May 25, 1888.] 



SCIENCE. 



251 



Stieler's Hand-Atlas. Gotha, Justus Perthes. V. 



It is now six years since the last edition of this great worl: has 

 Ijeen issued. Since that time the commercial development of cer- 

 tain regions, and the additions to our knowledge of others, have 

 been so great, that the atlas did not meet the demands of the day : 

 therefore the new edition, the first instalment of which has just 

 been issued, is highly welcome. It is hardly necessary for us to 

 dwell upon the fact that the technical ex'ecution of the maps is 

 artistic and accurate. In the last edition of the atlas a few of the 

 older plates, which were somewhat worn, and not as perfect as the 

 newer ones, were retained ; but these are now altogether elimi- 

 nated. We consider it a great improvement of the maps, that the 

 shading all along the coast which is intended to show the water 

 has been discarded, and that a delicate blue tint has been intro- 

 duced instead. The first instalment contains two new maps, — 

 one sheet of the new four-sheet map of Italy, and one of the four- 

 sheet map of Austria. The relief of Italy is presented here for the 

 first time in an atlas in a clear form, and, what is more, represent- 

 ing the real configuration of the land instead of the old conventional 

 forms. It is founded upon the surveys of the Italian Department 

 of War, which are rapidly being pushed forward. In order not to 

 disturb the impressiveness of the physical features, the railroads 

 are shown as formerly projected roads were generally shown. 



The first sheet of the map of South America may serve as an 

 example of the care with which corrections have been made on the 

 old plates. The sheet embraces north-eastern Brazil and French 

 and Dutch Guiana. The interior of the Province of Pernambuco 

 is entirely new, the northern tributaries of the San Francisco being 

 for the first time shown in their real form. While the old maps 

 showed a series of hills running from north to south, we observe 

 now a well-defined ridge forming the watershed between the prov- 

 inces of Pernambuco and Ceara. In other places, rivers which 

 were shown in solid lines on the old maps, are shown in broken 

 lines on the new ones, indicating that our knowledge is not so com- 

 plete as was formerly assumed. We observe this particularly in 

 the province of Grao Para ; and farther up the Amazon we see, to 

 our surprise, the course of the Rio Trombetas entirely changed, 

 although it was thought that its course was well known. The ad- 

 ministrative boundaries of the provinces of Brazil have also under- 

 gone important changes. 



The atlas, when complete, will contain ninety-five sheets ; the 

 map of the moon, and a few general maps of the old edition, being 

 left out in order to gain room for new detail maps. The following 

 maps have been added to the atlas : two sheets showing the east- 

 ern portion of Austria, a four-sheet map of Italy, a general map of 

 the Balkan Peninsula and four special maps of the same, a map of 

 Africa in six sheets, and western Canada. Besides this, the maps 

 of Germany, Austria, Denmark, and Asia Minor have been replaced 

 by new engravings. 



A Sy?topsis of Elementary Results in. Pure MatJieniatics. By G. 

 S. Carr. London, Francis Hodgson. 8'^. 



This volume of more than nine hundred royal octavo pages is a 

 handbook which must be extremely useful to every one engaged in 

 either teaching or applying mathematics. As its title implies, it is 

 principally a collection of results, more especially of theorems and 

 formulae. For example, the section devoted to the integral calcu- 

 lus, which comprises more than one hundred pages, contains a 

 complete synopsis of all the ordinary integrals, both definite and 

 indefinite, with brief indications of the method of deriving them. 

 The statements are models of condensation, being at once clear and 

 concise. Especial attention seems to have been devoted to the 

 typographical arrangement, which is extremely clear ; the words, 

 numbers, and formulse which are first to catch the eye, and are 

 principally to be used, being printed in large, bold type, while the 

 indications to be subsequently examined are in finer type. 



