January 8, 1897.] 



SCIENCE. 



61 



Mr. G. C. Curtis, Assistant in the Physical 

 Geography Laboratory of Harvard University, 

 from his own observations and after sketches 

 made by the writer. B. DeC. Ward. 



Harvard University, 



December 19, 1896. 



COMPLIMENT OR PLAGIARISM? 



Our attention has been called to a communi- 

 cation from Professor George Bruce Halsted in a 

 recent number of Science in which he says that 

 we ' took ' a whole block of problems and a long 

 note from Halsted's Elements of Geometry. 



If Professor Halsted had only printed in 

 parallel columns extracts from Halsted's Ele- 

 ments of Geometry and the corresponding 

 paragraphs in Beman and Smith's Plane and 

 Solid Geometry, his charge of plagiarism would 

 have fallen to the ground. For those, however, 

 who have not the two books at hand, it may be 

 worth while to make a few comments upon his 

 accusation. 



The 01-der of the problems : To bisect a peri- 

 gon ; to trisect a perigon ; to divide a perigon 

 into five equal angles; to divide a perigon into fif- 

 teen equal angles, etc. , is so natural that for this 

 Professor Halsted will surely claim no original- 

 ity. The same order may be found in Newcomb's 

 Elements of Geometry, an earlier book than 

 Halsted's. 



Does Professor Halsted claim that we Hook' 

 our solutions from his book ? A comparison will 

 show only such resemblances as are inevitable 

 when two authors are dealing with the same 

 material. 



It must then be the terminology, and especially 

 the word ^perigon,' which we have been guilty 

 of appropriating. A modern treatment of the 

 subject of angles requires the use of single terms 

 for the angle formed by a half revolution of the 

 moving arm and the angle formed by a com- 

 plete revolution. To designate the former the 

 term straight angle is now fully established ; 

 for the latter we had a choice among such terms 

 as round angle, circum-angle, perigon, full 

 angle, closed angle. After due consideration 

 we chose 'perigon,' a word given in both the 

 Century and Standard Dictionaries, and found 

 in several geometries, among them Faifofer's 

 (perigono). 



Finally Professor Halsted lays especial empha- 

 sis upon the long note which we ' took ' from his 

 Elements. We quote the two notes in full. ^ 



Halsted. 



Remakk.— From the time 

 of Euclid, about 300 B, C, 

 no advance was made in the 

 inscription of regular poly- 

 gons until Gauss, in 1796. 

 found that a regular polygon 

 of 17 sides was inscriptible, 

 and in his abstruse Arith- 

 metic, published in 1801, 

 gave the following : 



In order that the geomet- 

 ric division of the circle in- 

 to n parts may be possible 

 n must be 2, or a higher 

 power of 2, or else a prime 

 number of the form 2ni+i, or 

 a product of two or more 

 dilferent prime numbers of 

 that form, or else the pro- 

 duct of a power of 2 by one 

 or more difierent prime num- 

 bers of that form. 



In other words, it is neces- 

 sary that ?i should contain 

 no odd divisor not of the 

 form 2m-|-i, nor contain the 

 same divisor of that form 

 more than once. 



Below 300 the following 38 

 are the only possible values 

 of n : 2, 3, 4, 5, 6, 8, 10, 12, 15, 

 16, 17, 20, 24, 30, 82, 34, 40, 48. 

 51, 60, 64, 68, 80, 85, 96, 102, 

 120, 128, 136, 160, 170, 192, 204, 

 240, 265, 256, 257, 272. 



Beman and Smith. 



Note. — That a perigon 

 could be divided into 2", 

 3-2n, 5-2n, l5-2n eijual angles 

 was known as early as Eu- 

 clid's time. By the use oi 

 the compasses and straight 

 edge, no other partitions 

 were deemed possible. In 

 1796 Gauss found, and pub- 

 lished in LSOl, that a perigon 

 could be divided into 17 and 

 hence into 17 •2n equal angles; 

 furthermore, that it could be 

 divided into 2ni-|-l equal an- 

 gles if 2m -1-1 was a prime 

 number; and, in general, 

 that it could be divided into 

 a number of equal angles 

 represented by the product 

 of different prime numbers 

 of the form 2m +1. Hence 

 it follows that a perigon can 

 be divided into a number of 

 equal angles represented by 

 the product of 2ii and one or 

 more different prime num- 

 bers of the form 2m4-i. It is 

 shown in theTheory of Num- 

 bers that if 2m-|-i is prime m 

 must equal 22p: hence the 

 general form for the prime 

 numbers mentioned is 2-'P-f-l. 

 Gauss's proof is only semi- 

 geometric, and is not adap- 

 ted to elementary geometry^ 



Of course Professor Halsted is awjare that from 

 the days of Young, possibly earlier, in his Ele- 

 ments of Geometry, 1827, up to the present the 

 substance of Halsted's ' long note ' has been 

 given in the better geometries, as witness 

 Baltzer, Henrici and Treutlein, Chauvenet, 

 Newcomb. 



Professor Halsted's motive in making his 

 charges we leave for others to determine. 

 Beman AND Smith. 



VOLCANIC DUST IN SOUTHWESTERN NEBRASKA 

 AND IN SOUTH DAKOTA. 



Apropos of Prof. Salisbury's note on the sub- 

 ject in Science of December 4th, I would call 

 attention to the fact that the occurrence of 

 volcanic ashes in southwestern Nebraska has 

 long been known. At the same time, uotices of 

 present exposures are of value. The deposit 

 was at first called ' geyserite ' by Prof. S. 

 Aughey before 1880. References to the subject 

 will be found as follows : ' Sketches of Physical 

 Geography and Geology of Nebraska,' 1880, 

 by S. Aughey : American Geologist, Vol. I. , p. 

 877, and Vol. II., pp. 64 and 437; Proceedings 



