■ ■■^,'/-V 
 
 ' '- -i 
 
 ,* 
 V
 
 
 /
 
 
 A 
 
 E 
 
 Y 
 
 TO THE 
 
 COMPLETE PRACTICAL 
 
 ARITHMETICIAN. 
 
 CONTAINING 
 
 ANSWERS TO ALL THE QIJESTIONS IN THAT 
 
 WORK, 
 
 With the Solutions at full Length, wherever there is the 
 
 faialleft Appearance of Labour or Difficulty; 
 
 THE WHOLE INTERSPERSED WITH 
 
 Several ufeful Notes and Obfervations. 
 
 TO WHICH IS A D 1) E r-j 
 
 A N 
 
 APPENDIX, 
 
 CONTAINING, 
 
 A Synopiis oPLogarichiTiical Arithmetic, 
 
 Sliev/ing chcir Natu.c, Couilrrudion, and Uie, in the piaineft. Manner 
 poifiblc. 
 
 Tables of Comoo^ind Intereft and Annuities, 
 
 Kxlending froiu One to Foity Vears. 
 Alfj, Gciicral and univcynl Demon i>iatioiis of the principal 
 
 RULES IN TK^ COMPLETE PRACTICAL 
 ART r H M E 1' I C I A N. 
 
 The whole tog-etlier forming the moil: Complete Sydem of 
 Arithmetic extant, both in Thec--^- •^■■'•A V-^'i^^t-i^^e. 
 
 :riE-Xafy Inrro- 
 
 B V T_H O -M A S 
 
 Teichcr of the Mathematics, Author cr 
 ^ dudlon to ti:e Sc'.cnc- of Cii.oGR avh y, the 
 
 .^RJTHMETICIAN, ic<. 
 
 ^h '-V- 1— 4^..-^^- 
 
 -i: 
 /, 
 
 LONDON: 
 
 FRINTEO BY JOHN CROWDER, 
 F 'J K r. LAW, A V E - M A R I A - L A N E,
 
 4qo. 
 
 PREFACE. 
 
 A S this work makes its appearance with 
 ■*• ^ the approbation, and by the particular 
 defire of feveral eminent Mathematicians and 
 Schoolmafters, it would be fuperfluous for me 
 to fay any thing farther in its commendation, 
 than^ that I have paid every attention to 
 corredlnefs and brevity, confident with per- 
 fpicuity.— 'In working the feveral queftions, 
 I have, in general, made ufe of the Ihorteft, 
 and mofl fimple method of folution I could 
 devife, and where neceffary, have explained 
 the feveral fleps thereof — I have fometimcs 
 folved the queftions by the rules, and fome- 
 times by the notes, in order to exemplify 
 both ; this will be of cqnfiderable advantage 
 to thofe who are not proficients in Arith- 
 metic, and it can be no detriment to the 
 a 2 ingenious 
 
 f^JL ^-.v O. 'O / V/ 'if
 
 IV r R E F A c r. 
 
 ingenious Teacher, who is at liberty to ufc 
 what inethod of folution he pleafes, for I 
 have not the vanity to luppofe that my ib- 
 lutions are fo complete as not to admit of 
 improvement. 
 
 The Appendix commences with a Synopfis 
 of Logarithmical Arithmetic, which I have 
 had by me in manufcript above feven years. 
 This fynopfis I intended to prefix to a fmall 
 fet of rabies, on the plan of thofe publiHicd 
 by M. VAhhe de la Caille, The utility of fuch 
 a work was firft pointed out to me by Mr. 
 Landmann, ProfeiTor of Fortification and Ar- 
 tillery, in the Royal Military Academy at 
 IVoolwich, For, at that time, Sherwins Tables, 
 publifbed in 1705 and 17065 and Gardiner's 
 editions of the fame book, in 1741 and 1742, 
 were the only tables to be depended upon. 
 
 On the appearance of Dr, Hutton\ Tables, 
 
 which 1 believe are exceedingly corred, I 
 laid afide my defign. 
 
 In the year 1789 I publiihed the fub{l:ance 
 o{ xkixs fynopfis y in the fixft number of a pe- 
 riodical work, but by the unexpecled death 
 
 of
 
 l^REFACE. V 
 
 of the ingenious condudtor thereof (Mr, J, 
 Davi/cnJ, the fecond number was never pub- 
 lifhed. At the inftance of fcveral refpe^'^able 
 contributors to that work, xhxs fynopfts is now 
 republifhed, as containing the plained r.nd 
 mofl comprehenfive indru^lions extant, for 
 performing Multiplication, Divifion, Involu^ 
 tion, Evolution, &c. — Then follows, in order, 
 a complete fet of Tables of Compound Interefb 
 and Annuities, accurately calculated from i to 
 40 years. This was hinted to me as a mate- 
 rial improvement, by Mr. Hardy, an eminent 
 teacher of the Mathematics, at CottingbamyU^diV 
 Hull 
 
 I thought it quite unnecefiary to dem.on- 
 flrate every rule in the Complete Practical 
 Arithmetician, becaufe many of the opera- 
 tions carry their rationale along with them, 
 and an attempt to demonftrate a propofuion, 
 which is nearly felf-evident, fom.etimes occa- 
 fions obfcurity. 
 
 I have nothing further to add, than my 
 
 fincere thanks to thofe gentlemen, from whom 
 
 1 have received fuch liberal encouragement ; 
 
 a 3 and
 
 VI 
 
 Preface 
 
 and more particularly to thofe, who have fa- 
 voured me with remarks, tending to render 
 this work as complete and perfedl as pof- 
 fiblc. 
 
 THOMAS KEITH. 
 
 Londeriy 21th September^ ^79^'
 
 The C O N T E N T S. 
 
 The Contents of Part I. and II. follow in 
 the fame manner as in the COMPLETE 
 PRACTICAL ARITHiMETIClAN.- 
 
 THE CONTENTS OF THE APPENDIX^'- 
 
 THE nature and formation of the Logarithni^ 
 page i: 
 A Jhort and eafy rule for the conftruSlwn of Logarith?ns^ . 
 
 either common or hyperbolic^ tllujt rated by examples 4 
 The ufe of a table of Logarithnn — """ 7 
 
 To find the logarithm of a FraB'ion — ——8 
 
 To find the logarithm of a Circulating Deci?nal ib 
 
 Two or more nmnbers being giveny to find their produ^ 
 
 by logarithms — — — 9 
 
 T^o find the quotient of one number by another^ by loga^ 
 
 rithms — — — — 1 1 
 
 To involve a giv£n numd>er to any power — ] 5 
 
 To extracl any root of any given power — jg 
 
 The application of logarithms to Compoufid Inter ejl 22 
 
 The conjiru^ion of Table I. being the amount of il. from 
 
 'I to 40 years^ at 3, 3f, 4, 4^5 and 5 per cent. ib. 
 The conjirutlion of Table ll. being the amount of ll. for 
 
 The conftruSlion of Table III. being the prefent worth of 
 ll. for years — — — ib. 
 
 The conJtru5}ion of Table IV . being the amount of I /. per 
 annum^ or annuities for years — 23 
 
 The
 
 \\\i The Conten-ts of the Appentdix. 
 
 7 he conf.ni5iiQn of Table V. hc'ing the prefent ivorth of 
 \L per annum-t or annuities for years — 23 
 
 The conJlrvMion of Table VI, being the annuity which li.' 
 will purchafe for any number of years — ib. 
 
 ihe life of Table I. II. and HI. — — 33 
 
 The uf of Table IV, and V. — ^ '31 
 
 The tfe of Table VI. - — _ 32 
 
 I. De?nonJ}ration of the method of proving j^ddition^ hy 
 cajiing out the 7jines — — — 34, 
 
 II. DernoYiflration of the method of proving Aiultiplicatiouy 
 by cajVing out the nines — — 35. 
 
 III. Demon/haticn of the Rule of Three — ib. 
 
 IV. Demonp.raticn of the Rule of Five ^ i^c. 36 
 
 V. General princiyies and theory of FraBions 37 
 
 VI. ^emonjl rations of the rules of Circulating Decijtiah 40 
 
 VII. Demonfiratian of the ccnimon rule of Equation of 
 Payments^ on Cocker's, Hatton's, and Moreland*s, 
 or Burrow's, principle — — 42, 43 
 
 VIII. Demorifiration of the Rules of Fellowjhip^ by Air. 
 Adamfori — — — 44 
 
 IX. Demorflration cf the Rules of Lrfs and Gain 47 
 
 X. Demonj'iration of the Rules of Alligation -. 50 
 XL Dejmnjlration of the rulei of Pofition — 51 
 
 XII. Deimnjlraticn of the rules in Arithmetical Pro- 
 greffon — . — -- _« 53 
 
 XI II. Demoiifl ration of the rides m Geometrical Prc^ 
 grejpon — . — ^ ^6 
 
 XIV. Demonjiration of the rules of Variations 61 
 
 XV. Demorijlration of the rules of Combinations 62 
 
 XVI. Demonjiration of the rules of Siinple Inter ejl^ by 
 Dicimals — — — ^3 
 
 XVII. Demonjiration ^Malcolm's rule of Equation of 
 Payments — — — ib. 
 
 XV III. Demonjiration of the ru'cs of Compound Inter eji^ 
 hy Decimals — — •— 64 
 
 xix; J
 
 The Contents of the Appen"dix, ix 
 
 XIX. J general Demonftratlon of the rule of Equation of 
 payments^ at Compound Interefi^ on Moreliind's or 
 Burrow's principle \ alfo on Kerfey's, Malcolm's, 
 Cocker's, and Hatton's — — 64 
 
 XX. Demonjlration of the rules for Annuities in Arrears^ 
 at Simple Inter ejl — — — 65 
 
 XXI. Demonjlration of the rules for the prefent worth of 
 Annuities^ at Simple Interejl — — ib. 
 
 XXII. Demonftration of the rules for Annuities in Ar- 
 rears^ at Compound Intereji — — 66 
 
 XXill. Demonjlration of the rules for the prefent worth 
 of Aunuitiss in Arrears^ at Compound Interejl ib*
 
 BOOKS printed for and fold by B. LA\\''^ 
 
 At No. 13, Ave-Maria-Lane, 
 
 Price 3s. neatly bound, 
 
 THE 
 
 COMPLETE PRACTICAL 
 A R IT H M E r I C I A N. 
 
 CONTAl NING 
 
 Several new and ufeful Improvements, 
 
 ADAFTEB T© tHi V%t OP leHOQLI ANB r&IVATll 
 TUITION. 
 
 By THOMAS KEITH, 
 
 Teacher of the Mathematics* 
 
 * This work, which was underta'.cn with a vi6w to fur^ 
 nilTi a complete fyltem of Pradicai Arithmetic, for the ufe 
 of Schools, contains more ufeful rules and obferfationa 
 than are to be met with in any fyftem of Arithmetic ex- 
 tant, of double the fize and price ; and thefc rules are 
 fucceeded by near two thou (and ufeful and inftrud\ive 
 Examples; bcfides a variety of Bills of Parcels, Invoices, 
 Bills of Exchange, &c. &c. — The nature and praftice 
 of Circulating Decimals ; and the rules of Lofb and Gain,, 
 Fello".\fliip, Exchange, Sec. are here thoroughly confi- 
 dered, and treated of in a different manner to what tluy 
 have hitherto been.
 
 Books printed for B. Law, 
 Price IS. 6d. bound, 
 
 A 
 
 SHORT AND EASY 
 
 INTRODUCTION 
 
 TO THE 
 
 SCIENCE 
 
 O F 
 
 GEOGRAPHY. 
 
 CONTAINING 
 
 A concife and accurate defcription of the fituation, extent, 
 boundaries, divifion, chief cities, &c. of the feveral^ em- 
 pires, kingdoms, ftates, and countries, in the known v.'orld : 
 with the ufe of the terreftrial globe, and geographical 
 liiaps. Sec, 
 
 DESIGNED FOR THE USE OF SCHOOLS AND PRIVATE 
 TUITION. 
 
 llluftrated with the neceiTary engravings, and an accurate 
 map of the world, including the modern difcoveries. 
 
 The SECOND EDITION, correded and improved. 
 
 By THOMAS KEITH, 
 
 Teacher of the Mathematics, Sec, 
 
 ^^ The favourable reception which this little treatife has 
 met with from the public, has induced the author to re- 
 vife the whole with the greateft care, and to make fuch 
 alterations and additions as h,e conceives will be a mean 
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 former.
 
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 MISCELLANEOUS LESSONS on fynonimous Exprcf- 
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 By ELLIN DEVIS. 
 Price 2s. 6d. 
 
 A PRACTICAL ENGLISH GRAMMAR, for the 
 •ufe of Schools and private Gentlemen and Ladies ; with 
 Exercifes of falfe Orthography, and Syntax at large. 
 
 By the late Rev. Mr. HODGSON, Mailer of the 
 Grammar School in Southampton. 
 
 JFifth Editioa, with Improvements. Price is. 6d. 
 
 A NEW SYSTEM OF READING : or, THE ART 
 OF READING ENGLISH, praaicrlly exemplified in 
 almoft every word in ufe : and farther illuftrated from the 
 beauties of tl:ie whole Bible, arranged under different heads, 
 according to the moral virtues therein recommended, or 
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 and, by the help of whic , pupils, whether F^nglifh or 
 F'oreigners, may be taught to read Englifh in one-tenth 
 part of the time ufually devoted to that purpofe. 
 
 By Mr. DU MITAND, Author of a fimilar Syftem for 
 reading French and oth«M- grammatical works ; Teacher of 
 Greek and Latin, anA of the tciip 'ncipal turopcai living 
 tongues. Pxicc 2s,
 
 K E 
 
 TO THE 
 
 COMPLETE PRACTICAL 
 
 \ R I T H M E T I C I A N 
 
 PART 
 
 NUMERATION. 
 
 FORTY-NINE.—Seventy-five.—One thoufand and 
 feventy-five. — Three hundred and feventy-eight. — Four 
 hundred and thirty- fevcn. — T hree hundred and live. — One 
 thoufand and eighty -feven. — Forty-feven thoufand, three 
 hundred and eighteen. — Seventeen thoufand, three hundred 
 and forty-nine. — Ten thoufand, eight hundred and feven. — 
 Three hundred and fourteen thoufand, eight hundred and 
 fifteen. — One hundred and feven thoufand, and forty-eight. 
 — One hundred and forty-nine thoufand, three hundred and 
 eighty- feven. — One million, feventy-eight thoufand, four 
 hundred. — Thirty million, one hundred and eighty thoufand, 
 and feventy. — One hundred and eight million, three hundred 
 and feventy-four ihoufand, one hundred and eight. 
 
 (2.) 
 
 89. 750. 5,007. 10,087. 20,005. 685, .^60. 
 
 1,500,001, 27,365,000. 585,748,305. IT,000-f-I,J03 
 
 fji = 12,111, 50,000,0004-50,000-1- 5,coo + 50 =^ 
 50,05^,050. 
 
 B
 
 Simple Addition, Subtraction, kc. 
 
 SIMPLE ADDITION. 
 
 NL>t. Answers. 
 (I.) 26038 
 (2.) 180212 
 ij.) 1673039 
 {4.) 86S6587 
 (5.) 29S295769 
 
 NiTM. Answers. 
 (6.) 6898970405 
 (7.) 2079^-501 
 (S.) 81 Peterborough, 132 
 Lincoln, and 173 Hull, 
 
 SIMPLE SUBTRACTION. 
 
 KuM. Answers. 
 (I.) 8087801 
 (2.) 280007 
 (3.) 299S86 . 
 (4..) 212201 
 
 Num. Answers, 
 
 (5-) 33892 
 (6.) 45396 
 (7.) 4835656 
 (8.) 1344138 
 
 (9-) 
 
 Chriil: came into the world 4C00 years after the creation ; 
 hence 4000+ il^9=S'1^9 Y^^^^ ^^"^^ ^^^ creation. 
 (10.) 89 years old, 62 years I (11.) 34 years, and 46 years, 
 fincehedied. — 1789. | 
 
 (12.) 
 
 A + B + C won 97 
 B + C won 6q 
 
 A won 
 
 37 
 
 B -j- C won 
 B won 
 
 A + B + C won 97 
 A f- C won 62 
 
 B won 
 
 33 
 
 C won 
 
 60 
 
 35 
 
 25 
 
 SIMPLE MULTIPLICATION 
 
 KuM. Products, 
 (1.) 942694650 
 (2.) I ' 144SC221 
 (3.) 1R8194972 
 (4.) 28,^674870 
 (5.) 223C84518 
 (6.) 343353-^oo43 
 (7.) 297r949749<^ 
 
 Num. Products, 
 (8.) 42378429339 
 
 (9-) 5;H937H3o 
 (10.) 40872864296 
 (it.) 179245888S - 
 (12.) 2639559272 
 (13.) 989830464 
 (14.) 8172S6228
 
 Part I. 
 
 Simple Division. 
 
 Num. Products. 
 
 Num. Products. 
 
 (15.) 1132497912 
 
 (57.) 1 8765700000c oco 
 
 {16J 472087728 
 
 {38.) 191 69700000 
 
 (17.) 302020848 
 
 (39.) 15463500000 
 
 (18.) 244512951 
 
 (40.) 28610266500000 
 
 (19.) 3276600516 
 
 (41.) 93807952920000 
 
 (20.) 684S734752 
 
 (42.) 23190200250000® 
 
 (21.) 62244248^:^8 
 
 (43-) 330043 
 
 (22.) 10139541C480 
 
 (44.) 4459128 
 
 (23.) 2072213S22625 
 
 (45.) 8235460800 
 
 (24.) 320021 195962 
 
 (46.) 26156958360000 
 
 (25.) 556321146764 
 
 (47.) 2400355079025000 
 
 (26.) 17681 97301755 
 
 (48.) 137829543416436 
 
 (27.) 269510713983 
 
 (49.) 152415787501905210 
 
 (28.) 17529177479355 
 
 (50.) 1 21932631 11 26352690 
 
 (29.) 17464274403363 
 
 { 51 ) 86439657760572480000 
 
 (30.) 26885596435656 
 
 (52.) 777'5 
 
 (31.) 260232989070535 
 
 {53.) lolhort 
 
 {33.) 287523274925880 
 
 (54.) 160010532 
 
 i33') 2734390884446865 
 
 {^^.) 40 difference 
 
 (34.) 2r20503567i869797 
 
 {56.) 17624 fum, 76945744 
 
 (35.) 16651582419409925 
 
 produd 
 
 (36.) 21153742978114592 
 
 
 SIMPLE DIVISION. 
 
 Num. Qjjot, Rem. 
 
 Num. Quot. Rem. 
 
 (1.) 874671172—1 
 
 (18.) 35685129—126 
 
 (2.) 157116523—2 
 
 (19.) 10823637 — 12 
 
 {3.) 101776234— I 
 
 (20.) 4760566—38 
 
 (4.) 1408142S— 3 
 
 (21.) 364993—43 
 
 (5.) 28341563— 
 
 (22.) 407294 — 1080 
 
 (6.) 70534296—3 
 
 (23-) I3I95133--1S42 
 
 (7.) 38588S1342-7 
 
 (24.) 125139201 I3CIO 
 
 (8-) 4597301659—6 
 
 (25.) 269577255S82 — 5561 
 
 (9.V 7 100057 147—9 
 
 (26.) 14243757748— 35411 
 
 (10.) 3400653124—10 
 
 (27-) 15395919— I22I4 
 
 (ii-) 3925476122— II 
 
 (28.) 3COOICOO 6347 
 
 {It.) 2859/3914-9 
 
 (29.) 131809655 104990 
 
 (13.) 272202735—12 
 
 (30.) 300335575 — 273118 
 
 (14.) 5203802—61 
 
 {31.) 9948157977—81605 
 
 (15.) 11805558—53 
 
 (32.) 101489 62899 
 
 {16.) 39096821—64 
 
 (3 3-) 15655794—9075 
 
 {'7') 34297^82—58 
 
 (34.) 2667376—42700 
 
 B z
 
 S I XI r L E Division. 
 
 Num. Quot. Rem. 
 iSS-} 16871651—2944 
 (36.) 4C24416— 263149 
 (37-) 8909748—722934 
 (38.} 7264348958— 125715 
 {39.) 93191497743— 95257 
 {40.) 4087692937— 381715 
 (41.) 7943859—39 
 {42.) 1 19092— 12348 
 
 i)348 
 97 
 
 (5^0 
 2)194 
 
 97 
 
 471 greater J 
 
 I by 
 
 77 lers 5 
 
 note 6tli. 
 
 Num. Quot. Rem. 
 (43.) 59085714—84 
 (44.) 1258127— 1578— s 
 
 {45-) 123456789 
 (46.) 123456789 
 (47.) 714007 14374 
 (48.) 937143718740 
 
 149-) 1924530—652547 
 {50.) 119191753— 90107 
 
 (52.) 865 
 
 ,5J-) 942 
 
 (14.) 
 8117119^^865^ Tto'niJ+Tr4'-i^5-=4^4 Anf. 
 
 ider. 
 Then 5889454-237 = 589182 A 
 
 I '<^5 4 c = 237 remainder, 1185 X 497 ^ 58694I 
 
 niwer. 
 
 ^56.) i^ 
 I 2 
 
 2)144 a dozen dozen 
 6 
 
 S64 fix dozen dozen 
 7 2 half a dozen dozen 
 
 792 difference. 
 
 (57O 17^493745-^759 gi^'es 
 2:5946 for the quo- 
 tient, and 731 re- 
 mainder ; hence 731 
 —5001=231 exceeds, 
 Anfvver.
 
 Parti. Compound Addition 8c Surtr action. 
 
 KuM Answkrs, 
 
 COMPOUND ADDITION, 
 
 M. Answers. 
 509 a. 2 1. 1 8 p. 
 4797 a. 2 r. II p. 
 403 a. I r. I p. 
 
 12.) 
 (3.) 
 (4.) 
 
 (5-) 
 (6.) 
 
 (7-) 
 
 (9.) 
 (10.) 
 
 (II.) 
 
 (12.) 
 
 US') 
 
 (»4.) 
 
 ('5.) 
 (16.) 
 
 (17.) 
 (18.) 
 
 (19-) 
 (20.) 
 (21.) 
 (22.) 
 (23.) 
 (24.) 
 (25.) 
 (26.) 
 
 (27.) 
 
 (28.) 
 
 £' 
 
 779 
 
 976 
 
 1709 
 
 567 
 
 684 
 
 205 
 223 
 240 
 
 d. 
 
 H 
 o 
 
 10 
 4 
 5 
 3 
 
 12 
 
 6 
 
 » I 
 
 10 
 
 si 
 
 8i 
 
 23821b. 1 oz. i6d\vt. 
 41290Z. i9dwt. 20gr. 
 .4621b. 907.. i4dwt. 
 407 oz. i4dvvt. 5gr. 
 51 1 lb. II oz. 3 dr. 
 35280Z. 6 dr, I fcr. 
 3J02 dr. I fcr. 18 gr. 
 464 lb. 5 dr. 
 3833 ton. 7cwt. iqr. 
 3030 cwt. 1 qr. 27 lb. 
 370 qr. 151b. looz. 
 347 lb. 7 oz. 6 dr. 
 471 yd. 2 qr. 1 n. 
 484611. Eng. 2 qr. 
 3821 ell. Fr. 
 4768 cll. Fl. 2n. 
 489 lea. I m. 6 f. 
 4487 fur. 35P- 5y^^s. 
 47 08 p. I yd. 
 644 f t . lb. c. 
 
 Nu 
 (29 
 (30 
 (3' 
 (32 
 (33 
 (34 
 (3S 
 (36 
 
 (37' 
 (38, 
 
 (39' 
 (4a 
 
 (41 
 (42 
 
 (43 
 (44 
 
 (45 
 (46 
 
 (47 
 (4B 
 
 (49 
 (50 
 (5^ 
 (52 
 (53 
 (54 
 (55 
 (56 
 
 m 
 
 5P- 
 
 3209 tuns. 27 gall. 
 
 5422 p. 57g^^l- 2qt. 
 460 tier. 29 gall. I qr. 
 297 gall. 2 qt. 
 249 B.B. I fr. 6 gall. 
 323 A.B. 2 fr. 
 3i(foAhhd. 46gal. iqt 
 522oBhhd. 4gal. 2 qt. 
 529 ch. I b. 2 p. 
 3200 ch. iqr. 7 b. 
 3842 qr. 6 b. 2 p. 
 409 f. i2ch. 19 b. 
 4299 yrs. 8 m. 2 vv. 
 525 m. 4d. 
 ii27d. 2 hrs. 50 m. 
 4444 hrs. 23 m. 50 f. 
 X.2490 16 si 
 £' 397 I 4t 
 /•1729 19 3 
 >C- 551 15 3 
 £•^^^1 7 9 
 1241b. 10 oz. 3d\vt. 
 37 cwt. oqr. 17:5:1b. 
 255 yds. 1 qr. i n. 
 
 COMPOUND SUBTRACTION 
 
 Ni^M. Answers. 
 
 £' s. d, 
 
 (r.) 800 18 9^ 
 
 (2.) 3254 19 ii,i 
 
 (3-) 75 o 8f 
 
 (4.) 497 ^o 71- 
 
 Num. Answers. 
 (5.) 26 17 c| 
 
 (6.) 30 II IOy 
 
 (7-)^ 34 13 9i 
 (8.)* 32 13 'i 
 
 In this Example, fo. ii^-j £, read 747/. 
 
 B3
 
 Compound Subtraction, 
 
 Num. Answers, 
 
 (9 
 
 (iO 
 
 2-) 
 
 d. 
 oi 
 
 3i 
 
 £■ ■'■ 
 
 lOJ I 
 
 474 J 
 9524 8 
 in all. 
 62223 3 
 mains to pa) 
 8073 19 4| 
 out in all. 
 
 paid 
 
 7i 
 
 re- 
 
 laid 
 
 63365 
 
 12 
 
 Lr re- 
 
 mains in hand. 
 
 (15.) 
 (16.) 
 
 (17O 
 (18.) 
 
 (19.) 
 (20.) 
 
 (21.) 
 (22.) 
 
 (24 
 
 (25 
 (26 
 
 {28 
 (29 
 
 (30 
 
 2464 16 
 lance. 
 
 3 lb. ooz. i^dvvt. 
 90Z. 17 dwt. 20 gr. 
 1 5 lb. 3 oz. 1 6 dwt. 
 20Z. 18 dwt. 21 gr. 
 791b. I ooz. 6 dr. 
 
 1 2 oz. 4 dr. 2 fcr. 
 
 13 dr. 15 gr. 
 81b. looz. 7dr. 
 
 .) 12 ton, I7cwt. 3 qr. 
 .) 2 cwt. 2 qr. 261b. 
 .) 69 qr. 2 lb. 140Z. 
 .) 1341b. i4'Oz. 13 dr. 
 .) 134 yd. 2qr. 3n. 
 .) i24cllEng. 3qr. 3 n. 
 .) 96 ell Fr. 2 qr. i n. 
 ell Fl. I qr. 2 n. 
 
 Nu 
 (3'. 
 (32- 
 (33- 
 (34- 
 i3S' 
 {36. 
 (37- 
 (3«. 
 
 (39- 
 (40. 
 
 (41. 
 (42. 
 (43- 
 (44- 
 
 (46. 
 
 (47. 
 (48. 
 
 (49- 
 (90. 
 
 (51- 
 (52- 
 iS3' 
 (54. 
 (55- 
 (56. 
 
 (58. 
 
 (59- 
 (60.) 
 
 M. Answers. 
 1 7 lea. 2 m. 6 f . 
 I f. 34 p. 5 yds. 
 4 p. 3 1 yd. 2 ft. 
 
 oin. 1 b. c. 
 
 2 r. 1 8 p. 
 
 2r. 34 p. 
 
 I r. 26 p. 
 
 2r. 39 p. 
 
 ;oa. 
 
 37 a. 
 I a. 
 
 4a. 
 
 7t. 2hhd. 55 g. 
 67 pun. i4g. 3qt. 
 I tier. I g. 3 qt. 
 6 gal. 2 qt. I pt. 
 
 1 A.B. 3f. 6g. 
 
 1 07 B. B. I f. 4 g. 
 221 A. hhd. 1 g. 3 qt. 
 63 B.hhd. 2 g. 3 qt. 
 * 27 ch. ob. I p. 
 2ch. 3 qr. 2 b. 
 52 qr. 6 b. 3 p. 
 32 fcore, I ch. 13 b. 
 
 2 yrs. 1 1 m. 3 w. 
 127 m. 3 w. 6d. 
 147 d. 21 h. ^6 m. 
 79 h. 50 m. 5:4 fee. 
 
 3I. 9s. 
 
 I21I. 17s. 0;^d. 
 
 1500I. 2S. 6d. Dr. 
 520I. OS. 5d. Cr. 
 980I. 2S. id. ba- 
 lance. 
 1 80 1. II cwt. 1 2lb. 
 J y. II m. o w. 6 d. 
 
 8 h. 22 m. 
 difF.lat. 34° 35^ long. 
 165° i8^ 
 
 * In. this Example the 74 ch. ought to be uppcrmoft.
 
 Parti. Compound Multiplication, See. 
 
 COMPOUND MULTIPLICATION, 
 
 Num. Answers. 
 
 (l.) I 10 2 
 
 (2.) I 9 2 
 
 (3-) 2 o 5 
 
 (4-) -2 ' Si 
 
 (5-) 3 2 8 
 
 (6.) 5 6 Hi 
 
 (?•) 8 3 9 
 
 {8.j i6 8 7-i 
 
 (9.) 22 13 o 
 
 (lo.j 133 y^«' 3qr- 2n. 
 
 (ir.) 138 oz. 9dwt. i2gr. 
 (12.) 
 
 (13-) ^-3 19 4z- 
 
 (14.) 59 10 8 
 
 (15:.) 8 19 o 
 
 (16.) 60 14 pi 
 
 (17.) 208 13 9 
 
 (/8.) 154 12 3 
 
 (19.) 42 I 6 
 
 (20.) 123 17 9 
 
 fzi.) 819 6 o 
 
 (22.) 82 lb. 8 oz. iS dwt. 
 i6gr. 
 
 (23O 75a. 3^- 39P- 
 (24.) X.16 19 3 
 (25.J 19 10 81 
 
 Num. Answers. 
 /. s. d, 
 
 (26.) 12 18 A,k 
 
 14 
 
 8 
 
 (27-) 33 3 
 (28.J 83 2 
 
 (29-) ^37 7 3 
 (30.) 698 2 o 
 (31. j 743m. i^d. 6Ii. 
 (32.) 222 cwt. 1 8 lb. 
 
 (33.} 15 cwt. 271b. II oz, 
 
 7 dr. 
 
 (34-) ;^-343 18 7 
 
 (3S.) 2684 18 9 
 
 (36.) Z<^i 2 7i 
 
 (37.) 257 12 4 
 
 (38.) 62 I 7i 
 
 (39.) 15299 18 4 
 
 (40.) ^i 7 2f 
 
 (41.} 344 o 6 
 
 (42.) 1364 o o 
 
 (43-J 98 7 4f 
 
 (44-) I 14 2 
 
 (45-) ?66 5 3f 
 
 (46.) 17038 10 iii-| 
 
 (47.) 12422 2 7^-J. 
 
 (48.) 2875 o 7i 
 
 (49.) 1292 I 6|.i 
 
 (50.} 658 o \iy.x 
 
 COMPOUND DIVISION, 
 
 Num. Ans. Rem. 
 
 ^. .. d. 
 
 (I.) '3 9 I 
 
 (2.) 15 19 9^—1 
 
 (3.) I II 5^—16 
 
 (4.) o 15 i|— 15 
 
 (5.) o 4 7I-- 122 
 
 (6.) 65 18 7i-,3_ 
 
 (7.) 41 5 2 .>^ 
 
 (8.) 9 ,8 3I--92 
 
 Num. Ans. Rem. 
 (9.) 9 yds. 2 qr. in — I 
 (ic.j 41 a. 3r. I p.— 5 
 (11.) 8 lb. 7 dwt. 15 gr. — 45 
 (12.) 7 cwt. 3 qr. 6 lb. — 30 
 (13.) 2Eng. ell, 3qr.— 107 
 (14.} 8ch. 13 b. ip.-||ij 
 (15.) ioz.7dwt.9gr.~-|J| 
 (16.JX1 17 6 
 (n-) o o 6^-^
 
 8 
 
 Num. Answers 
 
 £.s. d. 
 (18.) o 17 III 
 (19.) 024 
 (20.) 7 j6 8| 
 
 (21.) O O 5I: 
 
 (22.) o o loj 
 (23.J o 4 9i 
 
 Compound D i v 1 s i o -x. 
 
 Num. Answers. 
 
 (24.) 1050— 175 -7- 150 = 
 
 X.516 8 
 (25.) 10 oz. ijdwt. 4gr.-i^V 
 (2f'.) I7cwt. 2qr. lilb. -^| 
 (27.) 36 fl. ell, I qr. 311. -/j 
 (2S.) 2a. or, i7i\-p. 
 
 (-9) 
 
 lb. oz. dw. gr. 
 2^)34 3 J I Hfilb. 30Z. i6dwt. 14^53 gr. wt. of each 
 — . wedge. 
 
 12 
 
 26)99(3 oz. 
 
 21 
 
 20 
 
 26)43 1( 16 dwt. 
 
 lb. oz. dw. gr. 02. dwt. gr. 
 34 3 II 14= 411 II 14 
 3f 
 
 171 
 
 24 
 26)374(i4gr. 
 114 
 
 m I 
 
 10 rem. 
 
 1234 14' 18 
 308 13 16I: 
 
 Guineas 1 543 8 lof 
 
 A nfwer 1 5:43 guineas, and 2 dwt. 6gr. 
 over ; or 1 54314^ guineas. — For if wc 
 divide 8 dwt. lOj gr. by 3I the quo- 
 tient will be 2 dwt, 6gr. = H guin. 
 
 (3C.) 
 
 Admit the weight of the drofs and hhd. to be t, then the 
 weight of pure fu^ar will be 13, and both together 14: 
 
 cwt. qr. lb. 
 Hence 14)7 3 14 whole weight 
 
 027 wt. of the drofs and hhd. 
 
 7 I 7 wt. of purefuL:^.r.
 
 Part I. 
 
 Reduction, 
 
 . (3'-) 
 il. 3s. lo^^gd. the value of a fhekel, and i^6oz. lydwt. 
 1 2 o-r. wt. of a talent. 
 
 (32-) 
 
 AnAver 
 
 REDUCTION. 
 
 Num. Answirs. 
 (1.) 1055. i26od. 5:04c f. 
 
 (2.) I200d. IOCS. 5I. 
 
 (3.) 380s. 4.56od. 1 8240 f. 
 (4.) 1 1 55s. r386od. ^544of. 
 (5.) 524.0J farthings 
 (6.) 37249 halfpence 
 (7.; 22736d. 
 f8.) ^726 twopences 
 (9.) ^767 threep. SjOid. 
 (lo.) 2879 groau, n|i6d. 
 
 4006^ f, 
 (11,) 4343 fixpences 
 III,) 216 crowns, 432 half= 
 crowns, loS'o Ihill. 
 ai^ofijtp. i296od. 
 (13.) 1493 s. 5975 ihreep. 
 
 7 1 700 f. 
 (14.) 288od. 24CS. 12I. 
 
 (15.; 74I. I OS. 
 
 (16.) 100 guineas 
 
 (17.) 105I. 
 
 (18.) 2410 cr. 602I. los. 
 
 (19.) 77760 groats, 25920s. 
 5184 crowns, 1296I. 
 
 (20.) 282 of each 
 
 {2f.} loi 1 69groats, 337235. 
 134H9J halt" crowns, 
 6744I cr. i6S6i^l 
 
 (l2.) 9968 farthings 
 
 (23.) 100320 grs." 
 
 (24.) 14 oz. 
 
 (25.) 184800 grs. 
 
 (26.) 13841b. 
 
 Num. Answebs. 
 
 (27.) 23 pints, and 3 oz. 
 
 i4dwt. over 
 (23.) 42 tea-fpoons 
 (29.) 7200 fcr* i4400ogrs. 
 (30.) 204 oz. 17 lb. 
 (31.) 86962 grs. 
 (32.) 3cilb. 40Z. 3drs. iicr. 
 
 (jj.) 69 of each, and jjgf. 
 
 over 
 (34.) 368801b. 
 
 14 tons 
 lb. 
 
 30 t. 13 cwt. I qr. 
 
 312 lb. 4OZ, 
 
 131 cwt. 2 qr, 141b* 
 
 14625 cwt. 
 
 168 parcels - 
 
 853 common lb. 
 
 350 great lb. 
 
 105840 lb. 
 
 (36.) 4207 
 
 (37 
 
 (38.) 
 
 (39-) 
 
 (40.) 
 
 (41.) 
 (42.) 
 
 (45.) 
 (4+.) 
 (45-) 
 (46.) 
 
 (47.) 
 (48.) 
 
 (49-) 
 
 (5^0 
 (530 
 
 27 II parcels 
 5024 nails 
 864 yards 
 78 f fing. ells 
 23940 nails 
 9996 yards 
 9160^- yards 
 36516! yards 
 3768 f. 1 507 20 poles 
 70 miles 
 
 88000 yd. 264000ft. 
 3 1 68000 in 9504000b. c.
 
 i« Reduction. 
 
 NvM. Answers. Num. Answers. 
 
 (5^0 
 m. f. p. f. 
 
 37 z 57 5 
 
 298 f. 
 40 
 
 1 1957 poles 
 16k 
 
 71747^6 
 ^^957 
 f978 6 
 
 197296 feet 
 
 C56.) i74of. 2 1 71 miles 
 (57-) 4755801600 barley. 
 
 corns, or 1 008 qr. 1 6 cpt, 
 
 5268 b. c. 
 
 (58. 
 
 (59. 
 (60. 
 (61. 
 (62. 
 
 (64. 
 
 (65. 
 {66, 
 (67. 
 (68. 
 (69. 
 (70. 
 (71. 
 (72- 
 (73. 
 (7+. 
 (75- 
 (76, 
 
 (77r 
 
 126720 times 
 12374 perches 
 108 acres 
 5726 perches 
 19 tenements 
 8064 quarts 
 13 tuns 
 
 4725 g'*^- 37800 pts. 
 970I half anchois 
 8 of each 
 14592 pints 
 15504. pints 
 297216 half pints 
 17Q0 gallons 
 1408 pecks 
 12 lafts 
 7200 pecks 
 2320 facks 
 31556935 feconds 
 30617316^ feconds 
 
 (78-) 
 
 1789 
 
 4004 
 
 5793 years fince the 
 365-^ creation. 
 
 2115893^ days 
 24 
 
 50781438 hours 
 60 
 
 3046886280 minutes 
 60 
 
 182813176800 feconds 
 
 (79*) 25395102 hours 
 
 y. d. 
 1815 o 
 1788 89 
 
 (8c.) 
 10 Feb. 
 31 Mar. 
 30 Apr. 
 1 8 May 
 
 89 days 
 
 (81.) 
 
 26 276 
 977 2| days 
 
 ft. lb. 
 
 234 4 
 8 
 
 908 
 
 22 
 
 23)41976(341 
 
 507 
 156 
 
 lb. 
 
 26 
 
 22 
 
 30 
 16 
 
 '5 
 
 23 
 
 33 
 
 341 of each, and 331b. over.
 
 Part I. Reduction, 
 
 (82.) 
 For 5s. 4d. (as printed) read 5s. 4fd. 
 
 II 
 
 I. s. 
 58 14 
 
 d. 
 
 6 
 
 a each 
 
 s. d. 
 
 5 4i 
 4 
 
 ' ■ s. d. 
 I 
 8 
 6 
 4 
 
 57 12 
 20 
 
 1152 
 
 12 
 
 6 
 ople 11 
 
 I 1 6 
 
 parifii. 
 
 2 6=30 d* 
 
 013832 
 
 461 pe^ 
 
 4 
 
 
 1S44 people, in all, partook of the charity. 
 
 (83-) 
 
 lb. oz. dw.gr, lb. oz, dw. gr. 
 
 371421X18 — 657718 
 
 From which fubtradl the drofs= o 7 i 14 
 
 oz. dw. gr. 65 o 6 4 =: 374548 gr. 
 
 18 14 10 ' ■ 
 
 19 15 II 
 
 24 10 14 Then 374548^43297=811411 of 
 
 412 o each, or 8 of each, and 58 oz, 
 
 22 II 14 I3dwt. 20 gr. 
 
 90 4 i=43297|;rs. 
 
 (84.) 
 
 365 d. 6 hrs. = 525960 minutes in a year. 
 500 X 100 X 525960 = ^.26298000000 the 500 
 men would count in i year. 
 2629 8000000 ) I, ooo,ooo.ooo>oQo [sS^ll^j years, Anfwcr. 
 
 Rem. 676
 
 12 
 
 
 
 R 
 
 EDUCTION, 
 (85.) 
 
 d. 
 
 h. / // 
 
 
 d. h. 
 
 365 
 
 600 
 5 48 55 
 
 
 2)365 6 
 1S2 15 
 
 
 
 II 5 
 60 
 
 66^ 
 
 
 24 
 
 4383 hours 
 60 
 
 
 
 
 262980 minutes 
 60 
 
 
 665) 1 5:77880c (23727111 years 
 
 
 
 
 Rem. 34; 
 
 
 
 
 (86.) 
 
 oz. 
 
 dw. gr. 
 
 lb. 
 
 07. 
 
 5)14 
 
 II 15! = 
 
 : I i 
 
 ivoird. 44-^ ) 1 2 in a lb. troy. . 
 
 
 6 ijf 
 
 wt. 
 
 of a halfpenny 5^w. lo^yygr. 
 wt. of a guinea 
 
 
 15KI 
 
 rs. difference 
 
 (87.) 
 
 I. s. d. 
 
 500 o o he left in cafh 
 
 /. 84 10 6 y. S =i 422 12 6 in bills 
 
 92212 6 he left in all 
 
 120 + 20 == 140 o o his debts 
 
 4) 782 12 6 the refidue 
 
 1 95 1 3 1 1 the eldeil fon's (hare 
 
 4 )586 19 4 1 
 
 1 46 14 10 each of the other's fharc
 
 rsrtl. The Role OF Three Direct. 13 
 
 The rule of THREE DIRECT. 
 
 If 2cwt. 3qr. 141b. : 61. 14s. 2d. : : I2cwt. 3qr. : 29I. ijs. 
 
 (2.) 
 If I2cwt. 3qr. : 29I. 15s. : : 2 cv/t. 3 qr. 141b. : 61. 148. id. 
 
 (3-) 
 If 61. 14s. 2d. '. 2cwt. 3qr. 141b. : : 29I. 1 js. : I2cwt. 34r. 
 
 (+•) 
 If 29I. 15s. : I2cwt 3qr. :: 61. 145,24!. : 2c\vt. 3qr. 14 lb, 
 
 (S-) 
 If 1 12 lb. : gl. i6s. :: i lb. : is. 9J. 
 
 (6.) 
 If lib. : jjd. : : 561b. : il. 63. lod, 
 
 (7-) 
 ]f jlyds. : 2I. i6s. ^d, :: 28Jyds. : 23I. 2S. oid. |, 
 
 If40guln. : 56 yds,:: 1135-1. los.; 1514yds. p 201 SFIcin, 
 eils, 2qr. 
 
 (9-) 
 From the 14th of May, 1 780, to the 1 1 th of December, 1783, 
 
 are 1 306 days. 
 If 28 days : 25s. :: 1306 days : 5SI. 65. o|d. ^ 
 
 (10.) 
 If IS. 6d. : I w, : : lool. : 1333^ ^* = 25 yrs. 8 m. i\ w. 
 
 (II.) 
 If III-. 5s. : 22oqrs. (ir 55yds.) : : looguin. : 2053jqri 
 ==410 Eng. ells, 3jqrs. 
 
 (12.) 
 
 1. s. d. 
 
 30 quarters coft 76 17 6 
 
 150 ^ 361 II 8 
 gain 20 
 
 !lfi8oqrs. ; 458 9 2 : : i bu. : 63. 4id. J|*
 
 14 The Ruli or Three Direct. 
 
 (.3-) 
 If I ell : 7s. 6d. : : 27 pieces each 34 ells : 344I. 51. 
 
 (14O 
 If il. : los. 6d. :: 475I. los. : 249I. 12s. pd. 
 
 (15.) 
 If 53cwt. 2qr. 51b. : 18I. 143. 9|d. :: i lb. : -'d. 
 
 (16.) 
 If 2547I. 14s. 9d. : loooiguin. :: il. : 8s. 2jd. 4^||.f| 
 
 (I7-) 
 If47i4l. IIS. icd. : 117]. 17s. 3id. :: il. : 5H ||HH 
 
 (IB.) 
 If 4714I. IIS. icd. : 235I. 14s. 7d. :: 350I. 14?. : 17I. los, 
 
 (19-) 
 
 Ifil. : 9d. :: 350I. 14s. : 13I. 3s. ojd.-j- 
 
 (20.) 
 
 If I cwt. : 2I. 14s. 9d. :: i4hhds, each 17 cwt. i qr. 
 141b, ; 282I. 1 25. ii^d. 
 
 (2..) 
 
 If 1 todd : 17s. 6d. : : 2 cwt. 2qr. i4ib. : 9I. 3s. 9d, 
 
 (22.) 
 
 If I cwt. : 5I. 5s. 4 : 157 foth. each 19^ cwt. : 16072I. 
 
 17s. 6d. to which add 5I. 5s. the fum is 16078I. 2S. 6d. 
 
 the whole expence of the lead. Again, 
 If 1 57 foth. : 1607SI. 2S. 6d. : : I lb. : i i^d. :rtKi Anf. 
 
 (23.) 
 If I oz. : 3I. : : 14 ing. each 31b. 11 oz. 15 dwt. 
 i3gr. : 2C06I. I2S. 9d. 
 
 (24.) 
 
 If4s. gd. : I yd. :: 722I. : 3040 yds. = 2432 ells Engliih, 
 
 which divided by 76, gives 32 ells Eng. in each piece. 
 
 (2v) 
 
 If 4 tons lefs 64 gall. : 247I. i;§, ;; j gall, : 5s. z^d. f*.
 
 Part I. The Rule of Three Direct, ij 
 
 (26.) 
 
 If I yd. : 17s. 9^. :: ii'ji yds. : 104I. 5s. -jld. value of 
 the broad cloih, which deduced from i 24I. gives 19I. 14s. 
 4td. value of the baize. 
 
 If 5 yds. : liyd. : : 11 7-'- yds. : 35! yds. of baize he bought. 
 
 If 35|yds. : 19I. 14s. 47^.' :: i yd. : lis. z^d. ^'y the vaitie 
 of a yard of baize. 
 
 I. s. 
 
 If I hhd, ; 12 guin. : : 59 tuns : 2973 12 
 
 The freight - - 47 i «> 
 
 Loading and unloading 7 10 
 
 Cudom - - 24 o 
 
 Charges of the cellar 3 3 
 
 Prime coft of the wine 3055 15 
 The gain 150 o 
 
 3205 j; 
 
 If 59 tuns : 3055I. 15s. : : 1 gall. : 4s. i *:d. -jV/^ prime coft 
 
 of 1 gallon. 
 If 59 tuns : 3205I. 15s. : : i gall. : 4s. 3id. ||f|hemuft fell 
 
 it per gallon. 
 
 (28.) 
 
 Firfl 1 5 ells Eng. zqr. 3n. = 3iin. 311 X 15X7X5 = 
 
 163275 nails. 
 If ^yds. : 4I. 7s. 9d. : : 163275 nails : 8954I. 12s. 3i'^d. 
 
 (29.) 
 If 2^ hrs. 56m. : 36odeg. each 691 miles : : i hr. : 1045 
 
 miles, 3 ^y^ furlongs, the inhabitants upon the equator are 
 
 carried per hour. Again, 
 If '23 hrs. 56 m. : 360 deg. each 37 m. 2 f. 37 p. 
 
 5t ft. : : 1 hr. : 562 ra. 18 p. 8 yds. lift.^Y* ^^^ v^^- 
 
 bitants oi Londofi are carried per hour, 
 
 {30.) 
 ^^ 3 + 3 + 3 + 2 =z II. 
 
 If II : 225I. los. : : 3 : 61I. ics. A, B, and C each 
 If 3d. : 61I. ics. :: 2d. : 41I. Dpaid. 
 
 C 2
 
 %6 The Rule or Three Direct, 
 
 d. '^-^ 
 
 I 20 -f- 3 = 40 
 
 1 20 -7- 2 ~ 60 If I ood. : 240 eggs : : ^d. : 12 eggs. Anf. 
 
 Tlie eggs cofl 1 00 
 
 Fir/?, While the hour-hand gees once round, the minute- 
 hand goes 12 times round, therefore the minute-hand gains 
 J I rounds eery 12 hours. 
 
 If II r. : izhrs. : : 2 r. : 2 hrs. loj-ymin. hence the hands 
 will be together loj^ minutes paft 2 o'clock. 
 
 SecoAaJj, While the minute-hand moves 5 minutes, the 
 hour-hand moves -j^^j of a minute, then 5 — ^K- = 4,\ minutes, 
 the minute-hand gains of the hour-hand every 5 minutes. 
 
 If4/jm. : 5m. :: I5ra. : 2 7y\min. hence the hands 
 Vrijl be 1 5 minutes (or 90 degrees) apart at 27-j\. minutes pafl 
 2 o'clock. 
 
 77?/>i//y, If J I r. : 12 hrs. :: 3 r. : 3 hrs. i6/,-min. hence 
 the hands will be together a fecond time at 1 6/1 minutes 
 paft 3 o'clock. 
 
 (33-) 
 
 IfzG.lps. : 3H. Ips. :: 3G.Ips. : 4l:H. Ips. 
 
 Here 3 greyhounds' leaps are equal, in diftance, 104!: hare's 
 leaps ; hence every 3 leaps the greyhound took he gained ^ a 
 hare's lenp, for while the greyhound took 3 leaps, the hare 
 took 4. 
 
 IfiH.Ip. 
 
 : 3 dps. 
 
 :: J 44 H. Ips. 
 
 : 864 G. Ips. Anf. 
 
 
 s. d. 
 
 (34-) 
 
 I. s. d. 
 
 If I p. : 
 If I yd. : 
 
 4 S •• 
 
 3 7 • 
 
 : i7sop. 
 : i749y^s. 
 
 : 3S6 9 2 
 
 ' 3^3 1 3 
 
 Shipped for Jamaica goods to the value of 699 1 6 5 
 
 I. s. d. L s. d. 
 
 trig. : o 6 9 :: 47f g. : 161 6 o^ 
 
 Ificw. : 3 15 7 :: 2-hd, each 7CW. 3qr. 151b. : S04 9 Jt-i^ 
 
 Received frcm y<7'/rff;V^ - - - 9^5^52 A 
 
 699 16 5 
 
 Balance 265 18 9 v'j
 
 Part I. The Rule of Three Direct. 17 
 
 (35.) 
 Ifiw. : 32I. 15s. :: 52 w. : 1703I. yearly expences. 
 3780 -T- 9 = 420I. land-tax. 
 
 3780 — 1703 -f- 420 divided by 20 = 82I. 17s. charit. dona. 
 If I d. : il. IIS. 6d. : : 365 d. : 574I. 17s. 6d. pock. ex. 
 
 Whole expences 2780!. 14s. 6d. this de- 
 duced from 3780!, leaves 999I, 5s. 6d. he lays up at 
 the year's end. 
 
 (36.) 
 s. d. g. I. s. d. t. hd. g. 
 
 If 3 7 : I :: 571 i 8 : 12 2 37|| quan. of wine bought 
 If 7 6 : I :: 419 no: 41 47^^ quantity of wine fold 
 
 8 o S'^zl^T quality of wine loft. 
 
 (37-) 
 cwt. 1. s. d. cwt. I. s. d. 
 
 If I : 14 19 7 : : 22i : 337 o 7^ 
 
 90 
 
 cwt. Is, d. cwt. 
 
 If I : 12 17 6 ; : 17-;- 
 
 ;f-427 
 
 7^ 
 
 : 222 
 90 
 
 I 10^ 
 
 £-312 
 
 I lOj 
 
 cwt. 
 If22i- : 
 
 If 17;- : 
 
 1. s. d. oz. 
 
 427 7I : : I 
 312 I o| : : I 
 
 d. 
 
 9I. 675 T 
 
 s. d. 
 If 5 6 : 
 
 (38.) 
 g. 1. s. d. 
 ; I : : 41 14 6 : 
 
 gall. 
 
 1 5 i-f\ of rum and water. 
 84 of rum. 
 
 ^Vtt of water put in. 
 
 C )
 
 I 5 1 K V 1 » £ E P R O F O R T 1 O v. 
 
 (39-) 
 guin. 
 If 47p. : 21 X 2 : : 62 p. : 55^9 guinea! worth at 17I 
 guineas per hhd. 
 
 guin. 
 I^ 55tU- : i7t X 2 : : 65g. : 43I. 2». 3id.i°. Anf. 
 
 (40.) 
 
 If i7cwt. 3qr. lolb. : i6cwt. 141b. :: ilb. : 140Z. 
 7/,/^5^dr. which fubtraded from ilb. leave loz. SJlf f dr. 
 loit in every pound. 
 
 If lib. :'Sid.-f i^d. :: I7cwt. 3 qr. lolb. : 81I. 3s. 4id. 
 the tobacco ftood him in. 
 
 If i6cwr. 14.1b. : 81I. 3s.41d.-j- lol. los. :: ilb. : is.cd. 
 -Jiilhemulllellitpsrlb. 
 
 INVERSE PROPORTION. 
 
 (2.) 
 
 If 20m. ; 6d. : : icm. : i2d. 
 
 (3.) 
 If2S. : Soz. :: 2S. 6d. : 6oz. 6]^ dr. 
 
 (+■) 
 
 If 3 m. : i8oof. : : i m. : ^400 f. 
 
 Then 5400 — 1 8co =: 3600 foldiers. Anfwer. 
 
 If I yd. ^qr. : jyds. 2qr. :: :Jyd. : 24iyds. 
 Then a+^yds. + :| yd. = 25iyds. Anfwer. 
 
 (6.) 
 If 20cl. : 12 m. :: 150!. : 16 m. 
 
 (7-) 
 If 22 yds. : 220 yds. :: 40yds. : 121yds. 
 
 If 1 m. : 72cm. :: 5 ra. : 144 mCD, which dedud^ed 
 from 720, leaves 576 men, Anfwer,
 
 Part I. T H E R u L E OF F I V s. 19 
 
 (9) 
 
 If (^ox. ; 7 colts. : : 2 ox. : 5-f colts. 
 
 If 7 colts. ; 87 d. :: 5-^ colts. : 105 days. Anfwer. 
 
 (10.) 
 yds. 
 If i;Jyd. : 1000X2!: :: |yd. : 4166yds. 2* qrs. 
 
 (II.) 
 
 From II Dec. 1786, to May 10, 1787, are i^o days. 
 
 From Sept. 5, 1788, to Chriftmas, 1789, are 478 days. 
 
 If 91 g. : 150 d. . : 661, 13s. 4d. : 214^! days. 
 478 — 2i4|§ = 263^Vdays. .1 
 
 If 661. 15s. 4d. : 2633-'od.. :: 4ol.^^438|-J days, hence 
 A is the perfon obliged , and mud lend Bij.ol. for438i-5-days. 
 
 (12.) 
 
 IfiSIb. : 100 ft. :: z+lb. : 75 ft. 
 
 The rule op FIVE, 
 
 (3-) 
 If 7 m. : 1 2d. : 126 a. 7X12X72 = 6048 dividend. 
 
 16 m. : * : 72a. 16 X i zS = 2016 divifor. 
 Then 6048 -7- 2016 nj days. Anfwer, 
 
 (4-) 
 If4it, : iSm. : 15I. 12s. (=3123.) 4JxiSx42o=:34oio dividend. 
 
 * t. : 72 m. ; 21I. os. (—420s.) 72 x 312 — 22464 divifor. 
 Then 34020 divided by 22464— I ^i^^ll ton. rr It. locwt. iqr. 4lb.-|-'^ 
 
 (;•) 
 
 Ifiool. : 12 m. : 4I. 100 X 12 X 20 = 24000 dividend. 
 * : 19m. : 20I. 19 X 4 = 76 divifor. 
 Then 24000—76=315^51. = 315!. 15s. 9id.-j|- Anfwer, 
 
 (6.) 
 1 1 cwt. 2 qr. = 1 288 lb. 15 cvvt. I qr. 22 lb. = 1730 lb. 
 and 61. 14s. 2d. = i6i6d. 
 If r 2881b. : 150 m. : i6i6d. 
 17301b. : 64m. : *d. 
 1730 X 64 X 1616 z= 178923520 dividend. 
 1288 X 150 1= 193200 divifor. 
 Thenr78923520-rj93200=926TVAV=3l»i7S'2iVTV'Anf
 
 20 Uniffrsal Proportiok, 
 
 (7-) 
 
 IfiSySfol. : 336 d. : 702 qrs. a2 536 x 112x702=: 1771S70464 div, 
 
 22536 fol. : ii2d. : *4rs. 1878x336 iz 63100S divifor. 
 Then 1771870464-a- 63 1008 :r 2808 fcldiers. Anfwcr. 
 
 100!. — 240c d 144I. 14s. 9d. — 34737 d. 
 If 2400 d. : 365 d. : 5I. 34737 x 495 X 5 r: 859740-5 dividend. 
 34737 : 495 d. : *1. 2400 X 365 rr: 8760)000 divifor. 
 Then S5974075-T-876C0001Z97 j^£^.7^1. - 9I. i6s. 3id.-f|f Anf. 
 
 (9.) 
 
 Ifiztaylors : 7 d. : 13 fuits. 
 * taylors : 19 d. : 494 faits. 
 12 X 7 X 494 =: 41496 dividend. 
 
 19 X 13 = 247 divifor. 
 Then 41496 -r- 247 = 168 taylors. Anfwer. 
 
 (ic.) 
 
 If ICO m. : 2s. 6d, : 20I. 
 
 * m. : IS. 9d. : 7I. 
 100 X 30 X 7 = 21000 dividend. 
 
 21 X 20 r:: 420 divifor. 
 Then 210CO — 420 zz 50 men, Anfwer. 
 
 UNIVERSAL PROPORTION. 
 
 Ifi261K : ioorr..Vs\^ 6s. 126 X 100X21 = 264600 dividerul 
 *lb. : 750 m. rN 21s. 750 X 6 = 4500 divifor. 
 
 Then 264600 -^ 4500 = 5S4lb. Anfwer. 
 (3.) 
 
 feive dtviillhd. 
 
 If24mea. : 3s.4d,Ly(i6m. t 6d. 24x40 (=3^- 4«I0 X48X4=i8432o 
 * mea..:2s.8d.^^48m. : 4d. 32(^=23. 8d.) X i6x6=::307zdivifo^> 
 • ferve 
 Then 18432c -7- 3072 =60 mcafurcs, Anfwer.
 
 Part I. Universal Proportion. 21 
 
 (4-) 
 
 eat 
 If 3600m. : 35 d. : 240Z. k^ I quantity. 
 * m. : 43 d. : 140Z. rS 2 quantity. 
 eat 
 3600 X 3^ X 24 X 2=:6o48ooodividend, 45 X r4~63odivifor 
 'ihen 6048000 -f- 630 zz 9600 men. Anfwer. 
 
 If36coom. : 24 oz. : 35d. k>^ i quantity of bread. 
 4800m. : * oz. : 45 d. rN i (the fame) quantity. 
 eat 
 -3600 X 24 X 3 5 =3024000 div. 4800 X 4? == 2 1 6000 divifor. 
 Then 3024000 -r 216000 z: 14 oz, Anfwcf. 
 
 (6.) 
 coft 
 If 150ft. : 3ft. : 40m. Kr>3l. 5:4X 8 X 25 X 3=32400 divid, 
 54 ft. ; 8ft. : 25m. rS*l. 150X3X40=18000 divifor, 
 coji 
 Then 32400 -r 18000 = i\^%\*zz il. i6s. Anfwer, 
 
 (7.) 
 
 If lool. : 365 d. y^ 5I. 
 1627I. 103.: zijd. ^'^ *\, 
 gain 
 32550 (=31627!. los.) X2I9X5 = 35642250 dividcnJ. 
 20C0 ( — locl.) X 365 =r 730000 divifor. 
 
 Then 3 5642250 -r 730000 = 48 ——-1. zz 48I. i6s. 6d. Anfwer. 
 
 73000 
 
 (S.) 
 
 3yv.l3. -^qr. 11 I5qrs. Ss. gd. n 105 d. 257yds. 2 qr. rr l03oqrs. 
 • c.Ji 
 1? 15 qrs. ^^^^ I05d. 1030x105 — 108150 dividend. 
 
 I03cf qrs. ^^S *d. Then 108150 -i- 15 .-=: 72iod. — 30I. 8:, 4d. 
 cofl 
 
 (90 
 
 drink 
 If 336 m. : 4d. : i pt. : 30]. k^ 20I. 
 ' 250 m. : *d. : i^:pt. : 24I. rS 500I. 
 
 drifik 
 336x4X30X500 zz 20160000 dividend. 
 2 50 X I i: X 24 X 20 =: 1 80000 divifor. 
 Then 20160000 ^ 1 80C00 zz 1 12 days. Anfwer.
 
 2 2 Universal P r o r o r t i o n» 
 
 (10.) 
 
 - (iig a trench 
 If^-^Sm. : 5d. : johrs. k^ 5 deg. : 70yds. : 3 w. : 2d. 
 s4om. : 9 d. : lahrs. j^^ 6 dcg. : * yds. ; 5 w, : 3d. 
 
 dig a trer.cb 
 24CX9X I2X 5X''CX3x"- — 544320CO dividend. 
 ■ 55<^X5Xiox6x5X3 = i5i2cooaivifor. 
 Then 5443204,0 -j- 1512CCO zr 36 yards. Anfwcr. 
 
 (II.) 
 huild 
 If 96m. : 48 d. k>( 1 wall. 
 *m. : 384 d. KS I wall. 
 huild 
 96 X 48 rr 4608 dividend. 
 Then 460S -T- 384 zz \i men. Anfwer. 
 
 (12.) 
 
 I cwt. = r 1 2 lb. il. 17s. 4d. = 448d, 
 ccji 
 If 1 12 lb. k^ 448d. 
 
 J lb. rS * 
 
 cofi 
 448 -^- 1 12 z: 4d. Anfwer. 
 
 (I3-) 
 hatter domon 
 Ifi2C. : iSpo. : 1 hr. k^ i caftle. 
 9C. : 24 po. : *hr. rN i caftle. 
 halter doixin 
 12x18 = 216 dividend, 9 x 24 — 216 divifor. 
 
 Hence tl:e nine 24 pounders would batter down the caftle 
 in the fame time. 
 
 (140 
 eat 
 If 1 2 ox. : 4 w. k^ 3 J a. 
 21 ox. : 9 w. ri * a. 
 eat 
 21 X 9 X 3-] =630 dividend, 12 X 4 = 48 divifor. 
 
 Then 630 -^-48 = 13^ acres, 21 oxen will eat in 9 weeks, 
 admitting the grafs does not grow ; but by the queftion 21 
 oxen will eat only 10 acres in 9 weeks; therefore, in 
 (9 — 4 = ) 5 weeks, 10 acres by the growth of the grafs are 
 ecjuivalcnt to 13I acres; hence 131 — 10 :=l ^\ acres, the
 
 Parti. Reduction op Vulgar Fractions. 
 
 increafe upon lo acres in 5- weeks. Now the real quantity 
 which 12 oxen eat in 4 weeks without any increafe was {^ 
 acres ; hence the /mvc" of increafe upon 24 acres is only 
 
 If 10 a. 
 24 a. 
 
 (18 — 4zr)i4weeks. Again, 
 h/crea/e 
 pv. k^ 3ia. 
 4W. rS * a. 
 
 24 X 1 4 X 3i = 1 050 dividend, 10X5 — 50 divifor. 
 
 Then 1050 -^- 50 =1 21 acres, the increafe on 24 acres in 
 14 weeks; hence 24 -f- 2: =45 acres, the whole quantity 
 of grafs which is to feed the required number of oxen 18 
 weeks. Therefore, 
 
 eat 
 If 12 ox. : 4 w. k^ 3|a. 
 * ox. : 18 w. rS 45 a. 
 eai 
 12 X 4 X 45 rr 2160 dividend, i8 X 33-21:60 divifor. 
 Then 2i6o-r-6o = 36 oxen. Anfvver. 
 
 Note. This queftion may be folved by feveral different methods, but 
 I apprehend the above Iblution to be the moft intelligible that can be 
 ^iven, for thofe who uuderftaiid Arithmetic only. 
 
 REDUCTION OF VULGAR FRACTIONS, 
 
 (^0 
 
 (3-) 
 (4.) 
 (5.) 
 
 (6.) 
 
 (7.) 
 (8.) 
 
 (9-) 
 (10.) 
 
 Common meafure 3 
 Common meafure 6 
 Common meafure 9 
 Common meafure 375 
 . ai6 ^ 
 
 Anfwer. 
 
 1080 540* 
 410 41 
 
 Anfwer. 
 
 10)-^ — ZZ-^. Anfwer. 
 '510 51 
 
 5) rr — . Anfw. 
 
 ^745 349 
 
 i?) zz--^, Anfw. 
 
 9747 361 
 
 •^'7143 2381 
 T26 
 (la.) 14 ir — . Anlvv'cr. 
 9 
 
 (13.) 15 = ^^' Anfwer. 
 
 3094 
 (14.) 34ir . Anfwert 
 
 3 
 
 (15.) ^-5^ = -^ 
 
 ro3 
 
 Anfwer. 
 
 (16.) 149 J — . Anfwer.
 
 24 Reduction op Vulgar Fractions. 
 
 94^ 375'^- -375+ 94 _o7:^^ Anfwer. 
 y9 99 99 
 
 (i8.) 
 
 543 _ i74;4^^ccc — 174^4 + '-^1 _ 174938^^49 .., 
 99999 5^999 99999 
 
 (190 
 
 473... ^- 9. 
 
 (20.) 
 
 1 6106 - - 
 1 - 80 ^ = . Ani wer. 
 
 ' ^' 9 
 
 (21.) 
 ^ = I. Therefore 7 7 -J = 78. Anf^ver. 
 
 (22.) 
 
 97vV Example is ivorked* 
 
 (23-) 
 ^5 )4790 I ^9^U = '9^1* Anfwer. 
 
 (24.) 
 )o8 } 1 5 12 { 14. Anuver. 
 
 999)37S9+i( 376 -9W Anfwer. 
 
 (26.) 
 349 ) 374517+ ( ^^ll^i^y' Anfwer. 
 
 (27.) 
 
 T'i'/V Example is fJJorheJ, 
 
 * Should the reader not immediately comprehend the manner in wliich 
 this and the following Example are worked, he may confult Note 6th, 
 p=gc 40th, of the Complete PraQlcal Arithmetician.
 
 Part I. Reduction- of Vulgar Fractions. 25 
 
 (28.} 
 
 (29.) 
 ^t_ = --S.= -i^^^ := ^\ =-^^-. Anrwcr. 
 
 (50.) 
 
 ; " I 5 5 
 
 (31.) 
 
 1789 'V^ SHXi7^9 9^9546 
 
 (32.) 
 394^^ _ 'IV! __^9^!i>l7i9_ 2 80985^0 
 
 894)!^-"' VtV 643333x^9^^89967' 
 
 (33-) 
 TZv'j Ex a '■/■pic is njoorkciL 
 
 (3+-) 
 
 4 9 lOI 13 4;. 2 ;< 101X13 IC5C4 
 
 41 ^ 41 X 4^ ''41 l6Sr ^ ^ 
 
 -— X— X~T. X -'^rr : ~ — -— . Anf. 
 
 no 19 /p'8 7 110x19X12X7 175560 
 12 
 
 (36.) 
 
 l^X ^ y, -J-^- X ^^z^ "9 X j; X3 X 7 _ 50J 
 S S^ ^c<X- 1 8x43 
 
 43 
 
 D
 
 26 Reduction of Vulgar Fraction's. 
 
 (37-) 
 
 ^Jx^x ^'-^ X -^^iz iiilIii-LiH!7= 970785: p^^^^ 
 1 0^ /3^ ^ 7x19x12 2261 
 
 ^9 34 
 17 
 
 (38.) 
 54_-^055: 
 
 37I 150+ 
 
 59 /^^ ^^4 I 59X376 22184 
 
 376 
 
 (39-) 
 TT-zV ExafHple is ixjorkeJ, 
 
 (40. 
 
