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CALCULATIONS MADE EASY. ClK Octimal $y$tm -OP- notation and numeration. COMBINING SIMPLICITY WITH THE GREATEST PRACTICAL UTILITY. GEO. H. COOPER. Nxw WmntnnTXB, B. C , Cakada, JmtMry, 1901, Pri»' 2S Cts. THC OOIUMIIAN eOMMNY, LTD. ♦ ♦ Cwenfietb Century Sysiem of nofaiion. CALCULATIONS MADE EASY. Cbe Octimal System -OF- Dotation and numeration COMBINING SIMPLICITY WITH THE GREATEST PRACTICAL UTILITY. GEO. H. COOPER. New West.minstki!, B. r , i'an.u>a, Jaiiiiaiii, I'.xil. ''^, ^ Gh^^-C^ PREFACE In i)Iacni- this l.itlr vv(,rk l.iforc the puhlic I iVc! Iliat I am Init carryinj,' out the pn-mptitiKS of an All- Wise Cna'.or whose laws arc all ilcfinitc in character and who for some inscrutable reason seems often to choose . sons occupying humble p. itiuns in life to hrin,tr int.. vv ce methods of working that tend to the higher de-el- opment of the human intellect and to a more thorou-h understanding: of li:> great work in the creation. I early observed that a'l phenomena were subject to the control of some law. whether understood or obscure, and when in the demonstration of it apparent aberrations occurred. They were due to the operation of another law of equal importance. Hence ^t becomes necessary to seek for causes of error in order to acquire a cijrrect knowledge of natural law. and also to bring into operation correct methods of acquiring the fullest amount of knowledge within our reach. The decimal system of notation is not a scientific method of making calculations, because it is not definite in character and requires the assistance of other methods of conveying ideas of proportional numbers. The octimal system seems to answer all requirements of mathemat- ical high science and if it leads to a wider knowledge and ts I shall feel amply repaid for my r attainmen effort in producing it. THE AUTHOR. INTRODUCTION IIk' present a-e is one of research and enquiry -Man lu.,ks uuh enquiry into the Past, the Present and tlio luture. X., field of experiment or discovery is shut om o. his crclc of investigation. Certain it is tliat this spirit o. enquiry has already resulted in large additions lo the tacts of cons -ious knowledge in the reduction of many compl-x forms to their proper elements, and in the sunpiihcation of methods heretofore laborious and cum- "-■rsonio. A'achines have taken the place of the human i.and and eye, where .lexlerity, exactness, or careful and accurate discrimiualion were demanded, the weakness of man s muscle being insuffcient for great works The .'.ctivity of man's mind has been called into operation wall the result that new forces have been found and new '-^^- nues constructed lor the accomplishment of mightv \v-rk. [n laboratory, in shop, in counting house, and in t.-H-l m every department of labor, the skill of the manipu- >^^;->>- and the discoveries of the investigator, have result- *<1 n, a system or method of operation which has greatlv i'HM-eased the elTicicncy of the laborer and the value o'f the work he has d-nie. i-.m while men have sought out r.ew inventions and f..llowed new methods or revised old ones it has i,een too ""^'" 'I-' oase that the simple has been sacrificed to the ""vel and t-ure has been foisted upon an unsuspecting P"''l"- many laborious and tedious methods of .tudv or "i^cl,an,cal operation simply to introduce the novefand peculiar or lo push the sale of some man's wares. Such can exist no longer for the spirit of the present age .lemands practical utility, produced in the simplest manner possible as the distinguishing characteristic of NOTATION- AND NUMERATION every systtiii or mctliod sockinrr a recogni;^c(l pla.-e in ihc operations of ilie busy life of today. To this end ail our appliances and nieiluxis now in use are bein- sub- jected t,. a r.-id inspection which will result in clcarin- them ot the u.^ of all useless and unnecessarv accretions" or discarding ihcm altuoeilHT, in ,>rder that place may I'C .q-.ven lo simpler forms and such as are better adapted to the demands of the present day. Such investigation, lookin- to simplicitv of method I-.as been ma:le already in many departments of Science and Art. but one field has scarcely been touched, namely that of the Science of Xundjers. It does not seem to have occurred to anyone that our present methods in numbers arc woefully deficient in their adaptability to even the simplest mechanical o])erations. and involve such labor to master as is quite out of reach of the ordi- nary uuhvKlual. or if such has already presented itself to any mmd the result has not been the presentation of anv- thui.i; snipler in Numeration. This may have been due to the !act ihat our present system of notation bein- based upon -iu- unit, it has been thought that there could be no sunpler basis, and such is true, but we find that whde we are to-day building upon such a simple basis smipbcity