Notwithstanding the general excellence of the book, it seems 

 susceptible of many improvements, both in its plan and in its de- 

 tails. It cannot displace the text-book, nor is it intended that it 

 should : hence it would have been well to omit all matter for which 

 the student would naturally go to his text-book, as well as that for 

 which no book is needed. This is especially the case with the 

 chapter on elementary geometry, and with large portions of the 



chapters on trigonometry, which might have been omitted or greatly- 

 condensed without diminishing the usefulness of the work. Not- 

 withstanding that the brief demonstrations are concise in the ex- 

 treme, many more are given than have any appropriateness in the 

 book. In most cases it is only the result, and not the proof, which 

 the person using the book will want, and when he does want the 

 latter he will generally know where to find it. More space might, 

 then, have been devoted to advanced subjects, which are not suf- 

 ficiently developed. 



In detail the defects are very numerous, considering the amount 

 of labor and care which seems to have been devoted to the work. 

 The astronomical and physical constants at the beginning of the 

 book are so far from embodying the latest results as to be worse 

 than useless to any one wanting precise values of constants. In 

 the factor-tables it seems almost ridiculous to see a mathematician 

 give zero as the smallest factor of a prime number. It should have 

 been unity, if given at all ; but Burckhardt's plan of indicating prime 

 numbers by a dash is much more convenient. Among the subjects 

 insufficiently treated are regular solids (no mention is made of sym- 

 polar relations), trigonometric series, and determinants. What is 

 given of the calculus of variations might as well have been omitted 

 entirely. 



The term ' eliminant ' being almost entirely replaced by ' result- 

 ant ' in mathematical language, the former should not have been 

 used to the exclusion of the latter. In Section 1628 an invariant is 

 described as multiplied by the modulus of transformation, when in 

 fact the co-efficient may be any power of that modulus. In Sec- 

 tion 1637, Cor. 2, it is stated, that, if any quadric is resolvable into 

 two factors, the discriminant vanishes. But this is not true of the 

 binary quadric, which is the most common one. 



We should naturally suppose that great care had been taken in 

 the printing : it is therefore surprising to see in equation (4) of 

 Gauss's trigonometric formulse, p. 190, 'cosAf,' printed in bold 

 type, instead of ' sin 4 c' 



These defects are not to be considered as materially detracting 

 from the value of a most excellent piece of work, which should be 

 welcomed by all teachers of mathematics. S. N. 



NOTES AND NEWS. 



One method of disposing of the surplus water of the Missis- 

 sippi River that has been proposed has been to construct an outlet for 

 the flood-water through Lake Boyne. Capt. S. S. Leach, Corps of 

 Engineers, formerly secretary of the Mississippi Commission, ex- 

 plained to the Senate Committee on the Improvement of the Missis- 

 sippi River, last Saturday, why this plan is not feasible ; in fact, he 

 characterized it as preposterous. He said that such an outlet would 

 increase the velocity of the river at New Orleans by at least twenty- 

 five per cent. Already it requires the best engineering skill to pre- 

 vent the banks at that point from being washed into the river. If 

 the velocity of the flow should be increased twenty-five per cent, he 

 said, no expenditure of money would make them retain their place. 

 Captain Leach also explained the plan upon which the Mississippi 

 River Commission is now working. He estimated that a system 

 of levees from the mouth of the river to the head of navigation, 

 protecting all points that need additional protection, will cost three 

 million dollars, and that thirty millions would be needed to establish. 

 * a ten-foot channel through the same length of the river. 



— The Hydrographic Office has received a number of reports of 

 peculiar colorings of the sea, of which the following are the most 

 interesting. The captain of the British steamer ' Kathleen ' reports, 

 that April 23, latitude 36° 25' north, longitude 48° 10' west, he 

 passed through about five miles of discolored water. It had the 

 appearance of sulphur floating on the surfafe. The captain of the 

 American bark 'John J. Marsh ' says, that April 27, in latitude 35'' 

 34' north, longitude 74° 50' west, his ship passed through a patch 

 of water as white as milk, the edge of which was distinctly marked, 

 and which was not phosphorescent. The extent of it was about 

 three miles m longitude and five miles in latitude. He found no 

 bottom by sounding "at thirty-five fathoms. The sky was clear, 

 and the stars shone brightly, at the time. The officers of the 

 British steamer ' Lero ' report, that April 25, in latitude 35° 04, 

 north, longitude 58" 16' west, their ship passed through a wide 



