# Full text of "Arithmetick, Vulgar, Decimal, Instrumental, Algebraical: In Four Parts"

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About Google Book Search Google's mission is to organize the world's information and to make it universally accessible and useful. Google Book Search helps readers discover the world's books while helping authors and publishers reach new audiences. You can search through the full text of this book on the web at |http : //books . google . com/| ^JJK:.,^c^^L!^:::>i-i^::5^ ^ ^...i'.-s /;:^.,..;x x™^\ /. , \/ ! ] ■J V ^ V 1 «.« •1 >i ■ t > f= / \ •s 4 r Qf\ "^ '^ ; :-■ * 9 • ' .•1 ^ . » > xmciE s Ann. - i ^''-^ ">• . E .'lit ff.:0i:t7tS ■ T III r > » r ' V. ARITHMETICK, VVLGAR, \DECIMAL, yNSTRVMENTALX (ALGEBRAICAL. } In Four Parts. • • By WILLIAM LETBQVRN. =, Ifclfc _ (7, , % T^tf Seventh Edition. Carefully Corre£ted ; and very much En- larged by the A U TH O R, An Account whereof is given in the Preface to the READER. LONDON^ Printed by J. Mathews, for Awnfflfom and "John ChurchillyZt the BlaekS^tuth^ Pater'- Nofier-Kow, M d c c: I . I I ' I ) ; \ /^ » s * X ^ t 'r ' 'k \ ' • « -I . V / ** ' , / * • V X . J A A !. ' ^ i ' K . t ' TO THE READER. THis Trestife oi Arithmetic k hath - paffcd fb many Imfre0ons that, (what with the fupine Negligence of the Printers^ and Cdrelejf- nefsw Jgmrame of thofe to whom the Revijkl tli^reof was committed for want of Application being made to the jMshor ; ) it hath contraftcd fo many Tj^tJgrafhicdl SfhalmAta^ as to disfigure its Beauty fo as to become a meer Str Anger to its firft Fdtent : However, in this Edition^ I have carefully Examined every Rule and Example^ and rendred them Perfe£l as at firft : And whereas, I have in fome places omitted fome things of Lefs Moment, I have in other places made a Quadruple amends, in Adding others in lieu thereof, in o* thet places : Of all whicli, I now come to give you an Account of the fevcral Parts of the Book^ and ia which fuch Additional Supplies are made. The whole Tr^^rtY^ is divided into four Parts. The F/ri? contains r^/^irr ARITHMETICK in Whole Numbers and Fractions : ^nd in every RuU there are Examples for Pra6^ice added, and ^efiions alfo wrought |w thofe fingie Rules : In Multiflication I have added divers Compendiurns ^ or hxveSvfzyioi Multiplying^ whereby Sums, (ha- ving 2 or 9 Figures in the Multiplier) may be per- formed^ withouj any biurthen or chirge to the Memory, more than inx)rdinary. MuMplicationy and yet no other (otzt moft very few) Figures fet Tothe READER. fet down but the ProduSi it felf. And in Divijion (which is the moft difficult of the four Species) there are two ways for the performance of it fo \ that every Man may make ufe of that which he { beft under ftands or fancies : And in the working of the Golden Rule^ &c. I have (to exprefs variety) made ufe fometimes of one kind of Divijipny and fometimes of another. The Second P4rt treateth of !DECiMAt Ar I t h m e t I c k : Which I have (in this £- dition) divided into Four Seiiions. In the Fir ft is taught how to ReduceVulgar Fracft^ ons into Decimal Parts ; and thereby to make Deci^ mal Tables to exprefe the feveral Denominations of the Coyns^ Weights and Meafures. (pi your own or other Countries^ in Decimal Parts ^znA how to make ufe of fuch Tables upon all occafions. The fecond Self ion contains Notation^ and how td work all the Rules of Jrithmetick (^treated of in the FirB Part) Decimally : And how to ExtraB the Square and Cube Roots ; in both which (in this Edition^ I have been very Copious and Plain. The Third SeBion trpateth of Simik and Qom^ found Inter eU — • Discount or Rebate of Money — jEquation of 'Payments — Pur chafe of Annuities ^ Valuation (k Leafes of Land or Houfes^ 8fc. With Tables of all of them ready Calculated. The Fourth SeSlion^ Teacheth how to MeafureJ Superficies^ as Boards^ GlaJ}^ Landj Pavement j &c. And of Solids^ as Timber y Stones^ Sphere s^ Pyramids ^ Cones J Cylinder Sy &c. --.— And alfo, of the Works of the feveral Artificers ^ rdatiiig to Buflding,; as Or- pentersy Bricklayers^ PUifier.^s^ M^JpnSy Pointer s^ ^oyntrsy Tc^f^f READER. JiTfnerSy &c. Whereby this Decimal Arithmetick will be as ferviceable to all of fuch Prbfeffions ; as Vulgar Jrithmetick (in the Firft Part) was to- Merchants and other Traders. The Third Part treateth of Instrumental Arithmetick; or Arithmetick It^rument^ idly performed in 2i Decimal wzy without the help ot Decimal Tables^ by which the whole work of ReduSiion is avoided, there being certain Scales of Englijb Monejiy Weights 2inAMeaJures^v\At6y and fo difpofcd, that by them (by infpeftion only) the Decimal Fraction of either Money ^ Weighty or Meafure^ may be fet down as exaftly, and in lefs time than they could have been taken out of the Decimal Tables in the Second Part of this Treatife. And on the contrary, any Decimal FraStion^ may be reduced into its^ proper parts of the integer with the fame facility, fpeed and exaftnefs: A Figure of thefc Scales is inferted at the beginning of this Third Part. Unto thefe Decimal Scales ^ I have now added T Wo other 5r 4/^ J, iovtht ExtraSlionoi Roots'^ By the one you may find thtRoot of any Number the Square thereof being given ; Or if the Root be gi- ven, you may find the Square Number anfwering thereuntt), and that by mfpcQ:ion only, without the help of either Pen or Compafs — And as this Line doth for extracting the Square Rooty ^he other doth the like for the Cube Root. And to make this Third Part of Infirumental Arithmetick y yet the more compleat, I have more largely (than in the former Editions) infilled up» on the Defer iption dXid Vft oiNe fairs Bones ; large- ly Tothe READER. ly treating of their Vfe in Multi^Ucathn^ Divifion^ and ^^trAdlon o^ Square and Cube Roots. And laftly. The Fourth Fart treateth of A l g e b r a. And , concerning that, whereas in former Editions^ there was infertcd an Abridgment of the Freceps of A l- G E B R A, written in French, by J^mes de Billy ^ J have now in this £^/>/(3»>!^ (ihftead of that^ added a more Comfleat Treatife of that Art^ rormerly Publifhed, (zt ray requeftj by my worthy Friend * Mv. Thomas Qwfony among other ^things of his, in a BoG^k called Syntaxis Mathematical whereinthe M^r/?^^ which he ;ther€ foltows is the fame asihv Mr. Harriot^ in fome places, that is, in fudi A^qtm^ tidns as are propofcd in Nun^ers : A6d as in Des Ciir^^^^ in fome other Piacesythit is, in fuch Mc[U4^ tions as are Solid and not m Numbers : Not that^the Book is talfen oat of them, neither does proceed coiitinually with thetn, but disjun£tly, as he thouglit fit to intermix them among bttjjer Things which are not in them. ,. . ;; Thus having given you a full Accibu'rjt of this TreatifeoiA rithmeTick, and of what is con- tained id the feveral F^^fs thereof; JT freely offer it to the ingenious Practitioner in the Art of Num- ber s^ deM&ghis Friendly acceptance, and par- don for. fueh Errors as may poflibly lidy e .efcape^ the Pr<?/}^ pr'my lelf (nOtwithftandm^ great Care i have taken for the prevention on either hand; And ifi fo doing, yod Will eiieourage him, who is ' '* ^ /'./...'.' -.^700.^'^'" • Mathi^ic4^:^ed, WilKam Ley bourn* } ^^■^^^•^mm ^ '"^ "TT-r THE CONTENTS. PART I. Of Vulgar Arithmeticki -r" NVmeratien^ , MStioft.' ' ^ >. r£w//yj Money. yiTr(ylVei£ht. ^ . J . f Little Wekht. C Liqnid. Of Meafmres < Dry, 0/ Time : , Of Amhecaries Wei^ts. The Troof of Addition^ 7\T h fS^^^ Denomination. ^ -^ \Divers Denominations The Proof of SHbftraaion. ' Qnefiions wrought by Addition and Subfira^ion^ Multiplication. Tm Tables thereof The Proof of MhlfifUcatiqn^ Comfendiums in Mnltiflication. Que fi ions ^ wrvHght by Afaltiflication only. Divijion. a . Page jt 5 6 8 . lo ibid IK ibid 12 ibid ibid 20 21 20 29 30 ^ The C 0*N T E N T S. The Me^ fir Worhin^ it. ibJi. ^feeorid w^ay of Divifioti^ and hm tQ Pre^vc the fame. 54: A Pofifcnpt to Dhijioti, . 38 CompehdihrTTs in Dhlflon, ^q 'Q^teftlom refolvedby Divijierj. ^i JleduEilon. v a2 Proareffwn, • ,45 .^Arithmetical. H)|(J iGeometricai. ^g The Golden Rule of Three, ' ■ <i JOireB ' - ibla The Golden Rule^ Compomdei of Five Niimbers. <q Of Fr anions. ' ^'^ Vi 5 Numeration. ^9 Divifiofj. . g Redn^lion. ^- \jdddition.' - gl . ' jSHhfiraBion. «q {The Golden Rule, ibid jRHlesdfPraBice. ' -^ PfTare^ Trett^ &C. ' ^^j !r/7tf ^«/tf ^ FelloivJInp. 70 ir/r/?o//^7^. Ibid 0/ Barter. ' g^ 0/ Inter eft J Simple and Compound. g^ .*;7' -u-^ {^ Medial. on , Albq-atton< ., y^ ^ lAkernate. p2 jHv i?i^/^ (j/ F^//^ Pofition. pg . \Tb^kid€ of Ceres ^;7<;^ Virginuin* 1 03 P'A R T The CONTENTS. \ PART II. * Of Decimal Ari^hmctick* T^t IntroduBi<m^ 115 To Hfdnce VulgaT &Mions into Decimal Pans. 116 Tv Exprefs EngUjh Coyn in Decimals. x 1 7' *" Englijh Coyn. Troy Weight. jivoirdufois Great Weight. Avoirdupois LittleWeight. Liquid Meafnres. Dry Meafttresk, Long Meafures. Time. yDozjLens. > The X)fe of thefe Tables. I ix The fever al RhUs of f^algar Anithmetich^ f er firmed ly De* TMes^ whereby to Reektce into * Decimal Parts. timals. CNotatiom \Addition. J. JStdffiradion^ ^ \MHltiflication* JComfendinms. {jDivifion. AtHffleinent u> Decimal Divifinn The RhU of Three in FraBions Vulgar and Decimal^^ Who RhU of Three Reveiftf. He RhU of Proportion cor^jling of iPlvf .NHwhers. CBarttr. ^jLofsdndGaim Exchanges* a 2 131 ibid 134 t3$ 138 14Z 147 149 15^ 158 161 I Si 166 170 Tables I ^ / / \ 173 i8i 182 183 ip2 193 The CONTENTS. Tables of Exchange. ' , , Of the ExtrafHon of Roots. The Genefisfor ExtraRing of the Square Roo^. The ExtraEl the Square Roop. The Genefisfor Extr ailing of the Cube Root. , To Extr^B the Cube Root. \ ^ ,^ Of Interest^ Simple and Compound: Difcountinr debate of Money : Of Equation of Payments^ &C. 20O Thh ConfiruHion and Vfe of the Tables of all of them, ibid Of the Menfaration of Suferficiei and Solids: And, of the Works of the feveraijiriificers relating to Building. 229 Of the Menfuration of all Plain Superficial SoUds: ibid Of the Menfuration of Solids. 23 $ Carpenters Bricklayers Mafons ' Of\ Plafierers Joyners Painters GUfiers >Worhff ^241 242 244 24S ibid 14^ ibid PART IIL Gf inftrumental Arithmetick^ 247 I.T3 T Decimal Scales. IJ Numeration* Addition. SuhftroElion. Ex$raBion of Roots^ {Square. Cube* 249 254. 2S9 2(Jq The CONTENTS. r II. JSyNepair'i Bones. ' 159 Of the Fabrick or Can^uElim ef theRiJs. . 260 To lay dorm any Number by the Rdds. 2^ MkltiflicaiiofJ. 2tfS Divifiofti ' \; ^ 2,75 To ExtroB ^Squate koot. 278 the iCnbeRoot. 288 Seme Vfes of the Square and Cube Rpets^ 2^4 11 - ■•■ -- • -- PART IV. Of Algebra. 297 ' A Troeme. . , 299 Xjl I^^finitions. ^ ibid Of Poteftates^ or Powers^ 300 ATable of them. 30 1 j Of CharaBtrs and Symbols. 302 Cjidd O 304 ^'^^^^S^^^ ibid CDivide J 30$ P/ t/E^quations^ Sirhfle and Mixt* 305 The RedaEHon of them. 308 Of the refolution of t/£qHations according to the general Methodof Mr. Tho Harriot. 31a Of Cubical ny£qHations^ Plain and Mix^d* pijt Their Refolution. ibid Of Devolution and^ticifotion^ 318 «f the iLeffir S , 31* hi Nm/Atrs, 324 a i General \ . V The CO NT E NT S^ C Rules. 3i< Gen€ral<Notes. \^ (.E^Mffples. a 20 nMim of SoUds. |,^ "Of Snrd Ndmbers. . „ -^ I'fumricalatidaemttric/ilFroblcms, Refihed. 339 t m^m» ADVER ADVERTISEMENT.' T^He Place of the Juthor*s Refidence is about X, Ten Miles from London Weft- ward, at a Place called S>outhd^ in the Road between Mo» ^nd VxhrUge^z.Vi& Three Miles from Brenrford: Where he intends to Readxht Mathematieks, and Inftruft young Gcrtlcmcn, and others : Ajid to Board fuch as fliaD be plcafcd to make a more clofe Application to thefe Studies : ^Where fuch Boarders, and others, (during their time of Re- fidence with hihi) (hail have the Ufe of all B<Mks Mifs^ Qlobesy moAothtx Mathematical Inftrumeuts as are neceflary for their InftrttStion, till they pro^ vide themfelves pf fuch as they fhall have occa» lion for ^erwards, A L SO, IF any would have their Land or Buildins Surveyed, or Meafured, and a Plot therecS" made ; or any Sun-Dial, or Dials about their Houfe or Garden (of what tipd foever. Fixed or Moveable) he will Prepare or Make for them fuch as fhfill be deHred. You may hear of him, and. have an Account of his Terms, . and manner of Proceedings: By Mr. Robert Mordea, at tlie Sign of the Mlat in CotenhiUy ne&r the Royal Ex^oftge, Gtoh-maker. m. Henry Wyn, in Chancery-Lave, over-againft the Rolls ^ Mathematical Infirunmt-maker. And where thefe Books are to be Sold. '••* ARTS MATHEMATICAL, Taught by the Author. In Whole Numbers , and Fraftions* . , In Decimals, and by Logarithms. Inftrumentally, by. Decitnal Scales, mpier's Bwes : and to cxtraft the Square and Cube ^, Roots by Infpeftion . 5 the Principles thereof J Praftice, and ^ I with the t Demonftratiqn. TbeDefcriptlonof the Circles of theSlphere, The Ufe of, the VCeleftial, ^nd Globes, \Terreftrial. To projeft theSpherc'/» Piiwro? Right, or upo^ any Circle, I ObliquQ. • jindufonthefeFomdatims^ the foUming SuferftrHaHres. fi^»iimerri^,r Heights, yCj^^^'^^fS^^^'; or the J Depths, C^ jMmes, Wells, Def- . ' Cpiftaj^ces,*? ^Cburches,Towers,??r. /^ Board* 'Arithmetic}, : Ceametric : j4firgnfmie : r •50 O S V ©f ^ S Flanometria, or tl^ejQlafs, C Or any other Menfurmon of ^Pavement, C Superficies. C Tiling, ^c. J • . ^, C Timber, growing or fquared. Ste^£ometmj or the ) ^^ regular or irregular. M£nfiirmm of ,^cask,commonlycalledGagein5, ^Geodafia, orthcMeafuring of Land divers ways, r Or, the Menfuration of 5 Plain and Triangles, both "i Spherical. , ^Geometry. 1 ^Aftronomy. Triaonometria: "S The Application thereof, in ]P Geography. the folution of Problems in SMavigation. ^Fortification . CDialling, ej*^. •ThePnnciple? t^iere- C The piam Sea-Chart, of, and die man-<-Mwtfwr's Chart, ner of Sailing by <^The arf hof a greatcirde C Sines. Jrhhmticiiyy by the Tables of < Tangents. I ^Logarithms. iGeometrifioUjS^czle^ and I hy |^Compal][es. ^ InftrumentA^ J by the SfeftcMr, Quadrants, Scales, S^. Of N4vigatien : florokgiogrofhia' Or Dialling ; ••5'^«-»~.- r (O Vulgar Arithmetick. PART L NUMERATION. NUMERATION, is accounted tlie Rrft Part of jirhhmetick^ aqd it is-to know how to read a Sunk of Figures exprefs'd in Writing; or to writedown any Sum to be expreffed* ' To the doing of which, there are Four Things ne- ccflary. Hrfl^ To know their Number^ which is Nine^ Secondly^ Their Jfedpei which are i. 2. 3. 4. ^. 6. 7. 8. 9, Of which, the iirfi: toward the left-hand ever iignifieth OnCf the fecond Thirdly^ To know th^ valtie of their places. Lufily^ Ho^ then: ]^ro^r fignifcdtion is altered thereby. The vaJue of their places is thus , when two, three, or more. figures ftand in one Sum, that is, without any Pointy Une, or Cont- ^ betwixt them, as 321, that place next the. Right-hand, where the figure i ftandeth, is called tfie place of Vnily^ or '^itesy and the figure i flandeth in that place onely for Oncy jnd the figure 2 when it is found in that firft place ftands only for Two • and^he like of the reft. But in the Sum 321^ above expreffed, the figure 2 is in the fccond pbce , and every place contains the value of that place before towards the Right-hand ten times , and therefore the fi- gure 2 doth not fignifie two, but (in this'fecond place) ten times ^0, that is Twemy^ and fo die figure 3, if it had been in that Pjace had fignified ten times Tfree, that is 7hiry\ but being n«re in the third place , it fignifies ten tunes Jhinj, that is Three B bundrei 2 NVMERJTION. hundred. And fo the whole Sum 321 1 is to t}e read three hun* dred Ttoemjf and one. • It is hereby feen, how their proper fignifications, which were Jlree^ troo^ and One^ are altered by being thus placed, and the Sum, which otherwife had been but Sixy is Tjree hundred troenty and vnej as before* In like fort if there had been more places, as Seven, the value is quite through increafed ten tunes, by being a place more to- wards the left-hand, as in the Sum i i i i i i i. The Figure i, in the fecond place, ftands for ten times one, ( that is Ten ; ) in the third for ten times ten , ( which is one Hundred ; ) in the fourth for Teft hundred^ (which is called One Thoufand'y) in the fifth, for Ten thoufand ; in the fixth , for ten times Ten tboufandy fwhich is One hundred thoufand ^ ) in the laft ( here the feventh ) place, for Ten hundred tboujandf which is called a Million : And fo on, if there were more places, obferving the fame order. Now to read this readily, make a prick over the place of Vni* g, another the third from it ; and over every third , ftill towards le Left-hand, for fo thofe points will be over the places of V^ nhesy Tboufandsj and Millions ; and fo beginning at the laft, that is, at the Left-hand, read One Million ; and becaufe the three fol- lowing towards the Right, fignifie properly One hundred and ek* ven^ but the prick belonging to them is in the place of Thoufands, call them One hundred and eleven thoufand , and the three remain- ing being under the point over Vnity ^ fignifie only One hundred and eleven \zxA all three points read together in one Sum, is One Million , one hundred and eleven Thoufand , one Hundred and * ^eleven* In like manner, if this Number 73598624, were given to be read, (according to the former direftions ) make a prick over every third Figure, beginning with the firft Figure towards the Right-hand, ( which is the place of Vnitj ) and then will your Number ftand thus, 73598 62 4 Then for the ready Reading thereof, (becaufe the third prick fig- nifies MiBions) call all the Figures towards the Left-hand , ftand- ing from that prick. Millions^ which in the Example are 7 and 3, fo then this Number contains 73 Millionst 598 Thoufand j [624} Six hundred twenty four ; Which in words at length we read. Seventy three Millions, five hundred ninety eight thoufand^ Six hundred troen^ y four. let NVMERJTION. 3 Let thus much fuffice concerning the placing of large numbers, for the ready reading of them ; only take thefe four Tables fol- lowing, for illuftration of what hath been hitherto delivered in words, the very fight whereof is better than a whole Chapter of information. The firft Table is thus to be read.l One in the firft Place fignifies One* One in the fecond Place fignifies Ten. One in the third place, fignifies an Hundred^ &a as in the Table. The fecond Table is thus to be read. 1 In this Table you (hall find the laft number thereof to confift of thefe figures, 357.846. 903. with a point or a comma betwixt every third figure, for diftinftion fake, and alio every three figures in their order are connefted together with this brace \y^\f>J^ which denominates thePlacesof AI/7/io»j, ThoufmdSy Hundreds^ fo that the laft num-r ber of this Table will evidently appear to be 357 Wllhns^ 846 Tboufandsy (^2) ^^^^ /hundred and three. The third Table J is only the Figures of the fecond, fet together, and orderly difpofed, having the fignification or reading of the fame Numbers in words at length to then^ annexed, and is here inferted for the better fatisfaftion of fuch as (hall doubt whether they perfeftly underftand what hath been before taught. The fourth Table] is much like the fecond, only it confifteth but of one number, and extends three Places fiirther then the greateft •number in the fecond t Table doth : v/^. to twelve Places j which figures are thus to be read, j '^6 Millions of Millions^ 842 Millions^ 708 Tboufandy (645) Six hundred forty pve. I. TABLE. r firft ' fecond third fourth fifth fixth fevcnth eighth ^ninth t*2 r I one 10 ten 100 a hundred 1000 a thoufand >'^-^ iccoo ten thoufand 1 00000 a hundred thoufand 1 000000 a Million lOQOoooo ten Millions v-iooooQooo a hundred Millions. B 2 II. TA- NVMERJTION. II. TABLE. 