288 Notices respecting New Books. 



The next step is directed to demonstrate, that in a series of equal 

 straight lines as before, if a straight line of unlimited length both 

 ways, be moved along the axis in the direction which removes 

 it from the vertex, keeping ever at right angles to the axis ; such 

 straight line shall never quit or cease to meet the series, without the 

 series having previously met the axis. And the proof is rested on 

 the fact, that whenever this moving straight line passes through 

 the extremities of two of the equal straight lines of the series, 

 there must always be two more of the equal straight lines, making a 

 smaller angle with the chord on the side which is towards the axis, 

 than in the case that last preceded; and consequently the only possi- 

 bility there is for the moving line escaping from the series by finding 

 no more of the equal straight lines over which to pass, is by some of 

 these equal straight lines either making no angle at all with the last 

 chord, or falling on the other side of it. But to do either of these, the 

 angle at the cusp must previously have exceeded the magnitude which 

 has been held forward as involving the necessity of the angular points 

 lying in the circumference of a circle, whose centre is in the axis at 

 a point between the travelling line and the vertex. And as this tra- 

 velling line cannot have quitted or ceased to meet the series before 

 it has passed the point in which the series meets the axis; it is con- 

 cluded that if it be ever found to have quitted or ceased to meet the 

 series, the series must have previously met the axis. 



To these preliminaries, succeeds the Proposition, that if the angles 

 at the base of the quadrilateral figure be less than right angles, the 

 angles opposite to the base shall be greater than right angles. And 

 the process for proving this, is by taking a quadrilateral figure of the 

 kind in question, producing one of its equal sides in the direction op- 

 posite to the base for an axis, and placing on each side of it a num- 

 ber of such quadrilateral figures side by side, whose bases will form 

 a series of the nature described in the previous Propositions. After 

 which, the proceedings are directed to establish, that if a straight 

 line of unlimited length be supposed to move along the axis till it cuts 

 it in a point beyond the side of the quadrilateral figure which was 

 produced to make the axis, and thence be still further moved forward 

 in the same direction, it must do one of three things. It must either 

 fall in with two of the angular points of the series, and there make 

 an angle at the cusp less than half the angle at the vertex or than 

 one of the angles at the base of one of the quadrilateral figures; or it 

 must make an angle at the cusp equal to or greater than this angle; 

 or it must be found to have ceased to meet the series altogether, 

 in which event it has been shown thn.t the series must have previously 

 met the axis. In the first of these cases, the demonstration is directed 

 to show, that the angles opposite to the bases of the quadrilateral fi- 

 gures cannot be right angles, because then the sides opposite to the 

 bases would be in one straight line, and there would be two straight 

 lines at right angles to the axis, meeting each other, which is impossible j 

 and that they cannot be less than right angles, because then they must 

 all lie on the other side of a perpendicular to the axis, and a fortiori 

 could never meet the other line. And in the other two cases, refer- 

 ence 



