60 Notices respecting New Books. 



words, throughout the greater part of the book Euclid's restriction 

 is tacitly reimposed, and this even in cases where, if the considera- 

 tion of reflex angles is to be thought advisable at all, we should 

 have expected to see them noticed. E. g. Cor. p. 49 [Euclid I. 32, 

 Cor. 2] is expressly limited to any convex polygon ; Ex. 2, p. 39, 

 Ex. 5, p. 40 are not always true if the quadrilateral has a reentering 

 angle. "We raise no objection to this but want of consistency. Se- 

 condly, with regard to the " reflex angle," it is merely said to be 

 greater than two right angles. Now take any Euclidic angle and 

 denote it by 6 ; are we to understand by the corresponding reflex 

 angle 2k— 6 or 2>i7r + 0? If the former, here is a restriction much 

 more arbitrary than Euclid's ; if the latter, we have a conception 

 proposed to a beginner which is well known to be one presenting 

 great difficulty to those who are begnining Trigonometry and have 

 therefore made some progress in Mathematics. Mr .Wilson has an 

 " axiom " which asserts that " an angle has one and only one bisec- 

 tor ;" but upon these terms here is an angle A B which has one, 

 or one of two, or two bisectors, according as A O B means some 

 one, or any one, or all of the series 2>i7r + 6. Thirdly, with regard 

 to the " straight angle." A boy comes to his teacher with the no- 

 tion that an angle is something resembling a corner of a square or 

 triangle; and without delay his teacher introduces him to a notion of 

 this kind : — Draw a straight line A B and take in it any point C ; at 

 C there are two straight angles, one on each side of the line — indeed 

 as many as we please, for there is no occasion to restrict ourselves 

 to a single plane. He comes with a notion that an angle is some- 

 thing between two straight lines that meet at a point, and he is 

 taught without delay that he has only to draw a line an inch long 

 and put a dot upon it to produce an infinite number of angles. 

 This may be all very well for the teacher ; but what is it for the boy ? 

 Not only, however, is it of more than doubtful expediency to put 

 the notion of a straight angle before a beginner, but Mr. Wilson 

 does it in a way which looks as if he had hardly considered all the 

 consequences that ought to follow from its introduction. We will 

 try to explain what we mean. It might perhaps be contended that 

 the theorem " all straight angles are equal " needs distinct proof ; 

 though it is not easy to see what the theorem would add to the 

 axiom, " two straight lines, which have two points in common, lie 

 wholly in the same straight line." Now Mr. Wilson does not deem 

 it necessary to give a formal treatment of this theorem ; but he does 

 deem it necessary to do what is much more surprising. He gives 

 Theor. I. " All right angles are equal to one another." Theor. II. 

 " If a straight line stands upon another straight line it makes the 

 adjacent angles together equal to two right angles." Theor. III. 

 " If the adjacent angles made by one straight line with two others 

 are together equal to two right angles, these two straight lines are 

 in one straight line." Erom Euclid's point of view this would be 

 very proper ; but from Mr. Wilson's point of view a right angle is 

 only half a straight angle. Now let us try the effect of writing in 

 these theorems, half a straight angle for right angle, and straight 



