530 Notices respecting New Books. 



Bell's previous definitions of mathematics and geometry ; except, in- 

 deed, he means to say that position is " a mensuriible quantity," and 

 yet not one of the objects of geometrical conception. Even then it 

 would be difficult to understand how this quantity which has no 7nag- 

 nitude is to be measured; except, again, its immaterialit)' should aid 

 its mensuration. 



His definition of a line is Euclid's : but he deduces from it as a 

 coroUartj (without showing any reason for his deduction), that " the 

 extremities of a line are points ; and the intersections [plural in 

 the text] of one line with another are points." Yet in the next line 

 he defines straight lines as those which " cannot coincide in two points 

 without coinciding altogether." Out of this " definition " he extracts 

 Euclid's tenth axiom ; and likewise the corollary to Euclid's eleventh 

 proposition as given in Simson. By what process he extracts them, 

 he has not, however, condescended to inform us. It would appear that 

 I\Ir. Bell's conceptions of the character of an axiom, a corollary, and 

 a definition are all equally muddy and confused. We will try to 

 ascertain presently : but we must first finish his first page. 



Defs. 4, 5, 6. " A crooked or broken line is composed of two or 

 more straight lines. 



" A line of which no part is a straight line, is called a curved line, 

 a curve line or curve. 



"A convex or concave line is such that it cannot be cut by a 

 straight line in more than two points ; the concavity of the inter- 

 cepted portion is turned towards the straight line, and the convexity 

 from it." 



We have always understood the word " crooked " to be a vulgar 

 term to express that a thing was not straight, without specifying in 

 what way it " divaricated from rectilinearity." A " crooked stick " 

 is generally " bowed " (scientifically speaking, according to Mr. Bell, 

 curved) ; a "crooked temper " is on the contrary often very " angular," 

 and sometimes " flies oft" at a tangent." A " shepherdess's crook " 

 is a long staflF with a graceful boiv at the top ; a " crooked spine " is 

 a portion of an inscribed polygon of a thousand sides ; and a " crooked 

 path in life" is one composed often thousand " doubhngs," a few rec- 

 tilinear but of erratic directions, and among which some are as com- 

 plicated in their " doublings " as the wreath of roses which Guido 

 Grandi presented to the Royal Society a century and a half ago. 

 This, we suppose, is one of the " precise " contributions made by 

 Mr. Bell to the Geometry of Euclid, and to the intellectual progress 

 of the "rising generation" of England. We cannot, still, bring 

 ourselves to think that this is anything more than a piece of sheer 

 vulgarism in reference to the study of geometry. 



Is Mr. Bell quite sure of the fact that a curve line cannot be cut by 

 a straight line in more points than two ? If his assertion be correct, 

 let our geometry be reformed ; and let his celebrated countryman, 

 Maclaurin, be called to account for leading even his Scottish com- 

 patriots astray on this subject. Let Newton too be arraigned, and 

 Des Cartes put in the pillory. Let the mathematical world no longer 

 labour under the hallucination that a curve of th.e wth degree may 



