NA TURE 



25 



THURSDAY, NOVEMBER 10, it 



AN IRISH ALGEBRA. 

 The NciJ Explicit Algelwa in Tlieory and Practiie : for 

 Teachers and Intermediate and University Students. 

 By James J. O'Dea, M.A., formerly Professor of 

 Mathematics, Natural Philosophy, and English Liter- 

 ature in St. P'rancis' College, Brooklyn, New York, 

 and St. Jarlath's College, Tuam. Parts I. and II. Pp. 

 .\ + 616, liv. (Longmans, Green, and Co., 1897, 1898.) 



WH.\TEVER maybe thought of the body of this 

 work, there can be no doubt that the preface, at 

 any rate, is remarkably e.xplicit. 



"The ' E.xplicit .-Mgebra' is the result of the Author's 

 earnest desire to facilitate, as much as possible, the labour 

 of masters and students in this department of Mathe- 

 matics, and to enable them to obtain the maximum 

 results at the minimum expenditure of time and trouble." 



Again, 



" The .Author has spared neither time, nor labour, nor 

 expense in his eftbrt to make the work every way worthy 

 of the object for which it has been intended : namely, as 

 a theoretical and practical text -book on Algebra for all 

 grades of Intermediate Education, University Matricula- 

 tion ! Pass and Honours), and First, Second, and Third 

 Class Teachers." 



Finally, having doubtless observed that a certain pro- 

 portion of reviewers derive the substance of their remarks 

 from authors' prefaces, Mr. O'Dea thoughtfully provides 

 us with a well-balanced appreciation of his treatise ready 

 to our hand. 



" The leading features of the ' Explicit Algebra ' are 

 fulness of detail, without being uselessly exhaustive ; 

 lucidity and conciseness of statement ; brevity and neat- 

 ness in the manipulation of examples, which are numerous 

 and \aried, together with copiousness and variety of 

 exercises methodically arranged, while the disposition of 

 the various portions of the work considered as a whole is 

 in strict logical sequence." 



In order that the reader may estimate for himself the 

 justification of this modest prologue, we hasten to give a 

 few illustrations. 



Page I, Definitions 3, 4, 5 : 



" The Symbols of Quantity are the letters a, i, c, 

 (/, T', K', .r, ji, s. These symbols are used to represent 

 numbers." 



" .An Algebraic Quantity is one that is expressed in 

 algebraic language, and is supposed to be known or 

 unknown." 



" .-V Kno\vn Quantity is that which contains a given 

 number of units of the same kmd, and is represented by 

 the leading letters a, d, c, d." 



Here is fulness of detail, without being needlessly 

 exhaustive ! 



Page 12 (the last of three pages devoted to addition) : 



" N.B. — When dissimilar terms which are to be added 

 have a common literal factor, which is called the Unit of 

 Addition, this factor may be annexed to the algebraic 

 sum of the others. 



" E.xample 5. Determine the algebraic sum of 

 2/i- >Ji +'c, y-4 ^'a + t>, 3a + d-2 Jc, ni 4- « - 3 v'^;. 

 NO. I 5 15, VOL. 59] 



'Arrange thus 



2a- si I' + '■ 

 -4^/3-1- /' -1- 3i- 



Sa+ rf - 2, 

 - 5y/a+ HI + n 



(5 V^ - 7) \''2 + < s'* -l)sjl> + (2 sjc - 1)2 s.'<: + <t + m + "." 



Considering that "literal" and "factor" have not 

 been explained, that nothing has been said about surds 

 except a scrappy definition of the " Sign of Evolution '' 

 only intelligible to those who know what a root is, and 

 that the student has actually been left to himself to find 

 out that ia is the product of 2 and a, this is a good 

 sample of Mr. O'Dea's ideas of logical sequence. As 

 another illustration, take the fact that the pupil has no 

 opportunity of practising the use of symbolical language 

 intelligently until he reaches problems on simple 

 equations, p. 165. 



The proportion of theory to practice (and such 

 practice !) in this remarkable book is perhaps one to 

 twenty, on a generous estimate. Here are tastes of the 

 author's quality, when he digresses for a time into the 

 barren wilds of theory. 



"When an algebraic expression containing .x- is divided 

 by .v-«, the remainder is the same as that which results 

 from suljstituting a for x in the original expression. 



"Proof: Let the expression a.x^ -\- bx- -\- c.v + d he 

 divided by .v-a until the remainder R, does not contain 

 X, and let the quotient be represented by Q 

 have 



ax' + b.x- + ex + d 



We then 



Q + R; 



X - a 

 " .\ ax^ + bx^ + ex + d = Q(.v - a) 4- R. 



" This relation holds for all values of x. Hence, since 

 R does not contain x, it will undergo no change what- 

 ever value be assigned to x. Substituting a for .v, there- 

 fore, we get 



a* + cfh + ac + d = Q(a - a) 4- R 



= Q X o + R = R. 

 Thus, 



R = a^ 4- a^i5 4- ac 4- d. 



This principle is called the Residual Theorem." 

 Observe here the charming vagueness of an " alge- 

 braic expression"; the ingenious substitution of R for 

 R/(.r - a), which is not a misprint, because the same 

 thing is done three times on a previous page ; and lastly 

 the use of the same symbol Q to represent two entirely 

 different things, namely the original quotient, and its 

 value when a is put for x. 



Here is the "demonstration" of one case of the rule of 

 signs in multiplication : — 



" - a •< - b = - a X 711 (assuming « = - b) 

 = a X - I X /« = a X -in 

 = 3X -[- b) = a V. b = ab ; 

 .-. -ax - b = + ab." 

 The petitio principii in the second line would be hard 

 to beat. 



As might be expected, Mr. O'Dea's discussion of the 

 theory of indices affords a magnificent display of his 

 peculiar gifts of "conciseness, lucidity, logical sequence," 

 and the rest of it. To give one instance, on p. 37 we 

 have 



" a" _ a X fl X rt . . . (a being taken m times) 

 a" ~ a X a X a ... (a being taken 11 times) 

 = a X a X a . . . {a being taken 111 - n times) 

 — a"'-" : " 



and this is immediately followed by 