g-.ves place to the greatest gomplexity when we yew the .structure that has been reared on 'such a basis. Xow while it is true that a too rigid simplicity may lack that efficiency necessary to any adequate sys- tem m any branch of knowledge, it must be conceded that if the same efficiency can be retained, the simplify- mg of any system is a desideratum. There is nowhere perhaps, greater need for simplicity than in the field of Mathematics of today. Our present system is so lacking in Its methods of diyisibility that it is of little use in many mechanical operations where measurements are required while many of its rules and forms of calculation are puz- zbng to young minds and often to those of mature THK OCTIMAL SYSTEM OF thouglit as well. Now a simpler system, with a greater atlaptation to the needs of practical life, and one which can be mastered in a much shorter time is required and such we believe has been found in the Octimal System of Notation. We do not hesitate to claim that this system is of more practical utility than any other, that it will render calculations comparatively easy and that its forms will be easier to retain in the minds of one and all. e^ Disadvantages of the Decimal System. Tlie Decimal System of Xoiaiicn. was ,k-vclupecl long i)ek.rc IVactical Science and Afatliemalics were fac- tors of any importance in the affairs of everv-day life Notiiing was required of the system but tliat it might be apphed conveniently to numerical calculations. Certainly our present system meets this requirement admirably but no better than, if as well as a system with a radix of c'ght as advocated in this treatise. When in later centuries these theoretical calcula- tions becamo of vast importance in almr.st every realm of human activity, it was found that the amount of labor entailed in their practical application w^as enormous. Sciences were developed individually and it was found very difiicult to express the results of calculations in one department, so as to make them of practical value in an- other, for instance, the unit of time, which is the ba- of Horology, bears no direct relaticm to the unit of circular measurement, which is the ba.sis of the science of \.-.yi- gation. nor do units of weight bear any necessary relation to units of volume. The adopiion of the French Met- ric .System would, it is true, reduce the al.oye-mentioned labor to a minimum so far as is possible with a decimal notation, but that the system is utterly unsuited to the demands of industry is proven by the fact that those who are engaged in mechanical operations, do. not and will not make use of it. and are opposed to its adoption since It IS deficient in its adaptability to their require^ ments. I. The majority of men cannot understand the di- vision of an rbject into parts other than halves, (piarters, THE OCTIMAL TYSTEM OF etc. It is only an exceptional individual who can, with any dc,tjrec of accuracy, divide a lent;th or a volume, into three, live or ten e(|ual i)ans, 'I'iie adoption of a decimal system of measurements tlien means simply this, that men will be unable by any ordinary means, at their dis- posal, to div.ile (jbjects into the number of parts neces- sary fur expressing their ideas or results in words or fig- uns, lor ins.ance, in such divisions as three-tenths, sev- en hundreilths or four Uventieths, it is difficult for men t( appreriale the relation of such parts to a whole. II. Jn sliii])-\v(irk in the manipulation oi mechanical tools the pri;ui])le underlying the Octimal System is in actual use and is the only reasonable and practical way of working. 'J"hrea(l> and sub-measuremeius are expressed in halves, (juar'ers. eiglillis. sixteenths or other even parts, and this ;s exactly the fundamental idea of the Octimal Xoiation. The ;ido])tion of tht- l-'rench Metric .^\-snm means, nic«.v>arily, -.he entire discardit-g of tlie mechanical a^)- pliances at present In use. and the manufacture of new ones with methods of division most, dilticult for practical api)iication and still of little improvement, even theoret- ically oil the present sysiein. This is the reason for the refusal of the I'.ritish manufacturers to adopt the Metric System. They have said: "We have altogether too much at slake to consider the matter until the advantages of the change pre more apparent." V\'e might suggest th.at tlifi-irists and edncr.t'on.alists aim to meet the de- mands of in;h:stry rather than to force visionary ideas u])o,n ]-)ractical men. CURRENCY. III. The currency system of many i is at pres- ent, is based upon a decimal system of nota. >n. Such a system of currency is found, in its ease of calculation, to be vastly superior, to that in use in the British Isles, so NOTATION AND NUMERATION. much so indeed, lliat the present Anurican system seems to he almost a perfect one; and its adv'anta,-,'es are cer- tauily very ^reat, hut its disadvantages tiiough corre- spon(hngIy great, are nevertheless unnoticed hecause the tlecimal coinage is ".s perfect as our present system of notation will al". )\v. However a moment's thought wdl Convince anyone oi the inconvenience and even ah- surdity of tiic use of coins whose values do not hear the same relations to one another as do the fractions of ar- ticles hough t and sold. Everyone has experienced the annoyance caused hy the ahsence of coins of the value of two and a half, twelve and a half, sixty-two and a half cents, etc. c)ur present commercial system calls for the values above mentioned, hut as yet it has been found im- practicable to circulate ccjins representing such values. 