IMAGE EVALUATION TEST TARGET (MT-3) V /. / / y. (/. ■^ 1.0 I.I 1.25 •" IB 112.2 12.0 |||||18 U nil 1.6 Photographic Sciences Corporation 23 WEST MAIN STREET WEBSTER, N.Y. 14580 (716) 873-4503 iV iV ^^ \ \ /^ 4^ <^. % '^O^ «■ <*. ^^^N ri? Lsignifie "A SUIVRE", le symbole V signifie "FIN". Maps, plates, charts, etc., may be filmed at different reduction ratios. Those too large to be entirely included in one exposure are filmed beginning in the upper left hand corner, left to right and top to bottom, as many frames as required. The following diagrams illustrate the method: Les cartes, planches, tableaux, etc., peuvent dtre filmds i des taux de r6duction di'f6rants. Lorsque le document est trop grand pour dtre reproduit en un seul cliche, 11 est filmd A partir de Tangle supdrieur gauche, de gauche d droite, et de haut en bas, en prenant le nombre d'images ndcessaire. Les diagrammes suivants illustrent la mdthode. rata elure. 3 12X 1 2 'r 3 1 2 3 4 5 6 , I FRAUDULENT OFFICIAL RECORDS OF GOVERNMENT. SECOND SUPPLEMENT TO THE (!ORRESPONDEN(;E WITH THE LATE LORD FREDERICK CAVENDISH, M. P.. PUBLISHED WITH THE CONSENT (^F THE RIOHT HON. THE .MARQUIS OF HARTINOTON, M. R, SECJRETARY OF STATE, WAR DEPARTMENT, JULY, ]88+. AN EXPOSITION OF thp: principles and methods employed in the fabrication . OF certain United States and Canadian ANNUAL TRADE TABLES, S^i'oin ISOT to 18S45, TOGETHER WITH THE MATHEMATICAL, FORMULAE ox WIIK II THK FABRICATION IS BASED; AS DERIVED FROM JAMES BERNOULLrs BY HENRY YOULE HIND, M. A., Brilhh Sciitilific WilnfSi at Ihf Halifax Fislitry Comti isioii, ami Official Compiler of tlu Analytical Indtx to the Documents of the Commission. ( Formerly Professor of Chemistry and Geology in ihe University of Trinity College, Toronto. } .olonistto the CANADIAN RED RIVER KXI'KDITION OK 1857.-111 charge of the CANADIAN ASSIN- BOINE and S.VSKATCHEWAN EXPEDIIION of 1X58.- -Author of Narrative of the Canadian Expeditions in iho North West, 1860. — Exijlorations in the INTERIOR of the LABRADOR PENINSULA, 1863.— Oftitial Report on the (ieology of New Hrunswick, 1 865. -Official Reports on Waverley, 1869. — Sher- brooke, 1870. — Mount Uniacke. Oklhani and Renfrew Ciold Districts of Nova Scotia, 1872, &c., &c., &c.— On the Fishing Grounds of the Northern Labrador, 1876.— Official Papers on— The effect of the Fishery Clauses of the Treaty of Washington on the Fisheries and F'ishermen of British North America; Parts I and II, 1877. — Recipient of Gold Medal and Diploina, Pahs Exhibition, 1878, for Maps and Chans illustrating the Fisheries of British North America and the movements of Fish in the Sea I if- 3;. •%' f \ ■•^^r. HI f { I i CONTENTS. Lctler lo ihc HonDrnlilc the Sccrclaty uf Si;\ic', \V:i',liiri(;tiin I^Hcr lo the Mcisl lliinnrahli: the Mnrqiii!i of I,nMscl.>«ni-. Covi'iniir ( iriural of (anncla •■ IM'RDDrc'TION. A Brief llistury iil tlie lit'tnuiilli Talilc The Interchnnijealili' ijuamilics in ihf Kerncnilli Talilc The Imerchanijcalilc Kignrcsof iho I'niii-d •^i.iius ami Canadian U.-ciinls nf Imeriialional Trailr The Tables I, li, 111 an.l IV What ha'i Iwoinic uf ihe DiilieKr The Maihemalical Kurmula I be Collon anil Woollen " Inipons" and " Knlrics for Home Consunipiion" in Caninlian Trade TablcR groU|«ul in Ihe form <>f an Kndlc^s Aritlinirtiral ''rogre«^^lll^ TAHI.l- r ' Kriini lames Hernoulli's Treatise iiK Aktk Tiimkii amu. Iternuulli> 13th I'roperty. 8th TAiii.i-: II. Statrment No. I. -The forged f'anadian Iniiwrls from iSrty to 1.S7;. Staiemrnt No. II. -The origin ■ f the " Fire-brick and Clay Series ;" so nanieJ from the .subsliiiilion of 11,184 i.ollais worl 1 of '* Fire-brick and Clay" for Fish. Statkme.nt No. III. Thi- ■• Kirc-lirick and Clay .Scries." STArE>lRNr No. IV. — The items grouped, shoning that all ihe larger icnna are stims of the smalh-r lerrai. Statf.mknt No. V. The " Fire-brick and flay Scries" in the form 0/ an indelinite .\rilhmetical I'rogression. I'Ai;r. V vi vH vii I Statrment No. Staiement Nn. VI. VII. TAWLK III. -Dr. F.dwaid Voung's I'nitcd Sales Ooniestic anil Foreign Fish F.\i'OKi> for the Fi-cl Vent 1.S72-J. -I>r. FMward Voung's Kximrl Fi.sh Trade Figures arranged in the order of Mtgnilude. Stclicii A'il IMI'rtUTS Irom llrilish .Anieiica for the Fiscal Vcar 1872-73. Stdion No. / -.Showing that Dr. Fd«aid N'oung's United Stales Inipiirl Fish Figures from llrilish America are maile up from sums of Dr. Fdward Voung's F..\l'OKT Fish Figures 10 Hrilisli America. SnUmi A'o. //. -Showing that Dr. FMward Voung's Cniicd States IMI'DKl Fish Figures from liriiish .Auieiica are sums of the terms of the Canadian " Fire Iwirk and Clay Series." -Dr. F'.dward Voung's llniled States KXl'OKTS of Fish, Shell Fish, Fish Ciil and I'roducis ol ihc Sea lo British America for 1S73 74, with the Canadian reciprocal siaienieni of IMl'OUTS from the United Stales, by J. Johnson, Canadian Commissioner of C'lisioms. Stiiiim Afti. I. -Showing that the United Stales Fiwh Trade Figures are siiniii of the Canadian Fish Trade Figures, and that Ixiih oi these are interchangeable with the " Fire-brick and Clay Series," from 1807 to 1873. CIIAI'TFK I. Statkmrnt No. X. The Canadian diflerc i lieiwecn " Im|wrts" and " I"ntries for Home Consumption," fi>r the year 1878, in respect of Coitons and Woollens —grouped. Stclien /.—In the form of a continuous Arithmetical Progression » SeditH //.—In terms ol the "Fire-brick and Clay' Series 3 Srclioti 111.— In TeriiLs of Dr. IMward Voung's Fish, Fish Oil, Shell Fish ami Protlucts of the Sm, Unitol Slates Exports to liriiish America in 1873-73 3 The Bernoulli Table 4 The Relation between Bernoulli's Column IV, Dr. Edward Voung's Fish Trade Figures of 1872-73 and the Canadian " WHri ences" for 1878 5 Section IV. — ] Sectioh \i.~ Exhibiting the relation,! 5 Section VII. — j An Important Properly of Bernoulli's Table ft li 'MK<'i!Hi^<«. • '/ ciiArrKk II. rin. KlXOkli IIV I'KOVINCKS. The Ucciiiil liy I'riiviiices as 'li>iiiii;iiislifil (mm tlic IWiinimipii Kcrtpnl year 1S78 . TAiii,:-; A. Com|inrisiin luMwcen llic sviins cif llic Diflt'ienccs . I'Ar.t. 7-16 16 (IIAlTKk III. The Krinlinn lielnffn Bernoulli'* {'nliimn IV. and tlie I Hflcrcnccs l>ei«cfn Coiion ami Wnullrn " Inipori.'." ami " H ilries fur Home t'( tisviinption,'' for the year 1878, and !hc Malheinatical Kurnuda shewing; iheir itrif^iii. The .Slims of ilif final I lillerenres lor Coiion'. and Woollens in Table A, in Icrms of Uornoulli's Column IV ThK MaIHRMMH AI l''i)KMl'l.A •■; IlluMrJIion of ihe fniiid Smie> Fi.sh Trade Figures in delail, »ilh Dcnuminaiions in ihe form of the larger (|uantilies being sums of ihc Mnallcr ipmnlilies 17-18 18 CIIAI'TKK IV. THE CAN.\D1.*N TRAljE TAUl.KS OK 1883. Provincial and I )oniiniitn IJifTerenires Comparison lieiween ihe Sums of (lie Differences . TAIil.K li, 1883. a3 CIIAI'TKK V. The I' The F The V Talile Tahle Tahic THE CANADIAN IKADE TABLE.S OF 1885. 'tovinCial and iMininion I)itTcrences for Cottons and Woollens 2$ irst Order of I hHcrencc.s for Coitons in 'he (onn of an Arithmetical Progression,— 1st — with 10 as a conui.on dilTerence 37 2d— .. 100 .. .• '8 3d •• l"X) . II ; • ^ •■ ^^ iist Order rjf OitTerences for W\«>Ilens in the form e>f an Arithmetical Pr.Tgression with 100 as a Common Difference 39 sliowinj; that the larger terms of the Firs: Order of Differences for Woollens are sums of the smaller terms .30 showing that the larger terms of the First Order of DilTerence^ for (Jottons are sums of thf- smaller terms . 3* showing that the 1-irsl Ordrr of DifVetcnces lor Cottons ate the equivalents of the terms ofthe " Kiredirick and Clay" Series 32 taulf; c . Coiuparini'ii lieiween the Sums of the I lilierences 32 CHAPTFK VI. The Monderful accuracy ofthe Figures in Ihc United States and Canadian Trade Tallies . 33 The alleged Custom House Finlries for the year 1878, 1883 and 18S5. in the form of an endless Arithmetical Progression 33 The Kipiit^leiit Canadian Dill'eiences tor 1878, 1S83, and 1S85, i.i terms of Dr. Kilwnrd Young's Fish Trade F'igures, taken from the United Slates lommerce and Navigation Keport lor 1873 36-37 The Fire lirirk and Clay Series in Terms of Bernoulli's Columns, Nos. Ill and IV 38 A Mathematical 1-ormuia lot obtaining any desirable Ratio between two Series 39 A general Mathematical F'xprcssion for the terms of .any one of ihe vertical columns in liernonlli's Table . ...' . 39 A .\Iathemaiical Process lot cuiivertiug Ikrnoulli's Formula into the " DiflerenlinI Method" of expressing the Sum of a Scries 40 A SlAIHMKNT AMI CoNCI.L'sroV 4I EliH-A-TA.. Table I. In the Formula lor l!erii,.ulirs Sib Pr.rperty insknd of 2'' read j"-' . Page 4.— Second line friuii b tlom. lor t liapter II read Chapter III. I. 5.— 15 hues from liotlom after the words "of DilVerenrcs " insert ** with their signs " 7. — IJ lines from top, for "al ihc end of ibis Chapter" iea I. A translation nf the three first Chapter-, (if tho seotiid part, or H.ioli, of Mr. j.imes Heriioiilli's excellent Treatise inlitlcil AkS CONJECTANDI, or " TMr AVT OF F.iRMISC I'tlOKSBLB ('oX;i^iTl'miS l\l\( tiUMNti EVKNTS THAT DtlrBNtl ON CtlAS"CH." Publiihcd in a small t^uart.i Volume ill Uasil. or Da^lo, in Switierland, in the year 1713. II. The l>,Kirine of I'eriiiiltalions, and Conibinatiuns, hcina an essential and fundamental iXlrt ot the IKK fHlNK t»K ChaniRs; ii.s it is ildivered hy Mr. James Bernoulli, in hii. exiellent Treatise on the Ooctrine of Chances, intitlcii *' An Conjectnniii," luid liy tho cclchrat.-d l»r. John Wallis, of Oxf.>rd. in a ITact intllled frtini lh« Subject, and publiKhed at the end of hii J'realise on Alijebra : In the former of which '1 racu is contained A Uem jn&lratiun of Sir Isaac N«wlon"s famous lllnol.iiial.. Theorem in the cases .>f Inte^'ral Powers, and of the Keciptucai^ of liUfgrat I'oweri, together with Some other useful Maihematirul J lacts. Pnhlishe.} l.y Kran< is Maseres, Ksq.. Cursitor Baron of the Court of Exchequer, l.ondon, 1795. JiflgCl?' ffJiBWWBB ^w^raeis This secret must also bo known to a ffw United States nnd f .maili.in suhordinato officials. It is a power which a few individuals ounlit not to [losscss unknown tn ihc general |ml)lic. who liavi- sutferfd, or do suffer, or may sulTcr, from its palpable abuse, as exhibited in these pages. Without liirthcr reference to the fundamental doctrine amoni; freemen, that all men are ecjual before the law, and the indisputable claims of justice and comity, may I very rcspei ifully entreat, in the interests of good neighbourhood between two kindred peoples, bound lojiethir by many a lie, such a just and open examination ijito this matter, that the well'.ire ot' Slates and I'rovinces, of t lasses and Communities, shall no lonj^er be subject to a secret process of wrong-doing known to very few individuals, but deeply afl'ectini; the bust uiterests of both nations, and at present conlinuing in imdisturbed stturily. I have the honour to be, Vour obedient servant. HKNKV VOn.K HIND, .\I. A., Officiitt Compiler ,. l.l-ri'ER TO THli GOVKRNORC.ENERAl,. CANADA. To THE Most rioNOl'MABt-K IHK. .MaRQUIS OF l.ANSlJOWNI.:, Goifinor Central of Canada My Lord, — The discovery by me of the mathematical proof of the fabiicated construction of the Canadian Trade Tables temoves the last objection which can be urged against public inciuiry into the whole matte It may be argued that as long as I failed to exhibit proof if artifi( ial construction, based upon mathematical formula;, it was luting that you should continue to receive the Annual Trade Tables from the Minister, just as the Minister continued to receive them t'rom the Commissioner of C'ustoms. The- Minister says : — " The undersigned has the honour to present to your Excellency the I'ables of Trade and Navigation of the Dominion of Canada, for the fiscal year ended ,^oth June, 1885, .as prepared from Olficial Returns and laid before him by the Cominissioner of Customs." I have now the honour to submit to vour ICxcellency the mathematical proof that these same tables, togahcr with those of many prior years, have been secretly manufactured by means of a mathematical fornuila, which converts them, in resiK'ct of the Eish Trade with the United States and in respect ol the Differences between "Imports" and "Entries for Home Consumption," into the ecpiivaleiits of the sums of selected co effii ients of the successive exjiansions of (i + t) to the power of n. Vour Excellency is aware, from numerous acknowledged < omnlunications addressed by me to your predecessor and yourself, that this subject has occupied my attention for many ye.-us, as an outcome of official work, and that I h.ive not failed in my duty in bringing it under your special notice, and assigning proper motives for the ,ut. It is with a certain feeling of regret that I am ikjw able to furnish your Excellency with the Formul.'C which jirove the deceptions and make it ini|)Ossible for the matter to be any longer evaded. I have always had before me the interests of millions of unsuspecting and loyal people, the well-being and good-neighbourhood of contiguous States, and the claims of honest dealing, which together outweigh all other considerations. I have the honour to be, Vour Ex( ('Henry's obedient servant, HENRY YOUI.E MIND. M. A., Official Compiler of the Analytical Index to the Documents of the Halifax Fisheries Commission. WiNDDOR, Nova Scotia, Dec i6th, 1886. ■■.-rTtST-'-rE- vM INTRODUCTION. A Hmif HisioKV OK iHE Bkrnoui.i.i Table. J.1IIH.S liirnoiilli was a Swiss hy t)irili. lie was txirii at Basel in 1654. In 16S7 he was appointed Professor of Mathcmatirs in the University ot Hasol. He was an exeelleni dassieal scholar, am] thoroughly ronvcrsant with the Krench ami (lernian languages. As a tnalhematitian ''he is well deserving of a place by the side of Newton and Leibnitz" (I'^n. Brit., 9th Kd.) His mathemaiifal works are - 1. JatoH /itrnoiilli /liisilunsis Ofifia,i.\ent:\\c, 1 744, i tf>ni. 4'° : — 2. Ars Conjtctandi, opus poitltumum : aaeiiunl tnutalii<: df Seriehus Infimlis, H t^stola (Galliee scripft) de Ludo Pilu Reticularh, Basilix- 17 13, i toiii. 