5 8. 7- 2 7 3 8 9 4 4 t 4 3* 7- 3* 9* 6. 6. 7 4 6 7 8 2 9 5 6 8 2 9 o 4 o PS A. 03 8 4 2 3 X 4 7 8 3 • I <A 8 (eight. III. TABLE. 82 1 54 rfSty fburr- •5^3 7^2 (feven hundred fixty two. 4 3485 (three thoufand four hundred eighty three. <=» 5 9762 1 (ninety feven thoufand fix hundred twenty one. v> a 6 1 243794 (two hundred 43 thoufand 7 hundred 94. •g 7 8749807 (eight millions., 749 thoufand 8 hundred »8 <75 1624.8 rfiftv feven millions. 3i<^ fhmifanrl. o>i: 9l and7» 57316248 (fifty feven millions, 316 thoufand, 248. . 357846903 (three hundred 57 millions, 84(J thouf. 903* 7 3 6 8 4 2 7 o 8 6 4 5 IV. TABLE. hundred^ ten CMillions of millions one J hundred <% ten J^Millions one J hundred^ ten S-Thoufands one J <»mm^ hundred'j ten >Hundreds one J A D D I^ / . A D D IT I N. AD D I T I O N, is the colleaing or gathering together of two or more fums, cither of one or of divers deno- minations into one fum, which is called the {jAggregaif\ [^ Totar\ or {Grofs Sum. 1 In Addition of Numbers of one Denomination , the Order Is, to fet the Numbers to be added one direftly under the other ; 5hat is to fay '^Vnhes under Vnitesy lens under Tens^ Hundreds uiM^er Hundreds J &c. RULE. Having placed your numbers to be added in due order^ one under another j draw a line under tbemy. and begin at the lowermoS figure forwards your right handy and add that to the next figure above ^ and the fum of them to the next figure above that \ proceeding in this order^ till you have aided the whole line together : which when you have dane^. confider how many^tens are contained in that line\ and for every tew, keep o^e Unite in your mind^ to be added to the next row \ but if there be any odd Digits, j'(7tt mufl fet them beneath theflroak . jufl under the line you added together. Having thus finijhed the Addition af one line^ proceed to the next\ and from thence to the third ; and ^0 forward^ be there never fo rhany. The examples fol- lowing will make this plain. Example I. Let the numbers given to be added together be 7S32, 5609, 376, 8547, having thus placed them in order one tinder another, as in the Margin is done \ draw a line under them ; then begin your Ad- ^ ^ dition at the lowermoft Figure towards your « "^ Right-hand} faying, 7 and 6 is 13, and 9 ^-^ en <« is 22, and 2 is 24: Now Cbeeaufe in 24 there ' ^ § S'S 13 two tens, and 4 remaining) I place the 4 H ffi H ^ under the line, and carry the two tens to the 7^3^ next Row of Tens j faying, 2 which I carried 5 ^ o 9 and 4 raake'$, and 7 makts i'3, and 3 makes 3 7^ 16 ; in which Row there is but one tea con- ^ 5 4 7 tained, and 6 remaining", which 6 I fet under tiie line, and carry the ten to the' next Row 22364 of Hundreds ; faying, i that I carried and 5 ^akes 6, and 3 makes 9, and 6 makes 15, and.S.m^kes. 23., in ;which 23 ten is contained two-times,* and three remaining ; the 3 I place under the line, and carry the two tens to, the next Row of Thoufands ; faying, 2 which I' catried and* 8^ makes rd, and 6 ADDITION, and 5 makes i$, and 7 makes 22; in which, ten is contained two times, and two remaining ; which 2 I fet under the line, and be- caiife there is never another Row to be added (to which I ftiould carry the two tens> I therefwe fet 2 down alfo under the line towards the Left-hand , as you fee done in the Margin : So the total or ffrofs Summ of thefe l^umbers, being added together, is 22364. * Example II. A Man hatb in bit Orcbard 136 Apple-Trees^ 76 Tear-trees J 107 Cherry-trees^ and 36 Hum-Trees^ and he depres rea- dily to know bam many trees he has in alL ' Place your Numbers one under another, as in the Margin, and then begin to add them together, at your Right-hand; faying, 6 and 7 make 13, and 6 makes 19, and 6 makes 25 ; place 5 under the line, and car- ry 2 to the next Row ; faying, 2 and 3 is 5, and 7 is 12, and 3 is 15, place 5 under the line and carry i to the next Row ; faying, I and I is 2, and x is 3 ; which 3 I fet un- der the line, and ( becaufe there was no Tens in that line) the Total is 355, and Apple-Trees 136 Pear-Trees 76 Cherry-Trees 107 Plum-Trees 36 Trees in all 355 fo many trees are in the Orchard, Other Examples for TraBice. 95432 [ 321 76100 1986 2570 23 832 I 107 Total 174934 Total 3437 9161 235 9 Total 9477 thus cha rafter Addition of Numbers of divers Denominations. I. Addition cf EngUJh Money. The moft ufual Coins ufed in England^ are Founds^ Shilling, feneCi and Farthings^ of which, 4 Farthings! 12 Pence >makc 20 Shillings J For a Farthing we ufe q. RULE. In the Addition of divers Denominations, this Order is to be obferv'dy irfj. Tlase aU Numbers of the Jjme penomination^ one di- ttSJif mifft mnher^ m Pounds twicr Pounds, Shillings,M»4er Shil- lings, '<»• 1 J: /. !• d. f. 37 l6 9, 3, 21 9* 8. I 13 12 9 2 ADDITION, 7 Kng^ Pence Kwirr Pence , and Farthings under Farthings. Tien irsw a line under thevHy and beginjour Addition vitb the leaS Deno- minmm fir St ; Obfervingj bow manf times the next greater Venomi^ nstion is contained in tbat lea^ : And for every time carry one Unite to tbe next Denowmation, e^ before you did tbe Tens^ jetting dawn the l(emainder^ if any be ; Then adding tbe next Denomination tcge* tber^ take notice bow many times tbe next greater Denomination U contained in tbe lejfer •^ carrying for every time\, one to tbe' next greater Denomination. Thus proceeding till you have gone over i& the Denominations^ be they never fo many. Example I. Let the Numbers to be added together be 37/. 16 s. gd» 3 J. — 21/. 9 J. 8i. I j. — 13/. 12 J, 9^. 2 J, Place the Numbers as in the Margin ; draw a line under them, and begin with the leaft Denomi- nation (which in this Example is F^rjjb/w^j) firft^ faying, 2 ^ • and iq. is 3 {• and 3 j. is 6 {• whicliisone Peny, and 2 j. remaining; which 2 g. 1 place under the line , and carry die 1 d. to the next Row, which is the place of Pence; 72 19 3 2 frying, li. and 9^. i$ 10 i. and 8^. is 18 i. which is I s. and 6i. (Now againft the 8, make a Prick with your Pen, for your better remembrance , to fignifie that there is ti u. to be carried to the place of SbiUings^ ) tlien go on, and fay, 6i. and 9^. is 15^. which is is, and 3^. therefore againft 9, make a prick with your Pen, and (becaufe that is the laftNum- ber) I fet down the odd 3 d. under the place of Pence ; and (be- ing 1 find two pricks In the line of Pence^ therefore) I carry 2 x. to the place of Shillings \ faying, 2s. which I carried, and 12 x. is 14 J. and 9 j. is 23 s. which is 1 /. and 3 x. remaining, make a prick againft the 9, and going on, fay, 3X. ,and 16 x. is 19 x. which ( being there is no more Numbers to be added, and being alfo lefs than 20 xj I fet under the line, and finding one prick in the line of Shillings, I therefore carry one to the place of Pounds ; faying, i which I carried, and 3 is 4, and 1 is 5, and 7 is 12; Tet down the 2 under the line ( as in Addition of Numbers of one Denomination) and carry i to the next Row ; faying, one that I carried, and i is 2, and 2 is»i4., and 3 is 7, which being the laft, I fet down ; and fo the Total or grofs Sum is 72 /- 19 x. 3^. 2 J. ExamplelL Let the Numbers to be added be 29 /. 16 s. 8 d. ""B2 /. 1 7 X. 9 ^. — 8 1 /. 1 3 X. ,1 1 d. Here in this Example the leaft Denomination is Pence^ therefore I begin with them, and fay, n d. and gd» is 20 d. which is 1 x. and 8i. make a prick againft tte 9, and fay, 8i. and 8 d. is x6 d* that is i x, and 4 A. make a prick 8 ADDITION. prick againft the 8, and fet down the odd 4:^. Then fbecaufe there are two pricks in the line of Pence) you muft carry 2 s» J to the place of Shillings •, faying, 2 s. which I J. 5. d. carry, and i3j, is ^5 j. and lyj. is 321. which 29 1 6* 8. is \U 12 s. make a prick againft 17, and fay,- 32 I7v 9* 1^^* ^^^ ^^^* iS'28x. make a prick againft x6,* 8i 1311 and (becaufe there is no more Numbers to be . : added) fet down the odd %s. under Shillings 144 8 4 and (being there are two pricks in the line of Shillings) carry two "to the place of pounds; fay- ing, 2 and I is 3 , and 2 is 5, and 9 is 14, fet down 4, and xarry i to the next line ; and fay, 1 and 8 is 9, and 3 is 12. and 2 is 14, which (becaufe 'tis the laft) you fet down 5 fo is the Totaly or grofs Sum, 144/. 8/. 4 i. Other Examples for praSice. J. s. d. «• /. s. rf. 29 18. 7 3 36 2 8. 63 II. 2. I. 29 2 229 4 2 31 16. '9- 3 7 10 I 6 2 5 326 I 8 3 103 2 o * II. Addition of Troy-Wtigbt. trofWeight is a Weight ufed in England^ by which is weigh'd, Breads Goldy Silver ^ pearly &c. The moft ufual Denomina- tions of. which Weight are, Pounds, Ounces, PenjWeigbts, and Grains ; of which, 24 Grains 20 Peny-We 12 Ounces 1 f I Peny-Weightl . , Tpw. igbt > make<i i Ounce f^^^^^^^^Ji tni. J li Pound J rafter dXi^, For a Grain we write gr. The Addition of Troy-Weigh Cand confequently of any other Weight or Meafure whatfoever, either Etomeftic or Forefgn) difFereth nothing at all from the Addition of EngUfl) Coin laft taught, if the Affinity of one Denombation to anotner be ^rft known; for whereas in Money, becaufe 12 d. make is, you therefore obferve how many twelves there are in the Addition of your^Pence, and for every 12 you add i j. to the place of Sbil" lings ; fo in the Addition of iroy-Weight, knowing that 24 gr. make one Peny-Weight, you muft therefore in the Addition of Grains J DD IT 10 N. 9 Qfsim of froy-Weigk^ obfenre how many times 24 yoo find Ui your line of Graim, and for every 24» carry one, to die place of ?tnyivi»^bts\ likewife,. in the addition of Penj^eigbtSf you muft coimder how many titties^ ±0 is cohtain'd in yoor line, and for every 20 carry one to the plae^ of Ounce f^ (becaufe 20 Peny-weighta maJce an Ounce). Alfb in l3ie Addition of Ouhces Troy, yoii liraft daferve' how many times 12 yon find in your linebf Oirifca, and for every 12 cany one to the plaoetyf 'Pe«»4f 5 Then Ikfkly^ Add your Founds together, as numbers of one denomination. ETULtttpU. Let the numbers to be added together be 7 /i. it ou* \lp9. i^gr. — 6 lb. Jou. 16 fw. '^9&'* '^^Ibi joU^ 9^. 6gr4 Place your nmntiers as in A44ition of Aloney, each under other, according to their refpeil^ve Denomhuvtions as in the Mdrginet then draw aluine under ttiem, and begin your Addition with the leafrDenomination ntft,v/^.iGrainsvftyiig^6gr.aad 19'^^ is^ 25 gr. which is qne J?f »^e^fe, and one Gr^ make a prick. agaioft 19, and car- ry the qdd Grain to the Number above ; fiyiug, i.gr.. and J9£n is sogr. which iS 2 19 20 (becaufe it is kfi than ow Peny-Weight J 1 fet under the line ; then finding one prick m the line of Grains, (I therefore) carry one to the place Of Peny- Weights ; faying^ i and 9 is la. and 16 is 26, Y^hlch is one Ounce, and 6frv, make* a prick againft 16, and fay, 6 and 13 is 19, which fbeing lefs than an Otince) fet under^the line j theh for the one prickj carry 1 to the place of Ounces ; ftying, t and 7 is 8, and 7 is 15, Which is one Pound, and 3^ Ounces ^ make a prick at 7^ and fay, 3' and 11 is ^4, which is i ih. 2 p». make a prick againft 11, and fetf down the 2 Ounces, and for the^ 2 pricls carry 2 pounds to the place of Fijundsj faying} ^ afid $ ts 5, and 6 is 11, and 7 is 18, which fet under the place of Pounds ? So is your Addi*-^ tion ended, and the Sum is, i%lb* 2ou. i9'>t9. 20 gr. 0tkef :Epc4mpUs for PraSkice^ ihi -eU. pTPffi ^K I 92 0. . 12 16 Ib^ eui pPf» ■ir- 7 IK 13 »9 6 7' 16. 19. 3 7 9 6 J32 9 17 ii. 6 34 8* 15. 8 to 4 9* 10 7 JbM^ 94 3 18 iS Ib^ cui ,pw. g^ o 10 17 ti O . 6» o X 6 o i9« 84 65 2 19 " f ■ i"n -!■ ■ r • Ii 1 16 19 15 lii. Addl* II JDD IT 10 i<r. j^ Pints 2 Quarts 2 Pottles 2 Gallops 4 Pecks 5 Pecks S Buibels 4 Quarters 5 Quarters j r I Quart ' iPotfle i Gallon } Peck i Buffiel Land-Mejfure I Bijfhel Water-Meafurc. X Quarter I Qialdroa i Wey yil. Of Limg Metfures. long Meafure is that by wliich is meafiired CM, land, Bodrd, Gltfs, Pavmmy Tapeftry^ &c. of which Meafures (according td the Statute of 33 MdWf i. and 25 £//V) a BarJey-Com is tlie 1^: " iSq tjiat, 3 Barlcy-pjrfts 12 Inches 3 Foot 3 Foot 9 Inches 6 Foot 5iyard,ori6iPoot' 40 Perches '■ 8 Furlongs 1, Inch Foot Yard £11 1 Fathom I Pole or Perch' I Furlong I EnglifhMile r^ . yni. Of Ttme. Time confifteth of Teiprs, f/lomh, Weeh, Dmsy Hms, and Mf^ ^mcs. So that, ;' ' ' f - 60 Minutes 24 Hours 7 Pays 4 Weeks »5 Months, I I)ay, 6 Hours 1 Hour I Day Natural ' I Week i Month of 28 Day$ I Year ^ JX. Of Jpotbecaries Weights. The ^eights ufed by 4mbecdrks arc Grms^ Scruples, Prams. and 0tfafW4 of Which, "^ • . ' 1 ?o prains I Scrupbs 1^ Ounces i ,Scruple 1 l^ranfc Cthus cha- 1 Ounce f^ rgflerM I Pound ^^elp of Khefc Tables, and the Rules and Cautions before ex- l|f>e^;i any Man may make Addition of any of the abovefak( * ' ' ' Meafures ADDITION. 1} Meafiircs (me with another, and therefore I ih^l fbcbeas to iHn* fttate them % Examples, but leave them t<> every Man's own pra- nce ; And thus I coaclude Adiitionk the ?fo(f of Addition. \ Having placed your >Iumbers in Qrder, and added diem toge-? ther, and fet the Total under the Line, Cut off die upper Num- ber by drawing a Line with your Pen betwixt that and the o» thers ; then add all the Numbers together except the upper- moft, and fet the Total of them under the Total before found ; Then add this laft Total, and the firft^ Number, which you cut off" with your Pen, together ; and if the Sum of thofe two Num- bers be equal with your Total Sum firft found, then, is your Work right, otherwife not. Example. In the firft Example of whole Numbers, the Sums to be added were 7832, 5609, 376, and 8547 ; diefs Numbers placed in Order, and added together, the Total or GroTs Sum of them is 22364. Now to prove whether this Total be true oc not, I cut off the uppermoft Number (to wit, 7852; with a. Dafh of the Pen, and I add the other three lumbers together, namely, ^609, 376, and S547, and die Total of them is 14532; ^hich Number being added to 7832, fthe JJumbercut off) the Sum of them is 22964, cxaftly agreeing ^ith the Total firft found ; clearly evidencing, that the Addition was truly performed : But if they had difagreed, then the Work had been erroneous. The like Gourfe muft be taken for the Proof of ^ofe Sums which fcive different Denomina- tions, as in Money and Weight ; as by the Examples following will appear. Other Examples proved' 7832 I Total 22364 2 Total i^ 32 Proof 22364 I. Example of Moneys /• s* d> ]• 37 1^ 9- 3" 21 9. 8. I 13 12 9 2 X Total 72 19 3 2 2 Total 35 2 5^3 I n il n il— .1 ' ..ii , n ■ ■■w n «< ?Voof "ji 19 3 2 II. Exampkof Troy-V/eigUt. lb. ou> pvy* ^'•• 33 9- 12 16 25 II 6 9- 34 11. 19. 17 9^' 18 18 60 n 6 " 2 94 8^ 18^ 18 Ther 14 SV EST RACT 10 N. There are other ways to prove MduiaPf by cafting away alt the Nines in the Number of one Denomination, and of all the twelves^ twenties^ suid M»«r, in Founds^ SbilUngSj Pence^ &c. but this, as the moft certain and eafie, I erdbrace ; And diu$ much of A4<litiont and the Proof thereof. ■■ 'M SUBSTRUCTION. SUBSTRACTION is the taking of one or more fmall Sums out of a greater, as js. out of it j. or 37/, out of 100 J. or 137 foot out of 983 foot, and the lik<^ As in Addition^ the Sums to be added may be either of one, or divers Denominations ; fo likewife they may be in SubftraHion^ and the manner of placing jthem is the fame ; for you muft kt * Vnitief under Vnities^ Tens under Tens^ Hundreds under Bun^ dredSf &c. Example I. Of Subftra&im of Numbers of one Denomination, Let it be required to fubftraft 234 out of 986. Place the Num- bers one under the other, as you fee done in the Margin, draw a Line under them, and begin with the Number given 986 firfl: Figure towards the Right-haBd,which Number to be"? is 4, faying, 4 from 6, and there remains fubftraftedj ^* 2, place 2 under the Line, and go to the ^^ --r— — next Figure, which is 3 ; faying, 3 from Remainder 752 8, and there remains 5, pilace 5 under the Line, and go to the next Figure, which is 2 ; faying, 2 out of 9, and there remains 7, place 7 under the Line, and your Subftraftion is ended ; and it is evident by fhe Work, that if you take 234 out of 986, there will remain 752, which you may thus prove. Add tlie 234 to 752, and you fcall find the Sum of that Addition to be 986, which is equal to the whole Sum from v^hich 234 was fubftrafted. Example IL Let it be required , to fubftraft 2975, out of 965-27. Place the Numbers one under another, as in the Margin you fee done ; then draw a Line under them, and begin- ning with the firft Figure towards Vol; Left-hand ; fay, 6 out of 7, and there remains i, place i under the Line, and proceed to the next Figure; faying, 7 out of 2^ cannot, c wherefore you muft always add f 10] to the Number above, which in this Ex- ample is 2, and it makes 12, therefore; take 7 out of 12, and there SVBSTRAeTION. 15 ftere remains 5, place f under the Line, and ( becaufe you ad- ded 10 ta the 2 to maKe it 12, you muft ) carry a Unite to the next Fignre ^ £iying, one^ which I carry, and 9 is 10, take 10 out of 5, which becauTe 1 cannot, therefore I muft add 10 to 5, which makes it 15, and fiiy, 10 out of i{, and 96527 there remains 5 } place 5 under the Line, and (becaufe 2976 you added 10 to 5, to make it 15, you muft therefore^ » carry a Unite to the next Figure; faying, one, which 93551 I carry , and 2 is 3, take 3 out of 6, and there re- mains 3 ; place 3 under the Line, and becaufe there b no more figures to be fubftrafled from the Number above, you muft fay, o from 9, and there remains 9 ; fet die 9 under the Line, and your Subftraftion is ended. " 7. Lent 57^2 l>ai4 ' 378 Other Examples foir VraStice. H^ams of Paper. Bought 9765 Sold 6529 Reftstopay 5384 ] Unfold 3236 Sheep. From xcxxD Take 394 Remains 6q6 SuhjiifuQion of Numbers of divers Penominatms. RULE. In Subftraftion of Numbers of divers Denominations ^ you muU ob- f&rve the fme Method as in Addition, namely ^ to place everv Num- ber in due Order^ rohh reffeSt ta the Denomination^ as Pounds under Pounds, Shillings under Shillings, ^c. the greater Sum always t^moS\ and drarving a^ Line under tJ^fHiy begin with th ka^Pc^ nomination firBf fubfiroBing it from the Line above^ and fitting the Remainder jn^er theUnti as in whole. Numbers ; but if the Pence or Shillings in the upper J($w , be fmaUer than tkofe in the lower ^yy(m wufiaii i2d. ^rjoj* io the fmalJer Number^ thatfo SubflraStion m^ be madey at k^, the Examples foUowitfgwfU more plainly 0fpear* Exampkli Let it jbe required to fubflra£t 382. 125. 8i. from 269 U lisi. ltd. Place yout Hvmbex^ as in the Margin v then, boning, witli the leaflc Denomi- • ^ ,: natioafirft, (which in this Example is r^ I. ' s. d. P«ifc)fgy, Srfrfrom iQrf.]a«dthi?rere- Lent' ^9-1^ 10 mains 2d» let the 2 ^« under the Line, Paid /^8 12 8 and proved to the next denomination, *-r ^^-^ — r— Wm is shiltings ; faying-, 12 1. .out of Rc& . 231 6 2 iS !• and t^ere remain^ ^s. place 6^* un- der 46 S V B S r RJC T 10 N. aer the Line, aad go to the Pounds ; faying, 8 out of 9, yu there remains 1 5 place i under the Line, and fay, 3 out of 6^ and there remains 3 ; then (becaufe there is no more Figures t< be fubftr^ed; fay, o out of 2, and there remains 2, whicL let under the Line. So is you Subftra£lion ^ended, axvd the Re- mainder is 231 /. 6 J. 2 i. Example IL Let it be required to fubflraft 2628 /.. 