 1 X 9 X 8 X 16 = 1152! 
 
 4 X 5 X 8 X 16 =z 2960 Kt 
 
 5X5X9X16ZZ 3600 f 
 
 II X 8 X 9 X 5 = 3960 J 
 
 ew numerators. 
 
 CX9X8X16ZZ 5760 Common denominatorji 
 1 152 2560 3600 3960 
 The Fradions are • — 7-, —r-y TTTl^ T^-^ 
 5760 5760 5760 5,00 
 
 (41O 
 
 238+8 4. 7 ^ zb 
 
 5 9 
 
 HcHce^he new Fraaions are ^, and --.
 
 Part I. Reduction of Vulgar Fractions, 
 
 r^ — ^i ^5 — 26 y.^- ~r 3 7 AS 
 
 a * 
 
 8x7X9X8—. 4032. Common denominator. 
 
 8)4032 7)4C52 9)4032 8(4032 
 
 504, 576, 448, 504, 
 
 41 26 37 53 
 
 20664 num. 14976 num. 16576 num. 26712 num.. 
 
 « „. %co6a. 1AQ76 16576 , 26712 
 
 Hence the Jradtioas are ^> < — - — , — — » and ^ . 
 
 4032 4032 4032 4032 
 
 (43.) 
 
 J. —A. 11j —III fi_ _il^ i 5J" _^ 
 
 15 "105' 94 ""470* 89|~7i7' 5f~~93* 
 
 105 . 470 . 717 . 93 denominators. 
 
 94 • 717 • 93 
 
 94 • 239 . 3; 
 
 5X3x7X94x239x311^73126830 the leafl: common denominator* 
 105)73126830 470)73126830 717)73126830 93)73126830 
 
 696446 1555S9 101990 786310 
 
 4 171 512 128 
 
 278 5784 num. 26605719 num. 5221S880 num. 100647680 num 
 
 Hence the new Fraftions in their lowefl: terms are, 
 
 7857S4 26605719 52218880 1006476.80 
 
 ■ and 
 
 73126830 73126830' 73126830 73126830 
 
 Had I not reduced thefe Fradlions, the refult of each numerator, Scc» 
 would have been 45 times greater than it is. 
 
 * To find the Icaft common multiple of two or more numbers, or to 
 find the leaft number that can be divided by two or mo.e given numbers 
 without a remainder. 
 
 Rule. Divide the given numbers by any number that will divide two 
 or more of them v/ithout a remainder, and fet the quotients, together 
 with the undivided numbers, in a line underneath j divide this fecond 
 line as before, ahd {0 on till there are no two numbers that can be divided, 
 then the continual products of the divifois and quotients will give the 
 multiple required* 
 
 D a x.
 
 28 
 
 Reduction of Vulgar Fractions. 
 
 (4+0 
 3X3X7X2 = 1 26 
 2X5X7x2 = 140 
 1X5X3x2= 30 
 1x7x3X5 = 105 J 
 
 New numerators. 
 
 5X3X7X2 = 210 Common denominator. 
 
 "5 
 
 Hence t!ie Fraftions are — -, — - 
 
 210 2IO 
 
 - — , and 
 10 210 
 
 (45-) 
 
 4 9 _36 8 _2SS 5 7 _35 8 _28o_ 
 
 7 9 03 8 504. 9 7 63 8 504 
 
 ._X^:rii25 7^63^44^ 524^9^76 Hence th. 
 
 S 03 504 8 03 504' ^ 504 504 
 
 Frafljons in their loweft terms are 
 
 0576 
 
 7^4" 
 
 1S9 2S8 2S0 i^4i 
 5C4' 5^* 5-4' 5^4* 
 
 {46.) 
 This Example is nvorhd, 
 
 (47-) 
 This Ex an: tic is <v:or'ked. 
 
 ExAMTLE. Find the leaft number that can be divided by the n'fie 
 • ,lti without a remainder. 
 
 
 I 
 
 - 
 
 3 
 
 • -4 
 
 .5.6. 
 
 7 
 
 • S 
 
 • 9 
 
 ' 
 
 ■ I 
 
 3 
 
 2 
 
 • 5 • 3 • 
 
 - 
 
 -t 
 
 • 9 
 
 I 
 
 • I 
 
 5 
 
 I 
 
 5 • 3 • 
 
 7 
 
 * ^ 
 
 • 9 
 
 
 
 • I 
 
 I 
 
 ^ 
 
 5 • I . 
 
 7 
 
 • 2 
 
 • 3 
 
 Then 2X2X 3X5 X7X 2#< 3 
 
 Anfw, 
 
 I have given this Ru!e and Example hc:e, becaufe th<gr maybe occa- 
 fionaMy ufeful in reducing fraf^.'ons to a coniinon denominatcr (as in 
 Example 43d). For by this Rule the ieaft common denominator may be 
 i MvA\ and thea the fevcral numerators, by the latter plrt cf the Ruie 
 tj Prop. Stn, page 29th, in the Ctirficte PraEilcal ArittiKetician.
 
 Parti. RfiDUCTioN of Vulgar Fractions. 
 
 29 
 
 Num. Answers. 
 
 (48.) 6H.-!- 
 
 (49.) 70Z. i^dr. 
 {^o } 70Z. 4d'vvt. 
 (51.) im. 6 f. 16 p. 
 (^2.) I r. 20 p. 
 (93.) 8yds. I qr. ifn. 
 
 ()4-) 3^1^^- 7 gall. 
 (55.) 29 gall. iqt. i/-j-pt. 
 
 Num. Ans\^ers, 
 
 (5:6.] 14 weeks. 
 
 (5:7.) 20 bufhels. 
 
 (58.) 5s. 4d. 
 
 (99.) 6cwt. 3qr. 61b. 
 
 (60.) 10 ft. 216 in, 
 
 (61.) 30 gallons. 
 
 (62.) This Example is njoorhed^ 
 
 i^S') 
 
 15s. lid. =r iQidr Then 
 
 191 
 
 = — ^I. Anfwer. 
 
 240 
 
 12 X 20 
 (64.) 
 
 5^d. =z ^M. Then -^ =i:|s. = i^s. Anfwer. 
 -'^ 9x12 108 54 
 
 I cwt. 2 qr. 61b. 3 oz. 8| dr. = 44600I z= *°^/°8 jj.^ 
 
 401408 
 
 4014.08 14 
 
 ^^^ — -^— —— cwu 
 
 9X16x10x28X4 25S048 9 
 (66.) 
 
 ^ ^^ ^ 5x16x16 
 
 6999 1- I8IQ87 
 40 5760 
 
 416 
 
 izSo 
 
 40 
 
 ■lb. avoird. 
 
 Hence -^ x 
 
 ^ ,, -lb. tro5'. Anfwer, 
 460800 
 
 (^7-) 
 
 3qr. 3^n. = 15^ n. 
 
 Tten — i^ zz-if^ -^ cUs Eng. Anfwer. 
 9 A 4 A J I Eg 45 ° 
 
 (68.) 
 .47 d. . jh. = 3;4.3 h. Then ^iil ~'J^ yrs. 
 
 8766 2920 
 
 155. 5-|d, =: 579 halfpence. Then 
 D 1 
 
 AnAv, 
 
 X I 2 X 20 400
 
 Reduction of \''uLGiR Fractions. 
 
 (70.) 
 V of a groat. Anfwer. 
 
 (-■•) 
 
 locwt. 1 qr. 12 lb. rr 296960 drams. 
 Sc'.vt. I qr. 25 lb. i oz. 7-A-dr. rz 24.2967 ^\ drams. 
 
 242967YV 2672640 9 
 
 296960 3266560 II 
 
 1 , 128 ^, 128 128 4 , . 
 
 4b. 2fp. = p. Then ir — =-qrs. Anf. 
 
 (73.) V'^ yd. Anfwer. I {-j^.) This Example is -ivirkcd. 
 
 {74.} -ir| a. Anfwer. j {'-jS.) This Example is ^Morked, 
 
 (77-) 
 
 — X— X-r, = = — -X Anfwer. 
 
 7 12 /p{ 7x12x4 336 
 
 4 
 
 (7?.) 
 
 — ;- X — X-- rz — of apennv. Anfwer. 
 £?;'P 11 + 
 4 
 
 (79-) 
 
 ^Tds. troy. Anfwer. 
 
 6 ii:5 "^12- 
 
 I I 
 
 X4X12 2SS 
 
 4 
 
 
 
 (So.) 
 
 I xi U 4 ^ 
 
 X— X— =~awt. Amwcr. 
 
 ;<p.o' I I 5 
 
 -^ X — X— = =: cwt. Anfwer. 
 
 5 2d 4 5 X 23 140 
 
 ^ X^ X^=^lb. Anfwer 
 ^l/^ I I 7 
 
 7
 
 Parti. Addition of Vulgar Fractions. 31 
 
 (33.) ' > 
 
 7 7 24 60 6o ^233600. , . ^ 
 
 -^ X-^ X— X-- X—zz:^^ feconds. Anfwcr. 
 
 71 I I 1 I 71 - 
 
 (84.) 
 
 — X -7- zi: of a hhd. Anfvver. 
 
 13 63 bi9 
 
 ADDITION OF VULGAR FRACTIONS, 
 
 T'Z/y Example is <worked, 
 
 587 280 2S0 
 
 (3.} 
 
 5 II 55 -8 8 
 
 Then -L +4|. ^2389^ i£l, ^^^^^ 
 55 8 4+0 44.0 
 
 (4.) 
 
 ^8 "8' 94.7_.~'ic4i' 314.1 "14157* ^" 9 "^ S' "^ 
 
 405 2140 __ 43^^^''^4S 4- ^9065^^10 + 186872404- i;94o64o ^ 
 1041 14157 ~ 3S'^'y9333 "" 
 
 i> -•« . Anfwer, 
 
 39299b3z 
 
 4of-^=-i., JorIl=^L, 4of^=^-l. 
 
 £^ 5 15 4 I 4 $ I 2 
 3 2 
 
 2 4 15 4 ^5 60 60
 
 Addition of Vulgar Fractioj:*. 
 
 (6.) 
 
 9 ^ i ^UL —112] 
 ffi ^11 "* 8 -176- 
 
 2 
 
 '76 
 
 {7-) 
 
 5 176 bSo Sto 
 
 s. d. 
 
 41. = II ii-/f 
 
 is. =: o 4T 
 
 -id. — o ot-il 
 
 Sum 1 1 6i;-i\ 
 
 (8.) 
 
 I. s. d. 
 
 ^ of ll. lOS. s= I 2 6 
 
 ~ of 5I. los. = 017 6 
 3^0 of icoguin. = 220 
 
 Sum 420 
 
 (9-) 
 
 oz. dut. 
 
 f lb. tr. = 2 8 
 ^ oz. tr. zz o 2 J 
 
 Sum 2 lOv- 
 
 (10.) 
 
 5 I 80—335 p«Tf — 
 
 9cwt. 1 qr. 6 J lb. AnC 
 
 -^ X-^ =-^ells Eng. Jj x-^=: -ells Eng. 
 
 2 3 
 
 Then -^ -f - =-y- ells Ene. = 2 ells Eng. 4—- qr; 
 236" 6 
 
 (12.) 
 
 I X T7oo = 450 mile. } X ^,Vo = Tjy^c mile. 
 
 -A + :i':c + izhc = llMlmile = 3^"^- 25?. 37yo->-ds. 
 
 rof¥=vf<- = 
 
 r. p. ft. in. 
 2 20 
 
 3 
 7 
 
 Sum, 
 
 2 20 U yi^T
 
 Fart I. Subtraction of Vulgar Fractions. ^j 
 
 (hO 
 
 Jx-^-3 hhds. Then 3 + - = 3-^ hhd. =: 
 4 ' 5 5 
 
 226— gallons. Anfvver. 
 
 4 
 
 ^-^ X— = 20 budi. to which add -^ b. and the anfw^r i 
 ' 7 
 
 20 -=^ bufliels* 
 7 
 
 (16.) 
 
 1 7 _ 7 J ' ' — ' J 
 4 I 4.5 24 ~" 120 ' 
 
 J. 4- -j — ~-i = 2d, 2 hrs. Anfwer. 
 
 4. 120 3 12 
 
 (17.) 
 
 JLx-ixiLii2d.-f x-^=id. 
 
 ^41 4 8 gf 24 
 
 852 //^ 6^0 
 8 
 
 21Q ? 273 106339 1 iQi . /. 
 
 4 24 640 1920 "^80 
 
 SUBTRACTION of VUL.GAR 
 FRACTIONS. 
 
 (I.) 
 
 T'^is Example is ^doorkcdt 
 (2.) 
 
 ■^-^4:=f' Anfwer.
 
 34 Subtraction of Vulgar Fractions. 
 (3.) 
 
 3 8 24* ^ 8 8 "-24* 
 
 Then -7-^^=12 =3!. Anfwer. 
 24 24 24 24 
 
 97 776* H5tt 799<^* 
 
 Theni^-ii^=i±llii. AnAver. 
 776 7990 3100120 
 
 (J-) (6.) 
 
 II 
 
 ^8 ^^8 ^^4 I 713 713 
 
 (7-) 
 
 S6^ = 3^4— • Anfweib 
 
 15 ^15 
 
 (8.) 
 
 m, 288 2 C18 ? A r 
 
 Then — -^ — = cn-^. Anfwer* 
 
 5 45 9 9 
 
 (9-) 
 
 -J- zz—. Anfwer. 
 
 8 5 40 
 
 (10.) 
 
 ^ 7-5 — ' 7tt 
 
 ^i — _ I o 
 
 ;t — 7.T 
 
 9i-*- Anfwer* 
 
 lof^zzl, Jof4:=i-^ 
 II 9 99 5 8 40 
 
 rru 5 21 33761 „2o8r . - 
 
 Then -^- + 9 =^^i-^ == 8 — -- . Anfwer. 
 
 99 40 39^0 3960
 
 Parti. Subtraction OF V^ULGAR Fractions, ^p 
 
 (12.) 
 
 s. d. 
 ■II ~ 15 o 
 
 4s. - 
 
 o SH 
 
 Anfwer 14 3^-^ 
 
 oz. dvvt. 
 4 lb. rz 10 o 
 1 oz. n — 1 2 I- 
 
 Anfwcr 9 - ^- 
 
 (14.) 
 
 — of -^ =z— lb. — ton. = 7 cwt, 2 qr. from which take !- lb. 
 ;S 4 2 S 
 
 2 
 the remainder will be 7 cwt. i qr. 27-i-lb. Anfwer. 
 
 Cv) 
 
 2 10 
 
 pt. 
 
 gal. pt. 
 |hhd. = 2S 1} 
 i of -"- ~ -5- 
 
 Anfwef 25 I 
 
 (.6.) 
 
 i X^ =-2 miles. -2 _ J =11 = , 1 miles. Anf^r. 
 J 1 5 5 8 40 40 
 
 (•7-) 
 
 -I of ~I— =— !^ days, -^of ^ of-r-, =z^ days. 
 
 ^ 4 12 ^ 7 16 /^ 896 ^ 
 
 r-T = — 7T^ = 202—7—0. =: 202 days, 
 
 12 896 2638 2688 
 
 21 hrs. 45 m. 32y fee.
 
 ^6 Multiplication of Vulgar Fractions. 
 
 MULTIPLICATION of VULGAR 
 FRACTIONS. 
 
 T'.Js JixaTnjle is ivcrhd^ ' 
 
 (2.) 
 
 ~ X — = — . Ajilvver. 
 
 7 A^ 35 
 5 
 
 ^ xj^=^ - 252 4. AnAvcr. 
 I ^^ 6 ^6 
 
 '6 
 
 111 il =1jS —111 — j^l Anfwer 
 27 
 
 '31 3 13 
 (6.) 
 
 Anfwer, 
 
 J X^ X-^ X^; xi- xi =^. Anfwer. 
 $ 7 ^ / II I 308 
 
 4 
 
 (7.) 
 4__._3. 7j'-_!i.. 35s __ 424? ^" 
 si ""4* 15 ""25' i29-j\ 15512* 
 
 ^hen-lx^X-^=:-^. Anfwer. 
 ? 3878 
 
 -^ X 1- X^' x^^ x^,^ =ii^^= 62-li. Anfw 
 5 II I ^^ $ 220 220 
 
 4 
 
 er.
 
 Part I. Division of Vulgar Fractions. : 
 
 (9.) 
 
 7ft. 9 in. = 2^ ft. 3 ft. II in, = ^, eft. 3 in. = — . 
 4 II ^ -^ 4 
 
 31 xi-7 x^'zzl^ = i59r'- Anfwer. 
 
 4 
 
 (10.) 
 
 12 ft. gin. = ~, eft. 7 in. = — . 
 ^ 4 -' ' 12 
 
 il^p-im=.,l. Anfwer. 
 4 - ^/ 16 '16 
 
 4 
 
 (ir.) 
 
 r . • 160 „ . 30 
 
 1 7 ft. q 1^ in. = . Q ft. Q in. = ^. 
 
 ' ^' 9 4 
 
 if xi'=i^=i73irt. Anfwer. 
 3 
 
 DIVISION OF VULGAR FRACTIONS, 
 
 (I.) 
 
 T/v's Example is ivorkcdt 
 
 (2.) 
 
 16 7 112 cc . _ 
 
 ^9 3 57 57 
 
 (3.) 
 3 
 
 CJ ; X -; = - . Anfwer. 
 
 ~ ; X — =: — -> Anfwer. 
 
 37 7 259 
 
 E
 
 3.3 Division of Vulcax FiACTiow*. 
 
 I?; X ^=z^ = .-. Anfwer. 
 
 (6.) 
 
 2 
 
 7/' 
 
 X 1^=^. 
 
 7 7 
 
 An{v,tT 
 (7.) 
 
 • 
 
 ■^X-^;xi: 
 7 9 $ 
 
 16 F28 . . 
 
 X— =-T~' Anfwer. 
 
 
 
 3 
 
 
 
 
 (8.) 
 
 
 20 
 
 
 
 
 9 ' 
 
 8 II 
 
 X -: X — 
 
 ^ 3 
 
 __i76o 
 (9.) 
 
 = 6s-^^ Ar 
 
 
 "! = ■? 
 
 1 54I 
 ' 93A 
 
 «.4763 
 8224* 
 
 239 
 
 7 ' 
 
 8224 
 
 X — z^ — 
 4763 
 
 1965536 
 33341 
 
 (10.) 
 
 - = 58^'^^'. 
 33341 
 
 ?'A 
 
 _ S6z 
 
 7' _ 
 
 568 
 
 95 ^045 H9f ^^95 
 
 281 239 
 
 Then i^i . yr^M. -tll31 - , ^!?I. Anfwer 
 
 209 284 
 
 (II.) 
 
 r^Jx^i'; xix^x-l=i^=,-il. ;tnr. 
 $ ^ gf ' I I 19 114 114 
 
 ^ 3 
 
 3
 
 Parti, The Rule of Three Direct in, &c, fg 
 
 (12.) 
 21 
 
 tc-i z=J, Tlien &; X -^ = 21. Anfwer. 
 4 4 4 3 
 
 (13.) 
 
 •^ ; X — ^ = — ^. ' Anfwer. 
 12 108 1296 
 
 (14.) 
 
 -~ ; X ~ X — X = -%-. Anfwer. 
 
 RULE OF THREE DIRECT im 
 VULGAR FRACTIONS. 
 
 Tlis Example is <wdrhd* 
 
 yds. 
 If I'V X I J 'OS* 2d.-TV : : I : 88. Anfwer. 
 
 (3.) 
 If|lb. : 7s. 9d. : : ^fMb. ; j^l. 5s. 6|d.-f, Anfwer, 
 
 (4.) 
 
 If A of |xV X| : 2Vt1. : : |x 'i^ ; 68lr 13s. lo^d.^. 
 Anfwer. 
 
 {$■) 
 
 If -fell. : 5s. 3fd. : : 5t X 35^ : J2l. os. 4|d.-t. Anf. 
 
 (6.) 
 quarts. - 
 If iqt. : 3s.3{d. : : i4fVX-ioo8 : 2405I. iis. iiid.-s\* 
 Anfwer, 
 
 E 2
 
 40 The Rule of Three Inverse, and 
 
 (7.) 
 
 ells. I. 
 
 IfjXjXl : |X| :: '^^dls. ; 199I. 153. 6id.f Anf. 
 
 (8.) 
 If I lb. : *jld. :: II hhds, each 4cwt. 3qr. 15-Jlb. ; 194L 
 los. 53.'^d. Anfwer. 
 
 The rule of THREE INVERSE 
 IN VULGAR FRACTIONS. 
 
 This Example is ^worked* 
 
 (2.) 
 U j^ift.br. : 27|ft. Ion. : : ^ft. br. : |73if. =: 
 1 9 1 yds. i ft. Anfwer, 
 
 M 
 
 If 4s. :d. : 57|yds. :; js. 8d. : i^j^^^yis. Anfwer. 
 
 (4-) 
 
 If I J yd. : n^ydr. :: lyd. : 39iyds, Anfwer. 
 
 (5-) 
 
 ]f jjcwt. : ijtm. :: 9i-cwt. : 6-y'^ m. Anfwer. 
 
 The RULE of FIVE in VULGAR 
 FRACTIONS. 
 
 (1.) 
 
 This Example is fworked, 
 
 (2.) 
 gain 
 If I col. : 1 vr. K^ 4I-I. 
 4S011. : 7iyr. /\ * 1. 
 gam 
 
 12^ X^Jx -2,X-L =m^l. = ,7,I.2S.„ld..^. Anf 
 442 100 3200^ ' *
 
 F^artl. RuLe OF Five in Vulgar Fragtioks, 41 
 
 (3-) 
 
 tra'vel 
 IfiF. : 7|d. : lalTirs. k^ *|+ m. 
 I F. : *d. : lojhrs. rN i47|m. 
 trauel 
 
 13 X ^^ X 1^7. ^i. X _L= ^ = 4lif^d. Ann 
 ^2 8 32 294 150528 i^o^-21% 
 
 equal 
 jf5o_oo^gals . i_5fj^ . 2iin. k> I 1 quantity 
 
 * deals : V f^* • i ^"- *^ i J wood. 
 
 of 
 
 200 
 
 c'0'p;:^ K 7 5^ 4 6ccoo „ ,^ . - 
 
 V^T^7' X ^^X~ = -^=857^?ft. Anf. 
 
 (5.) 
 
 coji 
 If 1 3I ells : f yd. k> 5I guin, 
 
 33*: X| ells ; I X|yd. rN *guin.- 
 
 cojl 
 
 -ii-X-^X-^X-^X-7; X -^ X-^ = -^guin, 1= 
 
 4 i 3 4 i 27 3 486 ^ 
 
 2 
 
 15I. 1 6s. o|d.-||, Anfwer. 
 
 (6 ) 
 
 'dig a tre7tch 
 lf*4«m. : 5id. : iihrs. k^ 7^. : 232iyd. :3jyd.:2ivd. 
 ym. : *d. ; 9^^^- ^ 4^1. : 337iyd. rsly^- = 3iyd. 
 dig a treJ2ch 
 
 X^ J^Zt 4 
 
 i^^X-rX— X — X^Xi3^x-^; X-7-. X — X 
 
 I i I \ i t i a ^ 
 
 E3
 
 4* Promiscuous Questions in 
 
 PROMISCUOUS QUESTIONS in 
 VULGAR FRACTIONS. 
 
 3ofi=-d. Theni-r5=i. Anfwer. 
 0^3 3*9 
 
 3 
 
 (2.) 
 
 s. d. 
 
 17 o 
 
 I of Vs. = 14 2 
 
 3 of Vs. = 129 
 
 jf . 2 511 This fum multi- 
 plied by 7 J gives 16I. 16s. 8Jd.-j. Anfwer. 
 
 (3.) 
 
 47A 
 141 = Hll 
 
 62t'^ Anfwer. 
 
 (4.) 
 1 1 200I. Anfwer. 
 
 i!£ 2I_ = i?l the part I have left. 
 
 16 2816 2816 '^ 
 
 80c 16000 2012CO, , ■ J 1 .V 
 
 2010 1 44 
 
 yaluc of the part I have left.
 
 Parti. Vulgar Fractioks. 
 
 (6.) 
 
 loa =11. io(L =2i-. 
 
 8 7 56 II 19 209 
 Then i|+li=^, and ii|5_ ±L= J^llii = 45 2211 
 
 56 1 56 56 209 XI7O4 11704 
 
 43 
 
 .Anf 
 
 As 6— d. » 
 2 
 
 i-td. 
 
 (7-) 
 
 work : ; id. ; — of the work A can do in x day. 
 13 
 
 work : : id. : — ■ 
 
 J? 
 
 ± X 'I 
 
 3 — d. : — work : : id. ; — 
 
 Then j- A + -1. — _Li of the work they all can do In i day 
 
 13 17 10 2210 ' 
 
 working together. 
 
 Tr '5^3 , 1 , J , 2210 687 , , . 
 
 If work : — day : : — work : — ZZ i days. Anf. 
 
 2210 I I 1523 1523 
 
 (8.) 
 
 In the fourth line of this Example, /or A, B, C, D, and E, In ii days, 
 read A, B, C, D, and E, in 12 days. 
 
 days, work dayi 
 
 19 
 
 13 : 
 
 IS .. — 
 
 l» : — : : X ; 
 
 I :-l 1 
 
 13 
 
 I : 
 
 Then, 
 
 "a+b+c+d+* 
 
 A + B + C+»+E 
 
 •* ' ■ 19 
 
 — yf^ A + B + * + D + E^ 
 A + * + C + D + E 
 
 14 : — : : j : — 
 
 J4 
 
 * +B+C + D + E 
 
 >
 
 44 
 
 Promiscuous Questions ii^ 
 
 Then — ^ H 1 +• — z; — ^— ^ part of the work they 
 
 13 15 la 19 14 311220 
 
 would do in 4 * days all working together, tonfequently they would do 
 
 1 109233 109233 
 
 — of ^ or -— - part of the work b 1 dav. 
 
 4 311220 1244880 
 
 109233 1 I 1244880 43317 
 
 As — ^-—- work : — day : : — work : -^^ = ii 
 
 1244880 I I 109233 109233 
 
 days, the time in which they would all finifh the work. 
 
 Again, 
 
 109233 
 
 20313 
 
 I244SS0 
 
 14 
 
 ~ 
 
 I244SS0 "^ 
 
 109233 
 
 1 
 
 - 
 
 43713 
 
 1244880 
 
 1244880 
 
 1C9233 
 
 1 
 
 12 
 
 = 
 
 5493 
 
 1244880 
 
 12448S0 
 
 109233 
 1244880 
 
 1 
 
 = 
 
 26241 
 1244880 
 
 109233 
 
 I 
 
 __ 
 
 13473 
 
 1244880 13 
 
 1244880 
 
 part of the work A could do In i day. 
 
 B 
 
 C 
 
 D ^ 
 
 E 
 
 Hence it appears that B would finifli the work the fooneft if left to 
 hixnfelf, the fra<aion oppofite B being the greateft. 
 
 • If the reader fliould not, at firft fight, be able to comprehend 
 whence thefe four days are got, he may caft his eye to the right hand 
 of the foregoing page, where he will ^txctivt four A^s, four Wsyfour C's, 
 four D's, iJidftur E's J confequently, they had each worked four da)'S. 
 
 (9-) 
 
 hrs. referv. hr. 
 As 6 : — : : I 
 
 8 : -^ : : 1 
 
 JO : 
 
 8 
 
 
 < B 
 C 
 
 Hence — + -^ + — = — of the refervoir will be filled in one 
 6 8 10 120 
 
 hour, if the filing cocks A, B, and C; are fet open together.
 
 Part I. 
 
 Vulgar Fractions. 
 
 Agaiil, 
 
 4i: 
 
 hrs. rei'erv. hr 
 
 As 9 : 
 
 13 : -- 
 
 £"2 1 Is 
 
 Hence — + — H — "^^ of *hc refervolr will be emptied i; 
 
 1287 
 tig cocks D, E, and F, are fet open* 
 
 9 I 
 
 2 hour, if the difcbarg 
 
 Then -i^ ^^ = iZi^ part of the rcfervoir will be filled 
 
 480 1287 154440 
 
 in I hour, if all the cocks A, B,.C, D, E, and F, are fet open together. 
 A, iliSi refcrv. : ^ hr. : , -1 referv. = iiiii2 hr. = 8 hrs. 
 
 154440 I I 17409 
 
 Ki m. 16 ^ f. Anfwer. 
 ^ 5803 
 
 * Gentlemen acquainted with the doftrine of fuiJi will readily per- 
 ceive, that this folution is on fuppofition that the weight of the column 
 of water in the refervoir, and the prelTure of the atmofphere, are 
 uniform during the influx and efflux of the water j if thefe were allowed 
 for, and the dimenfions of the ciftern given, the folution would be dif- 
 ferent, but thefe minutiae are ufelefs to one acquainted only with 
 Arithmetic. 
 
 (10.) 
 
 98. 9jd.-^. Anfwer. 
 
 Firft -of-? of c4 = 
 25 "^ 8 
 
 14 _ 112 
 95I" """763 
 
 (II.) 
 80 ' 
 
 Hi 
 
 94- 
 
 86 
 4.70 
 
 ^^T=f' 
 
 Thenii9^ 
 
 43 U ^1 
 
 ^ ^0 .. Z/;^^^ $^__ i2QX43Xt7 _ 94299 
 
 ^0'^^^fi ^0^ $ 20X94X109 20+920 
 
 40 94 109 
 
 20 
 
 the numerator of the new fradion.
 
 46 Promiscuous Questions im 
 
 Again, 
 
 ^71 792' i"~ 7 ' i5i""34i' ^8 » ' 
 
 7 5*9 
 _^55^^^^/^ ^^ 4^/J_ ^^^X7Xj)9 _ 2705'? 
 ^ 341 ^ 22x341 X 4 3000S 
 22 4 
 
 the denominator of the new fraftion. 
 
 3751 
 Hence '^^^^'^^'^ = 94299X3^^ _ 3?37'5549 ^he 
 VoVoV 2^ops>^iM^ii^ 6929241725 
 25615 
 fra^on required. 
 
 (12.) 
 
 Flrft, 
 Let — reprefcnt the number of marbles in the ring* 
 
 — of— = — A fnatchedoutofthe ring* 
 515 
 
 -I of -^ =: — B fnatchcd from A. 
 « 5 ao 
 
 at T 
 
 — 1— — z: — rcmamdcr in A's hands* 
 5 20 4 
 
 •1 of -i =: i. C got from A. 
 5 4 20 
 
 J. — -^ =: — A had then left, 
 
 4 20 20 
 
 123 
 
 — — — n: — A left in the ring. 
 1 5 5 
 
 — of ~ := — E got. 
 12 5 20 
 
 3 1 II ^ «. . , 
 
 — — — . — — D ran oft with. 
 
 5 20 20 
 
 2 
 
 Hence at the end of this fcuffle there remained with A —-, with B 
 
 20 
 
 — , with C — , with D — , and with E — 
 20 20 20 ao
 
 J*art I. Vulgar Fractions. 47 
 
 Second, 
 
 -Z- of — =: — A and C fnatched from Dt 
 II 20 20 
 
 20 20 20 100 
 
 -i of -Z = -Z-. B got. 
 5 20 100 
 
 2. ^ 
 20 
 
 7 28 
 
 -^ = E got. 
 
 100 100 
 
 ^ + 
 
 _L zr -^ B had in aU. 
 100 100 
 
 33 
 
 100 
 
 II 26 
 
 = C had in alt. 
 
 100 100 
 
 i-of 
 II 
 
 ,22 2 ^ 
 100 100 
 
 2 2 12 . 
 
 -\ — A had in all. 
 
 100 20 100 
 
 — — — 13 B had in all. 
 
 100 100 100 
 
 Hence at the end of this fcuffle there remained with A , with 
 
 100 
 
 ^20 . , « 26 . , ^ 20 , . , ^ 22 
 
 B , with C — — , with D , and with E ♦ 
 
 100 100 100 lob 
 
 Third, 
 
 J , 22 2 ^, 22 2 20 
 
 — of = Then = E had left. 
 
 J I 100 100 100 100 100 
 
 12 2 14 1 
 
 — — A rr ■ A had, after receiving — of E's. 
 
 100 100 ICO II 
 
 3 ^ 26 6 _. 26 6 20 
 
 J. of = Then = C had left. 
 
 13 100 100 100 100 100 
 
 ^14 6 20 5 
 
 '■ ■ ■ + — ZI ■ A had, after receiving ~ of C*s. 
 
 100 100 100 *' 13 
 
 Heace after their difpute was thus fettled, there remained with A - 
 
 100 
 
 with B , with C — , with D , and with E : fo that 
 
 100 100 100 100 
 
 there might be loo marbles in the ring at firft, and if fo, each boy put 
 in 20. This queftion is unlimited, but the above folution is the molt 
 natural that can be given, at leaft in tiiis place.
 
 48 
 
 Addition op Decimals. 
 
 ('3.) , 
 
 11 of — =: 1^ left the elder fon ; -i— li ^^ — the remainder. 
 79 I 79 I 79 79 
 
 35 
 
 44 _ J540 
 
 of:^ = 
 
 79 79 
 
 6241 
 
 left the younger fon. 
 
 3S r542 -.|l£i left to both the fons. 
 79 624.1 6241 
 
 -i «- 12£i — iiL rcfidue allotted to the Vfldow. 
 I 6241 6241 
 
 U ^ iil2 = iii2 of the Eftatc worth 500I. 
 79 6241 0241 
 
 If IliiE. : 5£?1. f. I22!e. : 7901- 4S. oJd..ti what was 
 6241 I 6241 * 49 
 
 allotted to the widow. 
 
 If 4^'e. : i^l. : : -^E. : 25471. 6s. iiW.-l the value 
 6241 I I •' 49 
 
 ef the whole Eftatc. 
 
 If 4w. : 57-l-yds. : 12/^ ft. : ifb. 
 
 ccj} 
 
 4w. : 57-l-yds. : 12/^ft. : if b. L^ 342II. . 
 4w. : 34|yds. : ii^ft. : 2^b. KS *l 
 
 cojl 
 
 By working this ftating, the Anfwer will be found 
 308I. 4s. 2|d..|||f 
 
 DECIMAL FRACTIONS, 
 ADDITION OF DECIMALS. 
 
 Num. Answers. 
 (i.) This is do7ie at length, 
 (2.) 27-2087 
 (3.) 88-76257 
 
 X 
 
 Num. Answers. 
 (5-) '835-599 
 (6.) 397*547 
 {7.) 31-02464
 
 Prirt I. Subtraction, &c. of Decimals. 
 
 49 
 
 SUBTRACTION of DECIMALS. 
 
 Num. Answers. 
 ( [ . ) This is done* 
 (2.) 51-722 
 (3.) 2-7696 
 (4.) 1571-8; 
 (5.) -6946 
 
 Num. Answers, 
 (6.) -89575 
 (7.) 603-925 
 
 (8.) i379'25922 
 (g.j 99'7o5 
 (10.) 17-949 
 
 MULTIPLICATION of DECIMALS. 
 
 Num. Answers. 
 
 ( r . ) This is -ojorkcd at length, 
 
 (2.) Ditto. 
 
 (3.) 26-99178 
 
 (4.) 10376-283913 
 
 (5-i 2'7S?39^"^65 
 
 Num. Answers. 
 (6.) '020621 1250 
 (7.) -28033797099 
 (8.) 175-26788396 
 (9.) -00043204577 
 
 (10.; 215-67436625 
 
 CONTRACTED MULTIPLICATION of 
 DECIMALS. 
 
 (I.) 
 
 This Example is ivorkcd at length. 
 
 (2.) 
 
 (3.) 
 
 54*7494367 
 
 475"7io5<54 
 
 357427-4 
 
 4946143- 
 
 21899775: 
 
 142713 
 
 3S32460 
 
 19028 
 
 109499 
 
 476 
 
 21900 
 
 285- 
 
 274 
 
 ^9 
 
 16 
 
 4 
 
 2^8*67756 
 
 62-525
 
 5^ 
 
 Division of Decimals, &c. 
 
 3754-+078 
 ^75437' 
 
 262S08546 
 
 11263223 
 
 1501763 
 
 187720 
 
 26281 
 
 2253 
 
 2757-89786 
 
 (v) 
 
 4745*679 
 9454-157 
 
 332^975 
 
 23728+ 
 
 4746 
 
 1898 
 
 237 
 
 19 
 
 4 
 
 3^66163. 
 
 DIVISION OF DECIMALS. 
 
 (I.) 
 
 T^is Example is 'worked. 
 
 Num. Quot. Rem. 
 
 KuM. Quot. Rem. 
 (2.) 487-74—37378 
 
 (3-) •13956—583 
 
 (4.) 1918;!. 528—696 
 
 (5.) -004735—0 
 
 (6.) '01666, &c. — 640 
 
 (7.) -006944, &c. 
 
 680 
 
 (8.) i-36832?i7— 3425 
 
 (9.) 12976*8062 — 474 
 
 (10.) '004958x31 — 69466 
 
 CONTRACTED DIVISION of 
 DECIMALS. 
 
 This Example is inorkai at length. 
 
 (i- 
 
 (3-: 
 
 (3 80C06 . 
 
 8864 495'783i69)i7493«407704962(35'2843 
 
 14-7 261990 
 
 662 14008 
 
 65 41S2 
 
 5 216 
 
 18 
 
 3
 
 Part I. Reduction of Decimals, (j i 
 
 (4.) 
 
 5-476 56 i8)98'i874^7poco( 1 1 '5834036625 
 i34aiS-lL^ 
 
 494525720 
 
 (5.) 
 
 706976300 
 28S5I3560 
 
 14-7349 
 
 5)47i94-375457(3202-8S6a 
 2989529 
 
 4^539 
 13069 
 
 lie I 
 ICZ 
 
 2 
 
 34216706 
 
 310459 
 
 56161 
 
 5303 
 
 ai; 
 47 
 
 IlEDUCTION or DECIMALS. 
 
 T/:is ExaiTiple is ivothd. 
 
 NwM. QuoT. Rem. 
 
 (2 
 (3 
 (4 
 (9 
 (6 
 
 (7 
 
 {« 
 
 (9 
 
 (10 
 
 ; .•57142^—4 
 ) '0041152263 — 91 
 
 ) 'IS 
 
 ) '20835, &^c.— 8 
 
 ) 15-38461—7 
 
 ) •^241379— 9 
 ) -026178010471 — 79 
 ) This Example is do?ie, 
 ) -37291666, &C. 1. 
 
 Num. Quot. 
 (ii.j -0062^1. 
 
 (12.) 
 ii'9- 
 
 12 7-818181, &c. 
 20 4-65 1 51 5, &c, 
 •2325757, &c. I 
 
 (13.) •12968751b, 
 
 (14.} -05 oz. 
 
 (15- 
 
 50Z. 4 dr. = •3281251b. avoirdupois. Now i lb. avoir- 
 dupois = 6999-5 gr» troy. 
 
 _. "■ 6qqq*c •328r2c „ „ 
 
 Hence -^-^ H^ ^= '39873453 lb. troy. Anf. 
 
 5760 
 
 (16.) 
 •168751011. Aafwer, 
 
 (17.) 
 
 -•6339285714 cwt. Anfwer. 
 
 F a
 
 52 
 
 Reduction of Decimals. 
 
 (i8.) 
 
 51b. I oz, 3dw-t. i3grs. =: 33685 grains. 
 
 Then 33685 -^ 6999*5 = 4'8i 24866061 86 lbs. avoirdu- 
 pois, which divided by 112 lb. give '04296863 cwt. the 
 decimal required. 
 
 Nu 
 (19. 
 (20. 
 (21. 
 (22. 
 (23- 
 (24. 
 
 (25- 
 
 (26. 
 
 (27- 
 
 (28. 
 
 (29. 
 (30. 
 
 i32' 
 
 M. Answers, 
 
 Num. Answers. 
 
 ) -3125 yards. 
 
 ) '55 ellsEng. 
 
 ) '0084359217 miles. 
 
 ) '48125 acres. 
 
 ) 'C099206349 hhds. 
 
 ^33') I qr- 23 lb. 3 oz, 
 
 13-44 drs. 
 (34.) 3 cwt. 2 qrs. 
 {3S') 15*04 drs. 
 [36.) 6dwt. 4-32 grs. 
 
 ) '1 171 875 chaldron. 
 
 (37.) 1-2 n. 
 
 } '07472051165 hrs. 
 
 (38.) 2qr. i-5n. 
 
 ) -12251. 
 
 (39.) i4p. 2 yds. 2 ft. 5 in. 
 
 ) -2685 acres. 
 
 i'i2S b. c. 
 
 ) -10:051136 miles. 
 
 (40.) 2r. 22 p. 
 
 ) This Example is nMorked. 
 ) 15s. I|d, -2 
 
 } 9-o522d. 
 
 (41.) 30 gall. 3qt. i-968pt. 
 (42.) 156 d. 12 hrs. i-^ m, 
 5 if. 36 thirds. 
 
 ) 7s. loid. 
 
 
 (430 
 
 '475 
 20 
 
 f. 9-500 
 
 *375 
 
 add 
 
 .b. 9-875 
 12 
 
 
 d. 10-500 
 4 
 
 
 f. 2-000 
 
 
 9s. lo^d. 
 
 A 
 
 (44-) 
 
 •75' 
 £ 
 
 ft. 2-255 
 
 3*036 
 •573 add. 
 
 . in. 3-609 
 3 
 
 b. c. 1-827 
 
 . V 
 
 2 ft. 3 in. i-82 7b. c. Anii
 
 Part I. C I » c o L A T I N o Decimals. jj 
 
 'AS-) 
 3 fur. 2j p. 
 
 (46.) 
 4cwt. 3qr. 41b. 1 1 cz. 4*224drs, 
 
 (47.) 
 3Cwt. 2qr. 261b. 12*8 drs. 
 
 - (48.J 
 70Z. i4dvv't. 2i'6grs. 
 
 CIRCULATING DECIMALS, 
 
 ___ 3 I 135 5 ~" 
 
 _ ^ _ ^ , an I ^ _ ^^^^ _ ^^. 
 
 .^ 6 2 , 162 6 , 769230 10 
 9 3 999 37 999999 ^3 
 
 999 39 99 i^ 
 (3.) 
 
 ' 999999 ^43 ' 99 ^^ 
 142857 1 
 
 (4.) 
 T^i^/V Example is luorhd, 
 
 ,,38. =IiizLi2=Iii=±; 7.54,3.= mirZi - 
 
 ■ 900 (;oo 37 990 
 
 746S_ 3734 — 7 ll2; 043'54' = ^^^^4 — 4 _. 435° _. 
 990 495 495' '^ c^yc^GO 99900 
 
 ^9 . .„..,> ^ 3754—371 _ 3379 _ 49 
 666 ' ^7 i4 — ^^ — ^^^ .— 37 ■^- i 
 
 990 990 330' ^^^^^—999990 333330 
 F 3
 
 ^4- Reduction op 
 
 (6.) 
 
 Tor '7' 5', read '75'; and/c/r '4'5', read '45'. 
 
 Th^ . / — ILZLl— ^— 11. .4V8' = lli^=^=:— 
 " '^ 90 90 45' 990 990 
 
 217 , 93 — 9 S4 7 47543 — 47 
 
 — ; •093' — ^zr rr — ; 4*75'43' n: ■ 
 
 495 9C0 9-0 75 ^ '^^^ 9990 
 
 ^74 134 .^^.o,., _ 9S7--9 _ j6?_. 
 
 ~ — r- rr 4—7- j •oc9b'7' zr rr -7 ; 
 
 1S5 185 99000 16500 
 
 90 so 
 
 (7-) 
 
 ^his Example is ivorkedm 
 (8.) 
 
 )2I0 3 ~2-f-3-T-2-r2 
 
 771^-76' ^^'"^ 16 =rS =:4 =2 = 1. Here the 
 denominator vanifhes after four divifions ; hence the decimal will be 
 
 21C 
 
 £nite, and conf.ft of four places, viz. — iz •1S75. ^This Ix- 
 
 1120 
 
 ample is the fame as the firft in Mr, Bcnrycajile'^s Arithmetic, page 154, 
 third ed tion, where, by inadveitency, he makes the decimal infinite, 
 and to confift of fix circulating fijjures. 
 
 4')-4- =— i tlien 
 /iibo 290' 
 
 {9-) 
 
 -7- 10 
 290'^ 290 — 29. 
 
 29 ) 9999» &^c. to 28 nines. Here 28 nines are made ufe of be- 
 fore o remains, and the denominator has been abridged once 5 hence the 
 tlcc'mal wiil c.nfift of one finite decimal, and 2S pure repetends, viz, 
 
 -It- rr •co'344Sz75S62o689655i724i3793i/ 
 
 IIOO 
 
 (10.) 
 \ 12 T 
 
 Firft, 12 — =-. 
 /132 II 
 
 J I ) 99 ( Here are two nines made ufe of } hence tlie decimal will b& 
 
 J2 
 
 infi.iite, and confiii of two pure repetends, viz. — - r: •c'9'.
 
 Parti. Circulating Decimals, ^^ 
 
 Secondly, 
 
 > 80 _ lb 
 
 V 1T5 "" 27' 
 
 '^7 ) 999 ( Here are three nines made ufe of before o remains j 
 hence the decimal will be infinite, and confift of three pure repetend>, 
 
 80 
 
 VIZ. m s'92'. 
 
 135 
 
 Thirdly, 9") — — — . Then 
 yi35 15 
 
 8 a^u„_ -^ 5 
 
 15 = 3 
 
 3 ) 9 ( Here is one 9 made ufe of, and the denominator has been 
 abridged once, therefore the decimal will contain two places, one finite, 
 
 , .... . 72 
 
 the otner mnnite, viz. zz "\V» 
 
 135 
 
 Fourthly, -^. Then f ^ "^"^ ^^ '^.! "^* 
 8544 8544 — 4272 — 2136 — 1068 zz 534 
 
 = 267. 
 
 267 ) 9999, &c. to 44 nines. ( Here the decimal will confift of 5 
 finite decimals, and 44 pure repetends. 
 
 Fifthly, 3)'-^=^; thenV "' "J 
 
 ■'/792 264' 264 ~ 132 ~ 66 — 33. 
 
 33 ) 99 ( Here the decimal will confift of five figures, three finite, 
 
 255 
 
 and two infinite, viz. zn "12JQ'6'% 
 
 792 
 
 ADDITION OF CIRCULATING 
 DECIMALS. 
 
 ('.) 
 
 77'/V Example h nuorked^ 
 (a.) 
 
 D'ljfimilart S'lm'ilar. Similar and Conterminous* 
 
 67-34'5' — 67-34'5/ = 67-34'54545/ 4545 
 
 9«6'5i' =: 9'65'i6' rr 9'65'i65i6' 5165 
 
 •2'5' — •25'2' — •25/25252 ..... 5252 
 
 17*47' = 17-47' = I7*47'77777' 7777 
 
 •5' =z -55' — -SSS'SSSS' 5555 
 
 I'he corrcB fum gs'^^' '^9^4-7' two to carry,
 
 55 A D D I T r o N o p 
 (3-) 
 
 DiJfi>nUar, Sitr.'ilar. Similar and Cc7iternnnous, 
 
 •4'75' = •47547' =: •47547547' — • 5475^ 
 
 3-754'3' — 3'754'3' = - 3*754'34343' •— 4343 
 
 64-7'5' n 64-757'5' — H"JSYS1^1S' •— 7575 
 
 •5'7' = .:75y = 'SlS'-^SlSl" •— 5757 
 
 i7'88' = •i78'87' =: •i78'S7887' .... SS7S 
 
 
 
 
 Sum 6q'742'o3ii2' three to carry^ 
 
 
 (+•) 
 
 DlJfrrMar. 
 
 5 
 
 nilar. 
 
 Similar ar.d C'^v.tcrmimui. 
 
 •5' = 
 
 
 •55' 
 
 = •55'55555555555' •— 555^ 
 
 4-37' , = 
 
 4-37' 
 
 = ^-hi'iiiiniiiii' '• in 
 
 49*45 7 = 
 
 49 •457' . 
 
 = 49'45'7 57 5757 575?' — • 575 
 
 •49' 54' = 
 
 
 •49 54 
 
 — -49-54954954954' .... 954 
 
 •7-345' = 
 
 
 •73'457' 
 
 = •73'45734573457' .— 345 
 
 
 55-62'0978c43-^5C-3' three to carry 
 
 
 is-) 
 
 DiJltr.Uar. 
 
 
 Similar. Similar and C»nt€rrr.irBUS. 
 
 •l'75' 
 
 ~ 
 
 •1/51 
 
 = •I7'5i75»' •••• 7517 
 
 4*-5'7' 
 
 \ — 
 
 42-57'5' 
 
 = 42-57'57575' — • 7575 
 
 •37'53' 
 
 — 
 
 •57'5: 
 
 s' = •??'53753' •— 7537 
 
 •59'45' 
 
 — 
 
 •59%5' = •59'45945' •••• S459 
 
 3 75V 
 
 
 3-75 4' 
 
 = 3*75 45454 •— 5454 
 Sum. 47'4-''544Si' three te carry^ 
 
 
 (6.) 
 
 Dijfnilar. 
 
 Similar, 
 
 1 65- 1 'C- 4' 
 
 ri i65'i6'4i' 
 
 147 -oV 
 
 — 1 47 -040' 
 
 4-9'5' 
 
 = 4-95'9' 
 
 S4-37' 
 
 — 94-37' 
 
 Similar end Conterminous, 
 n i65«i6'4i64i6, See* 
 zz I47«c4'c40404, See. 
 
 — 4-95'959595i- ^^- 
 
 — 94-37'777777, -^'C 
 4-7'i23456' — 4-7i'234567' — 4-7i'234567', &c. 
 
 •"aw 4i6'25'42S76i, &c. 
 
 * If thefe Girculatirg Figures were made conterminous, they ;v:.uld 
 .■^un cut to 42 p'aces of pare repetcr.d. 5 for the left common mukiplc 
 of th - number of figures contained in each circulating decimal (by the 
 Ruk, page 27th, o.'thisKeyj is 42. Thus, 
 
 The iff, 2d, 3d, 4th, 5th, Ciicu'aring decimal 
 Confiftsof2)3 ,2,2,1,7 piacs of pure repeteads* 
 
 Then 2 X 3 X 7 — 42,—— The aLove f^iution is carried far enough 
 for acy pra(^ical p urpofe»
 
 Part I. Circulating Decimals, ^-j 
 
 SUBTRACTION of CIRCULATING 
 DECIMALS, 
 
 (I.) 
 
 ^'^is Example is njoorked, 
 
 (2.) 
 
 Di[Jlm'dar. Similar, Similar and Contcrmlnsus* 
 
 47-53' = 47-53' = 47*53'33' .— 333 
 
 i-7'57' =: i7S'77' = i-75'77' —• 577 
 
 ■D'ff' 45*77'55' one to carry* 
 
 (3-) 
 
 ViJJimtlar, Similar, Similar and Conterminous, 
 
 i7-5'73' = i7-57'35' = i7-57'35' •••• 735 
 34-57' = H-57' = i4-57'77' •••• 777 
 
 Diff* 2*99'57' one to carry* 
 
 ■ ■! ■ 
 
 (4-) 
 
 ^ijjlmilar. Similar and Conterminous* 
 
 17*43' = 17*433' •••• 33?- 
 
 iz-345' = 12*345' •••• 555 
 
 Diff, 5-087' one to carry* 
 
 Is'-) 
 
 Dijffimilar, Similar, Similar and Conterminous* 
 
 l-i27'S4' = i*i27'54' = i*i27'54754754754' ••• 73^ 
 •47'3S4' = •473'247' = *473'S473S473S47' — 3^4 
 
 •653'70oi62So907' 
 
 (6.) 
 
 DiJJlmilar, Similar and Conterminous, 
 
 4-75 = 4750' •••• 000 
 
 •375' = '375' •— 555 
 
 DiJ^, 4*374' one to carry.
 
 58 Multiplication or 
 
 (7-) 
 
 VifftniUar* Simlar and Conterminous^ 
 
 4-704 rr 4*794o'oo' .... 000 
 
 .J7'44' ~ •i744'74' •— 474 
 
 Pif. 4'6i95'25' one to carry^ 
 
 DljfitnUar, Sml'ar amd'Corterminout* 
 
 J-457' = 1*45777' •••• 777 
 
 •3754 = '3754° •►" ^^^ 
 
 t>lg. i-og2<;7' 
 
 (9.) 
 D'ljfmilaf* Similar. Similar and Cttitermncuu 
 
 J-49'S7' = i-49379'37' = i-49379'37' .••• 93 
 •J47S zz '14750' "zz .147 5000' .... 00 
 
 Dlff. 1-3462/37' .-. 9S 
 
 MULTIPLICATION of CIRCULATING 
 DECIMALS. 
 
 
 
 {•■■) 
 
 
 
 This 
 
 Example is 
 
 ^jL'Gfked, 
 
 (^-> 
 
 
 
 (.3-T 
 
 •3754' 
 
 
 
 4-753' 
 
 ^4*75 
 
 
 
 7-437 
 
 18772' 
 
 ... 2Z 
 
 
 33273' ..• 35 
 
 26z8i'i 
 
 ... TI 
 
 
 14260'© ... 00 
 
 i5oi7'77 
 
 ... 77 
 
 
 i90i3'33 .. 33 
 
 3754'444 
 
 ... 44 
 
 sne tc carry^ 
 
 33273'333 — 33 
 
 5-537S05' 
 
 3 5 '3 50540' ^r.etoi
 
 Part I. 
 
 Circulating 
 (4.) 
 
 66 
 88 
 
 2Z 
 
 92 1 1 6' 
 
 2i4938'8 
 
 12282222 
 
 prod, '014523727' one ^0 curry. 
 
 (5.) 
 
 -14752' 
 •1497 
 
 303125 ... 55 
 
 i3259o'o ... 00 
 
 58928'88 ... 88 
 
 14732'222 ... 2'2 
 
 Prcd- 022054136' one to carry. 
 
 (6-: 
 
 337890' .. 
 
 i5oi73'3 •• 
 
 37543'33 •• 
 
 262So3'333 .. 
 
 00 
 33 
 33 
 33 
 
 •268397290' one to carry* 
 
 Th^ next Example is iccrked. 
 IS printed No. 6, by niijlake, 
 
 (7.) 
 
 zdpericd, 
 
 •37'54' — 754 
 17-43 
 
 ii2'64' ... 264 
 
 i5o'i9'o ... 190 
 
 262'83'28 ... 32S 
 
 37'54'754 — 754 
 
 ^•5445'37' one to carry. 
 
 Decimals. 
 (8.) 
 
 59 
 
 zd period, 
 4*73'5' "• 35 
 7-349 
 
 42 6 1 '8' 
 
 iS94'i'4 
 
 i42o'6'o6 
 
 33i4'7'474 
 
 .. 181S 
 .. 1414 
 .. 0606 
 •• 7474 
 
 34«8ooii'3' one to c-Hrry, 
 
 (90 
 
 id period. 
 4.i'42857' ... 142857 
 
 • 1797 
 
 29o'ooooo'* 
 372'857i4'2 
 290'cooco'oo 
 4i'42S57'i42 
 
 000 
 
 857 
 000 
 
 S57 
 
 '74447'i4-iS5' tne to carry* 
 
 (10.) 
 
 zd period, 
 7'H'93' ••• 493 
 5-43 
 
 2144'So' ... 4S0 
 
 2S59'73'9 — 739 
 3574'67'46 ... 74^ 
 
 38-8209'66' one to carry. 
 
 (II.) 
 
 zd period, 
 .40'705' ... 0705 
 7-345 
 
 203 525' 
 
 l62'820'2 
 
 35^ 
 820 
 
 I22'lI5'2I ... 152 
 
 284'935'493 «.• 549 
 
 2.98978'742' one to carry. 
 
 (12.) Thii Example is ivorked. 
 * See the note to the rule, p. 42.
 
 6o 
 
 M u 
 
 (130 
 
 1-475 
 1-754' 
 
 LTI 
 
 P L I C A T] 
 
 [ON OF 
 
 (MO 
 
 173-715 ■ 
 3-7545' • 
 
 9)5900 
 
 9)86.8575 
 
 6555' 
 
 7375 
 10325 
 
 1475 
 2.5S7805' 
 
 9650S3' 
 694860 
 
 86S575 
 1216005 
 521145 
 
 652«2226i83' 
 
 (»50 
 
 •37504 
 •7153' 
 
 9)112512 
 
 i25ei3' 
 187520 
 
 37504 
 262528 
 
 Prod. '26827861 3' 
 
 (16,] 
 
 5-" 534 
 •^735' 
 
 9)287670 
 
 319633' 
 172602 
 402738 
 57534 
 
 Prcd. -99553453' 
 
 (170 
 
 (20O 
 
 9)262451 
 
 291612' 
 187465 
 262451 
 
 Prod* '28411362' 
 
 54; 
 
 (18.) 
 ^I'li Exatrfle is iccrked* 
 
 (19O 
 *Tl''is Example is worked* 
 
 4-7157 
 
 3-754 
 
 3-7 540;rfuWJ</. 18862S 
 — 2^5785 
 
 330099 
 141471 
 
 17*7027378 
 
 17702 
 
 I 
 
 Prod. i7'7045'o82 
 
 •7378 
 . 770 &-C 
 
 ore tt/ 
 carry.
 
 Part L Circulating Decimals. 
 
 6x 
 
 ■007 5 
 00 
 
 (21.) 
 
 47-1937 
 •OC75 
 
 (".) 
 
 '•7 543S 
 
 I 
 
 4-37595 
 1-75434 
 
 'OOl ^neivmuh 2359685 
 3303559 
 
 •35395275 
 353952 
 
 75 
 
 3539 — 52j &« 
 35 — 39. &c 
 ... 35» ^^' 
 
 '•75434«^ww«//. 1750380 
 
 . 1312785 
 
 1750380 
 2187975 
 3063165 
 437595 
 
 Prod. •357528o'3' tivl? to 
 • • • ■- ■ — carry. 
 
 (230 
 
 371473 
 •7'53H' 
 
 1485892 
 
 371473 
 II 14419 
 
 1^5^365 
 26003 1 1 
 
 279771.17522 
 
 279771, Sec. 
 2, &c. 
 
 7'676(,t)4i23o 
 
 767690 
 
 7 
 
 (16.) 
 
 3-745 
 1.47' 
 
 9)262i8'8S8888S8 
 2913*20987654' 
 
 149S2'22222222 
 
 3745'555555555 
 
 4121 
 67 &c. 
 
 320 
 222 
 555 
 
 Prod. 279773-9'7295' 
 
 Pred. S'SIS°'9^7^5^2^' ore to carry 
 
 (24.) 7his Example is worked. 
 
 (25.) 
 
 4-57 
 2-45' 
 
 9)2288'8888S8S8 
 
 25432098765' 
 i83i'iiiiiiii 
 
 9i5'555555555 
 
 Prod. 
 
 432 
 III 
 
 555 
 
 (27.) 
 
 5-7195 
 1-788' 
 
 9)45756444444444 
 
 50840^49 3 827 16' 
 
 4 575^5 444444444 
 40O368'888888888 
 
 57i95'5555555555 
 
 ii-24o'98765432' ove to 
 ■ ' ■» carry 
 
 Prod. 10-23 1649'3827 1604' 
 
 ,.. 049 
 
 .. 444. 
 
 ,.. 88S 
 
 - 555 
 
 ore to 
 carry%
 
 Bt Multiplication or 
 
 (28.) {29.) 
 
 3-7S3*' 7H-32^' 
 
 •3425' 3-456' 
 
 9)i8766i'iiiiiiii 9)4^8593*33 
 
 2oS5i'2345679o' ... 123 4762i'48' ... 14.3 
 
 75o64'44444444 — 444 357i6i'ii ... iii 
 
 i5oi28'8888SSS8 ... 888 285728'888 ... 888 
 
 ii2596'6666666666 ...666 2i4296'6666 ... 666 
 
 Prod. i'2556S7i'2345679o' two to Prai* 2469«i748'i4' one io.carry, 
 (30.) Tb'ii Example is ivorked* (32O 
 
 zd period* 
 
 (31.) •4i'32.' ..13a 
 
 td period t 1*432' 
 
 ■573 4 — 34 
 
 1-57.3' 9)S2'64'264264 
 
 9)i72o'3'o303 9'i825T584' ... 91 
 
 — , i2396'396396 ... 39 
 
 i9i'i4478' ... II i65'28'5285285 ... 28 
 
 4oi4'o'4040 ... 40 4i'32'i32i32i3 ... 21 
 
 a867'i'7i7i7 ... 17 
 
 573'4'343434 ••• 34 Prod* •59i8'i3i46479' onetocarry. 
 
 Prod* •9022o'3367o' cm to carry 
 
 (330 
 
 2dper:od» 
 
 (34-) 3*7'534' — 7534 
 
 ^d period* -3757' 
 
 7*00430 ... 0^30 
 
 4.7005' 9)262'743' 2743, &c. 
 
 9)35o2'i5o' 21, &c.* 29'i93, &c. ro4ijfgtfrM 
 
 i87'673', &c. 
 
 3891278, Sec. 262'743'2, &c. 
 
 49C3'oio'3oi, &c. ii2'6o4'26, &c. 
 
 28oi'720'l720, &c. 
 
 Prod. I"4I04'7 circu!. conftjis of "ifi 
 Trod. 32 '9241 • • • • &c. fgurei. 
 
 * This and the following Examples would have taken up too much 
 room, had I continued them far enough to recur ; I have therefore cun- 
 tirmed them no farther than is neceflary in pradice, prefuming, that 
 when the youug itudent has attentively perufed what I have written on 
 this fubjed, he will be enabled to work any kind of circulating decimals, 
 with as much kwlJty as common or finite deciihals, without the help of 
 a tutor.
 
 Part I. Circulating Decimals, 6^^ 
 
 (350 
 
 2d pcn:d, (S^O '^'^i^ Example is vuorked, 
 
 5'437'i5' ... 715 
 • 37^5 3' (37' ) *I'hh Example is 'worked. 
 
 9)16311-47' 14, &c. 
 
 (38.) 
 
 1812385, &c. 
 
 27i85'78'5, &c. For I'lis' read I'l'i' s'' 
 38060' io'oic,&c.. 
 
 j63ii'47'i47i, &c. i'73'5' 
 
 -47053' 17 
 
 Frod, a'oi4647 . . . i;c. i"7i8 - - 
 
 —————— . 1*710 netu 
 
 376426' ... 66 "^ — muU 
 
 (39-) 47053'3 • • 33 I 
 
 54-7'53' 3'-9373'33— Zl Unetocarry, 
 
 3-4573' 5+ 47C53'333— 33 J 
 
 54.599 ■ 
 
 ■ 54*699 ruio •80837626' 
 
 3iii6o'..oo i ■ mul. 808376, &c. 
 
 3iii6o'o..oo I 
 
 ao744o'oo..oo i 
 
 i38293'333-33 \oneto 
 
 172866'6666..66-' 
 
 8083, &c. 
 ^)0, &c. 
 
 carry. 
 
 72866'6666.. 66-' trod. -81654 &c. 
 
 j89'ii2676o' 
 
 1891126, &c« 
 
 1891, &c. (40O ^'^'^ Examplt it worked* 
 
 Prt* i89'3oi97 • • Sec, (41.) Tiiis Example is worked. 
 
 (42.) 
 
 2d per lid, 
 4-57l'37' ..• J37 
 •148078 
 
 'M957'3' 
 
 3657o'97' ... 097 1495 
 
 3i999'59'9 ... 599 I 
 
 3657o'97'o97 ... 097 I. '148078 ntwmult, 
 
 i8285'48'5485 ... 485 V — _ 
 
 457J'37'i37i3 ••• lil-^ one to carry. 
 
 •6769i9529'92' 
 
 67691, &c. 
 
 676, &c. 
 
 6, &c. 
 
 P;W. -6837571 . . . ; &c. 
 
 G 2
 
 6+ Division 
 
 (43r) 
 2«/ period* 
 5-7U9Y — 93 
 
 4-7488 4*75'35' 
 
 47 
 
 457195 I ... 51 
 
 457i95'i'5 — 15 / 4-748« kcw ps'J^. 
 
 228597'5'75 .,. 75 L 
 
 4ooc45'7'575 ... ti^X two ts carry. 
 
 ^^8597'5'7575 ... 7$'' 
 
 27-13910419 3 
 
 2713, &c. 
 
 2, &c. 
 
 Prorf. 27-16627 • • . • ii04 
 
 DIVISION OF CIRCULATING 
 DECIMALS. 
 
 Tlis Example is fworked^ 
 
 ^his ExampU is ^worked* ■ 
 
 (3.) 
 For 7-54'o3, r^/7^7-54'o3'. 
 
 Then 14*25 ) l'S^'03%03^03, &c. ( -52914669, &c. 
 Rem. 7 8 
 
 (4.) 
 •075 ) 4-3i73'33* &c. ( 57-564' 
 
 Rem. 33 
 
 (5.) 
 
 178? ) I4-937V7575* ^'c C 8-3543+» &^- 
 
 Rem. 1583
 
 Part I. Circulating De-cimals. 65 
 
 (6.) 
 •00456 ) 43-s'75'57557^' &c. ( 955^*'^472* ^<^' 
 Rem. 343 
 
 (7.) 
 
 Th's Example is ixorhd, 
 
 (8.) 
 This Example is fworked* 
 
 M 
 
 quotient 
 i'7'59' ) 47*34Jooo ( 26-90424, &c. 
 
 •J 4.7345 
 
 neTrdivifor 1*758 ) 47-297655 new dividend 
 Rem. 1 1 8 
 
 (10.) 
 
 quotient 
 
 3*7S'3') -35^780^ ( •0937> ^^' 
 37 35^8 
 
 new divifor 5-716 ) "3482622 new dividend 
 Rem. 730 
 
 (...) 
 
 quotient 
 •4'5678' ) 17*34200000 ( 37*965, kc. 
 17342 
 
 new divifor '45678 ) i7'34i82658 new dividend 
 Rem. 17388 
 
 G 3
 
 65 Division or. Sec, 
 
 (12.) 
 
 quotient 
 1 V^j*;' ) '374530000 ( -26309, &c. 
 ' 37453 
 
 new divifor 1*4234 ) '374492547 new dividend 
 Rem. 10241 
 
 (13.) 
 .- this Example, for I3*5'j69533', read iS'S^'^S'iSS'y '^'^ 
 i^e Example is ^worked right. 
 
 ^his Example is ^worked. 
 
 This Example is <^csrked. 
 
 (16.) 
 This Example is ivorked* 
 
 (17.) 
 
 For 357V, read'SSl'i'- 
 
 Then -357'-*'-= •357'^72'* and 49'5'735'. = 49'573'557 
 
 quotient 
 49'573'557' ) 'SSl'^l^' ( -0072, &c. 
 4957 35 
 
 new divifor 49-568600 ) 'S'il^}'! new dividend 
 Rem. 34308 
 
 (18.) 
 
 S — ^ SS' , 
 '555 ) '1 5400000 ( 'i-]7AT quotient 
 
 Rem. 41 5
 
 Part I. Practice. 
 
 "3' ^ 'Si's' quotient 
 
 5 3 
 
 new divifor '^O '33° new dividend 
 Rem. 1 9 i 
 
 6^ 
 
 (20.) 
 The divifor nxjants a point before it. Then '7' = •7^7 77' 
 
 quotient 
 •7'777' ) 4'5'732' ( 5'8799» &c. 
 4 
 
 new divifor '7777 ) 4*5728 new dividend 
 Rem. 177 
 
 PRACTICE. 
 
 This Example is nvorked. 
 
 Num. Answers. 
 d. 
 