4'". " Like another Ardiimedes, he requested that, as a monument of his labours and an emblem of his hope of a resurrection, the loj;irilhmic spiral should be engraven on his tombstone, with these words— KAPEM MLTAIA RHSUKdO. Nearly one hundred years since a portion of Bernoulli's Treatise, Ars CoNjEciANnr, was published in English hy Francis Maseres Ksq., Cursitor Baron of the Court of Kxchequer. The part published related to the properties of the Table which is designated 'I'able I in this Kxposition of the structure of the Canadian and United States Records of the Industry of the people, for the years named and the articles of trade specified. The .iinomial Pheoicm in Its application to all powers, positive, negative, integriil and fractional, was discovered by Sir Isaac Newton about the year 1665, but 111 this extended form the jjroof does not appear to have been published until 1685. Mr. Henry Briggs, the computor of the Logarithms bearing his name, used this theorem with respect to the jKJsitive powers of (a+b)" and jjublished it in his Arithmetka Ltixdrit/imua in 1624. I'rior to the year 16S5 Dr. John U'allis, I'rofessor ot (leometry in the University of Oxford, published an " Arith- metick of Infinites," in which is a very curious table, possessing in some particulars properties similar to the Bernoulli Table. When the series in this table are read in a sloping direction they are found to be the co-efficients of the succes- sive expansions of the binomial ( 1 + 1 ) to the [lower of n. This table was reproduced in 16S5 in u published " Discourse of Combinations, Alternations and Aliquot Parts," by Dr. John Wallis. In describing his Table Bernoulli s.iys : — " Habet hicc tabula proprietates plane eximias et admirandas ; prteterquam enim qu6d Combinationum mysterium " in illft latere jam o.steiidinius, notuni est interioris geometri.i; peritis, prtecipua etiani to'ius rcliqua; matheseos arcana " inibi delitescere." The NlASKRhs translation is as follows : "I'he pro|ierties of the numbers exhibited in the foregoing table are truly curious and surprising, for it not only contains in it (as we have seen in the foregoing pages) the clue to the mysterious doctrine of combinations, but it is also the ground, or foundation, of most of the imiiortant and abstruse discoveiies that have been made in the other branches of the mathematics, as is well known to those persons who are skilled in the higher parts of geometry." TnK In iKRCiiANoKAiiii-; Propkrtiks ok thk (JiANriTii:s in ihk BERNOUiit Table. The leading property of the (|uanlities in the Bernoulli Table is their interchangeable character. .Ml the ipiantilies present in the Table can be put in the form of other quantities also present in the Table, and these again in the form of other ((uantities present, and these again in the lorm of others, and so on down to the natural figures. This is the leading property of the Canadian and United States Records of Trade submitted and ftnaly^ed in the following pages. Their praclUal identity, as far as properties are concerned, with the Bernoulli Series is shown, and consetpiently their fabricated < haracter proved. In order to present the relations between the official United States and Canadian interchangeable figures purporting to represent 'Tr.id; Records, I have carried out Bernoulli's 'Table to 60 terms, in other words I have construe ted a 'Table of the sums of the coefficients of the expansion of the Binomial (i+i) from the power of unity to the power of 6a On page 5 the successive coefficients of the expansion of(i + i;iVom unity to the power of 48 are given with respect to the 4th term of e.ich successive expansion -thus forming column I\'. of the Bernoulli fable I. The intercalated figures 6, 171 and 969 are |)rinted in italics. Any person can form this table without knowing even the first principles of algebra. Each succeeding number is formed by adding together the number above it and the next horizontal number to the left, lor instance 6188 being the 18th term of the \lth column, is equal to the sums of 4368 and 1830. The quantity 14,310 being the i8th term in tiie !.\tli column, is equal to 12,870+11,440, and so on in every ca.sc. .Among the many important properties of this table the following may be enumerated : — The quantity 24,310 is equal to the sum of the Series above 11,440, being the figures covered by the movement of a Castle in chess to the extremity of the board The quantity 24,310 is also equal to the Series above it, viz., 12,870 and the sloping column to the left, or the figures covered by the movement of the Bishop in chess, always to left and to the extremity of the board, or to c oluum I. This equivalency holds good for e.ach and all the quantities in the 'I'able. Tlte,'i*ifl3 ' Agnin, lakinR for an cx.inipli' any (iiianlity. 37. ij*. being tin- sotli icrni (if ('nhiiiwi \'n. This iiiianlity is also found to occupy tht place of ihr joih term in Column XIV. It ran, thircfore, be represented by the following series : J7,i3i =■ i8,5(>4 ■¥■ 8,568 = 8,568 + 18,564 27,133 is equal to the sum of the following series : I 6 ai n» •4M I •3 9' 455 1820 6188 18564 J7.'3' 43«S 6188 8568 »7.>3» It follows from these pro|)crlies, that if any one of the 20 horizontal columns be moved one s(|uare to the left, the figures in e.ii:h s juare are the sum of the entire series above it. I'his holds j;ood fur any niiinlier of vertical and horizontal columns, 20, 40, 100 or 1000. It will be observed that every one of the ijuantities given in the above series is also the sum of a scries preceding it For instance the quantity — 8568 is the sum of the series in Column V, beginning at 2j8o 6188 " " 1820 4368 •* " 1365 3003 U U ,QQ, And so on to the top of the column. And this character holds Kood for each and all the figures in liei.. ulli's Table. E.ich and all after unity are sums of precedmg series of linures given in the table. Hence the applicability of Hernoulli's legend, the conception of which he derived from the Logarithmic .Spiral and applied to himself— IvADKM MUTATA RKSIJROO. If figures i . .tny square be selected, such as 19,448, being in the 18th horizontal and Vlllth verticil column, then the sum of the figures covered by continuous movement one square to the left and one square upwaids, always to the left and to the extremity of the Hoard, will be e(jual to the ((uantity in the second sepiare below the square from which the start was made — less unity. Hut if the square oci:u;)ied by the figures denoting the number of the horizontal column be occupied by cyphers and one step more be made the sum will be equal. Example. .Startinj; from 19,448, being the 18th term in the Vlllth column, the .Series is— 19,448 6.188 12,376 2,380 1 8,ooS 1,820 4,368 . 5O0 3.°03 411, ■ 1.36'; *•§:■ 1. 00 1 j),!- 286 13 78 I 66 I II 11,628 the 20th term column VI. I So>387 The aoth 'erm in the V'lIIth column is 50,388. Numerous other pro|)erties are pointed out by Bernoulli, and mathematically proved. .Mso m l-'rancis Ma.seres translation many curious features are noticed and subjected to mathematical analy.ses. At the close of Chapter VI, I have introduced a formula which brings Hernnulli's formula and Table within the range of any one familiar with the elements of algebra. But it is the Interchangeable property possessed by the quantities which gives them jiresent importance. The following Formula is derived from Hernoulli's 1 2th Property. It develops some remarkable relations, and is especially Uicful for obtaining any desirable ratio or approximation to that ratio in the form of two series of numbers — The application of the letters is given in Table I. s 1 X n Therefore S X a = I X n And S : n = I : a • .' ix which, heing interpreted, is: The sum of the Scries Is to the niiinhor of terms in titc Scries, including cypher", a« the last term of the Scries is to the nmnbtT of the vtrtic al roliiinn. The otiicr proportions arc sflltvidcnt, ami whrn two Scries arc taken the n|)plication of these projxirtions comes prominently into view. The lNTKKrlIAN(iK.Ahl K rKOI'F.RriK.S OF Al.l. IHI' I'lMlll-s IN IMK L'NITKI) STA IKS AND OaNAIHAN RKCORPS OF Tkaiif. roK AkTici.KS and vkars si'kcikiici) in llIK K0I.tOW|N<; l>A(;^^. It appears from an analysis of licrnonlli's 'I'ahlc that its leading fchtlireis are the intcn hamjjvnblc properties of the (juantities or co effi< ients produced hy the expansion of ihe liinomial ( i + i ) to the power of n. These ake the leading KEAruKKs ok the (Canadian and United Siates khohds ok Iniehn ationai Thade, to which reference is made in Ihe following pages. I'liey jnintlv cover the ve;irM frnui 1867 to |8H^. The f'ANAiUAN I'isM Iuahe Rkiokds kkom i«07 10 1H73. If therc.ider will turn to Table II he will find that the alleged representation of ('onadian Kish Trade with the United Stales from |H(>^ to 187,5, is, in reality, nothing more than an artificial series of figures possessing the properties belonging 10 the figures in Hernoulli's Tahlc. The most impoitanl property is that ihe difiVrences between tlie agrc gates given, when dissecteil and traced to the original denominational figures in ihc annual trade returns, (01 m, when properly grou|)ed, an indefinite Arithmetical I'rogrrssion. When fiirther analyzed, these original dcinjiiiinational figures are tound to lie nothing nioie than the sums of the terms of Hernoulli's Column No. Ill, as shown in Cliapter \l. These original denominational figures are also found to possess the property of Ihe figures in liernoulli's Tables "hii h enables all the larger (pianlilies to be put in terms of the smaller (luantities. In other words the larger (juantitie, unis of the smaller (piamities, as in Hernoulli's Table. It is to bo remcmbereil that all the figuies possessing these properties in common with those of liernoulli's I'abli! are denominational cpiantities, and are su(>posed to represent Custom House imports, on w'hich duty is paid, fir Custom House exjiorts to the I'nited States, Canada, or other countries. The title of this Table is : — Table II- "To illustrate the Principle and leading Properties of Rcrnoulli's Tables as reproduc .d and applied in the Manufacture of Canadian Annual Trade and Navigation Tables, signed H. S. M. liouchette. Commissioner of Customs, and J. Johnson, Commissioner of Customs ; also as reproduced and applied in United States Annual Commerce and Navigation 'i'ables. Signed, — Edward Young, Chief of Piireau. Referring to 'I able III., we find that the United States Records of I'rade, under the siiiiervision of Dr. Kdward Young, also consist of figures having like properties, and, as shown in Table IV., they are interchangeable with Canadian official figures. The United States Records of Trade with Canada in Fish, Kish Oils, Shell Fish and Products of the Sea, for the year 1872 3, are specially introduced lo exhibit their artificial character and their relation to Canadian figures. It is important 10 note how accurately these figures fiillow properties of the Hernoulli .Series, of one of which they arc the ecpiivalents to a certain number of terms, the form being alone changed. A careful inspection of 'Table III. will suffice to satisfy any one respecting their artificial ch;vracter. The title of T.ible III. is: Dr. Edward Young's Official Figures of United States Fish, Fisli Oil, and Products of the Sea Exports to British America in 1H72 -3, grouped. /•irsZ-ln terms of the Canadian "Fire-brick and Clay" Series. Saotid — In tabular form, showing that his larger quantities are successively and continuously sums of his smaller quanti- ties. Third In the lorm of an Arithmetical Progression, identical with the .Arithmetical Progression of the terms of the " Fire brick and CI .y" Scries, The whole being properties belonging to the Bernoulli Series, obtained by the expansion of { 1 + 1 ) or ( 1 - i ) to the power of n, arranged in the form of the Bernoulli , 'Table. In Table IV., the United States Imports troin Canada are given and an.alyzed with like results. Their inter- changeable relations with Canadian Official 'Tr.ide Figures are well represented. In fact it may be said that both United States and Canadian details of Trade there represented are nothing nuire than the visible result ot mutual agreement between Dr. Edward Young and Mr. Commissioner liouchette, and do not represent the Trade or industry of two nations. It is important to note that the greater portion of these Imports are dutiable, and the record of duties being a [)ercenlage on these fabricated figures, that record is necessarily fabricated. 'The ciuestion arises, what has become of the duties ? How is it possible that duties received from scores of Custom Houses can form an arithmecical progression, and be put in terms of Bernoulli's 'Table? This question derives larger importance from the proved fact that the 'Trade Records of a vast number of other dutiable articles are subsequently recorded in such forin that they also can be put in terms of an arithmetical progression, or proved to be interchangeable with figures in years far apart, extending tq.theyear 1885. They can also be put in the equivalent forms of terms of Bernoulli's Table. . " ' The title of Table IV is.— • ■ . Dr. Edward Young's official Figures of United States Fish and Fish Oil Imports from British America in l87^-73; also, his official Figures of United States Exports of Fish, Fish Oil and Products of the Sea to British America in 1873-74. Showing, /"/«/.• -That his dutiable Import Figures of 1872-73 are nothing more than sums of his Ex|)ort Figures. Second —Thai his dutiable Import I'igures of 1872-73 are nothing more than sums of the terms of the Canadian "Fire-brick and Clay" Series. 7%»rrf.— That his Export Figures of 1873-74 are ■vv noiliiii- mure than sums of llic Can.i.lia,' (liitlahlo ami Vn-c I'ish ami l-isli Oil 'in|M)iis from the United States; also llial llicy arc sums of the tirms of i.'ii- ran.icli..ii " Kirchrlck ami Clay " Scriis. Fourth: Ihat all the l-inures arc iiitcrihanjiiMl.lL- ami icrivx-il (as siilisciiurnlly provcct in ciclail) from licrmmlli's Column No. UI. anil llcrnouli.'s C()lim\n No. IV carriod out to \C tfrms. WiiAP riAs iiKco.Mi. or niK Drnts.^ Wli.U has lnMome of the duties? and Ahat is the nature of the artual Trade which has taken place betwren Caii.ida .ind ih.- I'nited State' in respect of I'rodiuts of the Sea, as cnmiiared with the artifii ial Record ol Mr. Kou(hcitc, .\lr. Johnson and Dr. Young.'' These questions are of overwh|;lniing imporiancc at the present tune. There can he no doubt whatever that the United States Records of Covernmeni m relation to this Trade with Canada h.ive been jjriatly falsifietl, and there (an be no doidit that the Caii.idiaii Records of Trade with the United States ha'.e also been greatly falsilied. The (igmes pur|Kirtir.g to rcoord this Trade are malhemalii ally related -a thing im|)(issible in the ordinary course of Trade. 'I'lie i|Uesii,)n of Duties is, beyond .ill others, .i (lueslioti ol vital interest to both nations. The iiuestion of the l''isheries is another s'.ibjctt of great importame. Hut these are overshadowed by one of still greater moment to Canadi.ins, which is involved i;i the proved statement, that .\Ik. CommissioNKR John- son COSriS't'liS THK S^MK I'KACTICKS IP lO TIIK PKK'iENr llMK. The Chapters which suet eed the Tables prove beyond (piestion that in other branches of Trade between Canada and the United Stales, and Canada and fireat Uritain, the s.ime method of fabrii:ation has been pursued up to the last issue of Canadian Trade and Navigation 'Tables in the year of Crace iS«6. /Date of the letter, Dec. 22nd, 1885.) Cha|i! proves that the Canadian Trade Tables of iS;S were fabricated, with respect to Coltons and Woollens, according t • method «hi. h had been pursued foi so m.iny years with respect to 'Trade with the L nited Slates in the Products ol ilic Sea. In Chapter I, it is proved that the dilTerenc es between "Imporls" and " iMitries tor Home (.Consumption, 'with respect to Canadian Trade in Cottons and Woollens with the United Stales and (ireat Hriiain are, like the lish Trade Figures, capable of being |)Ut — • First: Into the form of an .■\rithmetii:al Progression. Second: — In terms of the " h'lrebrick and Clay" Series. Third:— \n terms of Dr. Kdwatd \i)migs United States Tisb, Shell I'ish. lish Oil and Tioducts of the Sea 1-Ap.)r!s to I'rillsli .\merica in 1.S72 _v Fnirt/i : In terms of the Differences between Uernoulli Column No. 1\'. to .pS terms, and Dr I'.dward Youngs I'ish Trade fr^ures before specified. (See page 5 ol this book.) The Table on pa^e 5 is a startling lei-nrd. Subtract Dr. Kdward Young's United States Fi.sh Tr.ide ligures of 1,^(72 7,i from James liernoiilli's Column IV'. to 48 terns 1 70 years old —and you get the Differences between Can.iiiian 'Imports'' and " Kntries for Home Consumpiion " for Cottons and Woollens in the year 1.S78. Wh.at a tale of deiepiion this Record unfolds ! V it has been the motive underlying it and Mistaimng it ? The duties are levied on " I'-nlries f )r rinine Consumptioi .uul these Kntrie.; are all forged. 'The Industry of a nation is misreiire- sentcd, and the Records of Trade with two ii.itions fabricated. Tor what purpose? In ('hapter II. the method of obtaining the proof of the faOrications with respei I lo I'rovinies is fiillv outlined. Kach imported article is stated as given in the Trade Tables. 'The lirst order of Differences shows the visible magnitude of the deiepiion, and the degree to which eai ii Province has been misre|iresenled in two i lasses of goods only. The accuracy of the Ki^jures is surprising, and ih.it al< ne is sutiieient lo establish inleientially. a so-called "cooking" of ai co:int.s. \',\n when attention is given to Cha|)ler III., the buret is exposed to view. 'The title .if this Chapter is: The Relation between I'.ernoidlis Column No. I\'., and the Details of the DifTjrences between Cotton and Woollen "Imports " and "Kntries lor Home Consumption '' ligures in the year 1878, and the Mathematii.il I'ormula showing tneir Origin. . ^ ■ Here let me again cnll atteillion to tin; great Difference between Provincial and Dominion Difl'erer.i es. For the sake of brevity, the Dominion Differei.ces have .done been presented in an analvtical lorm. On page 17 the Discriminating Differem es lor llie Dominion are alone given. 