Review of the Cambridge Course ofMathemalics, 313 



nique analitique" of Lagrange, and the " Mecanique 

 celeste" of Laplace, which, doubtless, at present, 

 form the limit to which human ingenuity and mental power 

 have been extended. By having elementary works com- 

 posed with reference to the higher books, and leading nat- 

 urally to them, much of the strength of an enterprising 

 scientific man will be spared, and reserved for some other 

 valuable purpose, which would, otherwise, be exhausted 

 and mispent upon treatises less fitted to guide him safely to 

 his ultimate object. 



We now proceed to notice particularly the volumes be- 

 fore us. It is important to remark, that the arithmetic will 

 be of little advantage to any, who are determined not to 

 take the trouble of thinking, and who have nothing of the 

 spirit of enquiry and investigation. At the same time the 

 book is calculated to awaken and cultivate this spirit. The 

 author first occupies himself with some general remarks on 

 the different kinds of magnitude or quantity, on the proper 

 idea of number, and on the natural mode of forming num- 

 bers. From the observation, that there are no limits to 

 the extention of numbers, he takes occasion to explain in 

 a very luminous manner, the construction of the numerical 

 nomenclature, by which numbers to any extent, are ex- 

 pressed by a small number of terms. This, again, gives 

 him opportunity to illustrate the written numeration, and 

 the fundamental law of it, " that a removal of one place 

 towards the left, increases the value of a figure ten times." 

 This method of expressing all numbers with ten characters, 

 by giving them at once an absolute and a local value, is 

 extremely ingenious, and appears to have originated in In- 

 dia.* (Laplace, Syst. du Monde, p. 368.) It is from this 

 construction, that the extreme facility of all our arithmet- 

 ical calculations is derived, by which the modern system of 

 arithmetic is rendered so much superior to the ancient. 

 We shall have some idea of the merit and difficulty of the 

 invention, by considering that it escaped the genius of Archi- 

 medes and Apollonius, two of the greatest men of antiqui- 

 ty. There is a verbal difference between enumeration as 

 given by Lacroix and other French writers, and as stated 

 in English and American arithmetical books. In both 

 methods of reading numbers, the seventh figure from right 

 to left, is denominated millions. In the English method, 

 the 13th figure is billions, the 19th trillions, and every addi- 

 tion of six places, receives a new denomination, while in 



