June 10, 1910] 



SCIENCE 



919 



toe, their habits are chiefly arboreal and the diet 

 insectivoroug-frugivorous. Tupaia retains many- 

 skeletal features that were characteristic of 

 Eocene unguieulates, e. g., long humerus and 

 femur, humerus with entepicondylar foramen, 

 femur with third trochanter, radius and ulna and 

 tibia and fibula separate, flexible carpus and tar- 

 sus, semiplantigrade, five-toed manus and pes 

 with divergent digit 1, free centrale carpi, as- 

 tragalus without trochlear keels and with a 

 rounded head, vertebral formula 0. 7, D. 13, L. 6 

 or 7, S. 3, Cd. 23-26— and many others. Other 

 features distinctly foreshadow the primate type, 

 e. g., relatively large brain case, broad forehead, 

 large, posteriorly closed orbits, and especially the 

 structural details of the auditory bulla and os- 

 sicles, dentition and astragalus. In Ptilocerous 

 the skull and dentition is even more distinctly 

 lemuroid but the rest of the skeleton is unknown. 

 It is of course possible that these lemuroid 

 characters are entirely due to convergent evolu- 

 tion, but the provisional conclusion is that the 

 Tupaiidse are descended from the Insectivore stock 

 that gave rise to the primates. Attention was 

 called to the resemblances between Ptilocerous and 

 the lower jaw from the Bridger ik)cene described 

 by Mathew as Entomolestes grangeri. The only 

 difi'erences are such as frequently separate more 

 generalized forms from their descendants. 



Fourth Journey of J^xploration in the South 



Seas: H. E. Ceampton. 



The speaker gave a brief account of- the new 

 results obtained in the course of a journey of 

 seven months' duration among the Society, Cook, 

 New Zealand, Tongan, Samoan, Fiji and Hawaiian 

 islands. The organisms forming the material of 

 investigations were terrestrial snails of the genus 

 Partula — a strictly Pacific group. The species 

 differ when a comparison is made of forms occur- 

 ring in neighboring but isolated valleys of one 

 island, in different islands of the same group, and 

 in different groups of islands. The uniform prin- 

 ciple of distribution summarizing the observed 

 facts is, that the degree of geographic proximity 

 of any two comparable regions is correlated with 

 the degree of biological differentiation of their 

 species. 



A description was given of two active volcanoes, 

 namely, of Savaii in Samoa and Kilauea in 

 Hawaii. Other older islands of volcanic nature 

 were brought into relation with these examples, 

 as later stages in the production of deeply-fur- 

 rowed land masses like Tahiti, where conditions 

 are such that isolated valley stations are found 



to be the homes of separate colonies of snails. 

 Regarding the relation of such islands to other 

 weathered peaks like Borabora, to coral atolls 

 and to islands of uplifted coral limestone like 

 many examples in the Cook and Tonga groups, 

 the Darwin-Dana doctrine was contrasted with 

 the views of Agassiz. It was pointed out that 

 the phenomena of distribution in the case of 

 species of Partula gave unquestioned support to / 

 the Darwin-Dana doctrine of a major process of 

 subsidence, although secondary sporadic examples 

 of the reverse process of uplift may be demon- 

 strated at diff'erent points of the South Pacific 

 Ocean. L. Hussakof, 



Secretary 

 Amebican Museum of Natural Histoet 



the philosophical society of washington 

 The 680th meeting was held on May 7, 1910, 



President Woodward in the chair. Three papers 



were read. 



A Method of Precision for Computing Square 

 Roots of Numhers: Dr. R. S. Woodwakd, of the 

 Carnegie Institution of Washington. 

 This method depends on the identity 



ab = i{a+ 5)=(1 — {a — h/a + &)=}. 



Let N be any positive number and write 

 N = ab, wherein a and 6 are any two numbers 

 whose product is N. Write also for brevity 

 x^ {a — J)/a + 6)^ 



Then 



VN=^Vjab) =i((i-f 6) (1 — a!)i 



= i{a+b){l-ix-icc^ )• 



It is seen that if the numbers a and & are 

 properly chosen the series in x will converge very 

 rapidly. They may be so chosen in fact that a 

 high order of precision will be attained from the 

 expression 



i(a + b){l-ix). 



It is seen also that the calculation by means 

 of the latter formula will be simplified if a and 

 6 are so chosen that {a — 6)/(a+5) is the 

 reciprocal of an integer n, or if x~^ = n^. This 

 applies especially in ease ii is one of the natural 

 numbers 2, 3, 5, — When n is an integer the 

 following relations hold: 



0= = Zf . re + 1/ra — 1, b' = N ■ n—l/n+1. 



When the approximation is limited to the first 

 term in x, the exact value of the remainder, or 

 error of the calculation, is 