 U 
 (3 
 (+ 
 (5 
 (6, 
 
 (7 
 (B 
 
 <9 
 (la 
 
 (II 
 
 (12 
 
 ('3 
 
 o 
 
 o 
 
 17 
 
 7 
 
 8^ 
 
 TT'M /f ivorked, 
 
 IS 12 5 
 
 3 
 
 31 
 
 7 
 29 
 
 43 . 
 77»;V is ^worked, 
 
 29 
 
 5 2t 
 
 Num. Answ 
 
 (14.) 
 
 I 
 
 
 
 (I?.) 
 
 16 
 
 7 
 
 (16.) 
 
 I 
 
 7 
 
 (17.) 
 
 66 
 
 7 
 
 (18.) 
 
 23 
 
 13 
 
 (I9-) 
 
 8 
 
 7 
 
 (20.) 
 
 27 
 
 6 
 
 (21.) 
 
 13 
 
 S 
 
 (22.) 
 
 32 
 
 ^5 
 
 {23.) 
 
 74 
 
 3 
 
 (24.) 
 
 3 
 
 12 
 
 (25.) 
 
 3 
 
 3 
 
 ERS. 
 
 1 1 
 
 n 
 li 
 
 lot 
 
 1 
 
 lol-
 
 68 
 
 P R A 
 
 Num. Answers. 
 
 NUM 
 
 £. s, d. 
 
 
 (26.) 85 14 7i 
 
 (68. 
 
 {^7-) 35 15 4j 
 
 (69. 
 
 (28.) 19 iO Ji 
 
 (70. 
 
 (29.J 47 3 Oi 
 
 (71- 
 
 (30.) 9 16 33. 
 
 (72- 
 
 (3I-) '5 19 ' 
 
 (73. 
 
 (32.) 11 5 Hi 
 
 74. 
 
 {33.) 171 II lO^ 
 
 (75- 
 
 (34-) 53 4 iij 
 
 (76. 
 
 (35.) 49 18 8 
 
 (77- 
 
 (36.) 25- r4 II J 
 
 (78. 
 
 (37-) 167 2 7i 
 
 (79. 
 
 (38O 63 13 lOX 
 
 (80. 
 
 (39.) ,85 15 6 
 
 (81. 
 
 (40.) 275 6 oi 
 
 . (82. 
 
 (41.) 19 II I 
 
 (83. 
 
 (42.) 15 3 loi: 
 
 (84. 
 
 (43.) 19 12 6 
 
 ^ll' 
 
 (44.) 160 3 ii| 
 
 . (86. 
 
 (45.) 217 6 il 
 
 (87. 
 
 (46.) 227 16 9 
 
 (88. 
 
 (47-) 283 13 3t^ 
 
 (89. 
 
 (48.) * This is ivaried. 
 
 (90. 
 
 (49.) 16 H 4l 
 
 (9i-> 
 
 (50.) IS 8 Hi 
 
 (92- 
 
 (51.) 31 J ^o. 
 
 (93- 
 
 (52-) 37 5 3f 
 
 (94- 
 
 (53-) 278 7 7i 
 
 ■ (99- 
 
 (54.) 32 17 8i 
 
 (96. 
 
 (55.) 28 17 6 
 
 ■ (97- 
 
 (S6.) 339 5 4f 
 
 (9^- 
 
 (57-) 22 8 3i: 
 
 (99- 
 
 (S8.) 289 14 3i 
 
 (100. 
 
 (59-) 232 8 9, 
 
 (lOI. 
 
 (60.) II I li 
 
 (102. 
 
 (61.) 306 14 it 
 
 (103.; 
 
 (62.) 24 10 lOi 
 
 (104. 
 
 {6i.) 33 4 
 
 (105:. 
 
 (64.) 251 9 4i 
 
 (106.J 
 
 (65.) 392 ^6 9, 
 
 (107.J 
 
 (66.) 34 9 6a 
 
 (108.J 
 
 (67.) 265 13 »i 
 
 (109.J 
 
 * lo-tbis Example, ^ 
 
 'or 4576, 
 
 Answers, 
 £' s, d. 
 
 10 I 
 6 
 
 10 
 
 II 
 
 4 
 12 
 
 o 
 
 18 
 
 15 
 
 S 
 
 11 
 
 5 
 
 12 
 
 3 
 
 35 
 
 27 
 
 428 
 
 371 
 28 
 
 44 
 
 293 
 37 
 29 
 
 38 
 47 
 39 
 
 42 
 
 32 
 455 
 
 33 
 33S 
 
 44 
 545 
 549 
 
 40 
 
 ,51 
 
 689 
 
 532 
 69 
 
 493 
 483 
 367 
 
 lOt 
 
 3t 
 
 Ik 
 
 I I 
 
 J. 
 
 II 
 
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 11 
 
 15 
 16 
 
 4 
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 17 
 
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 17 
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 2+ 
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 3 
 
 2 
 
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 10 
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 4 
 
 4 
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 47 10 o 
 
 56 17 
 350 16 
 
 447 o 
 536 14 
 144 J 8 
 
 2165 
 
 3237 
 172 
 436 
 
 .256 
 06 
 
 4 
 6 
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 14 
 
 4 
 
 ^7 
 
 read ^7 $6*
 
 Part I. 
 
 P R A 
 
 T I C E. 
 
 69 
 
 NUM 
 
 . Answers. 
 
 , , /• 
 
 J. d. 
 
 (no.) 2^9 
 
 14 
 
 (IM) 371 
 
 5 
 
 {n2.) 2992 
 
 16 
 
 (1 13.) 674 
 
 18 
 
 (m4.) 444 
 
 12 
 
 ([15-.) 35:2 
 
 
 
 {116.) This isiuoyked. 
 
 [W].) This is rworked. 
 
 (n8.) 82 
 
 10 
 
 (n9.) 160 
 
 4 10 
 
 (120.) 616 
 
 
 
 (lii.j 237 
 
 6 8 
 
 (122.) 192 
 
 
 
 (I23-) 495 
 
 11 
 
 (124.) 841 
 
 14 6 
 
 (125.) 78 
 
 I 2 
 
 (126.) 213 
 
 9 
 
 (127.) This is 
 
 ivorked. 
 
 (128.J This is ^worked. 
 
 (129.) 3091 
 
 8 
 
 (130.) 1910 
 
 13 
 
 (131. j 1370 
 
 16 
 
 ('32.) 3833 
 
 10 
 
 (I33-) 2349 
 
 6 
 
 (^340 J82 
 
 18 
 
 (135.) 2764 
 
 16 
 
 (136.J 1043 
 
 
 
 (163.) 
 
 1. s. 
 
 d. 
 
 4 'I 
 
 9 
 
 
 6 
 
 , 27 10 
 
 6 
 
 
 3 
 
 qr. lb. 82 n 
 
 6 
 
 2 
 
 ^ 4 II 
 
 9, 
 
 I - 
 
 r 2 5 
 
 10^ 
 
 ^ 7 , 
 
 \ I 2 
 
 I'i 
 
 ° 4, 
 
 ^ 5 
 
 81 -2^ 
 
 
 3 
 
 3i -28^ 
 
 
 91 I 
 
 • o| -53^ 
 
 Num. Answers, 
 
 
 ^. - ^. 
 
 (I3-.) 
 
 900 
 
 (138.) 
 
 877 4 
 
 (I39-) 
 
 691 4 
 
 (140.) 
 
 1632 3 
 
 (141.) 
 
 2348 8 
 
 (142.) 
 
 1023 
 
 {143J 
 
 1333 i^ 
 
 ('44.) 
 
 580 18 
 
 (Hi--) 
 
 894 18 
 
 (i,6.j 
 
 ' 7461 6 
 
 (147O 
 
 This is ivorked* 
 
 (148.) 
 
 2702 17 o|- 
 
 (149O 
 
 10023 18 7J: 
 
 (^50.) 
 
 1906 8 it 
 
 (i?i.) 
 
 1940 ^ w^ 
 
 (1,52.) 
 
 81637 15 3 
 
 (I53-) 
 
 4004^9 13 loj 
 
 (1 54-) 
 
 105668 4 4I: 
 
 (I55:-) 
 
 T^/> is ivorhed. 
 
 (156.) 
 
 13B5 9'-f 
 
 (157.) 
 
 24309 8|-i 
 
 (158.) 
 
 4125 4 8 1 
 
 (159-) 
 
 22799 4|-.'- 
 
 (r6o.) 
 
 11247 '5 lol-i 
 
 (16..) 
 
 8732 13 3 1^. 
 
 (idz.) 
 
 This is nvorked. 
 
 
 (.6+0 
 
 I. 
 
 s. d. 
 
 2 
 
 ♦ 8 
 
 
 S 
 
 lb. 
 
 I 
 
 n 
 
 3 
 
 4 
 4 
 
 i- 
 
 44 
 
 13 
 
 4 
 
 Add 5° 3 2;--,4^,&c. 
 
 < o o 4^- '142, iScx. 
 
 Sub. 037 '28;, &'c. 
 Anf. 44 9 8|. -714, &c.
 
 10 
 
 o 161^ 
 
 P R 
 (165.) 
 I. 5. d. 
 I I 
 
 A C T 1 C fl, 
 
 (.68.r 
 
 qr. lb. 1. s. d. 
 2 oi II 12 5i 
 
 36 
 
 
 
 
 11 
 
 9 
 
 
 
 38 
 
 3 
 
 3J-I 
 
 (166.) 
 
 lb. 
 i6i 
 
 I. s. 
 I 18 
 
 9 
 
 10 
 
 19 7 6 
 
 6J:'7i4, &c. 
 
 S;^ '214, Sec, 
 
 o 6 2|:'928, &-C. 
 Anf. 19 I 5|: '071, &c. 
 
 (167.) 
 
 cw.qr.lb. 
 10 o 
 
 I. s. 
 14 15 
 
 4 
 
 J 
 
 oz. 
 
 5 16 
 2 18 
 
 ' 9 
 
 o 8 
 o 2 
 
 O I 
 
 o o 
 
 ^i 
 
 , -5 
 
 3i-5 
 of '62; 
 Oi -812, Sec. 
 3 '453* &c. 
 
 10 15 
 
 o^ '640, Arc. 
 
 (169.) 
 
 1. s. d. 
 o 4 11* 
 6 
 
 d\v. gr. 
 
 2 c 
 
 o 12 
 
 To 
 
 I 9 
 
 9 
 
 12 
 
 t7 17 
 9 
 
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 51-8 
 
 •9^ 
 
 18 7 6^-7- 
 
 dw 
 10 
 
 o 8 
 
 10 
 
 o o 
 
 33 I 9 
 
 7 7 10^ 
 
 I 16 11^ 'J I 
 
 o 18 si -25 
 
 o I 10 '725 
 o o 7I -687, &c. 
 
 (170.) 
 
 s. d. 
 
 7 
 
 6i 
 5i-4S 
 
 '43 
 
 7- ^ 3 7 5i -95 
 
 7 7 •l62,&C. ■ ■ ^— «
 
 Part I. 
 
 
 P> R A C T 
 
 I 
 
 C E. 
 
 
 (11 
 
 I.) 
 
 ' (' 
 
 72.) 
 
 1. 
 
 S. d. 
 
 1. 
 
 s. d. 
 
 3 
 
 II 9iXi 
 
 2 
 
 11 9 X ; 
 
 
 
 10 
 
 
 
 10 
 
 35 
 
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 17 6X2 
 
 
 10 
 
 
 10 
 
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 i 258 
 
 15 
 
 
 
 5 
 
 16 - 
 
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 7 
 
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 5 
 
 251 
 
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 10 
 
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 19 ioi-5 
 
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 4 
 
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 16 
 
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 7l*o^ 
 
 
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 &c. 
 
 &c. 1877 
 
 3j-i:2; 
 
 
 2051 
 
 13 10 -92, 
 
 10 9 -97; 
 
 (■73 
 
 
 
 
 
 
 (174.) 
 
 1. s. 
 
 
 
 
 4 10 
 
 
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 d. 
 
 10 
 
 
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 8 
 
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 5 
 
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 10 
 
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 229 10 
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 348 14 
 
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 233 ^ iri 
 
 
 
 
 
 71
 
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 8 
 
 411 4 
 25 14 
 12 17 
 6 18 6 
 
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 qr. n 
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 1 o 
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 c t. 
 
 s. d. 
 
 17 91 
 
 8 I£>| 
 
 4 5i-5 
 
 IS 61-25 
 
 458 o si '25 
 
 (176.) 
 
 qr. n. 
 
 1. s. d. 
 
 1 i.il I II 9^ 
 12 
 
 19 
 
 I 6 
 
 10 
 
 190 15 o 
 7 18 III 
 7 i^i '^ 
 
 199 I lol -5 
 
 qr. n 
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 (1770 
 
 3 19 lit 
 4 
 
 15 19 4 
 
 4 
 
 63 19 o 
 3 19 ii| 
 
 9 iii*25 
 
 3 31*875 
 
 69 
 
 3 -125 
 
 (178.) 
 
 s. d. 
 10 
 
 2 
 1 
 
 To 
 
 -L 
 
 349 
 174 
 
 17 
 8 
 
 10 
 
 9 . 
 
 10 6 
 
 18 4i 
 
 qr. n 
 I 
 
 2 
 
 1 
 
 I 
 3 
 
 ± 
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 I. s. d, 
 III 6 
 
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 • 6 
 
 10 6 
 
 
 551 
 
 11 IO-5; 
 
 
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 47 S 
 4- 
 
 (»79.) 
 qr. n. 
 
 3 2 
 
 
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 380 
 
 3 2 
 
 at I 14 9i 
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 <: 
 
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 »73 »7 I 
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 139 I 8 
 
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 661 17 3 '3 
 
 
 
 (I So.) 
 
 
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 qr. n. s. 
 
 d. 
 
 ?75 
 
 • 3i- 
 
 18 
 
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 5 
 
 4)i8;6 
 
 469 
 9 
 
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 ■2 
 
 3, 
 
 
 2 
 
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 4 
 
 4i'87.- 
 2 •937>^'^^- 
 
 7 '46$, &c. 
 
 i| -281, &c 
 
 — — 
 
 «— 
 
 ■'- ■ -« 
 
 s. 422 
 
 2l , 
 
 9i' 
 
 id 
 
 d. 
 
 
 
 
 75 
 
 '44) I 
 
 9 9H 
 
 a.; !- 
 
 4 i| -281 
 
 44? 5 4i -281 
 
 II
 
 Tare and Tret. 
 
 TARE AND TRET. 
 
 (I.) 
 
 This Example is njcorlied. 
 
 Num. Answers. 
 (2.) apcwt. 3qr. 13^ lb. 
 (3.) 418 cwt. 1 qr. 10 lb. 
 (4.) 143 cwt. iqr. 251b. 
 (^,) 26 cwt. oqr. 10 lb. 
 (6.) 6ocwt. oqr. 12 lb. 
 {-,) 76 cwt. 3 qr. II lb, 
 
 (■3-) 
 
 cwt. qr. lb. 
 
 I 3 18 
 
 7 
 
 4 
 
 53 2 o 
 I 3 18 
 
 55 I i8lb. grofs 
 
 3 3 23 '^H 
 I 3 25 -642 
 
 5 3 20 '928 tare 
 
 49 I 25 -071 neat 
 
 lb. 
 
 14 
 
 I 
 
 Num. Answers. 
 
 (8.) 1 15 cwt. 3qr. 141b. 
 
 (9.) M243lb. 
 (10.) i8Qoolb. 
 
 (^ ^O 357 c^^f* 1 8 lb. 
 (12.) 'This is <ivorked. 
 
 cwt. qr. lb. 
 97 2 15 
 
 12 O 22 '87 
 0313 -63 
 
 13 o 8-50 tare 
 84 2 6 '49 neat 
 
 ,9X35=665 lb. grofs 
 
 Ha 
 
 7315, 
 332* 
 
 112)7647!: 
 
 68 ^V lb. tare 
 
 596 * -5 -^ lb. neat
 
 Part I. 
 
 (i6.) 
 cwt. qr. lb 
 4 3 H 
 
 1' A R E AND Tret. 
 
 ( I 7.) T^is w 'zvcrhil, 
 
 tiS.) 
 
 /!> 
 
 2 O 
 
 4 
 
 cwt. gr. lb. 
 12 "3 19 grof: 
 1 g tare 
 
 78 o o 
 .4 3 14 
 
 lb. 
 
 o I 26 -23 tree 
 
 82 3 i4gror3 
 
 12 oil '76 neat 
 
 10 
 
 ^ I r 
 
 I I 29 -7^- 
 i_ l-j o 2 26 •875' 
 
 14, o 6 '625 tare 
 
 68 3 7 '375 neat 
 
 (19') 
 cwt. qr. lb, 
 7 3 19 
 9 
 
 7» 
 
 lb. 
 16 
 
 (2C.) 
 
 cwt. qr. lb. 
 
 4 i »4 
 
 9 
 
 59 ' H 
 
 2 
 
 78 3 o 
 4 I H 
 
 lb. 
 
 42 2 6 
 
 7 3 rg 
 
 150 I 2j grofs 
 
 i *^ 10 2 27 -78, &c. 
 5 113 -89 
 
 16 o 13 '67 tare 
 
 83 o 14 grofs 
 
 2 s I ' 314 
 
 I i 1 I 26 •25' 
 
 o 2 27 '125 
 
 O O 20 -781 
 
 14 I 4 '1^6 tare 
 
 26 j68 3 9 •S43 tret 
 
 2 216 '532 futtle 
 
 26)134 I II '32 futtle 
 5 018 '74 tret 
 
 129 o 20 '57 neat 
 
 66 o 2 I '3 1 1 neat 
 
 (zr.) T;^.- ;^ .c-;n:../. 
 
 H 2.
 
 7C 
 
 Tare and T r e t; 
 
 (22. 
 
 lb. 
 
 (23.) 
 
 cwt. 
 5 
 
 41 
 
 qr. lb. 
 
 3 17 
 
 7 
 
 1 7 grofs 
 
 cwt. 
 
 7 
 
 qr. lb. 39 
 2 11 X5= 2 
 
 26)36 
 
 I 
 
 qr. lb. 
 
 3 II 
 
 5 
 
 27 grofs 
 3 27 tare 
 
 1 futtle 
 
 I i6'i5 tret 
 
 2 
 
 
 
 3 22-5 
 2 26-62 
 I 13-31 
 
 
 6-43 tare 
 
 4 
 
 168)34 
 
 
 3 1 1 -84 fee. f. 
 23-24 cloff 
 
 )37 
 
 I 0-56 futile 
 
 I I 20-48 tret 
 
 ■168)35 3 8-07 fecond futtle 
 o o 23-88 cloff 
 
 35 2 12*19 ^^^^ 
 
 34 2 1 6*59 neat 
 
 (z+. 
 
 cwt. qr. lb. 
 
 19 I 27 
 
 ' ' 9 
 
 18 o 18 
 
 1. 
 
 s. d. 
 
 5 
 
 4 
 
 X18 
 
 m 
 
 
 ID. 
 
 
 '6;j.9o 
 
 6 
 
 
 
 14 4 
 I 9i 
 
 01 
 
 2 li 
 
 73 
 
 9 3 
 
 J64 ri 4i- Anfwer. 
 
 
 cwt 
 
 . qr. lb. 
 
 
 
 1* 3 19 
 
 grofs 
 
 
 
 1 20 
 
 • tare 
 
 
 12 I 26 
 
 ^ neat 
 
 
 I. s. 
 
 d. 
 
 q^ 
 
 .lb. 5 17 
 
 8 
 
 I 
 
 
 16 
 
 4 
 
 I 
 y 
 
 
 12 
 
 
 
 70 12 
 
 
 
 
 
 7 
 
 4. 
 
 I 9 
 
 5 
 
 
 
 3i 
 
 i 
 
 16 
 7 
 
 9i-S 
 4i 
 
 
 
 3 
 
 8 -5 
 
 
 73 9 
 
 3 '3
 
 Fait r. 
 
 Tar 
 
 A* N D T R E r. 
 
 (26.) 
 
 lb. 
 
 cwt. qr. lb. 
 10 I 14 
 
 ib. 
 16 
 
 cwt. qr. lb. 
 
 4 3 n 
 
 7 
 
 54. I 7 grofs 
 
 7l-/-gji78 3 24-5' grots 
 
 II o 20*781, d'c. tare 
 
 ' -'-- 4- 3 n 
 167 3 3-718 luttle qr. lb. o r 6-/^ 
 
 6 I 22*758 tret 2 o break 
 
 315^- d ama. 5 o 2 3 ^ tare 
 
 161 I 8-960, ^jc. neat 
 
 1 i ifl 29 on f-g- futtle 
 
 ' I I I5i|l^reak, 
 
 &c.. 
 
 If I cwt. : il. 1 2gS. : : i6j cwt. 
 J qr. 8*96 lb* : 259l.cs. 5|d. 
 
 13 j 5 neat 
 
 lb. 
 
 (27-) 
 
 ^wt. qr. lb. 
 
 3 3 i> 
 X 29 
 
 Sj,-V|ri2 I 14 grofs 
 
 8 o 
 
 tare 
 
 26)104 I II futtle 
 4 o It tret 
 
 If iCAr. : il. 175. 6d. :: 27 cw. 
 2qr. 2341 : 51I. 19s. 3d. Anf. 
 
 (28.) 
 
 1. s. d. 
 
 414 6 neat value 
 211 4 cuftoms^ &c, 
 I I 6 freight 
 059 factorage 
 
 8 13 I 
 
 168)100 I 10-5 fecond fut. If rcwt. : 1 1 1 lb. :: 1 1 cwt. 
 
 o 2 10*595 
 
 clofF 
 
 ^99 2 27-604 neat 
 
 If icw, : il. I IS. 6d. :; 99 cw. 
 2qr."2 7*6o4!b, ; 157!. 2s. 
 Anfwer, 
 
 iqr. I 51b. : i cw. oqr. 1 5ilb.. 1 
 cwt. qr. lb. 
 I o 15V tare 
 
 lo o 27f neat 
 
 Jfiocwt. 27^ lb. : 81. i3.<;. 
 id. : : f cwt. : i6s. lo^d. 
 ^ °*| the fugar flood him in. 
 per cwt. neat. 
 
 f 3
 
 7S 
 
 S 1 M P L 
 
 (29.) 
 
 cwt. qr. lb. 
 
 lb. 
 
 N T E R E S T. 
 (30.) 
 87. 
 
 57.. 
 
 14UJ25 1 14 grofs 
 
 m 3 o I9-J 
 I 2 9I- 
 
 4 2 oj tare 
 2C 2 13I neat 
 
 If 7ilb. : 1 gal. . . ^yj^y>^. 
 2qr. i3slb. : 307!^ gal. neat, 
 which at 5s. 4d. pergal. amount 
 to 82I. 2S. oid. 7^. 
 
 I049S9 g^^^- gro^^ 
 450i\ tare 
 
 45o8y\ gall, neat 
 
 338ii-A-lb. neat 
 
 If 1 12 lb. : 5s. 3d. :: 33811/ylb. 
 : 79I.4S. io|d. 
 
 20CWt. 
 
 SIMPLE INTEREST. 
 
 T^is Example is n.vorled. 
 
 Num. Answer*. 
 (2.) 9-71. 13s. id. 
 (3.) 128I. 17s. 4id. 
 {4.) 87I. I2S. old. 
 (c.) 42i-+s. 5id. -75 
 (6.) 282I. 2S. 9d. 
 
 Num. Answers. 
 
 (7.} ic61. I IS. 6d. 
 
 (8.) J17I. OS. ii^d. 
 
 (9.) 17I. 8s. gid. -75 
 (10.) Sn6/> ;> (worked. 
 
 ('0 
 
 As 365d. : 81. 15s. i^d. '64 :i 3i5d. : 7L 11s. ifd, 
 
 70^. X 1+.^ X 4I -^ 36^00 = 13111111. z= 13I. IIS. jii. 
 (13.) 
 
 As 3^500 : 4 4L iz?. i< d. X 5 1 : 29 X 7 '• J?^' i5»- i<^<^' 
 
 Cr, 7300 : 4,41. iz.» Kd. ; ; i^X 7 • 'S'* 'S* i^«^' •Ac^-
 
 Part I. 
 
 Simple Interest. 
 
 7.9 
 
 (T4.) 
 As 36500 : 347I. los. y 4 : : iS X 7 ! 4I. 163. 2jcl. 
 
 From Jan. ift, to Sept. 22d, are 264 days. 
 As 36500(1. : 540I. los. X 4 :: 2646. : 15I. 12s. S^d, 
 
 (.6.) 
 
 (■7-) 
 
 d. 
 
 400 60 
 3 i 
 
 1200 I 
 
 200 
 
 14,00 
 
 2i 
 
 28 
 
 7 
 
 i 
 
 14 
 
 
 
 
 2 
 
 6 
 
 8 
 
 ^p 
 
 
 
 9 
 
 2 5 101 
 35 o o 
 
 d. 
 
 3lil294- 
 
 2o)73_ 6 
 Anfu'er 3 13 6 
 
 Anf. 37 5 loi (18.) This is nxjorkcJ. 
 
 .20)500 
 4+ 
 
 - 1-24)93 15 
 20 
 
 s. 18)75 
 12 
 
 d. 9)00 
 
 I9-) 
 
 m. 
 
 6 
 
 X 
 
 
 3 
 
 X 
 
 d. 
 
 
 30 
 
 i. 
 
 3 
 
 3 
 
 TO 
 
 I. s. d. 
 
 24 18 9 
 
 5 
 
 124 13 9 
 
 12 9 42: 
 
 6 4 8|: 
 
 2 I 61 
 
 145 9 4i: 
 o 4 i| 
 
 '5 
 
 H5 5 21 -5. Anf.
 
 So 
 
 Simple 
 
 (20.) 
 
 INTEREST. 
 (21.) 
 
 From May 15th, 1787, to 
 Sept. 2 2d, 1789,316 2yrs. 4111. 
 icd. 
 
 5.15)00 
 
 I. 
 
 500^ 
 
 m 
 4 
 
 d. 
 
 10 
 
 ± 
 3 
 
 1 
 
 1. S. 
 
 iS 15 
 
 d. 
 
 
 5i 
 
 15C0 
 375: 
 
 57 10 
 ro 
 
 
 
 
 
 5 
 
 .18)75 
 20 
 
 Anf. 44 5 
 
 ^ 
 
 I. 
 
 100 
 4 
 
 1. 4)00 
 
 ID. 
 2 
 
 d. 
 IC 
 
 6 
 
 J 
 
 1. s. d. 
 400 
 
 o 13 4 
 
 ToO , 4 
 
 i 10 O 2 
 
 ICO o 
 
 ['66 
 
 rremiuni o 10 o 
 
 Anf. ici 7 jJ-33 
 
 y. m. d. 
 
 2100 
 ^8 7 3 
 
 2 4 27 
 
 t's 10 
 
 1-43) 77 »o 
 20 
 
 £.15)50 
 12 
 
 d. 6)00 
 
 22.) 
 
 
 
 
 
 in. 
 
 I. 
 
 s. 
 
 d. 
 
 4 
 
 i 
 
 43 
 
 '5 
 
 6 
 
 2 
 
 d 
 
 
 
 
 
 30 
 
 
 87 
 
 II 
 
 
 
 
 
 14 
 
 II 
 
 10 
 
 jVol 
 
 3 
 
 12 
 
 i:i 
 
 
 1C5 
 
 '5 
 
 9f 
 
 
 
 7 
 
 3i'2 
 
 
 l^'S 
 
 8 
 
 5l *8 
 
 IS 
 
 10 o 
 
 980 18 5^-8 
 
 (23.) 
 S'ZvV Example is ivorkeJ,
 
 Paft I. SimpleInterest. 8t 
 
 (24O 
 
 100 -f 4 X 5 = 120I. 
 
 As 120I. : lool. : : 570I. i6s. 6d. ; 47 5I. \p, gd, Anf. 
 
 100 + si X /\.i— 1 1 61. 17s. 6d. 
 
 As 1 1 61. 17s. 6d. : lool. :: 205I. iis. 7|d.-f : 175!. i8s, 
 Anfwer. 
 
 (26.) 
 
 100 -f 4t X 3| = 1 1 61. 17s. 6d. 
 As 1 1 61. 175, 6d, : igol. : ; 350I. 12s. 6d. : 300I. Anfwer, 
 
 Tiif Examph is ^worked* 
 
 (28.) 
 
 1. s. d. 
 
 465 8 3 amount 
 344 15 o principal 
 
 120 13 3 intereft 
 
 As ^44!. 15s. : 120I. 13s. 3d. :: tool. : 28I. 
 Then 28-7-7 =4^' percent, Aufwer. 
 
 (29.) 
 For 175I. los. read I "^^h iSs. 
 1. s. d. 
 205 II 7 1 'I amount 
 175 18 o principal 
 
 29 13 7^ 'I intereil 
 
 .As 175I. iBs. ; 29I. 13s. 7jd. 'I : : 100 : 1 61. 17s. 6d. 
 Then 16I. 17s, 6d, -J- 3.J = n-I, ics. An(\vzx,
 
 Bx Simple Interest. 
 
 (30.) 
 I. s. d. 
 350 12 6 
 300 o o 
 
 50 12 6 
 
 As 300I. : 50I, I2S. 6 J. :: ico : 16I. 17s. 6d» 
 Then j6I. 17s. 6d. -^ 41 =1: -J. 15s. Aufwer. 
 
 1.7191 II 
 
 20 
 
 s.jSjji 
 
 12 
 4 
 
 1. s. 
 475 n 
 
 (31.) 
 
 d. 
 
 9 
 
 4 
 
 7>?7.r 
 
 is 
 
 370 16 6 amount 
 475 13 9 principal 
 
 I. 19!C2 1^ 
 20 
 
 s.o'si- 
 
 12 
 
 d. 6!6o 
 4 
 
 
 9^ 2 9 intereil 
 
 As 19I. cs. 6i^. 'I : I y. : ; 05I. lu 
 
 2(40 9d. ; 5}TS. 
 
 I. s. i^cr 17 5I. I OS. rf ^^175!. I Ss,^ 
 
 175 i3 
 
 4^- I. s. 
 
 ■ 2C5 I! 7I •* amount 
 
 703 12 175 18 o principal 
 
 87 19 ^ 
 
 29 13 tI *t infcrelt 
 
 As 7I. 78s. 3f-TVo •• ly- •• • 29I. 15s. 
 
 «iSS 7^<i- 'f • 3|yrs. Anfwer.
 
 Tart I. Simple Interest. 
 
 83 
 
 300 
 3l 
 
 900 
 
 1. 11I25 
 20 
 
 (34.) 
 
 J. s. d. 
 
 350 12 6 amount 
 
 300 o o principal 
 
 50 12 6 intereft 
 
 5. 5I00 
 
 As III. 5s. : lyr. :: 50I. 12s, 6J.. 
 : 4iyrs. Anfvvcr. 
 
 (35-) 
 As 160I. : icol. : : 264.16 
 100 X 12 :=: 1200 
 
 12 
 
 60 
 
 ICO 
 
 1. 160 
 
 27616 
 
 ICO 
 
 160 ) 2761600 
 
 1. 17260 Anfwer. 
 
 (56.) 
 
 1. 
 
 2ift Oa. 1782, lent 20 
 
 2,2d May, 17J54, borrowed 150 
 
 Ea;. 130 
 30th Ju!y, i7Si^> borrowed 15c 
 
 s. 
 400 
 
 2600 
 
 Bal. 280 rz 5600 
 21ft JuJ^', 1785, pa-d 15 18 
 
 Bal. 264 2 rz 5282 
 21II Aug. 17S5, paid 40 
 
 B.il. 224. 2 iz: 4482 ... 
 ftifl O^. 17S5, jra'd 50 
 
 Bal, J74 2 ^ 34S2 ... 
 
 Days. 
 579 
 
 69 
 
 356 
 
 31 
 
 61 
 
 115 
 
 Produft-s. 
 2316CO 
 
 179400 
 1993600 
 163742 
 25^.3402 
 400430 
 
 Carried oyer*
 
 8+ 
 
 Simple Interest. 
 
 1. S, S. 
 
 Bal. brought over 174 2 n 3482 
 13th Feb. 1786, paid 9 12 
 
 Bal. 164 io ~ 3290 
 13th June, 1786, paid iii 
 
 1070 
 
 Bal. 
 13th Jan. 1787, paid 
 
 53 10 =: ic 
 So 
 
 Bal. due to me 
 
 Intereft due to my friend 
 
 26 10 
 
 23 6 i^ 
 
 Bal. in my favour 
 
 3 3 lof . 
 
 Days. 
 
 214 
 
 Produfts. 
 394800 
 
 Sum 3634354 
 Subt. 231600 
 
 7300)3402754 s. 
 
 3 lof Anf. Intereft due 23 6 i^d, 
 
 500 gui 
 
 {37.) 
 1. s. 
 700 15 Amount 
 525 o Principal 
 
 [75 15 Intereft 
 
 The intereft of 52 5I. for one year, at 4^ per cent, is 23!. 12s. 6d, 
 Then 
 
 23I. I2S. 6d. : I yr. : : 175 15s. : 7 yrs. 16036yd. 
 
 Now by counting 7 yrs. 160 days back, from the 25th September, 
 3788, the bond will be found dated April iStii, 1781. 
 
 (38.) 
 
 1. s. 
 .11)38 10 
 
 310 rate per cent, or intereft of lool. for one year. 
 
 Now to anfwer the conditions of the queftion, it is evident that th( 
 intereft muft be four times the principal, or 400I. Hence, 
 As 3|1. : 1 yr. t: 4CCI. : ii4yyrs. Anfwer. 
 
 (39-) 
 
 The intereft of 50ol.,fjr 4} years, at 5 per cent, is 11815 c 
 The intereft of 2 50I. fu" 9I years, at 2^ per cent, is 59 7 ^ 
 
 Difference 59 7 ^
 
 Part I. 
 
 Brokerage, 
 (40.) 
 
 !. s. d. 
 
 To a bin dated ift Auguft, 1786, 7 „ 
 payable lit Oaobcr, 1786, J ^^^ 
 
 5tli Odlobcr, 17S6, received 94 17 o 
 
 Bal. 863 I o 
 
 27th November, 1786, received 47 19 6 
 
 Bal. 815 I 6 
 i^h December, 1786, received 105 o o 
 
 Bil. 710 I 6 
 tft January, 1787, received 55 I ' A- 
 
 Bal. 654 10 2 
 J5th March) 17S7, received 101 14 o 
 
 Bal. 552 16 2 
 [4th May, 1787, received 110 50 
 
 BjI. 44Z II 2 
 
 T9:h Auguft, 17S7, received 140 2 6 
 
 Bal. 302 8 8 
 
 1 1 ;h September, 1787, received 50 6 6 
 
 Bal. 252 8 2 
 ijth March, 178S, received 2!;2 8 2 
 
 Days. 
 4 
 
 53 
 18 
 
 17 
 73 
 58 
 99 
 23 
 
 185 
 
 Sum 
 
 Prod'jas. 
 
 3^31 la o 
 
 45741 13 o 
 
 14671 7 o 
 
 12071 5 6 
 
 47779 3 ?' 
 
 32064 17 8 
 
 4^813 5 6 
 
 695s 19 4 
 
 46695 10 IQ 
 
 253<322 13 o 
 
 I. S. 
 
 253622 13 X 4 _ , 
 ^^^'" ^6^0 = '71. 15^. lold. Anr.vcr 
 
 'BROKERAGE. 
 
 This Example rs iMorked, 
 
 Num. A^'swERs, 
 (2.) 20I. los. 3-025(1. 
 (5.) 2I. 8s. 5i-27d. 
 UO 37^» 4s» ii*4ud. 
 
 Num. Answers. 
 (9.) 4* MS. icil. 
 (6.) Ill IS. oj • J9'?
 
 86 Commission, I n s u r a n c e, &c. 
 
 COMMISSION, 
 
 Num. Answers* 
 (2.) 28I. 8s. 3|d. -2 
 (3.) 61. 17s. 8|d. -22 
 (4.) 51L 8s. 8id. •+ 
 
 (I.) 
 This Example is nxorhed. 
 
 Num. Answers. 
 (5.) 74I. IIS. 3ld. -28 
 (6.) 965I. 6s. 8id, "9 
 
 INSURANCE. 
 
 Num. Answers. 
 (i.) i^. ics. 4fd. 
 (2.) il. 6s. lod. of '35 
 (3.) 8504I. 3s. lod. of '44 
 (4.) 557I. 8s. lid. -84 
 
 Num. Answers. 
 (^. ) 1 157I. I2S. 6d. 
 (6.) 7s. 6|d. -08 
 (7.) 6;1. i8s. iiid. -45 
 
 TURCHASING STOCKS. 
 
 - ^Ti/V Exam-pie is <v:orked^ 
 
 (2.) 
 Tl^/V Example is nvorhed, 
 
 (3-) 
 If zool. : I25|l. :: 7575I. i^s- : 9J'7>- ii^--i- 
 
 (4-) 
 If loci. : Sg;i. :: gcel. : S03I, 5s. 
 
 (;.) 
 If lool. : >96^l. :: 1759!. i8s. gd. ; 3+53!. 175. 6fd.-;.
 
 ?art I, Purchasing S t o c k ai 87 
 
 5000 
 
 85I 
 
 8 150 loo 
 
 
 6 5 
 
 4268 15 
 
 brokerage 
 
 4275: 
 
 the purchafe 
 
 or) 
 
 7159 10 
 20 
 
 ' 
 
 1 1190 
 12 
 
 
 10180 
 4 
 
 
 3I20 
 
 
 1. s. 
 8|7 11 
 
 d. 
 
 10| *2 
 
 18 
 487 19 
 
 1 1 1 '4 brokerage 
 6|-6 
 
 488 18 
 
 6| the purchafc 
 
 425-000 
 
 : - ^8 75 
 
 4268I75: 
 20 
 
 1 5 loo 
 
 I. s. 
 
 759 ^o 
 8 
 
 6076 o 
 8 
 
 48608 o 
 189 17 6 
 
 487197 17 6 
 20 
 
 19I57 
 
 12 
 6.90 
 
 4 
 
 3160 
 
 61 — ^61 = 4I: per cent, the difference, or lofs, 
 * 700C0 8 )7oo|oo 
 
 4i 
 
 87 10 brokeriTge for buying 
 
 280000 87 10 ditto for felling 
 
 350CO 
 
 3150I00 
 J7J 
 
 175I. the ^o»^ broker gained 
 
 3325I. the gentleman loft 
 
 I 2
 
 £S 
 
 3) I s c o u 
 
 DISCOUNT. 
 
 5 15 
 2 
 
 (I.) 
 TI:is "Example is iV2riiJ» 
 
 ' it-) 
 
 ;)»i 10 
 
 
 3 16 
 
 ICO 
 
 8 
 
 As 103 16 8 : 100 ; : 594I. 14s. 9d, : 572]. i^^> 
 6 ill. 
 
 im. 
 
 ij 8 
 
 
 
 
 u 
 
 
 i 13 
 
 4 
 
 4 13 
 
 4. 
 
 100 
 
 
 As '104 13 4 : 4J. 13s. 4d. ;: 915I. 178. ; 40I. 
 6s. S-iVyd- 
 
 6 m. 
 
 
 5 
 
 2 
 
 10 
 
 
 im. 
 
 I 
 
 
 
 8 
 
 4 
 
 
 
 7 
 
 18 
 
 4 
 
 
 I 
 
 CO 
 
 
 
 As 107 18 4: 100 :: 75I. :691.9s. ii;d.-,yg. 
 
 (>■•) 
 From 22 Sept. to Chrillmas, arc 94 days. 
 
 As 365d. : 61. :: 94d. : il. ics. lojd. intereftof iccl. 
 for 94 days. 
 
 As loil. los. ic|d. : lool. :: 900I. : 8861.6s. 2-^y^\%d. 
 Anrvver.
 
 Part I. 
 
 I> I S C O U N T« 
 
 H 
 
 (64 
 
 If36;d.: 7-1-1. :: J7d. : d. 3U gld. 
 
 If loil. 3s. 9^d. : 15000I. : : il, 3s. gfd. : 176I. 9s. 
 
 in 5 6Da ^ 
 
 (7.) 
 
 From the 5th of July, 1788, to the 24th of June, 17893 
 are 354 days. 
 
 If 365 d. : 711. :: 554d. : 7I. 3s. ofd. 
 If 107I. 3s. old. : looh : : 1797I. lof/ : 1640!, 33^. 
 10 s^Vji^* Anfwer. 
 
 
 
 
 
 
 (^0 
 
 
 
 
 
 
 
 3 } 747 
 
 18 
 
 
 
 R. 
 
 d. 
 
 249 
 
 6 due 
 
 immedi 
 
 r. s. 
 
 ately, <Src,- 
 
 I. 
 
 
 
 d. 
 
 6 m. i 
 
 8 
 
 12 
 
 6 . 
 
 = 8^1. 
 
 6m. 
 
 4.m. 
 
 3 
 
 8 12 
 
 6 
 
 1 m. 6 
 
 4 
 
 6 
 
 3 
 
 4. 6 
 
 a 
 
 
 
 H 
 
 4i 
 
 
 
 z 14 
 
 2 
 
 S 
 
 
 
 7l- 
 
 7 
 
 5 
 
 100 
 
 
 
 
 
 
 J 
 
 100 
 
 
 
 IQS 
 
 
 
 71- 
 
 
 
 7 
 
 5 
 
 I. S; d. 
 
 ff 105!. OS. 7^d. : lool. :: 249I. 6s. : 237 7 i-| 
 
 If 107I. OS. 5d. : lool. :: 249I. 5s. : 234 10 ol 
 
 249 6 o 
 
 The prefent worth 
 
 1 3 
 
 72
 
 D 
 
 - ^ 
 
 (9-) 
 
 ^O O 
 
 ^ 0\0C VO 
 
 O «^ ■* O 
 
 O c^ " O 
 H >-< I-. c-n 
 
 ^ 
 
 00 
 
 t^ 
 
 CO 
 
 t 
 
 
 • 
 
 
 Add 
 
 
 
 
 ^'i-O 
 
 CO 
 
 VC 
 
 
 O r-.oc O 
 
 I 
 
 8 
 
 
 N M O O 
 
 -1"-+* 
 
 go 
 
 
 VC 
 
 00 
 
 1 o ^ o 
 
 
 
 
 
 H 
 
 c 
 
 •^i- 
 
 p.1 M - o 
 
 
 oo 
 
 
 H)C> »-.icl M 
 
 
 
 
 
 r^ 
 
 l 
 
 - vO ro 
 
 :r I H* SIS 
 
 
 
 oo 
 
 oc 
 
 ri- 
 
 
 VO 
 
 VO 
 
 
 , 
 
 IH 
 
 -* 
 
 
 
 
 
 o 
 
 
 o 
 
 o 
 
 o 
 
 -o 
 
 1- O 
 
 2 
 
 
 
 
 J 
 
 5 rt 
 
 
 
 1. s. d. U s.- d. 
 
 Firft, If ICO i6 8 : O i6 8 : 
 
 Secoad, If loi 5 0:1 5 o : 
 
 Third, If 103 15 o : 3 15 o : 
 
 Fourth, If 104 ij 8:411 8 : 
 
 Fifth, It 10- o 0:5 O O : 
 
 1. 
 
 s. 
 
 d. 
 
 1. 
 
 s. 
 
 d. 
 
 200- 
 
 4 
 
 : 
 
 I 
 
 ^3 
 
 I 
 
 133 
 
 9 
 
 4 : 
 
 1 
 
 12 
 
 "J 
 
 114 
 
 8 
 
 : 
 
 4 
 
 2 
 
 S| 
 
 300 
 
 6 
 
 : 
 
 13 
 
 3^ 
 
 ^i 
 
 5^ 
 
 8 
 
 8 : 
 
 2 
 
 9 
 
 I If 
 
 The Difcount 
 
 .3 I ic|
 
 Parti. Discount. 91 
 
 (10.) 
 I. s. d. 
 576 = 5|I. 
 3 
 
 |=:Jof3Ji6 2 6 365:d. : 5I. 7s. 6d. :: 41 d. 
 
 o 12 oj 
 
 100 o o 
 
 IX I2S. o|d. 
 
 120 15 2-5: : 100 ;: 1789I, 19s. lod. : 1482I. 5s, 
 
 7d* Anfwer. 
 
 (II.) 
 
 The difference is 500!. The intereft of loooL for 2oyrs, 
 is loool. but the difcount of the fame fum is only 500I. 
 
 (12.) 
 For 5 per cent, in this queftion> read 6. 
 
 1. 
 
 I. 
 
 1. 
 
 
 I. s. 
 
 d. 
 
 106 : 
 
 100 : 
 
 : 100 
 
 
 94 6 
 
 9i '20 
 
 112 : 
 
 100 : 
 
 : 100 
 
 
 89 5 
 
 8t-28 
 
 118 : 
 
 TOO : 
 
 : 100 
 
 
 84 14 
 
 lol -92 
 
 124 : 
 
 100 : 
 
 : 100 
 
 
 80 12 
 
 ^^l'3S 
 
 130 : 
 
 : 100 : 
 
 : 100 
 
 
 76 18 
 
 5t*i4 
 
 The prefent vakie of the debt, al- ") 
 
 lowing each perfon a difcount i 425 18 9^ 
 of 6 per cent. 3 
 
 Now, 
 100 + 1064- 112 4- 1184- 124 = 560!. the money which 
 may be made by receiving the payments as they become 
 due, &c. 
 
 Again, 
 The intereft of 425I. i8s. 9^d. for 5 years, at 6 per cent, 
 is J27I. I js. 7-id. which added to 425I. i8s. g^d. gives 
 553I. 14s. 4|d. this fubtrafted from 560I. leaves 61. 5s. 7id. 
 advantage, by receiving the debts as they become due.
 
 92 Eq^uation of Payments* 
 
 EQUATION OF PAYMENTS. 
 
 This Example is ^worked. 
 (z.) 4I months. | (5.) 6 months* 
 
 (4.) 6^ months^ 
 
 (5-) 
 1. s. s. d. 
 
 J 50 o X 2 m. = 3000 X 60 = 1 80000 
 
 147 17 X 74d. = 2957 X 74 =z 2 1 881 8 
 
 137 18 X 95 d. =r 2758 X 95 = 262010 
 
 65 OX 5 m. z= 1300 X 150 =: 195C00 
 
 10015 ) 855828 
 
 85^. tWiV" 
 or 2 m. 25 days. Anfwer* 
 
 (6.) (7.} 
 
 J 2 90 
 
 = 6 X 
 
 4 = 24 
 
 i = 
 
 18 
 
 =: 3 X 
 
 ^ = 15 
 
 4 = 
 
 15 X 25 = 375 
 
 z: 2 X 
 
 7 = H 
 
 X 
 
 9 
 
 10 X 90 = 900 
 
 I X 
 
 10 rz 10 
 
 
 47 X 137 = 6431 
 
 12 
 
 )63 
 
 
 90 7706 
 
 *^ 
 
 — ■ 
 
 
 
 
 Ji 
 
 months. 
 
 85-11 days. 
 
 (8.) 
 
 I. s. d. f. d. 
 
 50018 oxfyr. =: 480864x1X21= S7757680 
 
 9C0 17 6 X I yr. i'4d. zr 864.840 x 479 :=: 41^258360 
 
 1700 18 4|X 2|yr. =: 1632883 x 91^1= »450<^05737i 
 
 »978527 ) 199202^7771 
 
 668 d. 2325661! rem* 
 
 Cr I yr. 303 days.
 
 Part I. Compound iNTtREST^ 93 
 
 COMPOUND INTEREST, 
 
 (I.) 
 
 This Example is ivorked, 
 
 m 
 
 (2.) 
 
 I. s. d. 
 
 5I. i'-o ) 700 1 8 o principal 
 
 35 o loj '2 intereft forthe iftyear 
 
 a'<? ) 735 ^8 io| -2 amount for ditto 
 
 ^6 15 / If -36 intereft ftr the zd year 
 
 "s'o ) 772 14 10 -56 amount for ditto 
 
 38 12 8J '628 intereft for the 3d year 
 
 ^'^ ) 8ii 7 7 '188 amount for ditto 
 
 40 1 1 4f '2094 intereft for the 4th yeaf 
 
 8p 18 1 if '3974 amount for ditto 
 700 18 o principal 
 
 , 151 o II J "3974 whole intereft 
 
 (5.) 
 1. s. d. 
 41.^*5- ) 1057 17 6 principal 
 
 42 6 3 1 '4 intereft for the i ft year 
 
 25 ) I ICO 3 9i: '4 amount for ditto 
 
 44 o 1 1 '296 intereft for the 2d year 
 
 25 )ii44 3 11^ '696 amount for ditto 
 
 45 15 4+ '187 intereft for the 3d year 
 ♦ 
 
 -25 }ii89 19 3|- -883 amount for ditii 
 
 47 II Hi •67^- intereft for the 4th year 
 
 25 )J237 'I 3i '559 amount for ditto 
 
 Carried over.
 
 54 Compound In teres t» 
 
 Brought over 
 
 1. . s. d. 
 
 25 ) 1237 1 1 3i '559 ^i^o^^t for the 4th year 
 
 49 10 Oi '462 intereft for the 5th year 
 
 25 )i2S'] I 4 '021 amount for ditto 
 
 51 9 7|J|B intereft for the 6th year 
 
 1338 10 ii|*4oi amount for ditto 
 
 1057 17 6 principal 
 
 280 13 5I '401 whole intereft 
 
 (4.) 
 
 In this and the following Examples, the amount oreach 
 payment, (or refult of each operation) is only put down^ 
 the whole work would have taken too much room* 
 
 Anfwer 
 
 Anfwer 
 
 I. 
 
 s» 
 
 d. 
 
 522 
 
 2 
 
 S-i amount for the firft year 
 
 544- 
 
 6 
 
 61 ditto for the 2d 
 
 567 
 
 9 
 
 2J: ditto for the 3d 
 
 591 
 
 II 
 
 6i ditto for the 4th 
 
 616 
 
 H 
 
 4| ditto for the 5th 
 
 T. 
 
 s. 
 
 d. 
 
 733 
 
 5 
 
 amount for the ift year 
 
 768 
 
 
 7 ditto for the 2d 
 
 804 
 
 II 
 
 3 ditto for the 3d 
 
 842 
 
 ^5 
 
 7 ditto for the 4th 
 
 882 
 
 16 
 
 2|- ditto for the 5th 
 
 924 
 
 14 
 
 io| ditto for the 6th 
 
 968 
 
 13 
 
 5 ditto. for the 7th 
 
 The three Elkmples following are folvcd by the Note 
 to the Rule of Compound Intereft..
 
 Part h Compound Interest. 
 (6.) 
 
 Teaf/y Faymettts* 
 1. s. d. 
 52^ o o amount for the ill year 
 5:51 5 o tlitto for the 2d 
 
 578 16 3 ditto for the 3d 
 Anfwer 607 15 q\ ditto for the 4th 
 
 Half-yearly Paymefits, 
 1. s. d. 
 
 512 10 o amount for the ift payment 
 525 6 3 ditto for the 2d 
 538 8 10 J ditto for the 3d 
 
 551 18 I i ditto for the 4th 
 ^6^ 14 o| ditto for the 5th 
 
 579 16 1 1 ditto for the 6th 
 594 6 loj ditto for the 7th 
 
 Anfvver 609 4 o^ ditto for the 8th 
 
 ^larterly Payments, 
 
 I. s. d. 
 
 506 5 o amount for the ift payment 
 
 512 II 61 ditto for the 2d 
 
 518 19 8:1: ditto for the 3d 
 
 525: 9 5i ditto for the 4th 
 
 532 o 9I ditto for the 5th 
 
 538 13 9| ditto for the 6th 
 
 54 J 8 6 ditto for the 7 th 
 
 552 4 loi ditto for the 8th 
 599 211 ditto for the gth 
 556 2 85 ditto for the loth 
 
 573 4 ^1 ^^^^o ^o'' ^^^^ ^^h 
 
 980 7 6^- ditto for the 1 2th 
 
 587 12 7^ ditto for the 13th 
 
 594 19 6~ ditto for the 14th 
 
 602 8 3i ditto for the 15th 
 
 Anfwer 609 18 10^ ditto for the 1 6th 
 
 (7.) 
 
 1. s. d. 
 730 3 I of: amount for the ift payment 
 74S '+ 24 ditto for the 2d 
 761 II I A ditto for the 3d 
 777 »4 9i <iitto for^thc 4th 
 
 95
 
 96 Compound Interest. 
 
 1. 5. a. 
 
 794 5 4 amount for the 5th payment 
 
 811 2 io| ditto for the 6th 
 
 S28 7 7I ditto for the 7th 
 
 845 19 SI ditto for the 8th 
 
 863 19 2-| ditto for the gch 
 
 882 6 5 ditto for the I oth 
 
 901 I 4^ ditto for the nth 
 
 Anfwer 920 4 4 j- ditto for the 1 2th 
 
 (8.) 
 
 this E 
 t 
 
 xara 
 
 iple, _/i?r 4I per cent, r^-/?^ 4 per cent, 
 
 748 
 
 s. 
 6 
 
 .u. 
 2 amount for the ift payment 
 
 7S5 
 
 15 
 
 10 ditto for the zd 
 
 ^^3 
 
 7 
 
 ditto for the 3d 
 
 770 
 
 ^9 
 
 8 ditto for the 4th 
 
 778 
 
 13 
 
 loj ditto for the 5;th 
 
 786 
 
 
 'jj ditto for the 6th 
 
 794 
 
 6 
 
 loj ditto for the 7th 
 
 802 
 
 5 
 
 9 ditto for the 8rh 
 
 810 
 
 6 
 
 27 ditto for the 9th 
 
 818 
 
 8 
 
 3i ditto for the loth 
 
 826 
 
 1 1 
 
 lit ^itto for the nth 
 
 834 
 
 17 
 
 3^ ditto for the. 1 2th 
 
 8+3 
 
 4 
 
 3 ditto for the 13th 
 
 891 
 
 12 
 
 )o|- ditto for the 14th 
 
 860 
 
 3 
 
 2 ditto for the 15 th 
 
 868 
 
 IS 
 
 3 ditto for the 1 6th 
 
 877 
 
 9 
 
 0^ ditto for the 17th 
 
 886 
 
 4 
 
 6 ditto for the i8th 
 
 895 
 
 I 
 
 9 ditto for the 19th 
 
 904 
 
 
 
 9^ ditto for the 20th 
 
 9^3 
 
 I 
 
 7 ditto for the 21ft 
 
 922 
 
 4 
 
 2| ditto for the 2 2d 
 
 931 
 
 8 
 
 7A ditto for the 23d 
 
 940 
 
 14 
 
 1 1 ditto for the 24th 
 
 950 
 
 3 
 
 0} ditto for the 25th 
 
 999 
 
 13 
 
 1^ ditto for the 26th 
 
 969 
 
 5 
 
 0^ ditto for the 27th 
 
 978 
 
 18 
 
 lol- ditto for the 28th 
 
 giS 
 
 H 
 
 8 ditto for the 29th 
 
 998 
 
 12 
 
 5 ditto for the 30th
 
 Part I. Single Fellowship. 97 
 
 1. s. d. 
 
 1008 12 1 1 amount for the 31ft payment 
 
 1018 13 10^ ditto for the 3 2d 
 
 1028 17 7^ ditto for the 33d 
 
 '039 3 3 J ditto for the 3 4.th 
 
 1049 II 2\ ditto for the 35th 
 
 1060 I li ditto for the 36th 
 
 1070 13 i^ ditto for the 37th 
 
 740 18 o principal 
 
 Anfwer 329 15; it intereft 
 
 SINGLE FELLOWSHIP. 
 
 This Example is 'worked* 
 
 (2.) 
 J. s. d. 
 500 17 10 A's (Took 
 735 o o B's rtock 
 
 rem. 
 
 1235 17 10 fum 
 
 i. s. d. I. s. I. s. d. !. s. d. 
 
 1135 17 10 : 300 15 : : 500 17 lo : 121 17 9I— 19<;484 A's iTiare 
 
 1235 17 10 : 300 15 : : 735 o o : 178 17 21—101130 B's iharc 
 
 300 15 o proof 
 
 (3-) 
 1. s. d. 
 5075 18 o whol^ ft ock 
 
 574. i6 o A's ftock 
 
 947 18 6 B's — — 
 
 3044 17 o C's 
 
 4567 
 
 §08 6 6 D's Hock 
 
 K
 
 98 
 
 S I N G L 
 
 ELL O W S H I 
 
 ]. s. I. s. 
 
 io-5 18 : 1358 18 
 
 5075 18 : 1358 18 
 
 5075 18 : 1358 18 
 
 5C75 18 : 1338 18 
 
 1. s. d. 1. s. d. rem. 
 
 574 16 o : 153 J7 7I — 8825S A'sflia 
 
 947 18 6 : 253 15 5|— 77710 B^s 
 
 3044 17 o : 815 3 il — 6586 C's 
 
 508 6 6: 136 I 81— 30502 D's — 
 
 [358-18 o 
 
 Or thus, by Note I. 
 
 I. s. d. 1. 
 
 574 160— 574-8 A's flock 
 
 947 18 6 = 947.925 B's 
 
 3044 L7 o i:: 3044*85 C's 
 
 508 6 6 zr . 5c8'525 D's 
 
 5075 18 o — 5075'o whole ftock 
 
 5075-9) i35S'9coooo ( -267716, &c. con.mon multiplier. 
 
 5^4-8 x -267716, &c. 
 r47v 25 X •267716 
 3044-85 X -267716 
 508-325 X -267716 
 
 1. 
 
 
 1. 
 
 s. 
 
 d. 
 
 
 I53-SS3I568 
 
 — 
 
 153 
 
 17 
 
 7^ A's 
 
 fhare 
 
 253-7746893 
 
 — 
 
 253 
 
 M 
 
 5iB's 
 
 
 815-1550626 
 
 — 
 
 SI5 
 
 3 
 
 Il C's 
 
 
 I36.0S67557 
 
 ±1 
 
 136 
 
 I 
 
 S|D's 
 
 — 
 
 (4-) 
 
 This Exar^i 
 
 /, ;^ 
 
 -'.'"-'•■■(/ /^' Note !_/?. 
 
 1. s. d. 
 
 
 
 540 14 
 
 — 
 
 540-7 A's debt 
 
 770 18 
 
 ziz 
 
 770-9 B's 
 
 4005 14 
 
 — 
 
 40057 C's 
 
 975 18 9 
 
 'JZ. 
 
 975-9375 ^'^ 
 
 3150 
 
 zn 
 
 3150. E's 
 
 9443 4 9 ~ 9443*2375 fum of the debts 
 
 whole debt. elrefls. 
 A:9443-2375l. : 7I74-71- 
 per pound. 
 
 il. : '75977 12 J 2I. zi 15s. 2|d. 'I nearly 
 
 p 410-808294 
 
 X I 585-707627 
 
 75977i2i2< 3043-415543 
 
 — I 741 489217 
 
 1^393-27931'* 
 
 1. 
 
 s. 
 
 d. 
 
 =: 410 
 
 16 
 
 H 
 
 = 585 
 
 14 
 
 li 
 
 — 3043 
 
 8 
 
 3^ 
 
 — 741 
 
 9 
 
 9i 
 
 = 2393 
 
 5 
 
 7
 
 Part I. 
 
 Single Fellowship, 
 
 99 
 
 Otherwife, 
 
 After you have found whst the 
 Bankrupt's efleds- will pay in the 
 pound, (as above, or by the Rule 
 cf Three) then each particular part 
 may be found bj^ the Rules of Practice, 
 fufHcieiitly exaft for common pur- 
 pofes. — The above cftcdls will pay 
 15s. 2[d. 'I per pound.- 
 
 540 14 
 
 
 
 270 7 
 
 ^5 3 
 
 4 10 
 
 
 6 
 1I.6 
 
 II 
 
 c 4 
 
 3 -7 
 
 2i-8 
 
 410 16 iiA'5 fiia. 
 
 . I. s. d. 
 
 7754 17 
 15749 14 
 3497 16 
 5754 i« 10 
 3775 19 
 37497 19 S 
 
 — 
 
 (5.) 
 By Not& I. 
 
 7754-85 = A's flock in the Cargo 
 
 -j/lOV'S — C'2 "■ ■ - ■ 
 
 
 
 37497*9S3' = F's 
 
 74031 4 6 zi: 74031-225 fum of their ftocka 
 
 ■Loft 7975^3 
 
 Grtifl 7347 
 
 Dlft*. Whole loft 7x403 
 
 74031 '22 5 ) 72403'00, &c. ( '97800624 com. multiplier 
 
 7754-85 
 
 15749*7 
 3497-8 
 5754-9416' 
 
 3775*95 
 37497^.983 
 
 X 
 
 » •97800624 
 
 1. s. d. 
 
 7584-2916 := 7584 5 9I 
 
 15403-3048 = 15403 6 I 
 
 3420-8702 n 3420 17 
 
 5628-3692 zz 5628 7 
 
 3692*9026 rz 3692 18 
 
 *• 36673-2616 — 36673 5 
 
 (6.) 
 
 tons 
 
 500 A put on board 
 
 340 B 
 
 94 C 
 
 934 
 
 934 
 
 0-4 
 
 150 
 150 
 
 5QO 
 
 340 
 94 
 
 K 
 
 t. hd. 
 80 I 
 
 54 2 
 15 o 
 
 2 
 
 4l 
 4i 
 o 
 
 2i 
 
 A's lofs 
 B's — 
 C's — 
 D's — 
 E's — 
 F's — 
 
 g. rem. 
 
 12 510 A'a lof-: 
 
 26— 160 B's — 
 
 24— -264 G'; —
 
 ICO 
 
 Single FsLLou-sHif. 
 
 1 4 
 
 (r) 
 By note 2d. 
 4- 3-I-4 + 54-6ZZ21 fum of the numbers 
 
 21 
 
 1680 
 
 21 
 
 1680 
 
 21 
 
 . 1680 
 
 21 
 
 • 16S0 
 
 27 
 
 16S0 
 
 21 
 
 1680 
 
 80 the id number 
 160 tlie 2d 
 240 the ^d 
 320 the 4th 
 400 the 5th 
 480 the 6th 
 
 . (8.) 
 7 -f 8 4- 9 = 24 fum of the proportional flocks 
 
 24 : 357S4 1:7: 
 
 24 : 35784 '•' ^ ' 
 24 : 35784 : : 9 : 
 
 ^C437 
 11928 
 
 13419 
 
 s. I. 
 
 s. 
 
 3S7?4 ' S^o : I 
 357F4 : 500 : : 
 .15784 : 500 : : 
 
 10437 
 11928 
 
 U419 
 
 
 (gO 
 
 3 + 4. 
 
 + 7 
 
 J4 : 292 : 
 1 . : 292 : 
 14 ; 292 : 
 
 • 'i- • 
 
 s. 1. s. d. 
 
 521 17 o ] Their 
 
 596 8 o / refpeclivc 
 
 67c 19 o ) Itocks 
 
 ]/ s. d. 
 
 145 16 8 I Their 
 
 166 13 4 > feparatc 
 
 187 10 oj gain» 
 
 r4 fum 
 
 62 y gallori 
 83 -]^ ditto 
 ,D ditto 
 
 4- 3 + 5 = ^o1. 
 
 10 
 
 19090 
 
 10 
 
 19090 
 
 10 ; 
 
 19C90 
 
 I. 
 
 3818 A's part 
 
 5727 5's 
 
 9545 ^^
 
 Part I. 
 
 Single F e l l o w s h i Pi 
 
 10 1 
 
 (II.) 
 
 689, 5 ■ 50, ,, 3 
 
 129I. ns. Ad. ir — ^I. 5— =: — '• A'spart. 4 -i 
 
 ^ ^ ^ 3 9 9 7 
 
 = 111. 
 
 n' . ^ _ 37, „ 2 
 
 B 3 part. 4 — n — 1. 3 — 
 
 9 9 3 
 
 AslZ 
 
 IC23 
 
 "257 
 
 3 
 C's part. 
 
 50 31 1023 90650 72261 64449 227360 
 
 5 7 259 16317 16317 16317 16317 
 
 By reje<Stlng the common denominator 16317 (vide Note 4th, Prop. 
 8tli, Vulgar Fradlions) we have the following proportions : 
 
 1. s. d. 
 
 227360 
 227360 
 227360 
 
 6S9 
 
 3 
 689 
 
 3 
 6S9 
 
 3 
 
 rem. 
 1. : : 90650 : 91 II 4I 203840 A's (hare 
 
 I. : : 72261 : 72 19 iq| 80640 B's ■ • « 
 
 1. : : 64449 -65 2 o|— — . 170240 C's — — 
 
 To 
 20 
 
 (12.) 
 
 > I 
 
 > 4 
 
 15 
 
 6^' 
 
 12 
 
 To* 
 
 + — +— 4-~ — — fum. 
 60 60 60 60 . 60 
 
 By rejedling the common denominator, &:c. 
 
 1. s. d. rem. 
 
 30 : 194 16 I 76 A's fliarc 
 
 20 : 129 17 4I 25 B's — — 
 
 15 : 97 8 o\ 38 C's 
 
 77 18 5I i5D's— r- 
 
 77 
 
 ^ool. 
 
 77 
 
 : 500I. 
 
 77 
 
 : 500I. 
 
 77 
 
 : 500J. 
 
 12 
 
 (13.) 
 
 1. s. 
 
 57 18 A's gain 
 
 29 14 B's ~— 
 
 28 4 dlfterence of their g.ilns. 
 
 28I. 4s. •. 51!. IIS. 6d. : ! 57I. i8s. : 105I. 17s. lo^d. '^y A's flock, 
 from which fubtraft 51I, lis. 6d. the remainder is 54I. 6s. 4{<J. '-^y 
 B's ftock, 
 
 K3
 
 r02 S r N G t E F F n t o w s K r p". 
 
 Firil, 
 
 it* I cwr. : 2I. 16s. :: 2^ocwt. i qr. 22 lb. : 701I. ^s, 
 A's adventure, excljfive of charges. 
 
 Ifiool. : 91I. :: 701I. ^s. : 651I. 2s. 6d. value of A's 
 remaining part, which dcduded from 701 1, ^s; leaves 70I. 2s. 
 6d. A"s lofs, exclufive of charges. 
 
 Again, If 4I. 4s. : i cwt. : : 631I. 2s. 6d. : 150CWN 
 
 1 qr. 2 lb. the quantity A faved, which deduded from 
 2C0Cwt. I qr. 22 lb. the quantity he bought, leaves Joocwt» 
 20 lb. the quantity of fugar A loft, or caft overboard. 
 
 Hence, icocwt. 2olb. x 100 zz i 0017 cwt. 3 qr. 12 lb. 
 their whole cargo. 
 
 Secondly, 
 
 If z^ocwt. I qr. 22lb. : loocwt. 20 lb. :: iooi7Cvvtr 
 3qr. 1 2 lb. : 4007 cwt. 16 lb. the whole quantity of fugar 
 caft overboard. 
 
 1^33 ' ^i '- ' 4007 cwt. 16 lb. : 4508 cwt. 41b. the 
 w hole freight of A and B. 
 
 Then 4508 cwt. 41b. — 250cwt. i qr. 22lb. = 4257 cwt. 
 2qr. 10 lb. B's cargo. 
 
 Ifiooi7Cwt. 3qr. i2lb. : 4007cwt. i61b. :: 4257cwt. 
 
 2 qr. 10 lb. : 1703 cwt. 41b. quantity of fugar B loft, or taft: 
 overboard. 
 
 Then 4257 cwt. 2qr. lolb. — 1703 cwt. 41b. =:2554cwt. 
 2 qr. 6 lb. the quantity of fugar B faved. 
 
 If I cwt. : 2I. 6s. 8d. :: 42^7 cwt. 2qr. lolb. : 9934I. 
 -;. 6d. value of B's adventure, exclufive of charges. 
 
 If I cwt. : 2I. 6s. 8d. : : 2554 cwt. 2 qr. 61b. : 5960L 
 12s. 6d. prime coft of the fugar B faved. 
 
 If 100 : 120 : : 59601. 12s. 6d. : 7152I. 15s. the ad- 
 vanced value of the fugar which B faved. 
 
 Then 9934I. 7s. 6d. — 7i52l' 15s. = 2781L 12s. 6d, 
 B's lofs, exclufive of charges. 
 
