fixed by the writer of the text-book for the purposes of exposition, have a tendency io mislead. Zum Gebrauch an Gymnasien, Realschulen u.

Udgivet med bidrag af videnskabernes selskab i Trondhjem. Less than a third part of the exposition is devoted to determinants pure and simple, but the little elementary knowledge thus acquired is utilized to the utmost in the remaining pages. In some respects it is not an advance on similar booklets already in use. The general question which he raised, however, was both timely and appropriate, and was certain sooner or later to be amply discussed by intemperate advocates of a novelty and by natural opponents of change.

Falk's publication, though meant for the same kind of readers as Diekmann's, is of a quite different character, about two-thirds of it being occupied with the establishment and illustration of twelve carefully formulated theorems, including three on Jacobians, and the remainder with forty exercises, worked and unworked, culled from well-known sources. Guldberg's is still less of an elementary introduction, being indeed a reversion to an earlier type and intended apparently for such readers as Brioschi and Baltzer had in view.

It makes no pretensions to freshness of manner or matter. Tract No. The Analyst, iii. Casopis pro pestovdni math. If Todhunter's sketch in his ' Theory of Equations ' had been published separately, it would have been much preferable to Wright's; the latter, however, is somewhat more extended in range. The others are very elementary. Archiv d.

Casopis pro pestovdcni math. Taken together they occupy thirty pages of one and the same volume of the Archiv. In the former, which is the more ambitious, the question of the sign-factor is made a rather serious matter. Most of the points worth noting, however, have already been attended to in dealing with the same author's paper of Trudi is followed in making the sign of a term of a determinant dependent on the combined number of inverted-pairs in the two sets of suffixes; but Janni shows in addition that the interchange of two double-suffixed elements will make no alteration in the sign as thus determined; that therefore the order in which the elements may happen to be taken in the formation of the term is of no real consequence; and likewise that this order may be made such as to necessitate the counting of inverted-pairs in only one of the two sets of suffixes.

Trudi's theorem regarding a partitioned permutation is established only for the case in which the permutation alag The application which he afterwards makes of this in connection with Laplace's expansion-theorem will be readily guessed. Annales de Math, 2 xv. The problem is solved, and the relation between two solutions discussed. He first gives an interesting proof of the fundamental theorem regarding the evanescence of an array.

For better means of comparison between it and Dodgson's proof, let us apply it to the same simple case, namely, where the array to be proved evanescent is aL bl c1 dl el a2 b2 C2 d2 e2 a[3 b3 c3 d3 e3. Frobenius then. We may note, however, that this may be best viewed as an extension of the theorem that if a determinant vanishes, any two rows of the adjugate are proportional, and that a proof on the same lines is readily devisable. What follows on this mainly concerns skew determinants, and is therefore dealt with elsewhere. Times, xxxi.

Times, xxix. Tanner's is of considerably more interest. If the determinant be a15 1, and the sign of al3a21a32a45a54 be wanted, he writes the row-subscripts of the determinant and under them the columnsubscripts, both in their natural order, thus 1 2 3 4 5 1 2 3 4 5;. No justification of Tanner's rule is given. It is not difficult, however, to see that it is an old friend Cramer with a new face, what is counted being the number of inverted-pairs in Papers, x.

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C3 C4 C5 C6 alb2c31 a4bsc. Instead of following the matter up so as to include Sylvester's theorem of , he ends his note by recording the fourteen other like identities connected with a 3-by-6 array, and remarking that only five such relations between the ten products are independent. See under Skew Determinants. Zweite durchaus umgearbeitete, vermehrte und durch ein Aufgaben-Sammlung bereicherte Auflage. Zweite Auflage. Tome i. The historical sketch is improved and lengthened by insertions in reference to Hindenburg, Reiss, Grassmann and others, and by giving deserved attention to Studnicka's views of the preceding year.

The bibliographical references at the end of this chapter are also largely increased; the same, indeed, is true of the whole. The Collection of Exercises pp. The second German edition of Salmon follows closely pp. Serret devotes a chapter pp. The latter are the more interesting, the subjects dealt with being the discriminant still called ' l'invariant' of a quadric, the transformation of a quadric, Sturm's theorem and its application to Lagrange's determinantal equation.

In Polish. The fourth I have not seen. Prag , pp. The application of it, to obtain a result in continuants and another in skew determinants had already been made Brioschi, Trudi, Download preview PDF. Skip to main content.

### Introduction

Advertisement Hide. This process is experimental and the keywords may be updated as the learning algorithm improves. This is a preview of subscription content, log in to check access. Google Scholar. Paris: Ph. Binet, J. Boyer, C. History of analytic geometry. New York: Scripta mathematica.

Bourbaki, N. Elemente der Mathematikgeschichte , translated by A. Brioschi, F. La teorica dei determinanti e le sue principali applicazioni. Padua: Eredi Bizzoni. Cauchy, A. II, 1, pp. Paris: Gauthier-Villars, II, 3. II, 12, — Cayley, A. On a therorem in the geometry of position.

Cambridge Mathematical Journal 2 , — All page references are to The collected mathematical papers , vol. I, 1—4.

Cambridge: University Press, Reprint New York: Johnson, On the theory of determinants. Transactions of the Cambridge Philosophical Society 8 , 1— I, 63— CrossRef Google Scholar. On the theory of linear transformations. Cambridge Mathematical Journal 4 , — I, 80— Cramer, G. Geneva: Cramer and Cl. Edwards, H. In The history of modern mathematics , D. Rowe and J. McCleary, Eds. I, 66— Boston-San Diego etc.

Fontebasso, D. Treviso: L. Frobenius, F.