NA TURE 



537 



THURSDAY, OCTOBER 4, 1888. 



DE TERMINANTS. 



Teoria Elemental de las Determinantes y sus principales 



aplicaciones al Algebra y la Geometrla. Por Fdlix 



Amore'tti y Carlos M. Morales. (Buenos Ayres : 



Jmprenta de M. Biedma, 1888.) 

 " Q? O ME books," says Bacon, " may be read by deputy, 



*~-} and extracts made of them by others ;" and, at any 

 rate so far as English readers are concerned, the work 

 now under review belongs to this category. A very con- 

 siderable portion of it is taken up with translations of 

 selected passages from Muir's "Treatise on the Theory of 

 Determinants," of which the following is a sample : — 



" Teorema III. — Toda determinante centrosime'trica del 

 of den in\esimo es igual & la diferencia de los cuadrados de 

 dos sumas de determinantes menores del orden m< ; "' w " 

 formadas con las m. primer as jilas." 



" En efecto ; el producto de las dos determinantes 

 factores, por ejemplo, D y D', en el caso del pa>rafo 37, 

 es igual a 



MD + D'P-KD-D'PJ, 

 y si D y D' se desarrollan en funcidn de determinantes de 

 elementos monomios (22), las determinantes de una de las 

 expresiones son iguales a las de la otra. Luego queda 

 demostrado el teorema." 



The above is almost word for word the same as § 138 

 of Muir's " Treatise," which we subjoin for the sake of 

 comparing the translation with the original : — 



" A centro- symmetric determinant of the 2m'* order is 

 expressible as the difference of the squares of two sums of 

 minors of the m"' order formed from the first m rows. 



" The product of the two factors, D and D' say, in 

 the first case of § 137 is equal to 



{i(D + D')} 2 -U(D-D')P, 



and when D and D' are expanded (§ 29) in terms of 

 determinants with monomial elements, the determinants 

 in the one expansion are in magnitude the same as those 

 in the other : hence the theorem." 



We notice that Muir's formula is incorrectly printed in 

 the translation ; but it is only fair to add that such 

 inaccuracies are rarely met with in the volume before us, 

 which is more free from misprints than the first editions of 

 mathematical books usually are. The translators do not 

 appear to have caught the exact meaning of the words 

 " are in magnitude the same as," which they have changed 

 into " son iguales a\" Quantities which are the same in 

 magnitude (though differing, it may be, in sign) they call 

 equal, and are consequently forced to translate the words 

 " are equal " by " son iguales y del mismo signo " as they 

 have done elsewhere in more than one place. But a much 

 worse mistranslation (also from Muir) occurs on p. 85, 

 where the single word " es " is used as the equivalent of 

 " contains the term." A worse mistake than this one 

 could not have been committed, even by those who, 

 according to Hudibras, "translate, — 



Though out of languages, in which 

 They understand no part of speech." 



The above extract is taken from the second of the 

 three distinct portions, or books, into which the " Teoria 

 Vol. xxxviii. — No. 988. 



Elemental " is divided. The first of these books has to 

 do with determinants in general, the second (consist- 

 ing mainly of translations from Muir) treats of deter- 

 minants of special form, and the third is reserved for 

 algebraical and geometrical applications. The nomen- 

 clature adopted in the second book differs in some par- 

 ticulars from that employed by Muir. Thus our authors 

 do not follow him in substituting " adjugate " for the more 

 euphonious and more familiar adjective " reciprocal," and 

 they agree with Scott and others in calling those deter- 

 minants "orthosymmetrical" which Muir names " per- 

 symmetric." We think that their name " determinante 

 lirmisimetrica " is a distinct improvement on the old 

 " zero-axial skew determinant," but we cannot see any 

 special reason for speaking of determinants in which all 

 the elements in one row are equal to unity as " deter- 

 minantes multiples," and we do not consider that the fact 

 of the equality of all the elements in the principal diagonal 

 of any skew determinant is of sufficient importance to 

 necessitate the use of the distinctive appellation " pseudo- 

 simetrica " to denote such a skew determinant. 



The second book contains most of the principal 

 properties of the various kinds of symmetrical deter- 

 minants, and of Pfaffians, alternants, circulants, and 

 continuants, but not of compound or functional deter- 

 minants : these are mentioned, but their properties are 

 not investigated. The short chapter devoted to them 

 merely defines compound determinants, Jacobians, 

 Hessians, and Wrouskians, and then concludes abruptly 

 with these words : " Por mds interesantes que sean estas 

 formas, la indole de esta obra no permite entrar en el 

 estudio de ellas, para el cual se recomienda especialmente 

 el notable tratado del profesor R. Scott, • Determinants/ 

 Cambridge, 1880." 



Here our remarks on the second book (which finishes 

 with this sentence) would come to a close if we did 

 not wish to correct a mistake into which the authors 

 have fallen as to the origin of the name continuants. 

 These they say (see p. 112) " se denominan continuantes, 

 por sugestidn del profesor Sylvester." The real facts of 

 the case are these. Prof. Sylvester was the first to 

 discover the forms called continuants, to which he gave 

 the name of cumulants. It was Muir who suggested the 

 name continuant "as an exceedingly suitable and 

 euphonious abbreviation for continued-fraction deter- 

 minant" and as a " short literal translation of the 

 equivalent term Kettenbruch-Determinante ; which is the 

 received name in Germany" (vide American fournal of 

 Mathematics, vol. i. p. 344 : letter from Mr. Muir 

 to Prof. Sylvester on the word continuant, September 4, 

 1878). 



Of the third book we have very little to say. It is nice 

 easy reading for young beginners, and teaches them how 

 to solve systems of linear equations, how to perform 

 eliminations by means of Euler's, Bezont's as modified by 

 Cauchy, or Sylvester's dialytic method, and how to cal- 

 culate the roots common to two equations or the double 

 roots of a single equation. There is a short chapter in 

 which some of the most simple properties of the resultant 

 of two equations are explained. The last chapter in the 

 book is the only geometrical one ; its principal contents 

 are determinant expressions for the area of a triangle, a 

 quadrilateral, and a polygon, in terms of the co-ordinates 



A A 