16 s.io di out of 9320/. 10 s. jd. Place the Numbers in Order, and be-' ginning with the Pence, fay, 10 d. out of 7 i. I cannot ( there* ' fore I muft add i2d. (whkh is: is.) Lent 9320 10 7 to 7^.. and it makes 19 d.) but 10 d.out Paid 2628 16 10 of 19 d. aifd there remams 9 d. fetdown ' the 9 d* under the Line, and ( becaufe I, Refts 6691 13 9 added i2i« to jd.^ I muft therejR)ret| carry one . to the Place of Shillings ; fay- ing, IS. which I carry, and i6/. is 17s. but 175. from 10 s» I cannot, therefore I muft add 20|X. (which is i h) to 10 x« and it makes 30 x. then 1 7 x. out of 30 x# and there remains 13 x« let 13 under the Line, and carry one to the Place of Founds y fay- ing^ one which I carry, and 8 is 9, take 9 out of o , I cannot, but 9 out of 10, and there remains i ; fet 1 under the Line, and carry an Unite to the next place ; faying, i, which I carry, and 2 is 3, now 3 out of 2, I cannot, but 3 out of 12, and there remains 9 ; place 9 under the Line, and carry i to the next place ; faying, i which I carry, and 6 is 7 ; 7 out of 3 I cannot, but 7 out of 13 and there remains 6 ; place 6 under the Line, and carry one to the next Row; faying, i and 2 is 3^ tajcc 3 from 9, and there remains 6, place 6 under the Line. So is your Subftra^tion ended, and the Remainder » 669 1 1. 1 3 x. 9 ^<r Example III« Suppofe a Man had lent to anotho- Man 1000 U and that the Borrower had paid thereof at one time 1 27 /• at ano- ther time 490/. 10 x. and at a third Payment 50 1{. and die Croihx)r would know what he hath- received, wA how much is owing by his Debtor. Place the Numbers as here you fee, firft the Sum of Money l^t, and draw 'a Line .under it i theh • fet the Sums pyid at feve^ ral times, one uj^der another, and draw- a Line under zthota 't Them add all the Sums w^Ckbave been '^pal^ at fe?eral' Times ^together, ^Ai^hifch make 66jt. tos, which is the Sunk Whidi the Debtor hath {«ald-i% di i tbeft-^iUbaft this 667 h Money Iqit 1000 Paid at it "b ^V yeraltiffl^ J ^^^ Paifl in fSl 667 R^fb to pay 332 xo M o o 10 a tmmmmtmm 10 o d^ o o SVBSTRACTION, 17 i6:iU 10 T. from icxxd/. and there will remain 332/. 10 f. and Co moch is ftHl owing tx> the Creditort. Otter Examples for PraBice, Lent Paid 260X 13 98 7 2503 5 d. 6 9 9 Owing in all Paid in all Refts to pay /. 100 36 X. lO d. 6 Refts 63 - 9 6 f Lent /. i. d. 3625 16 8 • 3 ' Paid at feve- ral times ^ ICO 6 J 336 10 } 39 12 9 C 100 6 2 2 . Paid ia ^n 5 7<^ 3.40 Refts to. pay 3049 13 4 3 The Phof of Subfirgaion. . , . The Proof of SubflraBion is performed by Mdiuon, for adding the Number to be fubftrafted to the Remainder, the Sum of tliem inuft be equal to the Ni?mber given , if you have truly wrought: As in the fir ft Example or>{umbers of one Deno- mination. The Number given is — — ^ The Number to be fubftfeOed 986 234 The Remainder is 7^2 ^ A >44i Proof 986 Add the Number to be fubftraflfed^ 2 3 4, to the "Remainder, 752, the Sum of^thom js' 986, equal to the Number ^vcn. . . • I • I • I .• ■ I . D' • Sx.wt' •i-i . iS SVBSTRJCTION. Lent Paid Refts Proof Examples for PraSice proved. 62 18 9 37 19 /^ z' 24 19 3 ^62 18 9 I Borrowed Received Due Proof 100 o o 3^ 13 4 63 6-8 100 o o Other Examples in Weigh and Meafure. EXAMPLE L Cf Tr(ff-Weigk. lb. <m. pw. gr. Of Silver 7 11 13 i9 Sold 5 7 3 5 Unfold Proof 2 4 10 14 7 II 13 19 EXAMPLE IL Avoir dupoit Little Weight, lb. ou. ' dr. Bought 84 12 13 Sold . 26 8 II EXAMPLE IIL AvoirdupOff Great Weight. C. q. lb. oum Bought 37 3 22 II Sold 13 I 23 6 Refts Proof 24 27 37 3 22 II Refts Proof 58 84 12 13 EXAMPLE IV. of time. days ho. m. From 364 23 50 Take 76 9 22 Refts Proof 288 14 28 364 ;23 50 y'j^* Note. Becaufe that Addition ]s eafier than SubfirdSion^ I will (hew yoii how to perform both by the Method of the former : jThus, Begin with the loweft Denomination, Cwhich in this Example is Pence) obferve whether the uppermoft Figure be greater than the lower , and confider what other Fi- gure being added will make an Equality ,^ _ , between them, which note down, for Subftra£l 79 19 10 the Diilerence; faying, as in this Ex- aniple, 10 ^, and is. is iif* (et down (he X ii and prgcccd to the sexti where I ~ *" find From /• S. dm 143 17 11 Piflference ^3 18 SVBSTRJCTION, 19 find the uppermoft Figure is not greater but lefs than the under- moft, and in fuch Cafe you muft borrow Unity fronf the next Denomination , and confider what Number mH^es an Equality between the undermoft figure and that Unity ;' which Ndtnber add to the upper Figure, noting down the turn for your Diffe- rence, always carrying the Unity forward again to its proper De- nomination, as in the reft of this fexample, fayihg, 19 j. and i j. is 20 5. this 1 J. and 17 j. is 18 x. put down the 18 j. and'fay, I /. that I carry, and 9 is lo j now 10 is equal to 10 , the next greater Denomination, tlierefore note down the 5, and fay again, 1 that I carry, and 7 is 8, and 6 makes 14, place down the 6, and fo you have the Difference between 143/. 17 s. 11 i. and 79/. 195". 10 d. appearing to be 63 /• 185. id. — This with PraQice will become much more agreeable and ready, than the ufual Way of Subjhaftiony the Work being done and proved at one and the (ame timf . Queflions performed by Addition and Subftraftion. Queftion i. What kumber U that which bein^ added to l'l6^fiuU nrnk 1000 ? Subftrad 376 from loco, the Remainder is 624, the Number fought. ' . •' • Queftion 2. What Number of Pounds^ Shillings^ avd Pence muH ^9 ddsd to 36 1. 17 s. 3d. to make that Surn'up 100' I? Subftraft 3<^/» 17 J. 3 <:/. from 100/. the Remainder is 63/. 2j. pi. which added to 36/. 17J. 3 i. makes 100/. Queftion 3. In the T^ar of our Lord 1440, the famous Art or %/?f7 of Printing roofi invented^ 1 would knew how long it u fince ^^it Time to thpf Tear of our Lord^ 1699. From 1699, fubftraft U40, the Remainder is 259; and fO many Years are Expired fince Printing Was invented. , ' ' . Queftion 4. jin Army confifling of 13721 Horfe^ 26850 Foot\ i^ an Jngagement there w^te Jlain 3760 Nor fe^ and 7523 Foot ^ ^^' Queftion isy Bow mavy were flaiti in allf an^ h^w many Horfe^ ^d low many foot efiaped ? From tjie 13721 Horfe that went o'Jt, fubftraft the 3760 that were flain, there remains 9961, ^^ To many Horfe efcaped ; Alfo from the 26850 Foot which went out, fubftraft the 7523 which wjere flain, and there' remains ^9327» the Number of Foot whfch cfcapepi ; and by adjding the 37<^o Horfe which were lUiin, to the 7523 Foot that were Jiain, Nr Total is 11283, and fo many were flain in all. D 2 MV LTL (.iO) MULTIPLICATIO N. M ULTIPLICATION is that Part of Arithmetic, which teachetb how tp increafe one Number by another, _ fo that the Number produced by their Multiplicmm^ ihall contain one of the Numbers multiplied, fo many times as there are Vnites contained in the other,. Multiplmtim may fitly ; be termed a Compendium of Addition , for that it performeth at^ one Operation the fame, which to effeft by Additionj would re^ quire many. For inftancc, if it were required to know how much 7 times $ is ? To perform this by Addition, I muft fet feven fives, or nve fevens one under another, and adding them together, I fliall find that either of their Totals fhall contain 35 ; but this by Multiplicstion is performed with fer more Brevity, as^ by Examples hereafter (hall appear. - Before you enter upon%be Prafticeof Multiplication, it is ne- cefiary to remember the Produft arifing by the Multiplication of any one of the nine Digits, by any other of the fame ; as readi- ly to know that 4 times $ is 20, 6 times 7 is 42, 2 times 9 is 18, 7 times 9 is 63, 8 times 9 is 72, ^c. which this Table fol- lowing will plainly declare^ and mufl be perfeftly learned by he^t> before you attempt to multiply greater Numbers. The l/luJtiplication'Table^ i tims< ■ i- makes < \ 9 time^ ) 9 | makes 81 7i^ MVtr IP LIGATION. 21 Tie Vfe of the Table of Multiplication, and the Maner bm it J* to be read. This Table ftieweth what the rum of any two Digits multiplied one by another doth amoant unto, and is' thus to be read, 2 limes 2 makes 4, 2 times 3 makes 6, 2 times 4 malccs 8 ; Alfo 6 times 4 makes 24, 7 times 8 makes 56, 8 times 8 m^kcs 64, 9 tipies 9 makes ^i, ^c. Mother Tibk of MuhiplicMiov. 1 9 8 7 6 5 i 2 r' 2 18 . 14 12 10 t t ■ ? ^7 »4 21 18 '5 ' 32 40 28 31 24 so 20 25 4 3« 5 45 6 54 4B 42 36 7 «3 ,» 49 8 72 .4 — ' — ■ 8 8.| This Table is thus to be read ; In the firft RJiw, or Column, towards the left-hand, and alfo at the top of the Table, you have the nine Digits in bigger Figures than the reft ; the Figures in the firft Column beginning with 1, and fo proceeding by 2, 3, 4. ^c. 109. 'Thofe at the top of the Table, beginning wii.li 9 towards the left-hand, and fo backwards, by 8, 7, 6i iSe. ' to 1, at the iight-hand. ■ JJow if by this Table you i^ould know how much 8 times 7 is, f nd 8 apiong the great Figures at the head of tiie Table, and look iown that Row or Column, till you come againft 7 of the great Figures ip tlje firft Colj*w% againft which you fliall find 56, arjd (0, much is 8 times 7, or 8 intiltigiied by 7, ' '^' -■"■■■■■■■ - In 22- MVLTIPLICJTION. In the Tame manner may you find that 7 times 9 is 63, 5 times 6 is 30, 3 times 4 is 12, and fo of any two of the nine Digits. In Muitiplicatioh there are three terms commonly uf^^ that is to fay ; The Multiplicand^ The Muhiplicr^ The PreduB. The Multiplicand is the Number to be multiplied. The Mult^lier is the Number by which the Multiplicand is Multiplied : And, The ProduB is the Number which is produc'd by the Multiplica-r Hlon of the Multiplicand and the Multiplier together. Thus, if it were required to multiply 8 by 7, here 8 is the Mul-^ tsplicandf 7 the Mult^Uerj and 56 is the product, for S times 7, of 7 times 8, is 56. In Multiplication it mattereth not which of the two Numbers is made the Multiplicand^ or wl\ich the Multiplier^ for the FroduSk produced by eitlier, will be the fame ; but the ufual way is to make the greater number the Multiplicand , and the lejfer number the Muhtplier. x RULE. the Kuihbers to be multiplied muft be fet one under another^ viz. the Multiplicand (or greater number) above^ and the Multiplier (or lejfer number) belowy the laft figure of the Multiplier under the loft figure of the Multiplicand, tte» dram aline under them^ and (having learned the preceding Tables perfeSlyJff heart) multiply every Digit if the Multiplier, tnto every Digit of rfc Multiplicand, fetting the feveral Prodms under the line\ then having finijhedyour Multipli- qition, draw a line and^add all the ProduBs together^ and tbefum jjf thofe ProduBs is the general Produ£l of the whole Multiplication^ as by the following Examples will appear. Examplel. Let it be required to multiply 736 by 7. Firji^ 1 write down 736 the Multiplicand, and under it 7 the Multipli- er, and under them I draw a line ; then I multiply 7 into every Digit of the Multipliqand ; faying, 7 736 Multiplicand times 6 is 42, place 2 under the line, 7 Multiplier under 7, and for the 4 Tens keep 4 ■ . in mind ; then Jay again, 7 times 3 is ■ 5152 ProduB 21, and 4 which 1 kept in mind is 25, place 5 under the line , and keep the 2 tens in mind ; then fay again/ 7 times 7 is 49, and 2 which I kept in mind is 51 ; place i under the line, and the 5 Tens kept in mind (1>ecaufe there is no more figures to be multiplied; I fet down under the line, fo isthe^vork ended, and the Produd of this Multiplication is 5 1 5 2. Example MVLTIPLICJTION. aj Exmple IL Let it be required to multiply 3417 by ?. Waoe the Numbers one under another ^ and draw a line under them, as in the Margin ; then begin your Multiplication, faying 5 times 7 is 3$, place 5 under the line, and keep the 3 tens in mind; then fay again, 5 times i is 5, and 3 which 3417 I kept in mind is 8, place 8 under the line, and Cbe- 5 caufe it is lefs tlian 10, 1 keep nothing in mindj then ■ fay again, 5 times 4 is 20, place a Cypher under the 17085 the Line , and keep the 2 Tens in mind : Laftly, fay, 5 times 3 is 15, and 2 which I kept inmiadis 17, which 17 Cbe- ^ ing the laft number) I place under the line, and fo is my Multi- plication ended, and the Product is 17085. f You may be fatisfied of the truth of this work, if you will take the pains to fet down the Multiplicand 3417, five times one under another, and add them together, as fo many fe- veral fums, fo fliall you find the Total of that Addition, to be 17085, exaftly the fame with the Produft of this Mul- tiplication. Exmple III. In the two fore-going Examples, the Multiplier coafifted but of one Digits we are now to fhew how Multiplication is performed, when the Multiplier confiftsof more than one figure, therefore in this Example, ^ Ln it be required to Multi- 5704 Multiplicand l>/r5704 ^j'37. Place your 37 Multiplier numbers, and draw a line f ■ under them as you fee in 399^8 the Margin. Then begin 17112 your Multiplication in this ■ ' - ' • ■ — mamier ; Say ing, 7 times 4 211048 Produft is 28, fet 8 under the line and keep the 2 tens in mind , then fay 7 times nothing is no- thing, but the 2 tens in mind is 2, -fet 2 under the line, then ray 7 times 7 is 49, fet 9 under the line , and keep 4 in mind, then laftly, fay 7 times 5 * is^ 35, and ^"m mind is 39, which be- ing the laft Numbers to be multiplied, I fet down under the Ime •, foistheMQltiplicatipn of one of the'Digitsfriatnely 7) finilhed. Then begin to multiply the-fecond digit, faying 3 times 4 is ij, piace 2 in the fecondline, one place more towards the teft hand, >nd keep Vm, mind, then fay 3 times nothing is hbthing,^but 1 in mind is i, fet d,awn i by the 2 m-tbe fecond Ijft'e.i thirdly, 6y 3 times 7 is ai, place . i in the fecond line, "^rid Iteep the 2 to in &md. Laftly, fay 3 times 5 is 15, and 2 i^s 17, which 17 (kecaafe there is no more figuresW be multiplied) I'j^late'm the Iccoad line alfo.. « . Havmg 24 MV LTIP Lie JTION. Having thus done, I draw a line under them, and add tbefe two lines together, as in common Addition of numbers of oned^nb'mi- nation ; Saying 8 is 8, place 8 under the line; then fay 2 and 2 is 4, place 4 under the line ; then fay i and 9 is io, place aCyphet under the line ; and carry i to the next place ; faying, 1 aftd i is 2, and 9 is 11, place i under the line, and carry i to the next Row \ faying i and 7 is 8, and 3 is 1 1, place i under the line, and carry i to the next place ; faying, i which I carry and 1 is 2,' place 2 under the line; and fo is your Multiplication ended^ and thePrpduft is 21 1048. Example IV, Let it be required to multiply 57325 by 4032. Place the Multiplicand and the Multiplier one under another, and draw a line as before ;* then proceed 57325 Multiplicand to the Multiplication, as formerly ; 4032 Multiplier faying, firft, 2 tiines 5 is 10, fet *■ ' down a Cypher, and keep i mind ; 114650 then 2 times 2 is 4, and i in mind 171975 is 5, place 5 imder the line ; dien 2293000 2 times 3 is 6, fet 6 under the line ; ■ then 2 times 7 is 14, fet down 4 and 231134400 Produft keep one in mind; then 2 times 5 is 10, and i in mind is 11, which 1 1 rbeing the laft ) I fet down. The Multiplication of one of the Digits being finiflied, pro- ceed to the Multiplication of the next; faying, 3 times 5 is 15, fet down 5 in the fecond Line, one place more towards the Left- hand, and keep i ; then 3 times ^ is 6, and 1 kept is 7, fet down 7 ; then 3 times 3 is 9, -fet down 9 ; then 3 times 7 is 21, fet down I, and keep 2 in mind; then 3 times 5 is 15, and 2 in mind is 17; which being the laft, fet down alio. Two of the figures of die Multiplier being finiQied, proceed to the third, which fin this Example ) being a Cypher , you may wholly negleft, and proceed to the Multiplication of the fourth Figure; only remember to remove the Produft of the fourth Fi- gure one place more to tiie Left-hand, as in the Fxample you may fee, for the Cypher, though it be not written down, yet it muft keep its place, and the Figures following muft be reaio- ved a place arther. Then for the Multiplication of the fourth and laft Digit ; fay, 4 times 5 is 20, fet down a Cypher, ( under .9 ). and keep 2 jn mind: Then 4 times 2 is 8, and 2 in mind is lo,- fet do^^a Cypher, and keep 1 in mind ; then 4 times 3 is 12, and i is 13, fet down 3, and carry 1 ; then 4 times 7 is 28, and 1 kept is 29, fet do^trn 9, and keep 2 ; then 4 times 5 is 20, and 2 i^ 22 \ which (becaufe the Mvltiplicatjon is ended) letdown alfo. Having MVLTIP LiqJTION. 25 Having thus multiplied all the Digits feverally, draw a line un- der their Produfts, and add them altogether , as in the former Example ;. fo fliall you find their general Produft to be 2311^400. Other Examples for PraSice. 73260 45003. ii II •^ 219780 366300 293040 3296919780 I 50762 4567 355334 304572 253810 203048 I » . 2318^0054 The proof of Multiplication. The moft certain Proof of Multiplication is by Divifion; but becaufe Divifion is not yet known , I will here Ihew a near Way, by whith Multiplication may be proved . Which is thus ; RULE. MdkeaCroCs, of in the Margin \ then any Sums being multipUei you mdy prove the Truth of your Work in thps manner ; ( I ) CaH atva^ all the Nines which you can find in the ^ JF Multiplicand, what renaineth fet on the right Side of V Jfe Oofs. ( 2 ) Caa aroay alfo the Nines in the MuU ,/% tiplier, and what remains fet on the left Side of the ^^ofs. ( 3 ) Multiply the Figure on- the right Side of the Crofs^ by ^ on the left Side^ and out of that Prodaft caS away all the Nines alfo^ fetting the Figure remaining, over the Croh: Then (4)' «5 away all the Nines in the Prodaft ; and if the Fmvre remaining °^ the fame with that which Jiandeth over the Crofi, then if your Multiplication truly performed^ otherwife notn ^ * Example. Let it be 'required to prove the Sum in the Margin. 4324 . I; Caftaway ^11 the Nines in the Mul- 23 2 tipUmd ; faying, 4' and 3 is 7, and 2 Is ' ■ . '\X • 9) which being rejefted, there remains 4, 12972 5 jC 4 ^Jich I fet on Uie righ^t Side of the Ccofs, 8648 /V • *hen> . 2 2. Caft away all the Mnes in the 09412 mtipiier i fayins, 2 and 3 is. 5, (which E being / 26 MVLTlPLlCAriON. being lefs than 9J I fet on the left Side of the Crofei Then, 3. Multiply 4 by 5 ; faying, 4 times «5 is 20, from which caft away all the Nines^ which are two, and thefe remains 2, place 2 over the Crofs. And, 4. Caft away all the Nines in the Produ^ ; faying, 2 and «; is 7, and 4 is II, caft away 9 and there remains 2 : Which exaft- ly agrees with the Figure over the Crofs, and demonftrates that the Multiplication is truly performed. Conlpendiums in Multiplication. 3257 2600 19542 • 6514 846820X3 1. if the MuhipUcr confifts of Cyphers in the laft place or pla- ces, you may omit the Multiplication of them^ and place only the Figures of the Multi- plier under the Multiplicand : Thus, if it were required to multiply 3257 by 26cci ; Place the Numbers as you fee in the Margin; Then mul- tiplying 3257 by 26, the Produft will be 84682, to which if you add two Cyphers, (becaufe tliere were two Cyphers in the Multiplier ) it will be 8468200, which is the true Produft of the Multi- plication. 2. If it be required to multiply any Number by 10, 100, lOGO, 'loooo, ^c. You have no more to do, but to' add fo many Cyphers to the Multiplicand^ as there ai"e Cyphers in the Multi" plier': Thus, if you were to multiply 365 by 10, the Produft will be 365Q ; or by 106, it would be 36500 5 or by" 1000, it would be 365000 ; or by loooo, it would be 3650000. 