 Thirdly, 
 
 iooi7Cwt. 3qr. 12 lb. — 45o8cwt. 41b. = 5509cwt. 
 
 ^qr. 81b. C's cargo : loccwt. 2olb. -f i703Cwt. 41b, 
 
 -= 1 803 cwt. 241b. A's + B's lofs of fugar. 
 
 40C7Cwt. i61b. — 1803 cwt. 241b. = 2203cwt. 3qr. 
 rolb, quantity of fugar C loft, or call overboard.
 
 Part I. Double Fellowship. loj 
 
 Then 5509 cut. 3qr. 8 lb. — 2203 cwt. 5 qr. 2olb. nr 
 330^ cwt. 3qr. i61b. the quantity of fugar C faved. 
 
 701!. 5s. -f 9934!. 7s. 6d. ~ 1063^1. 12s. 6d. A's + B's 
 adventure, exclufive of ehargcs. 
 
 15778L 2.S. 6d. — 1063 5I. i2S. 6d. = 5142I. los. C's 
 adventure, exclufive of charges. 
 
 If 5509Gwt. 3qr. 81b. : 5:142!. los. :: 33OJ cwt. 3 qr. 
 1 6 lb. : 308 5J. I OS. prime cort of the fugar C faved. 
 
 If I cwt. : 4s. 8d. :: 330^cwt. ^qr. i61b. : 771I. 7s. 
 6d. C's gain by the quantity he faved. 
 
 Then 3085I. lOs. -f 77 il. 7s. 6d. = 3856K 17s. 6d. ad- 
 vanced value of the fugar C faved. 
 
 5142I. ros. — 3856I. 175, 6d. ~ 1285I. I2S. 6d. C's lofs^ 
 exclufive of charges. 
 
 Laftly, 
 
 cwt. qr. lb. 1. cwt. qr. lb. 1. s. d. 
 
 If 100 1 7 3 iz : 525 r: 250 i 22 : 13 26 A's part of the charges 
 
 If 10017 3 12 : 525 : : 4257 2 10 : 223 2 6 B's 
 
 If 10017 3 12 : 525 : : 5509 3 8 : a.%Z 15 o C's — 
 
 1. s. d. I. s. d. 1. s. d. 
 
 70 2 6-fi3 2 6rr 8350 A's who'e lofs 
 
 2781 12 6 4- 223 2 6 rr 30^4 15 o B's 
 
 1285 12 6 + 288 15 o zr 1574 7 6 C's————. Anfwer 
 
 DOUBLE FELLOWSHIP. 
 
 (I.) 
 
 T'/jis Example is *worked, 
 
 L s. d. 1. s. d. 
 
 547 19 6X7 = 3835 16 6 prod, of A's flock and time 
 
 475 18 0x9= 4-S3 2 o B's 
 
 1747 14 0X41:: 6990 16 o C's — 
 
 Sum of the prod. 15109 14 6 
 
 !• s. d. 1. 1. s. d. 1. s. d. rem. 
 
 15109 14 6 : 225 : : 3S35 16 6 : 57 2 4I 2769444 
 
 15109 14 6 : 225 ; : 4283 z o : 63 15 7 2725S48 
 
 15109 14 6 : 225 :; 6990 j6 o ; 104 2 o — 1757376
 
 [04- Double Fellowship, 
 
 Or thus, by Note 2d, 
 
 547*975 X 7 ::: 3*^35*^-5 fi"°^* °^ -^'^ ^^'^'^ ^"^^ ^'■'^s 
 475*9 X 9 rr 4283'i B'& 
 
 1747-7 X 4 = ^;-9^'"S C's 
 
 Sum of the prod. i5io9'72 5 
 
 35io9'725 ) 225'ooooc, &c. ( •0T4891 common multiplier, 
 
 1. s. d. 
 3835-825 
 
 4: 
 69c 
 
 iS35-S25') X r 57-11927 =r 57 2 4I A's ftare 
 ,283-1 ^-014891^ 63*77964 ir 63 15 7 B's — . 
 >9cc-8 3 ^^ L 104-1 rr 104 2 o Cs — — 
 
 (3-) 
 7 X 13 = 91 prod, of A's oxen and time 
 
 Q X 14 = 126 B's ^ 
 
 II X 25 n: 275 C's 
 
 »5 X 57 = 55^' I^'s 
 
 1047 fum of their produds 
 
 1C47 : 21 :: 91 : i 16 6 216 A's (hare 
 
 1047 : 21 :: 126 : 2 10 6t 158 B's 
 
 1047 : 21 :: 275 : 5 10 3! 135 ^'s 
 
 IC47 : 21 :: SS5 '■ ^^ ^ 7t 558 D's 
 
 (4.) . 
 weeks 
 
 26 time the family lodged 
 
 26 — 14 n 12 the firft 4 
 
 12 — 3 m 9 ■ ■ " thefccond 4 
 9 — 3 m 6 — the third 4 
 6 — 3—3 — the laft 4 
 
 Sum of the produfts 380 
 
 380 w. : 261. 2s. 6(?t : : I w. : is. 4^3» the fum paid per week by 
 each ledger. 
 
 Then, 
 
 s. d. L s. d. 
 
 260 X i 41 n 17 17 6 for the family to pay 
 
 48 X I 4i = 3 6 o for the li^ four 
 
 ^6x1 It = 2 9 6 for the 2d four 
 
 24 X I 4f = 1 13 o for the 3d four 
 
 12 X I 4t =; o 16 6 for the lad four 
 
 ;6 X 
 
 10 zz 260 
 
 12 X 
 
 4= 48 
 
 9 X 
 
 4= 36 
 
 6 X 
 
 4 = 24 
 
 3 X 
 
 A-— rz
 
 Part I. Double Fellowship. 
 
 lo; 
 
 (5-) 
 
 150 X 7 = 1050 
 
 ~.o 
 
 ■ ICO X 5 = 500 
 170 
 
 270 X 6 = iCio 
 
 Prod, of A*s flock 1 
 
 31^ X 6 = 1890 
 
 KO 
 
 165 X 9 = 14S5 
 500 
 
 665 X 3 = 1995; 
 
 Prod. ofC'sft-ock ? 
 and lime j ^ ^ and time j -^'^' 
 
 ioj X 5 == 1C25 
 no 
 
 315 X 4 = 1260 
 150 
 
 165 X 9 = 1485* 
 I377Q 
 
 Prod, of B's {lock 
 and time 
 
 3170 
 3770 
 5370 
 
 1 23 10 fumoftheproc 
 
 I2JI0 ! 450 : 
 12JI0 : 450 : 
 12310 ; 450 : 
 
 I. 
 
 I. s. d. 
 
 115 17 7I- 
 
 137 1^ 3k" 
 196 6 oi- 
 
 By, Note 4th, Piop. I. 
 
 3170 
 
 3770 
 5370 
 
 fern. 
 
 • 1740 A's n»ar« 
 
 .2380 B'l «-— 
 
 J. s. d. 
 
 29 1^. 7 "9 
 
 I, 
 500 X 3 
 
 400 X 2 : 29 12 79 '. ' 350 X 5 
 
 1. 
 
 s. 
 
 d. 
 
 
 
 
 ^s 
 
 II 
 
 ^^ 
 
 A' 3 
 
 ga 
 
 n 
 
 64. 
 
 16 
 
 3^ 
 
 B*s 
 
 — 
 
 
 29 
 
 12 
 
 75 
 
 C'3 
 
 — 
 
 
 [50 o o whole gain. 
 
 (7-) 
 
 By Note 5th, Prop. II. 
 
 J- s- d. 1. s. d. 
 
 4+40 X 6x12=3182 8 o A's gain X by B's and C's time 
 
 42 16 9f X S X 5i ^n 4112 12 5I- B'sgain X by A's andC's time 
 
 79 II Zy X S X 6=13818 17 75- C'sgain X byA'saiiJB's time 
 
 11113 18 4f- furn of the produds
 
 io6 
 
 Double Fellowshi?. 
 
 I. s. d. 1. 
 
 11113 18 4t : 227 
 
 111 13 18 4f : 227 
 
 iiij $ iS 4| : 227 
 
 J. s. 
 
 d. 
 
 1. 
 
 3182. 8 
 
 
 
 65 A'aftock 
 
 41 12 12 
 
 9l 
 
 : 84 B's 
 
 3818 17 
 
 7i 
 
 : 78 G's 
 
 (8.) 
 
 By Note 2d, Single Fellowfhlp. 
 A. B. C. 
 
 2 + 3 + 4 = 9 ^""^ °^ ^^^^^ proportional gains. 
 1. 
 
 2 ; 52 A's gain ) So far Meff. Hili and Bidt 
 
 3 ' 78 B's — ;. 
 
 234 
 234 
 254 
 
 are right ; the reft is to- 
 
 4 : 104 C's j tally falle in principle 
 
 Again, by Note 5th, Prop. II. 
 52 X 5 X 7 =: 1820 A's gain X by B's and C's time 
 78 X 3 X 7 =z 1638 B's gain X by A's and C's time 
 104 X $ X ^ =z 1560 C's gain X by A's and B's time' 
 
 5018 fum of .tiic produfts. 
 
 50 Id 
 fol8 
 
 1. 
 
 1. 
 
 s. 
 
 3822 : : 
 
 1820 : 1386 
 
 4 
 
 JS21 : : 
 
 i6$% i 1*47 
 
 11 
 
 $Hi I : 
 
 1560 ; nS8 
 
 3 
 
 844 C'i 
 
 T?7 
 
 (9.) 
 
 By Note 2d, Single Fellowfiiip, 
 X. Y. Z. 
 
 2 + 3+4 = 9 ^"^ of their proportional gains, 
 
 420 
 420 
 
 420 
 
 1. s. 
 
 93 6 
 T40 o 
 iS6 13 
 
 1. s. 
 
 93 6 
 
 140 o 
 iS6 13 
 
 8 X's gain ^ So far Mr. ^'j/ts' fo- 
 
 o Y's > lution is right; the 
 
 4 Z's J reft is totally falfe 
 
 in principle. 
 
 Again, by Note 5th, Prop. II. 
 
 d. 
 
 8 X 6 X 9 ZZ 5040 X's gain X by Y's and Z's time 
 o X 4 X 9 iz: 5040 Y's gain x by X's and Z's time 
 4 X 4 X 6 — 4480 Z's gain X by X's and Y's time 
 
 14560 fum of the produfts»
 
 it I. 
 
 D y B 
 
 LE Fellow 
 
 SHIP, 10 
 
 
 I. 
 
 I. s. 
 
 d. 
 
 14560 
 14560- 
 14560 
 
 : 4262 : 
 : 4262 : 
 : 4262 ; 
 
 : 5040 : 1475 ^ 
 : 5040 : 1475 6 
 : 4480 : ijri 7 
 
 11-.'/.°^ XV (lock 
 
 il-iW. Y's 
 
 S^iW^ Z's — , 
 
 (10.) 
 
 By Note 6th, Prop. III. 
 
 1. s. 1. 1. s. I. s. 
 7815:210X9:: 76 4 : 1828 16 pr. of A'sftockandtime 
 
 7815:210x9:: 5710:1380 B's 
 
 78 15 : 210X9 :: 105 0:2520 C's _ 
 
 1. s. I. s. 
 Then 1828 16 -^ 6 = 304 16 A's ftock 
 
 
 1380 -r 5 = 276 B's ■ 
 
 
 2520 O-r- 12 ~ 210 C's: . 
 
 
 210 D's » 
 
 
 in.) 
 
 
 By Note 7th, Prop. IV. 
 
 1. s. 
 89 S 
 9^ 15 
 38 10 
 
 s. 1. s. s. 
 
 — 17S5 A's ftock. 7.!^ 10 IT 510 A's gain. 
 
 — 1855 B's 37 2 — 742 B's 
 
 -1 770 C's 24 4 — 484 C's 
 
 510 X 1855 
 
 742 X 1785 
 
 Produa of 
 X 770 = 728458500 A's gain X by B's aad C's ftock 
 X 770 — 1019841900 B's gain x by A's and C's -— 
 
 484 X 1785 X 1855 — 1602608700 C's gain X by A's and B' 
 
 3350909100 fum of the produdls. 
 
 33509P9JC0 
 ^350909100 
 3350909100 
 
 23 
 23 
 
 23 
 
 728458500 
 1019841900 
 1602608700 
 
 5 m. A's time 
 
 7 m. B's -. — r- 
 
 ii m. C's — - 
 
 (12.) 
 
 By Note 8th, Prop. V. " 
 1 12 5 X 1400 X 1800 zr 28350C0000 B's gAiii X by C's and D's ftock 
 
 210 X 5000 X 1800 zi 1S90000000 C's gain x "y B's ind D's 
 ao2'5 X 5000 X I4<^0— 1417500000 D's gain x by B's daid C's 
 
 2835000000 t 
 2835000000 : 
 
 If 5000!. : 1125I 
 per annum. 
 
 : : 189000C000 
 : : 1^17500000 
 
 lool. : 22I, los. each Merchants' gain per cen 
 
 8 C's money was In trade 
 6 D's .
 
 loS Loss A :: D G a i x. 
 
 Here 315I. is to be divided into two parts, as 12 to 8. 
 12 -J- 8 zz 20 fum of the proportional nun)bers. 
 20 : 315 ; : 12 : i8q B's ftock, 1 For the gains are 
 
 20 : 315 : : 8 : 126 A's j equal, therefore the 
 
 flocks are reciprocally as the times. 
 
 (I4-) 
 a. s. d. a, r. p. 1. s. 
 
 If I : 18 o : : 394 3 34 •* 35^ 9*32^ val.of A'sprop. 
 
 If I : 19 6 : : 417 i 14 : 406 18-08125 B's 
 
 If I 121 o : : 714 3 o : 750 9*75 C's 
 
 Whole ^uant. 1527 o 8 1512 17*15625 whole value. 
 
 If il. 5;s. 6d. : : 1512I. 17W5625S. : 416I. 14s. 4jd. '25 
 value of the land allotted for the tithes. 
 
 If 19s. 9id. : 1 a. : : 416I. 14s. 4^d. '25 : 421 a. or. 
 16^1 p. quantity of land allotted for the tithes. 
 
 LOSS AND GAIN. 
 
 This Example is ivorked, 
 
 (2.) 
 
 I. s. d. 
 
 I5cwt, at il. us. 6d. percvvt. amounts to 23 12 6 
 
 i5twt. at 4-^d. per lb. — — 31 10 o 
 
 Diff. Gain 7 17 6 
 
 (3-) 
 s. d. 
 
 3 4 value of 1 20 eggs at three a penny 
 ^ o — 1 20 ■ two a penny. 
 
 8 4 prime coft of 240 eggs 
 
 8 o felling price of 240 eggs at five for 2d. 
 
 o 4 lofs.
 
 Part I. Loss and Gain, lOf 
 
 (4-) 
 
 12X2 — I =23 pipes faleable, 
 
 I. s. 
 1014 6 value of 23 pipes at 7s. per gallon. 
 907 4 = 75I, I2S. X 1 2 prime coll of the wine. 
 
 107 2 gain. 
 
 7s. 6d, — 5s. 4d. =: 2s. 2d. gain per yard. 
 
 Then 340 yds. at 2s. 2d. per yard, amount to 36I. i5s, 
 8d, whole gain. 
 
 (6.) 
 
 This Example is n/jsrked, 
 
 (7.) 
 If i7^d. : lool. :: i3fd. : 78I. ^s. 2|:d.-ff- which de- 
 duced trom lool. leaves 2 1 1. 14s. 9|d.-|;| lofs per cent. 
 
 (8.) 
 
 If 27yds. : 17I. i;3. :; i yd. : 13s, 2|d.-f prime coH; 
 per yard. 
 
 If 13s. 2td.-4 : lool. :: 9s. lod. : 74I. 7s. 4id.-J-I4- 
 which dedu<5led from lool. leaves 25!. 12s. 7;id.-T.T9 lofs per 
 cent. Or thus, 
 
 27 yds. at 9s. lod. peryard, amount to 13I. 5s. 6d. 
 
 If 17I. 17s. : lool. : ; 13I. 5s. 6d. : -4I. 7s. 4id.-H^ 
 &-C, as above. 
 
 {90 
 
 If6ol. : lool. : ; 75I. : 125I. 
 
 25 gain per cent, Mr. MaU 
 col/ns anfwcr by mi/lake (probably in i\v^ prefi) is only 5I. 
 
 (10.) 
 
 If 7s.6d. : lool. : : 6s. zH, : 82I. los. whichdevlufted 
 from lOol. leavca 17I. los, Icfs per cent, 
 
 L
 
 110 Loss AND Gain. 
 
 (II.) 
 ^hii Example is nMorked. 
 
 If lool. : 120I. :: iis.6d. : 13s. 9*d.-| pcryard. 
 
 (13.) 
 If lOol. : 94I. i: 6s. : 5s. 7-|d.-i| perbundle. 
 
 (I4-) 
 Ifiool. : 115I. : : 12I. 12s. : 14L 9s. g^^d.-lpercwt. 
 
 Cv) 
 If lOol. : 82I. los. : : 7s. 6d. : 6s. 2 ^d. per yard* 
 
 (16.) 
 This Example is ^^juorled, 
 
 (17.) 
 IfSs. 9d. : 112I. :: ics. 6d. : 13+I. 8§. 
 
 34 8 gain per cent. 
 
 (18.) 
 
 If 15s. : §61. : : 21s. : 120I. 'b%, 
 100 
 
 20 8 gain per cent. 
 
 (I9-) 
 If 20I. 93. 6d. : 112I. los. : : 17I. is. 3d. : 93I. 15s. 
 which deduced from lool. leaves 61. 5s. lofs percent. 
 
 (20.) 
 This Example is nvorked, 
 
 J. s. d. 
 
 If I lb. : 7fd. : : i5cwt. 3qr. i81b. : 55 U 9 ^^Id for 
 125I. ; icol. : : 55I. 13s. gd. : 44 ' » opnmecoft. 
 
 II 29 whole gain.
 
 Part L L o s 3 and Gain. i m 
 
 (22.) 
 
 1. s. d.- 
 coodeals,ati5d.perpiece,amountto 31 5 o felling pric<?. 
 If9il. : lool. :: 3il- 5^. : 3+ ^ 9HV pr- coil. 
 
 3 I 9i-vV whole 
 .. « lofs. 
 
 {23-) 
 
 I. 8. 1. I. s. d. L s. d. 
 
 If io6 10 : 100 : : 500 16 8 : 470 5 3IM C's pr. cofi, 
 
 38 II 6 B'sgain. 
 
 431 13 9^11 B's pr. cod. 
 
 If 104I. 6s. 8d. : lOol. :: 431I. 13s. 9i|td. : 413!. 15s. 
 '^\^Az\\% A's prime coil. 
 
 If i5p. : 4131. ijs* 2id..|i|^§ : : ig. : 4»»4i^' P" 
 gallon, Anfwer. 
 
 (24.) 
 ^kit Example is *Wdrked* 
 
 7 y 3?| y 15 = |7S748^ ^ jS^J. 7s. 54, prime eoft of 
 the cloth. 
 
 If locl. : 9cl. :: 1S6I. 7s. 6d. : 167I. 14s. 9d, felling; 
 price. 
 
 Then 186I. 7s. 6d. — 167!. 14s, 9d. = 18I. j2s. gd, 
 whole lofs. Or thus. 
 
 If lool. : 90I. : : 15s. ; 13s. 6d, felling price per yard. 
 
 Then 15s. — 13 s. 6d. = is. 6d. lofs per yard, and 7 pieces 
 each 35I: yards, at is. 6d. per yard, amount to 18I. 12s, gd. 
 lofs, as before. 
 
 * MefT. Birks ftate the queftion thus : 
 
 lool. : 61. los. : : 500I. 16s. Sd. t 32!. ice. 
 
 locl. : 4-|I. : : 429I. &c. making A's prime coft 41 il. Szc. — Mr. 
 Vyfe has folved the queftion in a fimilar manner, — here the gain (or lofs) 
 of lool. is the fccond term in the ftating, inftead of its amount j which 
 is tlic very remark Mr. Fyft makes \>pon Mr.. Dih':orth, 
 
 L 2
 
 JiJ Loss AND Gain. 
 
 (26.) 
 
 475 yards, at 10s. 6d. per yard, amount to 249I. 7s. 6d, 
 prime cod. 
 
 If lool. : 130I. : : 249I. 7s. 6d. : 324I. 3s. gd. felling 
 price. 
 
 Then 324I, 3s. 9d, — 249I. 7s. 6d. zz 74I. i6s, 3d. gain 
 in the whole. 
 
 (^70 
 
 If icwt. : 3I.cs.8d. :: 127X470^1. : 1733I. irs. prime 
 coft of the fugar, 
 
 £733!. IIS. 4- 52I. los. =: 1786I. IS. felling price of the 
 fiigar. 
 
 If 571I cwr. ; 1786I. js. :: ilb. : 6id.-4tf per lb. 
 Anfvver, 
 
 (28.) 
 
 I. 
 1400 cafks, at 2I. 5s. per calk, amount to — 3150 
 7C0 diito, at 2I. 15s. " ■■ ■ 1925 
 
 Value of the 700 remaining cafks 1225 
 
 Then 122^ 4- 70G =s iK ip. per cafk. 
 
 Or thu% 
 Since he gained 1 o% per caik by one half, he of courfe 
 muft lofe I OS. per eafk W the other, to make his purchafc 
 money ofi/j. Therefore zL 5s, -- iqs, =; il, 1554 as above, 
 
 (29.) 
 
 In order to gain as much by the v/hole quantity as four 
 yards are Md for, it is evident the merchant mull fell 96 yds. 
 at the prime coft of 100 yds. (for 100 — 4 = 96.) 
 
 Then 96yds. ; jizl. :: i yd, ; il. 3s. 4d. Anfwer. 
 
 (30.) ^ 
 
 If 80I. : looI. : : 20I. : 25I. prime coft of the fnake-root. 
 
 If looI. : 125K : : 25I. : 31I. 5s. the money he ou^i>t 
 to have made of it, which being fold for only zoITtislofy, 
 in point of trade, is iil. 5s.
 
 Part I, L O S 3 A N D G A I K, I J - 
 
 *ofi2olb. = 481b. of good tea, and | of 120 lb. =: 
 721b. of damaged tea. 48 lb. at los. 6d. per lb. amount to 
 25I. 4s. — Now in order to make a balance, the gain upon 
 481b. muft evidently be equal to the lofs upon 721b. viz. 
 3I. 12s. Therefore, 25I. 4.S. — 3I. t2s. zz 21I. 12s. the 
 prime coft of 48 lb. hence the tea coil 9s. per lb. 
 
 And 25I. 4s. + 3I. I2S. =1 28I. 1 6s. the damaged tez 
 fold for. 
 
 (32.) 
 800 X H^z 1 1 200 lb. at i2i;perlb. amount to i4ooood. 
 = 583I. 6s. 8d. the value of the anchovies, or what I re- 
 ceived for my 7490 lb. of tobacco. 
 
 If 117I. * : lool. : : 583I. 6s. 8d. : 4.98I. lis. 6j\\i, 
 the real value of my tobacco, which divided by 7490, givc^ 
 i^|d.-||-5|} the real value per lb. 
 
 (35.) 
 If lool. : 140!. :: 347^1. 15s. : 4866I. i s. flerling, the 
 Englifh goods fold for in France. 
 
 If lOol. : 85I. : : 4866I. is. : 4136I. 2s, io|d. the 
 French goods fold for in England. 
 
 Then 4136I. 2s. lo^d. — 3-l75l» 15s. =z 660I. 7s. lofd, 
 theEnglifti Merchants' gain, 
 
 (34.) 
 FIrfl 50C0 ells Flem. = 3000 ells Eng. = 3750 yards, 
 and 42 50 guineas ~ 4462I. 10s. Then, 
 
 ells. I. s. ell. 1. s. d. 
 
 I^ -^— + 3000 : 4462 10 : : 1 : 1 5 ^.^J the 
 prime coft of an Englifh ell. 
 
 el!. I. s. d, ells. I. s. d 
 
 whole piece Uood me in. 
 
 • Thrs (jueftlon is in Meflirs. Blrki and Vyf^i' Arithmetic, ,(cjtnd!cd 
 from Clare's Introduftion to Trade and BuUnefs) with a folution upon 
 falje principles, fimilar to the 23d qucftion ; their error may be fccn by 
 comparing their fclutions with the ftating marked wi;h ;hc alkrlfm.
 
 B X » r E (t. 
 
 BARTER. 
 
 CO 
 
 This Example is ^corked, 
 
 (a.) 
 
 e^ual 
 
 equal 
 
 93 X 400 :e 37200. Then 37200 -r 9 = 4^333 gallons, 
 AnlVver, 
 
 (3-) 
 5cvvt. I qr. 10 lb. at 2s. ^.Jd. per lb. amount to 70I. 7s. 9|d. 
 If 5s. 9d. : I b. : : 70I. 7s. 94d. : i^i^\\^ bufhels. Anf. 
 
 (4-) 
 
 equal 
 If 5000 yds. at i3l:d. k^ i value. 
 • at 1 8s, 6d. rS I value. 
 
 equal 
 
 500 y. 13! = 57500, this divided by 222 ( = i8s. 6d.) 
 gives 304 -jy yards. Anfvver. 
 
 (5-) 
 
 6 hhds. at 6s. 8d. per gallon^ amount to 12 61. 
 
 If 252 yds. : 126I. :: 1 yd. : los. Anfwer. 
 
 (6.) 
 
 288 ells, at [OS. 3d. per ell, amount to 18I. 
 
 If 19s. : I cwt. :: 18I. : i8cwt. 3^1, 2t/ylb. 
 
 (7-) 
 14 cwt. 3 qr. at il. 17s. per cwt. amount to 27I. 5s. 9d. 
 
 If 3s. 9d. : I gal. : : 27I. 1%, 9d. : 145 ^*j gallons. Ahf, 
 
 (8.) 
 3 cwt. 2qr. i61b. at il. 173. 4d. amount to 61. j6s. 
 If 5s. jd, : 1 2 lb, :: 61. i6s,- ; 315^-1^. Anfwer.
 
 Parti. Barter. tij 
 
 (9;) 
 This Example is ivorked, 
 
 fio.) 
 This Example is nxjorked, 
 
 (II.) 
 
 I. s. 
 
 700 gall, at 4s. 6d. amount to 157 ^o 
 Paid down — — 287 
 
 To account for 129 3 
 Ifiifd. : I lb. : : 129L 3s. : 2695^^3 lbs. Anfwer. 
 
 (12O 
 
 1. s. d. 
 
 57 qr. 61b. at il. lis. 6d. amount to 90 19 if- 
 
 i4cwt. 3qr. i81b. at 4I. 14s. 70 i 7^-^ 
 
 Balance 20 17 6 f 
 
 If 7d. : I lb. : : 20I. 178. 6d.4 : 7155J lb. Anfwer. 
 
 (13.) 
 
 1. s. d. 
 
 27 cwt. at il. IIS. 6d. amount to 41 6 o 
 
 ,25 pieces at il. 19s. lofd. 49 16 lof 
 
 Balance due to B. 7 i© lof- 
 
 If3lyds. : 15s. gd. : : 120yds. : 27I. valueof A'skerfey. 
 
 . If 7s. -f* 6s. 6d. : I ; : 27I. : 40 hats and 40 pairs of 
 (lockings. Anfwer, 
 
 1. s. d. 
 73 J yards, at 8s. 6d. amount to 31276 
 
 <r32 canes, at 3s. 79 16 o 
 
 1 6 pieces, at 4I, . 64 o o 
 
 Value of A *s goods 456 3 6
 
 ii6 Barter, 
 
 I. s. d. 
 B's cloth per yard — — i o o 
 Glafs manufadure, per lb. o i 8 
 
 Ditto finer — — 024 
 
 Value of I of each fort of goods i 4 o 
 
 If il. 4s. : I : : 456I. 3s. 6d. : 380/^ of each. 
 
 (16.) 
 1. s. 
 13784 8 value of 87 9 2 yards, at il. iis. 6d. 
 
 
 3446 
 
 2— 
 
 i 
 
 4- 
 
 ditto value ( 
 
 Df the fugar 
 
 
 
 
 1723 
 8615 
 
 I— 
 5- 
 
 JL 
 
 -1 
 
 ditto value of the j")epper 
 ditto value of the rum 
 
 
 
 
 1. s. 
 
 d- 
 
 C'.Vt. 
 
 I. 
 
 s. CWt. ( 
 
 IT. lb. 
 
 
 If 
 
 » ^S 
 
 6 : 
 
 r 
 
 • • 3446 
 
 2 : 1941 
 
 I 247* 
 
 
 If 
 
 7 3 
 
 9 : 
 
 ; I 
 
 : : 1723 
 
 I : 239 
 
 2 ^sh 
 
 '^i 
 
 If 
 
 5s. 6d. 
 
 : I 
 
 gal. 
 
 : : 861 5I. 5 
 (17.) 
 
 s. : s^szSl, 
 
 gal. 
 I. 
 
 s. 
 
 24 
 
 B 
 
 . puncheons, 
 paid in cafli 
 
 at 4s. gd. per gall, amount to 
 Balance 
 
 478 
 
 16 
 10 
 
 
 321 
 
 6 
 
 If 714 yds. : 52 lU 6s. : : i yd. : 9s. per yard, Anfwer. 
 
 (18.) 
 
 If lool. : 117I. 69. 8d. : : 6s. 3d. : 7s. 4d. A fold the 
 tea for per lb. 
 
 Then 7s. 4d. — 65, 3d. 1= is. id. his gain per lb. and 
 3s. 9d. — 3s. 4d. zr ^d. the gain upon i lb. of tobacco. — 
 The number of lbs. of tobacco is to the number of lbs. of 
 tea it2<verfely as 3s. 9d. is to 6s. 3d. or direSllj as 5 to 3. 
 Therefore the gain on the tobacco is to the gain on the tea 
 in a compound \2X\Ci of 5 to 3 and of 5 to 13, ox Jtmply as 2 j 
 to 39. But the whole gain is 81. ics. 8d. which divided in 
 the above ratio (per Note 2d, Single Fellowship) gives the 
 gain upon the tea 5I. 4s. and upon the tobacco 3I. 6s. 8d. 
 confequently the number of pounds of tobacco bartered were 
 160, and of tea 96.
 
 Parti, Exchange. 117 
 
 EXCHANGE. 
 
 This Example is wuarked, 
 
 [^■) 
 
 This Example is ivorked, 
 
 (3-) 
 As lojt : 100 : : 577 g, 14ft. : 546 g- 9tM P- Bank. 
 
 (4.) 
 As 100 : 105I : : 765g. 9ft. : 8o8g. loft. 2yV ?• 
 current, 
 
 (5.) 
 As i04|. : 100 :: 7570 g. 15 ft. : 2887 rix d. 26J||(^. 
 Bank. 
 
 (6.) 
 As i2;K : looU : : 797I. : 637L 12s. Bank. 
 
 (7.) 
 If 100 I u© i ! 790 J 9+8 ducati current. 
 
 (8.) 
 
 This Example is nvQi'^d, 
 
 (9O 
 If il. : 2450 fen. :: 4751I. 15^. : 11641787$^ fea. := 
 60634 marcs, 4 fols lub. j i|- fen. 
 
 (/o.) 
 If il. : 36s. 6d. : : 475I. i8s. : 208444 5- f^. Flemifli, 
 which divided by 2 zz 104222 -j-'o fols lub. = 6513 marcs, 
 I4jg fols lub. 
 
 If lU : 35 folsgr. id,:: 749I, 145, : 1893742! fen. =: 
 9863 marcs, 3 foh lub. lof fen.
 
 u3 Exchange. 
 
 (12.) 
 
 If il. : 3j|.foIsgr. glA, : : 754I. i8s. 9d.K.i889986j\; fen. 
 rr 3281 ri'x d. lofols lub. lo-j^ fen. Bank. 
 
 If 100 : io8|: : : 3281 rix d. 10 fols lub. io~ fen. : 
 2045909 fil^l fen. = 3551 rix d. 44 fols lub. 5*4 4 JH fen. 
 current, Anfwer. 
 
 ('3.) 
 If il. : il. 15s. 9d. ;: 757I. i8s. 7d. : 1354I. 15s. 
 1 J id.-^V Anfwer. 
 
 (14.) 
 If il. : 34 fols, 4fgr. : : 754I. us. 9d. : 7781 g. 
 13ft. io|pennings. Anfwer. 
 
 Us-) 
 
 If il. : 54s. 7-*:d. : : 479I. 148. : 199315/5 Flemifti pence 
 Bank, which divided by ico, gives 19933:^0 "J^- ^« == 
 1993 rix d. 7 ft. I o| pen. Bank. 
 
 As 100 : 104I : : 1993 rix d. 7 ft. ic-Jp. : 2085 rix d. 
 1 6 11. 1^ -iVcPo pen. current. 
 
 (16.) 
 
 If il. : 47icop. d. : : 547I. 191. lod. ! »69Sq|! cop. di 
 S a6o?9 copper dollars, * copper marcs, 3I ruoKks, 
 
 ('7') 
 If il. : 48 cop. d. :: 3749I. 14s. loid. : 1 799 87 -f^^ cop. d. 
 = 179987 copper dollars, 2 copper marcs, 6jrunftics 
 
 (18.) 
 77'/> Example is <woried, 
 
 {19') 
 If34i^ol»gr. : il. • "• 1788 rix d. 21 fols lub. : 414I. 
 14s. 2id.-if 
 
 (20.) 
 32s. 6d. = 2340 fen. ( X 12 and 6) and 747 rix. d. 2 m. 
 14 fols lub. ==: 430824 (X 3, 16 and 12). Then 2340 fen, 
 ; il, ;; 43o824fen. : 1S4I. 25. 3-rj. Anfwer.
 
 Parti. Exchange, 119 
 
 • (21.) 
 Firft, I rlx dol. ( = 3 X 16 ) = 48 fols lub. hence J rix 
 dol. = 30 fols lub. and 120 fols lub. 1= 20 fols gros, zz il. 
 Fl. ••• 6 fols lub. = I fol gros, and 8 fols gros, ~ a rix dol. 
 
 Then 104 rix dol, 30 fols lub. : 100 ; : 743 rix doL 4 fols 
 gros, : 7 10 rix dol. i m. 14 fols lub. 4 fen. Bank money. : 
 
 33s. gd. = 2430 fen. ( X 12 and 6) and 7 10 rix dol. i m. 
 14 fols lub. 4 fen. = 409324 fen. Then 
 
 If 2430 fen. : il. : : 409324 fen. : 168I. 83. ii-j\ller, 
 Anfwer. 
 
 (22.) 
 
 ioqI : 100 : : 1749 marcs, 13 fols lub. : 1^99 marcs, 
 j^ fols lub. 3 fen. Bank money, — 474^7.^/ dol. 2 fols gros, r= 
 ^48 marcs, 12 fols lub. (for 2 marcs zz: 1 J/et dol. and 2 fols 
 gros r= 12 fols lub.) 
 
 104I- : 100 : : 948 marcs, 12 fols lub. : 908 marcs, 15 foh 
 lub. 8 fen. Bank money. 
 
 mar. folsl. fen. 
 
 159.9 13 3 
 908 I J 8 
 
 2508 12 111= 481691 £en. 
 
 Now 120 fols lub. = il. Flemifh, and 12 fen. = i fol lub, 
 ••• 481691 fen. =: 334I. los. id. Flemifh. 
 
 If 35s. 8d. : il. : : 334I. los. id. : 187I. iis. 5^d. fter- 
 ling. Anfwer. 
 
 (23.) ■ 
 34s. 7d. : il. : : 1747I. 14s. 'jd. : loiol. 14s. 8^d.-JJ 
 ileriing. 
 
 ^ (24.) 
 105:4 • ^-o • • 7495 guild. 14ft. : 7 113 guild. 7 fc. ogr. 
 2 p. Bank money. 
 
 If 34fc. 4gr. : rl. ;: 7113 guild, 7 fc. ogr. 2 p. : 690I, 
 1 2s. 4d, fterling. Anfwer. 
 
 (25.) 
 
 If 34s, 3d. : il. : : 774I. iSs. ; 4;2l. gs. ii-iY^d. Anf.
 
 20 E X C H 
 
 A N G £, 
 
 (26.) 
 
 If 48! cop. d. : il. : ; 7i23Cop.d. i4run. : 146I. 17s. 
 6d. Anfwer, 
 
 (27.) 
 If 49 cop. d. : il. : : 5749 f. d. i cop. d, 2 c. m. 3 run. 
 : 352I. OS. 2|d,-||. Anfwer. 
 
 (28.) 
 TTjis Example is fworked, 
 
 (29-) 
 If 49 J d. ! irixd. :: 749I. i6s. : 3635 rixd. 2 m. 
 5 -i^j- fK, Anfwer. 
 
 (30-) 
 If 4s. 7d. : I rub. : : 7574I. 19s. : 33054 rub. 32 cop. 
 
 2 -i^ alt. Anfwer. 
 
 If4s. 9^. : 1 rub. :: 574I. i8s. : 2399 rub. 58 /j cop. 
 Anfwer 
 
 This Example is ivcrked. 
 
 If I rix d. : 48^ d. :: 9751 rix d. 4 m. 3 fk. : 1980I. i6s. 
 J4^'"tV Anfwer. 
 
 (34.) 
 
 If I rub. : 4s. 9d. : : 7454 rub. 4gr. 6 cop. : 1770I. 8s. 
 8 jl. Anfwer. 
 
 , (3v) 
 If 1 rub. : 4s. 7td. : : 7479 r. : 1729I. los. 4^. Anfw. 
 
 (36-) 
 
 If 1 24 cop. : 25^ guild. : : 4759 rub. 44 cop. : 9595"guild» 
 12ft. 14 pen. current. 
 
 If 103I : 100 :: 9595 guild. 12 ft. i4p. : 9271, guild. 
 
 3 ft. I p. Bank. 
 
 If 34s. 6d. : iL : : 9271 guild. 3 ft. j p. : 895I. 15s. 
 3^d. ftcriing.
 
 Parti. Exchange, 121 
 
 (37-) 
 If il. : 34s. 8d. : : 495I. 17s. 6d. : 206284 pence, FI, 
 If 52 ft; : I rub. :: 206284 pence : 7934 rubles. Anf. 
 
 {38.) 
 If 121 cop. : 2j: guild. : : 7437 rub. 5 griv. 24 cop. : 
 15367 guild. 4 ft. 10 pen. current. 
 
 If 103^ : 100 :: 15367 guild. 4 ft, 10 pen, : 14901 guild. 
 n ft. 2 pen. Bank. 
 
 If 34s. 7d. : il. : : 14901 guild. lift. 2 pen. : 1436I. 
 js. io|d. fterling. Anfuer. 
 
 (39O 
 If270gr. : il. : : 7947 Fl. : 883I. Flemifli. 
 
 If 33s. 5d. : il. : : 8S3I. : 528I. 9s. 6ld.-|J^ Anfwer. 
 
 (40.) 
 
 If il. : 34s. 8d. : : 749I. 17s. 6d. : 1299I. 15s. 8d. FI. 
 
 If il. : 2 74grofti. :: 1299I. 15s. 8d. : 356140]- ^ grolh. 
 d 3957 rix dol. lOy^ groHien. Anfwer. 
 
 ■f^ 273 groAi. : 6 guild. : : 4795 Fl. 4 ecu. 4 gr. : 
 3162 ^\ guilders current. 
 
 If io3|g- '• loog. :: 3162 /^ guild. : 3058 guild. i6ft, 
 I grot = 509I. 1 6s. id. Flemilh Bank. 
 
 . If 3i^- 7^* ' '^' • • S^9^' 16?. id, : 303I. 12s. i-^d. fter- 
 ling. Anfwer. 
 
 (42-) 
 If 3livr. : 3iid. ;; 636 li v. 3 fol. 9fd. : 27I. iCs. 
 Tl^'-H* Anfwer. 
 
 (+3.) 
 If I ecu. : ;5d. : : 57475 liv. 6fol. : 4230!. i6s. 4{d.^\, 
 Anfwer, 
 
 (44.) 
 If 54d. : I ecu, : : 759I. i8s. 9d. : 3377-^- crowns. 
 
 US') 
 If 3;49cr. 2 lir. 10 ol. : 343I. 14s. 3d. ; ; i cr, : ud. 
 p. ecu, 
 
 M
 
 122 EXCHANGP^ 
 
 (46.) 
 
 If4;^d. : ipla. :: 749I. 18s. : 396675^*^- piaflres. 
 
 (47.) 
 If I pia. : 47i-J. ; : 1347 pia. 2 r. 24m. : 266I* 13s, 
 2 id.- 3V 
 
 (48.) 
 If I pia. : 43ii. : : 9749 rials ; 21 81. 19s. 5i|d. Anfwcr. 
 
 ' (490 
 If I pia. : 41^.1. : : 7549r. v. : 87I. os. 5id.-|. Anf, 
 
 If I m. : 65(1. :: 7434cr. 347162. : 805I. 8s. lold.--^^. 
 
 ■ (,-■•) 
 
 If im. : S4td. :: 4756 m. 2 tes. 4 vin. lorez. : 1278I. 
 5s. old.-^h' 
 
 (52.) 
 If 66i:a. : i m. : : 17 881. 175. : 6456 milr 
 
 (53-) 
 'li'zcoom. : 6661, 135. 4d. ;; i m. ; Sod. :=: 6s. 8d. 
 p. mil re. Anfwer. 
 
 (54-) 
 ir I dol. : 53|d. : : 947 dol. : 2nl. 2s. o^d. Anfwer. 
 
 If47|d. : ipcz. : : 1749I. 17s. 6d. ; 8795 pez. 3 foL 
 S: den. 7ie(trly» 
 
 (56.) 
 if46Ad. : I pez. :: 747I. i6s.4d. : 387opez. 2 3-V\fol. 
 
 (57-) 
 If 1 pe/.. : i^y. : : 7439 P^^:. 9 fol. 3d. : 1499!. los. 
 3jd-Til- 
 
 ir,-4d. ; I pia, ;; -.,^^ ^T- '• 2^54 pia. 2 Uie. J3foK 
 4 den.
 
 Parti, E X c H A N G e.' 123 
 
 This Example is i<:odcd, 
 
 (60.) 
 Ifiool. : 105I. 15s. :; 694l.1Ss.6d. : 734^- ^Ts- 74'-io- 
 
 (6u) 
 If lool. : 176I. : : 917I. i8s. ; 46151. xos. o^dol^J'. 
 
 (62.) 
 T/jh Example is wori^d^ 
 
 (63.) 
 If 111]. los. : lOoU : : 6747 1. 14s. i 605 il. 14s. 
 
 (64.) 
 If 155I. : looL :: 475!, 14s. lod. : 352L 8-a\s. 
 
 Firft the commifli©n of gcol. at 5 per cent, is ^^]. Then 
 900 — 45 = ^55^' currency the Factor niuft remit according 
 to the courfe of Exchange. 
 
 If 125I. : lOoI. II 855I. : 684I. fterling the merchant 
 received. 
 
 Then 734I. 14s. gd.. — 684I. = ^oL 14s. gd. lofs. 
 
 Observatiok. The above feems to me to be the moft rational fo- 
 lu i m that can be given to this queftion. — A fwnilar queftion to this may 
 be lecn in fcveral books of Arithmetic, folved upon the following prin- 
 ciples. Firft 125 4- 5 ~ 130 percent. 
 
 If 130I. : looi, :: 9C0I. : 6921.64. iid.--,'j fterling. 
 
 Then 734I. 14s. pd. — 692!. 6s. l.|d.-j^j — 4il. 8s. 7d.-j\ lofs, 
 dift#ring from the above Anfwer by 7I. 6s. i^d.-jy. 1 fubmit to the 
 judgment of my reader vi^hich iolution i^ the proper one. 
 
 (66.) 
 
 T'^is Exatnple iifimiiar to the preceding. 
 
 The commifGon Gn7i5ol, 148^ at 27 per cent, is 178I, 
 1 5s. 4f d. Then7i;5ol. 14s. — 178I. 15s. 4|d. n 6971L 
 18.S. -^d. currency,, the Fadop niuft remit according to the 
 courlc of exchange 
 
 M2
 
 -'4 Exchange. 
 
 IfT35l. : lool. :: 6971L i8s. 7fd. : 5164I. 73. loid.-J 
 fterling. 
 
 (67.) 
 If 1^7!. : lool. :: 7549L iSs. 4d. : 4808I. 17s. 3ld,'y^^j. 
 An Aver. 
 
 (68.) 
 Ths Example it of the fame kind as the 65th and 66th. 
 The commifTion of 1757I. at 25- per cent, is 43I. i8s. 6d. 
 Then 17^71.-— 43I. 1 8s. 6d. = 171 3I. is. 6d. fterling, 
 to remit to Jamaica according to the courfe of exchange. 
 
 If lool. ; 157L :: 17131. is. 6d. : 2689I. los. 6td.-||. 
 Anfvver.. 
 
 (69.) 
 
 ^his Example is nvorked. 
 In the Anfwer,. for gain per cent, read lofs per cent. 
 
 (700 
 7his Example is ixJoAed, 
 
 In the Anfvver, for kfs per cent, read gain per cent. 
 
 (71.) 
 1^37-9-5. : lool. :: 33s. 4d. : 89I. 1 6s. 4jd.-.Vy 
 'I hen iQo — 89I. 1 6s. 4jd.-TV7- = lol. 3s. 7d.-7^\ lofs 
 per cent. 
 
 If 33s. 4d. : looI. :: 35s. 6d. : 106I. los. 
 
 Then 106I. los. — icol. = 61. los. gain per cent. 
 
 (73.] 
 Jr34s. i^. ; iQol. :: 33s. 6d. : 97I. x6s. ^\^*-Y^T* 
 Theiiiool. — 97I. i6s. zid.-fjy = 2I. 3 s. 9id.-jY7 lofs 
 per cent. 
 
 (74-) 
 If ;s. 7id. : rool. :: 5s. ^\^. : 92I. tis. io|d. 
 Then lool. — 92!. us. io|-d. =: 7I. 8s. i {d. lofs per 
 
 cent.
 
 Part Tv S'lNi^t^LB Asbitra'Tion OF Exchange. 125^ 
 
 ■ ' ' (750 
 If 5s..6c!r,:' lool. : : j's. lid. : 9:?^. ^s. 7-^d.-/y. 
 Then fcol. — 93!^ J3. 7'd.-/-r =± 61. i6s. 4^d.^V lofs 
 per cent. 
 
 (76.) 
 If4i|d. :.. jpoi. r: ^jfd. ; 127I. i^. 2;^d.-|f 
 Thea 127L r^s. 2fd.-^| — ^^^ — ^1^* '5^' 2itl'-t? 2'^^" 
 |>er cent. 
 
 SIMPLE ARBITRATION ok 
 EXCHANGE. 
 
 {77-) 
 Tihis Example is ivorked. 
 
 . (78.) 
 If 3-35. 7d. : 24od. : : 51 ^d. : 3oJ|5d. for 20 vintins^ 
 Of 400 rez. 
 
 Then 3o|-7^ x 2-1- = 76 J^fd. fterling, p. milre.. 
 
 (79.) 
 
 (This Example will be more properly exprcfTed, if' in the thiid line, 
 inltead of that Corrcj'pcndent, you read Rotterdam ; and in the feventh 
 line, inllead oi Parisy tczd Rot terdam. •^•'Thc '^z pence Flcm. Jhouid be 
 ^z piUings.) 
 
 Then inuerfely ^^A, : 32s. : : 54d. : 31^73. Flem. per 
 pound fterling. 
 
 (go.)- 
 In-ver/elj ^od, : 4.zd. :: 53id. : 39 -j'/^^d. per dollar. 
 
 (81.) 
 First, By remitting to Amfieidam and drawing tipon 
 
 If 52d. Flem. : 3i^d. fter. : : 34s, 5 d. Flem. : 9924if. 
 r: il. OS. 8fid. the merchant made of iL fterling.. 
 If il. ,: II..OS. 8|id. : : looI. : 103I. 8s. 3ld.-f[. 
 
 Then io3J..8s..3id.-A^ — icol. := 3I. 8s. 3;d.-f{ gain 
 per cent. 
 
 M3,
 
 126 Simple Arbitration of Exchange* 
 
 Secondly, By remitting to Paris and drawing upon 
 Amjierdam. 
 
 If3ijd. fter. : ^id. Flem. :: 24od. fler. : 399 i^ pence 
 Flem. 
 
 If34s. 5d. : il. :: 39Q-j^d. : 19s. 4~'^d. the merchant 
 would have made of il. fterling. 
 
 Ifil. : 195. 4^VJ^3d. :: lool. : 96I. 13s. nid.-^. 
 
 Then looI. — 96I. 13s. u Jd.-i|l = 3I. 6s. o Jd..|lf lofs, 
 per cent. 
 
 (82.) 
 
 If I due. : ^o^d. : : 9C0 due. : 4^225 pence. 
 If I due. : p d. : : 900 due. : 45900 pence. 
 
 d. pia. d. 
 
 If 41 : I :: 45225 : iioj^t piaflres. 
 If 42J : I :: 45900 : 1086/^*9 piaft res. 
 
 Lofs i6|4f^piaftres. 
 
 (83.) 
 d. gu. 1. 
 
 ]f 18 : I : : 500 : dSd^^ guilders. 
 ifiOQ : 3 :: 6666 j ; 22^ commiflion. 
 
 Diff. 6644I to remit. 
 
 If3ig. : icr. :: 6644|g. : 1 898 1| crowns. 
 If jcr. : 55d. :: 1898?,^ cr. : 43 5I. is. o|:d.-|f. 
 Then 500 — 43 5I. is. Oad.-|f = 64I. 18s, iijd.-|| the 
 ir.erchants' real lofs. 
 
 I. s. d. 
 I ool. 1 -i 1 64 18 II i-ll whole lofs. 
 
 12 19 9 J-|J the lofs percent. Anfwer. 
 
 OflstRVATicK. If the Faftor could have remitted to Bourdeaux 
 accordiiTg to the merchant's ord»r, he would have gained 7I. lis. i^d.-^y, 
 Jo that the variat'on in exchange nude 7x1. 103, ^^^i■g^^ ditVcrencf, or 
 comparative iof-.
 
 Parti. Compound Arbitration of Exchange. 127 
 
 (84.) 
 If I04gu. : 100 gu. :: 3000 gu. : 2884/3- gwilcl. Bank. 
 Ifgo-id. : 1 ecu. :: 28841^3 gu. : 127401. 58s. i-Y_5Lden. 
 Ifi05:gu. : 100 gu. :: 30oogu. : 2 857-} guild. Bank. 
 If 89?d. : I ecu. : : 2857! gu. : i276cr. 56s. r-^Y-den. 
 Then 1276 or. 56s. id. — 127401. 58s. id. =: i crown, 
 jSfols, the Paris merchant gained. 
 
 COMPOUND ARBITRATION 
 OF EXCHANGE. 
 
 (85-) 
 
 7his Example is ivorked. 
 
 (86.) 
 
 An^cedents. 
 
 I FrecK^h crown 
 
 65 Eng. pence 
 
 100 Stampt crowns 
 
 105 Ducats Banco 
 
 I Piaftre 
 
 * Flemilh pence 
 
 Confequents^ 
 rz 30 Engli/h pence, 
 rz I Stanapt crown, 
 
 rz 142 Ducats Banco, 
 rr ICO Piaftres. 
 rr 87 Fie mi Hi pence j 
 — 7547 cr. 15 fols. 
 
 4oyS4o||| 
 
 6 29 
 
 ^g^Xi42X/';^;;^X$^X7?4.7i _ 1 864774^3 _ 
 
 iX0^X/^0X/0^ ■" 455 
 
 ^3 35 
 
 Flem. pence, zz 10246 guild, o ft. 4*||: pen. at Anv. 
 fterdam. 
 
 Then, 
 
 If I ecu. : jid. :: 'jU'J cr. i^fol. : 9622 guild. 14ft. 
 1 4 pen. by the dired exchange. 
 
 Hence io246g. oft. 4||^p. — 9622g. 14ft. 14 pen-. =2 
 623 guild, 5 ft. 6 511 pen. gained by the circular ex- 
 change.
 
 128 Compound Arbitration of Exchange; 
 
 (87.) 
 
 Antecedents. 
 
 56 pence fteding 
 
 100 French crowns 
 
 T40 Ducats Banco 
 
 IJ5 Stampt crowns 
 
 * Pezzos 
 
 25 990 
 
 Confequcnts. 
 I French crowTit 
 60 Ducats Banco^ 
 100 Stampt crowns. 
 ia<; Pezzos. 
 i32i6od. iz. 7591* fterllng. 
 
 ^$:5X/g^^XX/^XX$//^p' _ 5 X^;X 990 ,^ 74250 _ ^ rs 
 
 X/pfSX^40X^/^ 
 
 7 7 ^3 
 
 pezzos by the firft method 
 
 Anteeedents. 
 ll. fterling 
 165. Flemifh 
 J I P.ix dol. 
 J40 Ducats Banco 
 J 1 5 Stampt crowns 
 * Pezzos 
 
 Again, 
 
 Confequents. 
 ~ 33s. Flemifh. 
 zn t Rix dol. 
 
 — 12 Ducats. 
 
 — 100 Stampt crowns.- 
 
 — 125 Pezzos. 
 
 — 759I. fterliog. 
 
 5 
 
 25 Z^ 
 
 /^a//x^^P^x//^ 14 1-4 
 
 4 1 iz 
 
 2 
 
 13 2:65 r y-J: by the fecond method. 
 
 Then 2651 \\ — 1515^5 = io86|f-| pezzos,. advantage 
 by tlie fecond method. 
 
 (88.) 
 
 AnteccdentSi 
 
 95^ Piaftres :^ 
 
 I Ducat rr 
 
 272 Maravedies, or i Piaftre rr 
 
 i^oo Rez, or i CrutaJc zz 
 
 56 Flemifh pence rr 
 
 I Ecu l^ 
 
 * pence, flcrling ^ 
 
 Confequents. 
 ICO Ducats Banco. 
 521 Maravedies. 
 631 Rcz. 
 50 Fiemifli pcnc^.- 
 
 I Ecu. 
 3i|d. fterling. 
 I 'Piaftre. 
 
 /dp' X 32 1 X 63 1 X ^.d X jij _ 321x631 y:^xzr\ 
 ^^xifiixi^<^xi6 
 19 136 4 
 
 6380396 5 __ 133805 
 
 I9XI3-X4XS6 
 
 d. fterling, p. piaftrCr
 
 Parti. Compound Arbitration OF Exchange. 129 
 
 K s. d. 
 
 7J47pia{!res, at ^j/.^VWi^^- P- ?^^^^^> = ^733 3 J^l" 
 754.7 ditto, at 52d. p. piaftre, z= 1635 3 8 
 
 Whole gain 97 » 9 si 
 
 If 1635I. 3s. 8d. : 1733I. 3s. lid. : : lool. : 105I. 19s. 
 9?d. from which take lool. there remains 5I. 19s. gjd. the 
 gain per cent. 
 
 (39.) 
 Bj the Note. — For the Circular Exchange ^ 
 s 00 : t ; : i : '005 general multiplier for the commiffion. 
 d. cr. 1. 
 
 If 54 : I :: I'JSl'l'y • 7812*2' crowns. 
 For commiffion |: per cent. X '005 
 
 — 39*o6i' fub. 
 
 Cro^^ns to remit to Venice 7773-161' 
 
 cr. d. or. 
 
 If 100, : 56 : : 7773*161' : 4352*9702' ducats. 
 For commiflion \ per cent. X '005 
 
 — 2 1 •764851' fub. 
 
 Ducats to remit to Hamburgh 4331 '205371' 
 
 du. d. du. 
 
 If I : 100 :: 4331*205371' : 433i20*537i'd. Flem. 
 For commiflion j per cent. X '005 
 
 — 2 165*60265' fub. 
 
 Flemifli pence to remit to Portugal 430954*934 f 5' 
 
 d. rez. d. 
 
 If 4C : 400 '/ 4309^4-93445' • 3S307'0'534»23 
 For commiffion \ per cent. X *oo5 
 
 — 19153*5526:0 
 
 Rcz. to remit to London 3S 11 556*98 rjjj,--^'
 
 ijo Compound Aubitsation of Exchange. 
 
 If locorez. : 63d. :: 38115^6-981453 : 240128-089831539 
 = I cool. I OS. Sd, received in London, 
 
 Or thus. 
 
 Antecedents. 
 540. FIcmiih 
 100 French cr. 
 • I Ducat 
 45d. Flemi/h 
 loco Rez. 
 
 Cdn^cjuents. 
 
 i«Ecu zz 
 
 : 56;Ducats — 
 
 io4>d. Flemifti zr 
 
 406 Re z. iz 
 
 6;jd. llerling-cr 
 
 The product of the fccond confequcnts is 24 
 dividend ; and tlie produft of the antecedence Ji 
 quotient is 1000-5 3 3795, &c^ I. — icooi. lo- 
 
 fey the circular exchange, as above. The 
 
 found by dedufting the commijjion from the firft 
 ing them hi the ratio of zco : J99. 
 
 Sec. Confequenti. 
 •995 Crowns. 
 55'72 Ducats. 
 99*5d. Ficmi/h 
 398 Rez. 
 63d. 
 '757-75 
 
 I 
 
 .3i2969058o'i93;5, the 
 i 243000000 divifor J the 
 s. Sd. received in London 
 : fee nd confequenfs are 
 confcquenls> or diminilh- 
 
 Bj the DifeSl Excha?ige, 
 If 34.S. 7d. : il. : : 1757I. 15s. : 1016I. 10s. 7d. 
 Then 1016I. los. 7d. — loool. los, 8d. =: 15I. 19s. nd. 
 advantage by the dired method. 
 
 Antecedents. 
 lb. 
 
 100 Engh'fh 
 
 78 Rouen 
 
 69 Lyons 
 
 72 Geneva 
 
 121 Marfeilles 
 
 103 Hamburgh 
 
 ♦ Paris 
 
 (90.) 
 
 Confcqucnts, 
 lb. 
 
 88 Rouen. 
 9^ Lyons. 
 53 Geneva. 
 100 Marfeilles. 
 
 100 Hamburgh, 
 
 1 01 Paris. 
 
 1 London* 
 
 47X53x100x10 1 ^ 
 39X69X9X11 X103 ^ 
 
 // 47 
 $$X^-^X53X/jE^^Xiooxioi 
 
 /^5T7fx69X^x7?>loo3 
 
 39 9 '« 
 
 L^li?-o j^^ ^^ p^^.^ ^ ^ j^^ ^^ ^^^^^^^ ^^^^ ^ 25159100 
 27440127 27440127 
 
 2281027 7700245 , .1. r> • iu 
 
 n '- = 1 oz. c -!-!■ ^ drs. the Pans lb. ex- 
 
 27440127 •'27440127 
 
 ceeds the Englilh avoirdupois lb. Or the Englifh lb, is 
 
 to the Paris lb. as lOO : 109 nearly.
 
 Part I. Involution. 
 
 '3t 
 
 INVOLUTION. 
 
 This Example is 'worked^ 
 
 (2.) 
 
 '754 X 1754= 307^5:i'5- 
 
 (3.) 
 
 549 X 549 = 3^1401- 
 
 (4.) 
 3^1416 X 3*1416 X 3-1416 = 3 1 •006494199296. 
 
 (5.) 
 
 •7854 X -7854 X '7854 = -484476471864. 
 
 (6.) 
 
 I + I = 2. 
 
 57'*5 X 57*5 = 3306-23% 
 2 + 2 =1 4. 
 3306-25 X 3306-25 = io93i289-o62y. 
 
 (7-) 
 I + I = 2. 
 1*732 X 1-732 =: 2-999824. 
 
 24-2+1= 5. 
 2*999824 X 2-999824 X 1-732 zi 15*586171061650432, 
 
 (8.) 
 
 I + I + I = 3. 
 
 735 X 735 X 735 = 39706537?- 
 
 Z -\- Z + 3 = 
 
 397065375 X 397065375 X 397065375 = 
 
 9- 
 6260 1 689 1 5 5608 1 39974609375. 
 
 (9.) 
 ^l + I + I r= 3. . 
 
 365^ X 365 X i^^s — 48627125. 
 
 3 + 3 = 6. 
 
 -48627125 X 48627125 = 236459728.
 
 I J2 E V O L U T I O N. — S QJJ A R L R O O 
 
 E V O L U T I O N.— S Q^u A R E Root. 
 
 STit/V Example is muorked^ 
 7'>6/> Exa'mple h ^worked. 
 
 (3.) 
 
 . . . root 
 
 393129(627 
 
 (4v) 
 • . ... root 
 3272869681 ( 57209 
 
 122)33 
 
 107)772 
 
 1247)8729 
 
 1142)2386 
 
 1 14409 ) 1 02968 J 
 
 (5-1 
 
 , root 
 
 5241578750190521 (12345-6789 
 
 22)52 
 
 243)*4» 
 2464)11257 
 
 24685)140187 
 
 246906)1676250 
 
 2469127 )i948i4i9 
 
 24691348)219753005 
 
 216915569)2222222121 
 
 (6.) 
 
 root 
 
 57i32*oocooo(239*e23,&c, 
 
 43)'/' 
 469)4232 
 
 47802 )iioooo 
 
 478043 ) 1 439600 
 
 o 
 I + 8 
 
 o 
 
 proof 
 
 5471 rem.
 
 part I. EvoLUTio N.— S a,u a s e Ro o T. 1 3 j 
 
 (7-) 
 
 , root 
 
 75* :547 0000000 (8- 6 802 6497 29, &c. 
 
 166)113+ (8.) 
 
 1728)13870 ... ... root 
 
 i788-57'777777{42-29i5»&c- 
 
 173602)460000 
 
 82)188 6 
 
 1736046)11279600 7 + 8 
 
 8449 )77377 6 ^ 
 
 168906 • proof 
 
 84581)133677 
 
 863324 
 
 12662 
 
 See contracted r lo 
 7 
 
 845825)4909677 
 
 Divifion of 680552 rem. 
 
 Decimals. 
 
 (9O 
 
 •4325000000 
 
 root 
 ( -65764 
 
 L, &c. 
 
 125)725 
 
 1307 ) 1 0000 
 
 
 5 
 1+4 
 
 •X- 
 
 13146)85-100 
 
 131524)622400 
 
 
 proof 
 
 96304 
 
 rem. 
 
 
 N 
 
 /
 
 ji34 Evolution. — S <^u a r e Root, 
 
 root 
 
 5'3'3353333 ( 2-3094.. &<:. 
 
 4609)43335 ^ ^^ 
 
 43)133 2 
 
 o -f 2 
 
 X 
 
 46184)^85233 2 
 
 ' proof 
 
 497 rem. 
 
 (II.) 
 
 This Example is ^worked. 
 
 This Example is ivorkei, 
 
 567 81 V 8j 9 
 
 (14.) 
 
 or 
 
 , \/i4822C 38c f 
 
 27; X 539 = 14822: and 1 i = ^-^ =-^root. 
 
 ill =.15 ,,d /ir^-iroot. 
 539 49 V 49 7 
 
 (^50 
 
 45 X 94 = 4230 and 2LL^ = - ^ ^^ = 
 ^^ ^^ "^ 94 94 
 •6918984, &c. root. 
 
 ('6.) 
 
 -V/15I = >/i5'625 = 3*9 > 2^4 root" 
 
 (17-) 
 
 V29>5 = 'v/29-i6 = 5*4 root.
 
 Parti. Evo L u Ti o N.— Sq^u A RE Root. 135^ 
 
 (18.) 
 
 (2oL IS. = 2401s. and v' 240 1 = 49s. = 2I. 9s, 
 
 3oJ;d. = 121 farth. and ^^ 121 = 11 men, they fpent 
 each 1 1 farthings, confequently each man drank a pint and 
 a pennyworth, or it pint. 
 
 (20.) 
 
 £4 X 24 =: 576 =: A B x A B 
 
 18 X 18 = 3^4 = B C X B C 
 
 v'goo nz 30 ~ A C the. 
 length of the ladder* 
 
 (zu) 
 
 50 X 50 r: 2500 zr E C x EC 
 30 X JO iz 900 = B C X B C 
 
 ^1600 iz 40 — BE 
 
 50 X 50 = 2500 zr E C X EC 
 40 X 40 ZZ. 1 600 zz A C X AC 
 
 V500 zz 30 zz A E 
 
 Then ABzzAE + EBzz 
 4o + 3oz= 70 ft. zz23|yd5. the 
 breadth of the tlreet. 
 
 Firft, 
 
 ^•449 zz B G 
 64- —AD 
 
 B E 
 
 z6-249 iz E G 
 
 97 X 97 = 9409 zz D G X D G 
 
 26-249 X 26-249 — 689-01 — E G X E G 
 
 N 2
 
 156 E V O L U T I O N. S QJU ARE R O O T, 
 
 ^8719-98 — 93-3808 ri D E = A B the iiitsnce of the httom of 
 the tower A f-om ttat of B i 76 -f 93*3803 rr 169-3808 r: A C the 
 ci.lance of the bcrtcm of the tower A Irom tbat of C zr K K. 
 
 Secondly, 
 64—50—14 — DK=:AD — CH 
 ^ 369-3808 X 169-3808 — 28689-8554:1: KH X KH 
 14 X 14 — ^S^" 
 
 z:DK X DK 
 
 ^^28885-8554— 169-958 — DH the 
 dlftance of the t':b of the tower A from tLat of C. 
 
 Thirdly, 
 
 90*249 -^ 5c = 40*249 — LG — BG — CH 
 And HL —EC 
 
 76 X 76 — 5776 = HL X HL 
 
 40*249 X 40*2-49 =^ 1619*982 ::: LG x LG 
 
 v^7395-9S2 r:"85-999, &c. = 86 «jr/jfj 
 the diA»ace of the /op of the tower B from ibat of C. 
 
 '3-> 
 
 For that of C 
 50teet, -ead thit, 
 of C 28 feet. 
 
 D 
 
 Then, 
 
 The fide of aa 
 3 cq«l lateral trian- 
 gle divided by the- 
 fqaare root of 3> 
 gives the radius 
 of .t? circumicri- 
 bing circle. 
 
 50 
 
 v: 
 
 i*73Z,£L-c. 
 
 — aS-SSy feet ir AG the diftance of each tower 
 
 from the centre of the garden, and 
 
 30 + 54 + 28 __ 
 
 — 3o| the height 
 
 of a mean tower. 
 
 30^ X 3 of = 940*444444 
 
 a8-S67 X 28*867 — 833*;c5689 
 
 •f the ladder nearly 
 
 Vj773-748i33 =: 4Ji»ii5 = PD the length
 
 Part I. E V o L u T I n. — C ubeRoot, 137 
 
 1773-745133 — PD X PD 
 
 30 X 30 m 9<^'0« ~ AD X AD 
 
 n/S73-748i33 = 19-559 = AP 
 
 177^742133 = PD X PD 
 
 34 X 34 zr'iisO^ = BD X B D 
 
 v'617.748133 = i4'S54 = B P 
 i77r74Si33 rr PD X PD 
 28 X 22 iz: 784* = CD X CD 
 
 v/989-748i33 lt 31-46 = C P. 
 Observation. Had the height of A been 38, B 42, and C 45, 
 and the diftance from A to B z= 50, B to C zz 40, and from C to A rr 
 47 feet, the operation wouVi have been more difficult.— The length of 
 a ladder in this caft would have been 49*552j and hence, the didances 
 would have been found as above. 
 
 CUBE ROOT. 
 
 T^is Exa?n^le is <WQrked, 
 
 (2-) 
 
 122615327232 ( 4968 root 
 4 X 4 X 4 = 64 
 
 4 X 4 X 300 — 4800 ) 58615 relbl. 
 
 9x9X9= 729 
 
 4 X 30 X 81 — 9720 
 
 4800 X 9 zn 43200 
 
 53649 fub. 
 
 49 X 49 X 300 = 7203CO ) 4966327 refol. 
 
 6x6x6 zi: 216 
 49 X 30 X 36 = 52920 
 720300 X 6 zz 4321800 
 
 4374936 fub. 
 
