Jan. 7. 1854.] 



NOTES AND QUERIES. 



15 



regular curved polygon, having the two axes for 

 axes of symmetry. The process would then stop. 



I will, however, suppose that the original bound- 

 ary is everywhere rectilinear. It is clear then 

 that, after every cutting, the boundary is still 

 rectilinear. If the creases be at right angles to 

 one another, the ultimate figure may be an irre- 

 gular polygon, having its four quarters alike, such 

 as may be inscribed in an oval ; or it may have 

 its sides so many and so small, that the ultimate 

 appearance shall be that of an oval. But if the 

 creases be not at right angles, the ultimate figure 

 is a perfectly regular polygon, such as can be in- 

 scribed in a circle ; or its sides may be so many 

 and so small that the ultimate appearance shall be 

 that of a circle. 



Suppose, as in Mr. Ingleby's question, that 

 the creases are not at right angles to each other ; 

 supposing the eye and the scissors perfect, the 

 results will be as follows : 



First, suppose the angle made by the creases to 

 be what the mathematicians call incommensurable 

 with the whole revolution : that is, suppose that 

 no repetition of the angle will produce an exact 

 number of revolutions. Then the cutting will go 

 on for ever, and the result will perpetually 

 approach a circle. It is easily shown that no 

 figure whatsoever, except a circle, has two axes 

 of symmetry which make an angle incommensur- 

 able with the whole revolution. 



Secondly, suppose the angle of the creases com- 

 mensurable with the revolution. Find out the 

 smallest number of times which the angle must 

 be repeated to give an exact number of revolu- 

 tions. If that number be even, it is the number 

 of sides of the ultimate polygon : if that number 

 be odd, it is the half of the number of sides of the 

 ultimate polygon. 



Thus, the paper on which I write, the whole 

 sheet being taken, and the creases made by join- 

 ing opposite corners, happens to give the angle of 

 the creases very close to three-fourteenths of a 

 revolution ; so that fourteen repetitions of the 

 angle is the lowest number which give an exact 

 number of revolutions ; and a very few cuttings 

 lead to a regular polygon of fourteen sides. But 

 if four-seventeenths of a revolution had been 

 taken for the angle of the creases, the ultimate 

 polygon would have had thirty-four sides. In an 

 angle taken at hazard the chances are that the 

 number of ultimate sides will be large enough to 

 present a circular appearance. 



Any reader who chooses may amuse himself by 

 trying results from three or more axes, whether 

 all passing through one point or not. 



A. De Morgan. 



THE BLACK-GUARD. 



(Vol. viii., p. 414.) 



Some of your correspondents, Sir James E. Tennent 

 especially, have been very learned on this subject, and 

 all have thrown new light on what I consider a very 

 curious inquiry. The following document I discovered 

 some years ago in the Lord Steward's Offices. Your 

 readers will see its value at once ; but it may not be 

 amiss to observe, that the name in its present applica- 

 tion had its origin in the number of masterless boys 

 hanging about the verge of the Court and other public 

 places, palaces, coal-cellars, and palace stables ; ready 

 with links to light coaches and chairs, and conduct, 

 and rob people on foot, through the dark streets of 

 London ; nay, to follow the Court in its progresses to 

 Windsor and Newmarket. Pope's "link-boys vile" 

 are the black-guard boys of the following Proclam- 

 ation. Peter Cunningham. 



At the Board of Green Cloth, 



in Windsor Castle, 



this 7th day of May, 1683. 



Whereas of late a sort of vicious, idle, and 

 masterless boyes and rogues, commonly called the 

 Black-guard, with divers other lewd and loose 

 fellowes, vagabonds, vagrants, and wandering men 

 and women, do usually haunt and follow the Court, 

 to the great dishonour of the same, and as Wee 

 are informed have been the occasion of the late 

 dismall fires that happened in the towns of Wind- 

 sor and Newmarket, and have, and frequently do 

 commit divers other misdemeanours and disorders 

 in such places where they resort, to the prejudice 

 of His Majesty's subjects, for the prevention of 

 which evills and misdemeanours hereafter, Wee do 

 hereby strictly charge and command all those so 

 called the Black-guard as aforesaid, with all other 

 loose, idle, masterless men, boyes, rogues, and 

 wanderers, who have intruded themselves into His 

 Majesty's Court or stables, that within the space 

 of twenty- four houres next after the publishing 

 of this order, they depart, upon pain of imprison- 

 ment, and such other punishments as by law are 

 to be inflicted on them. 



(Signed) Ormond. 



H. BuLKELEY. 



H. Brotjnckeb. 

 Rich. Mason. 

 Ste. Fox. 



THE CALVES HEAD CLUB. 



(Vol. viii., pp. 315. 480.) 



The Calves' Head Club existed much earlier 

 than the time when their doings were commemo- 

 rated in the Weekly Oracle (Vol. viii., p. 315.) 

 of February 1, 1735, or depicted in the print of 

 1734 (Vol. viii., p. 480.). There is a pamphlet, 



