232 



SCIENCE 



[N. S. Vol. XXV. No. 632 



period of deformation for erosion to have 

 greatly lessened or almost destroyed whatever 

 unevenness of form the deformation produced. 

 W. M. D. 



A MATHEMATICAL EXHIBIT OF INTEREST 

 TO TEACHERS 



Foe the henefit of students and teachers of 

 mathematics who may be visiting Columbia 

 University, the department of mathematics in 

 Teachers College has arranged a permanent 

 exhibit of material available for the study of 

 the history and teaching of the subject. One 

 feature of the exhibit is a collection of mathe- 

 matical apparatus and models adapted to the 

 needs of the various grades from the kinder- 

 garten through the high school, including 

 games, mensuration blocks and models usable 

 in geometry and trigonometry. 



In addition to Professor Smith's library of 

 several thousand books and pamphlets upon 

 this subject, there is also available his collec- 

 tion of mathematical instruments — some da- 

 ting as far back as 1450 — of manuscripts, and 

 of engravings and portrait medals of eminent 

 mathematicians. 



The early mathematical instruments ex- 

 hibited include the following: an astrolabe of 

 Arabic workmanship; one of Italian work- 

 manship, signed by the maker, and dated 

 1509; another, a part dating from about 1450, 

 and the rest, including the four plates, from 

 the following century; and one of Paduan 

 workmanship, signed by the maker, and dated 

 155Y, a practically perfect specimen, with five 

 finely engraved plates. There is also a 

 quadrant of the sixteenth century, one of the 

 primitive instruments of trigonometry, bear- 

 ing the early names ' Umbra recta,' and 

 ' Umbra versa,' together with several leveling 

 instruments of the seventeenth and eighteenth 

 centuries. There are also numerous measures 

 of length and weight, of the seventeenth and 

 eighteenth centuries, including the ell and 

 some interesting sets of money changers' 

 weights ; several finely engraved protractors, 

 diagonal scales, and similar instruments ; 

 several sector compasses and compasses of 

 other kinds, of the Renaissance period; a col- 

 lection of typical forms of dials to illustrate 



the application of mathematics to dialling in 

 the Renaissance period, and several armillary 

 spheres of the sixteenth, seventeenth and 

 eighteenth centuries. 



The material used to illustrate the develop- 

 ment of mechanical calculation includes the 

 following: a collection of medieval counters 

 (jetons, reckoning pennies) of fifteenth and 

 sixteenth century woi-kmanship, partly French 

 and partly German, some with the figure of 

 the Rechenmeister seated at the abacus. 

 Books showing the process of calculation by 

 means of counters ' on the line ' are also 

 exhibited. There are also to be seen an Arabic 

 abacus, a Russian tschotii, a Chinese swanpan, 

 a Japanese saroban, a set of Napier's rods, 

 and a set of Korean bones (the modern form 

 of the ancient Chinese ' bamboo rods,' or the 

 Japanese Sangi). Some Japanese books of 

 1698 are exhibited showing the transition 

 from this latter form of computing to the 

 saroban, which took place in Japan about 

 that time. Besides these there are shown 

 several modern calculating machines, inclu- 

 ding the Goldman and Stanley arithmometers, 

 slide rules, and similar devices. There are 

 also available for study, in addition to those 

 displayed, several early treatises showing the 

 use of counters, together with numerous 

 works on the historical development of this 

 phase of arithmetic. This is also extensively 

 illustrated in a collection of stereopticon slides 

 belonging to the department. 



There are in Professor Smith's library about 

 two thousand portraits of mathematicians. 

 Of these it is possible to exhibit only a rela- 

 tively small number. About forty are framed 

 and can readily be examined, and visitors 

 wishing to examine others in the collection 

 are assisted in doing so. This part of the 

 collection represents the work of a number of 

 years and the repeated examination of the 

 stocks of many European dealers. It is par- 

 ticularly rich in the works of early engravers, 

 although containing a considerable number of 

 photographs and modern process portraits. 

 Reproductions of a number of the portraits 

 have been made for school and college use by 

 The Open Court Publishing Co., of Chicago. 



The collection of ISTewtons includes all 