 496 X 496 X 300 — 73804^00) 591391232- refol, 
 
 8x8x8 zz 512 
 496 X 30 X 64 rz 95232c 
 73804800 X 8 zz: 5904384C0 
 
 ■59139123Z 
 
 N3
 
 J38 Evolution. — Cube Root, 
 (3.) 
 
 41421736 ( 346 root. 
 3 X 3 X 3 = ^7 
 
 3 X 3 X 30^ = 51700 ) 14421 refol. 
 
 4x4x4 == 
 
 64 
 
 3 X 30 X 16 — 
 
 1440 
 
 2700 X 4 — 
 
 10800 
 
 12304 fub. 
 
 34 X 34 X 3C0 = 346800 ) 21 17736 refol' 
 
 6x6x6 = 216 
 
 34 X 30 X 36 =r 3672a 
 346800 X 6 zz 2080800 
 
 2117736 fufe. 
 
 {4-) 
 
 705'9i9947284{S'9jQ4 root«, 
 8 X 8 X 8 = 512 
 
 2 X 8 X 300 — 19200 ) 193919 refol. 
 
 9X9x9 = 729 
 
 8 X 30 X 81 r= 19440 
 J9200 X 9 — 172800 
 
 192969 fub. 
 
 S90 X 890 X 300 — 237630000 ) 950947284 refo].r 
 
 4x4X4 =: 64 
 
 890 X 30 X 16 zz 427200 
 237630000 ^4 — 950520000 
 
 950947264 fub. 
 
 Rem. 20, &■;,
 
 Parti. Bto L u Ti ON.— C u B E Root. 139 
 
 is-) 
 
 . . ; root. 
 
 i7*5:4ooooooo(2'598. Sec. 
 2x2x2 = 8 
 
 2 X 2 X 3C0 = 1200)9540 refol. 
 5X5X5 = 125 
 
 2 X 30 X 25 ^::r I ;oo 
 
 1200 X > -■= 6coo 
 
 7625 fub. 
 25 X 25 X 3C0 = 187500 ) 1 91 5000 refol, 
 
 9X9X9 = 729 
 
 25 X 30 X 81 =: 60750 
 187500 X 9 zz 1687500 
 
 1748979 fub. 
 
 259 X 259 X 300 = 20124300) 1 6602 1 000 refoL 
 
 8 X 8 X 8 = 512 
 
 259 X 30 X 64 = 497^80 
 
 20124300 X 8 = 1609944.00 
 
 1 61 492 1 92 fub. 
 
 Rem, V4528808
 
 1^0 Evolution. — Cube Root. 
 
 (6.) 
 
 254358061056000 ( 63360 rooti 
 6 X 6 X 6 =z aj6 
 
 6 X 6 X SCO n loSoo ) 3S35S rex"'!. 
 
 3X3X3 = ^7 
 
 6 X 30 X 9 = ^^^^ 
 
 icSoo X 3 == 3MQ3 
 
 34C47 fub. 
 63 X 63 X 3C0 => 1190700)4311061 refoU 
 
 3X3X3 = 27 
 
 63 X 30 X 9 = 17^10 
 11907C0 X 3 = 3572TOO 
 
 35S9137 fub. 
 
 633 X 633 X 300 m 120206700)721924056 refoU 
 
 6x6x6 rr 216 
 
 633 X 30 X 36 =. 683640 
 
 120Z06700 X 6 zz. 72124020a 
 
 721924056 fub, 
 
 ••• • ceo 
 
 (7-) 
 
 . . . root. 
 •573450000000 ( '830^, &c» 
 8X8x8 = 512 
 
 8 X 8 X 300 — 19200 ) 61450 reW, 
 
 3X3X3= 27 
 
 8 X 30 X 9 = 2j6o 
 
 19200 X 3 = 57600 
 
 59787 fub; 
 
 83c X 830 X 3C0 :^ 206670CC0 ) 1663C00000 refol, 
 
 8 X 8 X 8 rr 512 
 
 83c X 30 X 64 r= 15936C0 
 
 »o66700oo X 8 = 1653360000 
 
 1 6549 541 12 Alb. 
 
 8045888 rcmr
 
 Parti. Evolution. — Cube P. o o t, 141 
 
 (3.) 
 
 .... . root. 
 75:'3857ooooo ( 4*224, Sec, 
 4X4X4 = 64 
 
 4 X 4 X 300 = 4800 ) 1 138 J refol. 
 
 10088 fub» 
 
 42 X 42 X 300 = 529200 ) 1297700 refol, 
 
 1:6344^ Tub. 
 422 X 422 X jco zii 55425x00) 2342520C0 refol. 
 
 213903424 fub. 
 
 20348576 rem. 
 
 «^ (9) 
 
 root, 
 •785400000000 ( -9226, Sec, 
 9x9X9= 729 
 
 9 X 9 X 3C0 zz 24300) 564.00 refol. 
 
 49688 fub. 
 92 X 92 X 300 = 2539200 ) 6712000 refol. 
 
 5089448 fub. 
 922X922X300 z= 255025200 ) 1622552C00 refol. 
 
 1531147176 fub. 
 91404S24 rem.'
 
 14*'. E V o t u T I N, — C e B E Root, 
 
 (10.) 
 
 root, 
 
 517-375475000000 (8*0278, &c, 
 8x8x8 = 512 
 
 80 X 80 X 300 =z 1920000)5375475 refoL 
 
 3849608 Tub. 
 802X802X300= 192961200)1529867000 refoL 
 
 1351907683 fub. 
 fo27xSo*7X 300 = 19329818700)173959317000 rcfolr 
 
 I546<;396i952 fub. 
 19305355048 renit- 
 
 (n.J 
 
 roof. 
 
 20874107909304)27534 
 2x2X2 = 8 
 
 2 X 2 X 300 = 1200 ) 12874 refoU 
 
 1 1 683 fub. 
 
 27 X 27 X 300 = 218700 ) 1 191 107 refol, 
 
 1 1 1 3 87 5 fub. 
 
 275 X 275 X 300 = 22687500} 77232909 refol. 
 
 68136777 fub. 
 i7 S3 X 2753 X 300 = 2273702700 ) 9096 1 32304 refol. 
 
 9C961 32304 fub»
 
 Part I. EroLUTio n. — C u b s R o o t. 1 43 
 
 (,2.) 
 
 root. 
 
 1551328-2159785156^5 (115.7625 
 X X I X I = I 
 
 X X I X 3^0= 300)551 refol. 
 
 331 fub. 
 
 ji X u X 300 :::^ 36300 ) 2Z0328 refol. 
 
 189875 fub. 
 115 X 115 X 300 — 3967500) 30453215 refol. 
 
 27941893 fub. 
 
 3157 X 1157 X 300 — 401594700)2511322978 refol. 
 
 2410817976 fub. 
 
 I1576 X 11576 X 300 — 40201132800) 100505002515 refol. 
 
 20403654728 fub. 
 
 115762X115762X300—4020252193200)20101347787625 refol. 
 
 20101347787625 fub. 
 
 (13.) 
 *Tbts Example is ivorked 
 
 (14.) 
 
 7i54'i09i6753o ( 19*26921, &c, 
 1 X I X I = 1 
 
 jy, ZOO zz 300)6154109 
 
 19 "\20513 NoTi. The 3d figure In the 
 
 •\- 9 JiS root by the rule would be 5, but 
 
 ■ .■ by Involution I find that too big. 
 
 285 ) 1 5 1 3 —-See the Note to the Rule. 
 1425
 
 144 E V O L U T I O N.— C U B E R O O T. 
 
 7I54MC9I675300COOCO 
 IC2 X 19a X 192 ~ 7Q778SS 
 
 192. X 3CCC0 zz 5760000 ) 7682116753C000000 
 
 1926 \ 13337008251 
 4- 6 / 11556 
 
 J9329 \ I78IOO 
 + 9 / 173961 
 
 193382 \ 413982 
 
 -|- 2 / 336764 
 
 1933841 ) 272I85I 
 I933S4I 
 
 7SS010, &c. 
 
 (•5-) 
 
 Flrft, root. 
 
 8302348CCCC00, &c. ( 20248'8475, See* 
 2X2X2 = 8 
 
 2 X 300 = ^^° ) 3^^348 
 
 2C2 ) 503 
 
 99 
 
 Second, 
 
 S3C234S0COCOOOOOOOCO00000 
 
 2C2 X 202 X i02 — S24240S 
 SO^XSCCOCCO— 6060COOCO ) 59C4QOCOOCOOOOOOOCCOOGO 
 
 2024 ) 98910891089108 
 40288 ) I7950S 
 402968 ) I72049I 
 2029764 ) 965870S 
 20297687 ) 1 53965291 
 202976945 ) IIS8I4820S 
 
 J73263483, fcc.
 
 Fart I. Evolution. — Cube Root. 14J 
 
 (16.) 
 
 Flrft, . . • root. 
 
 I'oooooo ( i'25992io49S94, &c. 
 1 X 1 X I = I 
 
 I X 300 zi 3c>o ) 1000000 
 
 ^^ ) 3333 '^^''^ 3^ figure would be a 6, 
 
 • but that, by Involution, is to* 
 
 14^ ) 933 ^'§> ^ therefore take 5. 
 
 57 
 Second, 
 
 2 '000000000000000000 
 
 125 X 125 X 125 — 1953125 
 
 S25 X 30C00 n 375^000 ) 4637500COC0000000 
 
 1259 ) 1250000000Q 
 
 12689 ) 116900 
 
 ;269S2 ) 269900 
 
 1269841 ) 1593600 
 
 323759, &c. 
 
 Tiiird, 
 
 •^ 2 '000000000000000000000000000000000000 
 
 [259921] =r 1999999762590486961 
 
 i2.:;oc2iX 3000C00 \ c 
 
 _Y775;630ococo ) ^37609513039000000000000000000 
 
 12599I104 ) 6-863600982125069, &c, 
 1259921089 ) 12466759332 
 
 12599210988 ) II274695SII2 
 
 125992109969 ) ii9532702o35o 
 
 X25992I0997S4 ) 6I39803II2969 
 
 1100118713833, dtc. 
 This laft operation would give the root true to n;ar twenty placet of 
 ilLcimals, if neceflary. 
 
 o
 
 146 Evolution. — Cube Root. 
 
 (.7-) 
 
 Firft, .... root. 
 
 •0C0135700CO0 ( •05i387799i2> &c. 
 5X5X5 = 125 
 
 5 X 300 = 1500 ) 10700C00 
 
 51 ) 6666 
 523 ) 1566 
 
 Second, 
 
 I 357oooooooocooooocoooooooe 
 
 513 X 513 X 513 = 1350^5697 
 
 513 X 3CCCC00 — 15350CCC00 ) 6943040000000C00CC000000 
 
 5135)451139670110461, &c. 
 51467 ) 400996 
 
 5H747 ) 4072770 
 
 5147549 ) 469 541 " 
 
 51475581 ) 63617004 
 
 514755222 ) 1214142361 
 
 JS4630717, Sec. 
 
 (18.) 
 
 Firft, . . . root. 
 
 J3'666666 ( 2'39cS6o3o, &c. 
 a X 2 X a = S 
 
 a X 300 zz 600)5666666 
 
 23 ) 9444 
 
 ^69 ) 2544 
 
 123, &c.
 
 Part I. EvoLUTio n. — C u b e Root. 147 
 
 Second, . ........ 
 
 13-666666666666666666666666 
 
 239 X 239 X 239 = 13651919 
 
 239 X 3C00000 — 717000000 ) 14747666666666666666666 
 
 23908 ) 2056E572756857, &c. 
 
 239166 ) 1442172 
 
 23917203 ) 71767568 
 
 : 505957, Sec. 
 
 (r9.) 
 Firft) . . . • . root. 
 
 92398647506217 ( 45208*6846, &c, 
 4 X 4 X 4 = 64 
 
 4 X 300 zz 1200 ) 28398647 
 
 45 ) 23665 
 
 5C2 ) 1165 
 
 161 
 Second, • ....?. . . 
 
 9239?. 647 5062 1 70C00000000CO 
 
 452 X 452 X 452 = 9^345408 
 
 45^ X 3000000 =. 13560C0000 ) 5323950621700C0000COCOC 
 
 45208 ) 3926217272640] 
 452166 ) 30 ,5772 
 
 4521728 ) 3827767a 
 45217364 ) 210384804 
 452173686 ) 2951540801 
 
 2384- 8685, .^'C. 
 
 Observation. When the reader has cnce m ide this n-.eth- d of ex- 
 trading the Cube Root fam;i'ar to hitn^ i am peiluad-d he will ftnd it 
 not only th-.- <<i/T/, but the /^eft method hitherto made uie of. — I know 
 of none that converges iaftcr, or is Icls troublefome in tlic proceft, 
 
 O 2
 
 J48 E T O L U T I O N. C U B E R O O T. 
 
 (lO.) 
 This Example is ix:orkcd» 
 
 (21.) 
 
 This Example is ^worked* 
 J^ -^68 - -71638, &c. 
 
 3 /iL. -- 3 / ^7 _ J 
 
 V 375 V 125 5* 
 
 ^ /-^ = '82207, &c. 
 
 (26.) 
 
 y |i = -9859^ &c. 
 _^ (27.) 
 
 Asi,^ : •5236 : rTsl^ : 1767.15 ttet. Anfwcr. 
 
 (28.) 
 ^^31185 — 3 1*476 1, the fide required. 
 
 (29-) 
 ai50'42 X 60 X 8 zr 1032201 '6 cubic inches content of the fcia. 
 
 Thcav/io32ici'6 z: lOi'C? inches. Arfwer.
 
 Part I. EvoLUTio n. — C u b e Root. 149 
 
 (30.) - 
 
 lb. 
 
 — >., 3 /';coo X Sooo 
 ai7 : 2c)* : : 9000 : / r: 69*2294. length. 
 
 V ^'7 
 
 217 rTfl' : : 9000 : / — . '. r: 5i'922 breadth. 
 
 — PAI ^ /90OO X 5 '2 
 
 Z17 •. 8p : : 9000 : / — r: 27*69x7 depth. 
 
 FIrft, 
 
 V 3 X i^5;^ rr i-SV^S =: 180*28 the keel. 
 v/3 X ~T|'^ = 25\/3 = 36-05 miJA'p beam. 
 v^3 X T5I' n 15 \/ 3 ;= 21*6 depth of the hold. 
 
 Secondly, 
 V^:- X 12^^ 1= iz$\/^ rr 99-21 the kee!. 
 
 V/ I X 25I rr 25 V^l zn i9"?4 midfnip beam, 
 v/i X 15)^ — 15 V^'t^ r: 11-905 depth of the hoU. 
 
 (3^.) 
 
 Admit the folidity of the coine t? be i, then 
 
 I '."^ : : 1 : / -— = 20 ^T - 13.^672 the heigh 
 cf the (op part. 
 
 3 ibcoo 
 I7'47i6 -- i3*S672 — 3-6044 }-.e"ght of the middle par:. 
 
 ; ^°l - i = J ~~r = ^^ l/^ ~ 17-47I and 
 
 Alfo 20 — 13-8672 + 3.6044 — 2.5284 height of the bottom part, 
 ia tnchei, Anfwer, 
 
 This Example is nxorJic£i. 
 
 o 3
 
 150 
 
 E V O L U T I O N. A NY R O O T. 
 
 (34-) 
 
 . . . . root. 
 a«oooooo ( I "414, Sec, 
 
 "14,* z: 196 fab. 
 
 14 X 1 — -8 ) 40 fecond div. 
 14?!* — 198S1 fub. 
 141 X 2 — 2S2 ) 1190 third div. 
 1414I* rr 1999396 fub. 
 Rem. 604, &c. 
 
 i3S') 
 
 .... root. 
 5'ocooocooo ( I '709, &c. 
 X X I X I = I 
 
 I* X 3 1:1 3)4° fif^ <^*^* 
 
 TtI^ z: 49^3 ^^^• 
 T70]* X 3 ~ 867CO ) 870C00 fecond div. 
 
 "1705]^ = 4-991443S29 l"'-b' 
 Rem. S556171, &c. 
 
 {36.) 
 
 root. 
 i728*oooococooooo ( 6'447, &c. 
 6x6x6x6r= 1296 
 
 ""^^ X 4 m 864 ) 4320 firft div» 
 
 64*4 zr 16677216 fiib. 
 
 64^3 X 4 ~ X048576 ) 6017S40 fecond div, _ 
 
 "'v>44^* ZZ 172005949696 fub. 
 
 644!^ X 4= 1063359936 ) 7940503040 thrrd^iv. 
 
 ^447)4 r: 17275502185S8481 fub. 
 
 Rem. 449731411519, &c.
 
 Fart I. Evoi. UTI9N. — Any Root. 154 
 
 (37-) 
 
 . . . . rout. 
 
 5y54ccooo6cooooco ( i*2<-9, ic:r 
 2X2X2xaxz r: 3* 
 
 T|4 X 5 = So) 255 firftdiv. 
 22 1> -zi 515363a fub. 
 12^+ X 5 ~ 1 171280 ) 6003680 fccond div. 
 224^ S ~ 563949338624 fub. 
 "22^'^ X 5 — 12588154880 ) 1 1450661 3760 third dlv. 
 Z249V rz 57537008386886249 fub. 
 
 Rem. 2991613:13751, &c. 
 
 (38.) 
 
 . • root. 
 
 3 '141600000000000000000000 ( 1*2102, &C, 
 TXIXIXIXIXI = 1 
 
 1^X6 ir 6)21 firft div. 
 
 T2,6 — 2985984 fub. 
 7?|5 X 6 IT 1492992 ) 1556160 fecond div. 
 I2i\^ ZZ 3138428376721 fub. 
 
 .55l^476l6oc7oo}3'"«^3»790ocoooo tlnrd div. 
 
 12102^5 — 31415421541C9890900723264 fub. 
 Rem. 57845890109099276736, &c, 
 
 In order to make room for more ufeful matter, I rtiall only fct down 
 pirt of the following folutlons. 
 
 (39-) 
 
 root. 
 547'5oooooooocococoococoo ( 1*461, &c.
 
 '5^ 
 
 Duodecimals. 
 
 (40.) 
 
 root. 
 547»30ooooooocoooococcoocooo ( 2' 199, &c. 
 
 (4'.) 
 i-53i328ii557^5>5625 ( 1-05 r3«t. 
 
 DUODECIMALS. 
 
 (I.) 
 
 'hlj 'Example h ivorked. 
 
 (2.) 33 ft. II in. iTpts. 
 
 (3.) 81 ft. 6 in. 6pt5. 
 
 (-4.) 38 ti. 6 in. 2 pts. 
 
 (y) 8^6 ft. I in. 6 pts. 
 
 (6.) 92 ft, 2 in. 5 pts. 
 
 (-J 2 2:;2 ft. 4 in. 6 pts. 
 
 (S.) 4203 ft. 3 in. 3 pts. o" Ci"\ 
 
 (9.) 253 ft. II in. opts, ii'^ 
 
 1 ]'" 2IV IV, 
 
 {10.) 80 ft. I in. 7 pts. 7" 9'". 
 
 (II.) 
 
 f. in. pt5. 
 
 25 II 6 S" 7'' 
 
 25 II 6 8 7 
 
 649 OHIO 7 
 
 23 9 7 I ^o 5^^- 
 1 o II 9 4 3 6''' 
 1538 5 8 8yi. 
 
 13 T 8 II O ivii. 
 
 674 I i' 4" 7'" 11^^ I'' 8^^ 1^". 
 
 (12.) 
 24 ft. 9 in. 5 pts.
 
 Part I. Duodecimals, 
 
 . (13.) 
 
 S5 
 
 m, 
 
 ^ 'M Heights 
 
 ^ ^ f add 
 4 3J 
 
 16 
 
 115 6 heights together. 
 
 Mult, by 3 6 breadth. 
 
 Prod. 404 3 = 404*25 feet area. 
 
 Then 404-25 X 14*5 = 5861*625 pence, zz 24I, Ss. 
 5 [d. Anfwer. 
 
 ft. in. 
 
 24 5 length. 
 12 7 breadth. 
 
 9)307 2 II area. 
 yds. 34 I 2 II 
 
 (14.) 
 ft. in. 
 
 2 
 8" 
 
 3 4 
 
 5. d. yds, 
 3 4 lil 34 
 
 4i '777 
 i-962 
 
 •987 
 
 '370 
 
 d, 5f '096 
 
 5 '3 4 
 
 si add 
 
 (15.) 
 
 Firft, 
 Find the value of the whole court at half-a-crown a foot, 
 and the foot-path at fixpencej the fum of thefe values will 
 be th« whole colt.
 
 '54- 
 
 D U O D E C I 
 
 ft. in. 
 
 62 7 length 
 
 44 5 breadth 
 
 Prod. 2779 8 II area of the whole yard. 
 
 s. d. ft. 
 
 2 6 III 2779 
 
 6 
 
 347 7 6 
 o 1 loj 
 
 347I. 9s. 4:Jd. val. 
 
 c' 
 
 I 
 
 
 
 ^ 
 
 6' 
 
 
 
 
 4' 
 1' 
 
 6 
 
 4 
 
 s. d. 
 2 6 
 
 I 3 
 5 
 i: 
 
 of the court-yard at 2s. 6d, 
 And 
 
 I '333 
 •833 
 
 IS. ioJ.d. -166 
 
 ft. in. 
 
 62 7 length 
 Si 
 
 344 2 6 area of the foot-path, 
 in. 
 
 d. 
 
 d. ft. 
 
 6 l,VI 344 
 
 8 12 
 
 id. 
 
 812 I J value of the footpath at 6d. per foot. 
 347 9 4J; ditto of the court-yard at 2s. 6d. 
 
 356I. IS. cfd. whole value. Anfwer. 
 
 (16.) 
 
 230219796I. -s. gd. 1= 227828377*5:ii'qo476' guineas, 
 or fquare inches; theie divided by 1296 fquare inches, or 
 gu neas in a fquare yard, ^ive 175793*501166593, Sec. yds. 
 rr 99 miles, 7*0633, Sec. furlongs ; fo that the national debt 
 would pave a foot-^ath of a yard wide r/earh from London to 
 Baih»
 
 Part I. Duodecimals. ijj 
 
 (>7-) 
 
 ft. in. 
 
 22 7 = "-583; 7 Mult. 
 13 II = i3'9io j 
 
 9 ) 3I4.-28472' feet « 
 
 34.'920524, &:c. yards 
 Mult. Ill- 
 
 Prod. 401-586053 pence, r= il. 13s. ^fd. Anf, 
 
 (18.) 
 
 #fc. in. ft. in. ft. ft. 
 
 74 loxii 7 ==: 74*83'xii-5S3' — 866'Sic4' area ofthe whflle room 
 7 6x 3 9 Z= 7*5 X 3-75 — 2,2' 1250' area of the door once 
 6 8x 3 4X5rz6'6' X 3'3'X5 — lUMiii'dittoofthe ihuttersoncc 
 
 8 + 8 + 3*3' + 3-3' X I* 16' X 5 n 131-222^' ditto of the breaks once 
 
 Sum of tlie areas ii38'2777' 
 Dedud 6 ft. 9 in. X 5 ft. zr 33'75 ditto of ttie chimney 
 
 Area of the whole work iio4'5277' feet 
 
 122 7253 yds. X 81i. — 
 
 io43'j65 pence — 4I. 6s. iid. Anfwer. 
 
 (19O 
 
 ft. in. 
 
 31 5 o breadth of the building 
 
 15 8 6 half the breadth 
 
 47 I 6 breadth of the roof 
 57 7 o length of ditto 
 
 100)2713 7 4 6 area 
 
 27 fq. 13 ft. 7 in. 4pts. 6" at half-a-guinea per fquare, 
 amounts to 14I. 4s. ii'ii6d. Anfwer.
 
 (20.) 
 
 ft. in, 
 
 14 II height 
 
 
 •947 
 
 2 
 
 6 
 
 5 
 
 3) 
 
 4736 
 
 
 
 6 
 
 272) 
 
 1578 
 
 8 
 
 2 
 
 
 5 rods. 
 
 21 
 
 8 ft. 
 
 (2r.) 
 
 ft. 
 
 in. 
 
 ft. 
 
 in. 
 
 ft. 
 
 in. 
 
 
 
 28 
 
 10 
 
 28 
 
 10 
 
 ^5 
 
 8 
 
 
 
 20 
 
 
 
 20 
 
 
 
 28 
 
 10 
 
 8 
 
 4f42 
 
 576 
 
 8 
 
 576 
 
 8 
 
 45^ 
 
 8 
 
 10 
 
 
 5 
 4 
 
 
 *r 
 
 
 
 3 
 
 
 28 10 
 
 1883 
 
 2306 
 
 8 
 
 ^3SS 
 
 2 
 
 302 9 
 
 ft. in. 
 2SS3 4 content of the firft 20 feet X by i bricks 
 2306 8 ditto of the fecond 20 feet x by t ditto 
 1355 2 ditto of the i j ft. 8 in. x by -j- ditto 
 
 302 9 ditto of the gable X by t ditto. 
 
 3)6847 II fum. 
 
 2282 7 B content of the wall, reduced to theHandard 
 thicknefs. Then 
 
 If 272 ft. : 5I. i6s. : : 2282ft, 7 in. 8 pts, ; 48I. 135 
 5-2d. Anfwer,
 
 Part I. Bills of Parcels, &c. lyy 
 
 BILLS OF PARCELS, exercising COM- 
 POUND MULTIPLICATION, the 
 RULE OF THREE, PRACTICE, 
 TARE AND TRET, &c. ' 
 
 
 
 
 
 ♦ 
 
 (!•) 
 
 
 (2.) 
 
 
 (3.) 
 
 rames Lamb, 
 
 Efq. 
 
 Sir John Guchim. 
 
 Andrew Wines, Ef^, 
 
 1. s. 
 
 d. 
 
 I. s. 
 
 d. 
 
 ,. 1. s. d. 
 
 3 17 
 
 6 
 
 2 II 
 
 ^1 
 
 109 8 2i 
 
 lo 9 
 
 
 
 8 15 
 
 151 13 2 
 
 8 i6 
 
 Si 
 
 8 10 
 
 I 
 
 157 14 io| 
 91 10 9i 
 
 11 8 
 
 i 
 
 8 I 
 
 li 
 
 13 2 
 
 257 
 
 5 
 
 59 19 iii 
 
 * 15 
 
 
 
 90 5 3 
 
 284 
 
 1170 17 ol 
 
 50 9 I — ^— 660 
 
 (4.) (5.) (6.) 
 
 rge VereS; 
 
 ,Efq. 
 
 Hugh Abbot. 
 
 MifsEvltt. * 
 
 I. s. 
 
 d. 
 
 1. B. d. 
 
 I. s. d. 
 
 6 
 
 5i 
 
 651 18 8 
 
 9 6 J 
 
 9 
 
 °i 
 
 92 9 7.1 
 
 21 7 oi 
 
 14 
 
 8| 
 
 374 7 ?e-j 
 
 12 13 
 
 7 
 
 3S 5 ^i 
 
 13 15 9? 
 
 80 9 io| 
 
 7 
 
 ^t 
 
 35 10 5 
 
 5 
 
 5i 
 
 
 10 4 o| 
 
 169 10 5|- 
 
 (7.) (8.) (9.) 
 
 Mr. Crowl 
 
 J. s. 
 
 ^ 14 
 
 3 19 
 
 378 12 
 7 12 
 
 30 17 
 J22 S 
 
 :her, 
 d. 
 
 i 
 
 6i 
 
 Mrs. 
 J. 
 
 13 
 11 
 
 79 
 189 
 
 ^3 
 
 Mertown. 
 
 s. d. 
 
 8 8 
 
 19 6^ 
 
 7 II 
 
 5 6:- 
 
 19 7 
 
 Mr. Ochterlony 
 I. s. d. 
 
 405 3 4 
 
 369 IS 
 10 15 6 
 
 267 .4 4^ 
 So 10 5 
 
 
 318 
 
 I 3 
 
 3j.-; ^3 4 
 
 5^^ 5 
 
 Jl 
 
 1476 :i II J 
 
 * In the firft line of this bill, /or per ell, read per yard, 
 P
 
 158 
 
 Bills of Parcels, &c. 
 
 ' (10.) 
 
 (11.) 
 
 (12.) 
 
 lir George Lovell. 
 
 Mr. Meafure 
 
 ;well. 
 
 Mr. 
 
 Cudworth. 
 
 1. s. ' d. 
 
 1. s. 
 
 d. 
 
 h 
 
 s. d. 
 
 3 12 o 
 
 14 14 8 
 9 10 
 
 7 9 10 
 
 6 5 
 
 7 8 
 6 12 
 6 15 
 
 3i 
 6 
 
 70 
 
 135 
 
 66 
 
 13 
 
 6 5i 
 15 I 
 19 6i 
 
 40 14 
 ai 9 
 
 5 '1 
 29 16 
 
 ?! 
 
 2 
 89 
 
 8 1; 
 12 4 
 
 96 
 
 62 I-l 
 
 (13.) 
 
 Anthony How, Efq. 
 
 1. s. d. 
 
 122 8 o 
 
 a82 19 6 
 
 133 17 6 
 
 (14.) 
 
 (15.) 
 
 Dr. 539 
 
 5 
 
 
 
 *59 
 
 13 
 
 10 
 
 I 
 
 9 
 
 
 
 3 
 
 Cr. 539 
 
 5 
 
 
 
 Geo. Germaine, Efq. William Weft, Efq. 
 
 3 
 
 7i 
 
 7i 
 
 1. 
 
 s. 
 
 d. 
 
 71 
 
 4 
 
 9i 
 
 66 
 
 II 
 
 4i 
 
 15 
 
 2 
 
 4i 
 
 41 
 
 13 
 
 74: 
 
 88 13 5 
 56 II s\ 
 
 339 
 
 1. 
 
 s. 
 
 7 
 
 17 
 
 4 
 
 9 
 
 9 
 
 19 
 
 36 
 
 z 
 
 12 
 
 9 
 
 5 
 
 I 
 
 76 
 
 (16.) 
 
 Theodore King, Efq. 
 
 I. 
 
 s. 
 
 d. 
 
 12 
 
 18 
 
 3l 
 
 4 
 
 8 
 
 2 
 
 99 
 
 14 
 
 2 
 
 rl3X 
 
 17 
 
 3 
 
 7 
 
 5 
 
 
 
 126 
 
 13 
 
 4 
 
 {17.) 
 
 (18.) 
 
 Mr. 
 
 Torin. 
 
 Valentine Fawkes, 1 
 
 1. 
 
 s. d. 
 
 J. s. d. 
 
 29 
 
 15 10 
 
 55 17 7f 
 19S 3 H 
 
 37 
 
 4 H 
 
 17 
 
 3 iot 
 
 31 7 H 
 
 II 
 
 9 » 
 
 39 19 4 
 
 9 
 
 6 4| 
 
 '6 3 4i 
 
 8 
 
 I 6 
 
 34 5 31: 
 
 382 16 ai 
 
 J13 
 
 375 16 5I
 
 •art I. 
 
 BiLLi OF Parcels, &c, 
 
 iS9 
 
 (19.) 
 
 
 Mr. 
 
 Carpenter. 
 
 1. 
 
 s. 
 
 d. 
 
 i8z 
 
 10 
 
 7^ 
 
 66 
 
 s 
 
 2 
 
 63 
 
 8 
 
 3 
 
 14 
 
 9 
 
 10 
 
 98 
 
 
 
 
 
 424 
 
 13 
 
 lOl 
 
 (21.) 
 
 Mr. Cole. 
 
 Neat 
 
 we 
 
 ght. 
 
 cwt. 
 
 qr. 
 
 lb. 
 
 15 
 
 I 
 
 7 
 
 20 
 
 I 
 
 20 
 
 18 
 
 Z 
 
 9 
 
 XI 
 
 
 
 n 
 
 4 
 
 3 
 
 26 
 
 7 
 
 
 
 r 
 
 Value. 
 
 
 1. s. 
 
 d 
 
 53 " 
 85 16 
 
 i( 
 
 43 13 
 
 
 27 19 
 26 3 
 
 37 6 
 
 
 ( 
 
 20.) 
 
 
 
 Mr. 
 
 WiUet. 
 
 
 1. 
 
 s. 
 
 d. 
 
 
 158 
 
 14 
 
 7| 
 
 
 39 
 
 5 
 
 
 
 
 55 
 
 12 
 
 i°i 
 
 
 9? 
 
 II 
 
 4 
 
 
 148 
 
 10 
 
 4 
 
 
 214 
 
 
 
 9 
 
 
 711 
 
 14 
 
 i^T 
 
 
 
 
 (21 
 
 •) 
 
 
 Mr. George Lane. 
 
 
 cwt. qr. 
 
 lb. 
 
 
 
 29 1 
 
 27 grofs 
 
 
 
 2 
 
 9 tare 
 
 
 27 I 
 
 18 neat 
 
 
 
 and the value is 
 
 
 I 
 
 351- US. 
 
 4id. 
 
 274 
 
 (230 
 
 Meflrs. Langton and Co 
 Neat weight. 
 
 cwt. qr. lb. 
 1520 
 
 *3 
 
 18 
 
 8 
 
 8 
 
 I 
 
 2 oX* 
 
 I 25 
 
 3 7 
 I iX 
 
 1 3I 
 
 Value. 
 
 1. s. d. 
 
 75 u 3 
 
 123 7 111 
 
 28 o 4|: 
 
 27 I 2| 
 
 23 16 6? 
 
 8 17 4^ 
 
 286 14 71 
 
 Mr. Henry Chapman. 
 
 Neat weight. Value, 
 cwt. qr. lb- 1. s. d. 
 
 9 3 i3i' 
 
 II o 13 
 
 II 22 ^672 i6 o 
 
 8 3 21 
 
 9 i ^S{ 
 
 * In working this (and every one of the following Examples which do not 
 divide even in lbs.) I have taken the neareft ^ or I lb. and rejefted the reft, 
 frQlumingtbat fufficiently accurate for Bufinefs.— Thus 16 lb. \-j\ 27 cwt. 
 I qr. 191b. now the true tare would be 3 cwt. 3 qr. 18-7 lb. but the 5 
 that is over I multiply by 4, and then divide that produft (20) by 7, and 
 the quotient is 2 or i, lo that I make it 3 cwt. 3 qr. 18^ lb. Had the 
 quotient been 1, I (hould have called it i, or 3 iz !> Sec. 
 
 P Z
 
 tSy Bills of Parcel*^ 3tc» 
 
 (J5-) {2.6,) 
 
 FrancU Clarice, Efij^ 
 
 Neat weight, 
 ewt. qr. lb. 
 3 1 i^Jl Value 
 
 7 a 4t 
 
 7 2 26i > 70!. 
 6 X 19^ 
 
 8 I iSL 
 
 Mr. Amuitie. 
 
 
 Neat weight. Value 
 cwt. qr. lb. k s. 
 
 d. 
 
 II 2 14 58 19 
 
 X 
 
 1204 11 
 
 H 
 
 33 
 
 (27.) (28.) 
 
 J Granville -King, Efq. Mr. Jolm Grant. 
 
 Neat weight. Value. 
 
 cwt. qr. lb. L s. i, 
 o 3 iifl 
 o 3 "f I 
 o 2 2i|: S 23 5 iij 
 
 Neat weight. 
 
 Value. 
 
 cwt. qr. lb. 
 
 1. s. d. 
 
 18 18 — 
 
 91 2 li 
 
 4 2 7i 
 
 73 9 S 
 
 35 5t 
 
 15 4 
 
 71 13 4i 
 
 19 3 12 
 
 41 S 7i 
 45 7 7{ 
 
 a 3 2i| 
 
 
 357 iS 3i 
 
 o 3 22i 
 
 2 2l|: S 
 
 1 o 23 
 
 3 2:iJ 
 
 22^ 
 
 (29.) (30.) (31._) 
 
 Wiimer WiUct, Efq. 
 
 
 Mr. Fenton. 
 
 Sir James Lawfon 
 
 Neat weight. Value 
 
 . 
 
 1. s. 
 
 d. 
 
 I. s. d. 
 
 cwt. qr. lb. ]. s. 
 
 d. 
 
 2 10 
 
 3^ 
 
 10 19 jI 
 
 9 2 i3n 
 
 
 2 7 
 
 
 II 1 2^ 
 
 2 7| 
 
 9 ' 5i| 
 
 
 3 3 
 
 8 oj 
 8 2231J 
 
 54 I 17 
 
 i| 
 
 5 I 
 
 15 17 
 
 I 16 6i 
 
 I J3 5;: 
 5 5: ■ 
 
 
 24 13 
 
 
 53 n 
 
 '1 
 
 27 16 4|. 
 
 
 

 
 Part L 
 
 Bills of Parcels, Sec, 
 
 i6i 
 
 (3^0 
 
 (33.). 
 
 Mr. Adams. 
 
 Mr. L. ThirKvall 
 
 1. s. d. 
 
 1. s. d. 
 
 62 3 z} 
 
 61 14 31* 
 60 8i 
 
 9 9;; 
 
 016 2 ' 
 
 5 9l 
 4 16 9i 
 
 r83 18 
 
 (34.) 
 Lumley Tutt, Efq. 
 
 { 
 
 I. 
 
 s. 
 
 d. 
 
 
 
 19 
 
 4i 
 
 3 
 
 14 
 
 8 
 
 
 
 4 
 
 3l 
 
 
 
 19 
 
 7i 
 
 17 
 
 19 
 
 10 
 
 23 17 
 
 * In this ftating, for 9d. per dozen, rfj</ 9s. 
 
 (35- 
 
 (36.) 
 
 Sir Leonard Hurt. 
 
 Mr. 
 
 :fham. 
 
 Contents. 
 
 7 fqrs. 40 ft. I In. 
 214 yds. 3 ft. 5 in. 
 
 Values. 
 1. s. d. 
 
 53 9 5 
 72 15 ii| 
 10 4 lOi 
 
 136 10 3^; 
 
 Contents. Values. 
 
 1. s. d. 
 
 10 r. 96 f. 2 in. 10 p. — 60 J I 4^ 
 
 15 rods, 227I ft. — 103 14 Sl 
 
 3367 ft. 1 in. 6' — 49 2 oi 
 
 862 fqr. 31 ft. 3 in. — 1810 17 li 
 
 2024 5 o 
 
 (37.) 
 John Carttar, Efq. 
 
 Contents. 
 
 204ft. Sin. 3pts. 
 399 ft- 
 
 Nofe, 6 fcore 
 deals make a 
 hundred 
 
 Values 
 1. s. d. 
 
 — 1 18 
 
 — 29 
 20 14 
 27 19 
 64 17 
 
 (38.) 
 
 Meflrs. Mount and Sc 
 
 Contents. 
 
 Value3» 
 1. s. d. 
 
 4I 2062 ft. 5 in. 1 pt. —31 o 7^ 
 
 [J7 li 
 
 20 yds. 6 ft. _ 4 6 6| 
 
 15 r. 255 f. I in. 8 p. — 86 13 8i* 
 
 1 1 yd. loin. 11' 2" 8"'— 2 o 5I- 
 
 8 fqrs. 91 ft. 9 in. — 57 10 4I 
 
 181 II 8i- 
 
 * For 5s. 9d. in this bill, read 5I. 9^
 
 i6x 
 
 Invoices, 
 
 &c. 
 
 (39.) 
 
 Mr. Conftant. 
 
 Contents. Value. 
 
 - 1. s. d. 
 7 fqrs. — 47 19 
 
 68 yds. 4 ft. 8 in 63 7 7 
 
 120 make a hun- C 11 1 l| 
 dred. / ^ ^4 ^ 
 10 I jI 
 
 (40-) 
 
 Mr. Craven. 
 
 Contents. Value. 
 
 1. s. d. 
 
 6 fqrs. 79 ft. 6 in. — 43 5 2| 
 
 2C yds. 8 ft. 5 in. — 17 0'- 
 
 395yds. 3f. 4in. 6' — 4 2 4! 
 
 48 14 71 
 
 141 3 'Sl 
 
 
 INVOICES, &c. 
 
 (4'-) 
 
 Invoice from Dubliv. 
 
 1. s. 
 
 Value of the butter - ^3^ 3 
 Ditto of the pork - 6713 
 Sum of the neceffary expenccs, as 7 
 
 fpecified in the Invoice J 34 ^ ' 
 CommiiTion (on 43 81. 8s. i|d.) \ 
 
 at 2i per cent. ] '^ ^9 
 
 d. 
 
 4 
 3 
 
 449 7 3t ^un^- 
 
 If 112I. : lool. : : 449I. 7s. 3|d. : 401I. 4s. ^{A, fter- 
 ling. Anfwer, 
 
 (42.) 
 Invoice from Jargiaica. 
 
 The neat weight of the tobacco is 7 « j 
 
 82791b. and its value is j^^ ' -^ 
 
 Sum of the neceffary expaaces, as 7 ^ 
 
 fpecified in the Invoice j ^ 
 
 Commiffion (rn324l. 2S. 2|d.) at / /q j. ' 
 
 r I I O lO;^ 
 
 3i per cent, ' 
 
 Sum 
 
 S3S 9 i; 
 
 If 125I. : lOoI. : : 335I. 93, ijd. : 268I. 7s. jjd. Her* 
 ling. Anfwer,
 
 Part T. 
 
 I N T O I 
 
 E 8, &C. 
 
 F65 
 
 (43.) 
 Invoice from Amjierdam, 
 
 g. ft. p. 
 Value of the Holland - 587 3 10 
 
 Ditto of the Cambric - 412 13 2 
 
 Ditto of the Ghentifli - 108 13 12 
 
 Sum of the necelfary expences, as ' 
 
 fpecified in the Invoice 
 Commiffion (on 1178 g. 15ft. 7 2^ g 5 
 
 i- 
 
 5 o 
 
 29 9 
 
 8 p.) at at per cent. 
 
 Sum 1208 4 14 
 If34s. 6d. : il. :: 1 208 g. 4ft. 14 p. : 11 61. 14s. 9 Jd. 
 
 fterlinor, Anfwer 
 
 (4+.) 
 Invoice from Bourdeaux, 
 
 Value of the claret 
 
 lb. 
 
 liv. s. d. 
 75 o o 
 
 Grofs weight of 13 cafks of prunes 9768 
 Ditto of 1 2 calks 1061 1 
 
 Grofs weight of the whole 
 Tare zz 79ilb. x 25 z= 
 
 Neat weight 
 
 20379 
 
 I9b7 
 839 ^^ 
 
 1^ J> 529 JO 
 
 If loolb. : 2I. 17s. 7d. : : i839i^Ib. : 
 ^29 liv. ics. 5d. the value of the prunes. 
 
 1. s. d. 
 
 Cuflom, &c. of the wine 10 o o 
 
 Ditto of the prunes ^ 118 15- o 
 
 , Sledage, &c. - o 15; o )»I5'8 
 
 Ditto for the prunes - 1 1 ^ o 
 To the Ship Broker - 500* 
 
 Poor's box, &c. - 12 5 7* 
 
 Commiflion (on 7 62 liv. us.) at 2| per cent. 20 19 4 
 
 Sum 7^3 10 4 
 
 ^ If 3 liv. : (;4td. : : 783 liv. ics. 4d. ; 59I. 6s. i ^d. fter- 
 ling. Anfwer. 
 
 * The reader will obferve that the tonnage is taken on the whole grofs 
 weight, and that I have made ufc of the Englijh lb.
 
 i64 Invoices, Sec, 
 
 No. 45, 46, 47, 48, and 49, being drafts, &rc. require no 
 anfwer. 
 
 (50.) 
 
 If il. : 34s. 4d. : : 571I. 18s. : 981I. 15s. 2|d. Flemifh. 
 Anfwer. 
 
 (5>-) 
 
 If 35s. 4d. : il. : : 7494 g. 14ft. : 707I. os. iijd. fler- 
 ling. Anfwer. 
 
 " (52.) 
 If I cr. : 4s. 3d. : : 5000 cr. : 1062I, los. fterling. Anf, 
 
 (53.) 
 If I p. : 54id. : : 1576 : 357I. 17s. Sd. 
 
 (54-) 
 If54^d. : r due. :: 174.9I. iSs. : 7741 i|-J due. Anf. 
 
 Th^ reji require ?i$/olution.
 
 K 
 
 E 
 
 TO t. H B 
 
 COMPLETE PRACTICAL 
 
 A R I T H M E T I CI- A N. 
 
 - < 
 
 PART 
 
 II. 
 
 ALLIGATION. 
 
 
 
 ( 
 
 I.) 
 
 
 rf>:s 
 
 Example is nuorked. 
 
 
 (2.) 
 
 
 (3.) 
 
 qrs.- s. 
 
 
 
 lb. d. 
 
 16 X 5S 
 
 = 928 
 
 
 50 X iri- = n5 
 
 30 X S3 
 
 =: 15^90 
 
 
 4-0 X 14 =r 560 
 
 21 X 39 
 
 = 819 
 
 
 27 X 30 1= 810 
 
 I J X 34. 
 
 =2 510 
 
 
 87 X ^6 1=3132 
 
 82 
 
 )3847{4-6|i«. 
 
 204. }5:o77(24i,?^d. 
 
 "— 
 
 
 
 
 = 2I. 6s. 
 
 ioJd..|-^ 
 
 Anf. 
 
 = 2S. oJd.-|*. Aafwer.
 
 66 Alligation. 
 
 (4.) 
 qr. 1. 
 
 2-5X2 = 5- 
 4*5 X 1-2 = 5-4 
 5* X -8 = 4- 
 
 i2 ) 14*4 ( I'zl. = ll. 4s. 
 
 Sd. 
 
 Car. 
 19. 
 
 Ts-T] 
 
 Ti&/V Example is nvorhdm 
 
 (6.) 
 
 r= 6 ib. at lod. "J 
 rr 2 — at 6 i 
 
 — a— at 4 3 
 
 (7.) 
 
 5 I sofasT s r 
 
 2+i|3of2ol3l 
 I iofi8>.5^^ 
 
 I lof 17 I ^1 
 
 4 4ofi4J"L 
 
 24-4= 6 ib. 
 
 2 
 2 
 
 Anfwer* 
 
 Anfwert 
 
 Other anfvvers may be found by Unking the fimples drfferently, 
 
 (8.) 
 
 Firft, 
 
 4S — 
 36— 
 24— I 
 
 
 Anfwer. 
 
 8 
 
 8 or 2 
 
 4 
 
 4—1 
 
 4 
 
 16+4 
 28 
 
 4— I 
 20—5 
 28 — 7 
 
 Second, 
 
 4 
 28+4 
 16 
 
 Anfwer. 
 4 or I 
 8 — 2 
 
 4— I 
 32-8 
 16 — 4 
 
 Third, 
 
 
 
 
 Anfwer. 
 
 
 48 
 
 8 
 
 8 or 2 1 
 
 
 36—1 
 
 4 
 
 4—1 
 
 ao. 
 
 24-T- 
 
 8 
 
 8 — 2 
 
 
 16—! 
 
 16 
 
 16-4 
 
 
 12 
 
 28+4 
 
 32-8 
 
 or wheat 
 Of rye 
 Of barley 
 Of peas 
 Of oats 
 
 Ofvrheat 
 Of rye 
 Of barley 
 Of peas 
 Ofoate 
 
 Of wheat 
 Of rye 
 Of barley 
 Of peas 
 Of oats
 
 Part II. 
 
 ao. 
 
 A L 
 
 A T I 
 
 Fourth, 
 
 48 
 
 
 8 
 8 
 4 
 4 
 28 + 16 
 
 8 or 2 
 8— 2 
 
 44—11 
 
 24—, 
 16— 1 
 
 12 
 
 
 Fifth, 
 
 !1— I 
 
 28 
 
 i I 16+4 
 
 Anfwer. 
 
 4 or I 
 
 8 — 2 
 
 8—2 
 
 28 — 7 
 
 2Q 5 
 
 Sixth, 
 
 
 Anfwer. 
 
 8 
 
 8 or 2 
 
 8+ 4 
 
 12— 3 
 
 4 
 
 4— I 
 
 164. 4 
 
 20— 5 
 
 28 + 16 
 
 44—11 
 
 N. 
 
 Of wheat 
 Of rye 
 Of barley 
 Of peas 
 Of oats 
 
 Of wheat 
 Of rye 
 Of barley 
 Of peas 
 Of oats 
 
 Of wheat 
 Of rye 
 Of barley 
 Of peas 
 Of oats 
 
 167 
 
 The following Anfwers, in addition to the above, may be found by 
 linking the prices difterentiy. 
 
 7th, 
 
 4 or 1 
 4— 1 
 8— 2 
 44—11 
 4— I 
 
 8 th, 
 
 12 or 3 
 
 4— I 
 
 8— 2 
 
 44 — 11 
 
 32- 8 
 
 9th, 
 
 8 or 2 
 
 4— 1 
 
 12—3 
 
 20— 5 
 
 32—8 
 
 loth, 
 12 or 3 
 
 4— I 
 
 4— I 
 
 48 — 12 
 
 28- 7 
 
 Of wheat 
 Of rye 
 
 Of barley 
 Of peas 
 Of oats 
 
 nth, 
 
 4 or I 
 12— 3 
 
 4— I 
 48 — 12 
 
 16 — 4 
 
 15th, 
 
 12 or 3 
 
 8— 2 
 
 8— 2 
 
 28—7 
 
 48 — 12 
 
 r2th, 
 
 4 or I 
 
 4— I 
 
 12—3 
 
 48 — 12 
 
 4— I 
 
 1 6th, 
 
 12 or 3 
 8— 2 
 12— 3 
 32—8 
 48 — 12 
 
 8 or 2 
 8— 2 
 
 12— 3 
 4— I 
 
 48—12 
 
 17th, 
 
 12 or 3 
 12— 3 
 8— 2 
 44—11 
 48 — 12 
 
 [4th, 
 
 8 or 2 
 12— 3 
 
 8^ 2 
 16—4 
 48 — 12 
 
 1 8 th, 
 
 8 or 2 
 12— 3 
 12— 3 
 
 20—5 
 
 i 48 — 12 
 
 Of wheat 
 Of rye 
 Of barley 
 Of peas 
 Of oats 
 
 Of wheat 
 Ot rye 
 Of barley 
 Of peas 
 Or eats 
 
 More anfwers may be found by nnklng, and thelc may Ic infri'ttdj 
 increafed by obfervirg the Note given to this Rule.
 
 x6S 
 
 A L i I 
 
 ^ T I • M. 
 
 (9-) 
 
 ^his Example is icorked. 
 
 8. 
 
 -J 
 
 4 — 
 
 diff. 
 4 + 1 =: 5- 
 
 (lO.) 
 
 difF. gal. dlff. 
 
 rr 4- As 5 : So 
 n 4- 5 •• 80 
 
 : 4 : 64 ga! 
 : 4 : 64 gal 
 
 •I 
 
 Anfwer, 
 
 12 
 
 8-, 
 6—1 
 
 s. 
 12 
 
 diff. 
 
 I. 
 
 difF. 
 I. 
 
 2|. 
 
 5-" 
 I. 
 
 (II.) 
 
 diff. lb. diff. 
 
 As 5 : 28 : : 2I : 
 
 5 : 28 : : r : 
 
 < : 28 : : 1 : 
 
 14 lb. 
 
 5|Jb. 
 
 5|lb. 
 
 Anfwer. 
 
 diff. 
 As 1 
 
 I : aS 
 I : 28 
 
 Or thus, 
 lb. diff. 
 28 
 
 5 
 
 2S)b."-> 
 
 70 lb. >An 
 140 lb. 3 
 
 fww. 
 
 Orher Ajifwers may be obtained by linking differently. 
 (12.) 
 
 40. 
 
 d. 
 
 48 T 
 36-i 
 
 24 — 
 18 
 
 diff. diff. b. 
 
 224-44-16^=42 As 8 : 24 
 
 S rz 8. 8 : 24 
 
 8 — 8. 8 : 24 
 8 — 8. 
 
 (13.) 
 7his Example is nvorkej. 
 
 diff. 
 
 42 : 126 b. 
 
 8 
 8 : 
 
 : 126 b. 7 
 
 : 24b4 
 : 24b.i 
 
 Anf. 
 
 (14.) 
 
 j6. 
 
 d. 
 
 24" 
 ao- 
 12- 
 
 diff. s. diff. lb. 
 8 As 24 
 
 4 i4 
 
 4 24 
 
 8 24 
 
 diff. 
 
 : 72 : t 8 : 24 lb. *> 
 
 : 72 : : 4 : 12 lb. / 
 
 : 72 : ■• 4 : 12 lb. J 
 
 : 72 J : 8 : 24 lb. J 
 
 Anfwer. 
 
 Sum of the 'if. 24 
 
 Other Anfwert «iiy be oltained.
 
 Part II. 
 
 Single Position. 
 
 16'} 
 
 $• 
 
 
 
 
 
 (15-) 
 
 
 
 s. dlf?. 
 
 s. diif. gal. 
 
 diff. 
 
 
 8 . 5 
 
 As 14 ; 16 
 
 •• 5,- 
 
 5 f g^l 
 
 
 7— 1 
 
 4 
 
 14 : 16 
 
 • 4 •• 
 
 4yg^^ 
 
 5- 
 
 1-1 
 
 2 
 
 14 : 16 
 
 : 2 : 
 
 27 gal 
 
 10 i 3 
 
 14 : 16 
 
 : 3 : 
 
 sH'^i 
 
 Sum of the dift'. 14- 
 
 
 
 
 "*■ 
 
 Or thus. 
 
 
 
 s. 
 
 
 diff. s. diff. 
 1:19 As28 : 
 
 16 : 
 
 iiff. 
 
 8 i 
 
 5+4 
 
 . 9 • 5 
 
 7 1 
 
 
 5+4 
 
 = 9 28 : 
 
 16 : 
 
 9 • 5 
 
 1 1 
 
 
 3+2 
 
 = 5 =^S : 
 
 16 :• 
 
 5 •• 2- 
 
 n 
 
 _ 
 
 34-2 
 
 = S 2S : 
 
 16 :. 
 
 5 = 2- 
 
 Sum of the diff. a 8 
 
 Other Anfwers may be found. 
 
 H 
 
 
 (i5.) . 
 
 
 
 d;ff. 
 
 s. diff. 
 
 
 
 6+5 = 11 
 
 As 31 : 64- 
 
 : II 
 
 22|t carats. 
 
 10 n 10 
 
 31 : 64: 
 
 : 10 
 
 20jx carats. 
 
 10 zz 10 
 
 
 
 
 Anf. 
 
 ^Anf. 
 
 Sum of the diff". 31 31 : 64 : : 10 : aof? carats. 
 
 SINGLE POSITION. 
 
 T^is Example is 'worhd* 
 
 (2.) _ 
 
 Sappofe 12 to be divided according (o thefe parts, asbe'iig 
 the leaft whole number divifible by 2, 3 and 4, without a 
 remainder. 
 
 i = jcsi !-"• 
 
 cUm 13 fhoald be 20.
 
 I70 
 
 Sing 
 
 L E 
 
 Position. 
 
 I. s. d. 
 
 13 : 20 : : 6 : 9 4 -j'^.J^ A'sfliare) 
 
 13 : 20 : : 4 : 6 3 ol--.^ B's i^Anfwi 
 
 13 ^ 20 :: 3 : 4 12 3^4? C's ] 
 
 (3-) 
 Suppofe they could finidi it in 210 days. 
 days. work. days. 
 
 7 • 
 
 I ; 
 
 : 210 
 
 30 by A 
 
 5 • 
 
 I : 
 
 : 210 
 
 : 42 by B 
 
 6 : 
 
 I : 
 
 : 210 
 
 35byC 
 
 Sum 
 
 07 
 
 Hence it appears that A, B, and C working together for 
 210 days (the time fuppofed) can perform 107 times the 
 work. Confequently 
 
 work. days. work. 
 If 107 : 210 :: I : i4|ydays. Anfwer. 
 
 (4.) 
 
 Suppofe the army confided of 30 men. 
 
 Then j of 30 = 6 
 
 \ =5 
 
 TO — 3 
 
 Sum 14 And 30 — 14 = 16, but 
 
 — fhould be 4000. ' Hence, 
 
 If i6m. ; 4000 m. :.: 30 m. : 7500 men. Anfwer. 
 
 (50 
 
 Suppofe the cidern would be emptied in 1287 hours, 
 h. cif. h. 
 
 9 ■ 
 
 I : 
 
 : 1287 : 
 
 143 by D 
 
 II : 
 
 I : 
 
 : 1287 : 
 
 117 by E 
 
 >3 : 
 
 I : 
 
 : 1287 : 
 
 '99 by F 
 
 Sum 35:9
 
 Part ir. 
 
 Double Position. 
 
 17: 
 
 Hence it appears that D, E, and F fet open for 1 287 hours, 
 would run out 3^9 times the contents of the ciftern. 
 Ifjjgc. : 1287 h. :: ic. : 3||| hours. Anfwer. 
 
 See the 9th of die Promifcuous Examples, at the end of Vulgar 
 Fradions. 
 
 (6.) 
 
 Suppofc the fum delivered was 5'cr, 
 
 Int€reft thereon for 3 yrs. at 4 per cent, 6 
 
 The amount is 56I. but ihould be 
 
 17 61. 8s. Hence, 
 If ^6U : 116U 8s. : : 50I. : 157I. los. Anfsvcr.' 
 
 DOUBLE POSITION. 
 
 (I 
 
 •) 
 
 T^is Example ii tiiarlcd^ 
 
 (2.) 
 
 FIfft, Second, 
 
 1. I 
 
 Suppofe. the firft horfe worth 30 
 Chaife, furninare, &c. 150 
 
 I. 
 
 Suppofe the firft horfe worth 1 50 
 Chaife, &c. J 50 
 
 
 
 3)180 
 
 3 )3oo 
 
 Vali\? of the fccond horfe 60 
 Chaife, <«c» 150 
 
 Value of the fecond horfe 100 
 Chaife, &c. 150 
 
 2 ) 210 
 
 2)250 
 
 
 
 105 
 Should be 30 
 
 125 
 
 Should be 1 50 
 
 Error. — 75 
 
 Error. +25 
 
 CL2
 
 172 Double Position. 
 
 By Rule I. 
 
 itfertor, 
 ift fuppofit'on 30 Kvyl — 75 
 2d fup£of)tion 150 1^^ H-25 
 75 i.'S frrcr. 30 
 
 lJ^$o 750 
 
 ^5 + 75 m 100 ) J2CCO 
 
 izoi. value of the firft horfe, and 50I. 
 :^ value of the fecoci. 
 
 (5-) 
 
 First, Suppofe it was 8 o'clock in the morning, then it 
 v.as 4 hours after fun-rife, and 12 hours before fuu-fet; 
 ••; I + 4 of 12 = II, but fnould be 8, hence the error 
 is — 3. 
 
 Second, Suppofe it was 12 o'clock, then it was 8 hours 
 after fun- rife, and 8 hours before fun-fet ; •.* | -f i of S =? 1 - 
 but ihould be 1 2, hence the error is -f 2. 
 
 By Ruje I. 
 
 its error, 
 
 ift fuppofition 8 k>< — 3 
 
 2d fuppoiition 12 rS 4-2 
 
 3 //J- <f?Tjr. 8 
 
 36 16 
 
 3 -r 2 = 5 ) )» 
 
 iOj= loh. 24' 
 fo tliat it was 24 minutes pad 10 o'clock in the morning. 
 
 (4-) 
 First, Suppofe o* the vahie of a French crown, then 
 will 7 dol. be worth 4I. ics. lod. =: logod. and i dal. = 
 
 * It is often of advantage to make o and i the fuppofitions. l may be 
 made a conilant fuppcfition in ail queflionsj and in moit cales it i? pre- 
 ferable to any other nuaiber.
 
 Part 11. Double P o s i t i o k. 175 
 
 i55-5d. alfo 4 crowns -j- 3 dol. = 46-]-}d, Then ^S-jjd, — 
 42od. ( = il. I 5s.) zi — 47f the firft error. 
 
 Second, Suppofe id. the value of a French crown, then 
 will 7 del. be worth (1090 — 11) =: i079d. and i dol. = 
 it^4^d. alfo 4 crowns -f 3 dol. = 4d. -f 462^ 1= ^66}, 
 Then 466! —420 = —464 thefecond error. 
 
 By Rule II. 
 
 ^ ^ ^^^" = iii = 65d. thecorreftion. Then 65 -f 1 
 .477 --46f 5 
 
 r: 66d. = 5s. 6d. the value of a French crown, confequently 
 a dollar is worth 4s. 4d, 
 
 Otherwife, 
 1. s. d. d. cr. dol, 
 
 4 10 10 = 1090 =11 + 7 
 
 1 15- o = 420 = 4 + 3 
 
 3270 = 33 +21 
 2940 = 28 H- 21 
 
 530 = 5 + 
 
 ••• I or. = -^^ z= 66d, z= 5s. 6d. confeqoently i doU 
 zr 4s. 4d. as before. 
 
 ExpLAK ATioK. I multiply logod, and its equivalent nirmber of 
 crowns and dol. by 3, alfo the 4Zod. and its equivalent number of crowns 
 and dol. by 7, in order to cancel the dol. and fubt.a^ one produft from 
 tie other, fo that there remains 5 crowns rr 330J.— For if equal quan- 
 tit'es be multiplied by equal quantities, the prod^ifts will be equal 5 and 
 if equal quantities be fubtradled from equal quantities, the ren;lainder wiU 
 be equal. 
 
 •• 
 ' Remark, By multiplying the firft line by 4, and the fecond by 11, 
 and then tdking their "difference, the crowns would have been cancelled, 
 and we fliould have had 5 dol. ZZ- 26od, confequently i dol. i::! 3id, zz 
 
 First, Suppofe there were 6 children. 
 Now 6 X 2 z: rzd. hence he had i6d. 
 Alfo 6x3= iSci. and 18 — 16 = 2d. but fhould 
 be 10, hence the error is — 8. 
 
 Q.3
 
 174 Double Position* 
 
 Second, Suppofe there were 12 children. 
 Now 12 X 2 rz 24d. hence he had zSd. 
 Alfo 12x3=: 36d. and $6 — 28 = 8d. l)ut fhould 
 be I o, hence the error is — 2, 
 
 By Rule II. 
 Z2 — znz corredion. Then 12 + 2 ::= 14 
 
 8 — 26 
 crhildren. Anfwer. 
 
 (6.) 
 
 First, Suppofe^ ~ 3000, irs Jog. ::r 3*477i2i3, 
 Then 3000 x'137^'^ = ^05. 3000 + 6 x log.i37ar: zz'-^oiz^^^ 
 and 4''X 430^1^ — ^^S' 4 + ^ x log. 436S rz 22-4437556 
 
 Diff. •1425097 
 ihould be o. ••• The error is + •1425097. 
 
 Second, Suppofe^ ir 2900, its log. rr 3«46239So. 
 
 Then 2900 X 147^1^ ~ 1"S- -5^0 + 6 x log. 1472 = 22.469S448 
 ' Should be 22-4437556 
 
 Error — '0260392 
 
 3JtI7Il!lJri:£^i2!l4^2l££i! - .C0227S3 the correaion 
 
 •1425097 + •02006^2 
 
 By the Rule, 
 •0260S92 
 
 7^ ~ 
 
 Then 3*4623980 4- '0022783 :r: 3*4646763 the log. of the value of 
 », the neareft number anfwering to which is 2915-3, 1 therefore take g 
 — 2915*3 for a fecond operation. 
 
 Then 291 5-3 x'T456^|^— 1^6' ^9i5'3 + ^ X log. 1456-7 
 
 — 22-4449038 
 Should be 22'4437556 
 
 Error — 'Ooii/.Si 
 
 By the Rule, 
 
 ^•AeASS-ll — 3462'Jq8o X '0011482 „. 
 
 ? ^^ — ^ ^ — ^ — — '0001052 correflion. 
 
 •0260892 — •001x482 
 
 Then 3-4646832 + '0001052 zz 3^4647884 the log. of the value 
 %Fgy the neareft number anfwe.Ug to which is 2916, the true value of 
 g, Anfwer.
 
 Part II. Arithmetical Progression. 17^ 
 
 Note. I have inferted this foigt'ion at full length in orJer to eluci- 
 date the note to the fecond rule 5 I would have inferted the following 
 likewife, but for want of room. 
 
 (8.) 
 Anfwer i -o^* 
 
 (10.) 
 Anfwer 6'098. 
 
 ARITHMETICAL PROGRESSION, 
 
 ^his Example is 'worked. 
 
 (7-) 
 
 Anfwer 3. 
 
 (9-) 
 
 Anfwer i '05. 
 
 3 + io3 X V = 777 fum, 
 (3-) 
 
 I + 12 X '5 = 78 Anfwer. 
 
 (4-) 
 
 The perfon mufl travel the ground twice over to fetch the 
 100th egg. ••• 
 
 2 + 200 X '1° = loioo yards =z 5nules 1300 yards, 
 
 M 
 
 This Example is ^worked, 
 
 (6.) 
 
 108 — 5 
 
 3= 8A common diff, Anfwer, 
 
 14 — I
 
 >S Arithmetical Progression, 
 
 2^3 ___ 2 
 
 -: = '} common difF. 
 
 8 — 1 ^ 
 
 2 lea. firil day's journey ,- 
 
 2 -h 3 =r 5 2 — 
 
 5-^3 = 8 3 
 
 8 -f 3 =z II 4 
 
 11+3 =14 5 — . 
 
 14 -4- 3 = 17 6 
 
 f] + 3 =20 7 
 
 20 + 3 = 23 8 
 
 ^"his Example is n^orhed,- 
 
 (9-) 
 
 i2 — mi zz 21, and 21 + i = 22 the number of terms, 
 
 5 
 
 (10.) 
 
 ?i_ir — ri 7, and 7 4- i = 8 the number of terms. 
 3 
 
 9^/V Example is ^worked, 
 
 (12.) 
 
 22 1 X 5 r= 105^, and loS — 10 j = 3 theleail term, 
 
 (T3.) 
 (^ I X 4 — 20, and 40 — 20 = 20 the lead term, 
 
 (14.) 
 This Example is ^werhed* 
 
 (15.) 
 
 ,22 — I X ^ , 
 
 1221 -f- 22 = 555:, ^^ = 52 a- 
 
 And ^5i — 52i = 3 ^^^ leaft term.
 
 Fart II. Geometrical Progression. 177 
 
 (16.) 
 
 12 1X4 , T 
 
 300 -T- 1 2 = 25, ^ = 22, and 25 — 22 z= 31. 
 
 Aufwer. 
 
 (^70 
 This Example is lAJorked, 
 
 (18.) 
 
 22 X 5 +3 =113, and 113 — 5 5s 108 the greateft term, 
 (19O 
 
 JOG X I 4- 2 =s 102, and 10a — i =: lois. sa 5I, is. 
 Anlwer, 
 
 GEOMETRICAL PROGRESSION. 
 
 ('■) 
 
 This Example is ^worked, 
 
 1.2.3.4.5.6.7, &c. indices 
 2 . 4 . 8 . 16 . 32 . 64 . 128, Sec, leading terms 
 -y -f. -7 4. ^ 2= 19 index to the igth term 
 12S X 128 X 3^ = 524288 the 19th term, Anfvvcr. 
 
 (3.) 
 
 1.2.3.4, 5;. 6. 7, &c. indices 
 3 . 9 . 27 . 81 , 243 . 729 . 2187, &c. leading terra* 
 7 + 7 -f 6 = 20 index to the 20th term 
 
 2187X2187X729 = 3486784401 the 20th terra. Anfwer. 
 
 (4-) 
 This Example is <workecl.
 
 178 Geometrical Progression. 
 
 0.1.2.3.4,5.6, &c. indices 
 
 7 , 14 . 28 . 56 . JIT . 224 . 448, &c. leading terms 
 
 6 -f 6^ zr 12 index to the 1 3th terra 
 
 448 y 448 
 
 = 28672 the 13th term. 
 
 7 ' / 
 
 12 + 6 = 18 index to the 19th term 
 
 28672 X 448 „ „ ., , 
 
 -^-!-- =z 1835C0S the 19th terra. 
 
 7 
 
 (6.) 
 
 0.1.2. 3.4. 5^ 6. 7, &c, md'ces 
 
 4 . 12 . 36 . ic8 . 324 . 97s . 2916 . 874S, &c. Uading terms 
 
 7+7 =: 14 index to the I5rh term 
 
 SL 111- n 19131876 the 15th term. 
 