1 he inconvenience of such an imperfect system has led many communities to ignore many fractional values, substituting for the same tiie next highest or lowest whole number. h"or instance, an article whose value is I2;.,c, IS given for convenience the \alue of 12c or i ?c 1 ms. It IS evident, means a serious loss, in a sliort time, to the one party in any business transaction, wiiile it means a corresponding gain to the other. Supposing, however, that conis representing the above values were in circula- tion, the inconvenience of calculations involving such values would not be remcned except in special cases. Commercial life demands a coin whose value is i_"_.c; and yet such a coin would he an annoyance in numerical calculations. It is within the mark to say that the value of any article or part of an article, actually called for in practi'-", could be meL, conveniently in Octimal Currency Coir doreover the values of such coins would lend theni:. -cs readily to the needs of business calculations. Many other disadvantages of the decimal system could be mentioned, a few of which are referred to in a later paragraph. 10 THE OCTIMAL SYSTEM OF The Octimal System Explained. This system, as its name implies, will indicate to all who are familiar with the term "decimal," one which e .- presses numhcrs in terms of groups of eight, or powers of eight. In such a system, eight will become the radix in- stead of ten, as in the present system. Perhaps it may seem ckarer to say that, whereas the decimal system is based upon the recurrence of the unit from one to ten, the Octimal .*^ystem has but eight such units in its basis. This suggests, at once, a simplified system of notation as compared with the decimal system, inasmuch as the basis is reduced to a simpler form. Furthermore, any- one who is ,-it all acquainted with numbers, knows that the names and the number of figures used in expressing our successive numbers from one to ten, and even of many beyond ten, are quite arbitrary. The immense amount of labor necessary to the young student involved in the learning of our present system of notation, is evi- dent to any ihoughtful mind. This is due to the fact that the system Itself, in method and nomenclature, proceeds on lines that are contrary to recognized principles of human thotiglit. Moreover, the methods of teaching numliors now in use are contrary to the first principles of Pedagogy and must reniam so, as long as the present system is in use. for the methods used are the result of an irrational system to be taught and not to a defective knowledge of how *< each. For instance, when a child has mastered the meaning of the name "one" he is not helped by this knowledge in the understanding of the name "two." When "i" and ''2" are learned and in a succeeding lesson appear as "12." a new name is given, "twelve" wliicli in the child's mind neither bears any relation, nor has any similarity of sound to the terms NOTATION AND NUMERATION. 11 "one" and "two." We are not now speaking of numeri- cal relations for we do recognize tiiat such follow in proper order of thought development so that when one has learned the signitKance of "j" and "lo" as numbers, their numerical relation, of which the figures "12" are the expression, I'.illows in natural order. IJut the name "twelve" used to express the number "12" has not such similarity to the names "one" and "two" as suggests these to be the figures re(juired. As a result, the majority of students make their knowledge uecessi\e terms, rot only is tiier^ a lack of snjjgestivcness of the number i \pressed, but the first fitjiire sut^^^-cstod is a three or a four, whereas the first to be written is one. It tiikes the majority of ciiildrm .i year or more to understand the meaning of the iirms fp)m liiirteen to nineteen and the chanfje in method of nainim,', from nineteen to twenty-one. O.ie can see, a: a -lance, that the proposed system of nomenclature is more rational than that now in use and can be mastered in much less time than is required to master our present svstem. I'or the expression of terms that lie on the rij^ht of the Oel;mal point, a similarity of nomenclature su,i,^f?ests iiself. 