Notices respecting New Books. 139 



posed, they are but the same thing stated in other words ; or 

 rather they tend, by different methods, to a similar result : the 

 real contrast will be found to exist in the affirmation of a posi- 

 tive quantity, (as a X + ^,) and in the affirmation also {not the 

 negation) of a negative quantity, (as —ox + b ;) and this con- 

 trast is exemplified in the contrary results of the first and second 

 cases stated above. 



Two observations only, in the way of elucidations, further 

 occur to me: it should be remembered that in the multiplica- 

 tion, as in other processes of algebra, the signs only affect the 

 signs, and the quantities the quantities : in the multiplication 

 — a X — Z; to perceive the separate effect of the one sign upon 

 the other, let us suppose b equal to unity, and we shall find, ac- 

 cording to the foregoing principles, — nx— h=+a; then 

 taking any other value of /', as 10, the result will be — a x — Z; 

 =z + IQa = + ab. To show that the result of + a x + b 

 must be similar to that of - « x — Z-, it may be observed that 

 as a new result of the multiplication + a x + b \s produced by 

 the alteration of one of the signs, as — a x + b, so the original 

 result will again be equally and indifferently brought about either 

 by the restoration of the original sign belonging to a, when it 

 will stand, as before, + a x -h b ; or by the further additional 

 reversal of the sign of the remaining term ( + b,) when the same 

 product will be represented by — a x ^ b: hence + a x + b 

 = _ a X — b ; and hence a negative quantity multiplied into 

 a negatire quantity gives a positive result. 



Trusting you \vill excuse a degree of prolixity, and even of 

 tautology, which appeared necessary to elucidate a subject ex- 

 tremelv exposed to difficulty and misconstruction, 

 I remain, Sir, 



Your very obedient servant, 



Paddington Green, Feb. 20, 1815. JoHN DiLLON. 



XXVI. Notices respecting New Books. 



A Treatise on the Construction of Maps; in tvkich the Projec- 

 tions of the Sphere are demonstrated, and their various prac- 

 tical Relations to mathematical Geography deduced and ex- 

 plained, 'iysttmaticaUy arranged, and scientificnUy illvslrated 

 from Twenty-eight Plates of Diagrams ; ivith an Appendix 

 and copious Notes. By Alex. Jamieson. 8vo. pp. 202. 



JL hR claims of the author are modest : " In a science that has 

 outlived the vicissitudes of two thousand years, and become 

 splendid amidst even the riot of barbarism, originality is hardly 



t» 



