New Protractor for Trisecting Angles. 243 



not gathered at maturity. The mean of the autumnal months 

 was 49°-70, about three degrees lower than in 1847 ; while De- 

 cember was unusually warm, 42°-66, or seven degrees above that 

 of 1848. It was also very rainy, being 6£ inches; raising quite 

 a flood in the southern branches of the Ohio, and threatening to 

 repeat the freshet of last year. 



Art. XXIV. — On a new Protractor for Trisecting Angles ; by 



J. H. Alexander, Esq. 



* 



The problem of the Trisection of the Angle, as it has always 

 been termed, was no doubt one of the earliest to attract the efforts 

 of the ancient geometers. At least, it is very reasonable to sup- 

 pose that immediately upon the discovery of the method of bi- 

 secting an arc, would follow an attempt to investigate the modes 

 of dividing it into three or any number of unequal parts or, in- 

 deed, in general to divide it in any ratio. We have no evidence 

 to show precisely when this was first undertaken ; nor has the 

 name of any philosopher come down to us, traditionally connect- 

 ed with the announcement or the history of such investigation, 

 as is the case, for instance, with the rectifying lunules of Hypo- 

 crates or the duplication of the cube of Menaechmus. But as 

 there is one Geometer, Dinostratus, to whom has uniformly been 

 attributed the invention of the quadratrix, a curve capable of 

 being applied to the solution of this very problem (although from 

 the loss of all writings of his, we do not know the extent of the 

 aim with which he applied it), we are warranted in ascribing the 

 attempts at the trisection of angles to a period at least as early 

 as his. And as he is reported to have been the friend and collab- 

 orator of Plato, we cannot be far wrong in fixing the date not 

 lower than 400 B. C. ; and a fresh association of interest is thus 

 connected with the problem, in considering that for 2000 years 

 and more its terms and conditions have occupied at some time or 

 °ther the thoughts of all who have devoted themselves to pure 

 mathematics. 



The problem was further piquant; because it shewed three ca- 

 ses admitting of solution by strict geometric construction. These 

 are the trisection of the circle, of the semi-circle and of the 

 quadrant. One can, therefore, appreciate the peculiarly torment- 

 Wg nature of the obstacle which, after the possession of so many 

 steps, prevented the acquisition of more, and which might seem 

 created expressly to baffle, in what it refused, the curiosity that it 

 stimulated in what it allowed. Of all geometrical problems, this 

 was emphatically the coquette. 



But such an epithet would be unjust, if it could apply to the 

 individuality of the question itself; for the obstacle, in fact, la\ 