3. If any Number were to be multiplied by 5, you may ab- breviate your Work thus. Add a Q^pher to the Multiplicand^ take half that Number, and it (hall be the Produft required. Thus, if were required to multiply 8627 by 5, add a Cypher to the Multiplicand^ then it is 86270, the half whereof is 43135, which i& the Produft required. 7o multiplj by any of the nine Digits^ mthoiit bunhet^ ing the Memory. To multiply any Number by 2 ; Either double the Number In your mind, or add it, by fetting it twice down 5 fo 57325 , produceth 1 14650. ' ' To multiply any Number by 3 ; To the Number given add th6 double thereof,the Sum is the Produft ; fo 5 7325.pr oduceth 1 7iV75-' To multiply any Number by 4 ; Double the Duplicanoif in your mind J fo 57325 produceth 229300. To MVLT IP LIGATION, 27 To multiply any Number by 5. Conceive a Cypher added to tlie given Number, and in your mind take half thereof for the ^ Prodaft, thus a Cypher added to 57325 , m^eth it 573250, the half whereof is 286625. To multiply any Number by d. Add a Cypher to the gireiv Number , and take the half, to which add the given Number, thatfum ihall be the Produft. Thus 57325 produceth 343950. To multiply any Number by 7. Take half and add it to the doable of the former Figure , fuppofing a Cypher added as be- fore; So 57325 thus ordered, produceth 401275. To multiply any Number by 8. Double each former figure and fubftraft it from the following, fo 57325 produceth 45860Q. To multiply any Number by 9. Suppofethe Number multiplied by 10, then fubftraft each former Figure from the following, beginning with that next before the Cypher, the Remainder is the Produft, fo 57325, produceth 515925. Other Brief J(uJes of MuhifUcation* Having fliewed you fome Compendious ways, of multiplying by Anick'NumBersy as by 10, 100, 1000, ^c, and alfo by the nine ^gitSy without any trouble or charge to the Memory: 1 will n^w lew how you may expeditiouflv and certainly, multiply any Sum by divers other Numbers coitfming of two or three pla.ces, with- out fetting down many Figures but the Produft itfelf. To multiply any number bytu RULE. Set tie Multiplkmi dovm twicer removing it one place to the left^ W, add them together , thefum is the J'roduffofthe Number multi^, Pb^ihyiu Example, let it be required to multiply 97 by i.i. Set down 97^twice, removing bne of them a place more to the left hand^s you Jee here 97 JP the margiti, then add them together, the 97 funJls 1067, which is equal to the Product of ■ 97multipliedby II. 1067 to multiply any Number by 12, 13, 14, 15> i^> 17> 18, or 19. RULE. To cffeft this, You have no more to do, but to multiply ^ given number by i, 2, 3, 4 5, 6, 7, 8, or 9, and in yourMuU B 2 tipUcation^ 28 MVLTJPLICJTION. tiflicationy amtinuaUy to add, that figure of the Multiplicand^ which ftandeth on the right band of the figure you are multiplying bjy fitting dopm tbefumfor the figure of the ProduB. Example. Let it be required to multiply 3624 4^ 17. Multiply in this manner by 7, Saying, 7 times 4 is 28, fet down 8 and any 2 — '• then 7 times 2 is 14, and 2 which I carryed is 16, and 4 (the figure of the Multiplicand which, ftands on the right-hand of 2> is 20; Set down o, and carry 2 then 7 times 6 is 42, a'nd 2 which I carried is 44, and 2 (which ftands on the right-hand of 6) is 46 ? fet down 6 and carry 4 then 7 time^ 3 is 21, and 4 which I carried is 25, and 6 (which ftands on tiie right-hand of 3 ) is 31, fet down i, and carry 3, which 3 added to 9 (the left-hand ngure of your Multiplicand,) makes 6, which fet down ; lb the Produft of 3624 multiplied 17, is 61608. In the fame manner may you multiply* by 12, 13, 14, 0>f. as in thefe Examples. 3624 17 61608 62^ 16 lOOOO 4793 91067 To multiply any Number by 102, 103, 104, X05, jo6, 107, 108, or 109. RULE. Multiply the Number given by 2, 3, 4» 5> ^k 7» 8, or g^fetting the ProduB two Places towards the Htght-hand of the Multiplicand, that Produft and Multiplicand added together in the fame Order Si theyftand, fbaU be the Produd of tke whole Multiplication. Example. Let it be required to multiply 3924 by 106, Set tliem down as in the Margin ; ti^en 3624 multiply 3624 "by 6, it produceth 2i744> 106 which added to 3624, in the fame Order as >■ they.ftand, the Sum of that Addition will 21744 be 384144, which is equal to the Produft pf 362VnJultipliedby X06. ' *^ ^84144 ether MVLTIPLICJTION. 29 Other Examples for TraBice. a7<55 103 <5374 107 2?95 44618 78795 682QIS • 7b muhiflj any Number by 112, 113, 114 115, 116, 117, 118, or 119. Multiply the given Sum by ij, 13, 14, 15, 16, 17, 18, ^ 19, 4f ib^ib ^eew Jieww already , ywi»g ribe proittd^ two Places to the J^gk'band of the Multiplicand 5 then add the Produfl and the Mul- tiplicand together^ in the fame Order as thp fiandj fo JhaB the Sum ^ that Addition be equal to the Produft of the Multiplication. As IS evident by the Examples following* Multiply 4065 By 113 The Produft multiply'd by 1 3 52845 Real Produft 459345 Multiply 7632 By 119 The ProduQ multiply'd by 19 1450x38 Real Produft 908208 ^eftions performed by Multiplication only. Qucftionl. Jf a Piece of Lani be 236 Perches long^ and 182 • Perches broad ^ how many fquare Perches are contained therein f Multiply 236 the Length, by 182 the Breadth, the Produft is 42952 ; and fo many fquare Perches are contained in fucli a fguare Piece of Land. Queftion II. In a rear there are 365 JDays Natural^ and in eve- 7 I>ay 24 H^s J Jffow many Incurs be there in a Tear ? Multiply 3^5 the Number of Days, by 24 the Number of Hours, the Produft is 87^ } And fo many Hours be there in a Year. Queftion JO DIVISION. Queftion III. From London to Coventry u accounted 76 miles ; IioiV9 many Tards therefore is it from London to Coventry ? Multi- ply 1760 (which are the Number of Yards contained in one Mile^ by 76, the Product is 133760 j and fo many Yards are be- tween London and Coventry. ■*■ «i DIVISION. % I DIVISION is juft the Contrary to Multiplication, ibr that turns Small denominations to Greater,as Multiplicatioq turns Greater to Smaller : Or (in whole Numbers, of which only We yet fpeak) Divifion is the afldng, how many times one fum is contained in another ; and the number which anfwererh to that queftion is called the Quotient. And the Number containing, which is to be divided, is called the Dividend. And the Number contained, or by which the Diijideni is to be divided, is called the Vivifor. And as often as the 2?iv/i^e»i contains the Divifor^ fo often doth the Quotient contain Vnity. So that, As Multiplication is a Compendium of many Additions; So Divffion is but a Compendium of many Subftractions. There are feveral ways by which this Rule ' of Divifion may be wrought ; of which , fome are more eafie than others, to be performed, and the ^uJe made more intelligible to the Learner. Whereft)re, rejefting thef Old and tedious way, by often fetting down the Divifor and Cancelling of Figures : I will only infift upon Two or Three of the moft Plain and Eafie Ways ; whereby this Difficult I^ule of Divifion may be wrought with much eafe and perfpicuity. And whereas I faid before, in Muhtplication^ thsit the beft Proof thereof was by J)ivifion\ So I fay here, Tlwit the beft Proof of Divifion is by Multi Plication : I will therefore take for my Examples . in this Jjiuky theConverfewiththofe which were formerly done in Multiplication'^ whereby the Proof of each J^w/f will be evident. For the Working of Divifion the moft Plain and Eafie way, this is the RULE., I. Set down the Dividend ( or number to be divided :) And on tho Left-hand thereof fet the Divifor (or number b^ which you are to Divide ; mtb a Crooked Line between them ; Jjmy 01 the High* band DIVISION, ji hdniif the Dividend mdkeanotber Crooked Line, wherein the figmres of the Quotient are to he placed : So that^ If 162483 were a Nnwh ber given to be Divided by 1^21: The Numbers muft be fet as here ym fee ; And the Divifion ended^ the Quotient will be found to be 123 • So that 132 ( the D'lvKov ) will be contained in 162483 (tfc Diyi- dend^ 123 Times: for that is the Number in the Quotient. Divifor Dividend Quotient 1321) 162483 (123 II. Demand (or aflc ; how often the Divifor may be had in the Dividend , and place that number in the Quotient : Then^ Multiply the Divifor by the Figure in the Quotient; and place that FroduS under the Dividend, and Subfha^t it from the Dividend, ftting the He- mainder under the Produft, and draw a Line under iu Them make a Frick under the next figure of the Dividend and bring tha figure down to the Remainder ; And then proceed as before* Example I. Let it be required to Divide 5152, by 7. F^5, Set the Numbers down according to the former Dire£U- ons, and as you fee done in the Margin ; And becaufe 7) 5152 (736 7 the Divifor J is greater than 5 the firft Figure of • • • the Dividend^ make a prick 1 Prod. 49 under the fecond Figure of ^1 ^ Rem. the Dividend^ H|niely, un- — ___— der I. Then, 2 Prod. 21 Secondly^ Ask how many 42 2 Rem. times 7, the Divifor , you ■■' : ■■■* can haveNin 51, the two 3 Prod, 42 lirft. Figures of the Divi- 00 iend? and the Anfwer will ^ 7 times ; wherefore put 7 in the (^otient, and multiply 7 the Divifor f by 7 in the Quotient , and tlie Frodufl wijl be 49 ; which fet under 51- Then fubftrad 49 from 51 : and there wiU •remain 2, which fet under 9, and draw a Line under it. Tbirdlyy Make a Frick under 5, the third Figure in the Dividend^ and bring that Figure 5 , down to. 2 tlje firft I(emainder^ making it 25; And a& again, How many times 7, the Divifor^ can you have in 25? the Anfwer will be 3, which fet in the Quotient^ and Biultiply the Divifor 7 by it \ faying, 3 times 7 isi'i^foj: tli<; fecond FroduB^. which 21 fet under 25, and ifubftra^Ujk 2l from 25, the JS^matndcr '^iW be 4, whic^ i^c under 2j, ana draw i Line under it* - ...... ' fourthly'. 52 , DIVISION. Fourthly^ Make a prfck under 2, the laft Figure of the Dhidenif and bring that Figure 2, down to the Second Remainder 4, ma- rking it 42 ; and then afk, How many times 7, the Vivifor^ you . <an have iii 42 ? the anfwer will be 6 times, which fet in the -Qu^ienij and multiply the Divifcr 7, by it, faying, 6 times 7 is 42, for the Third E^emdndery which 42 fet under 42, and Sub- ftrafting one from the other, ( vi^. 42 from 42; the Remainder will be (00). So is your Divijion ended;, and the Quotient is 736, which ihews, that the Divifor 7 is contained 736 times in the Dividend 5i52;For if youMttfe/p/y 73^ by 7, xhtprodu^ will be 5152: As by the Firft Example in Multiplication, ' This Firft Example hath but one fingle Figure to the Divifor and fo is more eafie and clear, and will be a good introduftion to thefe which follow, . Example II. Let it be required to Divide 21 1048, by 37. Set the Numbers down as before, and as in this Example. Ahd becaufe 37 the Divifor ^ is greater than 21 the two firft Figures ' 37) 211048 C5704 ^ . . . « "^m X Prod. 18$ 160 . I Rem* I 1 I 2 Prod. 259 148 2 and 3 Renii 3 Prod. 148 "^ 04 Remaind. of the Dividend J make your firft Prick under i, the third Figure in your Dividend : Then, Firft y afk how many times 37, can you have in 211, the anfwer will be 5 times •, (or how many times 3 can you have in 2i, and. then the Anfwer will be 7, for 7 times 3' is 21 ; 'but then, although you may have 7 times 3 is 21, you cannot have 7 times 7 which is 49 in I ; and therefore you muft not put 7 in the Quotient^ but a lefs Figure, fuppofe therefore 6, then 6 times^3 is 18, from 21, and there remains 3 ; but then again 6 times 7 makes 42, which cannot be taking out of 31, and fo a leffer figure than 6 njuftbe put in the Quotient, as 5,) And now to proceed: Saying; ^ How many times 3 can you have in 21, the Anfwer will be 5 ^es; Put 5 in the Qmient^ and multiply 37 the Divifor by 5, the Product wiU be 185, which fet under 21 1» and fubffarafiing l8{ from 211, the J^iiMi»ier will be 26 ; whichfet under it. Secondly Makea Prick under o , the next Figure in the Dividend. and bring that' o down to 26, making it 260} and afk, How many D if^ IS TO ^r. n many times 37 caii you have in 3^0, (or how many times 3 in 26) The Anfwer will be 7 times J Set 7 in the Qumem^ and Multiplying the Divifor 37 by 7, the ProduB will be 259 j which fet under 260, andfubftraaitfrom 259, fo will the J^m^iV/ier be i. Thed^ . tbirdJjt^ Make a Prick under 4, the next Figure in the Dividend and bring that 4 down to i, making it 14, and draw a Line under it: And a(k how many times 37 the* Divifor^ can you have in 14 the l(emmder^ the anfwer will be never a time, ( it being lefs) wherefore put a Cypher in the Quotient^ and fourthly f^ Make a Pritk under 8> the laft Figure in the bivideni^ and bring that S down to 14, making it 148, and aik, how many times 37 can you have in 148^ (or how many 3 -in 14J the an- fwer will be 4 times; Put 4 in the Qumem^ and mnltiplying the D/w/or 37, by 4, the ProduB will be 148, which ^Qt under 148, , and fubftrafting it therefrom, there will remain o« So js the ^ivi^m ended, and' the Qumcnt is ^704^ and fo many linies is 37 the D/v/ybr, contained in the Dpuidend 211048: For if you fflnltiply 5704 the Quotient^ by 37 the Divifor\ the ProduEt will be 21104B, As in the Third Exampky of Multiplication, which proves put'Vivifion to be truly performed . Example III. Let it be required tq Divide 23 11 34400, by 57325. Set the Numbers down as you fee here done, making a Prick un* to 4, the Sixth Figure of the Dividend^ and begin your Work as ^e: faying, 57325; 231134460 (4032 t trod. ' 229300 '"' \ 183440 • 1 and 2 Hem* >*M« 2 and 5 l?rpd. 171975 . I 14650 3 Rem- ^h l_MMia*^i*a* 4 Prod* 1 14650 \ '•'■ ' •• • o ' '4 Rem* Kr/f, How manyTimes 57325 (tl)^ Divifor) can you have itl 231134, cparfof the' 'Mvidend) 6r more calie, HoWmaft^ times i^34> under which dr^w.a Line...... Secmdly^ Make a Prick under 4; the, next^tigure in the Divi* «»i, and bririgrthat'*4tiowilco 51834^ making jitTi 8^44; Apd jflcn^ik how many times 573-25 can you^havr^iOj i8^44>. t|ie Amwer is,:acyeri'a:timc^. it being. thfc.gce^ter-.Iftwmfctrrthere-: ... F fore 34 6 ly I S ION. fore put a Cypher in the Quotient, and making a prick under the next Figure of the Dividend, which is o> bring down that o to 18944, making it 183440; And then, ' TUrdij^ Mkhovf many times 57325 f the Divifor) cart you have in 183440, or (more eafie) how many times 5 can you have in i8» the Anfwer will be 3 times ; put 3 in the Quotient, and multi- ply 57325, (the Divifor) thereby, and the Pro^uft will be 171975, which fubftraft from 183440, and the Remainder will be 11465 ) under which draw a Line : And, Fourthly f Make a prick under the laft Figure of the Dividend, . which is o, and brings that o down to the laft Remainder, ma* ki&g it 114650 ; And aik how many times 57325 can you have in 114650, (or how many times 5- in 11 ) the Anfwer will be 2 times; put 2 in the Quotient, and multiply the DiVifor 57325 thereby, the Produft will be 114650, equal to the laft Remam- derl So is. the Divifion ended, the Quotient being 4032 ; and fo many times is 57332, the Divifor, contained in 23113440O) the Dividend : As will appear, if you multiply them together, as in the Fourth Example of Multiplication. Thus you fee how Divifion proves Multiplication, and how Multiplication is proved by Divifion. But this kind of Divifion may be proved otherways than by multiplying the Divilbr by the Quotient : By this RULE. tftcn your Divifion if endedy add all ffe Produfts refulung in the whole Work together ^ in the fame Order as they fiand dn the Work ; the Sum of them (adding the laS J^emainder^ if any be) Jhall be enual to the Dividend. So in this Third Example. . • • The prB Produft is 229300 Thejecond and third Prod. . . 171975 • The fourth Produft . . . 1 14650 The Sum equal to the Dividend. 1 31 1 34400 And thus much for this Way of Divifion : Another Way, not much unlike it, followeth. A Second Way of Divifion.' ExmfleM Let it be required to divide 768325, by 324. Erft, Set down the Dividend between two ParaM Lines ^ and Oe Divifor on the L^-bmit within a Crooked line^ and another Crooked DIVISION. J5 Crooked line on the High-band for the Quotient : All this as in the former way ; Then having made a Prick under the third Figure, 8, of the Dividend, as you fee done in the following Example ; You ©ay begin your DiviiioA thus, 121 llemaiu 445 2312 • 1203 • • r Divifor 324) 768325 (2371 Quotient • • • •. • • • 648 972 •• 2268 • 5^,4 Sum 768325 Equal to the Dividend. firft, Afk how many times 324, the Divifor, can you hare in 768, the>*three firft Figures of the Dividend, the Anfwer is 2 ; fet 2 in the Quotient, and multiply the Divifor thereby, and the Produft will be 648 ; which fet under 768, and fubftraa it there- from, fetting the Remainder 120, over 768^^ Secondly, Make a Prick under the next Figure in the Divi- dend, making die Remainder 120 to be 1203 ; Then afk. How miay times 324 in 1203, the Anfwer will J)e 3 times ; put 3 in the Quotient, and multiply the Divifor thereby, and the Pcoduft will be 972, which fet under 1203, "and fubftraft it tliercfroip, fettingtheRemainder 23J, over 1203. r v. r^ a a Thirdly, Make a prick under the next Figure of the Dividend 2, making the Remainder 231, to be 2312 ; Then afk, How manv times 324, can you have in 2312, the Anfwer will be 7 times • put 7 in the Quotient, and multiply the Divifor 324 thereby, the Produft will be 2268; which fet under 2312, Ihb- ftraaing it therefrom, and fetting the Remainder 44 over 2312. Fourthly, Make a Prick under 5, the laft Figure of the Divi- dend, making the Remainder 44. to be 445; 1 hen afk, How of- ten you can have the Divifor 324 in 445, the Anfwer will be 1; put I in the Quotient, and bv it multiply the Divifor 324, which will be the feme ;" fet' therefore 324 under 445rand lubltratt it therefrom, fetting the Remainder 121 over 445- . And thus is your Divifion ended, the Qiiotient being 2371 -, and fo many times i^ 324 contained in 7^^8325, and 121 remain- ing: Which you may prove, -by multiplying 2371 by 3^4^ ana S6 DIVISION. adding 121 to the Produft; for that Sum wiB be equal to 76832$ the Dividend. But the Proof of Divifionmay he done moreeafilyin this Way of Divifion. For If you add aU the ProduSs together^ snd the fan J^emainder^ tf any be^ the Sum of that Addkim wiB be e^ual to the Dividend. • Examples for Praftice in both thefe Ways of Divifipn x^Ay wrought do here follow ; • ^ r Examples of the Jpirn Way. Divifor) DivM. (QuoticAt, 2325J 7153258 (3^8 ^•*< ^975 <5575 4650 19258 ^8600 $58 Remaindep tppi«*iMm«i Proof' 7.^3258 Equal to the DivWc^ Another Example^ 70993) 42009876 (591 « • • 3549*5 ^51337 (538937 124006 70993 53013 ^ Remalii 43C30j87f ISxim DIVISION. Mxmpks of tbeSeami mj. 17698 L Remainders, 7447 -J wMf PIvjftr 5<J78) 2345"578 (413 Qpotlent • • • 22712* ••! 5678 • IProdufh I 7034 J 2945014 Hemaind, add 664 Proo^ mmi 37 234567S Equal to the Dividend; J)ivi(br 542) 76353 (140 Quotient 221 Itemaind, add i*^— ■■'• |f|8:}prodaa, 75880 475 76353 Eqnal to the PlyMpuJ, P0ST4 ' < . ^ g8 DIVISION. I ill I. POST-SCRIPT T O DIVISION: OR, A RULE. B Y Which you may certainly know what Figure to fet ia your Qpotient ; and nevtr to take one too great, w too littlCy but that, which will )uft ferve : AND AUb how to perform Twith Eafe zM Certainty j the mofl: difficult Sum that can be propofed in Divifiort^ without the Afliftance of MukipUcation ^ only by j4d^ Stian and Sittfirall^m : Not burthening the Memory at all. IN the fn&ke of Divifion, there is nothing more difEcult, than in large Sums (efpecially if the firft Figures of the Di- vifor be either i, 2, 3, or Cyphers, and the laft Figures 7i 8» or 9) to know certainly what Figure to put in the Quo- tient) when you demand how often the Divifor may be had in the DIVISION. J9 the Dividend j for the certain finding whereof ( a little Pains be- ing taken before you begin your Work; do thus; Sappofe you were to divide any Sum, as 1097909, by' ^09 J Firft, fet down the nine Digits, i, 2, 3, ?