'Ihe first p'a-e to the riqht would be called "eths." the secotid, "eiredth.-. :" the third, "elandths :" the fourth, "eths of etandths,'' and so on. Thus, the follow- ing mmd)er \V( uld be read as indicated: 654.654 is six etred livet f(/ur and six etrcd fivet four etandths. We would suirsesr, liowever. that a sy-tem of spelliufj be adopted for all terms ov tiie risht of th ■ Octimal point, tor instance, instead of "six etrcd fuel four etandths," say "point s^x five four, .6 now means six ei.c^hths or e:hs, a divisior, mT the unit nuire easily comprehen.lcfl than tenths. Iiei:\<"- also such a division, r.s can be readily made with ordinary mechanical appliances. The fractional names quarter, half, and three-quar- ters may still be used if desired, but will be smybolizcd thus .2 .4 .6, the fi,<,'urcs representing- the actual value of the term, and the ecjuivalent of 2-8. 4-S, 6-8. 1-16 in the present .system will be 1-20 in the Cimal. written thus: .04 for calculation, or four etredths e(;ual 4-100 for ver- bal comparison. Another savincr oi labor wiil be seen in that greatest 14 THE OCTIMAL SVHTrCM OF of (lifHciiIties for ycniiifj minds, the inultipliiation table. Instead uf tlcvoii or nine tables beinij recjiiircd for calcu- lation but seven are now to be used. Moreover the two most dit'ticiilt are done away with. We subjoin the mul- liplieaiinn talile a> it will a[)])ear in the Oclimai System: T.vieK IlIKRH Pour PiVK Six Skvkn 1 ;ht IlMKS TiMKS 'I'lMES TlMhS TIMK.S TlMBS 1 are 'J 1 iiie .S 1 are 4 1 are 5 1 are tl 1 ..re 7 I are 10 2 " 4 '_' " i; L' ■' 10 L' • 12 »> •• 14 2 ■ It) i 2 " 20 3 ' ti :■. • 11 ;i "14 :< ' 17 ;{ • ■ *».i :{ " 2.-) .5 '• :» 4 '• 10 1 "11 4 " 20 4 " 2» 4 •' :io 4 •' ;f4 4 ' 40 5 " VJ •■> •■ 17 5 " !'♦ r. • :u ft '• ;-it. .-) •• 4;; T) " 80 «1 " 14 li •• •_>!> f. " ;i() ti " ;j(i t> " 44 (1 •• r.2 tl " 60 7 " Iti 7 '■ -.'.-) 7 " ;i4 7 " y.i 7 " .VJ 7 " til 7 " ■ 70 10 " 2i, 10 " ;;o 10 " 40 10 " :>o 10 • tin 10 " 70 10 " 100 It will bf seen at onee that sneh a ninltiplication ta- ble as w« have outlined will be'ninch more easdy mas- tered by t.ie l)ei,nnner than that in use at present. .More- over for sneh as have already learned the table now in use the ehans^e to thai o*' tlie new system will be easily elTected. It is evicL-nt that a eonsiderable ehanfjc will be necessary in the whole realm of numbers, if the new system is adopted, !)ut inasmueh as such adoption will probably not be made for sometime to come, it could be anticipated by tt;e ,t:radnal niasterinu' of the new sys- tem and by its introduction at first within certain limi- tations. .\nd we would c\ en advocate its adoption at once, believitiij that whatever diftictdties miylit arise in the radical chanj^e that would ensue such would be more than counterbalanced by tiie ijfreater convenience and practical vabic of the new system. That it would ren- der necessary a change in an almost unliinitcd area of life is no arcfumont aijainst its adoption inasmuch as if it is a simider system and has an adaptability to the require- ments of life ecpial to tliat of the decimal system the claim of future sjenerations makes its adoption imperative. We hope to show, however in a later paragraph that it has NOTATION AND NUMKKATION. IS i I not only simplicity in its favor, but that its adaptability to every rk-partnu-nt of life uIktc caKulatioi's of any kind arc rcc|uirni. siirpassi> that of any other system that has yet !jcen iis>ed anmiiL; men. Many of our present most diftkutt ineasuri'inents. as well as the calculations in Iloroloj^y. Astronomy and Xaviijation. etc.. which are now possii)ie only to advanced sehulars will be found to appear as self-evident facts, witlmut the aid of any pro- cess of calculation. The System Applied. The extent of our present treatise will not ah 5 to enter into the details of the practical applicatioi ,i«r i>yslem to all departments of Mathematics. W iall, however, indicate >uch fiafres of its applicaliilit;- will assure our readers, that in this r^-snect the syste a is in no way deficient and jjcrhaps in so doing we ma. sUSJResi the methods to he pursued in cases of api)lica! on not here specified. We shall briefly outline the m> hod bv which the system may be applied to Weight- ^ i-asures. Horology, Astronomy and Navigation, sine. ese ib jects are of greatest practical importance, ;i ; of m- -n interest to the general pub'ic. We believe \ve can sh'^* clearly that ihe Octimal System besides having adva i ages in theoretical calculation, also surpasses all other systems in its harmony with the laws and principles f nature, as well as with the various operati(jns of the hu r.an mind. To pursue the investigation of such harmony would be a most delightful task, but the litnits of our space forbids. T. T!ie =ysfpm .ts .npplied tn Weights and Measures. (a) Weights — We do not find it necessary to pro- Tin; OCTIMAI. SYSTKM OF lM)>c a Lliaii,,^' ill tlic prismt units ul wcif,'ht. 