NATURE 



{May 1 8, 1882 



Part II. treats of relations between algebraic quantities 

 (functions, &c), the theory of numbers (also continued 

 tractions), the combinatory analysis (including proba- 

 bilities), series and the doctrine of limits, imaginary 

 quantities (operations with the imaginary unit and the 

 geometrical representation of imaginary quantities : note 

 our remarks above on this head under Byerly), the 

 general theory of equations. 



The second of Prof. Newcomb's works before us is 

 " Elements of Geometry " (New York, 1881). An article 

 in our columns (Nature, vol. xxi. p. 293), headed "The 

 fundamental Definitions and Propositions of Geometry, 

 with especial Reference to the ' Syllabus' of the Associa- 

 tion for the Improvement of Geometrical Teaching," 

 gives its readers a hint that some such work as the one 

 before us was even then in the author's mind — "A sum- 

 mary of my own, the latter [i.e. the summary] still in an 

 inchoate state." The remarks in this article showed that 

 their writer was well fitted to address himself to the sub- 

 ject of a geometrical text-book, and the execution is not 

 at all inferior to the promise. The ground taken up is 

 the Euclidian geometry as comprised in the treatises of 

 Euclid himself, Legendre, and Chauvenet. As with the 

 " Algebra," here let Prof. Newcomb speak for himself. 

 As he himself says, the question of the best form of de- 

 velopment is one of great interest at the present time 

 among both teachers and thinkers. The object not being 

 to teach geometry merely, but the general training of the 

 powers of thought and expression being a main object, 

 Prof. Newcomb considers it most important to guard 

 against habits of loose thought and incomplete expression 

 to which the pupil is prone. This he considers is best 

 secured by teaching the subject on the old lines. The 

 defects he finds in Euclid's system are (1) in the treat- 

 ment of angular magnitude ; here he makes two additions, 

 the explicit definition of the angle which is equal to the 

 sum of two right angles, and the recognition of the sum 

 of two right angles as itself an angle. He adopts, from 

 the "Syllabus," the term "straight angle," though he 

 himself inclined (NATURE, loc. cit.) to the use of "flat 

 angle," and considers the German "gestreckte Winkel " 

 to be more expressive. Then (2) the restriction of the 

 definition of plane figures to portions of a plane surface. 

 " In modern geometry figures are considered from a much 

 more general point of view as forms of any kind, whether 

 made up of points, lines, surfaces, or solids.'' In an 

 appendix, " Notes on the Fundamental Concepts of Geo- 

 metry" he returns to a consideration of this subject. 



Features of the book are (1) the practising the student 

 in the analysis of geometrical relations by means of the 

 eye before instructing him in formal demonstrations ; (2) 

 the application of the symmetric properties of figures in 

 demonstrating the fundamental theorem of parallels (cf. 

 German methods and Henrici's conguent figures); (3) the 

 analysis of the problems of construction, to lead the pupil 

 to discover the construction himself by reasoning; (4) the 

 division of each demonstration into separate numbered 

 steps, and the statement of each conclusion, where 

 practicable, as a relation between magnitudes; (5) the 

 theorems for exercise have been selected with a view to 

 interesting the student in the study, and the author has 

 endeavoured to graduate them in order of difficulty ; (6) 

 some of the first principles of conic sections have been 

 unfolded, more especially for the use of students who do 

 not propose to study analytical treatises on those curves ; 

 (7) Euclid's treatment of proportion is " perfectly rigorous, 

 but has the great disadvantages of intolerable prolixity, 

 unfamiliai conceptions, and the non-use of numbers. 

 The system common in American works of treating the 

 subject merely as the algebra of fractions, has the advan- 

 tage of ease and simplicity." But to this last system 

 there are obvious objections, and our author essays with 

 some reserve, a via media. In this part and in the fol- 

 lowing Prof. Newcomb submits his methods to the judg- 



ment of teachers. Feature (8) involves the treatment of 

 the fundamental relations of lines and planes in space. 

 " In presenting it he has been led to follow more closely 

 the line of thought in Euclid than that in modern works. 

 At the same time he is not fully satisfied with his treat- 

 ment, and conceives that improvements are yet to be 

 made." 



It will be gathered that the book covers most of the 

 ground passed over by young students in plane and solid 

 geometry, and conies in their school training. We cor- 

 dially commend both Prof. Newcomb's works to teachers 

 in this country, and we feel sure they will not regret our 

 having called their attention to them so fully in the 

 author's own words, as they will thus see in what way his 

 books are likely to be helpful to them. We have read 

 them with much interest, and feel sure our readers will 

 endorse our favourable verdict upon them. We need 

 only say that the author considers that the study of 

 geometry as here unfolded can be advantageously com- 

 menced at the age of twelve or thirteen years. The 

 volumes, with a third, which we have not seen, on Astro- 

 nomy, form part of" Newcomb's Mathematical Course." 



R. Tucker 



ELECTRICITY AT THE CRYSTAL PALACE 



I V.— Submarine Telegraphy 



IN the stall of the South Eastern Railway Company at 

 the Crystal Palace may be seen a specimen of the 

 first cable core ever submerged. It consists of a slender 

 copper wire coated with gutta-percha, and was prepared 

 at Streathamby Mr. J. Forster. On January 10, 1849, it 

 was submerged by Mr. Walker, at Folkestone, and a copy 

 of the telegram announcing the completion of the 

 work is still preserved. It runs : " 1 am on board the 

 Princess Clementine. I am successful ; 12.49 P-m." Next 

 year a cable was laid between Dover and Cape Grisnez 

 by Mr. Wollaston, but lasted only a few hours. Several 

 specimens of it are shown in the Exhibition by the South 

 Eastern Railway Company and the Post Office. The 

 gutta-percha core was quite unprotected, and it was 

 sunk by means of lead weights attached at intervals. 

 Next year a core, protected by hemp and iron sheathing, 

 was laid by Mr. T. R. Crampton between Dover and 

 Cape Grisnez, and proved so successful, that it is sill 

 working. Specimens of this cable, which has proved the 

 type of all subsequent one-, are also to be seen. 



There are now some 97,200 miles of cable at work in the 

 world, and before this year is ended the hundred thousand 

 miles will have been attained ; for the second Jay Gould 

 Atlantic cable is still unfinished, and the s.s. Sifvertown 

 of the India-rubber and Guttapercha Telegraph Com- 

 pany is now on her way to lay some two thousand miles 

 on the West Coast of Central America. Nearly all 

 this cable has been made in London, and the Telegraph 

 Construction and Maintenance Company alone has 

 manufactured 65,400 miles, and laid it in almost every 

 sea, in depths varying from shoal water to 3000 fathoms. 

 In 1863 the firm was resolved into the existing Com- 

 pany. Specimens of all the cables made by them are 

 exhibited in a large glass case, together with a large 

 map of the world, showing all the submarine and land 

 lines in existence ; those constructed by the Company 

 being marked in red. The most novel feature of their 

 exhibit is, however, a plan for keeping up telegraphic 

 communication between a lightship and the shore. In 

 1870 an attempt was made to establish a floating tele- 

 graph station in the chops of the Channel ; an old man- 

 of-war corvette, the Brisk, being fitted up, and moored in 

 deep water about sixty miles from the Land's End. It 

 was found, however, that as the ship swung with the tide, 

 the telegraph cable fouled with the ship's riding-chain, 

 and likewise became twisted into kinks, which crushed 