 4 
 
 14 4- 5 zr 19 index to the aoth terra 
 
 '^'^'^^^ >< ^'' = 4649045868 the to* .e™. 
 
 4 
 7680 X 64 r: 491520 barley-corns in a bu(hel, 
 barley-corns d. barley-corns 
 
 451520 : 30 : : 4649045S68 t 1182I. 6s. 3d. AnfMrw^ 
 
 M 
 
 This Example ;i Vfcrked. 
 
 (8.) 
 
 0.1.2.3.4, &c. indices 
 4 . 12 . 36 . loS . 324, &c. leading terms 
 44-2— 6 index to the 7th term 
 
 324 X ^6 , , , 
 
 ^—^ zr 2916 the 7th term. 
 
 4 
 
 Then — ^^— — ^ zr i456,and J456 4--JniG— 4--? fumof the term:* 
 6.1.2.-;. 4 . 5 . 6 . 7 
 
 J . 2 . 4 . 8 . 16 . 32 . 64 . I2S . ^, ); .C 
 
 8 -f- 7 n 15 index to the J 6th tt. 
 256 X 128 — 32768 the 16th term. 
 J5 4. 8 -|- 8 :=: i^ jrdex to the 320 r. 
 
 3=768x256 X 256 n 2147433,6.3 -- ^- --'-^ 
 
 2147483648 — I -f- 2147483648 rr4-9V o^ ^^ 
 
 4473!;24l. 55. s^d. value of ths fir"
 
 Part 11. Geometrical Progression. 179 
 
 Again, 
 
 , 3x768 — I -f 32763 rn 65535 ftrthlngs, — CSl. 5s. ^IJ. value -of 
 t^e fecond iiorfe— T he dillcrcncc ot their values is 47738561. Anlwer. 
 
 (10.) 
 
 I . z . 3 . 4 . 5 . 6, &c. Jnuircs 
 
 10 . 100 . 1000 . icoco . loocoo . lOOOOOO, Sec. leading terrns 
 
 6 + 51= II index to the nth term 
 
 1000000 X icoooo zz looooooocooo the nth term. 
 
 loooooooocoo — lo 
 
 — iiiiiiliiio and luiiliiiio + 
 
 10 — I 
 
 loooooooocoo rZ IIIIIIIIIIIO wh-a. . nv jr. Vi' '.vh.;i.- = 
 
 7680 X 64 — 491520 wheal-C'-ra: ' : .1 bu'hci. 
 1^491520 : 4s. :: iiiiniiiiio : 452111. 4s. 6|:J. Anfwer. 
 
 (II.) 
 
 0.1.2.3.4.5, &c. iid'czs 
 1024 . 1536 . 2304 . 3456 . 5184 . 7776, ^c. leauing terms 
 
 5 -H 5 n: 10 index to the nth term. 
 
 7776x7776 
 
 . zz 59041;}. the eldeft fon's fortune. 
 
 1024 
 
 59049 — • 1024 
 
 f rr 29C12-4!. which incieafcd by 59049I. gives 
 
 SS061I. los. the nobleman died worth. ® 
 
 (.2.) 
 
 This Example is 'worked. 
 
 in-) 
 
 T I 
 
 , — X — = 
 
 2 2 
 
 ium required. 
 
 
 (14.) 
 
 1 i_* ' ' '. '.* ^ \, c 
 
 "3 9*" "9* 3 I" 9' 9*9""'» ""^ 
 
 required. 
 
 JO «— 9 n I and 10 X 10 — 100, then 100 -r i n iccinilcs. Anf.
 
 I So Variations and Combinations. 
 
 VARIATIONS. 
 
 (I.) 
 
 This Example is nvcrked^ 
 
 (^•) 
 1X2X3X4x5X6x7x8X9= 362880. Anf\V-r» 
 
 (3-) 
 
 1X2x3X4X5X6x7 = 5<^40 variations or days, r: 13 years, 
 -95 <l'»ys« Aniwer. ' 
 
 (4-) 
 JX-2X3X4X':X6X7X8 X 9 )k 10 X 11X12 =479^^0^600 changes, 
 4- 10 m 47900160 rr.inuicsj 365 d. 6 lirs. zz 525960 minutes. 
 
 Then 4-900160 -f- 525960 n 91 years, 26 days, 6 hrs. ringing with- 
 out intcrnailfion. Annvcr. 
 
 (5-; 
 
 This Example is ivorked^ 
 
 __^) 
 
 12 X li-"—! X 12 — 2 X 12— 3 X 12—4 X 12 — 5 X 12 — '- 
 IZ i2XiiXic;X9X8x7X6 zz 39916S0 changes. Aniwer. 
 
 (7-) 
 
 26 X 26 — I X 26 — 2 X 26 — 3 X 26 — 4 zr 26 X 25 X 24 X 
 3 X 22 n 7893600 words. Anfwer. 
 
 COMBINATIONS. 
 
 (I.) 
 
 This Example is nuorhd, 
 
 (2.) 
 1X2X3X4X5x6x7X8x9X10 = 3628800 dlvifor. 
 
 100 X 100 — I X ICO — :■ X 100 — 3 X 100 ~ 4 X 100 — 5 
 
 X ICO — 6 X 100 — 7 X ICO — 8 X ICO — 9 = 100 X 99 X 98 
 X 97 X 96 X 95 X 94 X 93 X .2 X 9i = 6281 56 50955529472000 
 ijjvidend, which divided by 36:>.88oo, the divilbr gives fbr the quotient 
 17 3 10309456440 farthings zz. 18031572350I. 9s. zd. Anfw-er.
 
 i^art IL SiMFLE iNThSEST BY DECIMALS. iSi 
 
 SIMPLE INTEREST by DECIMALS. 
 
 T.his Example is ivorked, 
 (2.) 
 
 '^35 X yiS X •<^5 + 23^- =1 279-06251. = 279I. IS. 3d. 
 .maunt. 
 
 (3-) 
 550 X 5 X '035 = 96-251. =z 96I. 5s. iritereft. 
 
 ■ (4.) 
 
 700*5 X 5-25 X 'oi -\- 700-5 =:: 810*828751. = 810I. 
 1 6s. 63d.-}. 
 
 ^?-) 
 
 715-75 X 7-5 X -04.25 -f 715-75 ~ 943*S953i25l. = 
 943I. 17 s. lo^d.-^. the amount. 
 
 (6.) 
 
 •002739726 X 240 = -65753424. 
 71 5-75 X-05X -65753424 1=23-5315066141. =231. 1C3, 
 7 Id. -246. AniVer. 
 
 (7-) 
 •002739726 X.6!^ =''17808219. 
 
 357-5 X •I78082I9 X -05 1= 5-183219146251. = 3I. -y. 
 
 7]d. -89. 
 
 (8.) 
 
 '002739726 X 120 -f 5 = 5»328767i2. 
 
 510 X 5-32876712 X ^05 -f 510 = 645-883561561. =: 
 645I. 17s. S*055d. . 
 
 (9-) 
 This Example is ivoih\^* 
 
 R
 
 i82 Simple Interest by Decimals. 
 
 (lO.) 
 This Ex auntie is ivorked* 
 
 (II.) 
 279I. IS. 3d. = 279 0625I. 
 
 j'75 X '05 + I = 1*1875' divifo:. 
 Then 279-0625 H- 1*1875 = 235I. Anfwer. 
 
 (12.) 
 
 96I. 5s. = 96*251. 5 X -035 rr '17^ divifor* 
 Then 96-25 -r- '175 ^ 550I. Anfwer. 
 
 (13-) ^ 
 
 Siol. 163. 6-|J.-| = 810-S2875I. 5-25 X -03 -f I == 
 i-i;;75 divifor. 
 Then 810-82875-7- 1*1575 = 700-51. =: 700I. los. Anf, 
 
 (M-) 
 
 943I. 17s. io3d.-i=: 943-8953x251. 7-5 x-o+25 + I == 
 1-31875 divifor. 
 
 Then 94-3'S953i25 "^ 1*31875 = I^^'IS^- = 7^5^- ^S^^ 
 Anfvver. 
 
 (I5-) 
 23I. 10s.7id.-fl n 23*5315067 dividend. 
 •002739726 X 240 X -o; = -032876712 divifor. 
 Then 23-5315067 -r -032876712 =: 7i5-75l- = 7'$^* 
 15s. Anfwer. 
 
 (16.) , 
 
 3I. 3s. 7|d.-|i- = 3'i832r96I. dividend. 
 •002739726 x6^ X -05 = -0089041095. 
 Then 3-1 b32 196 -^ •0089041095 ~ 357'5l. = 357l. losc 
 Anfwer. 
 
 (17.) 
 679I. 8:J. 4y-7V = 679*417808221. 
 •002739726 X 120 + 5 = 5-32876712 the time. 
 
 5*328-6712 X -05 + I =z 1*266438356. 
 Then 679-41780822 -7- 1*266438356 =: 536*4791781. 3 
 536I. 9s. 7-oo27d.
 
 Part II. Simple Interest by Decimals. i?3 
 
 (i8.) 
 This Example is ivorked, 
 
 279]. IS. 3d. = 279'o625l. and 279*0625 — 235 :« 
 44."o625l. dividend. 
 
 235 X 3'75 = 881*29 divifor. 
 
 Then 44-0625 -^ 881*25 = '05 the ratio, hence the rata 
 is 5 per cent. 
 
 (20.) 
 
 96*25 -r 5 5o"r^ 5 := '035 the lotlo, hence the rnte is jl- 
 per cent. 
 
 (21.) 
 
 810I. i6s. 6|d.-| = 810*828751. 
 Then 810*82875 — 700*5 ~ 1 10*32875 dividend. 
 
 110*32875 -^ 700*5 X 5-25 z=. '03, hence the rate is 3 
 per cent. 
 
 (22.) 
 
 943I. 17s. I o|d.-f = 943*89^3125!. from which fubtraft 
 7i5'75, there remains 228*1453125 dividend, 
 715*75 X 7*5 = 5368*125 divifor. 
 
 Then 228*1453125 -i- 5368*125 = '0425 the ratio, hence 
 the rate is 4^. 
 
 (23-) 
 
 •002739726 X 240 = '65753424, 23I. 10s. 7 Id. z= 
 
 23-53125- 
 
 •65753424 X 715*75 = 470-63013228 divifor. 
 
 23*93125 -^- 470-6301-3228 HI *o49999, <S:c. zz. ^05, hence 
 the j^ate is 5 per cent. 
 
 (24-) 
 •002739726 X 61; z=. •17808219, 3I. 3s. 7-4d,-|-|- rz 
 3-1832196!. 
 
 •178082196 X 357*5 = 63^664385C7 divifor. 
 
 Then 3*1832196-7- 63^664385c-; = •05, lie nee the rate 
 is 5 per cent, 
 
 R z
 
 (no 
 
 In this Example y for 5I. 10s. read 510I. 
 
 •002739726 X 120 + 5 =1: 5-32876712 yrs. 679k 8s. 
 4-|d.-y3- = 679-417808221. 
 
 679-41780822 ■ — 510 = 169-41780822 dividend. 
 5-32876712 X 510 = z-jiyG-jiz^iz divifor. 
 
 Then 169-41780822 -r- 27i7'67i23i2 = -062339332 the 
 r.ulo, hence the rate is 6*2339332!. = 61. 4s. 8*i4396d, 
 [cr cent. 
 
 (26.) 
 T/:/'s Example is 'worked* 
 
 (2;.) 
 
 279I. IS. 3d. =: 279-0625, and 279*0625 — 235 =: 
 44-0625 dividend. 
 
 ^3S ^ '^35 ~ 8*225 '^i^'i^or. 
 
 Then 44-0625 — 8-225 — 5'3S1H» ^^' = SUsY^^- 
 
 Note. The five Examples followlng^are the fame with the 20th, zift, 
 22d, 23d, and 24th piecedii^g, except in the divifors, which are found 
 by muluplying the principal by t'ue ratio inllead of the time 5 I daou^ht 
 h therefore unacceffary to infert their folu'.ions at large. 
 
 (28.) 
 
 
 (29.) 
 
 5 years. 
 
 
 5 i years. 
 
 (30.) 
 
 (31-) 
 
 (32.) 
 
 7 k years. 
 
 240 days. 
 
 6^ days. 
 
 i33'} 
 
 The dividend is 169-41780822. Sec Example i^th. 
 And 510 X -05 = 25-5 divifor. 
 
 Then 169-41780822 -7- 25-5 1= 6-64383, Sec, yrs. Anf.
 
 PartlT. Discount atSimpleIntereit, Decimals. iS^ 
 
 DISCOUNT AT SIMPLE INTEREST, 
 BY DECIMALS, 
 
 ('•) 
 
 Ti'ls Example :'s 'VJorkcd, 
 
 (2-) 
 49j'9 X -083' X ^ X -0375 = 7'74^437S^- i^tereft or 
 the debt for 5; months. 
 
 •083' X 5 X '0575 + ^ ^ 1*015625 amount of il. for 
 ditto. 
 
 As 1-015625 : il. : : 7'74-375 • 7-629231. r:: 7I. 12s, 
 7 "o 1 5d. the difcount-. Anfwer. 
 
 (3-) 
 
 1507I. 145. gd. = 1507*73751. dividend,- 
 
 •083' X 7 X -05 + I = 1*02916' divifor. 
 
 Then 1507*7375 -4- 1*02916' = 1465*00811. r: 1465], 
 OS. i|d, '773 the prefent value. 
 
 (4.) 
 7147-7 X '002739726 X 175 X -0375 =i 128-511729 -f 
 intereft of the debt for 175 days. 
 
 •002739726 X 175 X *0375 -f r n: 1*017979452 amount 
 of 1 1, for 175 days. 
 
 As 1*017979452 : il. :: 128*511729 : i26-24i9676I. ~ 
 126I. 4s. 1 00 7 2d, the difcount. 
 
 (5-) 
 
 1789I. 13s. 4d. ~ 1789*66' dividend, 
 
 •002739726 X 29 -f 3*75 ■=. 3*8294^2 years. 
 
 3-829452 X '043' + I := ^''■65943 c^ivifor. 
 
 1 hen 1789-666' -i- r 76594-3 ~ 1554-9692621, r: 1534?, 
 195. 4td. -492 the prefent worth, 
 
 R3
 
 86 Ec^ATioN OF Payments, &c. 
 
 EQUATION OF PAYMENTS at SIMPLE 
 INTEREST, BY DECIMALS. 
 
 Ofi MALCOLM' s PRINCIPLES, 
 (I.) 
 
 This Exa?nple is ^workej, 
 
 (2.) 
 
 Firft, 
 100 + 105 zr 205 furrv of rhe debts. 
 
 so 5 
 ICO X 2 X '05 — 10. Then Y i 11:21 '5 the firft number found. 
 
 Secondly, 
 
 105 X 2 -r 100 X '05 zi 42 the fecond number found. 
 
 Thirdly, 
 
 VzT^i — 42 rr V'42o-25 rr 20-5, and 21-5 — 20*5 — i year, 
 reckoning from the firll payment, •,• the equated time is 2 years. 
 
 (3-) 
 Firil, 
 From Michaelmas 17S8, to Michaelmas 1793, are 5 years. The fum 
 of the debts is 522I. 
 
 32c X 2 X '05 — 32, and {- 2*5 n iS*3i25 the tail number 
 
 3^ 
 fuund. 
 
 Secondly, 
 
 :02 X 5 -r 32-- X 05 ~ 63*125 the fecond number found. 
 Thirdly, 
 
 \/ i8-8iz7^' — 63M25 — V'290*785i5625 rz: i7'0524, and 
 ^•8125 — 17*0524 zTl I'76ci year, r= i year, 277 days. Hence tht 
 jta fhould be received the 3d of July, 1790. 
 
 COMPOUND INTEREST by DECIMALS, 
 
 This Example is ixorked, 
 
 1*05 X 1*05 X 1*05 ~ i'i57625 araoufit of il. for 3* years. 
 And I-J57625X275 — 318-3468751. — 318J. ^%% ir^-:l, the amount
 
 Part II. Compound Interest by Decimals. 187 
 
 (3-) 
 1 -04 X I '04 X I -04 X I -04 X I -04 X I '04 X I -04 = 
 1 '3 1 5-93 1 8, &c. this mukiplieJ by 'joo-]^, produces 
 922-139208851. =:_922l. 2S. gid. -64 the amount; from 
 which dcduifl the principal 70075, there remains 22 il. 7s. 
 9^d. •64intereft. 
 
 (4.) 
 ! '05 X I 'O^ X I '05 X I -05 X r '05 X I •05' X I '05 X I '05 X 
 1-05 rr 1-5513282, &c. See Ex. ift. Involution. 
 
 1-5513282 X 8co •— 800 = 441I. IS. 3-oi4d. intercd, 
 
 (5.) 
 T/j/s Example is njjorked, 
 
 (6.) 
 318I. 6s. iijd. = 318-346875, and 1-05 ' = ri;7625. 
 ••• As 1-157625: il. :: 318-346875 : 275I. Anfwer. 
 
 (7-) 
 
 819I. 15s. 6|d. '2504832 zr 819-77838592, and 1*04 X 
 1-04 X 1*04 X 1*04 = 1*16985856. 
 
 As 1-16985856 : il. : : 819-77838592 : 700-75!. z= 
 700I. 15s. 
 
 (8.) 
 1241]. IS. 3'0i7467875d. — i24i»o625727S2oi25\ 
 r^9 — 1.551328215978515625. 
 
 . As i'55i32S2i597S5i5625 : il. : : i24i«o625727S23i25 : 8oo]. 
 Aiifwcr. 
 
 (9-) 
 
 This Example is 'worked, 
 (10.) 
 
 31SI. 6s. ii-;d. = 318-346875. 
 
 318-346875 ^275 = 1-157625, the cube root of which 
 is I -05, hence the rate is 5 per cent. 
 
 (II.) 
 
 819!. 15s. 6|d. -2504832 =819-77838592, this divided 
 by 700-75, gives 1-16985856, the 4th root of which is 1-04; 
 or the fquare root is i -08 16, and the fquare root of i 'oS 1 6 =: 
 1-04, hence the rate is 4 per cent.
 
 i83 DiscouKT AT Compound Interest, 
 
 (12.) 
 
 1241!. IS. 3»oi7467S-'cd. iz: I24i«c625727828iz5, this 4- 800 
 gives I'55i3282i59785i56z5, the cth root of which is 1*05, or tVe 
 
 uberootis i'i57625, and \/ i«i57625 — i^oc, hence the rate is 5 
 ;.;r cent. 
 
 7'his Example is ^worked, 
 
 (H-) 
 The amount -f- principal gives i'iz^^6it^ {'vide Ex. 10.) 
 this divided by 1*05, and that quotient by i'C5, Sec. till 
 nothing reir.ains, the number of divificns will (hew the time, 
 
 3 years. 
 
 . ^'^'^ 
 The amount -h- principal z=. J'i6c)^y9>ii6 (fide Ex, 11.) 
 
 this divided continually by 1-04, will fliew the time, 
 
 4 years. 
 
 {16.) 
 
 Theamonnt -r principal = 1 '5 91 3282 15978^1 5625 ("j/dt 
 Ex. T 2.) this divided continually by I'O^, will ihew the time^ 
 9 years. 
 
 discounTat cOxMpound interest. 
 
 ^his YjXawfh is ^jjorked. 
 
 1*05 X vo^ X 1*05 X roj =: 1-21550625 the amount of 
 il. for 4 years. 
 
 Then 4C0 -H r'2r 550^25 = 329*081 -f - 3291. is. 
 7 id. '76 the prefent w'orti^ which deduced from 400L leaves 
 70I. 1 8s. 4£d. '2 difcount. 
 
 (3-) 
 
 i-C5\fi in I '340095640625, and 643I. 4s, lid. — 643*24583'. 
 Then 643'24583' -r- 1*340095640525 — 4S0I. (nearly) t'le prefcir* 
 
 worth.
 
 i'art II. Eqjjation of Payments at. Sec, iSg 
 
 M 
 
 Ratio — : i«o6 ) ioo.( 94- 3 3962.2 prefent worth for the ift 
 
 year 
 
 I o^'^n i'i236 ) 100. ( 88*999644 ditto for the 2d year 
 
 I'obj ^zz i'i9ioi6 ) loO'C 83*961928 ditto for the 3d year 
 
 J -06 4ri: 1*26247656 ) 100. ( 79*209366 ditto for the 4th year 
 i-o6^5:^ 1*3382255776) 100.(74*725817 ditto fertile 5th year 
 
 The fqm is the prefent wortli 4^1*236378 — 421I. 4s. 8*73Td, 
 Anfwer. 
 
 EQUATION OF PAYMENTS at COM^, 
 POUND INTEREST. 
 
 Th's Example is •worked, 
 
 (2.) 
 
 ^•051^ X 3^*^ "^^ 408*4101 amount of the ift payment 
 96* laft payment 
 
 504-4101 fumofthe amounts 
 
 320 + 96 rr 416I. fum of the debts 
 
 _. /3P-. 504-4101 — /opr. 416 „ ^ 
 
 Tnen '^ ^ ^ ^ ^ . — -0836905 -r '021 1893 n 
 
 kg. 1-05 
 
 3*90496, hence 5 — 3*90496 zzL 1*09504 years, the true equated time. 
 
 (3-) 
 
 1-05]- X 100 -zz. 110*25 amount of the firft payment 
 105 laft payment 
 
 215-25 fum of the amounts 
 
 [CO 4. 105 zr. 205I. fum of the debts 
 henca 3 — 1 — 2 years, the true equated time. 
 
 /o.T. 215-25 — /o^. 205 - - 
 
 Then .^ i — l i^ 2 — .0211893 -r •0211S93 =: i, 
 
 /.^. 1*05 ^^
 
 1^0 Annuities in Arrears 
 
 (+■) 
 
 J. 05^4 X ICO r: 121*550625 amount of the ift payment 
 
 ditto of the 2d 
 
 ■*"<-'5i* X 3^° ^^ 35^*"5 '^'"o of the 3d 
 
 fee laA paynieRt 
 
 iiS3'825625 fum of the amounts 
 
 ICC -f- 2CO -f 3CC + 5C0 zi iiccl. fum ef the debts. 
 
 „, /»r. ii83'82!:62C — /rff-. iioo 
 
 Trcn -i i 1 - t — •031S950 -7- -0211893 r:: 
 
 h^g. 1-05 
 
 2-50554, hence 5— i*Soi*4 — 3*49475 years, the true equated time. 
 
 ANNUITIES IN ARREARS at SIMPLE 
 INTEREST. 
 
 77'/; Example is nxarled* 
 
 142 + 3+4+5 + 6+7 + 8 = 36 
 
 One year's intereft of Sol. =4 
 
 Whole intereft due - - 1 44. 
 
 80 X 9 =^ '20 
 
 The amount /. ^64 
 
 i + 2 + 3 + 4 + r+^ = 2* 
 
 One year's intereft ot 560I. zr 22*4 
 
 Whole Intereft - - 470'4 
 
 560 X 7 = 3920 
 
 £' 4390'4 = 4390' 
 8s. the a mo u at..
 
 Part II. AT Simple Interest. 191 
 
 (4.) 
 1+2 + 34-^ -i- 5+6 + 7 = 28 
 One year's intereil of 173I. at 2^ percent. = 3*937^ 
 
 Whole intereft 1 10-25 
 
 17^ X 8 = 1400* 
 
 15101. 5s. the amount, 
 
 is-) 
 
 1 + 2 + 3+4, <l-c. to 19 = 190 
 One year's intereft of 1 7'5h at 1 1 per cent. = '2 1 87 ^ 
 
 Whole intereft 41-^625 
 
 17-5 X 20 = 3 JO 
 
 / , £• 391 -5625 ~ 
 
 391I. us. 3d* the amount. 
 
 (6.) 
 T^j's Example is <v:orkcch 
 
 (7-) 
 
 S64X 2 zr 1728 — 2a thedividcnd} 9x9 X '05 + 9x2 r= 22-05^:: 
 
 •;- + zr, and -05x9 zi '45 ^^ tr. Then 172S -7- zz'Of — -45 :::: 80I. 
 he annuity. 
 
 (8.) 
 
 4390*4X2 :z:27So-8 ~ za the dividend ; 7 X 7 X -04. -}- 7 X ^ r= 1 5-96 
 
 — ttr + ZA, and •04X 7 — '28 — tr. Then 3780-8 ~ i5-<^o 28 z: 
 
 560I. the annuity. 
 
 (9-) 
 
 1510-25X2 = 3020.5 = 2 J the dividend J 8x8x -0225 + b' X2 r± 
 17-4+ = ttr^\'^ty and '0225x8 — .18 — t>; Then 302C'5 -^ 
 
 17*44 — '18 IZ 175I. the half-yearly payment, hence the annuity 
 i. 350I. 
 
 (/o.) 
 
 39^*5625 X 2 = 733-125 ~ xa the dividend j 20 x 20 x '0125 -f- 
 20 x2 = 45 :r ffr -f. zr, and -0125x20 — -25 nrr. Then 783-125 
 ~5" 45 •— • **5 -^ I7*5l« the quarterly payment, hence the annuity is 70!^
 
 192 
 
 Annuities in Arrears 
 
 (II.) 
 
 This Exn^rple is ivorhed, 
 
 •864x2 2/j 2 — -05 2 — r 
 
 -43-=— J _- ^ I - 19-5- — — i 
 
 So X '05 r.r -05 X 
 
 Then v/43- -{- 19-5'.* — i9'5 "TL \/ 812-25 — i9'5 =9 years. 
 
 (13.) 
 4390*4 X 2 _ ^ , _ f;f . - — '^4 _ , _^ 2 — r 
 560 X -04 ~ ^^^ "" ^r' -04 X 2 •+ 5 — ^^ » 
 
 Then v/ 39^ + -4-5r "" --'■*-5 = "' >'"^'** 
 
 JC1C-25X2 61136 la 2 — '0225 395'5 - — ' 
 
 175 X 02-5 — Si ~" 7r' '0225 X 2 ~ 9 ~ ^^ 
 
 Then /^^-^36 ^ 595->" 395" 5 _ 467-S_ ^9^5 7^ _, 
 
 /'s/Sx 9. 9 9 9—9 
 
 8 payments, hence the time is 4 years. 
 
 (i;.) 
 39,-56.5 X 2^33S,^^. i^:£!is ^ L,-^^. 
 
 17.5 X 0125 i.r -0125 X 2 ZT 
 
 Then VX3580 + 79-5' ^ — 79'5 rz 20 payments, hence the time is 
 
 5 years. 
 
 (16.) 
 This Example is ^worked, 
 
 ('70 
 
 S64— bo X 9 = 144 = ^— «r j 9 — I X 00x9= 576c = f — 
 X tr. j Then 144X2 -r 720 r= -os, hence the rate is 5 per cent. 
 
 (18.) 
 
 439c«4— 500 X l — L-iO'Xzza^nt^ 7 — 1 X 5^° X 7 = 23520 
 
 — ; — I X f« 5 470*4x2 -r 23520 :z: '04, hence the rate is 4 percent.
 
 Part lU AT Simple Inter Es«r. tg$ 
 
 (I9-) 
 
 ISI0'25--»I7$X« = no-25 rStf — nf ; 8^i K 175x8 = 9800 
 
 *^ f — I X ffi'i iiO'2 5X^ -f 9*00 it 'oaajh theintercftof tool, for 
 I year, hence the rate is 4^ per cent. 
 
 (26.) 
 
 391*5625 — 17-5X20 tfr 41*5625 :=ta-^itti 20— t X iJ'SX 
 
 tn 6650 -- f— I X f«i 4i-562SKi -r 6650 ;2: •oii^l. the iatsreft 
 of looi. for ^ year^ hence the rate is 5 per centt 
 
 The present vtoKtU ot ANNUlTIiES m 
 ARREARS AT SIMPLE INTEREST* 
 
 This Exam/)/e is ivorkedi 
 
 M 
 
 *Thc amount of this annuity (vide Ex. id of the precedift|[ 
 
 Art*) is 864!* and 9X*o5 + i d i*45 ^^ amount of lU 
 for 9 years. 
 
 Then 864 -r- 1*45 = 5:95;*862o6897l. &c, = 595I. 175. 
 i|d, •58621 the preicnt value. 
 
 1+2 + 3 + 4+5 + 6 = 21 
 
 One year's intereft of 560I. = 28 
 
 Whole intereft 588 
 
 560 X 7 ^ 3920 
 
 The amount 4508 
 
 7 X'05 + I = 1*35 the amount of il. for 7 year* 
 
 Then 4508 •+ r35 sz 333r^'i9'^' = 3339'* 5^. 2$d. 
 ♦he prefent value* 
 
 S
 
 194- The PRESENT WORTH OF AMNUltllS IN ' - 
 
 (4-) 
 , ..rXhe amount^of .this annuity (vide Ex. 4th cfthe^ preceding 
 
 ^ty) is)i5io'25L,;and4X*045 + i = ri8, the aniSunt of 
 il. for 4 years. .. • .' i; : -.1 ,.. 
 
 Ihen I 910-25 -i- i-i8 rz 1279-8728813, &c. n 1279]. 
 17s. 5'492d. the prefent value. 
 
 The amoi'int of this annuity (vide Ex. 5th of the preceding 
 
 Art.) is 39i*5625l. and 5 X '05 + i = 1*25, the amount 
 of 1 1, for 9 years. 
 
 Then^ 391 -^625 —1*25 zz 5 13*251. rr 313I. 5s. tlie 
 prcfent'^'Qrthfc 
 
 ■•■ ■ ' (6.) 
 
 This Example, is <vJorkcd. 
 
 iv) 
 
 Flrft, 
 595I. 17s. 4d. 3-58&2I ^ 595'Sd2o6897l. tliis divided by 9, gives 
 
 9>C.ax-o5 + 2 =: 2-9 z= 2^r + 2; 9 + -o5 + 2— -05— 2*4:^ 
 
 (8.) 
 
 rr -f 2 — r. Then — ~ X 66'2o6896552' zi Sol. the annuity, 
 2-4 
 
 Firft, 
 3339^* 5^' i-f'^' ^^ 3339'*'59'^' '^'^ divided by 7 gives 477'o'37'l. 
 
 = -^i 7X2X-05 -f2zz2-7=:2/r4 2j 7 X'os+i — 05=1 » 3 
 
 n fr + 2 — r; Then — - X 477'0'37' ~ 560I. the annuity. 
 2-3 
 
 (9-) 
 
 1279I. 17s. 5'492d. — I279»37288i3l. this divided by S zr 
 
 i59«.984iioi625 rr -y 8 x^ X •^^225 + 2 — 2«36.rz a/r-f 2 ; 
 
 2.36 . 
 
 — X 
 
 2-1575 
 
 ^59*9^41101625-— 175I. the half-yearly payment, hence the annuity 
 
 i^ 350^- 
 
 SX'0225 + 2 — '0215 — 2*1575 n rr+ 2 — r ; Then ■■
 
 Fart II. Arrears, AT Simple Int»erb8'T. 19J 
 
 (io.) 
 
 ■p ' 
 
 3T3'25 -r 20 = 15-6625 — : — "j 20X 2 X -0125 +2 =z a'5 n 
 
 2;r 4. 2 ; aox*ci25 + a.— •<?i25=: 2*2375 ~''' + a — '•J 
 "a-k *-'",,■■" 
 
 Then ^ X 15*6625 — IT'S', the quarterly payment, hence 
 
 2-2375 
 
 th* Annuity is 70I; . • - 
 
 ^hii Example it nvorked, 
 
 (12.) 
 
 3339I. 5s. 2fd. = 3339-i'59'i' 
 3339-i'59'Xa . , ., 'ay 56o~i66'o62'96^ , __ 
 
 ■ : • — 230*5 10 ~ - - J ' 2 1 " ■"* 2 — 
 
 50oX'C5 rn 56ox*o5. 
 
 X3-537 03' =: =^ — f 5 
 
 Then V 23S*5'i8' -f iTsTFos^^— I3-537'C3'= 20-537'o3' 
 *— i3*537'o3' iz 7 years {exa^ly.) 
 
 {is-) 
 595I. 17s. 2d. 3-8621 — 595'862o6897l, 
 595-86206897x2 5, 2/. 
 
 80 — 29'793ic34485 ^ w — r/> 
 
 ^— r-^-^^i-^^-il-?— -I = 12-05172414=: i — li 
 
 i>o X -05 ^ • rn 
 
 Then V 297*931034485 -f 12-05172414V — 12*05172414 — 
 9' yeai-s. 
 
 (H-) 
 , I2791' ^7^* 5"492d« r: 1279*8728813!. 
 
 1179-8728813X2 £ /: ^Z* 
 
 — : zz 650-09416193 iz -^ ; 
 
 I75X-0225 ^ ^^ ^^ r« ' 
 
 175—28-70713982025 . r o " — '/' T 
 
 ^ — J zz 36-63o8i)5i2 — ^~'; 
 
 i75X'0225 . J J :) ^^ .> 
 
 Then V 650-09416193 + 3^^^88512] ^ — 36-63088512 — 8 
 payments, -hence the tinae is 4 yearB.
 
 1^ The FKBiKK'i^ WORTH or Anwuities 
 
 i7'5X*oiz5 "" -iiBys "" 49 "" ra * 
 
 ^7-5 — 3-9'5625 _ ^ _ ^3-475 _ 43 ^'it _ n-^r^ ^ ^^ ^ 
 lTSXO}zs ^""•21875"" 7 ~~ rn *' 
 
 Then / ^OS^^ . 43^-2.Y 43'-g _ S?!'^ ^^^ 43'-^ _ 
 
 N/49 yj 7""7 7"" 
 
 ~ 20 payments, hence the time is 5 years. 
 
 (.6.) 
 
 This Exavt^le is ivorked* 
 
 (I7-) 
 595!. 17^. 2d. 3'5S6i ~ 595'862o6897l. 
 
 80x9 — 595*86206897 — i24»i3793io3 m fit-^fi 
 
 595-86206897x2 + 20 — 80X9 = 551-72413754= */ + « — »r/ J 
 
 Then 124/13793103 ~ 551-72413794 X "I = '05, hence the rate is 
 5 per cent. 
 
 (18.) 
 
 33391. 5s. 2|d. r= 3339-2'59'l- 
 
 560x7— 3339-2'59' = 58o.7'4o' =«?—/>} 
 
 Ts39*2'59'X2+56o — 560x7 = 33iS'5'iS' zz zp^»^ rt j 
 
 58o'7'4o' H- 33i8.5'i8' x y zz ii6i'4'8i' -J- 23229*6'a9' n '05, 
 hence the rate is 5 per cent. 
 
 1279I. 17s. 5«492d. n: 1279.87288131. 
 175 x 8 — 1279-8728813 = 120-1271187 =r vt — />j 
 
 1279-8728813 X 2 -h 175 — 175x8 ir: 1334*74576-6 zn zf-^t n 
 
 — w^ ; 120-1271187 -^ 1334.7457626 X |- n -0225, the intereft of il. 
 for I year, hence tlie rate is 4^ per cent. 
 
 (20.) 
 
 i7'5 X20 — 3i3'25 r= 36-75 =: rr— / ; 
 
 313-25 X 2+ i7'5 — 17-5x20 — 254 — 2;) + «--wf J 
 
 36-75 + 294 X a\ = '0125, the intereft of il. for^yeaj, hence the 
 rate is 5 per cent.
 
 Part ir. Rev£Rsion, AT Simple Interest. 197 
 
 The' prf.sent worth of ANNUITIES int 
 REVERSION AT SIMPLE INTEREST. 
 
 ^lis Example is ^worked, 
 
 (2.) 
 
 Fir ft, 
 i-^2-f-3-f-4+ 5 + 6-I-7 — 2^j fum of the Teries. 
 
 20X '04x28 + 20 X^ n 182-41. amount of the reverfion. 
 
 Ani i82'4 -=8^ '04 + 1 rr; \i%*ii%' the piefent worth of the 
 legacy were it to commerce immediately. 
 
 Secondly, 
 
 i3S«i'8' -7- 5X'04 + I — i^S'i's'^* — 115^' 3'o'3's« the value 
 of the legacy, 
 
 (5.)- 
 
 i + 2 + 3^4'ir:*lo, fum of the feries. 
 
 50 X "05 X 10 4- 50 X 5 rz 275l» amount of the reverfion. 
 
 And 275 -7- 5X'05 4- I r= 220I. the prefent worth of the leafe, 
 fuppofmg. it to commence immediately* 
 
 Secondly, 
 
 220 -~ 3X'05 4- I ~ i9i'3043478i6 zn 191I. 6s. i«C4 3478d. &c» 
 the value of the leafe. 
 
 (4.). 
 Firff, 
 T+2 + 3+4, &c. to iQ rr 190, fum of the feries. 
 
 ^icooX'OSX 190 -f- 1000x201:129500, amount of the reverfion. 
 
 29500 -7- 2CX'05 + I rz: 14'' 50 the prefent wor-th of the revcifioo, 
 fappofing it to oomnren«€ inunediately. , 
 
 Secondly, 
 T^IS^ -r 5X'05 + I — I iSool. the value of the leafe. 
 
 . (,'.) 
 
 ^his Example is 'VJorked^ 
 
 S.3 ...
 
 198 Annuities in Arreaas 
 
 (6.) 
 115I. 3-o'3's. =: iis-i's'i- = /> ^ = 8,7=: 5. 
 
 SX2X-04 + 2 X 5X-04+ I n 3'*68 — 2rr + 2 X Tr + i> 
 
 8x8x-o4 + 2x8 — 8x-o4=: 18-24 zi rrr + 2^— rr; 
 
 Then 3- 168 -r 18-24. X iiS'i's' = 20I. the legacy. 
 
 (7.) 
 191I. 6s. i'0435d. r: i9i'3043479 + = /> /=5>7'=:3. 
 
 5X2X'05 + 2 X 3X'05 + I zz 2-875 = a/r + 2 >^ Tr + i j 
 
 5X 5 X -05 + 2x5 — 5X'05=: II =ur + zt — rrj 
 
 2*87C 
 
 Then x 19»*S043479 + = 50I. the annuity. 
 
 (8.) 
 11800 = /, f~20, andTn5. 
 
 20X2X-05 X a X 5X05 + 1 — s — 2rr + 2 X Tr 4-15 
 aoX2oX'05 + 20x2 — aox*o5z= 59 r:/rr+ 2r— rrj 
 
 Then -^ X 11800 iz loool. the yearly rent. 
 59 
 
 ANNUITIES IN ARREARS at COM- 
 POUND INTEREST. 
 
 ('.) 
 
 This Example is fworked, 
 
 (2.) 
 l + i*05+'i-o5)*4. 1.05P + I05J4+ 1^5+ i:^t +1^7 + 
 
 t'^Srz '^^ "" ■4- ^05]' = ii»0265634, this X 80 produwa 
 -i 1.05^—1 
 
 t8zM25]. — 882I. 2S. 6d. amount. 
 
 (3.) 
 !• + 1*05 +1-05)* +■^05]'+ I^4+ 7^ J +ro5i6 = 
 
 V^^^ ^ +"i^o5J* = 8*1420076, this X 560 produces 
 4559-5242561, zz 4559l« xos* 5i^«
 
 Part II. AT CoMPoVND Interest. ig^ 
 
 (4-) 
 ^his Example is <worked, 
 
 M 
 
 Here «= 175, y=:i'0225, and /=3. 
 Log,r :=. log. 1*0225x8 = '0773064, the number an- 
 fwering to which is 1 '19483 := r ; 
 
 Log.r — I = log. '19483 = — i'2896j5:8 
 Log, « = %. 175 = + 2-2430380 
 
 + 1-5326938 
 Log, r — I =/c/. '0225 =z — 2-3521825 
 
 X<?g-. of the amount + 3*1805113 thenuro- 
 
 bcr anfwerirrg to which is 151 5'344l. = i $; 1 5I. 6s, 1 •o5d. 
 
 (6.) 
 Here «n 17*5, r= 1*0125, ^'^^ /=20. 
 
 Z'Tg'. r z= log, 1*0125x20 = '1079000, the number an- 
 fwcring to which is 1*282035 =: »• ; 
 
 Log, r — I = log, '282035 -~ ■"■ ''45^03030 
 Log, n^ log, 17-5 = -f 1-2430380 
 
 + 0-69^3410 
 Log,r — I zzLog,*oii^ = — 2-0969100 
 
 Log, of the amount -f 2-59643 1 o, the nun:« 
 
 ber anfw'ering to which is 394*849 = 394I, i6s, n|d. 
 
 (7-) 
 This Example is nnorhd, 
 
 (8.) 
 882I. 28. 6*04d. =882'i25:i6I. =^7; 
 1*05|9 — I = '551328216 = / — I ; 
 
 'iSijzTzla ^ ^82*12516 = Sol. the annuity.
 
 I '05 — I 
 
 •4-07 1 CC4 
 
 2C0 Annuities in Arrears 
 
 (9.) 
 45591. ics. 5d. 3*7444 = 4559*52473 + = ^; 
 I'^i^ — I zz •4071C04 = r — I ; 
 
 X 4559*52473 == 560!. the annuity. 
 
 (10.) 
 97v> ExamJ>/e is ^worked, 
 
 (II.) 
 1515I. 6s. I'id. -zz 1515-3451. = a-. 
 
 By the 5th Example, r — i = 1-02251^ — i rz '19483 
 .' Log. r— J =: log. -012^ —: — 2-3921825 
 log.a-Iog.isiS'HS =+ 3"i8o5iij 
 
 H- 1-5326940 
 ir?^. '19483 ZZ — 1-2896558 
 
 Sum -f 0-2430382, the num- 
 
 ber anfwering to which is 175I. the half-yearly payment, 
 hence the annuity is 350I. 
 
 (,2.) 
 
 394I. i6s. icid. — 394'8428I. =.a', 
 
 By the 6th Example, r — i = ioi25i'° — i=:'282035 
 Zcg..r — I ^zlog. -0125 =: — 2-0969100 
 Log. fl =:- /og. 354-8428 n + 2-5^)64242 
 
 4- 0-6933342 
 Log. '282035 = — 1-4503030 
 
 Sum -4 r24303i2,thenuni^ 
 
 ber anfuering to which is 17-5 {»earlj) the quarterly pay- 
 ment, heucc the annuity is 70I. 
 
 (13.) 
 ^his Example is worked.
 
 Part II. AT COMFOVND InTIREST, 201 
 
 (h) 
 
 882I. 2S. 6*04^. = 882'J2^l6l. =r <jr, 
 882'! 25 16 X*05 =44*106258 =5 r — 1 y.a\ 
 
 Then ^''°^'^^ + , = i-5f 132815 = 7^' hence 
 
 / may be found by repeated divifions (or rather by logarithms) 
 ;;= 9 years. 
 
 (15.) 
 4^591. losi 5d. 37444 = 4559'5H 73 + /=^' 
 4559-52475 X •05' = 227*9762^65 =r — I X a\ 
 227-9762365 _^ ^ _ 1.4071004=^051' hence may be 
 
 found as above, = 7 years. 
 
 (16.) 
 ?">&« Example is ^worked, 
 
 ill') 
 
 1515I. 6s. i-id. = 1515*3451. =5^; 
 
 i5i5*34-5 X 1-0225 — 1515-345 + 175 = 209'0952 = 
 
 ar — a -{- n; 
 
 Log. 209-0952 = 2-3203441 
 
 Log, nz=. log, i""!^ = 2-2430380 
 
 L'yg, 1-0225 == -0096633 ) 0-0773061 ( 8 pay- 
 ments, hence the time is 4 years. 
 
 (iS.) 
 394I. 165. lojd. ir: 394*8428 =: ^; 
 
 394- ^4^^ X 1-0125 — . 394-8428 + 17-5 = 22*43554 = 
 ar — a -\- n i 
 
 Log. 22-43554 = 1-3509366 
 
 Log, 17-5 ;= 1-2430380 
 
 Log, 1*0125 = -0053950 ) 0-1078986 (20 payments, 
 hence the time is 5 years.
 
 lot ThEPRTSEN-T WORTH OF AkNUITIES I" 
 ^h:i Examplt is ivoried. 
 
 . 882I. 2s. 6d. -0+ =: 882-12516 = c; 
 
 882'i2fi6xr Q 88'2-i2ci6 — So 
 
 : — — r^ Zi : now ov tnai 
 
 ^O .... ;•..! 8.Q. . '. • -' 
 
 and frror, as in Example 6th of Double Fofition, I find r ss 
 1 05, hence the rate h 5 per i:ent. 
 
 The present worth of ANNUITIES in AR- 
 REARS, AT COMPOUND INTEREST. 
 
 (^•). 
 
 7^/j Example is nvcrked^ 
 
 (2.) 
 
 The amount of 80I. for 9 years, at 5 per cent, is 882*1 25I. 
 
 (vide Example 2d of the preceding Article, and 1*051 9 = 
 1*551528216 + the amount of 1 1. for 9 years. 
 - Then 882-125 : il. •• i'55i3282i6 4- : 568*62564l. = 
 568I. I2S. 6-i54d, the prcfent worth. 
 
 (3.) 
 The amount of 560!. for 7 years, at 5 per cent, is 
 
 4J59-?242~561. (vide 3d Ex. of the foregoing Art.) and 1-05V 
 — ■ I -407 1004 the amount, of il. for 7 years. 
 
 Then 1-4071004 : il. :: 4559-524256 : 3240*368821. = 
 3240I. 7s. 44d. the prefent worth. 
 
 This Example is ivorked, 
 
 (5-). 
 
 Here « m 175, r n I '0225, and /rr S. 
 L:g. i*c2Z5x8«rr •0773064, this fubtradlcd from the%. of i rr o, 
 
 iji%es — I*9226q3'6 for the kg. cT — , the number anAvering to which 
 
 r 
 
 ■V36938G, hence, i — -— ::z: i — •8569386 — •163061-:. 
 r
 
 Part lI/'Ai^REA**, AT Compound -Interest. 20 j 
 
 Log. -1630613 ' =:: — i'2i23509 
 
 Log, n-z^ og. 175 - + 2-24303550 
 
 ,\ ,.\ i 
 
 + 1-4553289 
 Log. r — I •=: log. -022.$ ~ — 2-35-1^25 
 
 Leg. of the prcfcnt wortli -f" 3*1032064, the number 
 
 anfwcring to which is i268«254l. zz 1268I. f-oSs. Anfwer. 
 
 Here ^«?^ J7'»5, r— i-oizj, and f n 20. 
 L^l^. I'0I2 5X20 — •10,79000, this fubtratted from the Ys^. of t — o, 
 
 leaves — i •8921000 for the 7?^. of --, the number anrweriiig to which 
 
 r 
 
 h •7800097, hence i — i* — -7800097 — •2199903. 
 
 - r 
 
 r. . • . . , 
 
 lo^. •2199903 =—1-3424035 
 
 Log. n zzlog. IJ'S ZZ + 1-2430380 
 
 + o 5854415 
 . Lo^, r — I zz log. ^0125 zz — 2-09.09100 , 
 
 Lijj;. of the prefent wortli. 4-2-4835315, the number 
 
 anfwering to which is 307-986 3I. zz 307I. 19s. 8^-d. Anfwer. 
 
 (7-) 
 'This "Example is ivorked, 
 
 568!. 12s. 6-i77d. IT 568-62571. zi p\ 
 5681-6257x-t)5 =r28-43i2»5 zz r—i xp-y 
 I -05] 9 11: 1-551328216 = r J I — 1-7- i'55i3282ib ZZ 
 
 1 — -6446089 ir •3553911 ilr 1 } Then 28-431285-7- * 
 
 ' - r^ 
 
 'j'553911 — So^' the annuity. 
 
 2842!. 7s. g-oiSifd. =: 2842-38757 rr /> ; 
 
 *_a842«387'57X'05 = i42'ii93785 zz r — i x /> j 
 
 1 -05) 7^— ' i' '407 icro4 rr r j I — 1 -r 1-4071004 r: i — '7106813 
 
 = -2893187 = i'—_L} Then i42«ii937'S5 ^ •2893187 — 
 
 r 
 491 '2208561. 13 49 il. 4s. 5d. the annuity.
 
 to4 The TxisitfT wotra of AuMfftrtti tn 
 
 (ro.) 
 Thit Example it wjorked* 
 
 (...) 
 
 Utxtt nS, rz: I'oaaj, and 1268I. 5'o8s. 2^ I468»254J. ~/> j 
 
 By the 5th Example, l — — — — •1630613, 
 
 t 
 Log. r-^ t 1^ log. '02%$ r:— i*?S*>S25 
 
 Leg. f n log. 1168 '154. ir + 3 '1032064 
 
 + 1-455388-9 
 Log, »x63o6i3 rr — i*2i235o9 
 
 Diff* 4- 2-2430380, the number 
 
 •Tifvverlng to which is 175I. the half-yearly payment, hence the annuity 
 k 3501. 
 
 (12.) 
 Hercr^ao, rr:i*oi25, 307I. 19s. 9»6d. i= 3o7*99l»=^} 
 
 By the 6th Example, i — «-— -^z •2199903, 
 
 r 
 
 Leg. r— I :zz leg. '0125 zz — 2*0969100 
 
 Log. p zn log, 307-99 rr + 2-4885366 
 
 4- o 48 5446 6 
 Log, •S199903 :i — 1 •3424035 
 
 Sum + J '2430431, the number 
 
 •ofwcring to which is 17-5 {nearty) hence the annuity is 70). 
 
 (>3) 
 Ty?"// E-xample is nvorkeJ, 
 
 (14O 
 5fr81. 12s. 6.i77d. -jz 568«6257l. zr /* } 
 80+568-6257 — 568-6257 X I-C5 = 5i'5687i5 ~ w+j>— /r j 
 £0 -7- 5i'56S7i5 ir: i'55i3282 — zz. i-05\'» hence the vaVieof r may 
 be found by repeated divifions, or rather by logarithms, r: $ yeari.
 
 Part IL Arrears at Compound Interect. to^ 
 
 ('$■) 
 
 $842!. 7s. 9'oi83d. r: 2842'38757U -r /» j 
 
 $6o+2S42-38757— 2«42-38757Xi'05r: 4-i7-88o62i5i::b+/>— ^rj 
 
 560-^417-8806215 — 1*34000956 zz i-ojy hence the value off nKyr 
 be fouad as above, zH 6 year*. 
 
 (16.) 
 Ti^/j Example is nvorked, 
 
 (■7.) 
 
 1*681. 5-o8s. n 1268-254!. zz^i 
 
 175 + 1268-254— I268-254X i«oa25 — i46'4643— fi-f^ '— ^^ j 
 
 Lig* n ~ log. 175 — ?.-243038o 
 
 ii^. 146-4643 ~ zibsT^ii 
 
 Log. r ir 1.0225 rS •0096633 ) '0773064 ( ^paymcnu, 
 hence the time Is 4 years. 
 
 (»8.) 
 
 307I. 195. 9-6d. =^ 307-991- = /; 
 
 17.5 ^ 307-99 — 307-99 X 1-0125= 13-65 nr+^—^rr 
 
 Lo^. n -n log* 17 •$ zz Ji 4430380 
 
 Log. 13-65 — 1.13513x7 
 
 Log, Tzz log. 1-0125 — •005395 ) 0-1079053 (20 payments, 
 
 hence tlie time is 5 years. 
 
 Tiis Example ij n,jorifd, 
 
 (20.) 
 568I. I2S. 6-i77d. = 568-62571. =: /> ; 
 
 568-625 7 -i- bo 9 9+1 80 „ , «. 
 
 563.6257 ■' +'■-•'■ = j^TelTi' ^^ ''^"^"" 
 
 648-6a57 x '-9 — 568-6257 X r^"" :=. ''>o, by trial and error r miy be 
 found zz 1-05, hence the jate is 5 per cent*
 
 zo6 The irREsrNT worth of Akkwities in, 5tc. 
 
 The present worth of ANNUITIES in RE- 
 VERSION AT COMPOUND INTEREST. 
 
 Th's Exajfiple is fworked* 
 (2.) 
 
 5.515631*5, this X 50 producesi76*»Si562 5l. amount of die reverfion, 
 i'05^,5 rr i'i7628i6, amoount of il. for 5 years. 
 As 1-2760816 : 11. ! : 276-2815625 ! 2i6«47383, tlic prcfcnt 
 worth of the lekfe were it to Commence immediately. 
 
 As i-o^]^ ,= I»157625 : il. •• : 2i6'473S3 : 187I. (nearly) the 
 p.efent wortli of the reverflon. 
 
 (5-) 
 
 — V -. I -Of; ^9 — I . — ^. 
 i«4-i'05 + i-05i*-&c. to 19 — — -^ — + i«o5V9 =: 
 
 33'^)954'*» *^*^ X l^^co produces 33o65«9542l. the amount of the 
 reverfion. 
 
 As i"^05"*<==i 4*6531977 •• iJ. • •• 33°6S*9542 ' i2462'iio3l. the 
 ^refent ^v^rth of the leafe, fup^ofirtg it to commence immediately. 
 
 As 1^05^5 =: 1-2762816 : ij. : : i2462'Zio3 : 9764*46781. — 
 9764I. 9s. 4id. "088, the prefent worth «f the leafe, 
 
 (4.) 
 7 his Example is *workfJ, 
 
 M 
 
 187I. 0».©|<i.--4i76 = 127-003561.=:/} 1-05— I = -05 = r — i; 
 
 r WT^"* -7^8 —,.4774554, this X •05X187-00356 
 
 ~ 13.814471 =r— I xr X/i r — I =r I'osl — I=:'a76i8l6i 
 The» I3-SI447X -r '2761816 = 50I. the annuity.
 
 Part II. The rt'RCHASE OF Freehold IvjTATEt. ao? 
 
 (6.) 
 5764I. 9s. 4|cJ. .cSS ~ 9764-46781. —/ 5 1-05 — i=-o5— r — 1 ; 
 
 5764'4678 rr 1653-297669 ~r — ixr X/jr — in 105^^0 
 -* I rr i-653z.;77 ; Thea i653-z97669 ~- 1-6532977 rr loool. 
 (ricurly) the annuity. 
 
 The purchase of FREEHOIJD ESTATES, 
 OR. PERPETUAL ANNUIIIES. 
 
 (') 
 
 J. 06 — T = '06, the ratio lefs i. 
 Tfcen 250 -7- '06 =: 4.166-6' :::4i66l. 13s. 4d. theprefent 
 worth. 
 
 (3-) 
 voz — 1 =: '02, the ratio lefs r. 
 125 -7- '02 =z 6250I. theprefent worth. 
 
 Or thus. See the Note to this Example, 
 
 1*019804 — I ■=:: '019804 -f , t! e ratio lefs i. 
 
 Then 125 -^ -019804 = 6311*8^61 = 631 il. 17s. ijd. 
 
 the prefent worth ; exceeding the above anfwer by 61I. 17s. 
 
 i-|d. and of courre a more correct anfwer, for it is certainly 
 
 more advantageous to receive the rents half-yearly than yearly. 
 
 (4-) 
 1*01125' — I =:*oii25, the ratio lefs r. 
 Then 500 -^- -01125 = 44444-4'l. — 44444I. 8s. ic^d.-^ 
 the prefent worth; but the anfwer would be the fame if the 
 rents were received annually inftead of quarterly. Hence by 
 the fecond table 500 -^ •011065 —4.5187-52824- 45187L 
 los, 6|d. exceeding the above anfwer by 743I. is. 8d. 
 
 (5-) 
 
 ''Jhis Example is tvcrked, 
 T z
 
 »08 fivYINe IKD S.ELLINC Fl-EKHeLD EsTATft. 
 
 (6.) 
 4t661. 13s. 4d. rr 4i66«6'l. i*q6 — 1 r= '06, tlie ralio 
 Jefs J. Hence 4i66'6'x*o6 z:^ 250I. the annuity. 
 
 (7-) 
 1*045 — ' — '^iS> ^^^^ ^^*^o ^**s I. 
 1 760 X '045 =: 79'il. = 79I. 4s. 
 
 T/t// Example is loorked, 
 
 2^0^4166*6'— '06, and '06 + 1 z: i*o6 the ratio. Hence 
 the rate is 6 per cent. 
 
 (10.) 
 
 60 -f- 1200 = *05, and *C5 -f 1 z= 1*05 the ratio. Hence 
 the rate is 5 per cent. 
 
 The buyikg and selling FREEHOLD 
 estates, according to a number of 
 year's rent for the purchase-money. 
 
 0*) 
 
 Ihh Example is n/jorked, 
 I2C0 ~- 20 = 60I. 
 
 (3-) 
 7hiS Example is ^worked, 
 
 M 
 
 6iy -i- 2j = 2j years. 
 (5-) 
 
 This Exa?nple h iforked, 
 
 (6.) 
 45 X 2j =: 625I. Anfwer.
 
 Partll. The PUHCRASE Of Freehold EsTATts. 20Q 
 
 (7-) 
 This. Example is ivorked, 
 
 (8.) 
 
 22|-f- 1 4- 22| = 209 4- 200 = I '04^ the ratio. Hence 
 the rate is 4^ per cent. 
 
 (9-) 
 
 This Example is ivorked, 
 (10.) 
 
 I -f- *04 = 25 years, Anfwer. 
 
 The purchase of FREEHOLD ESTATES, or 
 PERPETUAL ANNUITIES, m reversion. 
 
 This Example is njoorked, 
 
 6o*5 -4- 'O^ = 1 210, value of the eftate if entered on im- 
 mediately. 
 
 As iToT''^ = 1*6288946 : il. : : 1210I. : 742 83504!. 
 r= 742I. 16s. 8;^d. the prefent worth of the reverfion, 
 
 (3-) 
 
 290 -r '04 = 72501. value of the eftate if entered on 
 immediately. 
 
 'As I -041* = IM698586 : il. : : 7250 : 6i97*33l. = 
 6x97!, 6s. 7d. the prefent worth of the reverfion. 
 
 (4.) 
 This Example is fworkedv
 
 210 The Purchase of Freehold Estates, 
 
 M 
 
 742I. 1 6s. 8:Jd. '8 = 742-8351. 
 
 i-o5Vo X 742*835; = 1*6288946 X 742*835 =: 1210 
 (nearly) the amount of the purchafe-money to the time when 
 the reverlion begins. 
 
 1 210 X '05 = 60*51. = 60I. 10s. the yearly rent. 
 
 (6.) 
 
 6197I. 6s. 5|d. = 6197*32292 -f £. 
 
 "i'04'|4 X 6197*32292 = n698586 X 6197*32292 = 
 
 7249*991 5 1 5> the amount of the purchafe-money to the time 
 
 when the reverfion begins. 
 
 7249*99/515 X '04 = 289*9996, &c, = 290L (nearly) 
 the yearly renl. 
 
 rhe End of th SOLUTIONS.
 
 A N 
 
 APPENDIX 
 
 TO THE 
 
 COMPLETE PRACTICAL 
 
 ARITHMETICIAN. 
 
 CONTAINING A 
 
 SYNOPSIS OF LOGARITHMICAL 
 ARITHMETIC, 
 
 Shewing their Nature, Conftrudlion, and Ufe, in the 
 plaineft Manner poflible. 
 
 TABLES OF COMPOUND INTEREST anb 
 ANNUITIES, 
 
 Extending from One to Forty Years, 
 
 A L S 0> 
 GZNZRAL AND UNIVERSAL DEMONSTRATIONS OF TKK PRINCIPAS, 
 
 RULES 
 
 IN THE 
 
 -COMPLETE PR ACTICALARITHiMETICI AN. 
 
 By THOMAS KEITH, 
 
 Teacher of the Mathematics, &c. 
 
 Mathejis mentis expurgatio, H I E R o C L . 
 
 LONDON: 
 
 miNTED FOR B, LAW, AVE-MARIA-LilNE* 
 
 MOCCXCi
 
 A N 
 
 APPENDIX, Sc. 
 
 $. I. The nature aad formation of Logarithms, 
 
 Definitioft,T OGARITHMS are a feries of numbers fo 
 
 J J contrived, that by thetn the work, of malti- 
 
 plication is performed by addition, and diviUon by fub- 
 traftlon. 
 
 If a feries of numbers in arithmetical progreffion be placed 
 jis indices, or exponents, over a feries of numbers in geome- 
 trical progreffion, the fum or diiference of any two of the 
 former willanfwer to, or Handover, theprodud, or quotient, 
 of any two of the latter. 
 
 Thus 0.1.2.3. 4* 5* ^» ^^' ^"'^5^' feries, or indices. 
 1.2.4.8. 16. 32. 64, &c. geom. feries. 
 a 4- 3 = 5» or 3 — I =: z. 
 
 4 X 8 — 32, or 8 -r 2 rz 4' 
 
 Now the arithmetical feries, or indices, have the fame 
 properties* as logarithms, and therefore fucb a feries may be 
 taken for their foundation. Conlidering logarithms as indices 
 to a feries of numbers in geometrical progreffion, it isevident 
 that there may be as many kinds of logarithms as there can be 
 taken geometrical feries; and it is equally evident, that if 
 any one fy. em of logarithms be once obtained, an infinite 
 number of others may eafily be derived from it. But, though 
 ail thefe different fyflems will be equally perfeft, if calculated 
 with equal accuracy, yet the moft convenient rank of con- 
 tinual proportionals is a decuple progreifion. 
 » Thus o, I, 2, 3, 4, 5, &c. indices, or logs, 
 
 I, 10, 100, 1000, 10000, 100000, nat. numbers. 
 
 But if the natural number be lefs than an unit, its index, 
 or log. will be negative, 
 
 ^c.— 4, --3, —2, —I, o, I, 2, 3, 
 
 &c. 'ooooi, •ooor, 'ooi, 'oi, i, 10, 100, 1000, 
 4, &c. indices, or lo^s. 
 10000, &c, natural numbers. 
 
 * Thefe properties of numbers, which arc the foundation of l.garltlrtst 
 are declared by Archimedci in his Arcnarius i but the invention ot'toga- 
 richms belongs to Lord Njpier.
 
 4 ASyncpsisof 
 
 From this feries-it appears, that the index to any logarithm 
 will always be an unit lefs than the number of figures in the 
 natural whole number to which it belongs; hence fiat index 
 will always determine how far the tirft figure of the ahfolute 
 number is diflant from the unit's place, of which 1 fhall 
 have occaiion to fpeak more particularly farther on: I now 
 proceed to the confrrufticn of logarithms. ♦ 
 
 ^. II. A Jhcrt and eafy RitJe for the Conjiritff ion cf Logarithmic 
 
 either common or loyferholiCy illujir cited by E.^amplei, 
 
 The Rule*. 
 
 1. Let the fum of the propofed number, and its next kfs, 
 be called n^ by which divide '8685889638, &-c. and referve 
 the quotient. 
 
 2. Divide the referved quotient by the fquare of zr, re- 
 ferring this quotient, which divide by the fquare of «r, as be- 
 fore. Proceed thus as long as divifion can he ma-^e, and write 
 the referved quotients under each other. 
 
 3. Divide thefe quotients refpeftively by the odd numbers'^ 
 I, 3, 5-, 7, 9, II, \^y 15-, 17, &c. viz. divide the firflf 
 referved quotient by i.thefecond by 5, the third by 5:, ^lC, 
 Let the quotients found by thefe divifions be written regu- 
 larly under each other, and added together; to this fum add 
 the logarithm of the whole number, next lefs than the pro- 
 pofed one, and the fum will be the logarithm required. 
 
 Note. If, inftead of the nuir.ocr •8685^^^9638, &c. we irake uii of 
 the number 2, ihe. rule will fervc fj/ the conftni^ion of hyperbolic loga- 
 rithms. 
 
 Ex. I, Required the common logarithm of 2. 
 Here the next lefs number to the propoft-d one is i ; hence 
 2 + 1 z= 3 ~ w, and ■i,y.'3^ IT 9 rr the fqu.ire of //. 
 
 •S635889638 ~3zz*2895296546-i- i =-28952965:46 
 •2899296946-r 9 — •032i6996r6T- 3 •0107233205 
 •O32i6996i6-7-9rz:*C035'74440i-i- 5 = *ooo7i4888o 
 •003 574440 1 -T- 9 == '000397 ■' 600-f. 7 = '0000567 3 7 1 
 •0003971600-^ 9ir-occo44i2 88-r- 9 = -cooc 049032 
 •0000441 288-1. 913-0000049032^ 1 1 = -0000004457 
 •0000049032-^ 9::^ •0000005448— 1 3 rz '00000004 1 9 
 •0000005448-^-911: '000000060^-7- 1 5 — '0000000040 
 
 Sum '3010299950 
 To which add the log. of i =: •coocoooooo 
 
 Common logarithm of 2 rr '30102999^0 
 
 ♦ An inreftigation of this rule was given in the original eflay, but is 
 here omitted as b.-ing of little or no ufe to tho yoiir.g Arithmeticiaa, who 
 
 cianoc
 
 Log ARi THMi c AL Arithmetic. 5 
 
 Ex. 2. Reaj/ired the €om?n:n logarithm of ^, 
 Here the next lefs nuinber to the prc-vjfed one is 2 ; hence 
 3 -f 2 — 5 nz^, and 5 X 5 := 25 zz ihc fquareof », 
 
 •8685889638 -r $ — '\l^l 111^,^1 -^ I := -1737177927 
 •173 ; 177927 -^ 25 rr '0069487 116 *j- 3n ooi3 162377. 
 •000)4871:6 -r 25 ii •000277,4^4 -~ $ zn 'ooo ,555896 
 •0002771,484-;- 25 — •ooonijji79 ^ 7 rr 6000015882, 
 •C000111179 -^ 25 — •000000-. j.43 -^ 91::; 000D000493 
 •0000004443 -r 25 rr '0000000177 -^ n tz: •0000000016 
 
 Sum •1760912586 
 To which add the log. 01 2 rr -3010299956 
 
 Common log. of 3 — : '4771212541 
 
 Ex.5. Required the common logarithm of /^, 
 Here 2 X 2 =: 4. *.• 2 X /<5i. 2 n %. 4." 
 Hence '301029995X2 = '602059990, corr-mon log. of 4. 
 
 Ex. 4. Required the common logarithm of ^. 
 Here 10 -r 2 := 5. •.* Log. id — hg. 2 ri log, ^, 
 I — '30102999; zz -698970004, common log. of j, 
 
 Ex. C. Required the common logarithm of 6. 
 Here 3 X 2 =z 6. •.' %. 2 -j- leg. 3 z= log. 6. 
 Hence '301029995 + '4771212542 = '7781512492, com- 
 mon log. of 6. 
 
 Ex. 6. Required the common logarithm of '^, 
 Here the next lefs number to the propofed one is 6 ; hence 
 •7 + 6= 13 z= », and 13 X 13 = 169 zz the fqiiare of;/. 
 
 •8685880638 -r 13 — '0668145356 -r I — '0668145356 
 •0668145356 -r 169 z=. -0003053522 -r 3 r: '0001317840 
 •0003953522 -r- 169 rr '0000023393 -f- 5 n: •ooccoo4r.78 
 •0000023393 -f- 169 ~ •0000000138 -r 7 n: '0000000019 
 
 Sum •0669467893 
 To which add the log. of 6 — •7781 5 12492 
 
 Common log. of 7 zz '8450980385 
 
 Ex. 7. Required the common logarithm of ^. 
 Here 4x2 — 8. •.' log. 2 + log. 4 = log..^. 
 Hence '301029995 + '602059991 zz '903089956, com- 
 mon log. of 8. 
 
 canrrot reafonably be fuppofcd to be acquainted cither with the method of 
 JiuxiarSi or the ftature and properties of the iyftrinlaf whence this ralr 
 is dedaccd.
 
 6 A Synopsis or 
 
 Ex. 8. Refuired the common logarithm of(^. 
 Here 3x^ = 9. *.• /o^. 3 -f log. ^zzlog.^; or 2 X tog, j 
 = V- 9- 
 Hence '4771 2 1 2 J47 X 2= '9 5424.25094., common log* of 9. 
 
 Ex. 9. Required the common logarithm of \o. 
 Here 2x5 = 10. '.• %. 2 + Ir.g. 5 = /o^. 10 ; but the 
 common logarithm of 10 is known to be i» 
 
 Ex. 10. Required the common logarithm of \\, 
 Here the next lefs number to the propofed one is ic ; 
 hence 11 -f 10— 21 = », and 21 x 21 n 441^ the Iquacc 
 of/-/. 
 