'llic pound avuirditp.iis vvkuM remain ilic >ianilar(l as al present. I'.iit uc vvuuld cliaii^e tiie (li\i-iur i)ri .-iiit ;i.'j pound.-, for wliicli we mi^!:l >\\i\ ix'iain liic teim "■loii. ' li is w^ll ln i> tou larj^e, lu mtvc as a convenieiU unit, and iliat il ;.-, luii tittiil to praitioal lite, inaMHiuli a.- it ii.ii> not npiiM'iit a lair wa,4un !■ .;d ni any of ilie 01111- niodilii's ill urem r I use. Mtuli vrvijii lias hrcn (.lainud for tlie l-"rencli Met- ne >y>uin. i>eeau?e nf the >o-eaI!ed eonneclioii !)el\veen its 111' ,1-iires iif \vc i^lit and of \.ilnnie. Tlie tjrain wliich i^ ll'-i uirt 'i \vei,L;!it.-. in tli,> sy-tetu is livid as the \\i.i;;!m of .vaier at a ti iii])iraluve of 4 (K\!,rrees C, which is coiHaiiied in one culiic (■'-■iilinutri'. tin- unit "I \'''inu'. r.;ii as watiT i-han,Lri'S its \vi--t;-ht, ma- '.eriallv. wiih a ciian;;e >■{ temperature, it is elear that \vc can air.ve at an appn'\inia:e idea only of the weitrht Oi a known \ohiiiU- of water. Tlie one eonvenience of the iinils ill tlie I'reii h System '-^ that one iv aide to calcu- late the wei;.'!it of a ki.owti \oInme of a suhstance by sim- ply imil'Jplyiiii; the number expressinij this volume hv the spc'-iiic ufravity of the substance. r.ut surely by a comparaM\ely small amount of work specific i^Tavities !nis.dit be expressed with reference to a Solution of a ceria-n streiUTtli, oiie cubic inch of which u-eitrhs one ounce. This would furnish r^ with all the theoretical advanta<.je of the I'rencli systctn and at the same time save us from the annoyance which tlie intro- duction of such a system would cause in its demand for a universal chamje in our iiresent units of weicrht. The new table of weights would appear as follows: NOTATION AND NIMKU ' TION, 17 1(1 KiHltiH I'.nkf I Mtatiiiii of i\\'\> tabli.' in mir wcitjlils in prcs- iiit u-M' wDiiliI lint 1.1' dirtuMilt as i1k' piuitul remains un- chani^i'd wlnlr tlir I'liiuc is simply (IiPitl)lc its pixscnt value. ( lj) .".Uasires - I. I.iiuar and surface mea-uremcnt. 'I lie nhj^et nf any ehant;e in t)ur system of measurc- niciil is ti) iML-reome the neei>>ity oi' multipiiealifins and "<(' I'll,! It'll- iflntini ill hi II ftrliiiiii' I -'• M I M -i I I I I 1 -^ TT. lO divisions in ehan^Miig- from one denomination* to a lower or liiq-licr one. To aecomi)lisli tliis it is not at all neces- sary to elianpre our units of measurement. The present units, tlie ineh and foot, are too closely connected with industrial appliances and pursuits to allow of a clianp^e beinj; made if it can in any way he avoided. For conven- ience of calculation let eifjht do) inches be called a foot and eic^ht {\o\ feet he called a yard; then our present two-foot r.iie will become a three-foot rule. This would make the cubic foot to contain etand (i.ooo), our present 512 cubic inches, and tlie stpiare foot, circd (100), our 18 THE OCTIMAL SYSTEM OF present 64 sciuare inches. Thus in square measure the first two places of an ordinary number would represent square inches and the remaining ones square feet. For instance, if we rtquire the number of square feet and inches in a cjiven rectanj^lc Bv the Octimal System this would bo determined as follows: 15 Length, 15 in. x Hreailth, 12 in. -^ 202 ^q. in. 15 =-^ 2.02 so. ft. 12 15 202 W'c arc ; luis relieved of the laI)or of division, for the determining: of the number of square feet and inches contained in a surface becomes merely a matter of in- spection, wlun the number of scjuarc inches has been as- certained by multiplicatiiin. ■ In the L-ase of cubic measurements, the first three (iL;ures would represent cubic inches, and the remaining ones cubic feet. If the larger surface measure corre- sponiling to our present acre is to be an even number of etands or etions of square feet, it will be necessary to change the size of the acre. But this is merely a change in ideas and docs not concern any mechanical appliance cxce])t the surveyors' chain. MEASURES OF CAPAQTY. It is verv confusing to have measures of capacity which bear no direct relation to units of cubic measure- ment. It so iiappcns that the new cubic foot will contain ncarlv 7]- quarts. Let the quart measure be slightly re- duced in size, so that eight quarts will equal one cubic foot, which might be called as a measure of capacity, a "peck." \Vc would recommend that the eighth of the fjuart be called a pint and that the largest measure of NOTATION AND NUMERATION. 19 capacity be ctand pecks, which might be called a "tank." The measure of capacity will then appear as follows: 10 cubic Inches equals 1 pint. 10 quarts equals 1 peck. 10 pints equals I quart. 