^c. one under another; and againft the Figure 1 1 1 309 fet 309, your Divifor, which doubled is 618, 2 61S which fet againft 2 ; thefe added together ^ ^27 make 927, which ftands againft 3 : Add the ^ 1 1236 Divifor 309 to 927, it makes 1236, which is ^ i^^^ againft 4 ; to this, add the Divifor, and it ^ ^g^^ makes 1545, which ftands againft 5. And y 2163 thus to every laft Number, ftill add the Di- g 2472 vifor, tiU you have gone through all the nine ^ 27^1 Digits ; then will they be as in the Margin. Having prepared this Table , fet your Dividend and Divifor down, as in the Firft Way of DivHion, pricking the Dividend, and drawing a Line under it, iis is there directed , and as yott fee here done ; 3C9) IC97909 C3553 ' 927 1709 XH5 1640 M45 959 927 32 Then laying your little Table before you^ look in it for io97» fte four firft Figures of the Dividend, which you cannot exadly finddiere, but the rieareft Number lefs, (\^hich you muft always take when you cannot find the juft Nuniber you look for) is 927, againft which ftands 3 5 fet 3 in the Quotient, and fubftraft 927 out of 1097^ and there will remain' 170, to which bring down 9^ the next' Figure of your Dividend, and it is 1709 ; Look this Number m yout T^We, Which you jcahrfot find , but the next lefs is 1545, againft which ftands 5; let .5 in the Quotl^t^ and fubftraft 1545 out of 1709, and ttieire will remain 1^45 towftich fang down the nejctf Figure of your Jpivideiid ^which i^e is o^ waking it 1640} Look this 1^40 i& the Table, which you can- not J" 40 DIVISION. iibt find) but the next lefs is 1545, agaiaft which ftands ^% (e€ 5 in the Quotient, and fubftraft 1545 out of 1640, and there will lemain 95 ; to which bring down the laft Figure in your Divi- dend, which is 9, making it 959 : .Look this Number in die Ta* We, or the next lefs, which is 927, againft which fends 3 * fee 3 in the Quotient, and fubftraft 927 from 959, the Remainder is 32. So Is your Divifion ended, and the Quotient is 3553 t^-. And with what Eafe and Certainty this is-efFefted, no'Multipfi- tation being ufed, I leave to the Reader to judgei * Compendium in Divifton. to divide my N'umber by 10, lod, or ;oob, &c» ^ RULE. // frond the Ifmber to be divided, you cut off fa inmy Fhures towards the Right-hand, as there are cyphers aper the Vnit m the DiVifor \ Then the Figures towards the Left-hand (haU be the Quo- tient, and thofe cut off, the Remainder ; So^ 4^5oi|divide<iby| xoSV^O-jf {S'j RemainsI I 791634 J. LlCDOoJ ^""^'^ I791J 1.53^ to divide any mmber retdily, by any of the nine Digits.^ ' Tirft, The Digit i, neither multiplies nor divides anv Num* Iier, but leaves it the fame : But, Secotdly, To divide any Number by 2. It is but talcing the ^n .i ^ ^^"i^l'^^V ^ 730 divided by 2, the Quotient .Will be 365 i for the half of 730 is 365. ^ ■ TKri^, To divide any Number by 3 j Take one third part of the Number for the Quotient : But if the Number eiven can* not be parted into 3 equally, the Digits remaining are Thirds of the Vntt : So if 1095 were to be divided by 3', the Quotient would be 365 ; but if 1097 were to be divided by 3. ^e Quo. tjent would be 365! ; for one third part of 1097, is>36<.Tnd there will 2 remain, which are two Thirds. ^ '' • Fombly , If you you would divide any Number by a 1 One fourth part thereof is the Quotient, and the Remainder r if anv be) are Fomhs : So ufio^ivided by .4, the Quotient i^ gl? ; but i4<53 «»J»ded by 4, wiU give for the Quotient 36s i. ' * ^ Fifthly, If you divide :any Number by < ; Double the Num- ber, cut off the Figure towards the Kight-bmd, and the Figures towards the UH-hind flull be the Quotient, and the Figurt cut ©ff the -Remainder, oiF^bs; So if you would divide 1^25 by 5; •the DIVISION, 41 the Quotient will be 365 ; for the double of 1825 is 3650^ and iht Cypher abated^ the Quotient is 365. Sixthly, If you would divide any Number by 6 ; Take half the given Number, and one third part thereof wiR be the (^o- tient : So if you would divide 2190 by 6 $ the half of 2190 is 1095, one third part thereof is 365, for the Quotient, Seventhly, If you would divide any Number by 7 ; double the Number given, and cut off the laft Figure towards the Right- hand ; then take the (eventh part of that Number, and double it, and fubftrad that double from the former Number,, the Re^ mainder will be the Quotient : So if you would divide 2555 ^Y 7, that Number doubled is 5 1 10, from which cut off the Cy- pher towards the Right-hand, and it is 511, one feventh nart whereof is 73, the double whereof is 146, and that (uBftraa^ from 511, the Reipainder is 365, and that will be the Quotient of 2555 divided by 7,. EigKthly, If you. would divide any Number by 8 ; T%ke half the Niimber given fucceiTiveiy three times, the third half (hall be the Quotient : So if 2920 were to be divided by 8 ; Firft, the half of 2920 is 1460, the half of 1460 is 730, and the half of 730 is 365 ; and that is the Quotient of 2920 divided by 8. Ninthly, To divide any Number by.9 ; Take the third part of the given Number twice fucceflively, the fecond Remainder (hall be the Quotient : So if you would divide 32S5 by 9 ; One third part of 32S5 is 1095, and one third part of 1095 is 3651 Which is the Quotient required. Quefli(ms perfotmed by Divifion cnlj. Queftion i. If A Viae of Land, lying in a Jwg Square or PardJ- hlognmy contain 4,2^ '^2 fqiiare Fercbes^ and one of the fides tbererf be 236 Perches longy bono hngmufithe other fide be? Divide 42952 by 236, the Quotient will be 182, and fo many Perches long muft the other (ide be. Queftion 2. Jn a Tear tbere are %']6oBGursy and in every natural J>ay there are 2^ H<mrsy /demand bow many Days be tbere in a Tearf Divide 8760 by 24, the Quotient will be 365 and fo many Days be there in a yean Ooeftion 3. The' dtftance from London to Coventry is 13376a r^irif, and in oH^ Mile there is contained 1760 Tards^ notv I would yktfb)mmany',MksitisfroinLonioRto Coventry 5 Divide 133760 by 1760 die Quotient will be 76; and fo many Miles it isiirom Londonto Coventry. Thefe Qjicftions performed \yj Divifion only, are the converfe of t&ofe tlac werepejfonnedby iBtf/tr7/>&ario», which Itheratber G make' 42 REt)VC T ION. make choice of, that the Reader might fee how MuJtiplkation aii4 JPivifion prove each other. ' There are one or two more kinds of Divifion^ foipething like there laft, but I (hall forbear exemplifying them 5 for Variety helps to ipake a Book rather great tliaa ufemU f Here is to be noted. That in the following Rules, where there is continual life of Divifion, I fometimes 'inake ufe of oiie Kind, and fometimes another: But ithe Praftitloner may ufe which he is beft Ikill'd in, for they ill produce the fame Effea. ' -^ • - / REDUCTION - » S tWQ(bl4 ; flrft, That which turns Great Denominations in- to Smalbr^ a^' Pounds into Shillings' or Fence \ this i^ doner by MuhipUcatim i As followeth. • ExmpU I. tn 729/* itx. 2918a ■ 7 Let it be a(ked,. H[ow many l^ence are contained Hrft, A Shilling is contained in ^ Pound 20 timers ; therefore multipiy 729 by 20, or f whic^ is the fame multiply but fomewbat Ihorter) by 2, and put 6 tb the Produft; as in the Margin, this fliews, that in 729/. there are ^i 14580 Shillings. To which add iiiw it makes 14591 Shillings. - • • • * " Shillings -■'> . muhiplj / Again ; !^caufe one Peny is con- tain^ in one Sl^iiUng 12 tunes, mul- tiply 14591 by 12, it produceth 175092, to which add the 7 pence : So the Sum will be 175099 ; and^(b many Pence are contained in 729 /• Mi ixmph ih let it be alked. How many Pints are contained in Uk 4 Tuns, I Hoglhcads, 'tod 27 GaDuns? ■' ^ REDVCTION. 4} tlrft, One Tun is equal to 4 Hogfheads, therefore 4 Tun is equ?! tu 16 HogCheack ^ to which add the 3 Hoglbeads» fo there is 19 entire Hogiheads. Agdin^ Becaufe I Hogfhead contains e$3 Gallons, multiply 19 by 69, itproduceth 63 1 197 Gallons, to which add 27, it gives 19 muh^ff 1224 Gallons. -■ ■ 1 Laftiy, becaufe every Gallon contains 567 S pints, multiply 1224 by S, it produ- 63 ceth 9792, anci fo many pints are con- . ■ tained in 4 Tuns, 3' Hoglheads, and 1197 27 Gallons. 27 ddi After the fame fort might dry Mea-* ■ fures be reduced^ as (Xiarters to Buihels, 1224 GdBof^s Pecks, or Gallons; and likewile Weights 8 nuUt^lj and Outlandifh Coins, of which the pro* ■■ ■ ■■■ portion of the greater to the leffer is 9792 (before) knowA' or given. Secondly, it is often rec^uifite to turn SmaBer denominations to Greater: This is done by Divljlon^ as followeth. Example^ I. Let it be aiked how many pounds are contained in 80976 Ihillings? Dividt 80976 by 20, the quotient is 4048/. and i6i. iemainin(» which is the true Aufwer. 20) 80976 (4048 pounds '. . • • . , r \ ■ ■■ ' L ' ■ I 91, ..... 89 ; ^-^ 160 ^ '• 16 ShiUifigi EkAwpkj tl. Let it be afked how many pounds are in io97$4i. ? became One pound contains Qne ihilling 20 times, and One (hilling contains One peny 12 times, therefore if 102754 be divi" d^firft by 12, the quotient fliallbe 9146 Ihillings and 2 Pence over; then if 9146 be divided by 20, the quotient is 457 pounds and fix ihilli(igs remaining) fo that 109754 peace is equal to 457/. 6s. 2d. G 2 - X2) l__l 44 REDVCTIOKT, 12) 109754 (914^ •WW- • • • • 108^ X7 12 55 48 74. 1 .'i- 20) 9146(457 Pounds^. • • • j^fc- Or if 109754 had been at fi3f "divided by 12 times 20, that h 240, (which is the number of p^ce contained in a pound) the quotienthad been 457 pounds and 74 pence remaining) which is all one with the former ; io 74 penc^ is equal to 6 (hillings and 2 pence. More inftances Ihall not iieed herein, bccauTe the Uiing of it felf is very clear. . ■ '^ , - » I V »i • > p ^.^ - - ' r f 45 i > ■ PROGRESSION IS alfo of two SQilt; the )!^of cmain Nutnbefs io Aittrng- tual PrtponhH fr6m i ; thit h ftich' as diiSer e^ully^ as t; b, 3r 4* 5> ^9 where the common Difieredce is i /a^ h eafity ieien) or i, g, 5, 7, ^9 ii^ whefe the coAimon diflfereiic^ is:^^ or any other, as i, S, 15, 22, 291 ^, where the a^ittumdiferetiGe is 7^ this is called Arithme^al Pngriffim. Secondly, of certain Numbers in CemtPfica PtofX^tfiaa from I'l that is, ftich as increaffc by a common Multiplitatioa^ as i» 2» 4» % i^f 3^i ^^^i^^ th6 coiitmctt Multiplier ii 29 tfoit is the iirft multiplied by 2 produceth the fecond, and the fecond by 2 producttfi^ the dilrd, and fo on* . Or as 3,9, 2^, 81, 243, where the coinmon Multt^ier is 3, this is called Geometrical Propeffum* fioth the common Diilerenc^ ( in the iirft ) and the common Multipllcatipn (in the latter ) iliall b$ (hortneg hec^ftcr be ci^cithe CoiAmofi tikcefi. Firft, howdf the firft ibrt, or JrUhmetUdl Prosi^Jfrn^ the prin- jifiihmftical • I cipal ufe of this is, I., ftbe number of Places and Common Excefs be given^,to fni tbe I/ft.nuniBer. . , 2. When tbe Number of Pluses M \bt Uft Nk0tr Is ^in^ to fU ibe'Jggr/^itteior total Sum of nt the Nhmn: 3. WSn tbe l^ Number^ and tbe Torai Sum is glven^ id put, tbe Nmiber of Pi(v:ts. i 4.' The lAimber ^fPlisce^ and f of id ittW tein^ gl'Otni tofnH iSe Jafi Number.^ - 5i TUldJi JfiMer^ and n^ ta'finttheJom' mm t^efii- ' ^fbe Ijfi^-Nuihb& and Comtn Exctfs <beiiig giv^n'^ topiitbe number, of Places. I will inflance in no mwe/feWOf thdfc ev^hapjpSinfii^to be ufed. ' For the firft of thefe, let there be given, , The Number of Places 100 The common Excefs i ■ ■ , ■ " ' i To find the laft number alfo > i 1 ■ ■■ ■ icx? RULE. ifi P ROGRESSIONi R U L Ei y MukiPtj tbe lumber of Places Ufs by i\ bj tie "Cawmm Exceftf mi to the Froiu& add tie frft Number: tbe Sum U e^udl to the laft Number. So here, multiply 99 by i^ the PrbduA is 99, (for i neither Multiplies or Divides) to this add the gift Number I9 it gives |po for the laft Number. . ' : [ Or let the Numbers be i, 7, 13, 19,25, 31, where the Com^ mon Bxcels is 6, and the Number of places alfo 6. Now if the Number of places lefe by i, that is 5, be liiultipli- j ed by the common excefs, which is 6, the ProduA is 30, to which^ ji adding the firft Number which is i, the laft Number jUi is there- by compofed. This is fo eafie diat it needs no proof. i. For, the fecond, which is, Tbe laft Number^ and the Numbed of plofies given t to find tbe Total fum of all tte Number u RULE. Mi the frft and laft numbers together^ and mult^ly tbe Sum by half tbe number of places^ tbe ProduS U equal to tb^ aggregate ^r Sum^ if tU the numbers added togeiber. So if the firft Numbo: i ^be.added to the laft Number 100, it gives loij which . multiplyed b j 50 (^whid|f is*J[|ilf ^ number of places^ produceth 5050, which is equal to all dbe hundred num" bers added together* ; And hereby may that vulgar queftion be anfwercd which is, A' amanjake up 100 ftones placed ayMd one fromanother^ all in dJ(i^t Line by one at atime^ and bring tbem ba^k one h one to, bis frftfiandingi ifow many yards dotb be go bachfo^ and forwards ^.v It is (hewed before that he goes forward 5656 Yards, and he muft needs come back juft as mi^ch, that is, in all loioo yards, which is 5 Miles and 3 quarters; wanting 20' Yards^ Or fecondly, fuppofe the numbers were i, 9, 17, 2$, .33, 41* Whereof the common excefs is 8, the firft ^nd laft added is. 4^ which multiplied by 3, (h^f the number, of Places) the Prqdua is 126, which is the fum of them all. 3. For tlie third thing, that is, By the laft number and tbe T^dl to find the number cf Places. • ' RULE '*> PROGRESSION. 47 RULE. , Add tbefirfl and Ufi J^umbers^ and bj the fim 421) 12^ (3 if them divide the totals the Quotient mU be eqiul * io bdif the Number cf places. / ■■ ' This 4s lb plain, it needs no clearing. 126 4. For the fourth. If the Total and l^umber of places be gtven^ f fnd the Ufi number. RULE. M * Pivide the total bjf f^alf the Number ef places^ the Qmiem is S ' f^her^ jrcm'robkh if i be takenj the reft it the lafl Number. As let the numbers be 1, 3, 5, 7, 9, 11, 13, 15, or any other (in jtrithmetical proportion J whatfoever. The fuih of thefe is to be 64 and 4) 64. fi6 the number of places is 8, the half * of it 4. Now if 64 be divided by 4 the qutftietit is 16, froiil which if 1 4 be taken' there remains 1^5 ibr the 24 laft number. — — — . * 24 5, Now for the fifith variety, If thelaft number, and the number of places be given, to find the common e?:Cefs ' .( ^ RU L E; From ike laft Nmber tale i^ and the remain JhaU be the Dividend \ then from '^he number of places aljp take i, and make ttis latter re- main the Divifor\ then the quotient of this Divifion Jhatt be the cowh mm excefs. -^- > ; * ' Example^ Let the numbers be 1, 4, 7, 10, 13, 16 from 16 take 1, remains* t^ for' the 'dividend, then from 6 (which is the Nunh ber of places; take alfo i, remains 5 for the pivBbr* Now when i«} isdhrided by 5,the quottentls 3. 5) 15 (3 And 3is'alfof the' common 'cxcefs, or diflference betweexi i and 4, or 4 and 7, &r- ■ ' 6. Laftlff Let the laft number; and the com- 15 jnon exceis be given to find the number of places. • RULE. 45 PROGRESSION. RULE. '<v- JTrdff ^hc l^ nutffis^ t^ i, /pti4ividi theremdinisrhy%he common exifefs^ then io the ^imient a4d i, ibefum is the number of places, A§, l?jt the numbers be i, 5, 9, 13, 17, 21, 25, 29, feom 29 take I, remains 2S, which divided by 4f which is the cxcefs) the quotient' is 7^ to which add i, the fum 4) 28 {j is ^ Which is (he num])cr of the plaas, ts tb-* Reader may eafily count. — _ 28 7& I is 8 Geometrical Progrejfion. I CbHjt^l not be fo large in this as in the former, becaufe thefe things are of litde ufe to the Arithmptician^ except where a &um- ber is to be soaoy times doubled, tripled^ or the like, which an* not be fo eafily abridged here as in the other, becaufe there the laft number arifing of many. 4iMons( of dn^ excels to i, was eafily found by one Multiplication: but here the laft number be- ing made by many Multiplicatims of the excefs, is therefore many times harder then the other. T^ TSkxueties here (^11 be but two u The common excels, and qutnber of places being given, to find the laft number. 7^ ThQ Ei^^eTs, apdlaft Mumber being given, to find the T(h T^ fici: qf tl^^fe may tbus. be fouDd. Let the Numbers, be 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, the excefs is*2, the places IP, fi^ ouf tbci fifth Nurrtier (wWich is c^y done for any one xwy*'reclqonrp.6r.byh^^rf)that:isherei6, and multiply 16 by 16, it produceth .25<f, w^ich is the ninth Number : Lanly, multiply 2($6 tf the excpfs 2, t)^^iu;e arifetb 512, the nuQiber defired. So if the plac^ had h^n x^re, as 72, having foumC the 9th Numhcc 256, multiply it by 25$ thence comes 65536 for the t7th Nunjiljer, which/mqltiplied by the excels 2, giv<» 131072 for the 18th place, which m«ltiplic;d by 131072, gives I7i79869i84y for the 35th place ; and that multiplied again by the excels 2 gives 1 34359738368, for the 36th place, that Multiplied by 34259738368 I the produA will be 11805916207741 1327424, for the 7iftplacc, which laftly multiplied by the excels, gives 2361183241434822- 654848, i PROGRESSION. 49 654848, for the 72 place, which is the laft Number of the Pro- greffion required to be found. I Perhaps this may feem (bmewbat tedious, but where things can- not be performed without Labour, the Reader muft content him« felf with fuch Rules as make it lefs ; for it is certain, th^t this way is much (horter than to have Multiplied ftill by the Excels 71 times, which elie he muft have done. ' Allthisnotwithftandingyhe is not bound toufethe fame Num- bers, much lefs in other Queftions where the Number of places is not the fame: But whereas I began from the place 9, he may begin at 8, 10 or 1 2, or where he pleafcs, fo as he remembers .ftill wheie he is ; for this is general. If the I^umber belonging to my fkce rohdtfoever be Multiplied by it felf^ the Produd jbdll be tbe Number belonging to twice Jo many places roa^t onepace. Now for the lecond thing, which is, to find the Sum of all the Numbers, RULE. Frm tbe laft Number take tbe firfl^ and Divide tbe Hfmain by tbe. Ixcefs want i , then Multiply tbe Quotient by the £xctfsy and to tbe freiuS add tbefirft Number^ the Sum of them u equal to tbe Sim of iHftfctf Numbers. Soif from the lafl Numbers, of 72 pUice, be taken i,theRemaia is 2961183241434^22654847, which (hould be Divided by i» (that is the Excefs want 1, for the Excefs is but 2) but becaufe I neither Multiplies nor Divides, that Labour is faved ; Now Multiply this Remain by the Excefs, the Produft is 472236648286^ 645309694, to which adding the firft Number 1, by making the ''igure 7, next the Right-hand to be 6, you have the total Sum of an the 72 Numbers. A Qtieflion refolved by Geometrical Progreffion. A Londoner fojouming in a Country Market Town in Winter^ ^de bimfelf d new Iree\ Suit and Coat , on which were fet 6 i^ of Buttons of Silk and Silver '^ a. Baker being in bi4 Company liked it fo welly be would buy it of him \ the Citiien confentedto let ^bave it J paying for tbe frfl Button a ftngle Barly-cornj for the faoni 2, for tbe third 4, and fo on doubling to tbe lafi. The Bargain was liked on both parts for the prefent , but; fcjrtly after revoked, for it could not be performed, and np Maa <^a be holden to an impofijbiUty« But why this could not be performed, maybe judged ; Firft, ^y inquiring the worth of fo much Barly in Mony : And fecond* , ly, the weight of it ; and how it fliouid be removed. ^ ^=^' * 5d P ROGRESSION. 