 •868588963R -^ 21 tr •0413C13792 -^ I iz •0413613792 
 •C41 361 3792 -T- 441 zr •0C00937S9.. ■4- 3 ir: '0000312633 
 •0000937899 -i- 441 ^ •ocOoot)2»26 -r 5 ±r •O«oc0ooo425 
 
 Sum •e4i39268 5X 
 To which add the log. of 10 :fi i-ooooooooo» 
 
 Common log. of 1 1 -r 1*041 3926851 
 
 I 
 
 In this manner may be found the logarithms of any other 
 rime numbers; it may be obferved that the operation wiH 
 e fhorter the larger the prime numbers arc; for any number 
 exceeding 350, tne firft quotient, added to the logarithm of 
 its next lefs number, will give the logarithm fought, true to 
 7 or 8 places of decimals. The logarithms of the compofite 
 numbers are found, by addition or fubtra6lion, as above; 
 hence it will be eafy to examine any logarithm in the tables 
 which we fufpecl to be falfe ; or, if any logarithms be torn 
 out or obliterated, we can readily calculate them, and fupply 
 the defea. 
 
 Observation. If JV~ any number, as 1, 2, 3, 4, 5, &c. then 
 
 will A' X ^ — 1» ^"'l A^ X 6 4- 1, coniVitute a feries which will contain 
 ail the prime numbers above tlie prime number 3. 
 
 Ex. 1 1 . Suppo/e a per/on ^working a mathematical problem ^ 
 nvhereiti he has occnjion to make u/e of the logarithm of ^-ji ; but^ 
 up'jn locking in the tables, finds that logarithm obliterated , or 
 fufpe^s it to be falfe, hoiu mufi he find jt ? 
 
 Here the next lefs number to the propofed one is 570 ; 
 hence 571 -|- 570 =: 1 141 = n, 
 
 •8685S89638 -i- 1T41 zr •0C07611520 -7- 1 rn •0007612520 
 By Sherwin's tables, the log. of 570 — »'7 558749 
 
 Su.-n 1$ the tr« log. of 571 :r 2*7 566361
 
 L O G A R I THMICAL ARITHMETIC. *J 
 
 §. III. The ufe of a Table of Logarithmic 
 
 The methods of difpofing the logarithms (when made) into 
 tables are various, and therefore no general direftions can be 
 given here for taking a logarithm out of thefn ; nor, indeed, 
 is it neceflary, fmce in every book which contains tables 
 there is, or fhould be, a rule given for that purpofe. — Before 
 I proceed farther, it may be proper to obferve that every 
 logarithm confifts of two parts, an integer, or whole number, 
 and a decimal fradion. The decimal part is always ajfr^ 
 mati'ue, but the ;>//^^^r, which is generally called the index, 
 or ckarafleriftic, may be either affirmati've or negati've ; for, if 
 the natural number be greater than r, the index of its lo^^a- 
 rithm will be affirmative, (or -f ;) but if the natural number 
 be lefs than i, the index of its logarithm will be negative, 
 (or — ,) as I have already obferved. This index is not put 
 down in fe.eral books of logarithms, as Sherwin's, &c. be- 
 caufe all natural numibers, (confifting of the fame fignif.cant 
 figures,) whether they be whole numbers, or pure or mixed 
 decimals, have the decimal part of their logarithms the fame, 
 the only difference being in the indtx, or whole number, as 
 in the following examples. 
 
 Nat. numb. Logarithms. 
 •94621 = — 1-0743320 
 •094261 - — 2-97433^(5 
 •0-9426! = — j.974.3320 
 
 •0-09^261 = — 4-9743 ;;20 
 
 •000094261 = 9*9743320 
 
 •0000094261= — ^'9743320 
 
 From the preceding Examples it appears, that, if the /irft 
 figrihcant figure of the natural number (r ckoning fr -rn fhc 
 lefi hand towards the right) be units ^ te\s, hundreds, ihjujands, 
 &:c. the index to its logarithm will be c, i, 2, 3, 6cc. re- 
 fpedively ; or, if the index to any logarithm be o, 1,2, ^,- 
 &c.'the firft figurt* of the natural nuaijcr will be Lnir, tens, 
 hundreds, thoufan-'.s, Sec. refpecflively. Likewife, if the 
 finl fignificant figure of the natural nuuiber be te,'it,b., J:un^ 
 dndths, tUufandths, Sec, the ind x to i ; Ic'garithm v. ill be 
 — i, — 2, — 3, &c. and, confe-]uentiy, \i ihc index to any 
 logarithm be — 4, — 2, — 3, -—4, — 5, &c. tho irrft i.gni- 
 ficant figure of the aumber anfwering is \Y.z jirJi,fscoiid, thirds 
 fourth, fyih, Sec, place of decimals. 
 
 u 
 
 Nat. numb. 
 
 
 uogariihms. 
 
 9-4261 
 
 =:= 
 
 0-9743320 
 
 94-261 
 
 ~ 
 
 1*9743320 
 
 942-61 
 
 ~~ 
 
 2*97433^0 
 
 9426-1 
 
 •~- 
 
 3-9743320 
 
 9f26ir 
 
 •*-* 
 
 4-9743320 
 
 942610* 
 
 rz 
 
 5-9/-43320 
 
 9426100 
 
 -z:z 
 
 6-9743320
 
 S ASynopsisof 
 
 Proposition i. 
 To Jin d the logarithm of a 'vulgar fraHion* 
 
 RcLi. Subtraft the logarithm of the denominator from the logarithm 
 of the numerator: if you carry i, add to it the index of the logarithm 1 
 of the denominator; then take the aitfercnces of the indxes for the ^ 
 index of the bgari:hi:n of the fraftion; prefix the fign of— or -f before 
 k, according as the fraftion is proper or improper. Obferve to reJucc 
 compound, mixed, &c. fraftions to limple ones. 
 
 In this and the following propofitions, whenever an index 
 is written without a fign, it is always affirmative or +. 
 
 Ex, I. To find the logarithm 5/" -Jl. 
 
 Log. of 93 = J '9684829 
 Log. of 94 = 1-9731279 
 
 Log, of II = —1-9953550 
 
 Ex. 2. To find the logarithm of •^-^* 
 
 48 
 
 ^6^ 36x3 + 2 no . , r ». » • 
 
 Here -^ = ^ ^^ = , the log. of which is 
 
 48 48x3 144 
 
 — rS83«302. 
 
 PROPOSIT40N 2. 
 
 To find the logarithm of a ciicuUting decimaU 
 
 RpLE. Reduce the decimal to its equivalent vulgar fra^ioa, and thc» 
 ikid it3 logantkm by prop. i. 
 
 1. Required the logarithm of *i^6^\ 
 
 Here •c6,/ = I^lHi^ = i£2, the log. of which is 
 
 ^ -* 900 900 
 
 ^17507655. 
 
 2. Required the logarithm ^5"3'54'» 
 
 Here ^3^5+' = ^^^^=^^ = -^^» ^^« ^^S- o^'^^'^^ ^ 
 ^^^+ 999 999 
 
 ^ 0*7287071. 
 
 3. Required the logarithm of 'C 6^2 S 803 . 
 
 Here -o^'ziio^' = '^iH£3 = 1^1^ ,he H- cf 
 9999990 3/0370 
 which is —i' 7975 1 50.
 
 LoC A R I TH M I C AL ARITHMETIC. ^ 
 
 4. Required the logarithm of •oo84'9 7133'. 
 Here -oog+'gy . 33' = il?IIl^ = -ll2I^ = 4, . 
 
 999999000 999999COO 9768 
 
 the log. of which is — 3-9292724. 
 
 Proposition 3. 
 T^ot':? cr worr numbers being given ^ to Ji?id their prcduf! by 
 logarithms. 
 
 Rule. Add the logarichnQS of t'le nurTibcrs together ; when you come 
 to the indices, add the affiimatives and wha: you carry into one i'am, and 
 the negatives (if any) into ani-ther, and take t!ie difference of the fums, 
 to which put the fign of the grer.tcr fum, and this will be the index for 
 the fum of the bgirithms, the correfponding number to which witl be 
 the produfti 
 
 E X . I . Required the produB c/ 84X^6X37X8. 
 log. 84 = 1-9242793 
 log, j6 = 1*7481880 
 % 37 == 1*5682017 
 log, 8 =: 0*9030900 
 
 Prod. 1392384 log, = 6*1437590 
 Ex. 2. 
 
 Reqm 
 
 'red the 
 
 ? produa (7/ '84 X ' 05 6 X 
 
 •37' 
 
 log. 
 
 •84 
 
 = 
 
 — 1-9242793 
 
 
 l;g> 
 
 •056 
 
 = 
 
 — 2'748i88o 
 
 
 log. 
 
 •37 
 
 = 
 
 — 1*5682017 
 
 
 Sum of the decimals = •\- 2*2406690 
 Sum of the neg. indices = — 4* 
 
 Log. of the produfl '=■ — 2*2^06690, the number 
 anfvvering to which is 'oi 74048. 
 
 Ex. 3. Required the produd of '37 X 426 X '5 X '004 X 
 
 •275x336. 
 
 hg. "37 n — i'56820i7 
 
 ^ kg. 426 — 4- 2-6294096 
 
 log. '5 — — i'698o70o 
 
 Lg, '004 n — 3 '6020600 
 
 /er.-27 5 = — i-43933a7 
 leg. 336 tz + 2-5263391 
 
 Sum of the decimals and affirm, indices n + 7'4643i33 
 Sum of the negative indices r: — 6- 
 
 Logarithm of the produdl: — 4- i*4643i3J> the numbef 
 anfwering to which is 29«i28i76. 
 
 U z
 
 to ASynopsisof 
 
 Ex. 4. Required the produS 0/ 56]- X /© X 'J X ^ X 
 
 47 
 
 
 Log, ^e\ = hg,^i^ - + 1-75076^:4 
 
 leg. 7^ =: — 1-8450980 
 log. 4d = — 1-9777236 
 
 L.g. f^ =z log. 11^ = -1-9778857 
 
 S'^raofthedecimais and affirm, ind. zr +5'3473528 
 Sum of the negative indices =1 — 4* 
 
 -Logarithm of the prod 11(51 rr -f 1-3473528, the num- 
 ber anAvcri.ig to which is 22*25 n 6. 
 
 Kx. 4. Renuir^d the produfi of •0594405' X •583'x 
 •OJ229I6' X -423571' X -I'S'. 
 
 Log. -0/94405' n: %. ;^^ = %. ~ =-2-774o829 
 
 995-^9^^ 2SO 
 
 Z.?-.-c8y=:%.li-^-^=^=%^^ =%.-! = — 1-7659168 
 
 * ^ "^ "* 9OQ "^900 ^12 * ^^ 
 
 , 322016— 32291 
 
 "* -^ ^ "^ 9COCOCO 
 
 1 8 
 Loy.'i'^'-hg. — = —1-2596373 
 
 99 
 
 Sum of the decimals = -42-9407508 
 Sum of the negative indices = — 7* 
 
 -Logarithm of the produci = — 5'94^7SO^« 
 the number anfwcring to which is •000087247.
 
 LOGARITHMICAL ARITHMETIC. 
 
 II 
 
 Proposition 4. 
 To find the quotient of one mimher by another, by logarithms. 
 
 Rule. From the logarithm of the dividend fubtraft the 
 logarithm of the divifor ; but take care when you come to 
 the index of the divifor to change it from affirmative to nega- 
 tive, or from negative to affirmative; then, if the indices 
 have unlike figns, take their difference*, prefixing the figa 
 of the greater index to it; but, if they have like figns, take 
 their fum, prefixing the common fign thereto, for the index 
 of the logarithm of the quotient. 
 
 Obferve, when there is an unit to carry from the decimal 
 part of the logarithm of the divifor, to add it to the index of 
 that logarithm, if affirmative, or fubtrad it from it, if nega- 
 tive, before you change the indices as above. 
 
 Examples 'wherein the indices of the logarithms of the divifon 
 and dividends are affirmati<ve, or -f • 
 
 Ex. I. Divide 785*925 by 25. 
 Log, 785-925 = 2-8g538ri 
 Log. 25 = 1*3979400 
 
 Quotient 51*437 /o^. =: 1 •497441 1 
 
 Ex. 2. Divide 250 by 78*5. 
 
 Log, 250 = 2*3979400 
 Log, 78*5 =: 1-8948697 
 
 Quotient 3*18471 log, n 0*5030705 
 
 Here the indices of the dividends are greater than thole of 
 the divifors; hence thefe and fuch like examples are ma* 
 naged like common fubtraftion. 
 
 Ex. 5. Divide 51*2^ by 125. 
 Log, 31*25 = ] -4948500 
 Log, 125 = 2-0969100 
 
 Quotient '25 log, = — 1-3979400 
 
 * The word d'lfferenct In this and the preceding rule is not to be un- 
 derftood in an aigebra\c fenfe, but fimply the difference of the 'wdic»i ^ 
 thus tbe di/Jerence between 4-5 and — 2, is not +7 or ^—7, but +3, 
 
 Us
 
 12 A Synopsis of 
 
 Ex, 4.. Divide 125 by 312^. 
 
 Log, 125 = 2 "0969 1 00 
 Log. 3 12 J = 3-49+8500 
 
 Quotient '04 %. = — 2-6020600 
 
 Here the indices of the divifors are greater than thofe of 
 the dividends, and when they are changed are negative, 
 ••• the indices of the quotients are negative. 
 
 Ex. 5. Divide 962*5 by 770. 
 
 Log. 962*5 = 2*9834007 
 Log, 'J 'JO = 2*8864907 
 
 Quotient 1*25 log, z= 0*0969 100 
 
 Ex. 6. Divide 317*55 by 870. 
 Zc^. 317*55 = 2*5018121 
 Leg. 870 = 2*9395^93 
 
 Quotient '$6^ log, r= — 1*5622928 
 
 Here the indices of the divifors and dividends are equal, 
 and the difference in the 5th example is o ; but, in the 6th, 
 there is an unit to carry from the decimal part of the loga- 
 rithm of the divifor, which mud be added to its index, be- 
 caufe it is affirmative ; then it becomes +3; changed it is 
 
 — 3 ; then the indices are different, and their difference is 
 
 — I. 
 
 Examples ^wherein the indices of the logarithms of the di'vifari. 
 end dividends are negative , or — . 
 
 Ex, 7. Divide '0565 by •25. 
 Log. '0565 = — 2*7520484 
 Leg, -25 = —1-3979400 
 
 Quot. '226 log. = — 1*3541084 
 
 Ex. 8. Divide '00379 ^V -0678. 
 Log. -00375 = —5 '57403 13 
 Log, 'o6-jS r: — 2*8312297 
 
 Quot. 1*05530973 log, = —2*7428016
 
 Logarithmic A L Arithmetic. ij 
 
 Here the indices of the logarithms of the dividends (not 
 regarding their figas) are greater than thofe of the divifors. 
 In the 7th example, the index of the divifor is — t, changed 
 it becomes -{- 1 ; then the figns are different, and their dif- 
 ference is — 1. In the 6th example, there is one to carry 
 from the decimal part of the logarithm of the divifor, which 
 mud be fiibtrafted from its index, becaufc it is negati\'e ; 
 then it becomes — i ; changed it is -|- 1 j then the figos are 
 different, and their difference is — 2. 
 
 Ex. 9. Divide -9072 by •0084. 
 
 Lo^. -9072 = —1-9577030 
 Log. -0084 = --3-9242793 
 
 Quotient 108 log, z=. -f 2 •0354237 
 
 Ex. 10. Divide •4606 by '0094. 
 
 Log. -4606 zr — 1*6633239 
 Log. -0094 == —3*937^279 
 
 Quotient 49 log. = +1*6901960 
 
 Ex. II. Divide '001404 by 'ooriy. 
 Log. -001404 = —3*^73671 
 Log, •00117 = — 3*0681859 
 
 Quotient 1*2 log. = 0*0791812 
 Ex, 12. Divide '3515 by •37. 
 
 Log, '37 = —1*5682017 
 Quotient '95 log, = — 1*9777236 
 
 Examples fwherein the indices of the dividends are qfirmathV4, 
 and thofe of the divifors negative, 
 
 Ex. 13. Divide 6543*9 by '27863. 
 
 log. 6543*9 = +3*8158366 
 Log, -27863 = —1*4450279 
 
 Quotient 23485*97 kg, r: +4*3708087
 
 14 A Sywopsis Of 
 
 Ex. 14. Divide 54498 by '09 5. 
 
 Log' 54498 = +4*7363806 
 Log, -093 = — 2-9684829 
 
 Quotient 586000 log, == +5*7678977 
 
 Ex. 15. Divide 83-1*4 by -00112. 
 
 Log. 834*4 =z +>. •9213743 
 Log. •001 12 zz — 3-04.92180 
 
 Quotient 745000 Jog. z=z +5*8 7 21565 
 
 Ex. 16. Divide 19-206 by '0194. 
 
 Log, 19*206 = 41*2834369 
 Log. '0194 = —2*2878017 
 
 Quotient 990 log. = +2-9956352 
 
 Ex. 17. Divide 9?5*68 by -oSSS. 
 
 Log. 985-68 = +2*9937359 
 Log, -0888 := — 2*94.84130 
 
 Quotient 11100 /(3g-. =z +4-0493229 
 
 Ex. 18. Divide 9890*1 by •00999. 
 
 Log, 9890*1 ~ +3-9952007 
 Log, -00999 = — 3*9995<^55 
 
 Quotient 9900CO log. zz +5*9956352 
 
 From the 6 preceding examples we may infer, that, when- 
 ever a logarithm with a negative index is to be fubtrafted 
 from one with an affirmative index, if there is nothing to 
 carry from the decimal part, the index to the logarithm of 
 the difference will always be equal to the fum of the indices ; 
 and, if there is i to carry, it will be equal to the fum lefs i, 
 ^d in both cafes affirmative*
 
 LOGARITHMICAL ARITHMETIC. I5 
 
 Zxamples 'wherein the indices of the di-uidtnds are negat've, 
 and thofe of the divifrs affirinativt. 
 
 Ex. 19. Divide 11^ by ^\\^. 
 
 Leg. \\^ = —1-6930450 
 Log. -\\- =: +2'0-oiSco 
 
 Quotient '00419628 log. r= — 3-6228650 
 
 Ex. 20. Divide 1- by 7^°- 
 
 Leg. 4 = — 1-744727^ 
 Log. 75^00 _ ^^.Q„,g.,3 
 
 Quotient •0005^92' log. = — 4*7727562 
 
 Hence, by a comparifon of thefe examples with fome of 
 the preceding, we may inter, that, if a:W logarith:n with an 
 atfirmative index is to be fubtracted i'rom one with a nega- 
 tive index, the index to the iogarithm of the diftereiice ^yill 
 always be equal to the fum of the indices, if there is nothing 
 to carry from the decimal ; but, if there is i to carry, it will 
 be equal to the Turn more i, and in both cafes negative* 
 
 Proposition ^. 
 71? involve a gi'ven nwnher to any poiver. 
 
 Rule. 
 
 Cafe I. Wheu the index to the logarithm of the frfi pi^vji^^ 
 
 or ront, is afirmnti--ve, {or -f-.) Multiply the l:)^arithm of 
 
 the root (rejedino- its Index) bv the index of the po\ver re- 
 quired, and referve the produ<!t ; then mmtiply the md:x ot 
 the logirithm by the index of the p'.)w^'r ; this prod uilit, added 
 to the former, will give the logarithm of the power required, 
 witia its index always affirmative. 
 
 CaHj 2. IVhen the index to the logarithm ^f the fir ft p<nver, 
 
 or root, is 7icgative [sr — .) Multiply th^; logarithm of the 
 
 root (rejeOi'ling its index) by the index of the power required, 
 and referve the product, which will always be affirmative; 
 the'i multiply the index of the logarithm by the index of 
 the power; this produ(^; which will always be negative, 
 muft he fubtraa-d from the former to obtain the logaritlun 
 of the power required.
 
 i5 ASynopsisof 
 
 Note. In fubtrafting the above produifls, when you 
 come to the indices, or whole numbers, their difference mull 
 be taken, and the fign of the greater (which will generally 
 be negative) prefixed, for the index to the logarithm of the 
 power. If there be an unit to carry from the decimal part 
 of the lower produ«5t, it muft be added to the index of that 
 produdl before you take their difference, as above. 
 
 I . Examples ivhcreia a 'whole number or mixed decimal is in-* 
 'vol'ved to any ponx'er. 
 
 Ex, I. Innjolve \\"]Z ta the third pQ^joer, 
 log, 14*72 = + 1 •1679078 
 Index of the power 3 
 
 Produfl of the decimal -f 0*503 7 2 34 
 
 + 1x3 = -f 3 prod, of the index. 
 
 Power 3189*506 %= +3*5037254- 
 
 VIZ. 147721^=3189*506 
 
 Ex. 2. Required the '^(i^th poiver of VOO^S"]* 
 
 Leg, 1*00567 zr -|- 0*0024555 
 Index of iLe power ^6^ 
 
 Pfodudl of the decimal +0*8962575 
 
 + X 365= -fc prod, of the index. 
 
 Power 7*875125 log. =1 -|-o*8962575 
 
 viz. 1*00567]^*' =7*875125 
 
 Ex. 3. IfTJohe 1*05 to the ^oth po-ivert 
 Log, 1*05 =: 0-0211893 
 Index of the power 40 
 
 Produft of the decimal +0*8475720 
 
 + X40 =. -fo* prod, of the index. 
 
 Power 7-0399887 log. zz +-0-8475720 
 
 Viz. ro5l+° =: 7-0399887, the amount of iL for 40 
 years at 5 per cent, compound intereft.
 
 Log AR iTH MIC A L Arithmetic, 17 
 
 Ex. 4. Required the $"] ^ pwer of i^"ig. 
 
 Log. 14-79 = +1-1699682 
 Index of the power 3*75 
 
 Produft of the decimal +0*6373807 
 
 + I X3-75 = +3*75 prod, of the index. 
 
 Power 24399-49%. =: +4*3^73807 
 
 Viz. 14-79] 3. 7 5 =24399-49 = 1479I** 
 Ex. 5. Required the 1*26' ponver of ']Z^, 
 
 ^°S' l^S ^^ +2-8948697 
 Index of the power i '26' 
 
 Produft of the decimal + 1 'i 33 5:0 1 6 
 
 + 2 X 1-26' = -^2'S333333 prod, of the index. 
 
 Power 4643*387 %. == +3-6668349 
 
 Viz.l87l«-*<^' =4643-387 =7'8?|H 
 
 Ex. 6. Required the '075 ponver <?/' 147*5. 
 
 Log, 147*5 = +2*1687920 
 Index of the power -075 
 
 Produft of the decimal +0*0126594 
 
 + 2 X *075 z= +0-150 prod, of the index. 
 
 Power 1*4543 ^og' = +0*1626594 
 
 Viz. 14^51 -^7 5-- ,.4^42 =i47*5i4:V 
 Ex. 7. Required the *3^'s^' p^^'^t^ of 94*75'* 
 
 ^^'g' 94*75' = U' ^W = +1-9766047 
 
 Index of the power '34' 5^' 
 
 Produft of the decimal +0*3373636 
 + 1 X-34'54' = +o-34'34V54 
 
 Power 4«8i736 lag, = +0-6828090 
 
 Viz. 9f77i'^*'**' =4-81736 =^4*771^5inK
 
 iS A Synopsis o * 
 
 2. Examplei luherein a pure decimal is invol'ved to any ftr.Ki<. 
 
 Ex. 8. iKvoh'e '0725/0 the third piywer^ 
 Log, '0725 = — 2*8603380 
 
 Produft of the decimal -|-2'58ioi40 
 
 — 2x3=: — 6* prod, of the index, 
 
 Power •000381078 log, •=. — 4-5810140 
 
 Viz. •0725] = -000381078. 
 
 Ex. 9. Required the 6'2!^ /cnuer of 'co^t. 
 
 Log, -0032 = — 3*5051500 
 ^ 
 
 Prodttfl of the decimal. +3*1571875 
 
 — 3 X 6*25 = — 1875 pr.ofth<f 
 
 (index. 
 
 Power '00000000000000025538 /^^.=: — 1 6-407 1 S75 
 
 Viz- -0032/-*^ = -0000000CO000CCC25 ^38= •C032V* 
 
 Ex. 10. Required the -625 ponver cf 'OO^l, 
 Log, -0032 zz — 3-5051500 
 
 •625 
 
 Produft of the decimal +0-3 157 1875 
 
 — 3 X -625 = — I '875 prod, of the index. 
 
 Power -0275879 /<?^. = —2-44071875 
 
 Viz. -0032 -^*' = '0275879, Arc. =: •0032,*" 
 
 Ex. l.r. Ini.-ol've -09475' to *34'54* ponver. 
 Log. -09475' = %. jViT = —2-9766047 
 
 *34 H 
 
 Prod u ft of the decimal +0-33736364 
 
 — 2 X •34'54' = — o-69'o-'(;o89o8, Scq, 
 
 Power '4^.0; log. n: — 1-6464727 
 
 Viz.^^^^-^-"^' = -++307 = ■5VA,"»'=
 
 LoGARITHMlCAL ARITHMETIC. I9 
 
 Proposition 6. 
 
 I'c cxtra8 anj root of any given po'wer. 
 
 Rule. Di\ ide the logarithm of the power to be extrafted 
 by the index e::prefling the root, the quotient is the loga- 
 rithm of the root. — If the index to the logarithm be negative, 
 and does not exactly contain the divifor, increafe it by fucb 
 a number as will make it exadly divifible, and increafe 
 the logarithm alfo by the fame number, before you begin to 
 divide » 
 
 Note. The method of increafing the -negative index is bjr 
 adding equal numbers to the negative and affirmative parts; 
 thus, the logarithm of '0711 in — 2*8 5 18696 is equivalent 
 to — 3 + i'S5i8696, or = — 4-f-2'85iS696, or zz — ^ 
 -f-3'8^i8696, or it is = — 365-f 563'S5i8696, 6cz. where 
 the latter part is entirely affirmative : hence the fquare, cube, 
 fourth, fifth, 365th, or any other root of 'O'jii., or any 
 number, may be readily extract ji^ 
 
 I. Ex<imples njjherein the poucer is a <vjho!e number or mixed 
 dccimah 
 
 Ex. I. What is the/quare-root of ^? 
 
 Log. of 3 rz -1-0*4771213 ; this, divided by 2, the index, 
 gives 0*2385606 for the log. of the root, the number an- 
 fwcring to which is 1*73205. 
 
 Ex. 2. Extras the cube-root o/* 3189*505. 
 
 Log. of 3189*506 z= +3'5037234; this, divided by 5, 
 the index, gives 1*1679078 for the root, the number an- 
 fwering to which is 14*72, the root required. 
 
 Ex. 3. Extra<flthe^6^throotofyS']p2^, 
 
 Log. 7*875125 =: -|-o*8962574; this, divided by 365, 
 the index, gives 0*0024595 for the logarithm of the root, 
 the number anfwering to which is 1*00567 ; 
 
 Viz. 7*875125']tJt = i •^00567. 
 X
 
 20 A b Y N' O r S I S OF 
 
 Ex. 4. ExtraH the yj ^ root cf 24399'4g. 
 
 Log. 24.399'49 = +4*3873807, which, divided hy 3-7$'. 
 gives 1-1699681, the logarithm of the root, the number an- 
 iVering to which is 14*79 ; 
 
 Viz. 24399-49~lrTT = 14-79 = 2^J99^i*T 
 
 Ex. J. Extrad the 1*26' root 0/ j\.6j{.y^S-j. 
 Log. 4643*387 = +3*6668348, then 3-6658348 + 1*26' 
 
 rz: 3*66683480 — -36668348 + 1*26 — -12 r: 3-3001 9 132 
 -^ I '14 = 2-894869 J, log. of the root, the niimber an- 
 fwering to which is 785, extremely iiear^ the fmall error ari- 
 fing only from the imperfedlion of the logariihms; 
 
 Viz. ^Hy^%-l\-^'\-^ — 78; zz 4643*3^^'^ 
 
 Ex. 6. Extracl the 'O"] ^ root of 1*4^4318. 
 
 'Log. 1*45^4318 zi +0*1626^93, which, divided by •075', 
 gives 2*1687910, log. of the* root, the number anfwcring to 
 which is 147*5 "(^^ffj'i 
 
 Viz. i-4543i'8l-^Ti = i47'5 = 1 '4543^^1 V 
 
 Ex. 7. ExtraB the "i^4^ !^Ji^ root of /^'%l'] -^6, 
 
 Log. 4-81736 r= +0-6828091 ; then 0*6828091 -f- 
 •34'54' rz '6821262909 +■ '3451 = 1*9766047, log. of the 
 root, the number anfwering to which is 94*75'. f^ide Ex, 7, 
 in lu'volutmu 
 
 Viz. 4-8i736i*3-4/T4^=:94-75'=: 4^81736] "J^tt 
 
 2. Examples ^wherein the panver is a pure dccirnaL 
 Ex. 8. Extract the fquare -root of *02^. 
 
 Log. '025 = — 2-3979400, which, divided by 2, the 
 index, gives — riQ89700 for the log. of the root, the num- 
 ber anfwering to which is -1581 1. 
 
 Ex. 9. ExtraB the cuhe-root of '000381078. 
 Log. .000381078 =: —4*5810139 = —6 +2*5810139; 
 this, divided by 3, the index, gives — 2 + '8603379 = 
 — 2 "8603 3 79, log. of the root, the number anfwering to 
 which is '0725 uearlj ; 
 
 Viz. •000381078)1 = '0725.
 
 L G A R 1 T H M I C A L A R I 1 H M E T J C 21 
 
 Ex. 10. Required the 6*25 root of '0000000000000002553804. 
 
 Log. '0000000000000002553804 = — i6'J^07i875 =r 
 — 31-25 -f 15-25 + •407JS75: = T^^'^'i + rr657i«75 J 
 this, divided by 6-25, the index, gives — 5 -("''S^S • 5^<^ ==^ 
 ■ — 3-50,' 1 500, log. of the root, the number anfwerin^ to 
 whieh is 'oo'^ 2, the root required. 
 
 Ex. 1 1. Extras the '62^ root of ^02758791. 
 
 Log. -02758791 = — 2*4407 187 = —18-75 + 1.^*7 J + 
 •4407187 s= — 18-75 + n* 1907 1 87, which, divided by 
 •625, the index^ gives — 30 -f 27*5051500 ~ — 3-5051^00, 
 log, of the root, the number anfwering to which is -0032, 
 the root required. 
 
 Ex. 12. Required the •l,\<^)J r.ot cf'xi^lo-^Of^, 
 
 Log. -4430705 "=■ — 1-6464727 zz -~3-4'5+' + 2 4^5+' -f- 
 •6464727 = — 3-4'54'-|-3-ioo92 7i5'44', which^ divided by 
 
 •34'54^ the index,( = 3-45] +3-0978262273 -f- -3451), gives 
 — 10+8-9766045 = — 2-9766045, log. of the root, the 
 mimber aniwering to which is '09475' [extretnely near), the 
 loot required. 
 
 Having now given examples of all the varieties that can 
 pofiibly happen m the practice of involution and evolution, 
 where the index of the power is a whole number, or a mixed 
 or pure decimal, I (ball add but one obfervation more to- 
 wards rendering the practice complete, viz. that, whenever 
 
 we meet with an expreffion with a fradional index, as 475!^, 
 the numerator denotes the power to which the root is to be 
 raifed, and the denominator (hews what root of that powtr 
 is to be extraded. 
 
 In fome of the preceding examples in involution the powers 
 are lefs than the roots, and in evolution the roots are greater 
 than the powers ; fhould the critic be difpofed to cenfure the 
 naifiC 1 have given them by aflcrting that '* powers are properly 
 dilHnguillied by the elevation of roots, or involution ; and 
 roots by the depreffion of powers, or evolution ;" 1 have- 
 only to reply, that I may as properly put fuch examples undtr 
 the above titles, as the product, or quotient, of a pure de- 
 cimal can be placed under the title of multiplication or di- 
 viiicn. 
 
 X 2
 
 22 The construction of Decimal Ta-bles 
 
 The application of LOGAPvITHMS to 
 COMPOUND INTEREST,. crV. 
 
 A .principal ufe of Logarithms, as the celebrated Dr. 
 Ilariey obferves, is to folve all the cafes of Compound In- 
 tereft, which are not v.ithout great difficulty attainable by 
 the rules of common Arithmetic. — In the Arithmetician I 
 have given Logarithmical theorems for all tlie different cafes 
 of Compound intereft and Annuities; I now proceed to the 
 conftrudlion of Decimal Tables at 3, 'i,\, 4^ 4^, and 5 per 
 cent. Compound Intereft, by means of which the ajifwer to 
 mo'l: queftions in Compound Intereft and Annuities may be 
 readily obtained. 
 
 The cojrJiruSIion of Table I. 
 
 /-:j. i'03z:o*or2S372 /o^. of the amount of il. for i year. 
 0-0128372 
 
 I -0609 - 0-0256744 log. of the amount of il. for 2 years. 
 
 •092727 r::o-0385i 16 /^^. ofthe amountof il. for 3 yrs. kc. 
 
 The CO ijlruilion of Table II. 
 
 The amount of il. for a Atcj at 3 per cent, will be the 
 36cth root of 1*03, by the univcrfal theorem, Pfage rzth, of 
 I. •. .irilhmeticiaJi. 
 
 L'jg. I 03 .— 0*0128372, this -^ -x^d^y giv^slog. of the 
 root, = 0*0000351, the number anfwering to which is 
 1-0000:09, the amount of il. for 1 day. 
 Then 0*0000351 
 0-000035 [ 
 
 o'ocoo702, the number anfwering to which is 
 1-0001619, the amount of il. for 2 days, &c. &c. 
 
 The confiru^iion of Table III. 
 
 The logs, of the fereral amounts in Table I. fubtraf^ed 
 from an unit, leave th* lo^>, of the prefent worths in this 
 
 table.
 
 FOR Compound Interest. 23 
 
 fable. Thus i — •0256744.= — 1-974325:6, the number 
 anfwering to which is '9425959, the prefent worth of il, 
 for 2 years, at 5 per cent, &c. 
 
 T^e cor^Jiruakn of Table IV. 
 
 This table fhews the amount of il. a-nnulty, and is cor- 
 ^ruded from the firft table thus, 
 
 I '0000000 
 
 1-0300000 amount of il. for r year at 3 per cent. 
 
 2*0300000 amount of il. annuity for 2 years. 
 1 '0609000 amount of the feeond year. 
 
 3*0909000 amount of il. annuity for 3 years, 5cc, 
 
 "^he conjiruahn of Talle V. 
 
 Thi? tabic (hews the prefent worth of i-l. annuity, and^ 
 is conitru^d thus, 
 
 Firft year, table III., at y per cent, is '9708738 
 
 Second year, ib. — — '9425959 
 
 Sum.. Second ycar^ table V. i-9i34/)97" 
 
 Third year, table III.. -(^i^i^iy 
 
 Sum- Third year in tabic V- &c.. 2*82861 14, 
 
 Th€ conflruaimi of Table VI. 
 
 This table is eonftruCled from the 5th table by fubtracling 
 the log. of the prefent worth of il. from an unit;, thus the 
 log. of 2*82861 14 is 0*4515732, this fubt rafted from i leaves 
 -=- 1*5484268, the number anfwering to v.hich is •3535304y 
 the annuity which il, will purchafe for 3 years at 3 per 
 cent, 
 
 X 3
 
 TABLE I. 
 The amount of il. for years. 
 
 •r.^i ■; per cenc. ] 3^ uer Cent. 
 
 I|l .0300000 
 
 2jI.o6o900C^ 
 
 I 0927270 
 
 [255088 
 
 1.194052;. 
 
 229873^ 
 
 1. 266770c 
 
 ^•304773^ 
 ^•3439^63 
 
 1-384233^ 
 1. 425760c 
 1.4685337 
 
 .5125S97 
 1.557967.1 
 
 i.6o47o6.:j 
 I 6528476 
 1.702433: 
 1.753506c 
 1. 8061 1 1: 
 
 1.0350000 
 1. 0712250 
 1.0871780 
 1.147523c 
 I 1876865 
 
 '29-555 
 1.2722791 
 1. 3 1 6809: 
 I 362897; 
 1.410598- 
 
 ^•4599^: 
 1.51 1068^ 
 
 1.563956c 
 
 I 61S694; 
 
 ^•67534^^ 
 
 1.733986. 
 
 1.794^755 
 I 857489: 
 I 92250J ' 
 1.989708;- 
 
 I 8602945 
 1.9 16 1034 
 1.9735865 
 2.0327941 
 2-093777^ 
 
 2.156591 
 2.221 2 89'j 
 2.287927 
 
 2-3565655 
 2 427262 
 
 2.5000803 
 
 2.575082-^ 
 
 2-652335 
 
 2.7319055 
 
 2.8138624 
 
 2.8982783 
 2.9852266 
 3.0747834 
 3.1670269 
 3.262037; 
 
 4 per ^,^{\i 
 
 2.059131. 
 2 . I 3 I 5 I I C 
 2.2061I4J 
 2 283328. 
 2-363244' 
 
 2.44595b 
 2.531567] 
 2.62CI7IC 
 2.71 1877c 
 2.8067937 
 
 2.90503 u; 
 
 3.0067075 
 
 3.1 II 9425 
 
 3.2208603 
 
 V3335904 
 
 3.4502661 
 
 3.5710254 
 
 j. 6960 II 
 
 3-82537^7 
 
 ? 9592597 
 
 1 .040000c 
 1 .0816000 
 1.124864c 
 1. 1698586 
 1.2166529 
 1.265319c 
 
 315931*^ 
 
 3685691 
 
 4233 1 1 c^ 
 
 1.4802445 
 
 1.0450000 
 1.092025c 
 1. 141 1661 I 
 1. 1925186 I 
 I 2461819 
 
 1539454^ 
 1.601032: 
 1 6650735 
 1.7316764 
 1.8009435 
 
 4|perCenr. 
 
 I 3022601 
 
 360861:8 
 
 I 4221006 
 
 I 4860951 
 
 ij_i29^9. 
 1.6228530 
 1. 6958814 
 1.7721961 
 I 8519449 
 1.9352824 
 
 1.8729812 
 
 1 9479005 
 2.0258165 
 2.106849: 
 2.1911231 
 
 2.3699188 
 24647155 
 
 2 5633042 
 2.665 8363 
 
 2.7724697 
 2.8833685 
 29987033 
 3.1 1 865 14 
 3-243397 5 
 
 3-3731334 
 
 3 50B0587 
 3.6483811 
 3-7943^63 
 3.9460889 
 
 2.022370 
 2.1133768 
 2.2084787 
 2.3078603 
 2 41 1714c 
 
 2.520241 1 
 
 2.633652c 
 
 7521665 
 
 .876013^ 
 
 3.0054344 
 
 3.14C679C 
 3.2820095 
 3.4296999 
 3.5840364 
 3-7453iSi 
 
 4-1039325 
 4 2680898 
 4.4388134 
 46163659 
 4.8010206! 
 
 5 p'. rCei.t. 
 050OOOC 
 .IO250OC 
 .157625^ 
 .2155063 
 .2762816 
 
 .3400956 
 .4071004. 
 
 •4774554 
 
 •55132 
 
 .6288946 
 
 •7103393 
 7953563 
 .8856491 
 .9799316 
 
 0780282 
 
 3-9138574 
 4.089981c 
 4.2740301 
 4.4663615 
 4^7147^ 
 
 . 37737"84 
 5.0968604 
 5.3262192 
 5.5658990,0 
 
 .1828746 
 .29^018^ 
 .4066192 
 .5269502 
 •6532977 
 7.^59626 
 .9252607 
 .071523! 
 2251000 
 3^63549 
 
 •7334563 
 9201291 
 1161356 
 3219424 
 
 •5380395 
 7649415 
 0031885 
 .2533480 
 .5160154 
 
 7918161 
 0814069 
 
 3854773 
 704751 
 
 039988-
 
 TABLE II. The amount of i1. for davs. 
 
 3 \ er Cent. 
 1 .OOO0B09 
 1 .00016 I 9 
 I 0002429 
 
 c.ooo324'j 
 I 0004050 
 
 1.000486c 
 1.000567c 
 1 .000648c 
 1 .0C07291 
 1 .0008101 
 
 3-^ v-crCcnC. 
 
 1.0016209 
 
 1.0024324 
 1.0032445 
 1.0040573 
 I.OO4870S 
 
 7 
 
 80 
 
 90 
 
 160 
 
 no 
 
 120 
 
 3 
 
 14c 
 150 
 1 6c 
 
 1-0000942 
 1 .0001 885 
 1.0002827 
 I 0003770 
 I 0004713 
 
 I 0005656 
 1 .0006600 
 1.0007542 
 I 0008486 
 1.0C09420 
 
 i.ooi'8867 
 
 1.00283 
 
 1.0037771 
 
 1.0047236 
 
 1.005671C 
 
 4 p;r Lent 
 
 I.OOCIO74 
 1. 0002 I 49 
 1.0003224 
 I 0004299 
 1.00053 
 
 1 .0006449 
 1.0007524 
 1 .0008600 
 1.0009675 
 1. 00 1 07 5 1 
 
 4.-^ [:er Cent. 
 
 l-OOOI2C6 
 
 I.COO24I2 
 
 1.0003618 
 
 I.COO4824 
 
 1.000603 
 
 1.0007238 
 
 1.000844 
 
 1.0009652 
 
 I .0010859 
 
 1.0012066 
 
 5 per C.:u 
 
 1.0056849 
 1.0064996 
 1.0073151 
 1.008131 1 
 
 1.0089479 
 
 1.0097653 
 1.010583^ 
 1 .01 1402'! 
 
 1 .012221 c; 
 1.0130415 
 
 17CI1.6138623 
 1.0146.837 
 1.0155057 
 I. 0163284 
 
 1.017151S 
 
 18c 
 19c 
 20c 
 210 
 
 22c 
 23c 
 240 
 25e 
 26c 
 
 28c 
 29c 
 SO 
 31c 
 
 32c 
 
 34'^ 
 
 36 
 
 1.0179759 
 1.0188006 
 1.0196260 
 1.0204520 
 1. 0212788 
 
 1.OC66193 
 1.007^685 
 1. 008 j 1 86 
 i .0094696 
 1.C104214 
 
 I 0021513 
 1 .003228b 
 1.0043074 
 1.0053871 
 1.006468c 
 
 1.0001336 
 1.0002673 
 1 .000401 1 
 1.0005348 
 1 .000668; 
 1.0008023 
 1.00C9361 
 1. 00 1 0699 
 1 .001203} 
 1. 001 3376 
 
 1 .0075501 
 I 0086333 
 1 .0097 1 77 
 I oio8®33 
 1. 01 1890C 
 
 1 .0024148 
 1.0036243 
 1.0048354 
 1.0060479 
 I 0072618 
 
 rToc84773 
 1.0096942 
 1 .0109125 
 1.0121324 
 I-OI33537 
 
 1.CI1374 
 
 1.0123279 
 
 1.0132825 
 
 1.0142379 
 1.0151945 
 
 1.0129779 
 1.014067c 
 1 .01 5 1 572 
 1.0162487 
 1.0173412 
 
 1.0221062 
 1.0229342 
 1.0237630 
 1 0245924 
 1:0254225 
 
 1.0262532 
 1.0270847 
 1. 02791^68 
 1.028749 
 i.o29q83c 
 
 1.0161516 
 1.C17109S 
 1.01806S9 
 1.0190288 
 I. 0199897 
 I. 0209515 
 1. 02 1 9 142 
 1.022877S 
 1.0238424 
 1.024807S 
 
 1.0257741 
 I 0267414 
 1.0277096 
 1.0286786 
 1.0296486 
 
 1. 0306195 
 1.0315914 
 1.0325641 
 
 1-033537^' 
 lo^^i^iz 
 
 1.002677c 
 I.OC40I82 
 1 .005 36 1 1 
 1.0067059 
 1 0080525 
 
 1.0094009 
 1.010751 1 
 1.0121031 
 
 1. 0144565 
 
 1.0 148*25 
 
 1.0145765 
 
 1.0158007 
 1.0170265 
 1.0182537 
 1-0194824 
 
 1.018435c 
 1. 019.5299 
 1.0206261 
 
 I.02I7233 
 
 1.0228218 
 
 1.0239215 
 1.0250223 
 1. 026 1 243 
 
 1.0272275 
 
 ^ 02833^9 
 1.0294375 
 IC305443 
 
 1 0316522 
 I.03276I4 
 
 I 0338717 
 1 0349832 
 
 1.036596 
 1.0372099 
 
 I 038325 
 
 1.0207126 
 1.0219442 
 
 1.0231774 
 
 1.0244120 
 I.02564S1 
 
 1 .026885S 
 1.0281249 
 1.0293655 
 4.0306076 
 1.03I85I2 
 
 1.0330963 
 
 I 0343429 
 I.03559IO 
 
 1.0368406 
 1.038001 
 
 i 0393444 
 
 1.040598.5 
 i 0418542 
 
 1.0431114 
 
 1.03944131-1 C44370C 
 
 1 .0161699 
 1.0175291 
 1 .01 88902 
 1.0202531 
 1.0216178 
 
 1.0229843 
 
 1.0243527 
 
 1.0257228 
 1.0270949 
 1.0284687 
 
 1.0312219 
 1.03^6013 
 1. 0339^25 
 I •0353 -^56 
 
 1.0367^05 
 1-0381373 
 
 i -^^395^59 
 1.0409164 
 
 1.0422087 
 
 1.04^629 
 1.0450900 
 1 .0464969 
 1.0478967 
 1 0402084.
 
 TABLE IIT, 
 The prefcnt worth of il. for years. 
 
 I 
 
 3 pciCenc. 
 .9708738 
 
 3ipeiCe.nc. 
 
 4 per Cent 
 
 41 per Cent. 
 
 5 per Cent. 
 
 .9661836 
 
 .9615385 
 
 .9569378 
 
 .9523809 
 
 2 
 
 .9425959 
 
 .9335107 
 
 9245562 
 
 .9157299 
 
 .9070295 
 
 3 
 
 •9i5H'7 
 
 .9019427 
 
 .8889964 
 
 .8762966 
 
 Mi^i-je 
 
 4 
 
 .8884870 
 
 .8714422 
 
 .8548042 
 
 .8385613 
 
 .8227025 
 
 5 
 6 
 
 .8626088 
 
 .8419732 
 
 .8219271 
 
 .8024511 
 
 .7835262 
 .7462154 
 
 •8374843 
 
 .8135006 
 
 .7903145 
 
 .7678957 
 
 7 
 
 .8130915 
 
 .7S5992O 
 
 .7599178 
 
 •7348285 
 
 .7106813 
 
 8 
 
 .7894092 
 
 .7594116 
 
 .7306902 
 
 .7031851 
 
 .6768594 
 
 9 
 
 .7664167 
 
 •73373IC 
 
 .7025867 
 
 .6729044 
 
 .6446089 
 
 lO 
 
 II 
 
 •744^939 
 .7224213 
 
 7089188 
 
 .6755642 
 
 •0439277 
 
 .6139133 
 
 .6849457 
 
 .6495809 
 
 .6161988 
 
 .5846793 
 
 I 2 
 
 .7313799 
 
 .6617833 
 
 .6245971 
 
 .5896639 
 
 .5568374 
 
 »3 
 
 6809513 
 
 .6394041 
 
 .6005741 
 
 .5642716 
 
 .5303214 
 
 H 
 
 .6611178 
 
 ,6177818 
 
 •5774751 
 
 •5399729 
 
 5050679 
 
 ^5 
 i6 
 
 .6418619 
 .6231669 
 
 5968906 
 
 .5552645 
 .5339082 
 
 .5167204 
 
 .4810171 
 
 .5767059 
 
 .4944693 
 
 .4581115 
 
 .171.6050164! 
 
 .5572038 
 
 •5133733 
 
 .4731764 
 
 •4362967 
 
 18 
 
 .5873946 
 
 .5383611 
 
 .4936281 
 
 .4528004 
 
 .4155207 
 
 19 
 
 .5702860 
 
 .5201557 
 
 .4746424 
 
 4433018 
 
 •3957340 
 
 20 
 21 
 
 •5536758 
 
 .5025659 
 .5855709 
 
 .4563870 
 
 .4146429 
 
 3768895 
 
 •5375493 
 
 •4388336 
 
 .3967874 
 
 .3589424 
 
 2;i 
 
 .5218925 
 
 .4691506 
 
 .4219554 
 
 .3797009 
 
 .3418499 
 
 23 
 
 .5066917 
 
 .4532856 
 
 .4057263 
 
 •3633501 
 
 •3255713 
 
 H 
 
 •4919337 
 
 '^119'^!'^ 
 
 .3901215 
 
 •3477035 
 
 .3100679 
 
 25 
 26 
 
 .4770056 
 
 .4231470 
 
 .3751168 
 .3606892 
 
 .3327306 
 
 .2953028 
 
 .4636947 
 
 .4088378 
 
 .3184025 
 
 2812407 
 
 27 
 
 .4501891 
 
 .3950123 
 
 .3468166 
 
 .3046914 
 
 .ze-jz^'^oi 
 
 28 
 
 .4370768 
 
 .3816543 
 
 ^^^^h^lls 
 
 .2915707 
 
 .2550936 
 
 2r> 
 
 .4243464 
 
 .3687482 
 
 .3206514 
 
 .2790150 
 
 .2429463 
 
 1° 
 31 
 
 .4119868 
 
 .3562784 
 
 .3083187 
 .2964603 
 
 .2670000 
 
 •2313775 
 
 .3999871 
 
 .3442304 
 
 .2555024 
 
 2203595 
 
 J2 
 
 .3883370! 
 
 .3325897 
 
 .2850579 
 
 .2444999 
 
 .2098662 
 
 33 
 
 •3770263. 
 
 .3213427 
 
 .2740942 
 
 .2339712 
 
 .1998726 
 
 34 
 
 .3660449 
 
 .3104761 
 
 .2635521 
 
 .2238959 
 
 .1903548 
 
 35 
 36 
 
 •3553834 
 •34503^4! 
 
 .2999765 
 .2898327 
 
 •2534155 
 ,2436687 
 
 .2142544 
 .2050282 
 
 .1812903 
 
 .1726574 
 
 37 
 
 •3349829I 
 
 .2800316 
 
 .2342969 
 
 .1961992 
 
 .1644356 
 
 38 
 
 .3252262! 
 
 .2705619 
 
 .2252854 
 
 .1877504 
 
 .1566054 
 
 39 
 
 •3157536I 
 
 .2614125 
 
 .2166206 
 
 .1796655 
 
 .1491479 
 
 40 
 
 .3065568- 
 
 .2525725 
 
 .2082890 
 
 .1719287 
 
 .1420457
 
 TABLE IV. 
 
 The amount of il. per annum^ or annuities for years. 
 
 3 per Cent. 
 
 n ?«-•-'■" 
 
 I ooooooo 
 2.0300000] 
 
 3 0909000J 
 
 4 1836270] 
 
 J-J2211SS 
 
 6.4684090 
 
 7.6624622 
 
 8.8923360 
 
 ^10.1591061 
 
 lOji 1 .4638793 
 
 12 8077"^ 
 -_ 14.1920296 
 
 13^5-6177904 
 I4I17. 0863242 
 
 IJ 
 16 
 
 17 
 18 
 
 »9 
 
 ^^•5989139 
 20 1568813 
 21.7615877 
 
 23 4H4354 
 25.116868^ 
 
 1 .ooooooo 
 
 2 O35OOOC 
 
 3. 1062250 
 
 42149429] 
 
 5 362+659 
 
 4 per Cent 
 
 1 OOOOOOO 
 
 2 0400000 
 3.12 16000 
 4.24.^464 
 5.4163226 
 
 6 5501522 6.6329755 
 
 7-7794075 7-8982945 
 9.0516866 9.2142:63 
 10.36S4958 10582795. 
 II. 7313931 12.0061071 
 13.1419979 13.4863514 
 14. 6019616 15.0258055 
 16.1130303 16.6268377 
 17.6769864 iS. 29191 12 
 1 9 29 56809 20.0235 ^ 
 20. 9710297J21. 8245311 
 22.7050i58j23-6975i24 
 24.4996913 24.6454129 
 26.3571805 27.67 12294 
 
 20-26 870.3745 28 2796818129.7780786 
 
 21128.6764857 30 269470713 1.969201 
 
 22j3o»5367.So3l3 2.3289022!34-24r^698 
 23132.4528837134.4604137136.6178886 
 
 2 r_, __■ ._' _'_'j ._^^ 
 
 ■ i. 04. 5:0003 
 
 4^ per Cent. 
 
 5 pir Csot. 
 
 1 .OOOQOOOl 
 
 2.0450000! 
 
 3.1370250' 
 
 4 278191 1 
 
 5.4707097 
 
 I. ooooooo 
 2.050COO; 
 
 3.1525000 
 
 4.310125c 
 
 5.5256312 
 
 6.7i689!7l 
 
 8.0191 58! I 
 
 9.3800156; 
 
 10.8021 142' 
 
 12.2S82094 
 
 4I34. 4264702 36 6665282139 082604 
 2 5 1 3M5 9i^3 j 3^-949^567! 4i-6|5 9o" 
 26|3a.553o.;..22;4i 3131017144.3117446 
 
 27'40.7096335|43.759o6o2i47-o^42i44 
 
 2.^142.9309225 
 29:45.2188502 
 30 47_5754i57 
 3 IJ50. 0026782 
 ^027585 
 
 34s 
 
 46.29o6273;49-96'^85< 
 48.9107993 52.9662863 
 51. 6 226773: 56.0 849-^77 
 
 54.4294710 59-32833'i2 
 57-334502562 7oi4C)87 
 
 3155.0778413:60 3412101 66 2095274 
 34157-7301765163.4531 524169.8579085 
 35!6o. 4620818 66.6740127173 6522248 
 
 3^|63-27T9H3 
 37|66. 1742226 
 "^'169.1594493 
 39!72. 2342327 
 4075.4012597 
 
 70.007603 2! 77 -59^3 '3'^ 
 73 4578693IS1. 7022464 
 77. 028894-J85.9703362 
 80.724906c 90.409 1 497 
 84.5502778I95 0255[57l 
 
 13.8411788 
 15.4640318 
 
 17 1599133 
 18.9321094 
 
 .20 784 0543 
 22.7193367 
 24.7417069 
 26.8550837 
 29.0635625 
 31. 3714228 I 
 
 337^3'368 
 
 363033779 
 38.9370299 
 41.6891963' 
 44.5652101' 
 
 6.8019128 
 
 8.142C084 
 
 9.5451089 
 
 1 1.0265643 
 
 12.5778925 
 
 14.2067871 
 15.9171265 
 17.7129828 
 19.5986320 
 21.5785636 
 
 47.5706446 
 50.7113236 
 53-9;33332 
 
 57-4^30332 
 
 61.0070697 
 
 64 7523878 
 68.6662452 
 
 72 7562263 
 77.0302565 
 81.49661 Ko 
 
 80.1639658 
 91.0413443 
 96.13S2048I 
 01.4644240: 
 07.030323 I'l 
 
 23.6574918 
 
 25.840366^ 
 
 28.1323847 
 
 30.5390039 
 
 33-065954 
 
 35.7192518 
 
 38.5052144 
 
 4>-430475» 
 44.5019989 
 
 47.7270988 
 
 51.1134538 
 54.6691265 
 58 402582^ 
 62 3227 J ig 
 _66__4388475 
 
 70.7607899 
 75.2^88294 
 80 o63:'7oS 
 85 0669594 
 90 3203073 
 
 95.8363227 
 01.6281388 
 07.7095458 
 14.095023; 
 20.799774-
 
 TABLE V. 
 TI\e prefcnt worth of il. per annum, or annuity fpr years. 
 
 yr.- 3 per Cent. 
 
 3i per Cent. 
 
 4 per Cent. 
 
 4^ per Cent. 
 
 5 per Lent. 1 
 
 0.9523800! 
 I.8594JC ' 
 2. 7232. > 
 
 3-54593^ 
 4-329470/ 
 
 1 0.9708738 
 
 2 I.9I34697 
 
 3 2.82861 14 
 
 4 3.7170984 
 J 4-5797072 
 
 6 5.4171914 
 
 7 6.25C2S29 
 
 8 7.0196922 
 
 9 7.7861089 
 
 10 8.5302028 
 
 11 9.2526241 
 
 12 9.9540040 
 
 13 10.6349553 
 
 14 1 1. 2960731 
 Li '^-9379351 
 
 0.9661 836 
 1.8996943 
 2.8016370 
 
 3-6730792 
 4.5150524 
 
 0.9615385 
 1.8860947 
 2-7750910 
 3.6298952 
 4.4518223 
 
 O.956937I 
 1.8726678 
 2.7489644 
 
 3-5«75257 
 4.389976; 
 
 5-3285530 
 6.1145439 
 6-8739555 
 7.6076865 
 8.3166053 
 9.0015^10 
 
 9-6633343 
 10.3027385 
 10.9205203 
 IJ.5174IC9 
 
 I 2. 0941x68 
 12.6513206 
 I3.1896817 
 13.7098374 
 14.2124033 
 14.6979742 
 I 5.167I248 
 15.620410; 
 16.0583676 
 16.4815146 
 
 5.2421369 
 6.0020547 
 6-7327448 
 
 7-4:533'4 
 8.1 1089;; 5 
 
 8.7604763 
 
 9-5850733 
 
 9-9^56473 
 
 [0.5631223 
 
 11.1183868 
 
 5.157872s 
 5.8927C09 
 6.595SS61 
 7.2687905 
 7.9127182 
 
 8.5289:69 
 9. 1 1 85808 
 9.6820524 
 10.2228253 
 ^0.7395457 
 1 1.234c I 51 
 11.7071914 
 12.1599918 
 12.5932936 
 13.0079365 
 
 5.07; 69 21 
 5-7^63734 
 6.4632128 
 7.1078217 
 7-7 2 17349 
 8.3064142 
 8.8632516 
 9-3935750 
 9.8986409 
 10.3796580 
 
 10.8377695 
 11.2740622 
 1 1.6895869 
 12.0853208 
 12.4622105 
 
 16 12.561 1020 
 
 17 13.1661185 
 '8 ^3'7S3Sn^ 
 
 '9 '4-3237991 
 20 14.877474F 
 
 11.6522949 
 12.1656680 
 12.6592961 
 13-13393^5 
 13-5903253 
 14.029158c 
 14.451 1142 
 1 4.8 568405 
 15.2469619 
 15.6220787 
 
 21 15.41 50241 
 
 22 15.9369166 
 
 23 16.4436084 
 2416.9555421 
 25 17-4131477 
 
 13.4047239 
 13.7844248 
 
 '4-1477749 
 14.4954784 
 14.8282089 
 15.14661 15 
 15.45,3028 
 15.7428735 
 16.0218885 
 16.2888885 
 
 12.8211 527 
 13.163002' 
 13.488573. ' 
 13.7986418 
 
 14-0939445 
 
 26 17.8768424 
 
 27 18.3270315 
 
 28 18.7641082 
 
 29 19.1884546 
 so 19.6004413 
 
 31 20.00042 8 ^ 
 3220.3S8765S 
 
 33 20.765791S 
 
 34 21.1318367 
 3r 21.4S722CO 
 
 36 21.8322525 
 
 37 22.1672354 
 
 38 22.4924616 
 
 39 22.80S215] 
 4c 23.1147719 
 
 16.8903523 
 17.2853645 
 17.6670188 
 18.0357670 
 18.3920434 
 
 18.7362758 
 19.0688656 
 19.3902082 
 19.7006842 
 
 20.ccc66i 2 
 
 15.9827678 
 16.3295844 
 16.663061 b 
 16.9837132 
 17.292031 S 
 
 '4-375^853 
 14.6430336 
 14.8981272 
 15.1410735 
 15.3724510 
 
 17.5884921 
 17. 873550c 
 18.1476441 
 18.41 1 1962 
 I 8.66461 I 6 
 
 16.5443909 
 16.7888909 
 17.0228621 
 17.2467580 
 17.4610124 
 
 15.5928104 
 15.8026766 
 16.0025491 
 16.1929039 
 16.3741942 
 
 2o.290493h 
 20.5705254 
 20.8410874 
 21.1024990 
 21-3550723 
 
 .8.9082S03 
 19.1425771 
 19.3678625 
 19.5844831 
 19.7027721 
 
 17.6660406 
 17.8622398 
 I 8.0499902 
 18.2296557 
 18.4015844 
 
 16.5468516 
 16.7112872 
 16.8678926 
 17.0170406 
 17. 1 590867
 
 TABLE VT. 
 The annuity which il. will purchafe for any number of years. 
 
 yrs 
 
 3 pc. C Bt. 
 I .0300000 
 
 3i per Cent. 
 1.0350000 
 
 4 per Cent. 
 1 .04000.00 
 
 4.{- per Cent. 
 
 5 perfent. 
 
 1.0450000 
 
 1.0500000 
 
 2 
 
 .5221)108 
 
 .5264005 
 
 .5301901 
 
 •5339976 
 
 •5378049 
 
 3 
 
 -3535304 
 
 •3569342 
 
 •3603485 
 
 •3637734 
 
 .3672086 
 
 4 
 
 .2690271: .2722511 
 
 .27549^.1 
 
 •'2-l'^l\ll 
 
 .2820118 
 
 5 
 6 
 
 .2I83S46; .2214814 
 
 2.' 46271 
 .1907619 
 
 2277916 
 .1938784 
 
 .2309748 
 .1970175 
 
 .IS45975] .1876682 
 
 7 
 
 .1605064' .1635445 
 
 .1666096 
 
 .1697015 
 
 .1728198 
 
 8 
 
 .1424504! .1454767 
 
 1485279 
 
 .1516097 
 
 .1547218 
 
 9 
 
 .12^4339 
 
 .1314460 
 
 .1344930 
 
 •1375745 
 
 .1406901 
 
 IC 
 
 .1172305 
 
 .1202414 
 
 .1232909 
 
 .263788 
 
 .1295046 
 
 u 
 
 .1080775 
 
 .1110920 
 
 .1141490 
 
 .1 172482 
 
 .1203889 
 
 I2 
 
 .1004621 
 
 .1034840 
 
 .1065522 
 
 .1096662 
 
 .1128254 
 
 13 
 
 .0940295 
 
 .0970616 
 
 .1001437 
 
 .1032754 
 
 .1064558 
 
 H 
 
 .0885263 
 
 .0915707 
 
 .0946690 
 
 .0978203 
 
 .1010240 
 
 IS 
 16 
 
 .0837666 
 .0796109 
 
 .0868251 
 
 .0899411 
 .0858200 
 
 .0931138 
 .0890154 
 
 .0963423 
 .0922699 
 
 .0826848 
 
 17 
 
 •075^525 
 
 .0790431 
 
 .0821985 
 
 .0854176 
 
 .0S86991 
 
 18 
 
 .0727087 
 
 .0758168 
 
 •0789933 
 
 .0822369 
 
 .0855462 
 
 '9 
 
 .0698139 
 
 .0729403 
 
 .0761386 
 
 .0794073 
 
 .0827450 
 
 20 
 21 
 
 .0672157 
 
 .070361 I 
 
 .0735818 
 .0712801 
 
 .0768761 
 
 .0802426 
 
 .0648718 
 
 .0680366 
 
 .0746006 
 
 .0779961 
 
 22 
 
 .0627474 
 
 .0659321 
 
 .0691988 
 
 .0725457 
 
 •0759705 
 
 23 
 
 .0608139 
 
 .0640188 
 
 .0673091 
 
 .0706825 
 
 .0741368 
 
 24 
 
 .0590474 
 
 .0622728 
 
 .0655868 
 
 .0689870 
 
 .0724709 
 
 ^5 
 
 .0574279 
 
 06c 674c 
 
 .0640120 
 
 •0674390 
 
 .0709525 
 
 26 
 
 .0559383 
 
 .0592054 
 
 .0625674 
 
 .0660214 
 
 .0695643 
 
 27 
 
 .0545642 
 
 .0578524 
 
 .0612385 
 
 .0647195 
 
 .0682919 
 
 28 
 
 •0532932 
 
 .0566027 
 
 0600130 
 
 .0635208 
 
 .0671225 
 
 29 
 
 .6521147 
 
 .0554454 
 
 0588799 
 
 .0624146 
 
 .0660455 
 
 3? 
 
 .0510193 
 
 /05437^3 
 
 .0578301 
 
 .G613915 
 
 .0650514 
 
 31 
 
 .0499989 
 
 •0533724 
 
 .0568554 
 
 ,0604435 
 
 .0641321 
 
 32 
 
 70490466 
 
 .0524415 
 
 .0559.^86 
 
 .0595632 
 
 .0632804 
 
 33 
 
 .0481561 
 
 .0515724 
 
 .0551036 
 
 .0587445 
 
 .0624900 
 
 34 
 
 .0473220 
 
 .0507597 
 
 .054314*^ 
 
 .0579819 
 
 .0617554 
 
 35 
 36 
 
 •0465393 
 
 .0499984 
 
 •0535773 
 
 .0572705 
 
 .0610717 
 
 .0458038 
 
 .0492842 
 
 .0528869 
 
 .0566058 
 
 .0604345 
 
 37 
 
 .0451116 
 
 .0486133 
 
 .0522396 
 
 .0559840 
 
 .0558398 
 
 3^ 
 
 .0444593 
 
 .0479821 
 
 .0516319 
 
 .0554017 
 
 .0592842 
 
 39 
 
 0438439 
 
 .0473878 
 
 .0510608 
 
 .0548557 
 
 .0587646 
 
 40 
 
 .0432624 
 
 .0468273 
 
 .0505235 
 
 •O54343J 
 
 .0582782
 
 30 The use of the Decimal Tables. 
 
 The ufe of the preceding Talks, 
 
 TABLE I. AND II. 
 
 Proposition i, 
 
 GI-VC72 the pHncipaly rate, and timey to find the amount. 
 
 Rule. Multiply the amount of il. (in table I or II.) at 
 tlie rale and for the time given, by the principal, and the 
 produd will give the amount. 
 
 Note. If the amount be required for any number of year:- 
 or days exceeding thofe in the tables, divide the given num- 
 ber of years or days into two or more fuch numbers as are in 
 the tables, and multiply the amounts anfwering thereto into 
 one another continually, and the laft produft by the prin- 
 cipal, which will give the amount required. 
 
 Example. Vvhat will 2Col. amount to in 6 years, at j 
 per cent per annum. Compound Intereft. Vide Ex\ I.. Co?n- 
 pound Intereji. 
 
 Agaiaft 6 y. under 5 per cent, ftands 1.3400956 amount of il 
 
 200 principal. 
 
 a681' OS. 4{d. ~ Prod. ^T. 268'Oi9i2oo amountrequired. 
 
 In a fimilar manner the amount of 70I. for 40 days, at 5 
 per cent, will be found to be 7o*37(j277l. per table II. And 
 the amount of 82-51. for 75 years, at 5 per cent, will be 
 found, by the note, to be 3203*69661. 
 
 Observation. Anfwcrs to qnellions in any of the other 
 three propofitions in Compound Intereft, may be eafily ob- 
 tainted by "a little attention to the preceding Rule and Ex- 
 amples. 
 
 TABLE III. 
 
 Proposition. 
 Any fum of money , due fame time hence , being given, to find 
 its prejent Kjalne to the creditor y difcounting at any rate per tent. 
 Compound Interefi, 
 
 Rule. Find the prefent worth of il. at the rate and for 
 the time given (by tabic III.) which multiply by the debt, 
 aad the piodud will be the prefent worth required.
 
 The use of the Decimal Tables, 31 
 
 Example. What ready money will difcharge a debt of 
 243I. 2^*55. due 4 years hence, difcounting at 5 per cent. 
 Compound Intereit ? f^'ide Ex. i. Difcuunt at Com^mni 
 hitereji, 
 
 Agaiuft 4y. under 5 percent, (lands •82270^5 prefent worth Oi' il. 
 243!. 2^V' = 243 '1012 5 the debt. 
 
 Produft £,- 2CO the prefent worth. 
 
 TABLE IV. 
 
 Proposition i. 
 
 Cii'eu the annuity ^ the rale per cent, and time, to find the 
 amount. 
 