1000 picks equals 1 tank. The calculation of the number of cubic inches in a substance, from its dimensions would then at the same time determine the number of pecks or ([uarts contained in it. I'or instance if a vessel contain 64532 cubic inches its capacity is 2 inches, 3 pints, 5 ciuarts and 64 pecks. Moreover, knowing the specific gravity of the sub- stance in lerms of our standard solution a simple mul- tiplication will give us its weight. It will thus be evi- dent that, by the Octimal system, all the advantages of the 1-rench Metric System arc secured to us, without the necessity of the changes demanded by the latter in weights and measures, which cliangc are so great as to be almost impracticable. CURRENCY. It is a well-known fact that there is a desire among civilized nations for a universal coinage. If such is real- ized it is more than probable that the lirilish "sovereign" will become the standard, as it is even now, practically a standard the world over. Now it so happens that eight (10) r.ritish half-crowns make one sovereign, thus giving us a coin conveniently large and as perfectly suited to calculations as the coins of a decimal currency. The British half-penny is not far from our etredth (our pres- ent 64110 of the half-crown. Thus we have a base coin readv until tlie i)erfect octimc or cent could be minted. Hence bv ceasing to use shillings in making calculations the r.ritish .leople ha\ e coin, already, which, in the new notation, wmild not demand those multiplications and divisions by twelve and twenty. .\ verv convenient arrangement and one in harmony with the present currency system of Great Britain would 'M THE OCTIMAL SYSTKM OF be lu make tlic half-crown serve as tlie American dollar, 'ihen eight ( loj dollars would make one sovereign; the iiall dollar would ihen be 40 cents, the quarter jo cents and for convenience we would have eight and four and two-cent pieces also, the present English peiniy being used as a two-cent piece. W e recognize that this would mean a considerable change in our present coinage, but it is such an one as is necessary and advisable for it is in harmony with British values and indirectly with those of most other countries. It is impossible to harmonize our Decimal coinage with the sterling .system, or the sterling system with a decimal notation, so liiat if .; universal coinage is sought a radical change in the systems of most countries is absolutely necessary. 1 he e! nge in coinage which we suggest is m no sense a serious matter as Ui> depreciation of values resul- nd lurther. it will be an easy matter to become accub med 10 the use of the coins suggested as they will be very similar, in appearance, to those now in circula- tion. EXAMPLES. 'l"o find the number of pounds, dollars and cents in any number of units, mark oft the last two places to the right for cents and the next one for dollars; the others are pounds. Thus: Then also tr- 1 . C (1 c 64.5.32 E # c 1 14.0.32 yds ft in. C45 . 3 . 2 ft in. 045.32 ft. in. (14.. "132 tk. pk (j.p. i. 17.4114.5.3.2 H>s oz. dr. sc jjr. 74n 4 5.3.2 davs lirs ni. s. 17 . 4!l . 45 . 32 R. ° ' " 17.4(1.45.32 British currency. American currency. Lineal measure. Square measure. Cubic measure. Measure of capacity liquid. Weights. Time. Circle. NOTATION- AND NUMKRATION. 21 CONVBNIRNT BRITISH COINS. Gold —£=Eight (10) dollars Present half crowns Silver —DoUar=Etred (100) cents " half pennys " —Half Dollar- Fouret (40) cents To be minted " —Quarter Dollar=Twoet (L'O) cents " " —Eighth Dollar -Eight (10) cents " Kiclcel— I'our (4) cents " Copper — Two (2) cents Penny " —One (1) cent Half Penny Florins and bhilliiij^s fi,4 and 3 penny pieces can be used to make np Dollars until witlnirawn from circulation. CONVIiNII.NT AMKKICAN CoINS Gold — Enn'c lu.L'ht (1(1) iloll.irs To be minted Silver — Doilar-r- Etred ( lO(l) cents Present Dollar " —Half Dollar-- Fourel i ID, cents " r,()c " —Quarter Dolliir -Twoet (L'O) cents " u'")C " — l';i.i,'htli Dollar- Kiglil ( Hi; ceiil.s To be minted Nickel — I'our ( -t ) cuts " Copper — Two ( 'J ] cents " " — One (li cent " Value of" CfoM Cii') in Oitim:il !'i-i>res: !? T) ])icces I'lve lo) iinuars fill pit CCS - l-:t-luo ( r_'i I oUars fJO pieces Tuoot Umr(2-i) ilollars HOROLOGY, ASTRONOMY AND NAVIGATION. 'I'o such a-- ar:' at all ac<|uaiiuccl with t!ic present nu-thiuls of raK-iilatiuit in Asti^ .m imy and Xavi.tralion, it will !)(.■ c\i(l(.-iit thai sunie s'nii)kT inelhcd js cicsirahle. McircDvei-, tlKfc is no uniformity l)clwcon the divisions of I'.io dial a ;d iln sl- of the circle. Xow all the al)Ove sciences are based upon the divisions of the circle and since they are so inter-related, that each is constantly in demand in (Klennininc;- results in the oth.ers, it is certain- ly advisable that, in their operati(jns they should proceed on similar nutlK.ds. It is a very diffictilt matter at pres- ent lo express time in terms of distances on the earth's surface, owivq- to the lack of direct relation between 21' THK OCTIMAL SYSTEM OF hours and -Utrrccs. The Octinial System at once sug- ^'csts the foUdwin;;- divisions of the circle: (ItHi) ctri'd sccoiiils iimkc I iiiiiinti-. ( ICO) cfrcil iiiiniitcs '• 1 hour or dctriii' (liHi) cfrcd liiiiir-* '• 1 (liiy orcoiiip.ftf ri'volutidii. .