July 5, 1888] 



NATURE 



219 



There is a large collection of examples, and the whole 

 pamphlet is " teres atque rotundus." 



Dr. Casey's " Treatise on Plane Trigonometry " is 

 quite independent of the " Elementary Trigonometry " 

 by the same author. It is a most comprehensive work, 

 and quite as exhaustive as any ordinary student will 

 require. Dr. Casey shows his usual mastery of de- 

 tail, due to thorough acquaintance, from long teaching, 

 with all the cruces of the subject. He has embraced 

 in his pages all the usual topics, and has introduced 

 several points of extreme interest from the best foreign 

 text books. A very rigid proof is given of the exponential 

 theorem, and a section is devoted to interpolation. Dr. 

 Casey approves of, but does not at present venture to adopt, 

 the practice of French authors who use log sin A instead of 

 our old friend L sin A, i.e. he would prefer 1-859 to 9*859. 



Chapters V. and VI., which are devoted to triangles and 

 quadrilaterals, are exceedingly interesting, and contain 

 quite a crop of elegant propositions culled from many 

 fields. Following the course adopted by other recent 

 writers, he gives a systematic account of imaginary angles 

 and hyperbolic functions. " The latter are very interesting, 

 and their great and increasing importance, not only in 

 pure mathematics but in mathematical physics, makes it 

 essential that the student should become acquainted with 

 them." We may remark that Dr. Casey adopts the follow- 

 ing notation : sh, ch, th, coth, sech, cosech, for sinh, cosh, 

 &c. ; and has gone further than his English predecessors 

 in introducing at this early stage the angle r, Hoiiefs 

 hyperbolic amplitude of 6 (r =* amh. 6). Numerous 

 illustrative examples and tables afford practice to the 

 student in this branch. 



The modern geometry has a small niche, and here we 

 note, as one of several small clerical errors come across, 

 in addition to the list furnished, that (440) should have 

 cosecants in place of secants. The special results, which 

 on Dr. Casey's useful plan are numbered consecutively, 

 reach 810. The book is rich in examples, and will be 

 sure to find for itself a place on the mathematician's 

 shelves within easy reach of his hand. 



The object of the author of " A Higher Arithmetic and 

 Elementary Mensuration " is to furnish a work suited to 

 " the senior classes of schools, and candidates preparing 

 for public examinations." A large number of typical 

 exercises are worked out, and the student, being left to 

 observe and think for himself, acquires, or should acquire, 

 a sound practical knowledge of the subject, which 

 the author rightly thinks will be more abiding than the 

 knowledge of rules and definitions obtained by the mere 

 committal of them to memory. For the benefit of 

 beginners, in many of the examples the steps of the 

 reasoning are given at some length, but the student is 

 advised, as he grasps the details, to shorten the work as 

 much as possible in the examples he subsequently works 

 out. The text covers all the ordinary divisions under 

 which arithmetic is discussed in the books, even our old 

 friend alligation having a chapter assigned to it. The 

 last two chapters are devoted to the mensuration of plane 

 surfaces and of solids. There are 400 exercises at the end, 

 in addition to a very great number scattered throughout 

 the book. The whole is a vast storehouse of well-put 

 matter, which should render a reader quite independent 

 of any other text-book, and, we might say, of a teacher. 



Book II. of " The Harpur Euclid" is on the lines laid 

 down in the edition of Book I., and the subject is handled 

 in an interesting manner. There is a sufficient number of 

 good illustrative examples, with assistance enough to enable 

 a thoughtful boy to work them out by himself. We are 

 glad to see a few examples on antiparallels and sym- 

 medians. These lines must soon force their way to a 

 foremost position even in a school curriculum. This is a 

 useful and handy edition brought out in accordance with 

 the Syllabus of the Association for the Improvement of 

 Geometrical Teaching. 



THE BOTANY OF THE AFGHAN 

 DELI MIT A TION COMMISSION. 



The Botany of the Afghan Delimitation Commission. By 

 J. E. T. Aitchison, M.D., F.R.S., Naturalist attached to 

 the Mission. Being Trans. Linn. Soc, Ser. 2, Bot. v. 

 3, pp. 1-139, tt- l ~\% 5 with two Maps. (1888.) 



OF this expedition Dr. Aitchison has already pub- 

 lished, in the Pharmaceutical fournal and Trans- 

 actions, Ser. 3, v. 17 (1887), a report on the drugs, and 

 he is preparing a report on the zoology to appear in the 

 Transactions of the Linnean Society. 



In several previous collections and papers relating to 

 the Punjab flora (" Flora of Jhelum," " Lahul, its Flora 

 and Vegetable Products, " " Flora of Hushiapore," 

 " Hand-book of Trade Products of Leh "), and especially 

 in his Report on the plants of the Kuram Valley, Dr. 

 Aitchison had shown himself an excellent collector and an 

 enthusiastic botanist ; and by the knowledge of the Afghan 

 flora he had acquired in this preceding work he was 

 eminently qualified to make the most of the opportunities 

 afforded on hasty marches and in rough camps. The 

 Secretary of State for India, who employed Dr. Aitchison 

 on this duty, may certainly be well satisfied with the 

 present botanic section of the Report. In 28 quarto 

 pages Dr. Aitchison describes the country traversed, and 

 the general character of the vegetation, interspersed with 

 many economic and agricultural remarks. The re- 

 mainder of the Report consists of a list of the plants 

 collected in order, with descriptions of the new species, 

 most of which are figured. There are about 800 plants 

 catalogued, whereof 53 are new to science. The whole 

 forms a most valuable addition to our scientific know- 

 ledge of an interesting frontier region. Dr. Aitchison 

 started from Quetta'on September 22, 1884, and proceeding 

 west struck the Helmund on October 19 ; following the 

 course of the Helmund and Harut, he was close to Herat 

 on November 4 ; the remaining nine months, up to Sep- 

 tember 1885, he was in Khorassan and Badghis, i.e. in 

 North Cabul. 



The dry region of South-West Asia extends into 

 Western India — into Sind, the Punjab, Rajputana ; 

 but in his " Flora of British India," Sir J. D. Hooker 

 accepts the political frontier of India as his western 

 limit. It is impossible in local floras to find natural 

 boundaries. Beluchistan and Cabul are thus excluded 

 from the " Flora of British India.'' They are included in 

 Boissier's l< Flora Orientalis " ; but Boissier had by no 

 means plentiful material for this frontier. The additions 

 now made by Dr. Aitchison are not to be estimated by 

 the 53 new species alone, but by the further light thrown 



