Feb. 28, 1889] 



NATURE 



415 



The following example of a shortening process is applicable 

 only to proposed even numbers. As such are divisible by 2, 

 it may not be of much practical use ; and only of interest to 

 show what can be done. If the proposed number be 328, 18 

 steps are requisite by general rule ; but 7 steps are sufficient, 

 thus :— 



It must be noted that at the 6th step the sum of additions is 

 248, or 8 less than the next higher square (256 = 16"), and at 

 the 7th step it is 297, or 8 more than the next lower square 

 289 = I7'-^). The mean of these consecutive roots is 16^. 

 The consecutive square roots, corresponding for the 6th and 

 7tli steps, are seen to be 24 and 25. The mean is 24^. 



Then 24| + j6| = 4i I therefore 41 x 8 are factors of 328. 



There are other short ways of working the process, varying 

 with different numbers. For high numbers, when the difference 

 between the factors is very great, I have not completed a 

 shortening process that is altogether satisfactory, but I hope to 

 succeed before long. 



The examples given above are worked out by means of 

 "increasing squares, roots, and numbers." Similar results may 

 generally be obtained by operating with " decreasing squares, 

 roots," &c., but I prefer the "increasing" method. To show 

 the process by "decreasing ;quares," &c., I give two simple 

 examples — 



85 + 4 above 9- 

 Deduct 6 X 2 - I (or 5 -I- .4) =r 9 



Mean 3^ " _ 



(4 + ^3) = 7 



5 from 9- 



Mean 9i 



Mean %\ 



69 + 5 above 8^ 



)educt 5 X 2 - I (or 3 + 2) = 5 



64 = 82 



Then 8 8 or 84 8^ or 9i 9i 

 + 2 -2 +3i -3i +5i -5J 



10 X 6 12 X 5 15 X 4 



Showing 3 sets of factors for 60. 



A table of squares and square roots, such as Barlow's, is 

 requisite for enabling the operator to ascertain readily, as his 

 work proceeds, when the sum of the additions to the difference 

 between the proposed number and the higher square, first used, 

 becomes a square ; and also to show, in connection with 

 shortening processes, to what extent they may differ, at any 

 step, from being square numbeis. 



Whether the principle of this method or rule be useful in 

 working algebraical or other problems I am at present unable 

 to say, but it can no longer be said there is not a direct rule for 

 ascertaining the factors of any number, and consequently of 

 showing whether it be a prime or not. It may be impossible to 

 devise an algebraical formula, but there certainly is this simple 

 arithmetical method, applicable to all numbers. 



17 Morden Road, Blackheath. Charles J. Busk. 



THE FORMATION OF LEDGES 

 ON MOUNTAIN-SLOPES AND HILL-SIDES. 



T T is well known that Darwin attributed to the castings 

 -^ of earthworms the principal part in the formation of 

 these ledges ; although he mentions in his book the case 

 of a valley in Westmoreland, where it was " in no way 

 connected with the action of worms." " It appeared," he 

 concludes, " as if the whole superficial, somewhat argilla- 

 ceous earth, while partially held together by the roots of 

 the grasses, had slided a little way down the mountain- 

 sides ; and in that sliding, had yielded and cracked in 

 horizontal lines, transversely to the slope." ^ 



Ledges of this description are exceedingly common on 

 the mountain-slopes round Caracas, and indeed almost 

 everywhere in Venezuela. They attracted my attention on 

 my arrival in this country, and they did so all the more as 

 it appeared to me impossible to admit the explanation 

 given by the people, that they were the result of the tramp- 

 ling of cattle, for there were no cattle grazing on these 

 slopes, nor could anybody give me trustworthy information 

 that such had been formerly the case. I was pretty soon 

 convinced that this peculiar feature of the surface was 

 due to a downward sliding of the superficial layer, and 

 after having read Darwin's book, a copy of which I had 

 been so happy as to receive from himself in November 

 1 88 1, 1 at once wrote to him about the ledges, stating that 

 I believed the real cause of their formation to be 

 what he had suggested in the passage quoted above, 

 (Having no copy of my letter, I cannot give the exact 

 wording of it.) The 3rd of April (only a fortnight before 

 he closed his great life), he answered me as follows : — 

 " Should you observe the ledges on the mountains, I shall 

 like much to hear the results, though I do not suppose 

 that I shall ever again publish on the subject. Since the 

 appearance of my book, I have become doubtful whether 

 I have not exaggerated the importance of worms in the 

 formation of the ledges. Perhaps they maybe due to the 

 sliding down and horizontal cracking of (the) whole of the 

 surface soil." 



Since that time I have given a good deal of attention 

 to the subject, and the result is, at least in this neigh- 

 bourhood, absolutely in accordance with this latter sug- 

 gestion. There is no reason for giving any importance 

 to the action of worms, these animals being extremely 

 rare in the soil of the slopes. I find in my note-book only 

 six instances of their castings having been observed ; and 

 three, when by the tearing out of plants a worm was 

 brought to light. 



Our mountains are mainly built up of gneiss, which 

 is rather easily converted on the surface into a kind of 

 sandy loam. This surface soil is covered by a dense 



' Dar«in, "The Formation of Vegetable Mould through the Action of 

 Worms " (London, i83i), p. 283. 



