$6 4 



SCIENCE. 



Date. 



Brightness. 



Semi-diam. 

 Mars. 



Aberration 

 Time. 















M. 



Dec. 2.0 









i«S 



7-3 



5-3 



8.0. 









1 .2 1 



7-5 



5-2 











1.24 



7 .6 



S-o 



20. 









1.26 



7-7 



5° 











1.24 



7 l 



5° 



Jan. 1.0--- 









1.18 



7.6 



—5-i 



From this it will be seen that Phobcs, even on the 

 most favorable date, will be only about 14" distant from 

 the limb of the planet. In 1877 this satellite was ob- 

 served with the \2%. equatorial of the Morrison Observa- 

 tory when only 7" distant. In the present opposition the 

 satellite will be much fainter, 1 ut on the other hand the 

 brightness of the planet will be considerably diminished. 

 It seems possible, therefore that this satellite may be seen 

 with glasses of moderate size. 



Washington University, Nov., 1881. 



ELEMENTS OF QUATERNIONS.* 

 By A. S. Hardy, Ph. D., Professor of Ma hematics, Dartn outh 

 College. 



The American press may be expected to teem for the 

 next twenty-five years with elementary treatises on qua- 

 ternions, and as this work of Professor Hardy's is, we 

 btlieve, the first of the series, it merits on this account 

 the more attention. The book has a quite neat and at- 

 tractive exterior, and the mechanical execution is very 

 fair, though a few defects in letter press and engraving 

 are noticeable. The experiment of printing small Alphas 

 with an oblique line through them seems to be a failure. 

 See pp. 45 and 60. 



We cannot think the title happily chosen. There is an 

 incongruity, if not positive impropriety, in assigning to a 

 scant text-book intended for beginners in the class-room 

 a name associated these fifteen years with the great and 

 classic work of Hamilton. This however, is a matter of 

 taste. One of the most important and difficult steps in 

 the logical development of the calculus of quaternions, to 

 which their inventor gave no little attention, is that of 

 assigning a versor power to a vector, or of representing 

 rotation by a symbol that had hitherto been appropriated 

 exclusively to vection or translation. This, in the book 

 before us, is disposed of in a few lines, when, even in a 

 treatise where brevity must be studied, it is well worthy 

 of as many pages. There is, also, throughout the work, 

 an unfortunate fondness for the plane, where quaternions 

 are often at a disadvantage, and where their real power 

 and usefulness cannot be exhibited. The author may 

 have intended to thus avail himself of the student's 

 greater familiarity with the geometry of the plane, while 

 introducing him to a new method ; but it ought to be 

 borne in mind that one of the chief claims quaternions 

 have on the teacher of geometry is that they are specially 

 fitted to free the student from the too prevalent restric- 

 tion of his conceptions to two dimensions. A curious ex- 

 ample of this tendency of the book is affordtd near the 

 end in applications to loci. Here the author systemati- 

 cally interprets equations as relating to the conic sections, 

 when in reality they frequently relate to quadrics of rev- 

 olution, the restriction to plane loci having been elimin- 

 ated in the process of their formation ; and when he comes 

 "to transform the proceeding equations into the usual 

 cartesian forms," instead of substituting a trinomial for 

 the variable vector, he imposes a restriction to two dimen- 

 sions by adopting a binomial, and of course comes out 

 with a plane section in place of the sutface itself. Not- 

 withstanding these imperfections, Prof. Hardy has evi- 



* 8°, pp. VIII, 230. Boston, Ginn, Heath & Co., 1881. 



dently studied his subject and written his book with some 

 care, and with a view to the requirements and opportuni- 

 ties of those for whom it is intended, and it will doubtless 

 prove useful as an introduction to quaternions. 



Alex. S. Christie. 

 U. S. Coast & Geodetic Survey, 

 Washington, November 11, 1881. 



LARGE TELESCOPES. 



Professor Edward C. Pickering makes the follow- 

 ing suggestion in regard to mounting a telescope on a new 

 plan. He says : — " The small amount of work accom- 

 plished with large telescopes has often been the subject of 

 unfavorable comment, This criticism applies with espe- 

 cial force in America, where there are nearly a dozen 

 telescopes having an apert ure of a foot or over, besides 

 two of the largest size now in course of construction, and 

 two of twenty-six and twenty-four inches aperture which 

 are unmounted and have been for several years perfectly 

 useless. Among so many it seems as if one might be 

 spared for a trial of the following plan, which, if success- 

 ful, would produce at a small expense far more work than 

 could be obtained with a mounting of the usual form. 



Suppose that the telescope is placed horizontally at 

 right angles to the meridian, and that a plane reflector in- 

 clined to its axis by 45 is placed in front of it. This re- 

 flector may revolve around an axis coinciding with that of 

 the telescope. Such a mounting has been used in transit 

 instruments, and gives much satisfaction in the meridian 

 photometer of the Harvard College Observatory. The 

 principal difficulty with a large instrument would lie in 

 the flexure of the reflector. This difficulty has, however, 

 been overcome in a great measure in reflecting telescopes 

 by various ingenious devices. In the present case, since 

 the reflector rotates only around one axis instead of two, 

 the problem is much simplified. A slight motion at right 

 angles of perhaps 5 would be a great convenience, as 

 will be shown below, and would probably be insufficient 

 to materially affect the flexure. It may be said that it is 

 more difficult to make a plane surface than one that is 

 curved. But the principal effect of a slight curvature 

 would be to change the focus of the telescope, the aber- 

 ration being much less than the effect of the varying 

 flexure. Let us admit, however, that the best definition 

 cannot be obtained, in considering the purposes to which 

 such an instrument could be applied without disadvan- 

 tage. 



Many advantages will be apparent on comparing such 

 a mounting with an equatorial. Great steadiness would 

 be secured, since the mirror would be the only portion 

 moved, and this would be placed directly upon a low pier. 

 Instead of a large and expensive dome which is moved 

 with difficulty, the mirror would be protected by a small 

 shed, of which the roof could be easily removed. It 

 would therefore be opened and ready for use in a very 

 short time, and would quickly take the temperature of the 

 surrounding air. The object-glass would be mounted 

 directly upon a second pier, and, as it would not be 

 moved, would be in very little danger of accident. The 

 tube could be made of tin or other inexpensive material, 

 as its flexure is of no importance. It could easily be pro- 

 tected from the changes of the temperature so trouble- 

 some in the tube of a large equatorial. If preferred it 

 might even be exhausted of air, or filled with hydrogen, 

 and the effect of the changes of temperature thus greatly 

 reduced. 



The eyepiece could be mounted on a third pier, and 

 would be so far distant horizontally from the mirror and 

 object-glass that there is no reason that it should not be 

 inclosed in a room which may be warmed. The comfort 

 in winter of working in a warm room will be appreciated 

 by those who have used a large telescope in a cold cli- 

 mate. The result is sure to be an increased precision in 