'The Discriminating DilVerem es for Provinces are much greater : for Co'o.ns more *:ian double, ,is may be seen liy re;''.re!i-'e to 'lable \ on page 16. The |X)sitive .ind negative signs of the Diflerences are ailiniportant. ' 'hange the signs and the lepresentation of the value uf the Trade < hanges with them. .\ reconl 01 liitics in excess econies a record it duties in defect and the entire record 01 Traile is reversed. -Ml of these i|uaiitities represenliiv; Dominion dis rimiii.i terms of Bernoulli's Column No. 1\'. Dillerences are represented on p.c-es 17 and 18 in sur])rising representation of Canadi.iii Tiade in 'The Mailiem.itical I'ormul.i which unites ihein all, foll(nv> thi Cottons and Woollens with the United States and (ireat Hritain. This formul.i may be thus indicated ; — Any term .\ is rei)resented by the expres))ion , a(r.+ 0(a+2)(a+3) to... ; a+ (n - j) ;■ 1, 2, 3, 4, 5, 6, 7 8, to (D— I) where a is equal to the number of the column in Hernoullis Table, and n is equal lo the imiulier ol terms in l!ie column inctudin;^ cyphers. 'The general lormula for die entire series is given in Chapter VI. The illustration of Dr. Kdward Young's Fish Trade figures, presented on page 20, is merely putting, with their denominations, a few of the h^ures properly grouped, so as 10 exhibit to the eye the real niiure ol their > haracte: .Ml of the United .States and Canadian denomin.itional ligures m relation to International 'Trade in the Products of the Sea, cin be put in similar form for several successive years. If we attach to the figures in Section No. 11, Table III, the denominations given in Statement No. VI, Table III, then the entire series oomes out from beginning to end, in the form of the larger (piantitics being sums of the smaller quantities, the principle on which Iternoulli's Table may be artifically constructed. In like manner, if we attach the denominations to the figures in Section No. t, Statement No. VIII ; also in section No. U of the same Statement ; also in Section No. I of Statement No. IX, all in Table IV, we get a clear view of this sLirprisin,n property of interrhangeahility. Hut further If the reader will turn to (Miapter VI, he will see that this interchangeable character continues up to the present lime, and that the Differences in Canadian Records between "Imports" and "Entries for Home Con- sumption," up to i.S,S5, are nothing more than the sums of Dr. Edward Young's United States Export Fish Trade figures for 1872 7,^ and these again are interchangeable with the "hire brick and Clay Series," also with the United States Import I'igures for the next preceding year i87^v74- AH of whi<:h is shown in Table IV. I'lii' whole may be likened to a knitted .sock — unfasten the knot and the entire fabric can be drawn out into the one single thread from which it was dexterously constructed. Or they may be likened to a chequered |)atchwork, whose pieces are fastened together by the cli.iin stitch which is used in sewing machines. Untie the knot which holds the thread and the whole may be disintregated by simply drawing out the thread. In subsequent chapters I show that this well-joined adjustment pervades Canadian Trade Returns up to the latest issue of 1885, all of which, as specified, may be disjointed md resolved into Uernoulli's wonderful scries — and then put together again - lOADKM ML'TAIA kliSURUO, The CoiKiN \vi> Wocji.i.k.'^ 'Imports" and "Kniriks kor Homic CoNSUMpnoN'' grouped anu put in the FORM OF AN E.NDLESS ArIIHMETICAL PROGRESSION. 'This illustration diii'i ted to the relation of the cyphers in solvinj; iinililem- No. 1 1 2 1 1 i I i ' ' I II I III I IV \ -_L L_„.i__ \i VII vm IX X I XI Ml XIII XIV XV x\i I 3 ti ! lO lO »5 20 I 15 6 I I 7 ;i .55 35 28 1 56 70 I 56 I 28 8 I I 7i() ^4 1 26 1 26 ' 84 .5'^ I o o II 1 I t* II', 10 45 '. 120 210 I -'5; 2\0 '20 45 j 10 ' I II 55 ! «6S I 330 i 462! 46i 3301 '65 55 11 '3 I.' 6() 220 , 495 792 92 1 792 495 i 220 (>(> 12 14 1 I i 13 78 286 I 715 12S7 1716 I7'<> '287 I 715 286' 78. 13 14 91 364 1001 2002 300,5 ,5^32 ,5003 2002 1001 364 m 16 17 '5 '05 455 '3'J5 3003 500^ 6435 f)435 ' 5005 3003 1365 455 18 19 14 I I 16 I 120 560 1 <20 4j68 ; 8oo3 11440 12870^ II440{ 80081 4368 18201 560 ___! J ! i i i . > _ 105 I '5 I ISO 16 17 ! 136 ()8o 2380 OiSS 12376 1944S 24310 I 24310 1944.S 12376 6188 2380 48620 I 43758 18 153 I 816 3060 8568 18564 j 31824 43758 19 1 171 969 3876 11628 27132 50388 7558* I 9*378 9*378 31824 I IS364 8568 75582 503881 27132 i 1 I 680 136 3060 1 816 11628 I 3876 The Squares from No. i to xil and i to 12 are taken from ncrnoulli's "Mrs Conjtctandi, opus posthtnnum: acctdunt tra reinainin« squares from xiii to xx and 13 to 20 are added. Similar relations belong to the Coefficients of (i -1)" where /;=o, 1, 2, 3, TABLE I. XJLl:.I'S TREATISE 3DE ^RTE CON-JEOT^Isri:,!. .. (■on,l,„u„„niln,s,. W „1, .,. Algchrai.: k.,„c..n.:„ion „f IU.,n„„ll,'s ,.,1, lTo,,ortv or Tluor.n,. :,n,l l,i. s.h ITopor.v ^111 XIV I XV XVI I XVII XVIII XX •3 9» 455 Powers I of Binomials. liKRNori.i.i's i;iii I'roi'Krtv. ( ' + o)" ' , ... I,ct «= mimlxi S 820 560 188 564 2380 8568 120 68c t6 '36, 17 3060 816 I 153 j88 27132 1 1628 3876 ; 969 (l + i)" {■ + •)" (i+i)" , , .V. a I. n 18 17' I 19 HeKNOII-M's Sill I'KOPKRTV. Let .9 equal the mud of the llori/ontal Series. I Then: 1 S=i' *'"'''' ' \Vhi( h is the e\|iris~ion for the sum of the co-effi<:ients of (.v + i')"* (i + i)" These are e Z 3 C c c c ^ CO CO r C •0 o CD a a CO c CO d T3 O a (/) CO 2 CO h o c en 0) s o u c- _ CO C 3 .2 c en C F 0) .•-1 m (> .>-> (J) (U •d (1) 0) x: 4-> o D O CQ • 0) (/} - • a cc a. CO T3 . "a 03 C -5 0) • c (U -J fTl T3 ■0 CO in i- (1) tn 3 CO a h o ^ c "5 3 c-. (U C u •n a c (U OJ CD <: r- 5 c 05 -.J SJ w in m u> 0) -5 CO c s m (D 5 in CO CO 3 u o h h t: S C 3 I n ■J A — " x. X. s ?S8 i R. I ; I J3S,t2 I a- |,'9,!}2 1^1 H^ "|i| . s ='■ , i , » »*:**' I |l 2"'* Ai jfs«.»2 |i|ir"-«'|i|F*'" !i{jg t: o - N N >; - r>. r^ !>. r-,>o I X :^ y; jc X - ? 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UJ < §. ^ r- ;-t3 -5 -J «> o " :^ '; «Ai "lO f-- **- 1^ ^ o »^ T ••* ■/:, " -r -r 'o - -r I. ^ i ?i 5 r^ -r -r "1 "X c 5- ^i- -r u-1 ""< 5 p c ■"> 1^ V V ■c >o I ■J t r : -> ^ i 3 + + •-, s + tn + '+ o< + + ca .5 ■S .!: ^7 3 Qi I ^ 1 1^ o 5 7. 'A D ;f; " j; 3 ' + +"+: M - N iO ■» + + + + + + . li-i "-r 1^ N r, "-. I + + + + ^ + + + f ''k ^: + ; + ;:4t^3.!t r r+f:?;: ;++ 4-1- + - ^ r^ -o "1 ^ +++ *r^ + + + 8. g- ^ - «" -^ + !S+5 + ++; .+ + +++ +++++++++ ++ m 'n t- - 1 .t; i. N - - •i 00 •«• ■O CC e r^ -f >n t^ M N » o r^ s -^ in s -f o O m « c^ (?■ a> r- ff^ t c e( r-- <« - - r- « - 30 O o •Jm <> N ^ t J£ '^ U ■■^a S i ^^ eo 2 » % ^,^. (J ?! •n ::' r ■ii TJ c c ,r c rt fi £ t— 1 n c rj •5 s •^ ) rs 1 , c . 3 > K ' « O 3 " V 1 '/•.Cs . X. X irs.s'- 5;''iBj-;.-ift i-f- .lJis.s.= t feS ^ 2 V V & ??; c^ ^ »C li*' n •■" •* .1 -3 *=■* i4i xl-Z 8 r.5 - 2 i E ■ ■ W ^ n o *» I .1.-6. t. = 1 Ji"3 o " "85 III 'pi .3 5 6 1= 1 1 i i :5r = ,:r = ::: = : = ; •? f ■1 " . c = i " 'J ^ iii11»s 1 oo -c f> t- * y 1 ;-P||i| ■* ■ 3 y ^r .. t .' ■S. 7 i I* - S !5 1 , , £ £ i t^:z s 5 X 13 C <0 ■• n N f Jiao r-<0 ■' ••0 ^ :f=:.3I :^ = hi " 3 c* ^ ^^ t £ f t 5 5 t -■ ■? -J!^ II Is. F' i c IS -•5 ^5, ■ n « ( r -t ' M 'f^ N CI " rn *^ v; n ri X SC < !>. f-.X - « - - r 1 f .n r N X + M c/: ■*"" N PI T '^ L S^^ t- + - N "IX Jt t- ^^t++++^+. ^ t- ■t- *- ■'■ + &&§IIIH2?1 -R i^ 1^ -r r^ XX - >' PI N •- r* r^ t>. t-. "' O +■ a^ PI ^ ± - M — - ^C fO -(- -r.. i N r^ PI * -t -T *" *^ •■TiX C. 0-' P* X *- 4 t- t- t - - a c- . * Is. r-s"^ X - X X PI •- PI PI ' 1- ^ Tf t ■ 5^: ^« II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II P*) 0^ — l-^C *" + + + + + -I - 4 h". r>. C o5g + + " 4- ■ + „ H ■o + 75 1 C" n 1*1 « o o pmc 3 «C f*^ r^ — X i I H M^ -o r-. rt I ■f + -f -t- 4- ■ o t- + O . 1 d- ■* "*. "^ 1 ^ Is. I ■ C - I f^ X « 4- -t- X X 2 '" -t 1^ "f t N f + Z + t 8 ao in o S^ i - cS N -r - — N '^1 ox-ocT'CC^'re-'rij'-r'f ^C 'C rs. ts, 1 , I » o ts. T I/-. fO "^ I PI trip* a\-c-"i ""i^-pix o ■i'^in— o^-ro^ "ico r^ ^ ^ -*■ w^x 5«'n5*- o&vOvPi ------ -•'I pipi", i-n"^* ♦'C ■'O t^ t-* is-x 0% — — po« iQ ^ — pn O* rO»^ PI c c * ■0 i = 8- vO + + 1^ ^ JO + + + N X X + + + + ' O tn ' + X + + T- + '+-»4-. + + 4- pn Sj + 4- 4- + ' c x "^ »r> r^ pj — p>t tt II II II II II II -^ pt — 4-+ 1 .5, J S? .4- 4- + 2,.; + ^4- 4- + + + X w^ rs ^ 4-4- -f- ts, -I- 1; I N i/^ • 'A , nl 0'^ + tt + in ++; l-s fs. _ O. C3N 2 tn iri ^ cf ■J' -^ .. .. _^ -f- 4- 4- % 4- 4-: 4-' + a p^ I ^."rt "1 - J. ; - c -= 4- SC ;^ - c- t , ■ • -r 3-4-4-' ,.£ -1- 9, in 1 l-s Is. 4- + r". rs. 4- p* a 'a 4- 4- 4--f- ©I iri vO ■* O ^ II II II II II M y X - -(- -;: '8 Ov •«■ • H II II 11 II 'A - -i I - f^ ■* lA pi' Q m' - H Q <5 - fl N P^ rO « «« C P-, ^A PO S^ '^ 8 PI O Ji ;C = ! II I \n H\ 4- II *-J^ ^ +4- * 4-4- + Ci-g r+4- 4- + + + s 4- £ o- p< P* 4- + 4- «" 4-4- 4- -8« iJ? £ 4-4- 4-4- S-S s + I ^4- n if- "> 10 "1 1^ PI ts. PO I "4-4-"; + -- 4- 1 ^ X " < in I -. 4-4-4- 8. g g, »n u^ in 4-4-4- N 0-> - - N -t 4- N 4--f 4--I- 2 c; ^ * - - N N ?-H?4- - 0^ o ^ ^4- +.4. ^ \o It '^ - 4- + 4- :j + fo -J PI ^ 4-4- '" + S, t ? 8; in I 4 I g-l; 4- + 4- ^■^ -. ^ i X -t- -(- + + N PI Is. Is. -t X X Z z - II II II II II II II II n II II II II II II II II II I I -. 'O * .0 - -- • ^ -tO>^0 O^|s.lr|fn'-0p ts. «. va - O C' * ^^■S5 5^^^ IT^Sil" ;•. \ Mmimimmmii^K»iii*-iaijttjj The Canadian Trade Tables of 1878 are Fabricated Records. Il is proposed to prove ihal the Canadian Trade Tallies of 1878. in respect of the Trade between (Janaria and the United States and Canada and Great Kritain, in Jotlon Goods of all descriptions, and Wixdlen (io'ids of all descriptions, are nothing nvire than a fabricated record, utterly misleading. Il will lie shown that they discriminate by means of Fabricated Figures between Oreat Britain and the United States in certain classes of goods, also that the record of .luties received on |;<«i.ls alleged >■• be importeil is necessarily a false record. It will also be shown that the Figures in the Despatch of .March igth. l8;o. addressed by the Marrpiis of Lome to Sir Michael Hicki Beech, are faiiricatcd tigurcs ami wholly unsleading wi'.h regard to the object for which ihoy are siaieil in the Despatch to have been iransinilled, vii :" A Memorandum of the Fin.ance Miiiisier. shewing how far. romparatively. Kngland is favoured in ihc new Tariff." This Despatch will be found in Sessional I'aper No 155, anno 1S79. It is dated "Oiiawa, March 19. 1879." The fabricate.l figures recording Trade with (ireal Hrilain and the United States in the ^oods selected for illustration. \ir , Cottons and Woollens, are types of prevailing misrepresentations which impugn and render worthless Canadian Trade Tables signed R S. M. !' luchelte. Commissioner ol^ Customs, and J. Johnson, I ommis- ioner of Customs (or many successive years. .No attempt is now ma.le to arrive at any conclusion respecting the magnitude of the Fabrications. The item "All other" encloses an aggregate so disproportionate to the entries in detail as to afford room lor enormous fabrications impossible 10 detect from the properties of the tigures used Hut it must not he for one moment supposed that the " Final DiKKEkENCi£s" represent the actual Diflerences between " F^nirics for Home Consumption" anti " Imports." The Final Differences arc the result of the successive reduction of the Provincial Differences to Ibrm the Dominion Record. The Provincial Differences are given further on, and greatly ■ -eeil the Dominion Record .\ttenlion is particularly called to the amazing accuracy of the figures in deiail This accuracy will be aos. IV. V and VI ) On page 5. 5th ■ -When these last named Differences are grouped and adiled they again produce the Canadian Final Diflerences between Entries for Home Consumption as compared with luiporis for ('olton and Woollen floods. {.No. VII.) *Jn page 5. ST.\TEMKNT No. X. The Dominion Rf.cokd as Distiniiuished from the Record hy I'rovinces hrkkakter given. TABLE showing the diffcronri.' hotween 1'': alleged values of items "Imported" and "Entered for Home Consump- tion," in the Trade and Navigation I'ables of ('anada for the Year 1878. The eoltmin showing "Imports ' being alone found in the "Despatch": the coliMnn showing values "Entered for Home (Consumption," being the basis of I'arifT Exactions. The arguinent relates solely to the construction of the columns of differences and the mis- representing artilice this construction develo|>es. COTTONS <»— YEAR 1878. tmportad Entered tor ' \ Home I Differenow. I t'oniumptlon. I f /*ai,^f j^j, 7*. ami A'. TaMes. ) Bleached ami Unbleached .... t treat Britain .. .. I Cottons, Uniled States „ Great Britain United Slates (treat Britain n United Slates (ireal Britain ■, United Slates j >, Great Britain > •> United Slates . . . . , I f. Great Britain ' n Uniled States | i, ■■ ■ . ■ • . • • . (ireai Britain ' Cotton Thread, on Spools (3) .... Uniled Stales | " •■ (ireal Britain . Cotton Warp, not coarser than No. 40 Uniled Slates •< Great Britain ^ Carfels of any maltrial, txaft IViVllen [^) Unlteti Slatm 1 n n u » " Printed. Painted, &c. Ginghanis and Plaids. . Jeans, Denims, &c. . . Clothing, &c. Another (2) .. $ $ 431.807 430.337 539.7«3 536,357 1,982,444 1,984,044 893.681 89^,633 20,205 20,385 4.463 26,929 4,363 28,528 137.49a I3S.'65 174.288 177,407 191.441 191,35" l.75».8oS 1,761,293 729,071 725.366 ■75.797 183.221 ».'33 2.133 692 692 ■4.674 96,000 14,074 96,562 8.058 7.900 $ -1470 -3406 -(- 1600 - 1048 -t- 180 - 100 + 1599 + 673 + 3II9 - 90 +8488 - 3705 + 7424 o o + 562 ■58 (1) N. B. -These items are dilTercnlly grouped In the Despatch. They will he found on pages 353, 352 and 350, ,Vc., of the Trade and Navigation Tables lor 1878. (2) In the Despatch this item is made to assume the form of : - "Manufactures of Cotton, all other" Great Hrilain $3,735,249 ,. ,1 United Stales 1,622,752 various details Iwing grou^MJ.! in this item. (3) Transposed in Trade Tables, page 353, see FCrralt. This item is omitted in the Despatch. (4) See page 350. This item is introduced into the Despatch as "Carpels of VVo.d and I'oilon," under Woollen CcmmIs, 1 if I ■■■■IIIMHi CHARACTER OF C.OOUS.-WOOI.l, ENS, YEAR 1878. lireal Brilain ■ ■ ■ I'llilert Slates (irca* Ilfilain ■• - Uuileii States Cirea' Uriiain •■ • L'liitcil Statfs (Ireat liiilain ■ . Uniloil Stale!. Creat Rritain '■• United Stales (ireat Hri lin . . Unileil Slates Citefri Britain ■ • • Unitnl Slates W(K)llens, viz., lUankets *' ('arpcts .... " Klannels .... *' Tweeds ... " Clothing, .Vc. •• Wursted aiiii \'arii Other .. KriteriHi for linfMrled. Horn* Consumption. Oifffreiii*- T' $ ■ $ 17J.J94 I9«.'24 +J5.S30 aS.OOK i.S.Xaa 176 651,49" 64**.W4 2.5^3 M..»5.i i_t.95i V.K> I0i,(>40 '59.35'* ■ 2,287 68,I><)S 67,651 1.043 9J3..1''7 916,684 6,68} I0.02() 9.507 r^ 75O.4.W 77>.245 + lI,§o6 12X.44<^| Il8,44*t + J Wl.O^l 65,209 842 5.655 5.655 S,'i".(>ii 5.' .7.773 + 7.'S0 147.014 t.S,304 + 750 It will he .ihserve.l iliat the values of the item " Wmillciis, I )ther "■ fireal H.ilain- $5. ! Jo.OiJ ; Unitcil S ate>. $147,614, vastly exceeds in magnitude the nggri-galc sum of all the other items entered fur (ireat Brilain whose details arc given, ami thus forms a Ilc)»ry or l)ump in which very many oilier " Differences'" may he coniprehciKled and concealed. It is ini|Hiiiant to note this entry " Other" or " All other," or " .Ml other, N. E. S.," (Not Elsewhere Specified.) r I ■ PROOF IHAr AI.I. IHK.SE I KIURKS .\RE EAliRIC.Vl EI). T.ABI.E prese.-itinx the Kin.1l or Dominion Differences in Statement No, X, arranged in Columns of rositive and Nct-ativc tcrin.s, showing the excess or defect of the values of (ioods entered .is "Imports" when compared with the values entered for •' Home Consumption," on which Duty is i)aid. COTTONS .\ND WOOLLENS. Po«ltiT« Ternm or RM.flW of Values entarsd for ** Hollis Onnsuruptioll'* oTcr Values of " Imports." Total 2 180 673 75" 1.599 1,600 3. "19 7.15" 7.424 8,488 11,806 25.830 . 