IN 



-rAI.KS 



SCALES OF NOTATION 



eight parallel line* at equal distances from each 

 other (fig. 2). From ita ends and |x>irit of bisec- 

 tion draw perpendicular* to tin- eighth "f these 

 parallel line*. SnUlmde the left half of the 

 eighth line ax the Muue half of the nimple scale 

 it subdivide.!, .loin the first subdivision of the 

 ui>peinii>-t line friini it* bisection ith the |>oiiit 

 of hiMftion of the simple scale, and draw lines 

 parallel to thin one from the other ni.ie point- of 

 subdivision. The space between the l.i-.-.-ting line 

 anil the diagonal neare*t it on the lir-t parallel 

 how* one furlong, the space aliove it two, anil no 

 upwards according to the geometrical principle 

 stated Sup|Hise 16 miles 8 furlongs were to be 

 measured, rut the one leu of the divider on the 

 right end of the sixth pjimllel line and the other 

 where the diagonal line sixth from the centre cat* 

 that parallel, and the length required U found. 

 The diagonal node* on mathematical rules are 

 generally engraved with ten parallel lines, so as to 

 give nUiTWOB f tenth-, thU lieing the most 

 generally useful proportion. 



The romfxtrntirf trnle involve* no new principle. 

 It U a Kale drawn in a different denomination to 

 the name representative fraction. A scale in leagues 

 comparative to the scale in mile* would lie three 

 time* as long itself and in each of it* MbdifWm : 

 and if the comparative scale were drawn to the 

 same nnmher of units it might ! inconveniently 

 ! ong or short. Thus, a comparative scale showing 

 20 leagues to the representative fraction given 

 above would he 7 '5 inches, which might be too 

 long for working pnrpcwes or to be exhibited in 

 print or on the plan. But it is not necessary that 

 the same number of units in the larger or smaller 

 denomination lie taken ; and the length of scale 

 for a convenient mimlier is easily found by propor- 

 tion. Thus, in a scale 4O Kiissian versts are repre- 

 sented hy I'.'i inches; draw a comparative scale in 

 Knglish mile*. Show 20 miles. Take the verst 

 roundly OK I HIT Knglish yards. There are 17GO 

 yanls 'in a mile. In 40 versts there arc 46,080 

 yards ; in 20 miles, 35,200 yards. Then 

 MHO i MOO s 1 4* i S* This line 3'4 represents 

 90 miles to the same representative fraction as the 

 Roman scale of verst*. and it can lie divided and 

 subdivided like the simple and diagonal scales 

 above. 



Mralrs of Notation, in Arithmetic, have to 

 do with the representation of numliers of any mag- 

 nitude hy means of a few symbols. We ordinarily 

 express numbers in terms of the first nine digit 

 symlMils ami the syiiilml known as the cipher i.e. 

 ten in all. The number 'ten' is then repii--.-mi-d 

 by 10, a romhination of the 'one' and cipncr sym 

 bols, and so on in the familiar manner. Mat he 

 matically there is no reason why ten should lie 

 phown in preference to any other number as the 

 radix of our common scale of notation. It- con 

 veiiieuce arises from the way in which it suits our 

 numeration or naming of numliers. Historically 

 ihe development of decimal arithmetic of which 

 our denary wale is the highest phase is intimately 

 connected with the fact that man has ten fingers. 

 The full significance of the denary 1 scale will lie 

 bait seen if we take some other number, seven say. 

 as the radix. Our object is now to express all 

 numbers in terms of the cipher and the first -ix 

 digit* only. The numlier seven itself will lie re- 

 proMnted hy 10, eight by II, twelve by 15, fourteen 

 by 30, forty nine, or seven time* seven, by 100, 

 and so on. In other words, 49 to radix ten Is the 

 same ntnnltrr as 100 to radix seven. As another 

 example take the minilier of days in the year, and 

 let it he expressc.il in terms of scales whose radices 

 are twelve, U-n. -en, and three. Kememlieriiig 

 that 308 to radix ten U a concise notation for 

 J x 10* + 6 * 10 f 6, we must, in order to express 



it in the duodenary scale, throw it into a similar 

 form with twelve sulwtituted for ten. We find 



368 = 30 x 12 + 5 



= (2 x 12 + 6) x 12 + 5 

 = 2 x 12* + 6 x 12 + 5. 



Hence 365 to radix ten is the same number as 265 

 to radix twelve. In the other case* we get 



365 to radix ten = 265 to radix ticelve 

 Ix7 4 + 0x7* + 8x7 + l t or 1031 to radix teven 

 = lx3 4 +lx3-t-l x 3 + 1 x 3 + 1 x 3 + 2, 



or 111112 to radix three. 



These examples show that the simplicity of having 

 a few symbols is balanced by the disadvantage of 

 having to use long expressions for large numliers. 

 The attempt to work in other than the denary 

 scale is moreover greatly hampered by our lifelong 

 habit of thinking and naming our numbers accord- 

 ing to a decimal system. 



The fact that there are twelve pence to the 

 shilling and twelve inches to the foot has often 

 -invested the introduction of the duodenary scale. 

 According to this scale twenty -four feet nine inches 

 would be represented symbolically by 20-9. To 

 use this scale we should be compelled to invent 

 two new s\ miiols for what we call ten and eleven. 

 Hut unless we altered our numeration so as to be 

 in accord with the symbolism, the method would 

 lie impracticable. Fur example, we should have 

 to rename fourteen and ttrcnty-six so as to bring 

 them into line with their duodenary symbols 12 

 and 22. At present in all calculations involving 

 shillings and pence or feet and inches we are com- 

 pelled to !/(/;/ partly in the duodenary scale; but 

 the numbers themselves are expressed Wli sym- 

 Ixilically and verbally according to the denary scale 

 and decimal nomenclature. As a matter of history 

 the denary scale is the only one that has ever been 

 used purely ; to establish any other would necessi- 

 tate a complete revolution in modes of thought and 

 habits of language. 



Very similar to the mixture of decimals and 

 duodecimals in the examples just given is the 

 method of sexagesimals, which still survives in 

 the subdivision of hours and degrees. There are 

 sixty minutes to the hour (or degree), and sixty 

 seconds to the minute. This method is of great 

 antiquity, and hod no doubt an astronomical 

 origin. To the early astronomers it offered special 

 facilities for calculation and for representation of 

 fractions. It was used extensively by Ptolemy 

 and the Alexandrian mathematicians, who em- 

 ployed for symUils the usual Greek numerals as 

 tar as the symbol for sixty (see NUMERALS). At 

 its best, however, the sexagesimal notation must 

 have I'crn very cumliersome, even when assisted, 

 as it probably was, by use of the Abacus (q.v.). 

 It is evident that it does not form a pure scale ; 

 to do so sixty distinct symliols including the zero 

 would lie required. The Alexandrians no doubt 

 liorrowed the system from the Chaldeans. In the 

 older Babylonian inscriptions there is found a 

 sexagesimal not At inn identical, in so far as it is 

 a notation, with that used by Ptolemy and his 

 school. The symbolism is of course c|iiit,c different, 

 all numliers 1 icing represented by appropriate com- 

 binations of two cuneiform characters. The num- 

 bers up to nine are represented each by the proper 

 numlier of the simple wedge-shaped character. 

 Ten is symlioliscd by the angle shaped character, 

 two of w'hich give twenty, three thirty, four forty, 

 and five fifty. Mrty, however, is represented by 

 the same simple character as one ; five times sixty, 

 or 300, by the same character as five, and soon. 

 The famous tables of Senkereh contain the square* 

 and cubes of all numbers from 1 to 60, expressed 



