74 



NATURE 



\Nov, 28, 1878 



Consequently the 2^" power of 16, or the 2" power of 2, 

 increased by i, is divisible by 7 X z^* + i 

 In the second case, we find the remainders 



0000 



9 9 9 7 

 8 2 10 9 



''s = - 4 



^-4 = - 7 



^5 = - 9 



rg = - 4 6 6 II 8 



^7 = - 2 2 3 12 10 



rg = - 9 14 I o 14 



I 10 



o II 



9 4 



8 II 



8 5 



6 10 



rg = - 6 9 14 II 8 II o 



^10 = - 3 14 4 6 IS I 3 



''u = - 7 7 10 s 12 10 4 



rj2 = - 7 12 I 4 5 I 4 



''is = - 7 3 712 8 I 2 



''14= - 5 I 13 9 4 6 4 



^■15= - 3 15 2 2 4 13 3 



»'i«=-i 5 6 9 I 2 6 



''17 = - 2 5 9 12 5 9 3 



'"18 = - 7 13 " 14 2 8 s 



^9 = - 8 3 o o 6 15 7 



''so = - 3 14 3 10 2 II o 



'21 = - I 



Consequently, the 2-^ power of 16, or the 2*' power of 2, in- 

 creased by I, is divisible by 5 x 2'-^ + i. 



The work involved in the first verification can be done by a 

 good calculator in less than half-anhour ; the second is, I think, 



not more than thrice as long, for the divisions are more easily 

 performed. 



Here is one of the eighteen staples of the work of the second 

 verification, namely, the getting of ;'g from r^. 



rs = 9 



2 to 9 9 4 



4 10 



5 4 



5 5 

 5 5 



rg = — 4 6 611 8 811 



The last four lines of the work are made up thus : — In adding 

 the parts of the square depress the last six places to line 2, 

 leaving the rest in line i ; then proceeding to the extreme left, 

 carry the tens figure, in this case o, six places to the right, for 

 subtraction into line 3, and depress the units figare (5) into 

 line 2. Multiply the just depressed figure (5) by 16, and add to 

 it what is found (10) in the place to the right of it in line I 

 (giving 90) ; again carry the tens figure {9) for subtraction into 

 line 3, and depress the units figure into line 2 ; repeat the pro- 

 cess, moving to the right, until line i is exhausted ; then the 

 difference between line 3 and the last seven places of line 2 

 gives line 4, the result required. 



For the sake of safety, before proceeding to calculate r^, 

 calculate r^ again from the complement of rg with reference 

 to the divisor, in this case from +7 13 5 6 6 13. If 

 the same result is again obtained, you may go on confidently. 



Hampstead John Bridge 



Vulcan and Bode's Law 



In the year 1778 — ^just a hundred years ago — the astronomer 

 Bode published an approximation to a law respecting the plane- 

 tary distances. He took the numbers 



o, 3, 6, 12, 24, 48, 96, 192, 384, 



each after the second being double the preceding ; t» these he 

 added 4, giving 



4, 7, 10, 16, 28, 52, 100, 196, 388, 



numbers which, with the exception of the last, agree very well 

 with the distances of the planets from the sun : — 



3-8, 7-2, 10, 15-2, (27), 52, 95-3, 191-8, 300-3. 



The publication of this law, at a time when the asteroids be- 

 tween Mars and Jupiter were as yet undiscovered, drew attention 

 to Kepler's speculation that a planet was wanting between Mars 

 and Jupiter. Twenty-one years after Ceres, the first of the 



asteroids, was discovered, and then others, until now there are 

 nearly 200, the average distance of the whole beiag 27, and 

 agreeing very well with Bode's number 28. All this is doubt- 

 less known to the majority of your readers. 



In calling attention to the law, while not wishing to attach 

 too much importance to it, I would point out one or two sug- 

 gestions which present themselves. If we place 3 before the o 

 in the first row of figures the line becames 



- 3. o. 3. 6, 12, &c. 



If 4 be now added the numbers are 



I, 4, 7, 10, 16, &c. 



The number 4 in this line represents the relative distance of 

 Mercury from the sun ; may not the number i represent the 

 distance of Vulcan, or more probably the mean distance of a 

 ring of asteroids, of which Vulcan is the brightest ? 



Referring now to the modified law, represented by the 

 numbers 



I, 4, 7, 10, 16, 28, 52, 100, 196, 388, 



if I represents the mean distance of the Vulcan-asteroids, and 

 28 that of the Ceres-asteroids, it is a fact that after the first ring 

 come four planets. Mercury, Venus, Earth, Mars, and after the 

 second ring four planets, Jupiter, Saturn, Uranus, Neptune, the 

 two sets of planets having marked differences as regards axial 

 rotation and density. 



What, then, is beyond Neptune ? The law seems to say, 

 a ring of asteroids at an average distance of 772. The 

 motion of Neptune does not lead astronomers to suspect a 

 planet beyond. Perhaps the optical instruments of the future 

 may help to answer this question, Is there a ring of asteroids 

 beyond Neptune ? B. G. Jenkins 



4, Buccleuch Road, Dulwich 



Irish Bog Oak 



Can you or any correspondent kindly give me the scientific 

 name of the Irish " bog-oak " (fossil) ? It should be either 

 Quercus pedunculata or Q. sessiliflora, the existing species, but 

 though I have seen many specimens, I never got hold of one 

 which would enable me to determine the species, and, for aught 

 I know, there may be some of both. W. F. Sinclair 



46, Guilford Street, W.C. 



OUR ASTRONOMICAL COLUMN 



The Total Solar Eclipse of January ii, 1880. — 

 The central line in this eclipse ends soon after reaching 

 the coast of California, where it is possible totality may 

 be witnessed close upon sunset. Tracing the previous 

 path of the shadow through its long course across the 

 Pacific with the aid of the Admiralty chart, it will be 

 found that the oily islands included within it are the 

 Coquille, Bonham, and Elizabeth Islands, lying near 

 together, between 169° and 170° E. longitude, and be- 

 longing to the Marshall Islands group. The eclipse 

 passes centrally over the largest of the Coquilles, as laid 

 down in the Admiralty chart of this group, according to 

 a calculation in which the moon's place has been made 

 to accord very nearly with Hansen corrected to New- 

 comb, which gives the following track : — 



Long. E. Lat. N. limit. Lat. Cent. line. Lat. S. limit. 



168 ... -f6 44."6 ... -1-6 28 'o ... -f- 6 ii"6 

 170 ... 6 203 ... 6 3-8 ... 5 47-3 

 172 ... 5 57-8 ... 5 41*4 ••• 5 24-8 

 So that the breadth of the shadow in the direction of the 

 meridian does not exceed 33'. Reading off from the 

 chart, it will be found that the centre of the largest 

 of the Coquille Islands is in about 169° 35''5 E. and 

 6°8'*5 N., and, calculating directly for this point, it ap- 

 pears that the total eclipse will commence at 8h. 41m. 253. 

 A.M. on January 12, local mean time, and continue 

 im. 1 6s., and this represents the most favourable condi- 

 tion under which the eclipse can be observed on land. 

 For any other point within the shadow in this vicinity 

 the duration of totality may be determined by the fol- 