6q,i8.t Ncfstlvs Temii or IHtfocI nl TalueM eiit«riHt U>t " Home Consumption" ss coiupwsd with Imports. 1 1 « a 3 ^ \ 4 i i I I 9 9 10 in II 1 1 la 12 13 '3 14 «5 Negative Tirmt, Toitl a4t35o II is ihe characlcr, rnnslruction ami relation* of ihoe I'ositlvc and Negative Terms which develop and prove their artificinl manufacture, nnd the fahiication of the ijuantitict from which ihey arc derived. PROPERTIES OK THE KIOURES EMPLOYED, No, I. TAHLE sho ing that the "Diffeiences" in Stalcmeni No. X. when properly grouped, form an Arithmetical Progression (roni 1000 to 25.000 and upwards, no single term being repeated in any group of terms. •si SUMS OP TilK "DIKFEKENCES" OUOUPED. 048 ISW aaSy a»e7 2*87 7«'4 74»4 I04« 7150 7«'< 66B., |*X^ 7«»4 74»< 7414 74'4 ai; 74»4 II.Bn« 7'S" 8488 84I8 M83 6M, s«. i6a> «sn X^ \'^ loo 10,8 ,M^ (^70 '470 J"9 7150 6683 IU4B 74>4 7150 71S0 74>4 74« (6. ^'« KU i;6 llo 7S0 1 ■ S8 1048 .0,8 i6do .7* "04t 1045 104^ H" 370s 3705 3705 7150 «bB7 l^, .<« iDo go "S' io«3 too >S« 56» ■7« 750 ts8 7«>' 841 1041 1041 119 I1387 ,Mj .>87 JOO 100 loo 67, 90 )sft lex. 10 loo 1^ 90 3 ICO 7 SO .s« '^ ^ Ml 158 100 !e 180 176 'is. loo 90 lun m8 <>8 loti 3BO sis 90 loo 90 1,1 100 176 V > lor. 100 iOO JO" ? too Ml V >5« 100 90 90 IOO tOQo »aaa toon 4000 yno 6qou 7000 Baoo gooii 10,000 11,000 11,000 i),oao I4 ■ SB "S3 3715 7434 4> 36 »6 S» sft 1^ «» tlio >,H HI! Icx) 3« »»• >l>5 180 • as ifi Sl» aa 71 jm sa !6a 3119 4' too 71 n to at) 71 ,,a to i» (l> aa 10 4» 4a 50 73 aaS; ■a! Q a9 4> 5a 8« a6 aa 5a 4» "043 90 ifJo ts« t76 t8o 3rt) «it9 56a 673 750 84i 1043 1048 t47o 159^; tftuo aa87 3533 1119 3406 3705 ()6»3 7t50 7434 8488 tt,Bo6 35,830 fl will lie .ili,eiveil thai liy successive sulisiiiuli.m my i.f the Difleteuces from 90 Id 25.»jo, can \ie reiKlcreil in Icrtus of the nuaniilies 9, 10, 22, 26, 42, 52 anil 50, which form the basis of the "Kire-hrick anil I'lay Series," as shown in Table II., Statement 1\., or ni terms i)f Herii.iulli's ("olumn III., of which the " Fire-brick and Clay Series" is the ei|iiivaleni. .No. 111. Tmk Fi.N/\i. or Domimds Dikkkrh-.m ks I'liK Ciirro.vs in 1(178 in Tkkms oi.- Dr. Kuward Yoi7nc.'s Fish, Fish on, SuKt.i. Fisii AND Products or tiik Ska, Unitkd Statks Kxi'Oris 10 Hritish Amkrica in 1872-3. VotlNli's Finn FlUl'RKN tOl I nx DlKFRRK.NCKS YotlNO's KlSH FlliURKS CorrON DlKPERENCES Yoi'Nii's Fish Fii;"nKs CoirON DlKKKRKVt KS 51 30 8 90 'i i 1048 ••■• ■ " 3 3705 6a 30 8 6530 830 5« 6 7424 62 58 30 8 158 1599 7SQ4 ■74 7h 6 S4K8 174 6 i8(«b. l6<» 90U 1188 989 459 58 108 350 350 SJ 76 ; loS 280 3" 62 1 62 »5S 8 30 1 5» '''i 1 6 30 76 1 8 3 258 l(}8 76 62 5» 6 ^477 iSS 240 76 62 () 3"9 33*> 2.S0 51 3 673 2830 350 lotl 58 52 S 3406 Jl Thk Kinai. Dikkkkknoes kok rt'o.ii.i.KNs IS 1878 IN Tkkms ok Dr. f;iiwari) Yoi!nK1>INH TO COVNTKIK.s, VKAR I»f7». COTTON (lOOM. qttSAT IRITAIN. TNITBD &TATKH. PotUiw Tcnm. 180 S6a 1599 1600 3110 7424 8488 N«(Uin Tmhu. Po«iUv« T«nni. Nf^aiivv 1 trma. 3406 1048 too 90 23,972 1470 673 S507 WOOLI.F.N GOODS. GBKAl HMirArN. UNITUl MATU. + 22972 — 1470 + 2 1 502 8507 -8507 + 67J -7»34 uiiv« Torms. NrKfUhrvTarau. Pohitiva Tcnni- Ncgstiv* I'trma. 25.830 2523 2 176 11,806 2287 750 too 7." 50 6683 84a 1043 44.7S6 "3>335 752 20)8 + 44,786 — '2,335 + 32,45' -2038 _+ ^52 -u86 These numlwrs, (vissesiiing the relatio.is poinlehowa the origin ni their labric.ition, ami discloses other reinarkahlt- relations. In order tu lieierniine llie ;ip]iarent relative amount of ,Iuty cxacte-l on these differences in excess, or remitted on the tlilferences in defect according to die entries for Home <'onsum|ition, reference must lie li.,d to the firit or Provincial order of Differences as deduced from the returns credited to the separate I'rovinces and e>hi!iitcd in detail in sulise<|uent pages. Then there is to lie considered the Dump or Dcinisittiry - Manufactures of Cotton— All other- (treat Itritam .. .. .. .. $.1,735,-49 „ ,( I, -United Sia'.es .. .. .. .. 1,612,752 Wotillens -All otber- -Oreal Britain ., 5,150,623 M ,. —United Slates 147,014 4 THE RF.RNOUI.LI TABLE And the pAnRiCATRD Trade Tables concerning Coiton Goods and Woollen Goods in 1878. The Differences between the forRcd Imports of Collon (;ole, sums one n( ihc other, and apply also to the negative Difl'erence*. No. IV, Nn. V. No. VI. hlRNUl'I'i-i'a SRHIVS IV, will) iiAf>, in Urutlatril, llie ftif- tircs in ilalirt. * ■■• rt 10 30 35 g :;:::::;:: lao •65 '7' ••■• 220 a86 364 % ;::::::::: 680 816 969 1140 1330 1540 1771 2024 2J0O 3600 »9»5 3276 3«>54 4060 4495 4960 5456 5984 ■ ■ ■ ^545 7140 7770 8436 9"39 9880 10660 52yj 11480 5S94 '2341 6530 '3»44 7894 14190 17929 '5180 17971 16215 29597 o .."^'S I»K. Yoirvo's KlsH Tnaiic Kioi-Ki-a 3 6 8 , 30 52 :.:: &::::: •••• 76 .... 108 ••• 174 240 ::;; ^ :.::::. 329 — ■ •• 338 •■•• 350 ■••• 353 •■•• 459 700 .... 8.JO .". • •. 900 933 989 .... 1188 ■• '338 >57i .... 1641 • •■ I7l» .... 1865 .... 18S4 •■•. 2144 ■•■■ 2354 3477 . .. 2598 2614 2«}0 3452 j8oi 4553 4790 5204 DtfrKNRNl Kft. ... — 3 ... — a ... - 3 .. — 30 ... -- 3» := % . + 8 . + 12 9 .. - 69 38 6 35 "17 210 327 357 269 no 24U 398 583 686 729 958 •213 '413 . 1770 + 191(1 + 2141 + 348.! + 2858 + SW + 37'S + 3(188 + J969 + .)8«3 + 4349 + 4(17" + 5.5'>7 + 5';^^' + 58"! + 5350 - 37. W - 2791 -13382 -4906S + + + ..+ .+ ..+ .+ .+ ..+ ..+ .+ .+ ..+ .+ ..+ + ..+ .+ ■95.726 195,736 or 138,366 .Sutn of Ncualivc or IKMitive Diffttrcnces. 3 3 3 6 9 2U 33 32 38 69 3739 3791 ■3382 49068 69183 The remaining po..i- tive Fitial OifTerences, necessarily amount I'l 69,183, anri the Joint Cotton anr N'oiint; Fish Figures of i)i72-73, and their hijnres arc sums of Ihovc nifTer- ences. Tlieipiantity 24.350. Iteing the suiti of the defect of values en^ tered for " Home ('onsH in pi ion " as compared with •' Inu potts." is easily obtain- ed l>y the pror(..;ses illustrated in Xo. VII. Sum of Difference., without regard to signs . ., ** ** according to signs 69183 .. 138,366 69iS3 = o I'HK KuukM IN THK DKSCArCH OF .MsRCH I9TH, 1879. iHk KlNAI. SVRilUI rOM l'oTr"SS .\N|) WilTerence.s. $69183 1 ... 90 3 ... 100 3 . 158 4 ... 176 5 .. 300 6 ... 519 7 7 ... 84J 8 8 ... 1043 1048 9 9 ••■ 10 10 ... 1470 II II ... 2287 13 , 13 12 ... 13 ... ... 2523 ... 3406 14 ... ■ 3705 15 ... ... 6683 $24350 No. VII. The Final OiflTerences in the Tragic T.iMes of 1872-73 atif! in the Despatch of March 19th. iSyt), are siim-i of the Differenres, with- out regani to signs, between Ur, Young'* Fish Trade Figures of 1872-73 and the liernoulli Scries in Column IV. lioi Vm 9 iTlvi; TmiMS. 20 2 ! IlIK NUOATIVE T«R MS. 6 23 2 1 6 2 35 2 32 23 1 32 9 35 25 139 551 38 327 JS 20 117 32 583 398 (•9 117 180 563 673 750 90 ■00 158 176 117 3 * 1 9 1 2J 2 20 12 269 20 ■2 «9 ' ,i-' 8 23 35 213 18 95» 240 .15 35 32 38 117 2141 2483 210 117 38 958 210 4349 357 729 .2.3 1599 1600 3119 7150 300 519 842 "043 9 8 8 2 1 n 2 9 2 12 12 («» 20 y- 20 20 38 1688 . 95S 357 69 35 35 117 2483 3715 2141 .5969 210 958 1413 2141 [5367 3688 240 3715 729 5367 5811 i 104S 1470 2387 352J 1 2858 3370 5811 I 13382 : 398 ' 69 i 38 1 327 8 729 117 20 6 -—.— .■— »■ — _ 7424 ' 8488 11.806 25.830 3406 3705 6683 1 It follows ai an arithmetical conseeiiarttie liKiirc^ of ihc Sctie. iii Cohimn IX anJ XI 1 1 iJ l«rin« is laUn, llic kum ..f Ihese l>itr.ieiice» «iih,.iil r.-(;.>r.l ii. sini.s is c,|ual n. .hmhle ihc ilifTcinuc lictwecn Ihc lul terms of each, oi with regard to ilgnii, equal to o. This relation is shown Iwlow: ■■ ^ ■ l.Jumn XI. l)ilftr.iK««. o O O Culiluiil IX. O ... o ... o ... o ,. ... .. o .. .. 1 ... 9 •• o o o o o o , o o o I II 66 386 3.00J '•<*" 6.435 h°°i ia,S70 8,oo8 a4,)io I9i44^ 43.75» 43.75? 495 1.187 o o o o o I : "i **\ '54 1 • 419 \^l6,7c;6 1,001 2,oi>a I t>43> t.861 4.862 ' o 75,5S3 9J.37* -l6,79<> 167,960 167,960 o or 33, $9a without regard 10 sigiw. The last term of C i!v,mn XI i» ')2,37>i The last term ui Co/iiiun IX i» 75.5*1 ' Di«fe ence 16,796 Sum 167,960 or the sum is equal to ten times (he Difference, an.l 33,591 < 5 = 167,960. Thu properly of -some of the .:oIurans of Bernoulli's Table is introduced here for future reference. :|; > ♦ * * OHA-FTEJR ix. THE RECORD BY PROVINCES - YEAR 187«. lan he Spci lal attention is din-f led lo the Mlowlnn paKfs. The stibje< t treai.'d relatcn to the Oftu iai Rprord of Canadi, Trade liy rrovmcts. _, All tin details are Kiven. I'he prixcss of arrivinj; at the results is displayed in full. T aeniMty of the fl^\lres '\s slrikinn. '"hey are nctcssat'ly all fabricated, hetause they form when |)ro|)erly grouiicd ist, .All arithiiietiral I'ronteHsinn <«•/• rt«/c;; jnd, Ihey ran he put in the form of the Kire Urirk ,iiul Clay Se.ies; ,Ud, They are nialheniatii ally related In Kdw.ird NduuK's' United .States Kisli Prade finures in United States OlVu iai Kerord.s; 4th, They are niatheinati.ally related to llernoulli's t.d)le ; -th, 'I'hey are inatheinatiealiy e<(»ivaleni to the sums of the Co-efliticnts of the successive expansions of the Hinoinial (i + i) to the powei of n, where n is successively etiual to i, 2, 3, 4, 5, \c., to any number of terms repre- sented by »; 6th, They are mathematically subordinate to ihe general formula given at the end of this chapter. The Record by Provinces, as distinguished from the Dominion Record — Year 1878. COTTON GOODS. COTTONS, BLKACHi:i> OR UNBLEACHED, &c* limnia tuoa Great Britain Unileil Sintes Great Britain United .Stato Ureal Britain Unilcil .Sii\teH Great Britain United .Slates Great Britain United Stato Manitoba ard OHer ol UlffartncM lurtlMf (roupwl. 4lh or Final omcr ol G. Britain + 3038 -4508 G. Britain — 1470 - '470 U. Statei. U. Statei. - 3406 + 269 - 3675 - 3406 - "936 • Tola 11,490 COTTONS, rklNTEl), I'.MNTED, OK COLOURED, Sec. Great Britain Ontario 7»5.88j 720.955 - 4928 G. Britain. + 8630 United Slates " 326,205 326,309 + 104 G. Britain. Great Britain United States Great Brii.iin United States Quebec Nova Scuiia 977.243 397.630 76.795 59.421 976,59s 396,924 76,758 59,383 - 648 - 706 - 37 - 38 4928 648 .1309 + 8630 — 7030 + 1600 U. States. G. Britain. + 1600 - 7030 Great Britain United Stales New llrumwick 159.764 78,447 159.656 78,03:' - 108 - 409 U. States. • + 104 - 1153 + 105 — 1048 U. Stales. - 1048 Great Britain Manitoba 13.844 22,474 -4- 863" 105 United Sum 1, 7,220 7,220 706 38 409 Great Britain British Columbia (1 4.657 5.393 3,348 5.394 Total - '309 + I + 2648 United States - 1153 16,918 * Where a Pruvince is omitted, Ihe Imports and Entries for Home Consumption are the same. ;| : f i ff^^VeAWHtW!- CilNC.HAMS AN1> IMAIDS. iVc. IHIMITIO KKoM (>ie*l Utiltin United Sum . a ... . I m porta. i.S5» 8JJ 10,844 5.588 i.3»<) 1.395 >.Si3 436 ' Knlertd for j Id Onltr I Hufii« of «'imitiini^.i'n. l>lff«rrn<IIT' .'vlit'ti. UlffvrulK^VB. 41I1 C>nl>r I "I , l>lltrr«lu-«a. $ ■ .6<4 IO.S44 (i>9 5..S«« i.3«'l 1.390 1.533 510 91 101 ■o« o U't Britain. + lot _«J IMS 5 + + 5 « • X3 2 ' Total ! I j()4 IKAN.S DKNI.MS AN'D URII.I.INCS, &c. U. Htatea. + a — lOJ G'l BriMin. + 1S5 S + 180 V. Statet. + » — loa (i'l Mrilnin. + iSo U. Slate*. - too 80 (ireal Hritnin „i,„ Ontario •.. Ji37» a.708 + 336 G't Britain. United State* .. Great Hriiain »... II ... Quebec 1 35.8»9 ... IS.«">5 35.4aa '5.392 - 407 413 + 3.?6 '.790 li'i Britain. Unite:! States Great Ilritain Nova Scotia 57.6J9 2,048 58.28<. 2.04H + 657 2,I2(> 413 + 2.126 - 537 G'l Uiiiaiii United Statea (ireat Britain n New Brnniwick 9.005 .. 1..38 9.005 1,138 Q t>4 - 5*7 U. Stain. + 657 19 404 + t.599 V. Stnlen. + l.ogo + 1.599 United .States „.,., Great Britain Maniiitlia 18.906 J51 18.906 >37 114 ~ 407 + 673 + fi73 United Sutei i.59> 1.610 + 19 1.080 (Jreal Britain Briliali CniunriMa 5.3"5 7.105 + 1.790 - 407 United .State? ... Tiiial .. j 14.4 '■; H.»2l + 404 + 9»6 4.140 2,272 nOTlllNi; AND WEAklNi; AlM'AkKI., \c Great Britain United Slates.. Great Britain United Stales.. Great Britain United State*. Great Britain United States . (jreat Britain United States. Great Britain United States. » lUliiHfID fKnH Hi Onlaiic Impart* 40,104 Kntfretl for riinBiim|it'n $ 4o,oir Ill Onlnr ol tHII*r«i«'M. M Orttr nl IHfferAtims. M Ordn MhOider .ii or I>ltrtrf'n<-«a I)|(lBr»iir..«. OmI Britain - 87 United States (1 ... 4«,307 48.149 - 158 G'l Britain. Great Ilriiain .... United States Quehee 44.9>6 *3.5S4 44.773 13.479 143 - 75 -t- a,443 1.058 G'l Britain, + 3.S00 - 381 U'l Britain. Great Briuin Nova Scotia »5.405 a $.405 + 3.500 87 143 «5' - 381 U. Statet, + 334 -♦-3.H9 + 3.119 United Stales (;reat Britain United Stales Great Britain New Brunswicic Maniloh. 31.048 39.860 66.316 «8.a73 30,967 39.709 66,316 »o,7i$ - 8. - <5' + 3,44a U. States. + 334 - 3M - 90 U. Slates. - 90 United Slates Great Britain Britiib Columbia 3.S7a 4.784 3."9«> 5.843 + ai4 + i.osS >$8 United States 1. Total ■ 3.013 . •il.'^''.. - 314 44>9 + 3.039 Al.I. OTHliR, NOT ... I Ontario ■ ■ . ■ Quel«c . • . • . • • • Nova .Scotia • ■ • ••■ i New llrunswicl< ELSEWHERE SPECIFIED. .Vlanilolia .■■■ ■ ■ • : British Columlna Total . 1,001,050 i,or)S,633 + 3.573 G'l Britain. 367.751 363.57a - 4. '79 + S53.4ai 550.084 - 3.337 3.573 59 37a G'l Britain. 171,306 171,340 + 34 + 11.835 G'l Ilriiain 1.037 6,784 - 3.337 105,716 105.775 + 59 + 11,815 + 8.488 + 8,488 79,003 79.003 3.337 U. Statts. 53.053 53.4a5 A- 37a U. Slates. + + 1.69^ - 5.401 U. SUtes. 65.997 65.079 - 918 1,663 - 3.70$ - 3.705 7.676 8.713 + 1.037 1.696 ia.435 14.097 + 1,661 4.179 918 304 19.564 16.348 + 6,784 - S.401 31,57a 3i,a68 304 32,359 - 4.783 1, 10 COTTON THRr:Al) ON SPOOLS, &c. larains Fioit Great Britain Uniicli Columlti.-i iDiporU. 9,755 473 >"3."43 369 32,. 30 K95 '(.W KnUfMlfor UotiHi Ooniunipt'n. lit Ordar o( IMKaranoM. IndOnler of Piffcrenoea. 9.755 473 "".399 + 8256 (i. Britain. + + S25f> 369 -693 - 139 it. 446 - f^n -83a S95 U. Slate*. qoo 130 qoM 3nl Onlar [MITervnoM. 4th Ordar o( Dffftranocc + 74»4 (1. Britain. + 8156 - 832 + 74»4 G. Britain + 74*4 U. State*. V. Statei. CARPF.TS OK ANV MATRRIAI,, K.XCKl'l WOOI.. \<. Great Britain United Statu Great Britain United .states Great Britain United Sutea (Ontario iQncbec \iiva Scotia . • 50.347 I 4.099 ...I 16,907 i '.59a ..^ 8,4i8 I J97 50,680 3.941 17,446 1, 59 J 8,118 »97 + 3.i3 <•• Briiuin.j - IS* + MO o 310 I 333 539 + S72 - 310 I'. .Statea. 158 J340_ G. Britain. + 87» - 3'° 0. Biitain. + S6a + 56a U. Sutea. U. Stale*. - 158 - >58 + 404 The ligurw of Prince Edward Island from Great Britain and the United Slates are recorded as showing no difference between "Imports and ** Entered for Home tunsimiption" in respect of C'»tt 372 404 539 657 '037 105K 1227 1662 1790 1811 244a 3573 6784 8630 8356 41.808 Total UifTerence.. Total Sum ... . 41,808 28,140 69,948 ■ 3,tiOH tireal Brilaiii (itL-at Britain United State* United State* Unite.1 States (ireni Britain I niled Slates Great Britain Grt-at Britain Great Britain Great Britain (.real Britain Unitel States Initeil Stale* United Slates lireat Britain United Statei United .States Uniteil states Great Britain (ireat Britain tireal Britain United .State* United States Great Britain (item Britain (ireat Britain United Stales (ireat Britain Unileil Stales (ireat Britain United States Great Britain 5 37 . 5? 87 tot lOl 114 139 ■43 '*! •s< ■ J'M . '^ ¥V 409 . n <>iS laai '309 • 1391 . iSgS >9.l6 3337 4"79 ' 4^ a8,t4« 11 ' Th< Fourth, heint; l/ie Dominion, or Finn! Order oj T)u Third Order oj Uijfermm Jor Cotfon Goods, 1S78. ; Diffcrenm for Cotton Goods, 1878. OocntT laroims Pmii. Pottv*TarnM. Cireal Hrilain. . . . I'nilfil Stales Gtc.il Britain .... Unilcd Slates Grcai Hrilain. . . Unitcil Slates Great Britain. . . United Suies Great Britain. . United States (ireal Britain . . . . United .Stales Great Britain. . . . Great Britain l.'nitetl States ... . Total .Stini! . . Oifference 3038 369 8630 105 ■85 3 2136 1080 334 11825 1696 8256 872 o Country 1mih)RT«i> Kboh. 4508 .1675 7030 '153 5 loz 527 ,07 SU .14 .'337 5401 «33 3>o 15s 28,140 i(ireat Britain . iGreat Britain ■United Stales tircal Britain iGteni Britain . '(ireal Biiiaili Great Britain . (ireat Britain Tol.1 PMitlva Teriiu. Negative Teruw. 