1. For the firft, allowing loooo Corns to a Pint ( which is more than enough) then 512000 Corns make a Quarter; and yet (for fihortning the Divifion) we will allow 10000000 Corns ' to a Quarter 5 by which Dividing the whole Number of Corns ' (which is done by cuting oflF the firft 7 Figures towards the right hand) the Quotient will be 472236648286964, and fo many whole Quarters there are, omitting the Rema'm, as in this cafe inconfi- derable. Now allowing Barly were to be fold at 1 ^i. theBulhel (whidi is cheap) it is ib many Angels \ and therefore dividing by 2, it is 23^118324143482 pounds Sterling; which is in words, Two hundred thirty fix millions of millions, one hundred and eighteen tiioufand three hundred twenty four millions, one hundred forty three thoufand four hundred eighty two Pounds, which I take . to be too mudi for any Tradef-man to get or keep. And reckoning Land for ever at twenty Years Purchafe, if this Sum of Pounds be Divided by 20, the Quotient is the yearly Rentbf 11805916207174 Pounds. • • * • And this Divided again by 564 (the Number of Days in a Year) the Quotient is 32344975918, that is above Thirty two thou- fands of Millions a Day for ev^r. So great a Vanity may be con- cluded on for want of a little premeditation. 2. Now fecondly, for the weight of it, if we* put 8 Bulhels to weigh 2 hundr«i weight, (for fure it doth weigh more) then the whole Number of Quarters Multiplied by a gives the. Weight of all to-be 944473296573928 hundred weight, and* if this be Divided by 20, (which is but cutting off one Pigure towards the Right-hand, and Dividing the reft by 2) or which is all one, cut off one figure fix)m the Number of Quarters, the Quotient 472^23664828696 is fo many Tiiris. And therefore it . will require 427223664828 Ships of a 1000 Tuns a piece to carry it: Add confequently, if every Nation in the World had above • loooo fuch Ships, yet there muft be abdve four Millions of fuch Nations ; which I fuppofe are not to be found in this World. And here I will leave this, having ufed this long £xample» Cwhich though it require more labour as all great Examples do^ yet the fame skill will do it , as tf the places had been Fewer) that the Reader being throughly Exercifed thereby , may the cafier leap over others which are fhoxtsu THE I 5^ THE GOLDEN RU L E, ,: OR, Rule of Three DireiSl^ THis is the jtnoft ufeful and moft eafie IfjiU in Arithmttklt^ and deferves a Golden Ndme: It is, nvhen there are three Numbers given or known, to find a fourth in propor- tion with them. But 4 Numbers are in proportion , and called ProponiiffM! , when, as the firft is to the third \ fo is the fecond to the fourth. As if there were given 3, 4, and 6j to find a fourth, which nay be to 4, as 6 to 3, that is double, and that fourth Number js 8 ; and tliis is called Prcpmion DireB ; and the Rule whereby it is done. The VireB K^Je. There is alfo ^another Proportion called Kf$iprocdl \ which is, when, as the firft is to the third, fo is the fourth to the fecond: As 3, 4,. 6, and 2, this is called The K^erfe J^ule. In DireB Proportion^ the Produft of the two middle Numbers Multiplied together, is ever equal to the Produft of the firfk and laft Multiplied together, which ferves not only for a Proofs but a ground of the Bfik^ which K^k fliall here follow : The i^erfc fyle being deferred till we have done with this: And for Wpr-> king thereof, this is the - RULE. Nluhipty the Second I'erm (or Number) bj the Third, and Di- vide the toiduft i^ the titfi ; the Quotient Jfctffl be the Fourth Num- ; ber defired. [ Example. Let the three Numbers given be 2, 6, 3, Multiply I 6 by 3, theProdaft is 18 •, then Divide 18 by 2, the Quotient is J 9, which is the fourth Nurriberin Proportion with 2, 6, and 3. For, as 2 to 3 ; To 3 times 2, which is 6 , is to 3 times 3 which is 9. And fo the Produflt 1 8 Divided Jby 2, and the Quotient 9, caufeth that the Produfl cf 2 into 9 ihall be alfo 18, and confequcntlt Hz if - 52 The GOLDEN RVLE. if 2 be the firft of the 4 proportional Numbers^ and 6 and 3 the two middlemoft, then 9 is the laft ^^ . ^ Otberroifi. Divide tbefepond by the firft ^ ondMuItipfy tbe third by the Quotiem^ the Product jhall be the fourth. So if one .Divide 6 by 2, the Quotient is 3, by which Multiply 3, the Produft is 9, for the fourth Nymber, as before. Otiicr- Nways this Rule might be expreffed, but where the firft Way is fo mort and clear, there many other ways would rather trouble than help the Peribn that fliould ufe them. In the firft Way (which here we mean to ufe, and no other) I tS the firft Number be i ; then the Produft of the fecmd and third gives the/owti, without any Pivifion : Or, if the fecondot third Number be 1, then there needs no Multiplication^ but Di- viding the greater of them by the firft, the Quotient (in whole Numbers, for yet we (peak of them) is the fourth Number which was fought. But, Note I. To know vohen to ufe the DireB^ or tbe ^erfe J^le* Confider, if More^ require More ; or if Lefsy require ftill Lefs, then ufe the Direl^ l(Hje* But if More require lefs^ or Lefsj More^ then ufe the I(everfe i^/cj this willbeeafily underftood when we come to Example. Note II. To hnovf how to place tbe three Numbers when they are confufeilf given. Remember that 2 of then! are always of one Denomina- tion, as both Pounds^ or both Sheep^ .or both Tardsy or jicres ; and the other Number hath another Denomination: Now know, that this fingle Number is ever the fecond Number in order. And one of the other two, nanlely, that which hath feme itelation to this fecond i^^i^the firft '^ and the other is the third Number, ^ whole relation is'fought for in the fourth \ whence it's plain, that the fecond and fourth are aUb of the fame Denomination. ^ And having Premifed thefe things, let us now exemplifie the Rule in fome Queftions. *f<f. Quefthm I. If three Yards of Clothcoft4 /. what fliall 21 Yards coft ? Set the Numbers in order, as in the Example. If 3 Yardj coft 4/. what 21 Yards ? Here you fee that the firft Number and the The GOLDEN RVLE. 55 the third Number, arebotib of oneDenominadoo, vtx* both Tgrdtp and the fecond Number is of another Dmominatioo, namehr Founds^ where&np tbe fourth Number which is fought for, mvSt be alfo Pounds \ therefore multiplying Caccording to the Rule be- fore given; thcfecond Number by the tbird^ and Dividing the Produ£l by the firfly the (^otient ihall anfwer the Queftion. Firft, 21 jyiultiplled by 4, (which is the third Number Multi* Sied by the fecond) produceth 84, which Divided by 3 the firft umber, the Quotient isaS/. and fomuch ihall 21 Yards coft; For 28 & to 4, as 21 to. 9, feeing each contains other 7 times* And the Work wHl ftand thus. Tardt U rards If 3 coft 4 what 21? 4 I) 84 (28 Founds. m • 6 24 Quefikn IL If 4 Men eat 2 Pecks of Com in a Week, how many Pecks ftail ferve 100 Men ? Place your Numbers as here you (ee, then Multiply 100 by % (that is the third Number by the fecond) and the Produft is 200^ which Divided by 4, the Quotient is 50, for the Number rf Pecks required. tUm Pecks Men If 4 eat 2 what icx>? 4) 200 (5« . . 2CX> Queftion III. If 20 Sheep coft 13 Pound 13 Shillings 4 Pence, what is that for every Sheep ? Turn 54 TAe GOLDEN RVLE. ynoL tbe ShiUiflg^ and Pounds into Pence ; thus. Multiply li J. by 12, the Produftis y ^ .156 And 13 /. DV 240 (1)ecaufc 240 pence is a Pound ■ 3120 To whidb and the 4 d. ■■ ■ '4 ■k It makes in all 3280 Then the Queftlon wiflbe. If 20 Sheep coft 3280 Pence, what Ihall one Sheep coft ? Sbeep d. If20 coft 3280, sheep what I? > 20) 3280 • • • (i64Pe»re« 4 20 128 12) 1^4(1 88 12 44 80 : 8 36 . 8i. Bjr the Rule before delivered, I ihould Multiply, the lecond Number by the third, but in this example, the third Number being i, it doth not Multiply ; I therefore Divide 3280 the iecood Number, by 20 the firft Number, and the Quotient 164 is the Price of one Sheep in Pence, which Divided by 12, the (Xiotient is 13 x. and 8 d. remaining, the price of every 9ieep therefore is 13 x. 8 i« ^ Quefiion IV. How many 10 iiidi Tiles will pave a Floor that contains 16 fquare Yards? Firft, remember there are 56 Inches in one Yard in length ; which Multiplied into 36, gives 1296, for the fquare Inches in one fquare Yard; Multiply 1296 therefore, by 16, thence comes 20736, the Sum of all die 16 Yards in Inches. SeccMidly, feeing every Tile is 10 Inches in length, and 10 in breadth, Multiply 10 by 10, itproduceth icjofor the fquare Indies in one Tile j then, by the Golden Rule (ay. If The GOLDEN RVLE, 5J . Jncks tile tiles If icx) require I, ; what 20736? 207.36 ^ Here, Epcaufe i doth neither Multiplj nor DhUif ('as YaA been feveral times intimated} therefore. Divide the tbki^ Number^ 20736, by the Firft 100, (which is done by cutting off two Figures to the Right-hand) and tiie Quotient is 207 and 36 remaining. So it appears, that 207 i$ too little, and 2o8 too much to do tto Work: The juft Number being 207-j44, butwefhall not tronbte the Reader with this till he know fomething of Fraftions, Suefiion V. If iopi. gives 6/. Intereft for a Year, how much (hall 750 1. &Ye? MoltipJr 75<3 by ^» the Produd is 45oo> which Dhrided by 100, the Quotient is 45 /• for the thing required. If 100 give 6, what 750? 6 * J 45.00 Pounds. Queftion yj; If ^50 /. gives 45 /. Intereft for a Year, what fhall 100/. give ? Multiply 45 by 100, the Product is 4500, which Divicfed by 750, the Quotient is 6 /. for the Intereft of a 100/. for a Year. /. /. /. , If 750 give 45, what 100 ? 45 • 1 1 " ■■' 750J 4500 (6 P<nm4fm .45.00 Many other Queftion^ might be added, but the Rule is fo plain^ &t it needs them not; and fo general^ . that he which un re^ [(dye one, may as well refolve any other: And for that reafon, and becaufe in all the Rules which follow, this Rule will bc coa*. ftaQtly madeufeof, I will fiiy no more of it here. T2v I ' 5« I The Golden Rule Reverje. m F 12 Workmen do any piece of Work in 8 Months, how many Workmen ihall do thefame in 2 Mondis ? RULE. MuUMj tie firft Term by the fecond ; £nd Hrvide the ProduS by the third, the Quotient it the Nkmbet defired. Here 12 is not the firft Number, though it be firft named ; but the l:hree Numbers placed in order, ftand thus, 8, 12, 2, for the middle Term muft always be of the fame penomination with that which is required. Now Multiply 12, by 8, the Rmlud is 96,'which Divided by 2y the Quotient is 48, which anfwers the Queftion, As in this Example. Mmths Men Months 8 12 . 2 8 HM ») 96 (48 • • 8 16 16 For, If 5 Months require 12 Men; then (z fourth Part of 8) 2 Months, ihall require four times 12, that is 48 Men. For here Left requires More ; that is. Left tim^ More hands \ and therefore it is wrought by the J(£verfe M^ik. Queftion IL How many Ells of Tapeftry willferve to hang a Room 3 Yards high, 6 Yards long , and 5 Yards broad ? Not regarding Doors, windows or Qlimney, but as if there were no fucn. Firft, Multiply 6 by 3, the Produft is 18, which doubled (bc^ caufe there are 2 fides called lengths) is 36 Yards for all the lengdi. Secondly, The GOLDEN RVLE. 57 S^pndly, (for thefaoie reafbn) Multiply 3 by twice 5, that is by to, the Produft is 30 Yards, for ^ the breadth ; whidi added to 369 gives -66 Yards» equal to all the length and breadth in Yards, But now becaufe Ells^ that is , lUmifi) Ells (for fuch Meafuie are Hangings fold by) is equal to 3 Quarters of a Yard, that is, their Ells is to our Yard as 3 to 4. Say therefore, if 4 giVe 66^ wliat3 ? "Multiply 66 by 4, it produceth 264 ; then. Divide 264 by 3, the Quotient is 80. Again, Multiply 88 by 4, and Divide theProduft Cwhich is 352) by 3, the Quotient is 117, and i re- maining, to which the Divifor 3 being applyed ; the Number joft- ly aofwering the Queftion is 117 £)ls, iind one third part or m Ell Note I. Becaufe here we had to deal with things which had equal length and breadth,that is fquareYards,and fquare Ells,therefore one Multiplication and Divilion was not fuftcicnt to propottion this; Bot if inftead of working by 4 and 3, we bad done it by their Squares which is 16 and 9, it might have bfen performed at once-; thus Multiply 66 by 16, the Produft is 1056, which Divide by 9) the Quotient is 1 1 7 f , as before, but I began not with this way^ fivlfuppofedmy Reader ignorant of Squares* ■ i^^ILlt might ftlfo have been done,by reducing all theTerms into Quarters of a Yard at the firft, and after the Number is foanil, reducing them again to Ells j but becaufe it is more proper to-- vork thus, till Fradions have been taught; I leave that, and proceed to another Queftion. Quefim 111. If I Clofe would graze 21 Horfes for 6 Weeks: (foppofing no wafte to be made^ how many Horfes would it feed fcr 7 WeeJK ? Multiply 21 by 6, it produceth 126, which Divfded by 7 ^ Quotient is 18. At that rate therefore it would keep 18 Horfes for 7 Weeks. Qjiefllon IV. If i Clofe will feed 18 Horfes for 7 Weeks, how long ftallitfeed 63 Horfes? Multiply (according to the Rule; 18 by 7, the Produft is 126, which Divide by 63, the Quotient is 2, therefore 2 Weeks it (hall fcep them. The like way ferves for Hay, Oats, or any other Provlfion for Man or Beaft ; which may be ot ufe in Garrifonsj and fuch *like cafes where fcarcity may be feared, to proportion either the Months to ^Meatf 01 Meat totheMowix. 1 Befott 5» Tie GOLDEN RVLE. Befiire I leave diis Rule, (becanle it comes not fo much in ufe and Praftice as the direS l(^k doth, and therefore may be more apt to be forgotten J I will, to exercire the Reader herein, propofe the fol- lowing Queflions, giving the Anfwers Of them, and leave the PraAice to the Reader to find out of himfelf, the better to fix it, the Rule in hjs Memory. Queftkn I. If 12 Men wx)uld raife a Frame in todays; in how many days would 8 Men raife the iame ? Here, becaufe the fewer Men would require the longer time, thou^ the Number be 12, 10, 8^ yet you (hall (by obferving what hath been already delivered in this Rule^ find the fourth pro- portional (which is the Number anfwering the Queftloni to be x<, and fo many Men will do the Work in 8 Days. Quefikn XL If 60 Yard of Hangings of three Quarters broad would hang a Room ; How many Yards/of half a Yard in breadth would ferve to Hang the fame Room ? Anfwer Ninety Yards, ^ftiou HL If a Board being 12 Inches in breadth do require 12 Inches in length to make a foot Square ; What number of inches in length will make afoot Square, when the breadth of the Board is 16 Inches? Anfwer 9 Inches. Queftim IV. If the Bafe or end of any Solid (zs a piece of Tim-' ber or Stone) being 144 Inches, do require 12 Inches in length of that piece to make a folid Foot ; What Number of Inches in length will make a folid Foot , when the Square at the end is 216 Inches. Anfwer, 8 Inches* • I will fay IK), more of thb Rule; Neither will I treat of the J^wble Jfjile of Thne^ as a Rule by it felf; but come to the J(uk of five Numbers^ which is aii Abridgment of the other. Thi S9 ■rfUbi The Golden Rule Compounded (f five Numbers. . SKfiim h YF a hundred pound weight ("that is xi2 pdttil I weight) carried 126 Miles coft 14 x. how much JL fliaU three quarters of a hundred fthatls SfpouidJ toft; being carried 4omiles ? kULE. Multiply tbetbrceUft Numbert wetmotnmhery ^4r«r) rfcdurd hy tfe fourth, mi thtt ProduS by the fifth; tbe UJl Produ3Jb4Bb$ tfc Dividend. Agate, Multif^ the two firft Nitmhtrs together*^ the ProduSfidl he tbe Diviror* This DMfioH being made^ Ae Quotient witt be the Number of SbiBings de fired. Example of the former Queftion« Firft, piace your Numbers according to die Tenor of the Qpeftion thus : , /. miei s. U Miles 112 120 14 84 40 120 \1 2240 28 112 14 13440 168 84 nA 672 J344 i** 14x12 40 IJ440) 56448® (42 -P*»^^ 53760 26880 X 26880 1 3 Your 6o The GOLDEN RVLE, Your Nmnbeis being placed in order, reduce the 14 ;• into Pence^nd it is 168 L then multiply 168 by 84, the Produd is 141 12, ..whicji Multiplied by 40, it produceth 564480 f^r ^he pividend. Tkea Multiply ixa by 120, if produceth 1344P §^r the JK- vifor. Divide 564480 by 13440, the Quotient; will be. 4^ Pence , which is 3 J. 6d. anfwer^the Queftibn. 5' In this Rule, the ^rj^ Number ^nd famK ajfo the fecond and and J^^ ; and the third and fixtbt are of like Denomination and Nature. fP0 Queftm II. If jLo/. for 6Mondis Yield 3 U Intereft, what Ihall 625 /, yield for 36 Months ? Place them thus ; 100, 6, 3, 625, 36. Multiply the three laft, as before is fhewed, the latter Produft is 67500 for tfie Dividend ; and the 2 firft Multiplied make 600 the DtvifoTy then Divide 67500 by 600 the Divifor (cp 675 by 6, which is all one J the Quotient will be 112 whole pounds j. and 300 for 3^ remaining, which becaufe it is half the Divifor, iignines the half of a pound \ that is 10 Shillmgs. So the anfwer to the QuefUon isii2/i 10^. 100 63 •6 625 36 3 • 60b 1S75 36 11250 5625 67500 600) 67500 X112I ' • • • ' 600 750 600 . 1500 , 1200 .300 Which might have been given in one Denomination, namely 2250 Shillings, if before the work the Pounds had been turned into Shillings , Multiplying them by 20, as hath been (hewed before, But fmce moft Queftions, except, fuch as are ftudied for the purpofe, are apt to end in fome FraQion, I iball next treat of Blazons. OAly The GOLDEN RVLE. 6t Only firft, having fpokcn of the double Rule of Three, this inay let you know, that all Queftions which are wrought at once by the compound Rule of Five, may be done at twice by the fingle Rule of Three \ and the doing of them fi> by tw;o Operati- oiK, is called, the VmbkKgle. ' As in our hift Qieftion, there are two things ooufiderable, die difference of Money; and the difference of Time* Firft, for the Money. Say, if loo/. give 3 /. what 625 // Anfwer 18 -xi^U Secondly, for the time. Say, if 6 mo. give. 1 8 ^14 ^* ^hat 36 mo. Anfwer 112 ^-?