Dec, 9, 1875] 



NATURE 



103 



then C is D, it explains what is meant by its contraposi- 

 tive (if C is not D, A is not B), by its converse (if C is Z>, 

 ^ is B), and by its obverse (if y^ is not B, C is not Z?). 

 This last term we have heard strongly condemned ; it was 

 substituted (see Fifth Annual Report) for the more usual 

 term opposite on the ground that, in logic, two opposite 

 propositions cannot be true together. The terminology, 

 however, to our mind, is a matter of no great consequence. 

 For proving converse theorems frequent use is re- 

 commended in the work of a " Rule of Identity" here 

 given, i.e. if there is but one A and but one B, then if A 

 is B, it necessarily follows that B is A. (De Morgan's 

 ■'lustration is given in Wilson's Geometry.) 



The Straight Line is the subject of Book i., and takes up 

 five sections. Angles at a point. Triangles, Parallels and 

 Parallelograms, Problems, and Loci. Here, in the defi- 

 nitions, we have two difficulties to meet, What is a straight 

 line ? what is an angle f The former is defined to be 

 " such that any part will, however placed, lie wholly on 

 any other part if its extremities are made'to fall on that 

 other part" The latter is stated to be a " simple concept 

 incapable of definition ; " its nature, however, is explained 

 and illustrated in some detail. Parallel straight lines are 

 defined as in Euclid, and Playfair's axiom is Axiom 5. 

 Theorem 21 (Euc. i. 27) is proved as the contrapositive of 

 Theorem 9 (Euc. i. 16) ; Theorem'22 (Eur. i. 29) by Rule 

 of Identity, using Axiom 5. Book ii. treats of Equality 

 of Areas (Theorems, Problems) ; Book iii. -is on [the 

 Circle. Here a novelty is the treatment of Tangents in 

 two sections, directly, then by the method of Hmits. 

 Some, if not all, of De Morgan's suggestions (" Companion 

 to British Almanac, 1849) on this subject' have been 

 adopted here. The Syllabus so far is not a novelty to 

 many of our readers. Those possessed of Mr. Wilson's 

 "Elementary Geometry" (3rd edition) will know that he 

 has in the main, if not altogether, adopted the lines laid 

 down in the Association's work, adding proofs in full, and 

 much interesting illustrative matter. It hardly needs 

 our saying that the method of superposition is freely used, 

 and that alternative constructions are indicated. 



We .come now to Books iv. and v., which cover pretty 

 much the same ground, except that in the former book 

 we have the subject of proportion and its application 

 treated in a thoroughly rigorous method, which is a 

 simplification of Euclid's mode of treatment by multiples. 

 In the latter book the same subjects are treated in a con- 

 fessedly incomplete manner (for commensurables only) 

 for the use of students whose capacities or time may be 

 limited. 



Similar figures, areas, loci, and problems complete 

 Book v. 



We shall conclude our notice by taking a few extracts 

 from the report made by the committee appointed by the 

 British Association "to consider the possibility of im- 

 proving the methods of instruction in elementary geo- 

 metry." 



" It seems advisable that the requisite uniformity should 

 be obtained by the publication of an authorised syllabus, 

 indicating the order of the propositions, and in some 

 cases the general character of the demonstrations, but 

 eaving the choice of the text-book perfectly free to the 

 teacher. , . . The committee recommend^that'the teaching 

 of practical geometry should precede that of theoretical 



geometry, in order that the mind of the learner may first 

 be familiarised with the facts of the science, and after- 

 wards led to see their connection. With this end the in- 

 struction in practical geometry should be directed as much 

 to the verification of the theorems as to the solution of 

 problems. ... It appears that the principle of super- 

 position might advantageously be employed with greater 

 frequency in the demonstrations, and that an explicit 

 recognition of it as an axiom of fundamental assumption 

 should be made at the com.mencement. . . . The com- 

 mittee think also that it would be advisable to introduce 

 explicitly certain definitions and principles of general 

 logic, in order that the processes of simple conversion 

 may not be confounded with geometrical methods." 



The Syllabus now published is under the consideration 

 of this body of distinguished mathematicians, who will 

 report upon its merits and discuss the advisability of 

 giving it the authority of the British Association. In the 

 mean time it will be of considerable service if teachers 

 \rill practically test it for themselves, and make known 

 their views of its adaptation or want of adaptation for the 

 end proposed. We may remark that Def. 38 (when a 

 straight line intersects two other straight lines, it makes 

 with them eight angles, which have received special 

 names in relation to one another) is not quite correct, for 

 the three lines may cointersect, and then six angles only 

 are formed. Introduce the words " in two distinct points" 

 between " straight lines," and " it makes." 



ESKIMO TALES AND TRADITIONS 



Tales and Traditiom of the Eskimo^ with a sketch of 

 their Habits, Religion, Language, and other Pecu- 

 liarities. By Dr. Henry Rink Translated from the 

 Danish by the Author. Edited by Dr. Robert Brown. 

 With numerous Illustrations drawn and engraved by 

 Eskimo. (Edinburgh and London : Blackwood and 

 Sons, 1875.) 

 T~\ R. Rink is probably the greatest living authority on 

 ^-^ all matters cormected with the Greenland Eskimo. 

 The high value of his contributions to our knowledge of 

 Greenland and its people is universally admitted. The 

 English reading public, and English ethnologists espe- 

 cially, will no doubt be grateful to him for having 

 put his " Eskimo Tales and Traditions " into an English 

 dress. The translation is perfectly idiomatic and alto- 

 gether creditable to the author. 



Not the least valuable portion of the work is 

 the introduction, treating of the Eskimo themselves, 

 in which, in a few short chapters, Dr. Rink pre- 

 sents a succinct and clear statement of all tthat is 

 at present known of these interesting people. For 

 his present purpose Dr. Rink divides the Eskimo 

 into seven groups, groups which, we think, have quite 

 marked enough distinctions to be regarded as convenient 

 for most other purposes ; they are as follows : — i. The 

 East Greenlanders ; 2. The West Greenlanders ; 3. The 

 Northernmost Greenlanders or Arctic Highlanders of 

 Sir John Ross ; 4. The Labrador Eskimo ; 5. The 

 Eskimo of the Middle Regions, from Baffin and 

 Hudson Bays to Barter Island, near the Mackenzie 

 River ; 6. The Western Eskimo, from Barter Island 

 to the west and south ; and 7. The Asiatic Eskimo. 