180 90 5'J2 too ''73 '5S 1599 ■048 1600 1470 3"9 I3406 7424 3705 S48S 33.645 9977 Third Oni,-r + 41,808 - 28.140 . . i.i.66K fi9.'M8 TJierefLrt'. 4 1, Sots 9.977 5".785 is tfjual to Final Onit + 23.645 - 9.977 Suiii . . . .. 13,668 Difference . . 33.632 18,140 33.645 5'.785 The I )ifferences arranged according to Countries will display other relations. These will be found at the close of the analysis concern- ing Woollen Goods according to I'rovinccs. The "Country iniporteil from, ' ..-fcrs only to the positive terms. The entire relation of the figuies in respect of Country is shown on page lb — Table A. The Record by Provinces, as distinguished from the Dominion Record — Year 1878. ^WOOIL.LB3Sr O-OOIDS. lnrokTiD rsoii WOOLLENS — BLANKETS, &c RiiUred lorT Great Britain United Stales ^'.reai Britain United States.... Great Britain United Slates.... Great Britain United States.... Great Britain United States.... Great Britp.in United States . . . . Ontario t^ludhec Nova Scotia New Brunswick British Columbia Manitoba Imports. 4Si3S9 11,106 6»,|^ 6,66a 8.M6 Hoam I let Order. i ODfMmm£jt|n. I % '■ 45.066 10,878 Ol,OJ4 6,66a 7.981 — 393 — ia8 - 153D o - 3*5 1.492 '9.917 4.367 ".49» j 19.659 ' - »58 4,367 j o 7.985 I 33,986 ' + 25,001 i.»33 '.«33 j o 36 }24 39.699 I + 3»75 537 : 579 I + it_ I 1 1 .003 9nd Order. Srd Order G't Britain. + + 25,001 3.375 28,276 +28276 - 2446 393 1,5.!° 363 358 3.446 IT Ui.l.. o U. Slnlei. + 4th Order. + 35.830 I 5» 338 + 5» - 338 176 19 , ! CARPETS. \c. iHmRTVl) Fnau (ireai Hrilain United Stales.. CIreal Miiuil. L'nitcii states.. CtTQM Ilritain United States.. Great Britain United Slates ■ Gtent Britain United States.... Great Hrilain United Slates.. .. Great Britain . ..• United States Great Britain .••. United States Great Britain United .States Great Britain •■.. United Slates Great Britain .••. United States Great Britain .... United States Great Britain .... United Miic< Great Britain .... United Slatea Great Hrilain .... United Sute* Great Britain • • . . United Stalei Iniporta. KntmJIu Hau takOMir. Conaiii«|ii'h.| I Onlar. Srd Orrtcr. 4th Order. Ontario . . . Quei>ec Nova .Scotia New Brunswick British Coltimliia Maniloita- JS9.439 I aS4.874 I 7402 ^07,143 4S6S 7102 — 300 'Gt Hrilain 308,447 + i<3°* 1,304 753 753 51,602 52,054 I + 452 33' 3i' 116,161 II5,8J2 4920 535' 40s 4920 5966 405 Ontario .... Quclscc . . . • Nov* Scotia FLANNELS, Ac. i .... ' 38,858 ! 14,048 -... I 148,401 ■ • 'S,»33 .... 28,985 ! 8,991 t New Hninswicl^ British I i'iiind)ia Maniuiln . . 3i,''37 »8,5»7 3.294 943 8.9'4 a Ouelirc Ontario Nova Scotia N«w Brunswick 440,490 '.909 75.'«7 543 88,779 6s> 39.354 '3.«>4 '47,8'7 14,530 26,919 8,991 30,91 3 38,517 4,080 943 8,719 87 TWEEns. &c .... 321,735 3»'.683 6,916 6,916 '.390 75.439 543 88.156 — 4» 434.640 — 5850 iSlrl&^ o 3'9 + 615 o 7.565 -t- 496 - 384 -584 — 7'3 — 206(1 — 7J4 o + -86 o - 195 +_M 6003 a.37> 4,565 ' 329 j 4.894 U. Statea. + o + — 300 o ! 300 - 1513 300 G. Britain. + 496 786 1282 ^84 2at>li 714 '95 3569 U. Slates. + 54 384 7'3 1097 + 1282 — 3569 - 5'9 — 168 o -6*3 o 7aoi + 54 — 1097 2287 — '043 I G. Britain. ' + 1 I ° 1 5*}° 168 I I 6*3 ' .6683 i 1). State*. + o 5«9 + o -6683 + or — 5>9 6683 — 5'9 \\ i;t iMPOATRD KBOM Great Britain I'nited State** (iront Mritiiin Vniteii Strttes (Jrcat Iliitnin L'nhcil States Great Britain United States (Jrcat Itritain Vnilcii Sialfs Great Dritain Unileii Slates Great Itritain I'nited Stales Great Mritain United S'ates iiclter ■.-. •... 102.744 105,665 +2921 ' 2921 I 6484 .... 10,855 >".44<) Nina Sciitin S.mic, Sanu-. — • 4C6 22j4 12.397 445 346 + 12.597 79' ii,So6 + 11,806 New litun»»ick •. .. 126,0^7 125,^82 445 791 .. i >a,8i3 1 13.368 liiuish (uliinil.i;! . .. 30.479 : .36,963 + 545 + 6484 U. Sinle«. + „ .. 5>i»50 1 S'.668 P. K. Islan.l .. . 17.947 ! '7.60I .. i 2,f)62 I 2,C63 i Manil.ilia .. .. 1 55.625 ' 57.S.S9 -582 346 + 22J4 545 1005 '550 560 406 582 1548 + 1550 - 1548 3 + 2 „ 1,898 2.903 + 1005 ' . ' 16,486 WOR.STED ANn Y.ARN. &c. Oiiiaiiii i 19.707 I — ' 2,267 19,244 2,267 - 463 G. Britain. + + -842 1 ! '. - 842 (,)viel)|.c 23.396 2j.'55 - 241 463 1 i.>i}i 1 2.S33 ?i H^in^h ( ..luniliia .... 1 2.490 61 ^.35^ "3 - I3X 84a S4. U. Slalcs. + .\l,I. OIHKK. K.it«reil lor lit Order id Order ad Order 4th Order Nh.HTFn Kr«.m IK Iniporta. l,6o2,7ji nomd ul 1,1 ot ot OoMUDipfn l,6il,tloI DUferencaa. + 9070 Dllforuii{.ei. DIfferenoee. niRerennja. Gr»»l Britain Onlarii) O't Britain. riiitcil Stales .. .•-*.. 78,984 80.553 + 1569 + 9070 (ireni Ilritain Quelle 2,363.^15 2,364.277 4- 662 662 4160 + M892 I'niteil State* „., .,. .. 26,014 i5.<-" 293 13«92 I1742 + 7150 --■ 7150 <'.rMt Britain .,.„. „„, Nova Sriili.i 406,713 *04.359 - 2354 2354. 3652. 7,36 L'nileil States 9,423 8,688 - 735 rireal Britain New Hrun-«icl< 545.325 541.673 - 3652 674» I'niteil States .1 "6.393 16,393 U. Stales. + 1569 (ireal ilritain Briliih Columbia !i8,8ia 32-97» + 4160 209 + 177S I'niteil Sm« " 8.737 8.737 1778 - loaS 750 + 750. (ireal Miiiain ManiioLa 48,02 4r.3» - 736 293 735 I'niteil Slalc» " 701 910 + 209 103K »J.440 • \ — frtis-: — - u KK( AI'ULM.ATKIX. Wnoi.i.rNs i>it1uri-fi>-fk IhI ontf r m.inliCts $ JI.OOJ t, ariicls 7.5'>5 (.'lolhing 16,486 Worsu-vl, ic. 0l All olhet JJ.440 $ 02.539 Ti>iai I>ifferenc(^ \ViK)llen (ioods COTTONS, I hrtcri'iu-e.'« In 1 t'oUi>n.s, Blearheii, \c Ci)lt(in», I'lintnl, &c ( im^hanis ,in-l I'laids $ ll,4 -")4 leans, heninis, At . , All oihcr Cotion Thrciil, ,Vc, . •t.140 22,151) .i,o,S8 *'.ir|iels ... I.J4" $ 69,94s $ 9a.539 69.94» $l')2.4S7 THE SECOND ORPER OK DIFFERENCES FOR WOOLLEN GOODS, 1878. United Slates United States United Slates Tircal Kntsin Orcat Britain United States Great Britain Oreat Britain (treat Brilain tircat Britain United Stales Cirent Britain United state* Great Britain (fre.lt Brilain Great Brilain (.treat Britain Great Britain Great Britain (irtat Britain />j|/ltv Tentn. Ti)t»l nifierenre Sum • . — 61, 85^ - 30,687 92.S.W JI,i6S 61,852 Great Brilain Great Britain Great Britain ("treat Britain I'uiied States (ircat Bri:ain (irc.tt Britain tireat Bntain Unilcil Stales I'nited Stales (treat Brilain (ireal Brilain t.ireat Britain United Slates U'nited Stales Great Britain (treat Britain Uniie.. Sum nf (^(itton disciiniinuting Differencen " Woollen $ 5K,4lS more air certain clauci ul guud^. '3.434 '• ;; 25,967 less " " 4.720 '• '• $ 94.539 Total $ tK),948 92.539 $163,487 ) t 16 THE IIIIKD DRIM'-.R OI I)IFFP;RENCES FOR WOOI-T.ENS IN 1878, s» 54 1282 1551 1778 J371 ia.597 13,89' 28,276 61,852 Isl or 3rd Order of nifferenccs for Cmtons in 1S78 *' " W.wllens " 300 519 79> K42 1028 1097 1548 244b 3569 4894 6683 674» 30,687 + 61,852 ,30,687 Difference 92.539 Sura 31.16.S -(- — ... 41,808 28,140 61,852 30,687 Sum .. 103,660 58.827 103,600 58.827 )ifTerences for Cottons and Woollens in 187K 162,487 (nd 103,660 less 58,827 gives 44,833 THE FOURTH liKINC. THE IKniINIO.N' OR FIN.M. ORDER OK DIFFERENCES FOR WOOLLENS IN 1878. .; '.~ -"--•- TH nii/itv Terms. Nt/^Un Tirm. United Slates United Slates Great Britain Great firitain ...... Great Britain 7.150 II,8o(J 25.830 45.538 Difference Sum 176 300 519 842 1,043 2,287 »,5»3 6,6X3 14.373 ■t- 45,538 14,373 59,911 • United Slates United Stales United States Great Britain United Slates (ireat Britain Great Britain Great Britain 31,165 Sum of I'osiiivt Terms of the Final Order for Woollens in 1 Cottons J78... .45,538 2.!.045 Total sum of I'ositive Term.? for Woo lens and Cottons *' 69,183 Sum of Negative Term of the Final Orrie for Woollens in Cottons 878 ■ 14.373 9.977 Tntnl sum of Noea live Terms for Wo< jilens and Cottons in 187 R ■ 24,350 19 THE I'INAl. DIKFERKNCKS FOR l(Vn(i^ (iOOUS AND WOOLLEN OOOHS. PTrSITIVK TERMS OR EXCESS Or VAU'RS Fntcre.! for '* Home (.'onsiiiuption/' over values of " Iniports.' Puithv Trrms - Final Difif1ercncc» ilcrivi-.l from llcrnoiilli'« I'.iliimn IV uvl Dr. Ivlwnnl Nourr's Kiih Trade (igure» of 1872 73. ^"W'AklSON BETWHEN THE SUMS Ol- rilE I Hh I- KliENl.RS, \VJ I 11 AND WirilOfl KE'.ARt) TO Stl^Ns, VKRAM.F.Ii A(.10RI>ISr, TO COINTRIES, AS lir.RlVKU FROM IHE I't'OVlMI Al, OR FlKsr ORHER OF IUFKERRM ES AM> 1 HE KiNAt Ok DoMlMON ()R[IER OK DlKKERESrES IN rilE VHAK IS78, TABLK A. i«7S Cotton Goods Vear 1S78:- Oieat Britain United Sttlei Woollen Ooo,U -Year 1S78 : Oreat Britain United Statei ' First Oriitr. + 38.43* - - 16,930 - + 3.376 - — 11,210 - 69,94« Ftr^t Ottifr. + 58.418 - ■- 15.967 - + 3.434 - - 4.7«> - 9».S39 Final Ortler. + 22,972 ..470 -f- 673 - 8.507 3J.6M Final OrJtr. + 44.786 - i».335 + "J 2,038 DifftrcHLii. 4- lf;,46.>\ 15,400 1 + 2.703 1 - 2,703/ 36.3^6 30,920 I r o 5,4.^) ot o 27,204 or o 59.91" riffcitn.C' + 13.63JI — I3.6j» ) 2.682 ) = 5-364 "■ o 3J.628 Total Difference lielvsccn isl Order of I)iflerenre» an.l final Order of I)iflerenc» for Cotton .ind W,K.ll.-n 1 imxls in 1878 ! Total Diflfcrcnce Total Sum Isl or Provincial Onler, Cotton* and Wcmllenk " Final or Duiniiilun Order " 36,326 32,618 68,954 Sum of Final f\>silivc Term-, wiihoni ri-gard 10 ruuniries Sum of Final Negative " " ... ^['.'^ T'«^J/69.I'*3 '»> ')«*ii sl>"wn on paRc 5 lo I* the sum ol half the Diflmnres lieiween lirriioulirj Cohimn No. IV and I)r tdwai-d \ uung > KJi Trade h igurcs for iVja 73. The relaiii»» of the other afxregnies have now 10 Iw shown. 162,487 93.533 DiHerence . .. 68,954 69,183 24,350 Sum •• 93.533 : Difference • 44.833 1? OH^s^PTBR III. 1"lir. kEIATION tlF.T\VFrN llFRNOIl.ll's ("olI'MN No, IV ANI> TIIF. DKTAIIS OI- TUP. blFFP.KF.NC Ks HF.TWFFN Cni lOS ANtl \Vo«Ll,r.N "IMMlk^^" AMI " KvTmFS Kok lloMF. IdNSIMITlON" HC.lRF.i- IN 1 HP. YKAR 1878, AM) THE \l VI HI M \iH Al lokMllA SHKWIM. lilKiK (iKIiilN. I. Il has heen shewn "111 [latjc 5 ihat whcrr the ItiRFfcnrcs .Trc inkt'ii tigore for figure, hetween l)r. Krlward N'^irng's Vnileil Slates Kish Traiic Kigurrs f»tr ilic yrni 1S73 71 arnt Hernttiilli's (otiitnn. N 4S 'erni.s. the rcsiili is the imsiiivc lertns of ihc ('anathaii l-inal Difl'er- enccs Iiciween •' F'Tiines fitr III.. "onsuniiiiKin" ntxl " lni|>mm" in lelaiitiii to ihe iraiie of (. anada in (^'otions and Woollens with tireat Britain and the I'nited States in the j ^nr IS7S. It remains to show iltat hke rctatiuns suhsisi l^ctwccn HcrnoulU's Coliinm IV and the other BKgregnles given in Tnlile .\. The sum of the Pi>siiive Terms for C'oltoiis an)Ieiis is I. Negative 1. ., The sum of 40 tcrnu of the Ilernoulli Column No. IV i.i Total 69,183 J4,35o Difference 2,143 Hill 2,I4J IS the sum of the Negative Teim.i lor (."ottons in respect of (ireat Britain, and the Positive Terms for Cottons in respect of ihe I'niied Stales, vii;., (ireal Hriiain 1,47° Iniled Stales 673 Ah cxemplihetl in TuIile A where the account stands;--- (treat Hrilain —Charge,) for certain classes of Cotttin Goods ... , L'niteil Slates .1 •J43 $ 1,470 less than imported f>7,l more than imiwrtcJ II It follnws thai till- rriiiaininK |iorti from (ircni liritain und the iJnito'l Slates in respet*( o( tiuiio on (lifiereni tIns!M!> »i ijniKi* are rf|iial in the Mini of 40 lenns prties, anil Ixith are mathenia- tically e<|ni\alent lo />r. Hihiuni YotiHi^'^ huM I tatir /•'tx^u/fi vj iS^J-^J' This scconil relatiiin is shown helow. 1. The jtiini of Curion pi>sitive Terms fepreHeniinn the t*x^.•e*^ (»f chorgCK uvcr Imports on certain claiises uf goods iigainit lireai Hrilain is $22,972 3. The sum of Worillen !*osiiive Terms repre^^enling the excess ofchaives on certain cla«es of goods against (Ireat Hritain is 44i786 ,V The sum of Cotlim Negative Ttrms rrpiesfniing the defect of charges on certain classes of goods in favor of the Unile'+<)ti+55+45+36-f- jK+Ji + IS-f-io-fo-fi-Hi but it IS also e<|ual to iX()+ the «lo|)ing terms oti+ii + i, and these may be subsliiuled. These remarks apply 10 each inilividiial membe, given lielow. The Si MS ni ihf Iinm Iufffhrni r.i for Cottons and Woolle.ns is Taiuk A, in Terms op thk Ukknoulm COI.l'MN IV. SKCIION I. /H< i/uttHiily f.t,v7*- The 40th Term of Bernoulli's Column IV " 39ih ■• " 3"" " " »«h " " nth •' " 51b " « SECTION IV. '/lit iiuatililji 7_fv. 9, IJQ The fsth Term of Bernoulli's Column IV 8,436 " I4lh " 4,060 " i)th " 969 " 8th " 364 " 6ih " 4 •' 4th " ^ i»,or» t 7S« IF 18 SICTIOS tl. MCT Thf •/Mantiiy 44,736. n, fv ' The jKih Tfr'.n of llernoulii'^ Column IV r.770 The 30lh T«rm ..f Heinr IV J,«J4 " 37th •• 7,140 " 39tll •' 3,176 " 36'h " 6.54S " lOih " .... 816 " JSth " " i»ih " ... 680 " .Mth " " ii'A " 5.4i6 4.O00 " «ih •• " 7'h '• 56 10 " .IJnd " 4.495 •• 5lh " 4 " J51'' " ),oi4 •• 4lh '• 1 " '4ih " , 2M> " 9lh " ..... " glh " ■: 54 JS 8.507 " 7lh '• 10 " 6th " to " 5th •• 4 " 4th •• 1 44.7«6 !ii(;riON III. IBCTION VI. T^f i/uaHJi/y J,i>^y. THt i;'e)!"l'^ Jn lonnpil present ilicniselie. in llernoullis CoLimn IV, anil aUj, lioni the con>iruciion of the f^ile, in Column III, anil in ilie sloping column*. It noi» remains to supply Ihe Kenertl mathematical formula liy mean* of which ,ill of ihent quaniilie!. in»y \k obt«in«,l in (q[iiliu nriler. THK MATHFM.ATIC.M KtRMIM \ ^ t Kach lepanle quantity in the foraKoinii Seciioii, I to VIII, innlu.iTe. c.iii he i^sprowwl liy the Malhematinil Kormula — .1 (» + 1) (» + 2Ma + J) to la-k-(n-i))^ 1, 1. 3, 4, 5, 6, 7, 8 to (n - 1) Where a i» equal to the iiumlier of the column in IJernoulIi's Tahle, ami n is equal 10 the numhcn.f lerms in the column imtHjiHg 'yf/Kn. KxtMPi.e : The quantity 5984 in Section II w the 35th term in Column IV, wiih three r)nhers Iherefore n «= 31 plus three cyphers, and a - 4. Substituting Ihese valn«.i of n ami a, vi.. 31 ami 4 in the .ihovc formula, ihe expresMoii liecomts- 4. 5. 6, 7, 8. 9. *c., to U + (3a - 3) I^ 4. S. 6, 7, 8, 9, Ac, lu n I, a, 3. 4. S> ^, &C., «» (.3* i» 1, a, 3. 4. $. *c., to jt 16 » 11 « 34 =, 5p84 »' 33. 34 J. ». 3 All the quantities in .Sections I 1,. VIII ijeing represeme.i lj> ihe general formula A. each separate section conilsis of Ihe «imis of A, A ■ , A«. " 3 , A 4 , ,\c. I he values of A, A ■ , A " , A i , A » . &c. , lieing determinerf hy the //*/ of the term in the column, tach aggregate in Sections I to VUI, tuch a« 23.973 i» represented l.y A -I- A ■ + A» + A %+ A<, &c. Where A i . A'= . 4, S. 6, 7. &t.. IB U + (37-a) \ I. 3. 3. 4. *f^, ^o (37-1) 4, 5. 6.7. &c, to ^4 + (36-a); 1. a, 3, 4. &c., In (,36- I) 4. 5,6. 7. 4c., to ^ 4 4. (»«-») ^ '.a, 3, 4, &c., tp (a«— I) 9139 (see Table), = II436 — 4060 .. ISM^^L'. \'X- I» A>. A"=.. 4. 5, 6,7, *c., to {4 + (i7-»)' I. 3, 1, 4. *e-. t" il-t) 4. 5. ft. 7. *c,, lo ,'4 +(12 -2); I, 2, J. 4, .^C, 111 (12-11 i 1«H 4.5.6. r. *c.,io J4 + (a-») !■ ^ I, 2, I, 4, A.C., in (2-1) »l Kach Mpsroif iionniiiy in tht fuiegDinu SiTiiuin N... I lo V[|l ini-liiMiT, U ihe sum .if «ll Ibc t«rms ilwvt it iu B«r90ulli'» Colmnn No. HI, an.I can «Ui. l.c rxprr^iscl t'\ ilu- fiptnuil.T /. u V - , — ■ a which conslitiilcs lh« Alcgelnaic e>|ire>iiiiin nl llermmlli's i 'Ih I'toptrly, For exampli-: The (|iiaiilily 2925 i; the m lerm in Srclioii VI. Hy rcffrring In erniuini'i rxicn Ifil lahle, thi< number is fininci lo Iw (he sum (if J25 + ,)oo + 270 + 25.5 + 2)1 -f 210 + 190 + 171 + 15J + 1 ;<> + ijo -f- loj + gi + 7.S + 66 -f 55 -♦- 45 + j6 + 28 + ai + 15+ 10 + 6 + 1. Similarly e.irh r|UHniity in the <.'i({ht seciiims can l«. cxprcsseil in a serie.s (iiunil really formed in ihe llernoi'lli Tulilc. But 2925 is aUii equal In ihe sum iif Ihe lernis in ihe slnpiiii; cnUimn whose base is next above il, or e 2&00 i 210 ! aa 1 29aj = itc, *c. Until thejf Anally resolve themselves into Bernoulli's Columns I and II. The quantity 2600 can be put in similar (orin, and all the quantittf* '« *seclions I t<:>,VIII po&.sft!i$ like properties. But these are similar to Ihe pro)>eities [mssessetf by I>r. Kdward Younp's Fish Trade rigurcs for several ycftrs, which folluw the same law HI. The Fish Trade Ficukis. Take, for example, S'oung's Fish Trade figures jiven in TA8t.E III, being Stfttement No. \1, showing the LhwM'ii Slates Exports of Fish, Shell Fish, Fish Oils and Trmlucts of the Sea to Itnlish America in 1872-73 : ^«("^.,.W^. .T,- » the KoifiRn iii-ni fJO.S-i; S«i'line» in Oil e\i>.>rlfil u> (Jiuhi. . OnlMio, \c , i> nwiir itj. ..( ihr mixed Kluirii iiiid iLmiesUt iifni*:- $I7,91I> 'ill nlh«r»' li> (>cl)cc, Oliliiio, \r. 6,530 Honifslic Kxpciti to Novn Siulia nntl N. H. 4,^5i .. I, 111 Ilniish Oiiiina. 