-. But this will be better underftood anon; and then thie; Readtx may ufe that which he likes beft. OF FRACTIONS. / THe vford FraBion (ignifesa bret^hfg ox breach of any intiic thing into parts ; and when a Number is broken fo, the parts (which muft needs be every one leis than the wholes and the whole is accounted bwit One or Vnitp) being lefs than Unity, are called Fraftions (that is, fragments or pieces) of Unity. Noir ^e Unite, or intire Number which is to be broken, may be aiif tbingf as one Voimd^ in refpefl of which, SbiUingi and Pence aaA farthings 2XQ FraBions\ or, one Shillings inrelpeaof whkh, Femt and Farthings are fra&ions^ \ or, one Pen^j kk refpeA of whidh Farthings zre PraSions •, and the like of Wdghn and Me^ureti, «r aay other thing to be broken intto Parts. In Fraiimsy we ihall treat firft of Kumermiony then of Nbtki- fHcofion and JDtv^ioUf then of KiiuBim \ and laftly, of MdUem ixA SuhflraEtion. . The reafon of this Order will foou be feen,; fox .MulnplkmoB 4nd Dhifion. are here much ealkr than Addiuim, &£• and. there^ %e ou^hc to be learned before thenu NVME- 62 NUMERATION. NVmersthn b nothing eife but the way of writing FraSions } and that this may be done, we muft confider that any Vnhy^ or Mmbcr reprefenting an Vnhe^ may be broken into two parts equal; and then each of the Parts is called onefc cond^ or balf\ or it may be parted into three equal parts, and then each part is called one thirds and two of them ^ are called mv ttirds^ and the like maybe underflood if it were parted int049 $| ^> 7> 8, 9, 20, 50, or ICO, or how many focver. Now to write thde ; do thus: ^ Writer f One half One third dne foittth ' One fifth One fixth One feventh One eighth One ninth LOne tenth ^►Thus I T JL 4 1 ■ar I y I In ev^ry one of th^fe 10 FraSiMs^ the Number below the line k called the Denovningtor^ and Ihews into how many parts the Vniu is broken. The Number above the line fhews how many of thole parts are tatei, or contained in the Fra&Um. and is therefore called the Numsr^cr : So in the FrsBkn | , the Denmnmor { (hews the Vnhe to bef broken into 5 parts ; and the Numerator 3 fignifies 3 of fuch parts to be contained in the FraSion ; which Fradion therefore is called Tbree Fifths. And here it is plain, that. As the Numerator is in proportion to the Denominator: fo is the FraOion to i, or Unit^, for | or -f | or any the like, is equal to u And therefore all FraSions are Quotients of lefler Numbers divided by greater, as^ fignifies 4 to be divided by 7, and as th<^ Dividend 4, is to the Divifor 7, fo is the Quotient -^ to Unity. And therefore this line of reparation which is drawn between theJS^ividend and JHtnfor^ doth properly fignifie Diviften. Hitherto MVLTIPLICJTION. 6j Hitherto we have fpoken only of fudi FraStims as are leTs tbcn I, and thofe are called Proper FraStiom \ but there are alfo 2 » 3 4> 5 T> ^ I > and the like mixed Numbers j which fo figniS mmdanbafy 3 4»43 qumers.five andafevemb, 6 md^Jfths. Thefe by the multiplying the whole Numbers, by the Denomina- tor, and to the produft adding the Numerators refpeflively are torned to |, «;., ^, >|, which are called Improper Fraami be- caufe every one of them contains more than Unity. T/^efe, neverthelefs may be Multiplied, Divided^ Added, or JW- fin&ed in the fame way as arc proper Fra3iofts. And this (ball ferve for Numeration of Fri&imu MULTIPLICATION. RULE. MVlttply aU the Numerators togethr, the lafi ProduB JhaB be the Numerator of tbe Produrt required : nkewUe iHuh tipJy aU tbe Denominators togetber, tbe la/l ProduaiboM be r& Denominator cf tbe Produ&fougbt. ^ ^ Example I. If .» be;to be Multiplied by J Multiply the Nmeratot ^ 3 by the Numerator 4, the Produd is 12, for the Numerator o^ the new Produft. Alfo Mul^plying the Denominator 5, by tlie Ver nminator 9, they produce 45,'fc)r the denominator of 'thedefircd Produft, fp thatProdufl: which was tequircd, is JJ. ; Example II. If x, |, .4^ j., and r^ were to be Muki|JUed all to- gether, begm with the Numerators, faying, once 3 is 3, and 3 times 4 is 12, and 12 times 5 is 60, and 60 times 3 is 180, for theiVa- meraa^ : Then Multiply the Denominators ; faying, a times 4 U 8, and 8 times $ is 40 ; and 40 times 9 is 360, and 360 times 1 1 IS 3960, for the new Denominator. So that the produO of all thefc is rJ'l^» Jhat is 1644 to^-, as (hallbe feen hereafter in JMuSion. And thus it appears, that proper Fraaions being lefsthan One- are ftiU made lefs by Multiplying: As here the produa^j- is much lefs then ^, which is the leaft Multiplier ; and the reafon hereof 1$ plain, for feeiilg Multiplication is but the taking of a Number a certain Number of times,, if that Number of times be more than 1, then the Number to be taken isincreafed by being taken more than once j but if the Number of times be i, it is not increafed nor ^4 DIFISION. ' nor diminifhedy bat is ftill the fame ; Laftiy, \i tliat Number of times be left than i, as J-, the Number not being taken once, but lialf of once,produceth a:Number lefs by half \ that is, the half of the Number to be taken \ and the like reafbn is of all others. Example III. Multiply tlie mixt Numbers, 3 !, 4f, and 5^: Firft (as hath been fhewn already) turn them to improper frm^ tctns'y thus, firft fay, 2 times 3 is 6, and i is 7. So the firft is J. Secondly, 3 times 4 is 12, and i is 13 : So the Second is *-|. Laft- ly, 4 times 5 is 20, and 3 is 23 : So the laft is * *. Now the Praftl- ons to be Multiplied are |, *f , and ** ; Firft, for a nt^ l^umerator^ fey, 7 times 13 is 91, and 91 times 23 is 2093, for a new Nume" rotor. „ ^ Then fay, 2 times 3 is d, and 6 times 4 is 24. So the new De- 9<mwator is 24. . And the pioduft of all thcfe FrMons is * •f [, that is, If leal D'l V I S I N. Dlvifim^ to Divide one Pra£Uon by another, is but the croS Multiplication of them ; that is, tlie Numerator cf the one, by the Denominator of the other, and hereby the proportion of one Fraction to another is feen. 24 kxampJe I. Divide <} by f, to do It, f^t them thus : XAnd Multij^y as the Crofs leads; Say-ing, 3 times 8i& jf 24, which fet over the Crofs for a new Numeratory and 6 times 4 is alfo 24: Which fet under the Crofs for a 24 new Denominator ; (b the CJpotient is aJ, that is i, which fliewsthe FraSions to be equal one to another. 27 Example II. Divide | by -J- Firft, fct them thus : XAnd (ay, 3 times 9 is 27, for a Numerator^ and 5 times^ -J 4 is 20, for the Denominator : So the Quotient is |-J» and fo many times is ^, contained in >f that is, as 27 30 . is to 20, fo is| to J, and fo is {i to i. In Drtfifion it is to be remembied, that the Numerator of the ettot/CTft ever arifeth rf the Numerator o{ the Divi4enii And th^ Denominator of the Quoiiem comes of the Denominator of theZ>^ D 1 1^ t S to N. 6 j videndf each being crofs Multiplied as before. And allb remem^ ber always t<;» fetthe Dividend oh the left hand of the CnoTs. if a Fradion be to be Divided by a whole Number ; Multiply the Detumdrntor by that Number, die Product gives tl^e new Dc* fiomnmor^ziidi the iSTtfmer^ar remains the faole^ So if ^ bediVi* ded by 9, (ay 9 tuhes 4 is 36* So the Quotient is j^. Or if 1 were to be Multiplied by 9, the Produ^ Cby Multiply* Ing the l4ttmerator by 9,) will be \^ that is, 2 > « EwnfU Itl. Divide **J by ^^, thusi Say 320 «88o times 9 is 2880, for a Numtrator \ And 8 times 4<; *^ 7^ is 360 for a Denominator* So the Quotient is »|^f ^*J It^ *i 360 tk>r ^^i is equal to 40, add ^j e^wli id 5i but 40 i^tains^ eight times. > < r Aiid fo lA the fecotid Example, It may be proved, tliat as 27 td so, fo ls4 to i. for iirft, Multifrty tbe two middle moft, thad iotiiitesi is *•, that is 12. Secondly, mttltiiflythe.firftsUid laft^ audtUeii 2 7 times; is <«J) thati$ai(bit« Wherefore by that which hath bedn faid in the Gtlitn fyU^ thd four T^unlbrn 27^ 20 1, ^, are pro{iertiotiai« i^«iMUlteiMMMMai^^lMlH*A^MA«^ri«^^'iAMM4rfMi*««Mi«i RE J>U C T 10 JSt. TJ Edualoil of trtftloiii is thrcfefoldi K Ttf reduce m Ffdimi (Hrhidi is Hot sttwSidy id tlitf 1<^) iohiJedSTerms. 2. rd re iKa jiw»; ItiAiolii of divers PetiomnitUHs^ to oHe HtH^ fnmMan* ^. td riim inf F^o^idnjroni onetfenominitM (^litH asimy be) t0 mj oiket IHneminAtioH iefvreii *", t. iJdr the firft rf thefe, f« uitui i tra£fioti ^d /w Uxjlt ttrmti ftivide both the iimsrix^ir and thiJ OinoiAinmr by the greate^ft , CoMni^m ^/w/or that you can think of; the two Qmknxi being filaced fcfpeftivdy in a FrafKon, that l^raftiont ftaH be <?9Ual ta» ^t^stm^ (ra&ioui and in IcflerTerms^ ^ 66 R E DV CT 10 N. So (In the 3 Examples of Vivtfion) to reduce- *f-||.J to|. Di- vide 1880 by 36b, the Quotient is 82 Then Divide 960 by 366, the Quotient is i, and the new Fraftionl^ is equal to the former Fraftion ^{1% and in lefs Terms, as you may fee* But to fifid the greatcft common Divifor, this is ^ The RULE. DhideAbe greater Term by tbeXtStT (I mean by Terms, the Numerdror and Denominator) and by the ranainder f if any be) ii- vide the Divifor, Snd if any thing ftill remains^ by that divide the Uft Divifor, continuing thU courfe tiUncthing remain greater than Unity- ' that Divifor vfhich is haft of allf u the greatefl Common Meafurc of both Terms f by mhich both being divided^ and the Quotient flacei Vke a Fra£liori, that Fradion fbaU be equal tg the former Fradion, and in the leaft Terms. Example. Reduce *^ to the leaft Terms; iirft divide 148 by 16, the Quotient Is 9, and 4 remains: Ag^in, divide 16 by 4, the Quotient is 4, and nothing remains ; wherefore taking 4, (the Jaft pivifor) for the greateft common Divifor, by it divide 148, the Quotient is 37, and by it Divide 16, the Quotient is 4. Thefe two laft Quotients placed orderly in a Fra£lion^ make '^, which is equal to ».f|, and in the lean terms, for no Number greater then I, will divide evenly Jxrth 37 and 4. Other ways there are of leffenlng Fra£Hons,asPividiiig the Terms (if they be even Numbers) by 2, and the Quotients (if even) a- gain by 2, or elfe by 3, or any other Number that will divide them both evenly, that is, leave nothing remaining, but the former Rule being general and eafie (hall f^rve for all. II. Now fecondly, To reduce many Denominations to one com- mon Denominator. Let the Fraflions be {, J, f , -J, -f, to be reduced all to one Denomination. RULE. Multiply all the Denominators together ^d the lafl TroduH JhaB be the oommon Denominator to all the Fra^ions Then Multiply every par- tieular Numerator into all the Denominators except hit own^ and the Jaft Vvoiu& Jhall be Numerator to that FraBion. Thus to reduce the forementioned Fraftions |, J, f , -J, 74, in- to one Denomination: Say, 2 times 4 is 8, and 8 times 5 is 40^ and 40 times 8 is 320, and 320 ^times 10 is 3200, this laft Pro- da A 3209 ihall be the common Denominator. Jhentoget Nur ■ ' - meratcrs HBDVCTION, 67 jnerators fcr every one of them : As firft, for the.firft, fay i times 4 is 4, and 4 tidies 5 is 29, and 90 times 8 is 160, and 160 times 10 is 1600. For the firft Numerator ; fo the firft. Fraftion reduced is *H*rJ* Then for the fecond Numerator-. Say. 3 times 2 is 69 and 6 times 5 is 30, and ^o times 8 is 240, ana 24.0 tfmes 10 is 2400. So the fecond Fraction reduced, is ftw. Afer the lame manner may thef other three be reduced to J J I J ^)rthethlrd:--^^^J forthe$)iuth; and ffAj. forthe )aft; Thcfe are feverally equal to the other, die firf^ to the firft, Cff . as flaay be proved thus. Let the Unity be a pound Sterlings thcQ .Tl|e|ofitis 10 ^nd lis ' 15 * and :Jis }6 dU and ^is 17 6*^ and ^ is 18 Jn all - 76J. 6 i. That is 3 whole Unites, and 16 u 6d. over; Turn i5x. €i, allto fix pences, it is 33, and becaufe 6 d, is the fortieth part of a Pound, therefore all the Fraflions arc equal to 3 J J-. Now add the newFraftions (which being all of one Denon^ina* tionj may be added like whole Numbers ; Thus, 1600 240Q 2560 SSCXD 2880 In all 12240 Which divided by the Denominator 3200, the Quotient is 3 li-Jv- Now f-j4j> reduced to the leaft Terms, as hath beei\ ihfewed how it may, will be .J4? fo the Sum of thefe alfo is 3 J-J, which is equal to tie Sum of the TraRions given to be reduced, and therefore they are equal in Sum, and might be thus proved equal Ifeverally, that is, the firft of them propounded to the firft reduced* Divide tiie Numerator 1600 by the Numerator i, the Quotient is 1600. Alfo divide the Denominator 3200, by the Denommator 2, the Quotient is alfo 1600: and fo may any of the* reft be pj[oved equal by the Equality of Quotients. But I leave it as plain enough already* ' K 2 m- Thirdhs e$ KEDVCriON. ni, Tiirdlf^'Anf Fraftipp being given, to change the JPenc^ minmon to any odier more re^diiite^ retaining ftill (as near as may be) the fai^e Value, RULE. Multiply the Nqnierator ^/vf», fytffe denominator rf|«/rfi; and iivUe the Produft hytbc Denominator given \ tte Qnoticnt Jhall be tte Numerator requirei* Exmplep Let the Fraftion given be ^l of a pound Sterlwg^ vlut is^that in the twentieth Parts or Shillings ? Multiply 7 by 20, the Produft is 140, which divided by 13, the Quotient 10 -J J, thatis,^ 10 J, and -5 J of a Shilling; which maybe brought to pence' thus, Multiply 10 by 12, PrcBuft is 130, which divided by 13 Quotient is 9 ^1 i* And ^gain, Multiply 3 by 4, the Produft is 12, whiph Divided by 13^ Quotient is ^ of a Farthing, folfcven thirteenths of a Ppupd is 10 j. oL and almof); a Farthing. But he wl^ch is refolved to have it in tlie fmalleft Coin, do it ;it firjl Work | for feeing aFartHing js the 960 part of a Pound,^ Multiply 7 by 960, they produce 6720, which Divided by 13. the Quotient is 516 feithmgs, and \\ of a&rtbing; Thefe farthings ; may be turned to ShilTii^Sy Dividing by 48, or to pence by 4, as in ^iKfiiow, This Rule though it be brief and pisun is of great ufe in Ar rithnetick \ either for turning natural and furd Fraftions into Decimals; or any « other defirtd J^^qminmon^ wttb fuch i^cility 9ad fpecd as may be >yiibe4. y. FraUion ff Fraflions, In Reduftlon of Fraftions, fome ma^e another, or more part^ as Fr Action of FraSions for one : Tliat is, when there is part of ^ FraWon ; or apart of a part of a fraSicn, ^c. to \^ valued! inone Fra^ion, • ^ — f ' ' , . = R U t E, Multiply aU the Numerators tjpgef fcer, tbe hfl Produft jlrf be tie Numerator d^firei : then Multiple all the Denominators iogetbep^ kndtbpf lafi ProduBpAU bctbepenommitQTfi^ Bxdmfile. Let the FriBibfis rf Fra^ion^ propounded, be f of l of l/jior Co they are ufuaily writ^nj aa4 lot the t^ifmm$9s ' ' -' ■ ■ "'^ • '■■ " '"' ■ ' ■ he ADDITION. . 69 be Multiplied: Saying, 4 times 3 is x2, and i2ti9ies i is 12, the Ar«f»er^iV' therefore required is 12: Then for ihe Pmomfitatiff fay, 5 times 4 is 20, aiid so times 2 is 40, for the Den^nmr rt- quired} and •;j.4 is equal to ♦? of J of -J, Tie froof\ Let the Unite be 40 x. one fifa of 40 Is 8, and therefore f is ;2, of which one fourth is 8, and{ is 24, of which one half is 12, and therefore lir is the juft Sum « all the Fraflions : This needs HP fiirther exemplifying, ' J D D IT I N. TO add many FraAi(Hi& into one Sum » conffder whether they be of one Denomination or divers : If of one, Then add all the Numerators together into one Sum, that Sum hdie new Nutnerator, and the Iknomnator^ in this cafe is not altered. ' 1 Let the FraSions to be added be J, }, J, « • Add the Nu- mefatM^; Saylkig 1 and 4 is 6, and' 5 is 11, and 1 is 12. So the Sum of them all is ».J, that is 3 Unites* . As, let the Unite be 201. one foiuth is 5 5. and \ is 10 x. and ^ 20 f. which added to 10 t. is 30 j. then {■ is 25 x. which added to thirty Shilling? gives 55 j. And laftly, '^- is 5 s. which added to 5 5 J. makes 60 1. that is 3 times 20 x. that is 3 /• or 3 Unites. But if tlie Frafllons to be added, be of divers Denominations ; 9$ let ^iem be *, J, f, -J, then (by the J^duSion afore-gi>ing> they muft be turned all into one Denomination, and then they win be ij^, » f J, Iff, and Jj?, and may be added likethofe be- f)re : Thus, 320 160 384 420 In an - 1284 So 70 SVBSTRJCTION. So the Sumof all is '-JJJ, or Hi, or that is 3 ^Iv, which if it be in Money, and the Unite x /• it is tiien 3/. i ;. and loi. as may be tryed thus, Firft f of a pound, is 135. and 4^. and ^is i%s. and 4 is 16 s. Laftly, J is 17 s. 6 d. Thefe all added toge- ther, the Sam is 3 /. 1 x. 10 d. SUBSTRACTION. IN Subflra&ion of one Fraftion from another, if they be both of one Denomination : It is done by taking the Numerator of one from the Numerator of the otiier, the remain is the new Numerator, and the Denominator the fame as b^re* So if l be fubftrafted from j, the remain is J, the like of all other. -^ ^^ But if tJiey be not of one Denomination, they muft firft be reduced to be fo j then that which is faid before is fuffici^ntt Concerning the Golden Rule in Fradlions. ^Tr"^He Golden J^Je in Frontons is the fame as in whole l^um- I bers, I will give you but one inftance. JL If l of a Yard of Tape coft- a of a Pem\y, yrftat Ihall one Inch, that, is, -^i of a Yard coft? Multiply the fccond by the third, the Eroduft is ^J, ^l^ich divided by {, the Quotient is -.^vj of a Penny, for the frifc pf -si of a Yard. Othervoife. Seeing { of a Yard maybe turned to 27 Inches: Say if 27 coft ; , what I ? liivlde { by .27, it makes ^*- for the Anfwer : Which is equal to -5-,-J, and. in the leaft Terms. - ' . And wherefoever this may be done, to have the firft and third Nunlbers of Fraftions of one Denomination, the beft way i5 ^^ work with their Numerators, not regarding their Denominators at all ; As, If j coft J, what J ? Inftead thereof write. If. 2 coft i, what The Rule of Pra^ice. 7 j J, what 7? Multiply { by 7, kproduceth ^J-, which divided bv2 the Quotient is H, and that is the anfturer in the leaft Terms. * And all this while it fliould have been noted that the Fraftioos are^i^er written in a fmaller figure then the whole Numbew* The Rule of Praaice. rf the bolien^Je^ or J(ule of Three Direct, I intimated : tliat if the firft of the three Proportional Numbers given were One^ that then the Prpduaof tht fecond and third Numbers gives the fourth Proportional Number fought without ufing of any lyivifioH ; Alfo, that if tile fecond or third of the Pro- portionals given were One^ then there was .1*0 need of Mulmluatm 2 but dividing the greater of them by the firft, the Quotient Ihaa bt the fourth Proportional fought for. And fix)m hence is firaifted this Rule of Praaice, (by fome called the Merchms J^le) which always hath One^ an ingredient in tlie Queftion, and it is no other but aa Abridgement or Compendium of the l(ule of Tbree^ when One a one of the three Proportionals given. And that fuch Queftions that are to be refolved by this Rule may be the m6re.rcadily aad eafHy anfwered fMoney cwjiraonly being one of the three Terms> it is expedient that he whi^ in- tendeth to make much ufCvOf this Rule, fhould have readily in his Mind the Even or jViiHQt parts of a Pounds of a SbilUnjr^ and of a Pen9ty. And alfo to have in Memory the feveral Product of 12 {the Number c^ Peme in one SbiUing) Multiplied into 2, 3, 4, 5f^» 7^8, 9,10, u^and 12. AUwhtcfaaic fat down in thefmaa r^/« following, which ought firft perfeftiy^o be learned by heart. before fertlierJProgrefs be made into this Rule. TA RLE !• Tie Jlqmor Even parts of a Pound or 20 SbiaingT^ Jb. ^d. I- 2- 2- 3- y 6- -o ■0 -6 -4 }>is tlie<^' ►0 •0 -8 iO- >■ ' oj _.j. One Twentieth -J One Tenth •J One Eighth i One Siith . \ One Fifth I One Fourth I One Third I One Half of aPouni or 20 X. TABLK 72 The Rale of Prdiice. T A d L B It Ittf Mijjut 0^ ivtn fans of a. ShitHftg* \ is the. I X One Twelfth I One Eighth 4^ One Sixth 'JjOne Fourth \ One Third iOne Half of a ShiHingi J 7 A B L E III« The fiver a FiHcc in i SUVing Muttlfljci bj tli 2'> 3 4 6 Pence MukipUcd by ^ la produceth ^ 9 so II X2J 48 6o S4 loS 120 Li44 Tor tbe working of the J^e df PrdSiee^ when the Price giten I9 bf equal Parts of a .fii/irnj thisis^ The R U L fi* XjMfing hfyowr TihUtAMfm cfsL SbUHngHU^ fvirhether -^ ,, <|, &cO itviiff fi^ Sumfrofmnded. by H^ and the SmicmJIHiU he m number of Shillings atfrnering yeur QuefiUmi Exmpk. At 6d. tbe ounce^ i»bdt 7625 duHtat %{% Pence is {T)y your Table). 4. of a Shfllii^, wherefore take one half (if 76251 and*it is 38125 and i remaming, which i\%6d. So that 7625 Ounces, will coft 98121. 6 i. which is reduced into Pounds, by cutting off the laft Figure towards the Right-hand of 38x2, and taking the half of the other Figures, whichtVil) be Pounds^ and if one remain, in taking of the half it is 10 x, So the Figure 2 being cutoff from 3812, the half of 381 is 190 and i remainingi which is 190 i» 121. So the price of 7625 Ounces will be. 190 /• X2X. 6d. And fo muft yoti do for all others^ As if the price be I, take}, if ;• take}, as by the Examsks fol]6Wing4 1%)M (0 At 6 d. the Ounce, what 7<Ji5 Ounces?