<^^'^. d^^' ^'o objection couhl be made to tliis division of the circh lor altlioui^h the new seconds as applied to the cir- cle would be about five times as larj^e as at present, yet the Octimal divisions would be as conveniently read by the new vernier scale as arc the present decimal divi- sions. The divisability of the circle by such a method will be at once apparent. It will now be possible to ob- tain by simple bisection any an.s^le required. This may be done witli mathematical precision and at the same time by the only method of division of anp^les known to ,J?eo- metricians. At present no angles expressed by whole luunbers except those of 15 desjrees, 30 degrees, 45 de- grees. 6o degrees can be measured without the use of methods that are practically equivalent to guess-work and which are wdiolly unscientific. Furthermore the method of the Octimal system makes the circle the exact counterpart of the .Mariner's Compass. The system outlined above presupposes one revolu- tion of the hour hand in our present time of twenty- four hours. This would necessitate of course a com- NOTATI' \ AND MMKKATIOX. 2:» plctcly new set of timc-piccis ur at kasi a luw set of works ill tlic old cases. 'I'lic liinc-pieocs at present in use coultl be gradually replaced hy tiiose wnli the new- marked dial. The new divisions c,f tlie dial would mark this period ol time off into loo divisions or hours, thus makinj; tiie hour nearly ihree-ei|L,dnlis of its present length. A considerable advantage will thus be gained in the reading of time, for instance the symbols A. .\l. and P. .M. will be unnecessary, for no confusion cati arise as to whether j o'clock means 3 .\. M. or 3 I'. .\l. I'.y the new divisions eight (,10) hours would signify our present 3 A. .M., while 50 o'clock would indicate our i)restnt 3 P. M., etc. li may be thought that sucii a number of di- visions on the dial will make the telling of time more dif- ficult; but if we make the hour-hand the longer one, the hour can be seen at a glance and the minutes need to be expressed but approximately, since they have become comparatively small. The determination of longitude and latitude beccjmes a very ^"inlc process instead of the intricate and difficult problem ..resents. Conversion of time into longi- tude is now -ni rely discarded and ail that is required is to add or subtract the necessary correcii(.ns to or from the sextant readings or tables direct. This tnethod is very much lo be preferred to our present system and has an incalculaijle advantage over the French .Metric System, wdiich will if adopted be a constant source of confusion to navigators. At present four minutes of time corresponds to one degree of longitude, while in the Metric System there is no integral relation between min- utes of time and degrees. The new second on a great cir- cle will be about 500 feet present measure. !■ 24 THE OCTIMAL SYSTKM OF Transformation from Decimal to Octimal Notation. I'or a CDUsidcrable time to come, books will be in common use, wherein the decimal system is used for all numerical values. Many of these will never he repriiited but, nevertheless, will remain as valuable works of ref- erence. It becomes necessary therefore that some meth- od of ready transformation should be known to every in- dividual. Durinj;^ such time as the new rU'iii is rcprcscnicd by 3,641, icX) in the ('ciiiiial SysUin. L'unii)aralivc tables, sliowiug numbers in ordinary use (say up -.o 500 or i.cxxj) expressed in the two systems, mij^iu be prepared and placed in all buuks of ordinary literature. This would mean no appreciable additit)n to the cost ot a book and wouid serve occasionally as a con- venient reference. Numbers hii^her than llie-e yivc only a \ery \ as,uu- idea, at best, to the huiiian mind; hence a le^^oll or tW'p on what we mii^ht call "ImaiLjinative pro- portions" Would be ample to j;ive anyone a sulticient idea of how any Ic.r^e number, expressed decimally, would be exjn-e.-^sed in the (X'tinia! System. I'or instance, 10,000 in cnir present notation would be -'.H-O in the new; one liundri(i ;ho;;sand l.)ecoines three etreds and three etands, two eired t,,iiret and so on. Thus any number of thou- sands (.r millions could be rouj^-jily expressed with but little nienl;;l cltort. rile t,-Teaicst objection that can be raised to the in- troduction 01 a new system of notation will be that all matluniatical books and tablo will have to be renewed at considerable trouble and cx])ense. However, the ex- pense v.ould I)e triHin^r comi)ared with the loss in the value o!' niachiuery entailed I'y ihe introduction of the Metric System and, moreover, it will siion be repaid by the ,y:a:n in labor and efficiency. The tnnilile will be borne by nu-n whose time is jCfiveii to educational work and not by Ijusiness men or mechanics. Then, too, it will be some years before either the IVench .Metric or an Octi- mal system could be introduced and by that time, new- editions of all the most useful books would be rccpiired and the change could be made as they were printed. Even the task of i)reparin,c: a new table of lo,G:arithms, with the multitude of double transformations necessary, is, com- paratively a small undertakinjj. 