1*0 Sotclinr* 111 It, \V. Inilict. b Iirifil Ki»h to «^utl«r. SJ9.59r $17, 919 i« mailr up nf ; — $i6,(Mi ('. S K»innl« 10 Kirtu Scoiiii tnil N. II. l.j^S all iiihcr«. 350 Sarilinc.^ 10 N. !^. 240 lietring !o Hritith (iuianii. $i7,qj9 All ulhen In l,>ucUc, Uniario, \r $16,001 i> made up of i — $7,894 I'ish iilher luteil hi Brilish Weil Inilicj. 0,530 . picklcil 111 " 1,571 Ncwrmin'llanil. 6 .. iliicil lo l.'uetifc, ic. $16,001 K«pl :— $16,001 Fnieign KxporH to N. S. and N. II 1, 338 '.ill iiilieis' 111 Hrilish Columliia. J5* Kish, Tickleil tn Bri'iih (iuiana. 240 Herrinv m 76 I tried Kish to Nova Scotia and N. II. jj I'ickled ,. 10 (Quebec, 4tt. '6 Diied ,. to $17,971 Ki.h. iltied. 10 H. W, I $5,^94 Mackerel lo Nova Srotia and N. H. is made up of; — $4, j53 " Ush, all oihtrs." to Nnva .Scotia and N. B. ',338 M M lu Britioh C'ulumliii. 3 Ki»h, pickled, li> $5,894. $5,204 Herrinc in Nnv.i Siotui :ind .Sewlnundl.init 1- made up of' $I8o AM iiihirs III British liuian.1. 258 Kish, pickled, 111 108 Iresli ti»h 111 Onlariii and i^uiliec. 6 Sniiiked tish .. „ 3 I'icklrtl lish to British Coluinlii*. $7,894 $6,530 'Fish, pickled, t" H. W. Indies' is made up of:-." $ 7b Smoked tish. 2,144 Fish, pickled. 1,712 Fish, other cured. 2,598 I iy-ler>. $6,530 $i.;;7t lish. pickkd. to .Ncwlniindl.ind is made upof;— " $980 Fish, n1) nihers, til British tiuiana. 350 .Sardines lo N, .s, and N, II. 174 Fresh tish tn U. \V, Indiea. 52 I'ickleil tish tn i,lueliec, t)nlario, Ac, 6 Sninked fuh in $1,571 $1,188 S.irilincs m I 111, IS in.'ide up of 1- - $280 Sardines ill llr. West Indies. 258 I-'ish, pickled. In Briiisti t iuiana. 240 Ilcrring ii> British iiuiana. 174 Fish. Iiesh. to 111. Wisi Indies. 108 ■• " to I.Uiebec, Ontario, M,iniliiba. 76 II snuikeil, to Nnva Scotia and N. II. 52 II pickled, lo t^uclicc, C ntaiin and Maritolit/ $l.l88 $1,338 'all others 6.5 Foreign F.xporls fioni Talde 2 108 6 I 7f e.'4A l,71J 2.59« 9*» 350 174 ^1 58 I /.»94 6.S30 .. FioRi (I) Fiohi |2| 74 r 6 1 1,571 ..' ffom (3) From (4) IbiOni ;. . 16,001 ■ From Ta'ole in Commerce and Navt|{alion Kcjioits. ( 35° i I 4553 1 5»4 I — I . 16,001 F;xpressed in languaee, these qnanlilies in I and 11 and indirectly in tll from the conitruciion of the Talile, are the equivalents of the auttw of the coefficients of the successive expansions of (I ♦ 1) 10 the [lower of n. Other formula, which may lie obtained from morlern alRchras iixed in schools and colleges can be rendered applicable. It i» curioua lo note that in ihe examples eiven in some of these algebras, the series forming the 3td and 4th columns of Bernoulli's Table are employed. Special reference will 1)« made to ihese fonaulic and examples in a succeeding chapter. 01 aH-A-FTElR IV. The Canadian Trade Tables of 1883 are Fabricatod Records. In a i.uWlsh-.l Iclior l,,.-.,inu ,l.,i, VV„„1«,, N.,v,i Sclu. ,\l.,y iMl,, iSS^, rl.o .laiaiU -.f ih» Tf.vlu iu C.n.m (i..o i^ nf.-ill .k.scriptir.n, hciwcc-h (. ,.na,U ,.vl Ihr I'n.ie.l -iiaic. .m.l Cann.l.. .,n.l (,r«l lltiNiiii, nvcc siil.inillcl lo ibo Kl. II. ,n. Sir fharls. W Dilkc, M. I'., ihfii rresiilfiii of ihc l,<»:iil (.ovoniiiuMii ll..,ir.l in l,oniliMi. Ihu v„luniim,u, a,.cinni-ni, ..I uliicii the |."ilil„l:f.l loticr »a» nn nl.Mrafl, waj rolurnf.l Ic, mo. Tho iliwiiMianl ooniaino.l Ihc riifTfri-n- cts l«l»ecn ihr allcg,.,! ^u|Mrl^ nt i;,,!.,,,, (;,„„|s |,y („„.„!., fruii, (ireai H,ii.,i„ ,ii,.i ilir Ih.iie.l Siaic, ai.-l ihv termi h,'lw:in ihc afhi^.it vjOu^ n/ 'U»i( *^lmport'> *'EHta«i for liontt Cottyumptum," in thf f'nvie ami Mtj^nuiafi /)tf>ff> of Canoiin fop fhe ymr tSSj. and CHARACPKR (W OOODS, COTTONS, -YKAR (J'offiM III Tnwtu Krnt y&vlfftUun Tal.l«*, flS T^ ^"i tnr)iulv«.) (^I'HTWf. 1883. lni(Hn1ctl. (trcftl Itrilaiu Cnitcci SlJtes (iToat Britiiin I'nited Stales iWeat Hritain railed StutrK ■ (ireat Itriiain Unilc'l Simen ■ iire.1t ItrJlain - fnitt'd Stale* (ireal Hiitain {%) I'nitet! States (ircat Ittitain rnited Stalen . (Ireiii Britain I'lillctl Stales *Jieal Hriiain L'nileil Statos • (irent Britain Cnilc'I Stale* < irent Britain Unilcd Stales (ireat Britain l'nile<{ SlalcB ( Iteal Biilain llniteii SiiUfs (ireai ItnUiii) United Stales ( ireat Britain United States ( ireat Britain United States • (■rent Britain United States (ireat Britain United Stntei (ircav ttritain United States (ireat Britain United Stales (ireat Britain United States (ireat Britain United Slates • (ireat Britain Unitnl Slaten - (ireat Britain Uniletl Stales (treat Britain Unitetl Slates - UotlniiN, (ireynnd Blenchod, iVf, M (iinghams and I'laidn (dyed) •• i>(.Miim\, l))dlin|£s, \c. . . . . M While or I>yed t'oiion Jynns, ike. II II It II . . - >• Wiiddjng, ilatling, \Vari»s, \c. (not dyed) ■ . Knitting Vam. i{o«ii ry do., \c. „ • • .1 Wadding. Batting. W.irps, \e. (dyed) , * n Knitting Varn, isc. '.dyeil) • ■ , - . II 11 .' I • • . >i Sennitess Bag^ ■ • » . . .1 Shins and 1 )i:iwors, \c. . . ,. ^ , .1 Sewing Thread on s|H>»)ls . , , , _ i. Sewing Thread in Hanks, Ac. . ♦ , . .. Diirk, tor Iwmts, \c. - * • II Med (.-iiinltirlerw, tVc • . « • , • .1 Cloihing 111- uihor nialrrial nui .iihcrvvise pniv'd fur, \c 11 RaK'' I'y the Needio ..... t. Netting for HuotJi, dec, .... It i'runello for Boots, Ac. (a) ■ ■ ■ • II Parasols and Uipl>retlas ..... Shawl: , Velveteens, &c. ..... M Winceys, Plain, Sic. II ...... „ Winceys, Checked, Ac. „ Winceys, Checked, &c., liver 25 in. „ At. I OfJIliK MANt.'K.vrURES OF N. K. S. (J) - J-4,545 i2o,.vyi H'Uiie I>iffiireii«i riiy . -0. 205 3^5969 3,i(X) I..J74.798 t 16,050 iO$,yii 3.636 15.. 58 725 29,505 1 • '.3.'i6 2 14.748 1 .).046 ! ■" 479 31,611 2ci,S89 i + 257 197 l-t.n-^J 4.J78 i(i,oOo 2,246 117.607 + 2,225 .i4.t»S 3.^4° i-'4.047 ,4- 12,612 f.>'.4,i4 qos 511 •)54 930 175.665 11,048 + 930 4.529 2.016 - 86 J't.'.SSl . 76 225.224 + 218 5.5J0 558 7.364 ^ 2,297 5.328 -f 239 471 r2,fjo7 + 5.308 866 189,956 2,487 + 668 + 66 16, u8 — 66 880 317.226 + 59 1,823 352.568 + 3>.o27 365 — 2Q 21,584 — «47 22 1 28,333 + 782 '65 3.473,878 + 8,561 fiii 421,224 —~«l?*|(Wj>,*-. mm } SUPriRMKNTAUV TAllI.E. (lint Hrilniii Unilril Slairn (irrni llrirnin Uniicil huiea N, Dcgii OHilaining Kinc Soil ' • Car(>cl», nm clwwhere »|)«iifi«.t ... I 11 S|xin«l attention ia ilirmc^l i" iIim Min'ltMiiriiinry lalile. B. The thtty i* !n*eil ii|*m the entric* Un 'Mltinu* < '(inHvtniir.lt tn." Knt«rnl fnr lln.nt' CoiMUfitptlOA. eo,4ir I0.SJ8 ' •- 159 •J yj + < I69.4J7 I il*).ilV> - l$i ... 01 a o llf prpeclli«l fuither on. Ill UL'Tfrrtu'c In the Trnilc Tnl'Ir*. |>n||f t>7. •.Iu-h'. ihn' Ihtii^h t '•limiliift diil nni imfKnt our ** Knti/inx Jii/f, .V*'.,'' h-imi <» In i!»>(j. lait ibrre nrr ciiirtril J ititllitr^' wnrih (or " Iltniic Ctrnt. Mptitm" Iroiii (Irrai Hrlmin. II i« nolrwurlhy ihnt witkoiil thi the " *%me'i for I'ntttin'* IfW iHKj " vttnilil lur tnciiin|tlclr. IS) $7.a(n *' lin|i*»Tlrthcr N. K S.'' it rcin.irkaMr, ln'caiue il is «o mnch ({rcaler ihan (lie %uin til by fat the tar|;er [Nirttoii iif eniinierjtrd. In ilir |iie*i*uc<' "f ih» " Sttin^ 'A Itlffcience*," il tiiay well lit* .isked ; Ol tthnl arlirlas N. K. S. tn»>t cltawhrre *\ 3.4S5,Jt7 lUillari' worth ciinsiM 1 le.il Rrltaln 1 t ilollaiti Ihc article* jtacitied) ila I'kOM.NCl.M. ANlJ DO.MINUN Dll' I Kkl.NCKS. Comfariim NlWHI Ihi hint or l^nituM Ort/tr oj lliffttinm aHii ikt b'lHal or lifmiMian (>ii/ei of />i/ftrtneft /or Hr Voir iSSj, COTTON COODS OK ALL IiKSCRM'TIONX. f Vttaa t88j. 1 Vpak 1883 1 FinI tr Prtvi.unl On/fr t/ Ihfxtkni. r /i«a/ «r Drmitiifm ihJtf ff IHfftttHitl. \ » real Uriiain Unilttl Klatcn. real Britain. Uniieit Staler. + •f ^ + __ + 7S4'? I2U 45» 'S 7633 r»s use 11248 'JSK* 25S!( 34S7 i«l iSS •il |iUI> lobt, US •7i 2CU9$ 930 36 J6 31" >4 l«>J 91 1 16050 9J0 Sl« •«4» "J37 114 017 1167 1 » 4P9 66 540 ■ 827 113 439 T. '97 'S 86 14102 i6o)l >57 1225 *& ttVil 25tl^> 1710 'al? S9I9 3*41 to q»2 r>5 <4 1 I6(l 50 t»t 3* 10700 '305$ 7ii» 193 17810 .1761 137 124 3 '19.234 20,MII 799' M.9SO 107.0? I 'iSl8 18J7 *ym Jf" 503 1 159 3 I4» 103 "44 "5 7993 li' 2824 II9,«S8 »I.445 »K.4SO 107,071 8828 n,i*i> The (|ui\iniiiek lie|„w the first laials are pl.iced in the Cnna.iian Trade Tablen far apart from the otiict items— they will lie found oa Ijafjes 7 ninl 43 ami nrc the 1 MfTorencea ill the items "Aofi " containing Fine Salt, anil "Carpets ' not elsewhere s|i«cifie'1. The felalioia of tlle-w r).4.it)ilt.e, nrr very nnlcworlby. WDOl.lF.N tlOODS ()!■ Al.l, DKSCRimONS. I'OMI'AtlNnN RCTWR».|I TIIK KIIMT IMU riNAL IIHOM Of UirrCRRNCKN ro* WtKMJJW OOMM. MM fin! tr PrtvimMl < >n*r «/ {Hffirmn. iHmi tr DtmMm Onlir */ Diftrtmu. VtMi »IW3. Veak laaj. YCAK itt}. Yka* 1883. (•rem Ilriiiiii. United SwiWi. ClrcM Bcitahi. United Stttet. ■f + + - „ + -. alio 4^8 167 50 176 14908 167 ll 114 S*o IM ISS ! iiii 17369 M4 Tto ■ei84 a79 : 11)11 vmi ^5! IIM 6ai5 ia6 95S6 776 ihx8 7379 »»9 «7*5 106 «S iirts 17I **i* IJIO io«9 1310 1089 114 41.11 27« 49* tht 3JK 45" '5' U} ji.a Jlj 2J5 ill lOIV 91 688 >9e 3179 iuro 1406 : 1139 649 MS 544 Coo >i« MM 4470 I4S4 4499 Mi 6so Ml 4> (71 ifM *S* 'i* 306 491 « I4S $03 103 J».dBb 33.118 U.J98 3>M '7$ 17,561 5tt8 = — 8,828 = + 3,.S29 = - M.78O = I42.5'4 first Qrdir. + 33.I.S8 -- 68,398 + 3.>95 - "75 1 04,92(1 Final Onttr, + 17.561 — 52,801 + 3.070 - 50 73.48« + 13,6l> — 12,617 + 5. '64 — 5. '64 35.562 + 15.597 - 15.597 + 125 — 125 3>>444 45,434 or o 10,318 nr o 31,194 ot O 250 or o ! Sum of Differences. / 3$.S6a \ 3>.444 67,006 Total sum of l-'irst or I'rnvincial Dnler, Collim? ond Woollens " Final or Dominion Oriler, " Sum of Provincial Positive Terms without regard to countries " " Negative " " " Difference Difference $283,ooi 215.996 67,006 $283,002 The Supplement«ry Cotuin Diflrrences derived from " Iiiii«iil5 «re so remarkttMc a» tu ilcsrrve si>ecial notice. The Ilfim ill ihc TinJe Tallies arc as follow*; — Carpeli not elwwhorr tpecifted. Ureal Hrilain. -Onlario ... f^uelxrc .... .... Nnva Soolia. . . .... Now lirunswick .... ... Manitolia .... British Columbia 1', E. Islan o a 1 a 39 8 i 41 4 -«» The Supplementary .Series is,— » ',,■■' J •5 (03 144 14a 151 306 50a 153» The levcral sums of these quantiiica are respectively equal to the turns of the Difference* between Detnoulli's colimwis— The h ;sh TtaiV» figures and the Cotton and Woollen Series. 1 « OH-A.FT.'EJI?, ^V. The Canadian Trade Tables of 1885 are Fabricated Records. Th«s« IHffercnres hava not been puMisheii. T.i. / are very remarknltlc. All ihe terms of iha first or Provincial orHcr of Oifler.- anees for Collons in 1S.S5, in respcci ol (Ircai Hritain, are sinis of eacli other, the larger terms heing siiuis of the smaller terms. It is necessary to show there relation-i, and also in exhiliit them in the (on.- .)( an .VitHinetical pro^'re"i-«i(in f»»r the p.irp-.ise of illustratin;^ the further reiiiark- ihlc reIationshi|i hctween ail the Tra-le Tables from 1867 ti> 18S5 "specially portraye. in the following pages. The Process for olMaininR the Differences is precisely sii.iJIar to those exhibilad in relation to Cotton (»oa U ami Woollen (ioods in 1878 nnil 1883. To introiluce the c!eiait> would unncces;jirily inciimlier this praliminary view of the fabricated nature of our Canadian Trade KecoriKs. The matter (iiust neces-sarily l)ecome subject to thorough iiivestii»alion with reference 10 (Canadian Trade in many other branches than those pertaining tc< Cottons and Wooll^n«. Attention is particularly directed tn the projwrty alxne named viz.—thai lh« larger terms of the [)ifi"erenccs are nr3 1 3 1* |i a 16 a *H •8 130 d 49 I3« (0^ 90 IW OS? iS lis <$17 itt *93 34S * 2088 55"^ 337 39* 170^ St' 401 ^ 4I« i*» 539 1*1 X S STs m 5i? 9$ 1066 •« :i£ ■004 M0$ , •99? JJIO 4044 rw» 600a l»9S3 -37.185 + 17.U3 f- 13.003 + 13.005 4- I7.>23 - 37.'«5 — 10,388 Sum — 14.180 Rum. + 6,735 Difference . . . . 60.190 Piffcrence . . 37-5" 60,190 — 14,180 a7.5>' ■¥ 6.T35 liiim . 87,701 - 7.445 Pifl'erence 3^.679 Jo,9i5 96 lOCl 1246 »49 5*3 1030 (6091 10,388 (•'INAI. 1)11 l-r.Rl.N'CKS~CCTTONS- i«vSs. 13 Ih a< 106 «39 ^\ 7»7 910 98r 1791 619} 7051 +18.94J l»iMt Krilairi. Unftid St»le«. -3J.«*» 7 3i.««« + 18,945 Sum.. Dift'trcnce i4,iSo $1,070 " ' 11,76* Saw 64,sjs Difference . 39,30$ + 975" + 9750 — 30'5 + 6735 ia.765 3 I24(> 341 349 1070 - 301s fIRST OkDER t)R pfeoVINCIAl, DIFFERKNCES.-VVOOLLKNS, 1885. 4- 112 171 107 1034 1740 3730 144 178 1589 tor 871 4»7 3>9 330 16 1 »334 8280 4*4 •o) 911 J760 i33 '349 190 4'47 4»7 + 34.144 GtemI Hriltm bifietence.. 66,412 5«9 I031 193* «4> «W |6oi 415 •sr no United Smies. -3«.»78 "54 ^ + 158 4333 8«B.. ilM + J58 4491 - 3971 M 2) I-'IKAI. OR I)f)MINl<)N DIFFERENCES -WOOLLENS, 1885. Ortfil Britain. + 97** 1921 9«19 3683 ijoa 4147 427 J 1, 470 kiS, «S9 19.6 1 J + 21,479 - 19.613 bifTcrencc . . 4i.o6i Sum 1866 United Slates. + '54 + »5fl Diflerence . Hum 4316 >5 - 423' - 4»3i + 358 4487 — 3975 THE ARITkMEttcAL pkOGRESSlON TABLE FOR THE YEAR 1885, OOTTOISrS. he First or Prot'm-ial Differences for Cottons in respect of Great Britain for the Year 188$, in the form of aH Arithmetical Progression, with w as a Common Differenct. 10 I i 106 110 12 28 170 I 16 »93 3'o 3 6 401 410 I 6 •3 '3 106 3 189 }}0 I 3 16 '3° 170 320 49 37> 430 >3 16 30 3 6 121 '3° 13 189 '3° 28 393 330 •3 16 401 430 12 28 40 I 3 6 '.30 140 106 13 340 49 5d «3 16 111 l6 16 28 60 38 132 "5° 49 189 350 .2 13 189 337 139 ~^o 350 28 3 411 6 441 440 4SO 160 121 •39 360 I 3 6 '3 337 360 ■32 •39 189 460 3 12 55 70 3 6 55 >o6 170 12 28 28 201 370 3 i!6 13 337 370 '3 16 441 4JO 12 13 55 80 t 3 6 \^a 180 I 6 13 90 170 280 3 13 28 337 380 3 106 37' 480 I 12 28 49 90 t 6 13 170 190 I 13 16 90 TO 390 t 3 49 337 390 49 441 490 t 16 28 55 100 i »3 16 170 I 6 »93 300 38 183 189 400 I 3 6 49 441 500 And so on continuously. ^mmammB^ms With /iX) as a CiDiimcM Difftrtnc/ I |6 tS 55 6 a8 1066 1 1 00 I And so on. 53° 403 55 12 <3 16 170 too I 3 '3° lo66 I 6 300 3 la ui •83 98. 1300 «99) 3 aooQ 28 '83 l8g 400 I la a8 »93 1066 140Q I 3 (1 49 441 300 '.59 645 7»o 1500 I 90 106 403 600 121 <479 i6qq I 3 90 eo6 700 3 (6 202 '479 1700 With fooo >is a Common Differ in(t. I 6 90 106 597 HOQ 3 106 710 981 ^Soo 2004 981 la 3 3000 1997 1820 i8.? 40.00 4510 441 49 500Q 4044 1820 13» 3 I 600Q 3 6 »93 S97 900 6 12 403 1479 6003 981 16 X 7Q0O I 55 «44 3 16 55 ■ 06 1820 ■ 2000 710a 777 121 8000 600a 2205 777 :6 9000 600 a 2205 1066 710 16 4044 6002 7102 45>° 45'o 45'° 1066 39» 37> "; 90 6 16 \ 3 lOOOO 1 1000 .\n d I JOOO so on. >»953 28 16 3 1 300Q "953 981 49 16 I 14000 W9S3 1997 49 I 15000 It is to be noted with regard to these Arithmetical Progressions formed out of the Differences between "Imports" and "Entries for Home Consumption" in the Canadian Trade Tables for 1875, that the next succeeding Table shows that the larger terms of the First or I'rovincial Series of 1 )iffercnces are nothing more than si,n\s ot the smaller terms. Therefore, the preceding progressions can be put in a vast variety of forms by substituting for values of the larger quantities the equivalent values in terms of the smaller quantities. EXAMPLft" Let it be required to put the quantity 400 in different forms of terms of the IUffercnces, , 28 28 t8 28 28 28) ■3 1 '3 13 '3 '3 183 -400- 189) 170 1 t89 - 170 - 3) 6( ._ 3 _ 6 3 6 170 V 55 i 106 I 55 '\ 400 16 i3\ go 1 I 40Q1 3 '3( 1 2 f. \ 3 28 55 49 106 / 55 '3 6 16 90 1 3 6 400 6 55 16 40a 4?) And so on down to the lowest terms. 40Q ^ *U*i-.- TJFj1 fTWBaSJ*t .. 39 THE ARITHMETICAL VROGRESSION KOR WOOI.I.ENS. - 1885. FIRST OR PROVINCIAL ORDER OF DIFFERF.NCES. (Siwu quantities are introduced in duplicate for special reasons.) 37 161 300 a 37 »i* 649 QOO «3 161 1316 1500 a 37 i6t «S7 913 1 03 1 3300 3'9 1031 1740 3100 <3 157 3730 3900 And so on. >S 141 144 300 t 178 207 400 171 329 500 144 >3 ' 161 427 310 415 3»9 677 424 900 000 1000 3 ! 161 3 37 190 '5 649 ' 349 1 91J 912 1 871 1 600 1700 ! 