710 



SCIENGE. 



[N. S. Vol. XIV. No. 358. 



" In the course of the last eight or ten 

 years this journal has published numerous 

 articles on Metageometry, written with as 

 much competence as good sense. We 

 counsel their perusal." 



It will be seen from our quotation, that 

 Professor Barbarin bases his exposition on 

 the method of Saccheri as the simplest. 



The same is true in the other new text- 

 book, ' Manning's Non-Euclidean Geome- 

 try.' (Boston, Ginn & Co., 1901, 8vo, pp. 

 v + 95.) 



Saccheri's first proposition is {American 

 Mathematical Monthly , June, 1894, Vol. I., p. 

 188): 



"If two equal straights, AC, BD, make 

 with the straight AB angles equal toward 

 the same parts : I say the angles at the 

 join CD will be mutually equal." 



On the next page is " Proposition II. 

 The quadrilateral ABCD remaining the 

 same, the sides AB, CD are bisected in 

 points If and S. I say the angles at the 

 join MR will be on both sides right." 



Professor Manning paraphrases these 

 two together on page 5. 



" If two equal lines in a plane are erected 

 perpendicular to a given line, the line join- 

 ing their extremities makes equal angles 

 with them and is bisected at right angles 

 by a third perpendicular erected midway 

 between them." 



Under the heading ' Definitions,' Saccheri 

 says: " Since (from our first) the straight 

 joining the extremities of equal perpen- 

 diculars standing upon the same straight 

 (which we will call base), makes equal 

 angles with these perpendiculars, three 

 hypotheses are to be distinguished accord- 

 ing to the species of these angles. 



" And the first, indeed, I will call hypoth- 

 esis of right angle; the second, however, and 

 the third I will call hypothesis of obtuse an- 

 gle, and hypothesis of acute angle." This 

 Manning paraphrases as follows, under the 

 heading ' The Three Hypotheses ' : 



''The angles at the extremities of two 

 equal perpendiculars are either right angles, 

 acute angles, or obtuse angles, at least for 

 restricted figures. We shall distinguish 

 the three cases by speaking of them as the 

 hypothesis of the right angle, the hypoth- 

 esis of the acute angle, and the hypothesis 

 of the obtuse angle respectively." 



Saccheri's Proposition III. is: "If two 

 equal straights, AC,BD, stand perpendicu- 

 lar to any straight, AB : I say the join 

 CD will be equal, or less, or greater than 

 AB, according as the angles at CD are 

 right, or obtuse, or acute." 



This Manning paraphrases as follows : 

 " The line joining the extremities of two 

 equal perpendiculars is, at least for any 

 restricted portion of the plane, equal to, 

 greater than or less than the line joining 

 their feet in the three hypotheses respect- 

 ively." 



In the same way is paraphrased Saccheri's 

 Prop. IV., the converse of III. 



Saccheri's corollary about quadrilaterals 

 with three right angles is given by Man- 

 ning on page 12. 



Saccheri's Prop. V. is : " The hypothesis 

 of right angle, if even in a single case it is 

 true, always in every case it alone is true." 



In giving this. Manning has : ' If the 

 hypothesis of a right angle,' etc., evidently 

 a slip for his usual the right angle. Of 

 course the Latin original, of which I have, 

 so far as I know, the only copy on this con- 

 tinent, has no article. 



Prop. VI. and Prop. VII. are combined 

 by Manning on p. 13. 



Prop. IX. is reproduced on p. 14. 

 Prop. X. is given on p. 9. 



In Prop. XI. Saccheri with the hypoth- 

 esis of right angle demonstrates the cele- 

 brated Postulatum of Euclid, thus showing 

 that his hypothesis of right angle is the 

 ordinary Euclidean geometry. 



Manning says, p. 27 : "The three hypoth- 

 eses give rise to three systems of geom- 



