3i6 



NA TURE 



[August 3, 1893 



It seems probable that the "pipes " frequently found in sand 

 will give us a clue as to its formation. In that case silica in the 

 form of quartz is held together by ferruginous matter ; here 

 water holding silica in solution must have passed through the 

 chalk like a vortex, and cemented together masses of chalk with 

 its enclosed flints, with the result that we have a " pipe " of 

 cretaceous matter held together by silica. 



Here also, as on the Norfolk coast, are to be seen rings of 

 flint on the shore, sometimes so placed as to form two or three 

 concentric circles. 



These instances, as well as others, point to the fact that 

 masses due to segregation often assume the form of rings or 

 cylinders. In flint this arrangement of growth is probably 

 much more common than is generally known. I have already 

 suggested (Geol. Mag., June, 1893) a theory to account for the 

 existence of these forms in flint, which, since Lyell's description 

 of them, have been an enigma to geologists. 



Tunbridge Wells, July 18. Geo. Abbott. 



Simplified Multiplication. 



The object of this note is to explain a process of simplifying 

 multiplication. To most people multiplication by 2, 3, 4, 5 is 

 sufficiently easy (and not worth incurring any trouble to make 

 easier) ; whereas multiplication by 6, 7, 8, 9 is decidedly more 

 difficult, so 'm sometimes worth while simplifying. 



Now, by admitting the use of negative digits (which may be 

 marked by a minus sign placed over the digit), we may write — 



6 = 10-4 = lJ, 7 = 10-3 = 13, 8 = 10-2 = 12, 9 = 10-1=11, 



This notation is pretty concise, being no longer than the 

 ordinary notation whenever the extreme left-hand digit is < 5, 



thus — _ 



17 = 23, 39 = 41, 278 = 322, 196 = 204, &c. 

 whilst it requires one digit more than the ordinary notation 

 when the extreme left-hand digit is not < 5, and is followed by 

 a negative digit, thus — 



99 = 101, 789 = 1211, 676 = 1321, 5678-13322, &c. 



The preparation of a number so as to contain no digit >5 (by 

 introduction of negative digits, each not > 5) will be called (for 

 shortness) Reduction : the process is so very simple that the 

 " reduced " number can always be written down at sight (a most 

 important matter). 



To form the product P of two given numbers M, N, either 

 one or both of the factors M, N may be "reduced" as a 

 preliminary to multiplication. If both factors be reduced, the 

 rule of signs of algebraic multiplication must be used, viz. 

 -f X -f = -)- , and - X - = -I- ; but -f x - = - , and - x -|- = - 

 This "reduction " of both factors is particularly useful when 

 many large digits occur in succession in both factors, in which 

 case the whole of the multiplication can often be done mentally 

 (without even writing out at length), thus — 



99== 101-= 10201= 9801 



999- = lOOP = 1002001 = 998001 



998^ = 1002- = 1004004 = 996004 



The following factors become particularly simple by this 



" reduction," viz. 



999...9 = 1000 1, 888. ..9 = mi 1 



777. ..8 = 1222 2, 666. ..7 = 1333 3 



When the results cannot be readily done mentally, the multipli- 

 cation may he done by writing out at length in the usual luay 

 (attending of course to signs), thus — 



89 =111 789= 1211 



89 =111 789= 1511 



_111 



m 



111 

 89= = 12121 =7921 



1211 

 1211 

 5422 

 1211 



•. 789= = l422521 = 622i;21 

 It will be seen that the ease of the above procedure depends 

 chiefly on the digits being so small (in both factors) as not to 

 involve any carrying from digit to digit in the multiplications ; 

 this will always be the case when no digit exceeds 3 or 3 (because 

 the greatest product 3x3 = 9 only). But when the digits 4, 5, 6 

 occur in either factor, this will usually involve carrying in the 

 multiplications (because 3 x 4 and 2 x 5 are both >9). In this 



NO. 1240, VOL. 48J 



case it is generally better to " reduce " one factor only, and by 

 preference that factor which has the greatest number of large 

 digits (i.e. 7's, 8's, 9's), and further to use this factor as "multi- 

 plier," keeping the other factor unreduced as multiplicand. 

 Further, it is often convenient in this case (especially when the 

 factors are large) to completely separate the positive and nega- 

 tive products, add them separately, and finally take the difference 

 of these sums ; this will be the required product : this procedure 

 (of using negative digits only in the multiplier, and then separ- 

 ating the -f and - products) has the great advantages that (1)' 

 no further attention need be paid to the signs, and (2) the final 

 line has all its digits necessarily positive, so is itself the required 

 product (in ordinary notation). 



£x— Given M = .34,892, N = 89,795 ; to find M x N. 

 Choose N as "multiplier," because it contains four large digits. 

 The work proceeds thus — 



34 892 =M 

 110 215 =N 



174 460 =5xM 

 3 489 2 =lxM 



3 489 374 460 =Positive sum=^ 



All 

 nega- 



i 



348 92 

 6 978 4 

 348 92 



= lxM 

 = 2xM 

 = lxM 



356 247 32 = Negative sum = « 



.-. j>...K = 3 133127140 = Product MxN 



It will be seen that this process requires two more lines than 

 the ordinary process (viz. the two lines /, k), but the actual 

 multiplications are far easier. 



It is obvious that the two lines/, « may be separately tested 

 by the usual processes of " casting out the nines, elevens, &c." 



The whole process above is so simple that it might well find 

 a place in works on elementary algebra immediately after the 

 explanation of the ruleof signs in multiplication ; it is thoroughly 

 practical, and having been much used by the author, can be 

 confidently recommended. 



The use of negative digits, as above explained, may also be 

 applied to the process of division, and in some cases with ad- 

 vantage. This application is, however, in general by no means 

 quite easy, so cannot be recommended as a practically useful 

 process. 



[This process — as applied to multiplication — is not of course 

 new ; but it seems worth while to attempt to revive it now ; as a 

 process, somewhat the same in principle, has just been published 

 (in the Annales des Fonts et Chausshs for April 1893, p. 790) by 

 Mr. Ed. Collignon. The orAy actual multiplication required in- 

 his process is by the digits 2 and 5 ; the elimination of actual 

 multiplication by 3, 4, 6, 7, 8, 9 is of course an immense ad- 

 vantage. To this end he first shows how to "reduce "any 

 number N to the algebraic sum (say Ni-fNj-Nj) of three 

 others, Nj, N,, N3, composed solely of the four digits 0, 1, 2, 3. 

 To multiply two numbers M, N, one of them, say N, is to be 

 "reduced" as explained: the products MNj, MN„, MN3 are 

 then to be formed in the usual way ; their algebraic sum, 

 MNi-t-MNj-MNj is the product required. The process has 

 two decided defects, viz. — (1) the "reduction" of N is some- 

 what troublesome ; (2) the forming and adding the three pro- 

 ducts (MNj -f MN„ - MN3) is a good deal longer than the ordinary 

 process.] " Allan Cunningham. 



Thunderstorm Phenomena on the M^tterhorn. 

 In 1888-1889 I witnessed some eight-and-twenly thunder- 

 storms on the Pampas of South America; and came to the 

 conclusion — 



(1) That there was no reason to suppose that the so-called 

 "sheet-lightning," or "summer-lightning," is anything more 

 than the glare of distant spark-discharge ; 



(2) That by far the greater number of discharges took place 

 between different layers of cloud, and not between clouds and 

 the earth ; . , 



(3) That the origin of these storms lay in the electrical 

 excitation due to the friction between opposed currents of air 

 (carrying cloud), upper and lower respectively. 



This year I was witness of a thunderstorm under very differ- 



