16 



KNOWLEDGE 



[November 1, 1889. 



t'riwcr of Cursire Slwrthand. (The Cambridge System.) 

 By H. L.' Callendar, M.A., Fellow of Trinity College, 

 Cambridge. London : C. .J. Clay & Sons, Ave Maria Lane. 

 1889. lieiiiUiKi Prm-tici'. Facsimiles (if Actual U'ritiil;/. 

 By H. L. C,u.LKNDAR, M.A., Fellow of Trinity College, Cam- 

 bridge. London: C. J. Clay & Sons, Ave Maria Lane. 

 1889. — Mr. Callendar has followed up the publication of his 

 " Cursive Shorthand," noticed in Knowledge for March, 

 by the issue of a sixpenny Primer, and two threepenny 

 Reading-books of eight pages each. These will be of 

 assistance to the student. The reading-books are to be 

 multiplied as occasion demands. In these days of short- 

 hand competition Mr. Callendar does well to give to those 

 learning his system advantages oifered by other authors ; 

 though his methodical treatment of the subject will pro- 

 bably save him from the labour of compiling, and his 

 students from the drudgery of studymg, a Cursive Dic- 

 tionary. For reporting scientitic language with ease and 

 accuracy, the systems that depend in practice upon con- 

 sonantal skeletons are wholly inadequate. Only a johied 

 vowel system, such as Mr. Callendar's is, can have any 

 pretensions to deal efi'ectively with technical terms. 



ILrttrvs. 



[Tlie Editor does not hold himself responsible for ihe op 

 statements of correspondents.] 



MORE PR0PERTI1':S OF NUMBERS. 

 I'd till' Kditar iif KxdWLEDGE. 



SiH, — I have just read, with considerable interest, Mr. 

 T. B. Russell's letter on the above subject in the Septem- 

 ber number of Knowledge. 



When the whole of the multiplication-table is dealt with 

 on the lines which he suggests, and the result tabulated, 

 the phenomena exhibited is very curious, and some of the 

 peculiar properties of numbers still further demonstrated. 



It will be remembered, by those who have read Mr. 

 Russell's letter, that he introduces the subject by a refer- 

 ence to the well-known " property which is possessed by 

 the mtegers of all multiples of 9, namely, that when they 

 are added successively until only one tigure remains, that 

 figure will always be 9." 



After applying the same process to three other numbers 

 of the table— 8, 5, and 6— Mr. Russell remarks that while 

 none of them possess the same property as the number 9, 

 " in each case some sort of order or progression was 

 exhibited by the sums of the integers'." 



This " order or progression " is more marked when the 

 process is applied to the whole table. 



I reproduce his experiments with tal)le 8 for reference : 



8x1 =8 



Sx2==lt!C-f-l r= 7 



8 X 3 = 24 2 -I- 4 = (! 



8x4 = 32 3-1-2 =r .5 



8xo = 40 4-t-0 = 4 



8 X G = 48 4 -t- 8 = 12 2 + 1 = 3 



8x7 = oli 5-(-6 = ll 1 + 1 = 2 



8x8 = 04 e + 4 = 10 1+0=1 



8 X i) = 72 7 + 2 = 9 

 and so on. 



Begmuing with the first number of the multiplication- 

 table and applying tlie same process to the first eight 

 consecutive numbers, and tabulating the result in each 

 i-aso, we get the variety of series 'placed in the veni>-{tl 

 cohumis of the following table, under each number respec- 

 ti\-ely. 



If the process be applied to the next eight consecutive 

 numbers of the table after 9, or to the eight consecutive 

 numbers which follow any multiple of 9, the result will 

 be a repetition of the same series in the same order, ad 

 intiHitum. 



It follows, therefore, from tliis, that if the process be 

 applied to consecutively higher multijjh'x of any numbers 

 in the above order the result wiU also be the same, ad 

 injinittim. 



The above table, therefore, exhibits the whole series of 

 numbers which would be obtained by an application of the 

 process to the entire multiplication table. 



Let us note a few of its phenomena. 



(a) The sum of each series is a multiple of 0. The 

 integers of the sum of each series when added each make 9. 



(h) The series in the first vertical column is the same as 

 that in the first horizontal column. That in the second 

 vertical the same as that in the second horizontal, and so 

 on to the end of the table. 



(c) The series in the vertical columns 1, 2, 3 and 4, are 

 the same, but in the reeerse order, as the series in columns 

 8, 7, 6 and 5 respectively, viz,, 1 and 8, 2 and 7, 3 and 6, 

 4 and 6. 



((/) The series m colunm 10 iwlien produced) is the 

 same as in column 1, 11 the same as 2, 12 the same as 3, 

 and so on. 



From (<•) and ((/) we can deduce two simple rules. 



From ((■) we note that any two numbers whose sum is 

 9 (or any multiple of 9) will produce the same series but 

 m the reverse order; e.<j., 9-=81, 22 + 59 = 81. There- 

 fore if the process be apphed to 22 times, and 59 times 

 any eight consecutive numbers ; they will each produce the 

 same series but in the reverse order. 



From {d) we note that if any of the numbers at the 

 head of the table (lugher than 9) be divdded by 9, the 

 reiiuiindcr will indicate the column in which the series may 

 be found which such number will produce ; e.(i., 12+9=1, 

 and 3 remainder. Therefore 12 times will give the same 

 series as 3 tuues. Again, taking a much higher number, 

 78424 + 9=8713, and 7 remainder. Therefore 78424 

 times any 8 consecutive numbers to which the process is 

 applied will produce the same series as 7 times. Any 

 number which divides by 9 without a remainder will, of 

 course, give the same result as 9. 



I suppose all the above are explicable on the same linos, 

 Yours obediently, 



Wji. Stanifobth. 



Upperthorpe, Sheffield : Sept. 19th, 1889. 



[I have to thank very numerous correspondents (26 in 

 all) for letters on the above subject. Mr. Staniforth's 

 letter, perhaps, puts the facts more concisely than any 

 other. The fact that the sum of the digits of any multiple 

 of 9 is itself a multiple of 9, may be made evident thus : 

 With the decimal notation we use any number may be 

 written as a series of powers of (9-1-1) multiplied by 

 digits ; for example, 3591 may be thrown into the form — ■ 

 3(9-M)H5(9fl)-2 + 9(9-M)+l 