1800 IS 3 17' >44 37 ■57 t4« 1 '7« 329 «S7 1 178 913 913 871 1031 103 1 329 9.2 — 3400 2500 2600 •7' 15 3"° 213 37 330 233 141 3760 3^9 «44 "55 303 3760 1 1 3300 3300 3400 15 •3 3 3985 »57 '5 190 37 »33 178 677 871 3730 912 3985 4000 5000 6000 190 303 307 600 190 »33 677 190 677 1031 1900 212 333 "55 2700 13 37 312 649 3589 35°° 3 «5 102 6881 7000 3>o 212 .78 700 '5 .78 330 677 3 37 178 871 912 2000 2 310 •54 3334 2800 3 '3 37 310 649 2589 3600 3 37 1031 2760 4170 8000 13 213 233 242 700 3 437* 871 1300 3 «3 15 33° 1740 2100 3 37 161 243 203 2255 2900 370° 677 »33 190 373° 4170 9000 161 310 329 800 310 .78 912 1400 >3 37 312 1938 37 203 2760 3000 37 161 203 310 3'9 2760 3800 » 13 «5 33° 1740 3730 4170 1 0000 ^imm^ to THE FIRST OROF.R OR I'ROVINCIAI, niFFF.RENCES. - WOOI.LEKS, 1885. THK l.AKdEK PRRMS ARE srMS Of TKE SMALLER IKRMS. 1 144 '3 >3 2 2 37 190 ;J7 »4i '44 15 15 141 102 1 "5; 1 144 '54 141 144 154 161 171 178 1 1 190 ■3 ' 2 212 2 ' 2 15 2 a 190 '5 '3 37 141 'S «3 190 15 203 ■54 141 144 »^3 171 171 -3 207 212 m 242 3'o 329 330 37 J03 2 2 13 190 312 «3 '7« 212 '3 >5 207 3^9 330 «S 178 '5 »33 '03 207 207 toa S19 t6i ■ 386 4'5 4J4 4J7 4J7 5'9 542 649 '3 >7> 178 2 '5 2 '3 37 <5 .78 3'o '5 107 f. 37 161 649 212 424 102 gi2 '5 329 1740 3'o 912 912 141 '44 '034 330 1031 677 871 1031 •034 '349 1740 '938 190 2 13 2 •3 '3 2 2 1031 427 330 207 37 '57 '5 4'5 «034 871 677 1031 161 161 37 373° 1034 «74o '349 427 »o3 3'° 329 330 3601 »334 2760 360 1 1 2760 "55 3760 2589 3730 3985 4147 '3 i^ 1 2 '5 330 «S 15 ■57 2760 677 33'^ 3985 360 ' 3601 242 3985 82S0 8280 4170 6881 8869 On page 32 the "Fire-brick and Clay Series" is put in terms of the Provincial Differences for Cottons for the year 1885. Any well-trained school boy can put the first order of Provincial Differcnics for Woollens in the terms of the "Fire-brick and (Hay Series." Thus reversing the o|ieration, but using Cotton Differences in one case and Woollen Differences in the other. I do not introduce thi.s form of the Differences for Woollens for the year 1885, because it can be so easily effected by any one who choMs to take that trouble. THE FIRST ORDER OR PROVINCIAL DIFFERENCES. -- COTTCNS, 1885, Af.I. THE TtRMs OF THi First or Trovincial Differencis for Cottons in 1885. in respect of Great Britain, Sums of each other, tnk Larger Terms being Si>m^ of the Smaller Terms. i 3 i 1 12 ! ' ! 1 6 , 3 3 i 3 16 ; 13 49 It 6 i 6 12 'A 1 28 49 •._ *■ — '•■ 6 11 u 16 a 49 SS 90 16 i 3 1 1 3 «3 U 13 90 12 b i '■ 6 ! 6 170 170 1 .189 106 121 i 1 6 1 16 ! 106 '3» , .^ N 106 111 I. 10 1 132 139 iro 183 i«9 am I 16 i 1 3 : 3 16 i >3 28 4<) 3 6 16 ^n «93 193 16 12 371 39» 37 « 40J j 345 •93 J.?7 345 371 392 401 403 41* I 6 1 3 12 3 3 i 18 12 i 16 S30 6 6 4" 49 441 t> 12 5. .8 5o« 53° 49 539 S97 *<;i S08 530 527 S4« 539 597 «o6 1 1 'J' '30 I I 12 1 1 3 6 64s 777 130 7'7 d «3 49 \l !l 55 •^5 16 908 907 ,1? 606 . 6*5 710 777 907 908 944 98. 1066 , > I 1 6 6 12 16 i i SI 16 12 '3 13 371 ar ■•I 508 <83 1820 441 2205 ,m 1479 l8«3 1479 2004 2205 4044 45 lo 1479 1820 1997 2004 2205 4044 45 '0 7i3 139 16a 6 la aS ■39 185 I I 1 i«> 1 1 IJ 1 3 1 6 3 6 16 as aS 3 6 a8 la la 6 16 38 ' 55 170 U 55 .^ 37 < 1W 54a 54* 530 710 1066 16 aoa 1066 1 aoa 1066 1066 3S« 55a 565 ;qo 7MIMON OROKK Of Differences for the Year 188^- Great Biiiain United Slates C.reat nritain United Slates TABLK C., 1885 — Conris Goons, Year 1885. firsl Ordtr. + 33,005 - 37.^liS + I7.ia3 .... — 10,388 ♦ 87,701 Woollen (;os, Year 18S5. Fit si Ordii. + 34.'44 — 33,378 + 2J8 — 4.aj3 Total sum Firjt or Provincial Order— Coltims and WoolleiH, It Final or Dominion Order, .... Sum of Provincial Positive Termt without refard lo countrin. Negative ., 70,913 r>iH'eren>:e, DilTMenre, Final Ottitr, + ■8.945 — 33. 'as + 9.750 — 3.o'S 64.835 Finai Ordet . + 31,479 — 19,613 + 256 — 4,23' 45>579 Differtmti. — + 40601 8, lao or 4060/ - 7373 / *''* 33,866 Differeiuis. 4- 12,665 — 13,665 + 2 — 2 35.334 a5.33oor 4 or o $ 158,614 110,414 48.»oo + 74.530 - 84.084 i5«.6m CHA-FTER 'VI. THE INTKRCHANOKAHI.E OFFItJIAl. FIGURES OF IWO NATIONS. In this (.'lia|itcr some illustrations arc (jivcn of tlio interrhjiin,'ealilc diaracter of the i|iiantities cmiiloyed under the siipcrvision of Dr. Edward \ oun)^ and Mr. ('(jmnii.'.sioner lohnson, in the Manufacture of ( anadia- and United States Records of irade. 'I'hcse illustrations cover the years froiii 1872 to 18K6, intlusive. Any person who (hooses to take the troulile ran put all the Canadian Entries for Cotton Coods and \V'oollen Cioods in the form of an indelinite Arillimetical Progression, bv ])ursuinf; the process mdi( ated in the followin({ pages, for the years iK;,^, i88j and 1885, and, 1 have reason to helieve, for many other years and many other classes of goods. Any [lerson can also take the differences between the "Imports" and "Entries for Home Consumption," and put them in terms of the Denominational Figures used by Dr. Edward Young to represent United States Trade in Fish, Fish Oils, Shell Fish and I'nulucls of the Sea. These again can be put in terms of the "Firebrick and Clay Series," and they can all be rediued down to the Bernoulli Columns III. and II. In other words, they are all the equiva- lents of the sums of selected Co cffi( ients of the F.xpansions of ( i + i ) or ( 1 - i ) lo the [lOwer of n, where n is succ^jssively e(|ual to 1, 1, 3, 4, 5, \-(., as far as it is found convenient to go. The work is done by means of a "Ready Reckoner." These Trade Tables are, in fact, practical illustrations in figures, of Bernoulli's famous words engraven on his tomb: EaDEM .MITAIA RliStJROO. To these illustrations I have added one of the etpiivalem forms of the "Fire brick and Clay Series," in terms of Bernoulli's Column No. III. .'\lso, First .\ mathematical formula for obtaining any desirable ratio between two (juantities, and the corres- ponding series in Hernoulli's I'ahle. Second — A general mathematical expression for the terms of any one of the vertical columns in Bernoulli's Table. Third — The mathematical process for converting Hernoulli's Formula into the "Differential Method" of expressing the sum of a Series, as given in modern algebra.s. I'ttK WONDERFUL ACCURACV OF THE FIGURES. It is time now to direct attention once again to the surprising .icruracy of the Figures employed. Nothing short of a joint ready-reckoner coulo have produced this .accuracy, (^istom House figures in 1S78, 1883, and 1885, referring to certain cLisses of goods, are found to be interchangeable with Custom House records relating to other classes of goods in prior years, 1872 3 or 1873 4, tor instance -and also interchangeable with the dutiable Custom House Records of a neighboring people, numbering til'ty-six millions, the greater portion of whom are more or less engaged in commercial transactions who.sc alleged records pos.se.s.s these properties. The illustrations given below are types only of a vast system THE AI.I.EC.ED CUSTO.MS HOUSE ENTRIES FOR THE FORM OF AN ARITHMETICAI. YEARS 1878, 1883 AND 1885, IN THE I'ROCRESSION. The following tables show how the .alleged Custom House entries themselves, for the years 187S, 1883 and 1885, may be grouped in the form of an .\rithii>etical Progression, with looo as a common difference. Vinr /SyS. Ecjiiivalcnl Differences, Cotlnns only. + 5<" -)- iSo - 158 THE yUANriTV lOOO, WITHOUT KEOARD TO .SIGNS. Yrai iSSj. E(|uivalent I^ifTerences. Col tons only. + 78» + 218 — 100 1000 Ioe» Yttir iSyS. --The corrc ponding entries with (heir negative signs changed : I'lrent Bril»in— Carpets of any intterial, except W.xilien, (jreat Hritain —Ginghams and Plaids, ... .... ... United Elates- Carpels uf any material, except Woollen, 1,'nited St*Ie< — Ginghama and Hiaids, .... ... $I294WS 128468 Ytar iSSs. Kquivaleni Olffercnces. Cotton.s only. + 403 + 54a - ss $96562 20385 8058 446J 129468 $96000 20205 7900 4363 1 2846* i . Ytar /SSj.—Tht roc rwpom ling cniiin with ncgmive iigns ch.iiigtrf. Cir.il Itiilain Wincryi, I lirikecl, .tc, Mvi-t J'i inclifs, ^... Unit«l SUIM.— Clothing, ot uihet m»lefUl n»i othi-rwis* (inviikil fm, $25.1557 »5a557 looo Ynr tSil.—Tht: CDtresimmlinu cninc* »ilh negative iignu ch»nntil: litMt Brilain. All other, ,.>. '" t "" "" (iteiil Britain. --Par«»i>l», ■ • • ■ •••• •■•• Grnt Btitun.— ColtoD», I'rinlcd oi Dyed, ... .... li842H $ JX.M.l 153557 $J7,'!5« »5».S57 4050-1 4U101 6l(iai 60946 37913 - 37J81 liUM U84»8 rtar isri. /^uivalenl I )ifreit nccb Cottons & \V\iu)lcn». — 100 + ito THE QUAS iirv JOOO. Yiar iSij. Kiguivalent Piffrrences. Cottons only. -90« -613 — 479 1000 Viar iHyS. Unitcil Slalen. - llin.fhams aiiil I'laiU.s, . .. United Sia:e». -Carpets, Oreat Britain.— Coltom, Primed, Pointed, 4c., $2,002,760 *,ooo,76o Yiar riSj. RquivalenI 1 >ifrerence». Cottons only. + "W7 1 4.4''.> "4.25J 1,984,044 1,00a, 7te $ 4..163 1J,95.' i,<)8a,444 a,ooo,76c.' Ytar iSSj. Great Britain. -Sewing ThnaO, in ILinksSc, United States. All other Maniilnciuri-i N. E. S., Cleat Hfiuin.— Wadding, Batting, \Var|», &c.. s,ooo $577,704 575.704 $ I52..?4i 411.837 3.5»5 577.704 i 1 5 '.434 421,224 3.046 575.704 Cleat Britain. Great Uritaio. Year /SSs. - Cottons. Bleached, Sc, -Winceys, itc. $44.6I« 42,6lK 20,429 IS.'89 44,618 $ 42.Cl!i !! Y,ar 1S7S. Bquivalent Differences. - 1048 - 842 -t* 561 - 158 - joo - 90 3000 THE IJirANniY jooa Ytar /,Wf. Kquivalent Uifferencea. + 1356 ■¥ 78J - 558 + 21K — 86 3000 Yta iSXj. ■jjuivttlen 1 lifTcrcnccs + i0O4 981 JOOO United States. —Cottons, Printed and Painted, — Great Britain. — Worsted ami Yarn, .... ■-.. Great Britain.— CarpeU of any material, except Woollen, t'uiled States.— „ .. 11 n United Slates. -Carpets, .... United States.— Clothing, &c., $ 1,270,046 1,267,046 $ 893,681 66,0s I 8.058 ■4.353 191,441 1,270,046 -.- 1,267,046 JOOO \'-\ ■i-w Kr'»lnSii'mmM/:VMiJ\ ' •M Vtar iMj. United Stii!««, — W.id.ling, Hailing, W.-\tp», &c., (not dyed) .. (iteal lltitnin. - Winceys, ('liccked. Sic, nut over 25 in., flreftt Hriloin. -Ha^;'* by the Needle, United Stiles. -(Nuliinu nr olhcr material, &c United Staien. - Iteil CnnifuiKrs $291,243 28)i,242 3COO Vtar iSSj. Great Hritain. - VVIvcteeni, ...... (ircal Mritiiin.— Ik-d (lomfnrier.^, Great Hrilain.— Jcins and C'uulillei, C'ireal llrilain. —All other, $285,671 282,67 ■ ]ooo 291,242 2S5,67i 188,241 $ $ 06,178 3.476 2,229 73,788 - 304,174 3,495 1,217 73.785 282,671 Vtar iSfi. Equivalent I )ilk'rcncc!. + "599 + 1600 519 + 180 -.. 100 + 3 40cx> Vtar IS78, (Ireat llritain. - Jeans, Penims, &c., (ireal llrilain. -rotions, I'rinled, I'-unted, &c., 1,'nited Stales. - Tweeds, (Ircal Hrilain. Ciughams anil Plaids, I'liiicd Siale.s.— .. ■. United Stales,— Clotliing, &c., THE cJUANTirV Vtar tSSj. 4000. IV|U valent Diftercnces, + 1356 - 908 - 558 - 613 = *?? 4cx» $2,175,894 1,171,894 4000 Year 188^ United Stales.— Wadding, Batting, .%c (Jteat Britain.— Sewing Thread in Hanks, &c., (ireal Hrilain.- Hags l)y Ihc Needle, V'niteil States.- Alliilhcr manufactures N. K. S., (ircal Hrilain. • Wadding, Hatting, Warpk, ic, United btales.- Bed Cunifoticrs Year I8)i5. (Ireal Britain. —(;, .lions, Ble.iched, (ircal Britain.— All iilher (ireal Hrit.ain. -Shirts, $615,380 611,389 4(XI0 $691,479 687,479 4IXXD And so on, continuously, up t^) 10,000, or 30,000, o. 50,000, &c., &c Vtar iSSs. Kiiaivaltnt liitTerenccs + 1907 "f lita - 183 4000 $ $ 28,528 — 16,939 1,984,044 — 1,982,444 10,026 — 9.507 20,385 — 20,205 4.41^3 — 4,363 128,448 — 128,446 3.I7S.S9-1 — 3,171,894 $ $ 39,505 28, 149 I53.i42 - IS'434 6,07s 5.520 421.837 - 431.224 3.535 3,046 Z,102 — 2,016 615.389 611,389 1 $ 39,439 564,379 97.<'7l 37,433 - 502,559 - 97,48s 6»1.479 — 687.479 In a similar manner all the Custom House entries answering to ihe Difierenccs on pages I and 2, 21, 22, 33, 25 »nd 16, can be Jiul ' - - Where ihe reduced Uominion differences form rences furnish the details for a common differ- in the form of an Arithmetical Progression with a common difference of 1000, or 100 9' 'o. only an Arithmetical Progression with toco as a common difference, then the Provincial diffe: ence of 100 or 10, as shown on pages 27 and 28, for the year 1885. rms o( every -Vrilhmetical Progression are subject to certain malhematical relalioni, and the grouped Custom House entries , of an Arithmetical Progression arc subject to the same relations. This is a point which need not be illustrated, for it reqmtiB 1 it is a mathematical certainty. It remains to show that the Equivalent Differences can all be put in terms of Dr. Edward Now, the terms c being in the form ( Y'oung'°"FiIh Tr"adc 'Figu"e5,"an'd"liuw con'nec'i'the deceptions going on now, with the deception* which have been succesjful, but Mill temaiu to be used again, as these Trade Tables fur 1885 foreshadow. Thi CquiVAi.iNT Cotton or Wooi.lin DivriRtNCU rot 1878, i88j anm 1885 in Tr.«M« nr DR. Kow^tn VnrNO'i FllH TiADi FiouRU TAKIN nii'M Hilt Uniikii Htatm C'ommircc AND Naviuation KuroRT, roR 18717.1. (Sm nti't III., SmitmiMl N; yt.,/tr DtHomimUtmal dtlails.} 1' S6a 180. 158 Sr\* 1878. F^luivaknl I)«nnmiii4tioit«l Fiiuin i.f Ynnng', V. 8. Kiih, I'ickleil, lo lltiliih (JuiaiM. .. Fresh, to Qurtx'i*. .. Siniikol, III Niiv» Scoun, n \Vh»le Oil 1.1 N.iva .scntia \ N«w Drumwick. I) Tickled. In (^'uelwc. n Iirieil, til .. I, Kre«ti. to liondura.«. .. Drieil, 1,1 (,iiirl)ec. Whule Oil 10 Nova Scolm ami New Uiuii'iwick. Spermaceti lo Quebec, &c. Sperm, 10 .. WhaletKinc lo Britinh Cnlumliia. VVlLile Oil to Nova Scoti:i nnd New llninowick. S|ierm to » •, WhalcUme to Hritiih Columbia I 1 78J ai8. YlAR IMJ. Kl(fcr«nce«. Colton* only. 403 $4* $5 340 108 Herring to British lUiiana. Fiih, Fresh lo (^ueliec. 52 „ I'ickleil, to ., 3 !• .. lu Hritith Columliia. 340 Herring to British Cuian.i. 174 Fish, ■ -'■ - "- ■ 76 YrAR 1878. F.quivRl«nt T1iff«rcncei (ottonft k Wolilten*. 5a 3 Fresh, to British West Indies. Smoked, to Nova Scoiia and New Bninswick. I'ickled 10 t.)uel)ec. lo British Columliia. 300 l6no Whale and Fish Oils lo N. S. and N, H. Spcrni, to l^ueliec, &c. Whalelxine to British Columbia. Fiih, Fresh, to Quebec. ,. Smoked to N. S. and N II. Spermaceti to Queliec. Fish, Tickled, 10 British rolunibU. >i I>ried, III Quebec. Oysters to Newfoundland. .Sardines 10 Nova Scotia and New Brunswick. ,. to British West Indies, Fish, I'irkled, to British (•ui.ina. .. Kre»h, to Bntisli West Indies. ,1 Dried, to Nova Scolia and New Brunswick. •• Pickled, III British Columbia. Veak i^i. KquivBlcnt Tliffercnre.1 Vkak 1K85 KquivsUtil Differences. C'Hions only, 240 258 908 6t3 479 I P. /'353 I 108 ; 61 ('(itliKi^ f>nly. Sanlinct, Ac, to Nov« ScntU and New Bi"nswick Herring lo Hriuith (tuiana, Ki^h, Pickled, to {British (fuiana. •• •• to Quel>ec, Onlariu. Whalebune tu Hrituh I'alumUia. Fish, Wricd, to Hritish (.Columbia. Fresh, lu <^ucUrc. Whale Oil to Ni>va Scotia and New Hrunxwick. Fish, Pickled, lo *)iiel>ec. .Sperm lo do. Whalebone to Hritith Columbia. Herring to British (iuiana. Fish, Fresh, lo i^ucbec. Dried, to Nova .Scotia. I'ickled, (o Quel>ec. H to brituh Columbia. 1997 1K84 Mackerel tu Priliermareti to <,>ucbec. 5a Fi»h, Pickled, tn Qucl)ec. ) .. .. to britiah ('olumbia. 3 Fiih, PickSed, m Britiah Columbia. 3: Ykar 187K Sanivmltnt iHffiirvnrt*. 84J 562 *». 90 3000 I. *J I 45') I .15" I ^ / «.« I hi !" 6 ! ( 340 I N I '' I S r.guivftlfnl (Icniintrtali'iriiil Funrci of VtHlill 4 U. H. Kl.h IriJa lta<:<>n)* ill ll;a )}. VVhtlelMinc 10 (judivr. Ike, Spernuirrii iii „ Kiih, I'icklcil lu .. .. Driril, In .. OyMrra III NrwfniinilUiul. Snriliiifi III N.ivfl Scniin. >>|i«rni 111 i^iurlirf. hull, I'icklcil III llniuli I'liltimliit. Ki»h, Until, HI lliiii.h I'nlijinliia. Kresh, m i,>ueli«c. Whale Oil III Niivn Scniin >ni| New Rruniiirick. Sjicini III i^luflifc. Kul), Dried, In i^'iicIh*!- 11 I'ii'kk'il, In MiMi'll ('iilumlNa. Merrill); in ltriii»li liiiiann. lull, rii'kicil, lu i,liltln:r. Whalflnilu' In llrillili rnlilinliit. Kmh I'irkleil, in l,turlirr. Sjieriti 111 i^hu'liec 'VhaliiUiiic lu llrili>li (.nluiuliit. \'nr iSg], KilUlviite IJiirirc IvRtem wt*. KiiaivalMN JOOO 115ft 7»i. 558 JtH 86 J003 I FigiHt* III Viiunf'ii V. |. ItrifiU uf i>7i • 7]- (' 700 While Oil In HtltUh (Aimnii. Jjo Surilintk 111 111! m N. S anuelier. 76 > Orieil. Ill Nuvn Sciiiia ami Ncy Qruniwkk. JO S(«nii In i^iiielwc. 6 Smnki'tl |''ii.h In l^uelirt-. iu8 S» 5i 61 8 6 »9W Ki>h, Kre^li In (,>iii'lirc. Sjieriii.n:rli In tin, I'li'Ui'il Ki'h 111 Owlier. Vkau iSHj. Killll*alriit niffircMi e*. K |iiiv. ill-lit lltii'>i«iiuii-)iiNl Kigurtt of Vimiiir't KilH'rni In i^luclier. IH Whalelinilc U> Uriti\h ('nlutifl^i^. 