286 Notices respecting New Books. 



the things concerned ; and others, into Corollaries from the rest. 

 That which declared that the whole is greater than its part, has been 

 omitted as amounting only to an identical proposition, that the great- 

 est is greatest.' And on the attempt to solve the vexata qaestio of 

 Parallel Lines, it is added, that ' the use made of motion or moving 

 magnitudes in the 11th and 12th Books of Euclid, has been con- 

 sidered as sufficient for the introduction of a straight line that moves 

 along another straight line keeping ever at right angles to it ; without 

 infringement of the assertion in the title-page, that no new principle 

 has been introduced. But if this should not be conceded, the impu- 

 tation is still only on the correctness of the title-page.' 



The process by which the establishment of the Theory of Parallel 

 Lines is sought, is by demonstrating that if at the extremities of any 

 straight line, two other straight lines be drawn towards the same 

 side, equal to one another and making equal angles with the first 

 straight line or base, and their extremities be joined, the angles 

 opposite to the base shall be equal to one another, and the side op- 

 posite to the base shall be parallel to the base ; and subsequently 

 endeavouring to prove, that if the angles at the base are right angles, 

 the angles opposite to the base shall also be right angles. After 

 which, it would be easy to establish the equality of the three angles 

 of a triangle to two right angles, and all the properties of parallel 

 lines. 



The way in which this demonstration is pursued, is by trying to 

 establish, first, That if the equal angles at the base of the quadri- 

 lateral figure are greater than right angles, the angles opposite to 

 the base shall be less. And this is done, by placing a number of 

 such quadrilateral figures side by side so that their bases join, and 

 proving that if these bases be severally produced, they shall cut off 

 greater and greater portions in succession from the side of the first 

 quadrilateral figure, and consequently, if the number of quadrilateral 

 figures be increased, some of them shall meet the series of lines 

 formed by the sides of the quadrilateral figures which are opposite 

 to their bases ; from which, (and its having been previously estab- 

 lished that in each of the quadrilateral figures the side opposite to 

 the base is parallel to the base,) it is inferred that the sides opposite 

 to the bases of the quadrilateral figures are not in one straight line, 

 but make with each other an angle that is less than two right angles. 

 It will be perceived that this contains the principle which was 

 brought forward by M. Legendre in the 7th Edition of the ' Elements 

 de Geometries for the purpose of proving that the three angles of a 

 triangle cannot be greater than two right angles, and was subse- 

 quently withdrawn in consequence of the imperfection of the proof 

 of the other point required, viz. that they cannot be less. The only 

 difference is, that the principle was there applied to triangles ; and 

 here it is applied to quadrilateral figures. The coincidence is noted 

 in the Preface ; but with an intimation that the application to quadri- 

 lateral figures was completed many years before the author had the 

 opportunity of being acquainted with the other demonstration. And 

 as the proof of this particular portion of the subject has been for a 



long 