 Rule. Find the amount of il. per annum at the rate and 
 for the time given (by table IV.) which multiply by the 
 annuity, and the produdt will be the amount required. 
 
 Example. What is the amount of an annuity of 80I. to 
 continue 9 years, at 5 per cent, per annum, Compouatl Ivk'* 
 tereft ? Fide Ex, 2. Annuities in An cars, 
 
 Againft 9 y. under 5 per cent, ftands ii'Qz6^(]^7, amour.t oi' il. 
 
 So annuity, 
 
 S82I. 2,s. 6d. — ProJud /. S82T251440 amount. 
 
 Observation-. Anfwcrs to queftions in any of the other 
 3 Vropofitiom in Annuities in Arrears may eaiily be obtained, 
 by paying a little attention to the above Rule and Example, 
 
 TABLE V. 
 Proposition i. 
 
 To^nd the frrfi:ni luorth of an antmity at Compound Interejf, 
 tee lime of its con'tinuance and the rate per ant, beijig 
 
 given. 
 
 RuLK. Find the prcfent worth of il. at the 
 far the time given (by table V.), which multiply by 
 
 rats and 
 ^ the an- 
 
 p:i 
 
 Y
 
 52 The use of ihe Decimal Tables. 
 
 Example. Required the prerent worth of an annuity of 
 Sol. to continue 9 years, at 5 per cent, per annum. Com- 
 pound Intered, Vide Ex. z. Pre/ent Worth 0/ Annuities. 
 
 Againft 9y. under § per cent, ftands 7 '10782 17 prcfent worth of il. 
 
 8.0 annuity. 
 
 568I. J2S, 6d. n Produft ;^. 568-6257360 the prefent worth. 
 
 Proposition 1, 
 
 ^ofind the frefent tvorth of mi Aumiity in Reaver/ion at Com" 
 Poiind Initreji, 
 
 Rule. Find the prefent worth of il. per annum, at the 
 given rate, for the time being, and alfo for that time and 
 the time in reverfion added together (by table V.). Sub- 
 tra<rt the lefs value from the greater, and multiply the re- 
 mainder by the annuity, the produd will anfwer the queftion. 
 
 Example. What ought a perfon to pay in ready money 
 for the re'verfion of lOOcT. a year, to continue 20 years on a 
 leafc which cannot commence till the expiration of ^ years, 
 allowing the purchafer Compound Intereft at 5 per cent, ? 
 t^ide Ex. 3. Annuities in Re-cerfo72. 
 
 The prefent worth of il. for 20+5 i:; 25 yrs. 7 ^ 
 
 at 5 per cent, is — — \ '4'0959443 
 
 Againft 5 yrs. under 5 per cent, ftands 4-3294767 
 
 Diff. 9-7644678, which X 
 
 jooo, produces 9764'4678I. zr 9764I. 9s. a^z\. Anfwer. 
 
 TABLE VI. 
 
 Proposition. 
 
 ^o find tuhat annuity y ta continue a ginjen number of years , 
 4i^y fum of money nvill furchafey at a certain rate per cent. 
 
 Rule. Find the annuity which il. will purchafe at the 
 rate and for the time given (by table VI.}, which mul- 
 tiply by the purchafe -money, the product will be the an- 
 nuity required.
 
 The use of the Decimal Table*. ^^ 
 
 ExAMPLF. What annuity to continue 9 years will 
 568*62 5736I. purchafe, allowing the purchafer 5 per cent, 
 per annum. Compound Interei^, for his money ? See Ex. 8. 
 Vrefent Worth 0/ Annuities, 
 
 Againft 9 yrs. under 5 per cent, ftand?; •1406901, the annuity which 
 i!, will purchafcj this x 56^'6a£736, giveb Sol. the annuity required. 
 
 Observation. The above Rules and Examples arefut*- 
 ficient to elucidate the ufe of the tables. 
 
 r t
 
 GENERAL AND UNIVERSAL 
 
 DEMONSTRATIONS 
 
 or THE 
 
 PRINCIPAL RULES 
 
 I N T H E 
 
 COMPLETE PRACTICAL 
 
 ARITHMETICIAN. 
 
 I. Demonji ration cf the method of proving Addition y hj cajiing 
 out the 7iines, 
 
 THIS method of proof depends upon this theorem, 
 ** any number divided by 9 will i^r.-.e the fame re- 
 mainder as the fam of its digirs divided by 9." Let N be 
 the number, a, by c, dy the digits which compofe it, then by 
 the nature of notation loooa -f loob ■\- 10c + d -zz N. But 
 
 I Geo y^azz. 999 ■{- 1 Xa-=za-\- a X 999 ; 1 00 X ^ = 99 -|- 1 
 
 X^ = ^+^X99; ioX<:zi9+i Xt=r -\- cy.g\ \' icoo« 
 
 -f- ioob-^ioc-i-d= gg^a -f-^ 4- 99^ + <^ + 9^+ ^ + dizN, 
 
 N icooa-\-ioob-\-ioc^d ggga ~{-ggb -i- gc 
 
 and — zz n: + 
 
 9 9 9 
 
 '■ ; but 999^ -f 99 5 4- 9 f is evidently divifible 
 
 9 
 
 1 ^^' -t, . 1 r -4 a-^b-^c-^d 
 
 by q. •.•—will leave the fame remainder as 
 
 - ^ 9 9 
 
 Now the excefs of nines in two or more numbers being 
 feparately taken, and the excefs of nines alfo taken out of the 
 fum of the former exceffes, it is evident, that the lait excel, 
 muft be equal to the excefs of nines contained in tlis tot.i'. 
 fam cf all thefe, the fum of the parts being equal to the 
 whole. ^. K, D,
 
 Demonstration of the Rule or Three. 3^ 
 
 II, Bemonjiration of the method of pryving Multiplication ^ by 
 cafting out the nines, 
 
 ** If any two numbers are feparately divided by 9, and 
 the two remainders multiplied together, and that produ(fl 
 divided by 9, the laft remainder will be the fame as if you 
 divide the produft of the two firll numbers by 9." 
 
 l^tti^A-^-a and <^B-\-b be the two numbers, a and b Hr^t 
 
 remainders. Their produft is ^y.(^A-\- <^Ba-\-<^Ab -^ ab, 
 but the three firft terms are evidently divifible by 9. •.• there 
 is no remainder but what is had by dividing; a X b by q. 
 
 Wha-t has been demonftrated refpefting the number 9, 
 holds good with refpe*^ to the number 3. A little attention 
 to the preceding demonftrations will render the method of 
 proving Divifion, Square Root, &c. obvious. 
 
 Note. The rule of Addition, v/hich is founded on this 
 fimple axiom, " That the whole is equal to the fum of its 
 parts ;" and the rule of Subtradion,. which may be deduced 
 from the fame axiom, feerti to require no demonltration. — 
 The rules of Multiplication and Divifion are likewife bell 
 explained by examples, the former by Addition and the latter: 
 by Subtraction ; the Compound Rules and Redudion carry 
 their rationale along with them,. 
 
 II I. Demotifi} ation of the Rule of Three, 
 
 Deftnition. Four numbers are faid to be proportional,, 
 when the firil contains the fecond,, as often as the third con- 
 tains the fourth.. 
 
 Viz. If^ = ?-, then J'.B;: C:D,orB:J:: D:C,, 
 which is the definition I have given of proportion. 
 
 Theorem. If fbur numbers are proportional ; A i B : : 
 C : D, the produd of the extremes is equal to the product 
 of the means, viz. A x D zz B x C. 
 
 A C 
 Demonstration. Let — = -^ = r, then A zz Br, 
 B JJ 
 
 and C=zDr, htncc AD=zBrD, and^C = BrD. •.• AD 
 
 — BC, i^ E. D, Hence we deduce a demonftration of the 
 
 Rule of Three, 
 
 Y3
 
 36 DEMO^'STRATION OF THE RuLE OF FiVE, ScC. 
 
 If J : B : : C : D. 
 
 Then .^ X D = B X C. No;v ./ .....1 i..n.';... 
 be divided by equal numbers, the quotients will be equal. 
 
 B X C 
 ',' D =z 7—, Nvhich is the rule, i^ E. D. 
 
 CoT.D= ~xC = ^xB=zC-^^--B^^. See 
 
 the notes. 
 
 Note. The Rule of Three Inverfe may be made a Rule 
 of Three Dired, by making the third term the firil, and by 
 proceeding forv.ard to the other two terms ; therefore the 
 above demcnftration will ferve for both rules. A fmall 
 attention to the notes annexed to the Inverfe Rule will render 
 it very plain. 
 
 IV. T>emoTijiratiott cf the Rule of Five, ^c. 
 
 All queftions that come under this rule may be fohed by 
 two ftatings in the fmgle Rule of Three, either both dired, 
 or one diretft, and the other inverfe. The reafon of this 
 rule will therefore appear obvious from the firll, fecond, 
 and third example. 
 
 I. Hen- the blank falls under the third term. 
 
 If 7m. :izd.^i.6a. ^ _,6X3XIZ6 
 
 7 X la 
 
 Ey fwo <Iire£^ ftatings, 
 
 .r * ^ c 126X16 
 
 If 7 m. * : 126 a. : : i5 m. : a. 
 
 7 
 126x16 16X3X12G 
 
 If 12 d. ♦ : a. : : 3d.: . • 
 
 7 7X12 
 
 By one dired, the other Inverfe. 
 
 If 7 m. : 12 d. : : i5 m. * : d. 
 
 16 
 
 ,r 7X12 4 , , 16X126X3 
 
 If -— d. ♦ ; iz6 a. : : 3d.: -» 
 
 j6 ^ 7X12 
 
 II. Here the blank falls under the frji term. 
 If 7 m. : 12 d. : 126 3. 
 
 3 
 
 : 72 I 3X"6i
 
 General principles and theory of Fractions. 37 
 
 By two dired^ flatings. 
 
 If 12 0.* : 126a. :: 3d.: a. 
 
 12 
 
 , i26x^* 7X12X72 
 
 Ir —a. : 7 m. : : 72 a. : -— . 
 
 12 3X126 
 
 By one diie<ft, the other inverfe* 
 
 1-2 y 7 
 
 If 12 d. : 7 m. : : 3d. * : m. 
 
 3 
 
 T- C * 7XT2 , 7X12X72 
 
 Ir 126 a. * : m. '. : 72 a. : m» 
 
 3 3X126 
 
 III. Here the blank falls wider the fecottd term* 
 7XI2X7S 
 
 If 7 m. : 12 d. : 126a 
 
 XI 
 
 16 : * : 72 
 
 16x126 
 
 By t\vo dire ft ftat'mgs. 
 
 Tr * /: ^ 16x126 
 
 If 7 m. * : 126 a. : : 16 m. *. a. 
 
 7 
 
 7X12X72- 
 
 12 m. : : 72 a. : — -r-in* 
 
 16x126 
 
 By one direct, the ot'uer inverfe. 
 
 Tf /: * 7X72 
 
 If 126 a.* : 7 m. : : 72 a. : r- m, 
 
 126 
 
 .. lY.^'i- , . ^ 7X12X72 
 
 If — -m. : 12 d. : : 16 m. * : — -• m» 
 
 126 16X126 
 
 Agreeable to the ruk, and whatever the qucftion may be, 
 it is evident there will be the fame reafon, for the blank muft 
 fall under either the firll, fecond, or third term : therefore 
 the rule is true, ^ E, D. The univerfal rule of proportion 
 is demonftrated from a fimilar manner of reafoning ; for the 
 firft example, which ccnfifts of 13 terms to find a I4.th, may 
 b(f eafily reduced to 5 terms to find a 6th. 
 
 V. General principles and theory of FraSIionf, 
 
 Proposition i. 
 
 The reafon of the rule to this propofition , may be clearly 
 fhewn from the firft example: thus, 24. meafures 192 fince 
 o remains i it alfo meafures 216, which is feme multiple of
 
 38 General principles and theory of Fractions. 
 
 192 with 24 over. For 192 X r -f 24 rr 216 ; for the fame 
 leafon it meafures 408, which is fome multiple of 216 with 
 
 192 over. For 216 X i + 192 =408, &c. v» 24.meafures 
 both 216 and 408. 
 
 It is the greateft common meafure ; for, if there were a 
 greater, then fince the greater is fuppofcd to meafure 216 
 and 408, it will alfo meafure 192 and 216; and fince it 
 meafures 192 and 216 by fuppofition, it ought alfo to m.ea- 
 fure 54 and 192, which would be abfurd, for a greater num- 
 ber cannot be contained in a lefs. Whatever numbers are 
 propofed, the manner of reafoning will be the fame, •.• the 
 rule is true,. ^ E.. D. 
 
 Cor-. Fknce if there are more numbers than two, we can 
 find the greateft common meafure ot two of the given num- 
 bers, then of that common meafure and a third number, &c. 
 
 Thus the greateft common meafure of 72, 60, and 28, is 4. 
 
 Proposition 2. 
 
 The rule to this propofition is founded on this axiom, 
 •* That dividing both the terms of any fraction by any num- 
 ber whatever, will give another fraftion of equal value with 
 the former." 
 
 Proposition 3, 4, and j, are felf-cvident. 
 Proposition 6» 
 
 r. 
 
 Let -^ reprefent a complex fra^ion in its moft fimple 
 
 'd 
 florm, to which form every complex fraftion muft be reduced 
 before it can be folved by this rule. Now every fraftion 
 denotes a divifion of the numerator by the denominator, and 
 its value is equal to the quotient obtained by fuch a divifion 3 
 
 n 
 
 '~^ n , N »D ^ ^ ^ 
 
 hence ^=;j -^F = Nd'^'-'^' 
 
 D
 
 General fp^inciples and theoryof Fractions. 59 
 
 Proposition 7, 
 That a compound fradion may be reprefented by a fimplc 
 one is evkfcnt, fince a part of a part muil be equal to fome 
 part of the whole. The truth of the rule may be iliewn thus. 
 
 a f. b , \ . b h 
 
 Let the compound fraclion be— of — , tnen — ot — _ — - 
 
 ^ n m n m in 
 
 b . r , ^ r ^ ^ ^ ^ 
 '^nzn — , andconfequently— ot — — — x «= S 
 
 if the compound fraftion confifls of more numbers than two \ 
 wc may firll reduce two of them to a fimple fradion, then that 
 one and a third, &c. ^. £. D, 
 
 Cor. 1. \iN~A-\-B-\-Cy$iZ, then— of A^ZZ— oi A ■\- ' 
 
 iof£+ iofC, &c. Ex.20— ii+8» s* J; ofac = iof li 
 
 + i ot i> zi 5. 
 
 Cor. 2. —Qi A — A X — of u Ex. | of 2=1 of iriaXiot i- 
 n n " 
 
 Cor. 3. — of » n -5 of I. Ex. \ of il. — | of il. 
 A A 
 
 Cor. 4. — of/?: zi — of ^. Ex. 4- of 2 zr ^ of 3. 
 n n ^ 
 
 CoR. 5. — of b zn — of axb. Ex. I of 5 zz |of 10. 
 n n 
 
 Proposition 8. 
 The numerator and denominator of each fraftion are equally 
 multiplied, viz. by the denominators of all the other frac- 
 tions, confequenily the fradions produced are equivalent. 
 Thus in the firll example, 
 
 ^7X11—303' 7 4Xii'~3oS'ii 7x4 3^^' 
 The fecond rule under this propofition is equally obvious, 
 
 for -^of 5o8r2-of3oSx3 = -X ^t!' ^'- ^^' ^' 
 4 4 - 4 7 >vii 
 
 Note. The following rules in vulgar I'raaions will appear 
 fufficiciilly plain, from an attentive obferNancc of the nature
 
 40 The THEORY of Circulating Decimals. 
 
 of the examples. — Tfee truth of the rules for working finite- 
 decimals will llkewile appear more clearly from the defini- 
 tions, and the nature of the operations, than they can be 
 rendered by a multiplicity of words. 
 
 \I. Demonjirathns of the rules of Circulating Decimals » 
 
 D E F I N I T I o K I. 
 
 ** It is afferted that if the denominator of a vulgar 
 fradion, in its loweft terms, is not compounded of 2 or 5^ 
 the decimal produced from fuch. a vulgar £ra<ftion will be 
 infinite*" 
 
 K 
 Demonstration, Let— be the given vulgar fraclion,. 
 
 tken fince D has not in its compofiticn r or ^, nor any mul- 
 tiple thereof, there may be found a certain number of 9'$ 
 which D win mdafure ; for by dividing 1000, &c. by any 
 prime number whatever, except 2 or 5, or fome multiple 
 thereof, the figures in the quotient will begin to repeat 
 ©?er again as foon as the remainder is iv And fmce 999,- 
 &c. is lefs than icoo by r. •.• 99Q,<i-c. divided by any prime 
 number whatever excepting thofe above, will leave no re- 
 mainder when the repeating figures are at their period.. 
 
 Suppofe the number of 9*8 which D will meafure to be 
 
 reprefented by 99, &c. take D : 99, Sec, : '. N : R, then 
 fince D meafures 99, Sec, N muft meafure R. 
 
 T, QQ, Sec. R „ . . . . TX IT 
 
 For^— — = — , •.• i^is ao integer. Again, D : N :: 
 
 CO, Sec. : R and — =1 , but — - — is evidentljr 
 
 D QC),^c. 99, &c. 
 
 a pure circulate, the repetend of which begins immediately 
 
 N 
 after the decimal point, •.* yr is a repetend, ^ E. D^ 
 
 Proposition i^ 
 
 Every pure circulate is, from the nature of a decimal 
 iiradion, a feries of decreafmg fra<f\ions with equal numera- 
 tors, and their denominators a geometrical feries increafing 
 in a tenfold proportion. Let R reprefent the repetend or 
 
 aiimerator of the fraftion, 1 o" = r the ratio,.
 
 The theory of Circulating Decimals. 41 
 
 R R R R 
 Then will — •{- —; A — T H 7i ^c. ad infinitum, reprefent the 
 
 decimal, and the fum of this feries by the laws of progreiTion ^ 
 
 ^ . ^ ^ ^ „ fi,^ which is the rule. 
 
 R^ ^R R L. ^ __ i? _ -R 
 
 — ^_j — 10"— I ~9'^'<= 
 
 ^ £. D. 
 
 Proposition 2. 
 Let /^ reprefent the numerator of the finite part 10^, its 
 
 o 
 
 denominator, then will :^ — be the vahie of the pure 
 
 g9,cvc. ^ 
 
 fcpetend by the laft propofition, ••• 
 
 lo" 99,<!s;c.xio« 9;,&c.xio" 99, &;c. x io» 
 
 Fx ^00, Sec. + R~"F ^ ^ _. ...... 
 
 rr the traction reprefenting the value ot any 
 
 99, &c. X lo' "^ " -^ 
 
 mixed rcpetend, which is the rule exa(5lly. ^ E, D, 
 
 The notes which follow are corollaries, deducible from the 
 preceding dcmonftrations. 
 
 Proposition 5. 
 
 N 
 Let "Tj l^e a circulate, D having fome other prime in it 
 
 than 2 or ^, then D may be reprefented by Ay.i", or J X 
 5'", otJxz" X 5'", &c. where ^ is a number which has no 
 power of JO, 2, or 5 in it. 
 
 „ iV 7Y iV I ^ r 
 
 Hence — = zr -r X ; but is 
 
 D .:^X2"X5'" ^ 2"X5'» 2"X5'" 
 
 a fraftion, haying no other prime than 2 or 5 in its denomi- 
 nator, and \- is reducible into a finite decimal which has 
 2" X 5'", or fonie power of 10 for its denominator; fuppofe 
 
 10 ; then -;; :;; may be expreffed by •
 
 42 Demonstration of the common rule of 
 
 Hence^=jX — =-^-,o ; Now — 
 
 may be an improper fradion, but it cannot be an integer ; 
 
 Na r ,N 
 
 for if it were, then — p -^ i o = — mufl either be an 
 
 integer or a finite decimal, contrary to the fuppofition : 
 
 Na 
 ••• if —r- be a proper fradion, the decimal equivalent will 
 
 be a pure circulate (prop. i.)» and if it be an improper 
 fraftion, it will refolve itfelf into a mixed decimal, the finite 
 part of which will be a whole number. Now if this decimal 
 quotient be divided by lo'', it is plain the figures will not 
 be changed, the decimal point will only be removed as 
 many places to the left hand as there are units in lo^, there- 
 fore the repetend begins after fo many places as are exprelTed 
 by ?•, and hence the whole is evident. ^ E, D, 
 
 Whoever underftands the preceding demonftrations, which 
 are the foundations on which the other rules for managing 
 circulating decimals arc built, will readily fee the reafon 
 of the feveral operations. — The rules of Pradi^ce, Tare and 
 'i'ret, Intercft, Brokerage, cSrc. are felf- evident. 
 
 VII. Demonftration cf lie common Rule cf Equalkn of 
 Pnjmtnts, 
 
 I. 0]i Cocker's principle^ 'vide note id to the rule, 
 
 J (■ .^ 1= a debt due a years hence, 1 and .v for the 
 ^ I -5 zz a debt due h years hence, ^ic. J equated time, 
 
 Kovv .V muft neceffarily fall between a and h, hence .v — 
 a zz time A is at inierelt, and h — x ■=■ time B is at in- 
 
 tereft : alfo x — a X Ar =. intcreft oi A fcr its time, and 
 
 X yi B r =^ intereil of ^ for its time. •.* x — a x A . 
 
 =z b — X X Brhy the fuppofition, oi Arx '\- B r x = Ara 
 
 r. , , Aa^ Bb ^. . . , 
 
 •\- Brb^ whence .v = -— which is the rule, and the 
 
 A -^ B 
 
 faTiC may be fliewn fcr any number of pajinenrs. j^. E, D,
 
 E c^u A T I o N OF Payments. ^j 
 
 II. 0/i Hatton's principle y <vide note ^d to the rule, 
 j^ J ^y4 = a. debt due a years hence, 7 and x = the 
 I B zz: a debt due 6 years hence, J equated time. 
 
 Then J -^-B x fx = interefl: of the fum of the debts from 
 tlie time of thcquellion to the equated time ; Jra:=: interefl: 
 of ^ for the time<7, and Brb =■ interefl of/? for the time b. 
 
 \' A-^-BXrx-zzAra-^rBrh by the fuppofition, or 
 
 Arx •{• Brx :z Ara -{-Brb as above, hence x :z: — — - 
 
 A -{- B 
 
 which is the rule, and the Tarns may be fhevvn for any number 
 of payments. ^ E. D, 
 
 III. G// Moreland's or Burrows' principle, 'vide note \th. 
 Let 
 
 A 1:1 2i debt due a years hence T 
 
 B zz a debt due b years hence ( ^ — ^^^ 
 
 C zz 2i debt due c years hence f equated 
 
 D rr a debt due ^ years hence, &c. J ^^™^' 
 
 d — ^ rz time ^3^ is at interefl:, 
 
 d — b 2Z time B is at interefl. 
 
 d — c zz time C is at interefl:. 
 
 ^ — d zz time D is at inierell. 
 
 A -\- A X d — a X r = amount of^ for its time, 
 B -{- B X d—b X r = amount of 5 for its time. 
 
 C -{- C X d — c X r zz amount of C for its time. 
 
 D -{- D X d — d y. r zz. amount of Z> for its time. 
 Now the fum of the amounts according to the fuppofitiou 
 muft be J 4- J X d — x X r, putting s {or A -\- B -{- C -{- D, 
 
 A X d—a X r -Y B X. d-^b x r + C X d—c X r -f 
 
 Z) X d — d X r = s X d — X X r, by di\'iding by ;• and re- 
 ducing the equation, we obtain A d — Aa-\-B d — B b -\- C d 
 
 ^Cc-{-Dd—Ddzzsd-^sx, then A+Bi-C-^-D^d 
 ^Aa-^Bb-^Cc^Ddzz sd-^s,\\ 
 
 Z
 
 44 Demonstration of the 
 
 Hence x = =: - — - — - — -- — -- — 
 
 ■ s - A'+B + C + D 
 
 tv-hich is the rule. ^ E. D. 
 
 Hence the rule is univerfally true according to any of the 
 fuppofiiions, anc^asthefe fcveial pi-inciples produce the fame 
 final fimple equation : it there- ore follows that they are ma- 
 nifeft confequences of each other, as I have aiferted at note 
 5th to the rule, 
 
 VIII. Demon/? rat ion of the Rules of FelloivJJjip, 
 
 BY MR. V/. ADAMSON, TFACHLR OF THE MATHEMATICS, 
 AT BURLINGTON, YORKSHIRE. 
 
 I. Shigle FellQnx>Jhip, 
 
 Merchants. Stocks. Gains 
 
 A o 
 
 B h y > Notation. 
 
 C c z. ) 
 
 Ey the general definition of Fellowlhip it muft be 
 As i7 : A- : : ^ : _>• : : t' .: z, hence by the dod\rine -of 
 proportion we have a-\-b\c : a 4-j-f 2 : : a \ x, ^E, D. 
 
 II. Double Fellonjc/bip. 
 
 chants " 
 
 I. n 
 
 > Notation. 
 
 Admit u4 
 
 . B . 
 
 c. 
 
 Sec. 
 
 merchants 1 
 
 Put in a 
 
 . b . 
 
 ey 
 
 &c. 
 
 ftoclis 1 
 
 For i 
 
 • P ' 
 
 9^ 
 
 &c. 
 
 months j 
 
 And gain x 
 
 • y 
 
 Zy 
 
 Sec. 
 
 pounds J 
 
 Now ;f ~ J's gain in the time /; and by Single Fel- 
 
 lowfl:iip, 
 
 bx _, .... 
 
 zz B s gain in the time /. 
 
 a 
 
 ex 
 a 
 
 •zz C's gain in the time /. 
 
 — '.: p : -^ = J — -^'s gain or lofs in his time/. 
 a ta 
 
 — '.'. q : -^-~ :zi z zz C's gain or lofs in his time y. 
 
 ia
 
 Rules of Fellowship. 
 
 45 
 
 Hence we (hall have their refpedive gain expreficd three 
 ways. 
 
 I. II. III. 
 
 A. x—x 
 
 B, y-x X — 
 
 •^ at 
 
 C K-=zx X ~ 
 
 tp 
 
 x—y X 
 
 c q 
 
 x—z X 
 
 ^—2; X 
 
 qc 
 
 a t 
 
 Let G = the whole gain, then wtf' fhall have the three 
 following equations. 
 
 bp 
 
 I. X ■\- x X— +XX -^ zzG-, or at + bp -^c-^ XX ::z:aiG. 
 at at 
 
 II. yx'r -fv+J x\^ — G, or at -i-op-^ca Xy=.bpG. 
 hp ' op 
 
 III. XX-- ■\-7^-k~-\-z — G)OVat-\-vp-^cqX^—cqG, 
 cq cq 
 
 Now let. the fum of thefe produds be put = s, and wc 
 
 *^ sx = atG '.' s : G :: ai : X ) 
 
 sy = bpG -,' s ', G -.'. bp ', y >^ E. D. 
 jz :ii c gG •.' f ; G :: C7 : z} 
 
 Proposition i. Note 4. 
 
 By the rule <)f Double Fellowlhip we have the following 
 proportions {x being given). 
 
 at + [>/> : X -^y :: at : -v 7 j_j^^^g 
 a t -^ c q : X -{- z, : : a t : X I 
 
 bpx •=! aty '.' at '. X : : bp :y 1 ^ j^^ ^^ 
 cqx zz atz. '.• at '. x :: cq : ~J 
 
 Proposition 7. Note ^. 
 ^ut .v-f-j-f 2 rz G the whole gain, and at^b p -^-c q-= m 
 the fum of the pr jdufts of each man's itock and time, per 
 Double Fellowlhip. 
 
 m : G 
 m : G 
 m : G 
 
 : at : x'i 
 
 : bp '. y^ 
 
 : c a \ ^\ 
 
 Hence ^, = - — ^ * ^"^^ 
 G X y ^ 
 
 from thefe equations we deduce the following. 
 
 Z a
 
 46 
 
 Demonstration of the 
 
 I. 
 
 A' a :n a 
 
 C. i — aX — 
 
 II. 
 
 a — by, 
 b- b 
 
 c-b X 
 
 
 9y 
 
 III. 
 
 qx 
 a-=.c X — 
 
 t s 
 
 Now putting s for the fum of the ftocks, we have 
 
 I. o-^uX — -^<iX — — 
 
 p^ 
 
 II. b X^- -^-b+bX^- - s. 
 
 1^ 
 
 'iy 
 
 Hj 
 
 III. cx'—\cx^^^rc~ 
 
 :3 px. 
 
 P i .V + ^ ?>+'f ~X:i IT / J .V i. 
 
 fiX^--tqy\tpz.Xo — t<iy 5. 
 
 ••• pqx Ar tqy Jr tpz, : t 
 ' ■ pq X -^ rqy + tp^ : s 
 
 ••• p3--< + r^y -i- 'f^ •• i 
 
 p q>:-\-tqy-\-lp^X C ZZ t p % t. 
 P 
 
 p qx '. al 
 
 t qy : b L'^ £. D. 
 
 Proposition 3. Note 6. '^ 
 
 By prop. I. f/e have hpx zz c/r, and cjx zzatz* Now 
 let a l'€ given. TliCn 
 
 A- : ^ / ; : y ; hp, and — = ^ 
 ;«■ ;<?/:: s : f<7, and — =z c 
 
 ^^./>. 
 
 Proposition 4. Note 7. 
 
 By prop. 2. we have — = -^^ = - , and from thefe e- 
 ;c J a 
 
 quations we deduce the following. 
 
 II. 
 
 I. 
 
 A. t — t 
 
 ^ ^y 
 
 C. q — t X — 
 
 bx 
 t—p X — 
 
 ay 
 
 P -P 
 
 b^ 
 qZZp X — 
 
 cy 
 
 III. 
 
 q X 
 
 'y 
 ? = ?
 
 k tf LE S O F L OS S A N D G A I N. 4^ 
 
 Xow by putting j for the fum of the times, we fhall have 
 
 -ay , z- 
 
 I. /+ X 7^ + r X - =15 
 
 II. p X — 4-/>4-/X — = 5. 
 
 ay cy 
 
 III y X — H ?x 7^3 + ? — i. 
 
 i^TA-rf 'it-y-j-ii ^ <i.X< 
 
 6 cx-\-a cy-\ub z XpZHa c y . 
 
 cy : 
 
 Proposition ^. Note 8. 
 
 •. • hex •\- a c y + «''''« : ^ •• : ^ f x : t') 
 
 '.' I c X ■\- a c y -f- a b z. : j : : rt c >• '. p >^ E. D. 
 
 ••• Zt^ + <2f_y + <^bx, : i :: abz, : qy 
 
 By prop. I. we have bpx rr city, and c^x = /7/^; 
 multiply the firft of thefe equations by c, ami the fecond by 
 b, and we (hail have cb^x rr c atjy and c bqx ::=. batj,^ 
 let / be given. 
 
 Then b c x : t : : c cy : p} ^^ p ^ 
 And ^ r .V : / : : ^ rt 2 : 5^ f "^* 
 
 The method of demonftration made ufe of above is very 
 general, and will be the fame let the number of partners be 
 ever fo many. 
 
 IX, Demoyiftration of the Rules of Lof and Gaiiu 
 
 Proposition i. 
 This rule is felf-evident. 
 
 Proposition 2. 
 
 p.' r Prrprime coft ) of an f to find G tliegain, 
 \ S — felhng price ) integer, [ L the lo's per c^n 
 
 or 
 nt. 
 
 r : d : : loo : — - — or f : loo : 
 
 I GO S 
 
 S : -— l,y 
 
 the 
 
 rule, and agreeable to reafon. — Mow if 6" be greater than F, 
 
 ioo5 .„.,,, , „ 100.9 
 
 — —- will evidently be greater than lOO. Hence — — 
 
 100 n G, for — - — is the amount of loolr 
 Z 3
 
 4$ Demonstration of the 
 
 Again, if 3" be lefs than P, then will — -- be lefs tha» 
 100. ••• ICO — — p— = ^ t"^ ^0" per cent. 
 
 Proposition 3. 
 
 ^. J P=primecoft of an integer 7 to find 5 the 
 
 I G=thegain,orZ = the lofs per cent. ) fellingprice. 
 
 It has been already (hewn that — p- — 100 1= G, and 
 
 100 — — ^ zi L, Hence 100 X S = P x G ■■\- 1 00, 
 
 nd 100 XS zz P X 100 — L. 
 
 ••• I CO : 1 00 -f G : : P : .S if gain, and 1 00 : i co — Z. 
 ; P : ^ if lofs. ^E,D. 
 
 Proposition 4. 
 
 or /the 
 given fei- 
 
 ^. V 5 rrfeUmg price of an integer 7 f"/"'^ ^ '^'.^'»'"! 
 
 Given -^ ^ — , • T ►u I r ^ * ?■ 'ofs at any other gi 
 
 |G:=thegaiii,Lir thelofspercent. ^,5^^ p^. J^^ & 
 
 From the equations in the preceding propofition, we have 100 4- G 
 
 looS ... 
 
 ! 100 : : 5 : P ZI if gain, or 100 — L : JOO : : 5 : P 
 
 lOO+G 
 
 "=°^ if loft. 
 
 100 — 1/ 
 
 Nov;, by propofition 2. we have 
 
 ^ lOQp , ioo/> 
 
 P : 100 \ '. p '. — — i i then — — i- — 100 zr ^, and loo — 
 
 — •' n /, in thefe equations for P, fubftitute its values ■ — -, 
 
 ioo5 ico-i-Gx/' 
 
 , -. Then w.U — 100 zz r, and 100 — 
 
 and 100 — L S ^ 
 
 zz 1} or joop 4- Gp — 100 5 4- Sgi and ico* — i/> 
 
 o 
 
 ICO S — SI. 
 
 •.• 5 : ICO 4. G : : /> : 100 + ? if gain, and 5 : 100 — • L : : /> : 
 
 JOO— /if lofs. i^£. £>.
 
 Rules of Lossand Gain. 49 
 
 Proposition ^, 
 
 ^. 5" S =the feUing price of an integer 7 ^^ ^"^ ^}'' wjiole grn 
 
 ^■'''"j G zzche gain, or L = thelofs per cent. \°' ^^^' by the fale of 
 
 (, 3 ^"y qi^antity ^ 
 
 By the firft dating in the l?.ft: propofitioa we have 
 lOO ^- G : lOO : : 5 : P the prime coil of an integer, if gnin, 
 loo — L t loo : : S : P , if iofs. 
 
 Now if S and P be equally Increafed hj any quantity P, 
 the proportion will evidently continue the Cun.c, 
 
 . . J TOO + G : loo : : 5 X ^: Z' X ^if gain ) ^ _ 
 • J loo — Z, : loo : : 6" X i^: /- X i^if Iofs | ^^"^ '^ 
 ^ ^is the whole value at which the goods were fold, and 
 P X i^is the whole prime cod.— ^Hence the difference l)e- 
 rveen S x i^and P X ^ mull be the whole eain or Iofs. 
 9. E. D, 
 
 Proposition 6. 
 
 ?^= che quantity of goods bought 1 to find the 
 P — the prime coil of an integer > whole gain 
 
 ^ G n the gain, or Z-zz the Iofs per cent. J or Iofs. 
 Here P X i^ =: the whole prime coft of the goods. 
 
 From the lail propofition we have 
 
 loo : lOO + G :: Px^: 5 X ^ if gain. 
 ICO : loo — I :: P X^: 5 X ^ if Iofs. 
 
 Hence the difference between S % ^ and P x ^ = the 
 whole gain or Iofs. i^ E. D. 
 
 The rules of Barter and Exchange will appear plain by 
 obferving the nature of the feveral Examples. — The rule of 
 Compound Arbitration of {'Exchange may feem to want de- 
 monltration; but whoever takes the trouble to folve any 
 one of the examples by two, or more flatings, in the fingle 
 Rule of Three, in the fame manner as I have proved the 
 Rule of Five bv ingle flaiings, will immediately fee the 
 truth and nature of the rule. 
 
 Any perfon who can extra*?l the fquare and cube-root in 
 Algebra, will not be at a Iofs to demonftrate the rules of 
 fquare and cube- root J and to thofe who cannot, a demon- 
 
 ilratioQ
 
 50 
 
 Demonstration or the 
 
 flration would be of little or no ufe. It may not be amifs 
 to remind the reader, that the 2d rule for extrafting the 
 cube-root is deduced from Mr. Ward's Mathematician's 
 Guide, page 151. Sch edition. 
 
 X. Demonjiration of the Rules of Alligation. 
 
 PROPOSITION I, 
 
 This rule needs no demoi\ftration. 
 Proposition 2. 
 
 Suppofe four fimples A, B, C, D are to be mixed ; let the 
 mean price be m, the price oi A =^ m -\- a, of /? = m -\- b, 
 of C = m — Cy and oi D z=l m — d. And let the quantities 
 to be taken of A, B, C, D, be x, j, z, nj refpedively. 
 Then 
 
 prices, quantities. 
 
 
 m -f n 
 
 mean 
 
 m -\- b 
 
 m 
 
 m — c 
 
 
 m — d 
 
 Now if each quantity be multi- 
 ^ plied by its price, the fum of the 
 produfts will evidently be equal 
 to the fum of a.l the quantities 
 multiplied by the mean price. 
 
 Viz. m -{- a X X -\- m -j- b X y -{■ m — c X. z. -{- m 
 zn X -\-/v -\- y -\- z X m. 
 
 dX 
 
 Let m -^ a X X -\- m — dX'vrzLX-^-HJXm, 
 
 And rn -\- b Xy + m — c X z z= y -\- z X m. 
 
 Then mx -f ax -f- m^v — d -z! =■ mx + m'v. 
 Alfo fny -J- by -f fnz — c z '^. my -\- mz. 
 
 \ A 1^— ^!*but here are four unknown quan- 
 tities and but two equations, •.* x and y may be taken at 
 pleafure. Take xzza and y—c, then will ^vzza and z=b, 
 and confequently the work will ftand thus. 
 
 m + a" 
 
 m + b- 
 
 m — c- 
 
 ^1 
 
 c /Which is the rule.
 
 Rules of Position. 51 
 
 Cor. Since ax = dn; and by = cz^ take x :=s m d and 
 y :=z uc \ then will <v z=: ma and z ■=. nb. Then by putting 
 md, nc, 71 b, ma^ for ;f , j', 2;, f refpeftively, iniiead of the 
 differences dy r, b, a, we lliall have m dy ncy nby ma, which 
 is a dired demonftration of the latter part of the note. 
 
 ♦ Pr ©position 3. 
 
 Here one quantity is given, and confequently all the 
 quantities «, b, c, d mull be increafed or decreafed in pro- 
 portion. Hence the rule is evident. 
 
 Proposition 4. 
 
 Here the lum of the quantities is given, confequently tlfe 
 other quantities muft be taken in proportion, fo that a -^ b 
 -\-c-\'d may be to the whole quantity as any of the differences 
 <7, by 8cc. to the refpedUve quantity required. 
 
 XI. Demo^i/I ration of the Rules of Fofitlon, 
 
 I. SINGLE POSITION. 
 
 Such queflions properly belong to this rule as require the 
 muhiplieation or dlvifion of the number fought by any pro- 
 pofed number, or when it is increafed or diminiihed by ii- 
 felf. or any part of itfelf a certain propofed numl^er of times. 
 ^-For in thefe cafes the refults will be proportional to the 
 fuppolitions agreeable to the definition. Thus 
 
 X s 
 
 — I X : '. — '. s 
 
 n n 
 
 XX s s 
 
 ^ + —y Sec. : X : : ~~ + — , Sec : s. Hence the 
 n -^^ m n — ?n 
 
 whole is evident. 9^ E. D. 
 
 II. DOUBLE POSITION. 
 
 This rule is founded on a fuppofition, that the firft error is 
 to the fecond, as the difference between the true and firil 
 fuppofed number is to the difference between the true and 
 fecond fuppofed number.
 
 52 Demonstration of the 
 
 'Let A and B be produced from a and b by any fimilar 
 operation, to find the number frorn which N is produced by 
 the like operation. Put % for the number fought, 
 
 ^' — A z= r, and A' — B z=.s, then by the fuppofition 
 r : s : : z — a : z — b •.' rz. — rb =■ sz. — j£, and by 
 tranrpcfi.tion rx — sz zn rb — sa, 
 
 Hence z = ; or if j be negative, or B greater 
 
 r — s 
 
 than A', then will z = I-iif which is the fiift rule. 
 r-i-s 
 
 Secondly, 
 
 Let s be the lefs error, being the error of b, and r the 
 correction ; then if B be lefs than A^, b + c ^ z, and c zz, 
 
 — A — ^^"~-^^ /.^ y3 — sa — rb^ ib _ b — gXjf 
 ~" r— / "" r — i r— i 
 
 But if 5 be greater than N, then b — r = s, and r =: ^ — « 
 
 , rb-\-sa rb-\'sh — rb — sa b — tfX/ ,.« • 
 
 =£> — = ; -zz , which IS 
 
 r-\-s r + s r+s 
 
 the fecond rule, ••• the rules are univerfally agreeable to 
 their principles. ^ E. D, 
 
 Scholium. It has been fhewn that the nuoiber fought 
 will come out exadly by this rule, when the errors are 
 cxadly proportional to the differences of the fuppofed num- 
 bers from the true one. It therefore follows, that when the 
 errors are nearly proportional to thefe diferences, that the 
 anfwer will come out nearly true. And thefe proportions 
 will be the nearer to an equality the nearer thefe fuppoled 
 numbers are taken to the true one. Hence we fee the reafon 
 of the fecond note to this rule.
 
 Rules in Arithmetical Progression, 
 
 XII. Demonjiration of the Rules in Arithinetical 
 Progrejfion, 
 
 ^Z 
 
 f/ = the leaft term 
 g =■ the greateft 
 . n = the number of ten 
 I J = the fum of the ten 
 [ ^/ = the common dirlei 
 
 Then will 
 
 /+/+^+/+2^+/+3^-f/+4^, &c. be In Arithmetical 
 Progreffion. 
 
 Now it is evident that the fum of the extremes will always 
 be equal to the fum of any two means that are equidiftant 
 from the extremes; and confcquently if the number of terms 
 be odd, the fum of the extremes will be equal to double the 
 mean ; 
 
 That is /+/-f 3^=/-f^+/+2./= 2l-\-2d 
 
 I-}-l+4.d=zI-i-2d+/-i-2d=2/+^d 
 
 l+l-^-zd—l-^d^l-^d— 2l+zd, Sec, 
 Hence Propofition i. s = i-\-gx~. For 
 
 2 
 
 r /4-/-I- d 4- 1 A- id -^rTTd ■+ 
 
 Add 
 
 V /+/+^ + i+zd +l-\jd -hI-\.^d,Scc.zz{um 
 
 I l^^d-\rTTTd +/+2^ -{- /+7 + / zrfum 
 
 2/+4^+2/ + 4^/-f 2/+-y-{-2/++^+2/+4^, &C. 
 
 *.♦ twice the fum is equd to as many times / -f / 4- ^d, or 
 I -\- g as there are terms in », confequently s rz l-{-g x 
 ^. E. D. 
 
 n 
 2 
 
 Inferences deduced from this equation 
 
 \ c T 7 2/ — 712 
 
 Interence I. / iz ^, 
 
 n 
 
 II. n = ^. 
 
 Tir i!—"^ 
 
 m. i = -;;-. 
 
 IV. s = "J±:i.
 
 ^4- Demonstration of th 
 
 Propofition 2. g — I zz n — i X d, 
 
 1.2.3 .4 • 5> &^c. to n, 
 
 l^TTd-^T+n-\rJTTd-\-JTTdl &c. to^. 
 
 Here the common difference d is evidently as often re- 
 peated as there are terms in the feries wanting one ; that is, 
 evexy terra, except the firft, is equally increafed by dy *,• 
 
 J-— / = «— r X^. ^E.D, 
 
 Inferences deduced from this equation. 
 Inference J. I =z g — « — i x d, 
 
 n. . = -^±^. 
 
 HI. d- ^^^. 
 
 IV. g=: I + n—l X d. 
 
 Kow by the afliftance of the above deraonllrations, if any 
 three of the iive terms /, g^ », s, d he given, the other tw© 
 are readily found. 
 
 Proposition i« 
 Given /, g, and n, to find s and d. 
 
 (Prop. 2, Inf. 3.) d= ^^^^; and (prop. i. inf. 4.) s =: 
 
 7 '^ 
 
 /+; X -. 
 
 Pro position 2. 
 Given /, g, and /, to find n and </. 
 
 {Prop. I. Inf. 2.) » =z -^, =. ^"^ ."" (prop. 2. Inf. 2.) 
 
 Hence we deduce ITieo. III. and IV, 
 
 Proposition 5. 
 Given /, g, and </, to find n and /. 
 
 (Prop. 2. Inf. 2; « = ^-~—, = > (prop. i. Inf. 2« 
 
 Hence we deduce Theo. V. and VI,
 
 Rules in Arithmetical Progression. $$ 
 
 Profosition 4. 
 Given /, «, and /, to find g and d. 
 
 Prop. I. Inf.3,)^ = ^~-^ = ^+^— ix^ (Prop. 2. 
 
 Inf. 4.) 
 
 plence we deduce Theo. VII. and VIII. 
 
 Proposition 5. 
 Given /, », and d, to find g and s. 
 
 {Prop. 2. Inf. 4.) ^ = / + «— I X^= (Prop, i. 
 
 Inf. 3.) 
 
 Hence Theo. IX. and X. are deduced. 
 
 Proposition 6. 
 Given /, s, and J, to find g and ^, 
 
 (Prop. I. Inf. 3.J ^ =z =/-f //— I Xd (Prop. 2 . 
 
 Inf. 4.) 
 
 (Prop. I. Inf. 2.) «r=-ii=rC±4ll^ (Prop. 2. Inf. 2.) 
 Hg d 
 
 From tbefe equations we deduce Theo. XI. and XII. 
 
 Proposition 7. 
 Given gy ;/, and s, to find / and d, 
 
 fProp. I. Inf. I.) / ~ 1-11^=: ^— ^117 Xi (Prop. 2, 
 
 h.i'. i.) 
 
 Plence v/c deduce Theo. XIII. and XI'v' . 
 
 Proposition 8. 
 Given gy 7/, and d, to find /and s» 
 
 (Prop. 2. Inf. I.) l—g-^n-^i Xdzz-- ''^ (Prop. i» 
 
 Inf. I.) 
 
 Hence we deduce Theo. XV. and XVI, 
 A a
 
 5-6 Demonstration of the 
 
 Proposition 9. 
 Given j-, j, and d, to find / and 11, 
 
 (Prop. 1. Inf. 2.) '^=;A^ = '^^^~^ (Prop. 2. Inf. 2.) 
 
 (Prop. 2. Inf. I.) / =:^— a— i X ^ z= J^^'^ (Prop, u 
 Inf. I.) 
 From thefe equations we deduce Theo- XVIL and XVIII. 
 
 Proposition 10. 
 Given ;/, /, and dy to find / and g. 
 
 Prop. 2. Inf. 4.) g =:i-i- n — i Xdzz (P^op* '• 
 
 Inf. 3.) 
 
 (Prop. 2. Inf. I.) 7=:^— ;J^ X^=: ^i^— ^ (Prop. i. 
 
 iQf. I.) 
 
 From thefe equations are deduced Theo, XIX. and XX. 
 
 XIII. DemonJIratlon of the Rules in G^oinetrkal 
 
 ProgreJJion, 
 
 Before I proceed to invefligate the Theorems, it may not 
 he improper to explain the rules to the firft and fecond pro- 
 pofition, page 95, — In the firil propofition, where the firft 
 term is equal to the ratio, the reafon of the rule is evident ; 
 for as every term is fome power of the ratio, and the indices 
 point out the number of multipiiers, it is plain, from the 
 nature of multiplication, that the produd of any two terms, 
 will be another term correfponding with that index, which 
 is the fum of the indices ftanding over tliefe refpedtive 
 terms. 
 
 And, in the fecond propofition, where the feries dees 
 not begin with the ratio, it appears that every term, except 
 the firlt and fecond, contains fome power of the ratio mul- 
 tiplied into the firft term, -.• this rule is equally evident 
 %\ith the former.
 
 Holes in Geometrical Progression-. ^y 
 La /r:the leaft teniT 
 
 ^r=:the greatell 
 
 vnthe number of" terms 
 
 j~rhe fum of the terms 
 r— :the ratio 
 
 log. iiilogarithm of any let- 
 [tcr, or term. 
 
 Then /, /r, /V% /r% /;4, 
 
 Geom. Progreflion. 
 
 [hen /, /r, /V% /r% /;4, &^'. -J 
 
 Or, /, — , ~, — , --, &:c. ( 
 r r^ i^ r^ \. 
 
 Kow fincc the above progreflion is compofed of two ranks. 
 
 The equals /, /, /, /, /, 1 .... , 
 
 Geometrical proportionals i, ^, rS r^ ^4^ J &«^ »t 'oUows th.t 
 
 the moft natural Geometrical Progrefiioii is that which bcg'ns wlr.h i,. 
 both incieafing and dccreatinjr. 
 
 Proposition r. 
 
 Demon. For / : /r : : s — g : s — /) By Geometrical 
 But / I Ir : : I : r 3 Proportionals. 
 "Whence 1 : r : : s — g • ^ — ^« 
 '.• s — I^ s — g X r. i^ E. D. 
 
 Inferences deduced from this equation. 
 
 Inference I. s =: ^^i^^ z= ^-^ + g. 
 r — I r — I 
 
 II. r zz .. 
 
 HI. / = s -h ^g — rs = rg — r — I X /» 
 
 ,-- sXr — 1+/ * s-i-/ 
 
 IV. g = =z s . 
 
 ^ r r 
 
 Pr OFOSITION 2. 
 
 r X / = g. 
 For /, ir, Ir^, lr\ /r^ &r. 
 
 And I^ = /r^-' ... //-^ =^. ^ £. Z>.
 
 j;8 DlMONSTRATlON OF THE 
 
 Inferences deduced from this equatidn. 
 
 n — I 
 
 liiference I. ^ = /r 
 
 T 
 II. / = -'^. 
 
 /; — I 
 
 r 
 
 , III. „ = !£I:f=i£I:> + ,. for/-=-f. 
 
 Jog, r i 
 
 IV. rzn-^p' 
 
 Hence by the affiflance of the preceding demonftrations, if 
 any three of the terms /, g, », ;, r be given, the other two 
 are readily found. 
 
 Proposition i. 
 
 Gi^en /, g, and », to find / and r, 
 1 
 
 (P/op. 2. Inf. 4.) r = 4i'*""' and (Prop, i. Inf. i.) ; = 
 
 ~ — . Hence we deduce Theo. I. and IT, 
 r — I 
 
 Proposition 2. 
 Given /, g, and r, to find » and r, 
 
 (Prop. J. Inf. 2.) r=: ^-^^, and (Prop. 2. Inf. 5.) » =r 
 
 ^'^ ^^ — {-/. Hence we have Theo. III. and IV, 
 
 l;>g. r 
 
 Proposition 3. 
 Given /, ^, and r, to find « and s, 
 
 (Prop. I. Inf. 1.) / = ^— z= -^ + g- 
 r — I r — I 
 
 /Prop. 2. Inf. 3.) « =: -^f ~-^?:- + i, the fame wirh 
 Theo. V, end VI,
 
 Rules in Geometrical Procressioit, 59 
 
 Proposition 4. 
 Given /, », and /, to find g and r, 
 
 (Prop. 1. Inf»4.) g n ^ = Ir (Prop. 2r 
 
 Inf. 1.) Hence we deduce Theo.. VIIT, 
 
 ft I 
 
 (Prop. 2. Inf. I.) ^ = ir (Prop. i. Inf. 2.) r = 
 
 J / 
 
 . Hence by fubftitution and redudion we deduce 
 
 Theo. VIL 
 
 Proposition 5*. 
 Given /, /?, and r, to find g and /• 
 
 (Prop. 2. Inf. 1.) g — Ir which is Theo. IX» 
 
 Hence r^ = /?•. 
 
 But (Prop. I. Inf. I.) ;=: ^^^llll/ =: ^llH^ bv fubai- 
 r — I r — I 
 
 r" — I 
 
 tution, V • y, 1 =z s, which is Theo. X. 
 
 r — 1 
 
 Proposition 6. 
 Given /, s, and r, to find'^ and //. 
 
 (Prop. I. Inf. 4.) g z=. . (which is Theoi 
 
 XI.) =: /r (.Ptop. 2. Inf. i.) Hence we deduce 
 
 Theo. Xir» 
 
 Proposition 7. 
 Given g, k, and j, to find / and r. 
 
 (Prop. 2. Inf. 2.) / = — ^ (Prop, i. Inf. i.) s = 
 « — I 
 r 
 
 -^ , by fubftitution and redudion we get j r r— x ;■ 
 
 != gr — g, and hence we deduce Theo. XIV. 
 Theo, XIII. is the fame as Theo. VIII. 
 Aa 3
 
 6o Demonstration: '. 7 run 
 
 Proposition 8. 
 Given g, », and /•, to find / and r* 
 From 'the preceding propofitlon we have Theo. XV. and 
 
 from the equation sr — sr =^^ — g* I" the 
 
 fame prop. \^e get Theo. XVI, by divilion. 
 
 Proposition 9, 
 Given g, s, and r, to find / and «. 
 (Prop. I. Inf. 3.) Izzrg + s — rs, which is Theo. XVIL 
 
 (Prop. 1. Inf. 3 and 4.}4 = ^ = ""^ 
 
 (Prop. 2. Inf. I.) which is Theo. XVIII. 
 
 Proposition 10. 
 Given n^ s, and r, to find / and g^ 
 
 (Prop. I. Inf. 4.) g = i^irjuH:/ -- // ^ (P,op. 2. 
 
 Inf. 1.) Hence we deduce Theo. XIX. 
 
 Eyprop. S, sr — sr =z gr — g. Hence by 
 divifion we eet Theo. XX. 
 
 Cor. In the original equation s — /= rX s — g, when. 
 
 the !eaft term /:= o, then s = -l^, which is Theo. XXI* 
 
 r — I 
 
 and hence the reft are deduced. In the demonftration to 
 
 prop. I. in tlie theory of circulating decimals, it is afferted 
 
 , n R R R , . ^ . R 
 
 liiat — -i — - -{ — : -] , &c. ad infinitum, rz . 
 
 r ?•* r' r+ r — i 
 
 Demon, x ~ r x s — Rj from above, hence by rcduc- 
 
 n 
 
 ti«nj=: . ^£. Z),
 
 Rules of Variations, Sec, 6i 
 
 XIV. DemonJI ration of the Rules of Variations, 
 
 Pro position i. 
 
 The reafon of the rule to this propofition is evident. For 
 one thing, as a^ is capable of bat one ppfition a. And if 
 there are two things, a and h^ they are only capable of thefe 
 two variations aby bay and thefe may beexpreffcd by i X2. 
 
 Suppofe there are three things, abc\ abc | be a 
 then any two of them leaving out the acb cab 
 tiiird will have 1X2 variations; and bac \ cba 
 confequently when the three are taken in, 
 there will be 1X2x3 variations. Sec, Sec, 
 
 PROroaiTION 2. 
 
 " The nunnber of changes of ^ things, taken » at a time, 
 is equal to m changes of «i — i things, taken k — i at a 
 time." 
 
 •* For fuppofe ^ things, a b c d e be given, firfl leave out 
 <7, then we (hall have the four b c d e, out of v/hich let there 
 be taken all the 2's bcy b dy &c. put 'v = the number of 
 variations of every two out of the four quantities bcde. 
 Now if <7 be put in the finl place of each of them, it wilt 
 znake abcy a bd, See. then will each confift of three letters, 
 viz. t; = number of variations of every 3 out of 5, abcde, 
 when a is firft, 
 
 "In like manner, if b, c, d, e be fucceffivcly left out, the 
 number of the variations of all the 2's will alfo be ru ; and 
 putting be de in the firft place to make 3 quantities out of 5-, 
 there will ftill be 'v variations as before. But thefe are all 
 the Tariations that can happen of 3 things out of 5, when 
 a, by e, dye are fucceffivcly put firft ; and therefore the fum of 
 aTl thefe is the fura of all the changes of 3 things out of 5-. But 
 the fum of thefe is fo many times ~o as is the number (A 
 things; that is, jo; or m'v zz all the changes of 3 things 
 out of 5. And it is evident the fame way of reafoning may 
 be applied to any numbers ;;;, ?i." This being premifed, we 
 deduce a dired demonftration of the rule, for 
 
 " Suppofe any number of things a b c d e fg\ here fn=z-i, 
 and let // = 3, Subtra(5l i from 3, and there remains 2; 
 fubtraiii 2 froii; 7, and there rcTi^iLns 5. Hence it is evi- 
 dent.
 
 6z Demonstration of the 
 
 dent, that the number of changes that can be made by taking 
 I by I out of 5 things, will be 5=-^. 
 
 ** Then when »i = 6, w =: 2, the number of changes = r/n/ 
 ^ 6 X 5 zz 1; a fecond time* 
 
 " Again,, when m = 'j^ n z::z ^, the number of changes 
 
 = m'v IT 7 X 6 y 5, that is rzTwX m — i x m — 2, con- 
 tinued to 3 or « terms. And the like may be ihewn for any 
 other numbers." ^. E. D. Vide Mr. E?nerfon^ permu^ 
 tations, prop. 2, and prop. 3. 
 
 XV. Demonjlratlon of the Rule of Cjmhi nations. 
 
 The number of combinations of 2 things out of 5, ah cde 
 are 10, as follow. 
 
 For with Ul V^'V;uV: 
 
 the different < c >are combined J •• v ^f, ^^, ''^^ 
 
 things 1^1 \— ''''/; 
 
 Hence the combination of 2 tilings in 5p ~ la 
 
 The number of combinations of 2 things in 5 is fhewa 
 to be 1 + 2 + 3 + 4, and by the fame manner of reafoning 
 the combination of 2 in 6 will be 1+2 + 3+4+5, hence 
 univerfally the number of combinations of m things taken 
 2 by 2 will be 1+2 + 3+4+5, &c. \q m — 1 terms, an- 
 fwering to a figurate number of tlie third order, the fum- 
 
 , . . m m — I 
 
 whereof is — X . 
 
 1 2 
 
 Again, the combination of 3 things out of 5 are 10, as 
 follow. 
 
 With a. With h. With r. 
 
 a b c, a b dy a b e \ ^ 
 
 • .., a c dy a c e \ 2 
 
 • . ., . . ., a d e\ I 
 
 bcdy ^f ^ I 2 I cde I I I 
 , . ,. bde\i
 
 Rules of Simple Interest, Sec, 6j 
 
 The number of combiRations of 3 things out of 5, are 
 fr.evvn to be i -f 3 -f 6 zz 10, and the number of 3 things 
 at a time out of 6, by the fame manner of reaioning, will 
 be i-f3 + 6-fiozz2o; this ieries anuvers to a figurate 
 
 number of the 4th order> the fum whereof is — X 
 
 I 2 
 
 — — , which is the rule, P. £, />• 
 3 ^ 
 
 XVI. Demoriftration of the Rules of Shnple Interefly 
 by Decimals. 
 
 Ufing the notation prefixed to the rules, we have, by 
 proportion, i : r : : p : rp, the inferell of / for one year ; 
 and I '. rp :; t I prt, the intereH ©f p for the tiir.e /;. 
 alfo p -\- prt HZ the arrear at the end of the time /. 
 
 Hence prtzzi, and prt -\- p = a, from which equations 
 
 the rules are deduced. The rule of Difcount is the fame- 
 
 with the rule to prop. 2, putting D in the place of a. 
 
 XVII. Dsmonf} ration of Malcolm'j Rule of Equation 
 of Paymerits. 
 
 The principles on which this rule is founded are given 
 in the note to the rule. 
 
 Let / n the firft payment, P = the fecond, / = the 
 time between the two payments ; r zz the ratio, or rate per 
 cent, divided by 100, and ,v zr the equated time from t!fte 
 firlt payment. 
 
 Then prx = the intereft of/ for the time x. 
 
 And zz difcount of P for the time / — .v. 
 
 rt — rx -\- I 
 
 Hence, according to the fuppofition on which the rule is 
 
 Pr* Prx 
 
 founded, /r A- = 1. Now by reducing th& 
 
 rt — r.v 4- I 
 
 equation, Scq. we get xzzt^ 4- — + / ^^ + -H — I" 
 
 but it will be found, upon examination of the problem, that 
 only one of thefc values will aufwer the conditions of the 
 queftion; and by a fynthetical demonftration that value is 
 
 < + /^-5-. + — — l-l, which is the rule. ^E.D^
 
 64 , Demonstration of the 
 
 XVIII. Demonfiration of the P^ulcs of Compound 
 Inter £ji^ by Dtcimals. 
 
 Here we fliall ufe the fame notation as is prefixed to the 
 rules. 
 
 r zz the amount of il. for i year,, and by proportion,. 
 1 : r :: r : r^ amount of il. for 2 yeari;. 
 
 I : r :: r^ : r^ ditto • for 3 years, 
 
 I : r : : r^ : r"^ ditto for 4 years. 
 
 And if the number of years, or payments, be denoted by 
 
 /, the amount of il. for / years will be r , hence it appears 
 that the amount of any other principal fum / for'/ years,. 
 
 will be /;- for I : r : : / : pr ,\- ^r z= a, and hencc 
 all the rules are deduced* 
 
 XIX, Demonfiration of the Rule of Equation of Pay- 
 mentis at Compound Interej}, 
 
 Let Ay By Cy D debts be payable at the end of a, hy Cy d 
 years, x =: the equated time from the firft payment. Then 
 d — X zz. the time from the equated time to the laft term. 
 j4-\-B-\- C-^- D = s. Then by Moreland's or Burro'vj's prin- 
 ciple (vide demon. 3 of Equation of Payments at Simple 
 
 Intereft,) we get ^/~''-f b/~~^^ c/'^' + D =z P = 
 sr ; by dividing by s and extrafting the root by loga- 
 rithms, we get x = d Qg- — og.s ^ which is the rule 
 
 log. r 
 exadly. ^ E. D^ 
 
 Cor. I. If we divide the equation by r we get yr ZZ Ar -f- 
 
 —b ——c D i 
 
 •Br •\- Cr + — :, hence by the nature of negative indices — zr 
 
 A , B C B V 
 
 ' + — 7 + + — ; ^P, ••• Pr^ = 5, and x — 
 
 ra / ^c ^d 
 
 log. 5 — lo g. P , . , . ^ ^ ,. , 
 , which IS Kerfy% rule. 
 
 Cor. z. By Malcolm'^ prindple j fuppoling x tKe equated time 
 
 to fall between the fecond and third payment, we have At — 
 
 X — ^ r D 
 A+Br — B fum of the intcreft zn C — H -^ "— '. 
 
 Sum of the diCcov«t$. By tranfpofition ^r'^"^^+ ^r*"" +■ 
 
 II
 
 Rules for Annuities in Arrears at, &-c. 65; 
 
 CD X 
 
 h -— — J-\-B4-C-^D zz ^J divid:; by r and we havs 
 
 rr— X d — .V 
 
 y r 
 
 ^a ^b C D S S yj B 
 
 Ar + 5 r ^ + = > or — ^ (r 
 
 j,c ^a ^x J.X ^.a ^o 
 
 — - A , the very fame exprefllon as we deduced from Ktrfty'^ 
 
 principle. 
 
 Cor. 3. Hence we may obferve that the rule is univeifally tin?, 
 whether we argue from Burrc'zv''s, ATer/Vy's, or Malcolm^ s principle. 
 But if we argue from Cockex^^ or Hattons principle, we deduce a different 
 theorem. For in the former cafe we get 
 
 ^/""'— >^+Br'"~*— J^zrc/"""^— C+Z?/""""— D, and In the 
 
 By proportion < 
 
 latter y/+5+C+^ X^ — Ar ^Br +Cr +Dr - 
 
 Scholium. From what has been demonftrated, Art. 7, 17, and 19, 
 it plainly appears, that all the rules are true according to the fuppnfitioni 
 on which their refpeftive authors have founded their demonitrations ; 
 and the reafon why they differ, arifes only from the injuftice of Simple 
 Intereft. I fhall only remark, that what ^'fr. Todd has written on this 
 fubjeft, in his folution to his own problem, at page 39 of the Ladiei' 
 Diary for 1789, is a compofitioa of error arid inconfifliincy. 
 
 XX. Dejnonjlration of the Rids s for Annuities in Arrears 
 
 at Simple Intercfi, 
 
 Here we ihall ufe the notation prefixed to the rules. 
 
 J- I : r : : n '. rn the fecond year's intereft, 
 
 I : r : : 2« : trn the third year's intereft. 
 
 I : r : : 3« *• 3r« tlie fou.th year's inteteii. 
 
 I : r : : 4" : \rn the fifth year's intereft. 
 
 1 : r :: t — JX« : r—iX»'' the rth year's intereft. 
 
 J 4. 2 ^ 3 4.4, &c. to /—I years x '«> gives thcwho'e intereft due 
 orv the annuity agreeable to the rule to prop, i, or ttj -p 2 rn + 3 r« -t- 
 
 4r.«, &c. to /— 1 years zz — ^"j the rules of progrefiion. 
 
 Hence -f" '" — ^> ^"*^ hence all the theorems are deduced. 
 
 XXI. Demonji ration of the Rules for the prcfcnt worth 
 
 of Annuities at Simple Interejt, 
 
 By Art. i 5, ^r r +/' = <»> and by the preceding article '— 
 
 4- t rznay ••• prt ■{■ p zz + tity and hence al the rules 
 
 arc<4c>luced,— The truth of the rule to prop, i, is evident.
 
 Demonstration of, ice. 
 
 r> r- L . t:r-\-Zt — tr , 1 
 
 Cor. From above we get p zr X «» and h — ■ 
 
 putting T in the place of /, and a zz l, then '-^ " ■ x 
 
 r+3 Tr + 1 
 
 ZZ p, the prefent worth of an annuity in reverfion at' Simple Interell, 
 and hence the tules are deduced, 
 
 XXIL DemonJiraUon of the Rules for Annuities in 
 ylrrears at Compound Inter eft. 
 
 Here we fliall ufe the notation prefixed to the rules. 
 « rr the money due at i years end. 
 » + r» rr arrear at 2 years end. 
 « + rfi + r*«rr arrenr due at 3 ye^rs end. 
 n -^ r >i •\' r^ n ■]- r' n ZZ. arrears at 4 years end. 
 
 K+rw-fr^w -}- r's + &c. to «/•' ' rz:. arrears due at the end of t 
 years. By the rules of geometrical progreflion the Turn of i -|- / + r^ 
 
 4- r^, Sec, ... to r'""'ziLlZi ^^^ .unount of il. for / years. 
 r — I 
 
 rf I 
 
 *.* X «Zl^j and hence air tlie other theorems arc deduced. 
 
 XXIII. Demonjlration of the Rules for the prefent worth 
 cfAtinuitiei at Compound interejl^ ^c. 
 
 By Art i?. ^/ zz^j and by the preceding Art. X « — 'J* 
 
 / »■' — I 
 
 '*• t r zz. X «> from this equation all the theorems are deduced. 
 
 1— V 
 Cor. I. From the above equation we get X nznf, by 
 
 ■dividing the former part of this equation by r we get 
 
 ,_ix.-^+^ 
 
 X rZZ py which" is Theo. I. of the prefent worth of Annuities in Rever- 
 fion at Compound Jnteieft j and hence we deduce Theo. II. 
 
 Cor. 2. From Cor. i. we have X « — ^5 but when t is 
 
 infinite, r^ is infinitely greater than i, or i is infinitely fmall in com- 
 
 parifonwithr ••• ?— i ZZ r , and hence X n ZZ py from 
 
 which equation all the rules are deduced for buying and felling Freehold 
 
 FINIS.
 
 .m 
 
 I IP lamiv ""^^-^O^-
 
 -Vl^-' 
 
 .h:' 
 
 
 '%■ 
 
 1^* 
 
 
 ^ 
 
 ^■\ 
 
 ^' ■:-•.■ ^■■:.^: ■'/:'''■' ^■^' 
 
 ":','•. ; . / "