3 K11I1, I'lcklcd, to I. ,. 1| b ]c\»\\, Drieil, In i^iirli«c. J II I'ickleti, In Dritinh (.'uluinhia. 6 II Drieil, in (iiieliec. JOOO JOOO And to un fgr luccceilmi; it>nii«. The tlHive illuMraiiiw for ihe year lUj is itiir«ihioeil lo niiuw thM these Equivalent Uifrcrencrs, nlthmt^h they ^rnduce ihe re(|tlired iiiiin* lier JOOO for cotlnnii, yet cannui be pu( in teritis uf l>r, Kdward Vuung'f lish Traile Fifuren for 1872-73, the qiianlity 86 nnt lieing iiitei- chaneealile with thein. Ttwrefore uther K<)ii>valent Differences mux be la\;n, — Micb aj> ihe fullowiiiK, which aiunxr nil (he cuudi/iuoi : 782 = 318 908 = 350 280 6a 53 30 8 loS I 108 = 58 I 5» 909 8 f 12 6.3= I ( 280 I tun JOOO jam On pa^e 29 11 is staled that "some i|uaniilics aie intru>Uiced in d«pU<|iU for »p*:ial reasons." The reasuna arc ikal anieu regard l> hod 10 the duplicate nr triplicair fnrms 111 which the dilTi.'reni leriiiH nf the Arithini^lical I'rugressinn luay 1>C put, eKurii to ulilain ihc ojiiivalent diflerenccs in leriii r triplica siif^ln. Kdward \'inini;'s Ki^h Trade Figures may fail. The reader wilj Ijeaf in mind that fhe foregoing "Denominationai Figures of Young's United States Fish Trade Records of 1872-73," can t)e imt in terms of an Arithmetical I'rogression, as in Se<;tion III, Statement VII, Table III; also, that the larger tcrnii are sums of the smaller terms ; also, that these ie denominational figures can be put in terms of the United .Slates Imports from British America, as in Section I, Statement VIII, Table IV; also, that they can be put in terms of the "Fire-brick and Clay" Scries, as in Section I, Statement VII, Table III; and if the reader will turn to the suceeding page, he will find the "Fire-brick and Clay Series" in i'irnts of Bernoulli's Column No. III. The relationship Is thus carried back from 1885 '" i'^^?! ^"'^ 'l'^" '^^<^'' '° ''^^ Bernoulli Table, first published in 1713, or 173 years ago. S^rtwi*5:^^ l'fMS(M^9WWHIMIp^».»-.-J f I. 81 THK "FlRK-BRtCIv .\NU CLAV SKRIKS IS \'Hh 1:ijLIV.\LP:NT UK BKRNOUI.MS COLUMN No. III. (It will Ih.' reiiu'iiilH.-l'i.'d that the name "Kirehriik ami Cl.iy" Scries is derived from the substitution of 11,184 dollars' worth of "Fire-bricks and Clay " for "F'.h," in the rcndeniig of the Canadian Trade Tables cif 1867.) .V T.AHt.E II, SlATEMKNT I, for the Orixin 0/ this Kcmarknblt Sfries. It I-. slidwn Ml Tabic II, Statement IV, that the "l-irebriik and (^lay Series" ronsists of the sinus of the (iiianlities 9, 10, 2.>, 26, 4." .iiid jt), lieihu Its >ix l(]«e^t terms. I'hesc quantities are the ecjuivalents of the grou|)eoseless. The ])rni)ertv of the Hernoulli Series is th;it c.v h term of any series is the sum of two preiedinij terms of the s.ime and iie\t adjoinin;^ i ohinin to the lel't. 'The " Fire brick and Clav Series" can therefore be put in small groups of Column III and IV, of which the following is one of a vast number of eipiivalent forms. 'The endless variety of these forms of the lnr>;er terms is a notable feature. Thk "Fire Brick and Ci.av Seriks" in ihe Terms ok Bernoulli's Con mns III ani> IV. 1 ' ; 3 « t 6 ! 1 10 1 ' 6 »5 1 '5 10 88 I. 28 S6l I90 1696 6 28 9' 4 10 78 794 2^ 78 •42 I '27.') 1276 26 3 6 162 6 20 '27.^ 1301 4i 10 ■85 1326 '.13^ 6 10 i(> 5» I 6 .?5' 358 3 6 45 •378 • 43- I 10 45 56 3 31 528 ■ 10 66 '378 '455 aS 45 73 I 3 561 56s 4 310 '378 1 60a 4 703 2* 6 1 20 3 1128 i t 66 1176 3 J 5 55 loSi iqo '3 75 i 6 300 '54° 1 176 '43^ 2600 1326 300 ijafi 1225 3654 1879 '893 J462 2l»33 »793 3729 3961 3 I 1 3 2S 3 1 3 3 ' 3<' '5 496 3^5 6 595 45 190 45 10660 1 1 480 1 20 1 9600 4060 4060 9SK0 17296 4114 41R7 9943 '7.4JS 20,198 11,184 ii,.So8 It must be borne in mind that ir\ order to api)re< iate the full value of the artificial construction of the "Fire-brick and ("lay Series " this remarkable series is not only the equivalent of the sums of groups of its first si.t terms, but it is also the equivalent of the fore);oing sums of the terms of the Bernoulli's Columns III and IV, each term of which is itself the sum of a prior series either vertical or sloping in Bernoulli's Table. l-urtlier, the "Firebrick and (lay Series" has been shown in Table II to jxirsess certain pro|)erties ; in 'Table III it rqiresenis the terms of \ouiig's United Stales Kxport Fish Trade figures, in Table IV, Young's United States Imimrt Fish Trade h.9X4.?,7S** 75.SXJ 75.S«-' 831,407 43.758 '9 ,?93.«''2 4.V75« 831,401 Ilernoulli expresses this remarkable relati^m in tlu- followint; (|iiaiiit languagf, as ijiven in the translation published by I'kANcis .Masirks, Cursiior Hamn of Ihi- Court of K\ehe<|uer in 1705: "The sum ol any number of terms in any of the vertiral eolumns 1 ontained in the forcijoinH table of combinations is t.. uie sum of the same number of terms all e< 21 29j,9jo /x ff S = S - In detail, the calculation is as follows: a ' \l\ H - 10 quantities, then, 13, 13 12, IJ, 14 12, 13, 14, 15 12, 13, 14, 15, 16 12, 13, 14. 15, '6. '7 , '». '3. '4. '5. '6, 17. '8 a. 3. 4 2. 3, 4, 5 2, 3, 4, 5. 6 12, 13, U, 'S. t(>< I?. iX, 19 12, 13, 14, 15, 16, 17, iS, 19, 20 S I + 12 + + ■ * *,3 2, 3. 4. 5. 6. 7 2. 3. 4. 5. 6. 7. Ii 2, 3, 4, 5, 6, 7, «, 9 :^ I + 12 + 78 + j(.4+ i.j(.5 + 43!i6+ 12,375 + 31,824 + 75,582 + 167,960, which is the Xllth Column in Bernoulli's Table to 2 1 terms, and each of the quantities in the series is etpial to the sum of the series in the column next precedilig if to the left and beginning with the quantity one square above it. Thus, 75,582 is the sum of all the quantities in Column XI from 43,758 u|)wards, and there are 19 terms in that t olumn, including cyphers; or 75,582 is the sum of the Vlllth column, beginning with 31,824 and thence upwards. 3i,82.( is the sum of the VTIth column, beginning with 12,376 and thence upwards ; or ol the .\lth column, t)«ginning with 19,448 and thence upwards, and so on for all the (|uuntilies in the series. It IS to be noticed thai the sloping column to the left of 75,582 up lo No. 1 1, consists of the same figures 83 the vertical column No. XII over 75,582. ThiS rule holds good throughout, together with numerous i>thcr relations bct(»tfcn columns and |>arts of columns, which it is not necessary now tu puiiu out V^ ! i 40 ^ III. THK KFrECT OK POSH ION IN IJSlNd RKRNOULI.l S TAIILI;. The effect of position in Bernoiilli's Tabic is rfniarkablc, niui must always be attended to. The value of n varying with each column. When using any of the formulas given in school or college algebras for determining the sum of a series by the Differ- ential Method, such as that given on iKige 5,^7, of Hind's Klements of Algebra, 5th Kd., and applying it to Bernoulli's Table, the value of ;; must bo made e<|ual u> { h - a t- i ) on account of the cyphers which Bernoulli's formula alone includes. The etjuation then becomes (I) V: 1. J. i. 4 /am (1) .V A. Ex.\.MPi.F- I.ct ii be re()uircd to fmd from "ither formula the sum ol 40 terms of tin- series I -H 4 + 10 -t- 20 + 15 + 56 + 84 + \c., 10 91 j9, which is the 40th term in the Bernoulli Column IV'. including cyphers, (40 i) (40-2) (40- 1) (40-0) J7 X jg X 39 X 40 Th*i> from ID .V = Kn.ni (2) V I. J, ,), 4 0139 X 40 I. 2. .1. 4- 91J90 01300. Many other striking features and projierties of Bernoulli's reniarkab! table are pointed out in the Ars Con/ti- tandt, or in the M.aseres translation, or .ue easily deduced by any one familiar with the elements of Algebra. Thk M.\(;NiTi;nK of thk iMceisruRF.. It will be understood why no attempt is made in this Ex|X)sition to arrive at an approxim.ition to the extent to which the use of the artifice |)ortrayed affects the visible record of Canadi.m Trade since 1878, when it i'. stated that the combined Pkovinciai, 1)ifh-:kkncks in rhe year 1883, for Cotton Coods and Iron (ioods in relation to tr.ade with Great Britain and the United .States, amount to no less than $377,000. If the si^ns onlv of these Differences be changed, the resulting sum would represent a difference in the record of the Tratle, tVom what it now is, of $754,000 for these two articles alone in one year. But besides Iron Cioods and Cotton (ii>ods and VVix)llen (joods there are numerous other commodities in which the recorded " Different es" between "Imports " and " Kntries for HonR- (Consumption " are verv large TI.ese figures have no relaiion to the "Valuator's" estimate, or to "Bonded (foods." or to " Drawlncks,'' or any other Custom House contingency. They are the grou|it-d terms of a (ontinuous .Vruhmetical Progression, interchangeable with United State-' prior Custom House entries, anc i»ortion of both natio is. As a conaequence of the succe.ssful practice and continued maintenance of this conspiracy, we have bow to con- front and diacuss serious international complications, to the detriment of good neighborhood and the psoatotion uf ilU wilL Therefore, it is necessary to present the fai ts fairly before the public, in order to lessen these evils; not to du so, would encourage multiplying troubles. The necessary Annual Official Statistics of the United States in relation to the Fish I'rade, during the period when .lh«e records were under the supervision and control ol I )r. Edward Young, were fabricated throughout, and made 41 siihordinatL- to reii()rociiI Canadian stalistics by a mathematical |)rocess pursued in rominon at Washington and at Ottawa. The method and its results are sulTiriently described in this hook, but suscepiible of much more ample demonstration. This secret process has subsecjuently lieen continued by Canadian officials up to the preSL-nt lime, and in such a (jross form that the figures of the National Records of the Trade of ("anada in their latest issue, are interchange- able with those of the earlier Trade Records of her powerful and independent neighbor, and mathematically related to them. They can all be put in the form of an endless .\rithinctical Progression, with subordinate consequences. In glaring contradiction to the solemn averments made in the Canadian Parliament (i), and re-echoed it the Imperial Parliament (2), some \ears since, this unexampled abuse of trust is mathem,">ticallv proved in the pages of this book to have been going on then, as stated, and also to be going on now, but with tenfold greater ( ontingcnt burdens. The responsibility resting on those who favored this subterfuge, and permitted the secret scheming to continue, has yet to be measured. If pa|)ers are railed for in the House of Representatives, it w ill be found that I have not been remiss in comnuini- eating the grave delinquencies of Dr. Kdward Young to the Ciovernment of the United States, even so far back as i88j (3). 'These printed communications point out the artificial construction of Dr. Kdward Young's Trade liguresin the United States Commerce and Navigation Reports for six years. 'They notice also his misrepresentations of ('anadian OtTicial Returns in the "Monthly Reports." I now find that the special features then enlarged upon are nothing more than the subordinate consequences of the continued use of selected co efiii ients of the successive expansions of the Uinomial (1 . i) lo the power of 11, in order to represent United States Trade in the Products of the Sea. The absence 01' mathematical proof of the fraud may account for the absence of notice being taken of the state- ments made. But the permitted i ontinuame of these practices in Canada derives special force from the fact '' with the exception of the formul.1, the whole was described by me in a communication to the Rt. Hon. Sir Charic? \V. Dilke, (4), when that statesman occupied the |)osition of President of the Local CiOvernment Hoard, in May, 1.SS4, and a synopsis was previously published by nic and circulated in Kngland. and sufficiently so in Canada (5). The title of the voluminous paper addressed to Sir ("harles Dilke and ret,urned to me, was .is follows : "TlIK C.\»ADIAN OfFICIAI, FRAl:nS CONTINliOlM.V PRACTICKt) IN RKI.ATION TO THE CANADIAN ProI KCIIVK. 'Tahikk and Canadian Trade with the United Kinc.dom." 'This v.ast fraud, as continued since 1877, unequally afTects the international 'Trade Rel.ations of m.iny millions of people on this continent alone. It has favoured the industries of cme class and abused the privileges and rights of another class. It has unecpially and inequitably represented, by means of forged figures, the several industries of these cl.asses, and their commercial rclation.s. Page j of this book embodies an indictment no one can dispute ; Chapter VT further confirms it. 'The hidden influence this fraud exercises may at any moment be called into action. Unless exposed to public view, it may .again suddenly become an unsuspected controlling power in legislation, in the adjustment ol trealies. in arbitration, and in the administration of the law. 'The conspiracy has already become a rooted and a growing evil, which must be extirpated at any < ost, or it will lead to grave international bickerings, and internal unrest, if not disaffection. It IS not very long since that men wondered at the saying imputed to a distinguished personage, ex, Ited in rank and high in public estimation, that "Representative institutions are now on their tri.il." It looks as if this saying had a great deal of truth in it. 'The irony cast by the process 1 have outlined, on solemn arguments based on the interchangeable figures which are its outcome, is uns|Kakable. (1.1 8m oIIIi'IoI r<;|>t>r« ■>( the Dlmiaiiion li the (an ili«n St'iiatc »ml th« IVnvllan Hotiie n( Cmiimuim. In .l«nimr.\ nnd Fehnmrv, \»*\. rpii|Hic'llni| the <'hi>n;F> Dwcid 1.) tnt ; >l»>. Mr ('lmr Whiti'hrr » »hlt«wuhili|| lueiiKiraiKlllin (U.M W Ihi.na, M I* . I>re«iendl«h." publUIio-i »Uii tin- coiia.-nt of Uiv Uaniuiii I'lhoiiWiioa b« very jratalul If >oo aoiil.l aulhorlle oie Ui |.olili«li my lottar of May loth to youraoll. arlth the aililaiiUa, oiiilMliiu «uih |««»«K«ii, if any , yoo miifht ntlno«i1 atmrloua erlmna. Ju«tJo« to iiilllliHta of tha iinluatrlal olaaaoa of our oountryinjn ; an4. Wr, In rauly to your latter, 1 ain illreotaii liy Sir Charloa IHlkt to mo thai he never under any olreuioalMieaa iMn«ant» to the |.unll.atlon ol o,.rre-i».ii.leiH ■ «llh I am vonr olia'llant aarvant, II. Y. HOD, R.I I"'!"""" •' ' « " •'■•^ Ooiiuaminit a letter lo the Ht Hon Nir Chailee W IMIlio, M. I'., I'roaldenl of the lo-al (l.rterninanl lloanl. rvlatlnir lo the farf* Ttwia Tat-toa at Itf hUn. Don (t.) Oon I 'nlon o( (luuuia for the ) earn l«7H and twts, with an llliBilrallon o( one oietho.1 o( fmnd. May I'Hl ,*f»i.- I ■♦a Consiik-ring that I have rercived the consent of the Rt. Hon. the Marquis of Hartington (i) to the publication of correspondence with the late Lord Frederick Cavendish, ri-iating to the first discovery of the conspiracy, now more (ully developal, I am not stepping lieyond the hmits of respectful bearing if I |x>int to certain conditions urtd siiggest a (luery. The highest human function is the administration ot justice. To this end vv- are governed by direct or delegated authority. Can it be the function of the .Minister of (Customs, or of the Covernor-Cieneral, to receive manufactured Records of Ciovernment which display their own one-sided falseness, and turnish an unanswerable indictment against their compiler and his evil methods? It now remains for those who seek to mould publit ojiinion. or profess to guide and protect public morality, to take the matter in hand, and aid in sustaining the principle on which our liberties rest, -that all men are equal before the law. The independent press of both coimtries can do infinite good by persistently calling attention to this matter, inquiring into its insidious partiality, and denouncing a practice which can only lead to discontent and destroy the blessings of good neighl)ourhood. HENRV VOUL.K MINI). (I J KmihlulcTit Offlfii.1 KcM-onN ot Uuvvminent (\>rrii(iioiKl«iict' with the l«te Ijonl Krodorlck r^vendiiili, H V. {*ubtt(thetl with the eonsoiit of the HI. H«i Th*" HaniuUol HMtiuykMl, M. )'., Bvrratary ol HU(«. War l>u|Mu-tiui;iit, Jul;. 1864. u43i[£kL. i •'tmm mmm >m i