26 THE OCTIMAL SYSTEM OF In the transition period it would be necessary to have a working acquaintance with both systems. This would mean some additional work and perhaps l.ewildfrment, thougli neither would be unreasonably great compared with tliu confusion arising from a complete change in weights and measures. CONCLUSION. \\ c have now outlined in brief for our readers the Ociimal System of Notation and Numeration and have sliown its application to a few departments of knowledge, more especially to those in practical life. There are many matters ui detail that must necessarily be omitted in a pamphlet such as the present one. We trust we have made it clear, however, that the system, as far as notation is concerned is perfect and admirably adapted to all the needs of caUulation. Changes have been made, but all such have in their favor the claim of simplicity and in no case is anything lost in accuracy. Such a system, whose simplicitv makes it possible for the young student to master it' in a much shorter space of time than is required to master our present system, has a claim for recogni- tion by every fair-minded man. Again the transformation from the decimal to the Octimal system can be so readi- ly accomplished that no book at present in our libraries will lose anything of value, since all its numerical ex- pressions may be at once converted into terms of the Octimal System. Furthermore, the Metric System is being thrust upon us more or less forcibly, a sy-tem whose advantages we must recognize but whose disad- vantages are also most apparent. The Octimal System, as we have said before, furnishes us with all the theoret- ical advantages of the French System while at the same time leaving us free from the great annoyance of a com- NOTATION AND NUMERATION. plcte change in our present weights and measures and the meclianioal appliances involving them. We therefore otTer the Octimal System to the public without any doubt as to its effciency and with the belief that it is better adapted to the necessities of Commercial life than any system heretofore published. t^ 2S THK OCTIMAI, SYSTKM OV WEEKLY COLUMBIAN. New \V»'8tnilT.8tcr. R. C. Kcbrunry J(», laoi. THE OCTIMAL SYSTEM. Whit Columbian CoWtgt Professors Say About th< N«w System of Notation. Tu tin Sir. Ill ii It" I lit is-siif ((( your vuliiablf i)a|» r, the I'lii ulty of (•ohinihiiiii ('tJll<'K<' WU8 i.f. rird to in an lutiilf. •Matlniiiuilcal benefit di' your many riatleis we >iiibjulii IMitor of The Culumlilan: th. Uevoliition." '•''II' a biief statement i .. ■ einiiiK tlie new HVhti m of notation. '1 Ik- orlKin of tli- HyKteni. and Its applleution to many de- liartmeiilh of niatlieniatU s and Keieiicc, iiior>. etipeeiully to the i-eli i.-es of horology, atitrontjiny ami naviRalion. are the wo'k <.f Caiii'iln t' )o|)ei. of this city. Tli print iple of the system was 6ulin;itte'l to uk some nioirths a^o. diiriiiK wliieh time we have souph* (,) llnd ^ow widely e.\tended its appliration mlKht be to ail d.paitniiilts where ealciilations are neccn^aiy as well a.s to ascertain whether any defect could !)-• discovied wlij'h would nnder it an imperfect system of netation. We are now piments of the system to dls- ■ over at every turn, how Its adoption will ^imiilif" our present method.^ of mathematical operation, and render -iunificant to the most ordinary intelligence, terms and exiirt'ssicis hereto- fore almost meanin.i;l( ss or of little practical vain ■. Manv of 'ii>r I iborious tasks, (specially such as require the u^'e of vulRar fractions, methods of deduction, etc.. will be practi/*'"" m ""'"'"•'• ^■^«'»- «••'• l.Ugl.al an.l Inip.! ^. 'i, '7, ^ 1 L .' '""*= ' ^f"««l«'"« '"<■"-. w. wm „.,t .V,,: 'i, „•';;'■ ■"•••'i •'; '"• '-i>> ■■■nn.v^ h«.nu.nu*V;Ja;'^^' ?.."';;. ';:;r';;;.*;: '"""-"^ "- •-" - thf «iil,.s.:.n.-,.. " "'""" ^^"'' ""• «l''finc gravity of mal s.vs„.„, ,„• „,„at,on """"■""'• '"'vantaR.- .,f u.Ing th,. .Ktl- applianc, .s. **" '"" I"<'«"'t giaduafd mechanical vant^;^:,!';;;^' .::::?;;- s:.r r ;..:!;, "r '*". '"• ."" "■•'■^'-' »''■ to all o,n ,n .hanlral app a n..^ «, n '! T '''"' ""' ^••^■^'>' fun,iani..,ital .hang.. In , , ,(1 n.^ ""'"' ^'"""" ""V S'Hh a .yst. ,n. u. f.-H ass ,•,■ vii ,, ""•"«"';-""nt of wHght. fitur... Inasnuuh as It s th ^ L 's/l ?"''' ^'"' ""♦"^'"" "' 'lu- evfi- disrovored. Amplest and mcsi i-ractlral method W. J. SIPPREI.L, n.A., BD Principal Columbian Co'lleeo J. O. DAVIDSON. B.A., Rfglstrar.