A treatise on plane and spherical trigonometry

288 BIBLIOGRAPHICAL NOTICE. ./1 Treatise on Plane and @herical Tr'i~onometry. By W1LLIAMCHAtYWNI.--r, A. M., Professor of Astronomy and Navigation, United States N a v a l Academy ; Third Edition. Philada.: Lippineott, Grambo & Co., 1854. Great Circle Polraclor. By Prof. W. CHAUVE~CET. The subject of Spherical Trigoaometry is by t:ar the most difEcult o n e which comes under the notice of a student of mathematics; and the intrin. sic dit'tlculties of the subject have been increased heretofi~re by the f a c t that the treatises were either extended to unwieldy length, as well l)y includil~g matters of detail which are seldom of practical use, as By tedious and intricate modes of demonstration; or else in th6 e[lbrt to ;~void this dif~[iculty, fell into the opposite error of too great brerity by the omission of intermediate steps of the demonstrations, to an extent which renders it di[Iie~dt for the student to follow them, and the neglect of matlers whieh, though not formerly used, are now among the m o s t i,nportaut applicati~ms of the science. We have seen no text-book w h i c h has so well hit the happy ,neao between these opposite errors, as that of Professor Chauvenet. The w,rl< is comprehensive, and includes the case of the General Spherical Triangle, of whose properties, so much advantage has been taken in the labors of" Professors Gauss and Bessel ; it is perspicaous~ and as tier as the nature of the subject allows of it, easy--so that we hope fewer candidates ma'~' he discour~ged fl'om their pursuifs than formerly; it is neat and simple in its language, leading the student into no digressions from the plain course of the reasoning. We have no doubt that it will form a very efticient text-book in the best class of our colleges and academies, and hope to see it largely adopted. The Great Circle Protractor, of the same Professor, is a neat and ingenious practical solution of a p,<~blem of great importance to navigators : namely, to find without calculation or elaborate plotting, the shortest distahoe between two points on the surthce of the earth ; that is, the are of a great circle which passes thro,Jgh these two points. Now, this, u p o n an ordinary chart; is a matter of time and dilticulty~and many able mathematicians and practical navigah~rs have enrteavored to attain a solution of sufticient accuracy to render it valuable in practice, while it w a s simple enough to be within reach of" every commander of a "vessel. But though this has been for years laboriously sought for, we believe P r o fessor Chauvenet's plan is the first one which insures success, and a mere inspection of it will suflSee to show; that while its accuracy is sufticiently correct for practical purposes, its simplicity leaves nothing to be hoped for or desired. It consists simply of two superposed charts; the upper one being of transparent material, so that the lines on the one below ma.y be plainly seen }hrough it ; on the lower chart are described the meridians and parallels m stereographic projection ; on the upper, or transparent sheet, a similar set of circles, of which, however, the roeridians represent the systetn of great circles while the parallels serve as circles of distances. It is h a r d l y neeessary to follow out the description and show how the angle w h i c h the great circle at any point of its course makes with lhe meridian, a n d the distances run and to be run, may be at once read on the appropriate lines, as the chart is accompanied by a plain explanation, and directionS, and suffieientlyglucidated by examples to make it clear to any comprehension.