* ^ ^ i ' 38112 — ■• — 6i4 190 12 ■■■'■■6^* (2) At 4i« the yard, what 3^1 yards? f ^2017 6o//« 7J« oa. (3) At 3 j« the Gal1o^, what 989 Galloos ? ^ i 2417 . ' 3^^* (4) . At 2d. the Pound, what 6760* Pounds ? i ii2[6 * 8i, 56//* 6j. 8rf. (5) At 1 i. 2 J. the Ell, what 9^23 Ells? ^ . 12012 10 i. 2 1* 667. 2T. loi. 2|< (6) A£ i i. the Oun^, wk(t $72 Qitnces? 'f i)us haza you Examples when the price is f i^e^ >Mi of a StiiSngi It wbe4 they are Koexfe;^ pni of a Shillings as 5 ^ 7 i. or the {ike, then you n^o^ do the w<^kiat two or, three Operations, tho^ in the faine manner, as Pence* ^ fence , 4* i . . 3 and 2 4and3 iTthe J 8C _, ^ i* V ^4 and 4 frice b0 ^ 9r *** V and 3 6and4 6and3a[ad^ txmptes ef ibefe uHeveH Pmi of d shiUmg* fi) At 5 ^ the Gallon^ what 6254 Galtons? . i^'is'lL , . 1563 3^< |.is2i# . * 1041 4^* t , w*; 74 r/&« ^"^ <f Pra^ke, (2) Atfd. the Ounce, wbat 9271 Ounces? |is4i« 3090 4dm -Jisgrf. 2317 94. 7 54o|8 idi 270/j. 8x. , lif. rs) At 8i. the Yard, what 79^2 Yard&? •■■■■■— "^i* |i$4i. 2651 8(f, I is4i. 2650 8i. .8 53ofi 41/. 265//. 11; 4i. (4)At9i. the ED, what 3769fins^ |is6i* 1884 6d. |:iS3i. 942 gi 9 282|d 9^. 141 li. 6 s. 9(<. (0 At 10 d. the Dozen, what 625 Do^en? 4is6rf. ^ 312 _ 6i. |is4i. 208 4tf. 10 52|o 10 i. 26 /#• ox* io4« (O At iii. the Pound, what 6952 Found? \\&6d. 347^ •5:is3i. 1738 iis2i. 1158 8i. II 637I2 %d. 318//. 12 x« 8i» (7) At 12 rf. or I u the Ounce, what 9871 Ounces? •^ of 20 f • therefore \ of 987I2 is (493//. 12/. If the price of the Commodity is in Farthings, or Halfjpence, bring the Sum into Pence, and work as in the preceeding Quefti- ons, and according to the following Bxamj^les. The Rute tf frtStke, 7$ (iMt I f tbe Pound, what 6392 Pound t tI- »3 3 af, iUf lit* 2i. ' (2) At 2 {. the En, what 362$ Ellt ? Ty Ml* 7 h, II f. piL 2|; * (3}At3^ thcOimcc,what732iOanm? ... • 30$ 8^ 3f ■ < 11 45I7 This is the manner of working for the eren ports of a Peay^ butif tbey be uneven parts ; As two pence 3 fiurtfaingi, firepencp I farthing or the like, W0rk, firft for the e?eA part of a MBHg^ indtfaen tor thcfittUngs^ which adde4 tbs Work is4one» Asii| tbefe Examples. (i) At i i. 3 {• Ae BO, what 817 Ms? i 204 31% 8 A 3|j, I 204 12 It. I'ii what 7138 Po 1189 1 189 148 a ag^in 1189 '8 .i 148 *|^^ 126 /i . 8x. odi I' 2 For l6 OfTf(e^Trett, &c. • For the even parts of z^Ppuvdj you muft take the parts as you find tliem expreffed in the TaWe j as for lO ^. the -f, for 4/. the f J as in Example, CO At 2 J. 6 i, the Eil, what<5294. Ells? -f. ' 786//. 15 X, {2) At 4 J. th6Ream, whafc 735 Reams ? Ift in this Rule J at any time the Queftion confiffc oJF the part pf an Ell^ rariy Pounds Ounccy Grofs^ or the like ; you, muft deal with the whole Ells^ rardsy Ounc^Sy &c, firft, arid afterwards idd the price of. the -J, -J, j, or what other Part foever it be. And thus much fliall fuffice for this ^k of PraQice. A I tf I* ^ » ■■ Of Tare,-Trett, &c, % IH Merchandise there is ah <Mowancey made by the Merchant to the Buyer, for the Wfeighf of the Casky Bogy Chc&y Frealy ^c. in. which anv GopdVaS^e put: And this Allowance is caHedv ^.4 Jt -^ » which befttg dedufted from the Grofs Weiik C which is the Conm$ditj znd Ca^ky &c. together^ the,JlemaWer is the- Weight pf the Cottimodity'b^ly ; ktii is* cfalled' J^Elt- There ' is alfo an AHcnfente jWadebV Mercharnf^W the Busfci; for i^tt/^otlF^r^if that may bcf'ihiXcdwfthtbetL'btrtriiddity ;• a$ mt MoatSy &c. Cas in Commodities Garblabky zsSpices\ &cO and this Allowance is called TusfT/fi .which is alijFsgfs^^^^ Jin the jm- 4rsd Weight ; But the Allowance for Tare is various. In fpch Commodities where^;(^^ is allowed, the Remainder after fiich Allawance, is called SVrTtLE-WEJGH7yivAo\xtoi that the Allowance for Tr^ttUsmade ; and when that is deduded, the Remainder is called NE7TrWEIGH% 1 - • • Exemplary Queflipns mM rrnfke tffefe KJJ.l E S plain. QueflioH I. There are ^Cbsfis oi Sugary the Grofs Weigh of all which js 44 C I jr. i3//>.-.^nd thef^^ allowed* for each Cheft 37 Jib. What is the NettrW^igk of Sugar in all die. 4 Chefts ? C jr. lib. Froip ipr 1 13 Tot^l Grofs Weighty Subftr. 1^8 Tot^lTkrtf, Remain 43 o 5 Total A'i?tt-J^«r/gfe. Queftion Of Tare, Tmt. Btc. 1. TheGrd/y-ITe'jfglr 99oCy 3jr- 21 /ii. reduced into Pmxir WiDbe 110985 A». ' * 2. Ssij by the J^k <f fUree VirtS^ / As iia/i*. : to;i4//fc :; Sp.iio985f; to 138;^^. . . .- Wbctdfiirey ■-••. . 3*' FiOm - ^f 10985 "GHfs^ekbt^ > SubHf* '■ 13«>H ttndtrare. Which reduced is equal to ^yC,[J^lib» Note, That vhen th^ Number of Jfi^mds to be abated perC. for r^<?, are ;»i;>rfiqto»: part of u:aW/fa iii tiie former Exam- ple, where it is 14 lib. ^ki^ is equal to 4 of 112 ; Then . the Propotf iop. ^aj ber, , ^,^ . On c. c. c.\(A:.Mb: c. tit. lib. As 1 : to 4 :: So 9jl^ ? 21 : to 123 3 . 13I. ;990^i r ^ --^ r|i3 3 .00 Hritc^ o itr$ 00 2|- Total Tare 123 5 .13*4 I^)?w» III. A Merchant buys 1 1 75 //^. Weight. # a/Commo- dity, (as Cloves, Nutmegs^ or th^. }iJf^ fo|-.,which he is. to be al- lowed for Tretty 4 W- in the irmdfed Weight : How many p<?ttifii Weight Ought he to receive ? Then fay by the J^k of three DireSt, Jib. Jib. lib. lib. As 100 : is to 104 ;: So is 117$ : to 1222. Qtt^im IV. A Merchant hath 1222 lib. Weight of a Commo- dity', part whereof he bought at a certain Rate per lib. and the reft was allowed him as an Overplus ; after the Rate of 4 lib. Weight in the 100 lib. Weight, which he bought : I demand how many Pounds Nett-Weigtt did he buy ? / Say V' 7« Of Tdr^^ Trett^ kc. \ Say by the J^fle tf Ikcc DinS j I fibp m. lit. Ui. ' i ' As 104 : is to icx) :: Sq is 1222 : to 117^. Thfe Qacftlon is but the'Reverfe of the fbrHlcr} amd (hews the way whereby to make abatement for Tret$* Queftkfi V. If from $^C if.of Gro&Weisht, Tare is to be SuUmfted after the rate ei i6per Cent, and from tbe remaining Trntj is tt> be abated after tbe rate of 4hper 104/. The Quezon is, what the Nen Weigk is worth in Money, ^Stist the rate of 8 /. Ux, for every C Cor 112 /a#) ? i^ The Grofs Weight in Pounds is, 6 1 88. Then, lib. lib. lib: lib. As II? : is (0^$ : : So is 6188 : to 884, .. From tfi88 the Grofle Weight, Snhftr. X84 the Tore iLemaiA 5304 theTrett^ Then, Vk^ lib. Hi. m. ^ As 104 : tp 100 ^ : So 5304 : to 5100^ . And then, ' A$ii3tK*.Wei^; Ist68/.8x. Sob5ioolf&* Tb382/. lox. Whidi is the worth of the Cpfnmodityf TH]^ 79 THE R U L E OF t FELLOWSHIP. m I r'l IHis Rale b nlefulfcr TMrrdhni^ and all fuch as Trade in V I ^ CM^jyiei, with a J6ynt ftock ^ and muft (hare a propor- JL tional part of the gains, or lofi \ every one according to his flock which he laid in. The Rule is two-fold, withe^fiM/ time; or with ixirqiu; time. That which is with cmuUtime^ is commonly called, die Kuk of FiOmMprmbmrtmc. ^ ^ Qfthis we will firft fpeak, T-HE RULE. . ^ the Vfbok^im Stock u to aUthegm or lofs: SqU each nuns finkular Stock, to bit fm oftbigm, or hfs^ : Exmffhl* Twp Purchafers^andAbuyTOo/. a year Land for ever, (when numey is at 8^ CmtO for 14000 A of which jf. paid fooo/. and Bi 6q9oL after 5 years (money being bllta to6fer Cnrr.) they fell it for 18700/. fo there is gained 4700/. hc)3v much of this mafl: A. have, Firft for A. Say, if 14000 gain 4700, what 8000? Anfwer, 258$ ij;;!. Then for B. tf 14000 gain 4700, what 6000? Anfwer, 2014^};^;. As by He ^llowifig Operation da& appear. ' ' 8o The Rule of Fellowjbif, * a) For ^ U h w If 14000 gaia 4700, what 8000. - 4700 ria 14000) 37600000 (2885^5 • • « • 28000 96000 mmmmmmmmmmmam 84000 ' 120000 S 12000 80000 7000a 10000 ai) For R If 14000 gaia 4700, what 6000? 4700 . 14000) 28200000 (2014 7^ • • • • 28000 20000 14000 60009 -i 56000 4000 Remainder* Here note, That this Work might have been much abreviatsed^ if fipom each of the tiiree Numbers you had cut off two Cyphers toward? the right Hand, as hath been formerly Ibewed in the Com* pendiums of MultiplUmon and JPivifim* "~\ Now for the Proof heretf. . - you aiU 268$ ffm^ irhich is tbe iam^M A. gained; ' » . : To uL i aoH-rtJAg The rum which B. gainecT; the run of theat \sry 4700 Which is equallD the total Gain. :f And according to 4f proportion of thefe t^ l^ifinbefs: That i^ as 8 to 6, or^n^ t^ %. So they ought to have ^parted the yearly Rent alio, all thetinethey received it : That is. A, ought to have 4op/. Yearly; andJ^^oo/. ExamfJeti*'A»B.md C-joyn their moneys to make aftock of 25000/. of which j^.1aid in loooo /. B. 8000/, and C.put in 7000 1. with this rafter a eertaintime inr trading) they g^uned 7500 /• how onifttbis be parted fT FirftfiJT'^'^ Say, if 25000 gain 7500, whafe5'o€xi)o? Or fhorter, if 25 get 7 J, what 10? Multiply 7^ by 10, it produceth 75, which*dirided by 25, the quotient is ^| that is^ (reftorin^ th^thnee Cyffeersy^o^o/. for^. Then for B. K « Say, if 25000, gain^T^fip what 8000? Or fiiorter, if 250 yt 7;, what 80? Multiply and divide :j|9 the OcJden J^k requires, to the quo* tient reftore the tV^Clnh^, then it will be 2400/. for B. {4;tfly,forC Say, if 250 giv^ ^^ what 7e>? Aafwer 21, to which put the two Cyphers, it makes 2100 for C. And thefe three 3000, 2400,' and 21Q0, being added toge- ther, make« 75cx3.* 'And have that proportion as the particular ftocks had : And therefore the Work is right. U (I) for a ■J 1. TbeRulehf:Feg0»(l^^ 71 t . * t J If X 2 If «i g3|a 7t. what; io ; ..;. ,. „. . . , '.1. '.• •; . ■. . '. ,„* 11 ■,!..) ^ / . / • a « » 4 N. • •- ^ 1 s • « (IDforB-. 2$ogam75, whatSo? ....... 75 , ; -I « 2iQ) 6ooa (H'!l • V . 1 l> 1 ' V 1006 • 1 1 1000 24oo/»forJ« - (III) for C. II 210 gain 75 whaj 70? 7Q - . 250) $250. («!••. 2100 500 250 2<0 //forC. And .J And if inftead of gaining j^cx) /• whereby every one is fup- f6(^~ to have his Stock, and pant €£ the ^ins ; they had loft 7506 /• then, thiit particular Stocks had not been doe to them, buf lb mudi4B wotiU i)e left after their proportional I^artsof the lo(s were abated. « * J Example III. ^^B.and C. with a jdytit ^tock of 2 $000/. gaiil 7 $00*:* cf -which -iit gets jooOj \?. i4<xS C. 2100 j what was their Stock ? 'tbfi is but tli^ RfeVerfe of the fortncfr, therefore fay, if 7500 require 2^ebo, what dodi $000 re^tlire ? ioooo for ^^ and fo work for the other two. > ' ^ Many Examples are of little Ule* (except to load fhe Readers memor^ Where thp Riile is To Ifaort af|d- plain ; 1 . kill theiefere add no more to this part of the Kute but immediately coine to the Rule of RUowftiP with Time. iSitmitammi^mmtmmmmmmmmmm hf'^ - T H £ - ^>> ■ ^ I i :f L RULE *♦ f: J . . . . ^ .^i-.TJII. r : > » < ( :i F E I^ L m^S HIP With ' I • « .1 THis Rule is to bc^ufc4 whjg^,-the Times of the continuance of the partjojlar/^it^^b are ungual, and differ v. fo that here thei/J^r|»^<? of Ttme^ and alfo the differenc^'^ Stock heingboth to be conriderAd.5 it can be' done no better w5y 'than by taking the Power of them both to be" the particular Steel \ and sffl thofe Powers added, to be thfe wWe Stocky that whidi I 'call the Power^ is thePrWfi^ of iHie Money of every one, multiplied^ by htsTimf } And th^: - ^ ' • « ' 'THE r:uLE. jts thefm of^p Produfts, is ta the wboU Cd}kiy pi js eab particular Produ£i, to its part of the Gain. ' M 2 Quejiion !• TheRuk-of:R8ewfi^, a)fi>rwf.-'-- -- -- i If 45gi>ifl7T. whatiip': ,.,. \ •t 1 1 t y i X 2 * .'. .. 10'(^5 > .^: t t, -^ . ... « I V. •>v i. 25) f 75 .(300C1I. • ■ * • • r ■ ^ ' • « • »-f- S • * m 75: .. . (njforB* . 1 1 » ( If 2Sogaia75, what8o? « t . ^ • ■• 75. : ... V « 2^0) 600a (H'*l • V • . / $00 / 1000 • 1 I / • " » - - ' • 1000 , 24oo/»f6r#* k - (lip for C. » If iio gain 75 whait 70 ? . 70 . , * 250) .5250. (?I,.. • • 2100 500 250 2<0 UtQtC. And TbifiRiile^of: Feffo^fbi^mtb Time. g; And if Inftead of gaining .7500 /• whereby every one is fup- ^6fM~ to have his Stock, and patit rf the ^ins ; they had loft jp&h then.th^ir partkular Stocks had not been doe to tiiem, bur fo much 4B wo^ld be left after dielr proportkmal Parts of the Jols were abated. Exajitple IIL ^^B.and C. with a*}(>yttt5todc of 2jooo/. g^in 75oor of -which 'jit gets 300O1 -?. -i^ocS C 2100 j what was their Stock ? /thfe is bitt tlj^' Rfciterfe of the fonher, therefiwre lay, if 7500 require 25;obo, what dodi 3CX)0 re^tlire } loooofor A^ and fo work for the other two. ^ ' Many Examples are of littte nfe' (except to load fhe Readers fflemoi^p Where thp Rule is foifaort andptelQ; I MM thefeiR>re add no more to this part q^ the Rule but immediately come to the Rule of JcUowftip with Time. . • ■ ■ r '«■•''! i : •-"* . I h < r.*^ ' • » • • ■. ><A ~ T H E . -iC;.. RULE < . L r ■ ci^r-^"- FEI^!LOW-S H l^ With Time. • i .■■ <■: t-' ■ v: - - • - ■ ■ - . V . X . 1 . . « . THis Rule is to bc^i fed wbJji .-the Times of the continuance of the partiojhir/^^^^tx , are unequal, and differ ;.. fo, that here th^^differejtue of Tirm\ and ri^ the diferetK^of j^ock being both to be confiderAd.5 it can be' done no better wSy 'than by, taking the Pawer of them both to be the /jrr/Va/tfr Stcch\ and sfU thofe harpers added 1 to be tluk t^Mt Stocky that whidk I 'call the Power, is the TroiuS. of the Mimey of every one, mukipUed: by hrs Timi And then>^ - ' " .,.;'* J '' ^tpE RULE. As the fiuin of tlfoff 'PtoAn&s^ is ta the wb»U Gain;* jSikei^t t^ticukr Produft, to its part of the Gain. ' M 2 Quejlion Z6 Wfti B A RT E R. f l"^ Barter is to exchange one CommiKiity for another, the . I Nature whereof wBl beft appear by the refolving of ■' JL fome Qucftions. . *• , <■ Queftion I. Two Merchants Barter, One hath Sugar at 4 L the C. ready Money, but in Barter he will have 4L i^s.^i. The other h2th French Wine at i^/, the Hogfhead ready Money ; at what price muft he rate his w ifle, to equalize the others advance of his Sugar in Barter ? ..-.?--. Say, by the Rifle of Three direft, If 4;. in Bsuter, require ig'x. /^i* aidvance, what jpbaU xj./« ii) Barter require ? - what 13? ^ ./ . c ; V 4) ^80 ($^:d.::. . -:, • • • • J t 12) 520 42.ft !.: . 160 Pence. 45 _ ., rt 40 n 2080 • i^ Pence.) That is 43 4, or 2 3 4 the Arifwcr. Cueflion II. Two Barter, one hath 3 C{. of Ginger at 13 i. ^ per Pound. The other hatli Sugar at 15^. -^fer Pound. How much Sugar muft be delivered for the 3 Qi ot Ginger, Firft, by the Rule of Three (or Praftice) find what tJie 3 C a of Ginger comes to at 13 rf. iper pound, which will be found to be 22 /. 1 8. For, Tf Of Inter ^ Simfk 4»A Ctm^ounJL gy If iC*. coft i-jA i what J Cf crfk? AnTwer, %%L is. Secondly, fiiy, If i)iL } bay i lib. of So^i what (haO 23/. II. buy? .^ Anfwer, 347 ty* (tfefiioH in. Two Barter, One hath Tobaccaat 14 i.^^ pound, which he wiO Barter for Sugar at 10 d* per h bow much Tobacco ffloft be given for 8900 lib. of Sugv? . Firft, the 8900/. of Sugar at' loi. [er pound, comes (0 370JL Then, If i^.L buy % lib, of Tobacco, what Number of pounds win 370/. i6s. id. buy? Anfwer, 6357 Pound, and Co many pounds of Tobacco at 14 i. mpft be given mr 8900 pound of Sugar at 10 d* ' ft Qiuftim IV. Two Barter, One hath broad Qoth at i^s. the Yaid ready Money, for which in Barter he will have 16 x. 3 id lie other hath Wool it2uiod.per pound ready Money ? What {rice muft his Wool be fet at in Barter to equalize the advance which he puts upon kis Cloth. Say by the Rnle'of Three diileft. If x$f« ready Mdney require is. id. in Barter} what fliall ;2i» ioi»ready money xequire? , ,• Anfwer, ai.-^ 3J. f So that he muft rate his Wool at 3 x. 3 1. \ of a farthing far pound. * . - , K * • 4 • - I» •AliAHAMiii^paAaM^iM* • J OF I N T E R E S T • - / Simple and Compound. « ' * ' • IN the Second, Fan of this Booi^ I have TibUs of Ccmpound imereSj l(ebate or JPifiomt of Money^ -Punbace of Uafis and Annuities, wbofe ConJiruBion and Vfe are There Exemplified hi' Refohring of Queftions fuitableto each Table^ as by having re- ^urTe thither will appear* As for Simple Ihtejreft, any Queftion there- 88 X^ Iwt9r^ Sinifit sMCmfwnA\ thereunto relating. Inch nm be I^elfaifod bv" the A^gk or Gmh- fwmi IMss of Tropmim bene tmght in this Book ; of which aUb wxExmfU ^ two I will heve tofert? A&4 wii(: (kifing any thing of the other, tiU I come to (peak thereof in die Sectd Queftian L tf 109/* in 12 Moatbs gain 61. what Ihtfl 625A gain la 3 Years or 36 Months? The Proportion is, As 100/* is to 6/. in a Tear, So is 62; X. to 112I. i6$^ In a Year. Wbvefbre mttjtiply 625;. by 61. and divide the Predaft% 100, (by cutttog oflF two figures) the Quotient, will be 37 rH tliat is, 39/. lOf.and tfai$ bein|( Multiplied by 3, gjttA xi2/.ioi. as by the Work appears* h h I. k ■ ■ 37150 Or 112— 10 ii2/$o (2yf/li(>tt n. If iQo/. in 12 Montto gain 61. 4tet wiB 23^ 10 A 5i. gain in 16 Mon^? -^ . The Proportion. .; As 100 /• is to 6/« in a Year, So is 236/. lox. <^i. * ,To X4i. 3 J. 9<'* 3 !• taa Yeari . „ So is 236/. 10s. %i. to 14/. 31. 9i. 32* ina Year. Multiply 236— 10— —5 by 6, The Produd is 1419— 2 6, ' This diride by 100, which is do^e by'-cpttine ^ two iff^ of the Integer, leaving i^Uon titet-efrhaild of the ^ine. iToc figures on toe FJgbt-harid multiplied by 20, and the -figures (or remain^ again by 12; and lafUy by 4; Ihall in all give 14/* 3^* ad. 3 J. ' "^ ' ^ r : ' J Which divide by 3, and add that iJiirdTArt to 14 /. 3 J- 9«' 3 J. thefum will be x8/. i8x. $4. oj.as bythe Workappearetn. 100 1 In I of a Year 4 14 7 x in 16 Modtte iS ig 5 o I. I4|i9— 2-r-i ^- 3J8? . , • .y 9 {r «0 14 . 4 //. i i. uiofleYear 15 01 ot ,lu ti. lU i; d. ioe^'i-^-6-— 417— -1 x«-i-o8 o 25(05. / 120 10' 02 |t3 , J la two Tears 50 oa dl 1126 I ♦ Tilts thcfe Qtteftions are wiroiriit bjf die J/^^/^ A/# ejf rtrrfi i^t they ma; be otherwife wiougjbt by tlie Gol£n K^U C^nbMi V i mmbiirU Ot whieli in tiiat i^/f you tare an Mt^vtiglk a ithk J.. <9o) K « \ — .'i A L L. I G A T t U. « « T His J^e takctli its ZsTiw^ from Bindwg^ Xying or Vnhingi m^xvf Tmiettlars in one A!|/i «ir .»<mi And ir^is iJithef Meiul or Alternate. Examples in botb.whlchfQllow ": I. of AIUgatimJAeiidh . ' " : • ^ s ^/?i»i>rMeif ^Hs ; When having the feverali^iW/r/ej and ^« of diverfc Jim/?ftx propounded, we do difcover the Mean I^eof a Mixture Compounded of thofe Simples. As, hariirg ipBuJheUoi J^4r at 4 Shillings (or 4» Pence) th< Buthel, 40 Bujhejs otlf^Jfe, at 3 Shillings (or 36 Pence the Bujbel:) iWeiwf, win. tell- ypU:^e/Me<«» Pwcof mtMi/f/i^^. Ana for the . performance tiierebf this is " > The R U L E^ ■I _ ' - , • . __ ' ^IfrVt^ Sumnptbe given <lpaiitities : Tben find the Total Value $f gJitk-SiSi^^cs: Wbieb done. Th e Proportion will, b^. As the Sum of tfe^ i C«tf» wiej, Is to the Total Yalue of the Simples^ i ' So is any ^itjoLibsJAixtwre propounded, To tlie required iWe^» i^^te, of Pr/tf^, 'oFthat Piarf, Example i T n fh^ forcrm ention'd G^r^fej, I demand how much one Bujhel of that M^ing is worth ? ;. ^ , NowtheSiQi of ^hcgiyenjjy^w/^fejof Bk/J^/x, fvi^. jo, 40, ,50, '^0) h 120 Bufiels : htA IO-) r Wheat") (-48 p r 480 * 50 r """i^" '^'^ Barley C" ^ 24 C to ^ 1 200 lyjJ Coats ^<!' W2^ C 240 k ■' ' ' ■ • ■ . . 120 InaH " Peace in all 3360 -Thea Then fty by the GoJdeH Kjin Vkc^^ \i 120 Biifhehf gire 3360 Pence i What i? ' ' !>/ 33<56^ (2«* Pence/ * s« ■^ t »« •« « V ' 240 or u i. 2 4 In lik^ mannQF; ii^ it had been deoiaBded what i.BujheUy^ One Quarter of ii^ttMiftling is wq):t]i"'j.the Anfwi^ W9UI4 ha?<p teep, ^24 Pr»<?^, wfiich is i B j. 8 i.§t tjxe Pri#e o^ j^g gj<y^y* , ,.' 120— -3360 ■ 8 ^' 126;; aTSSSo (224 • • • 24< • 288 ?4Q 4^9 I ;2) 224 Ci«x, 12 104 96^ (Si. The Proof of this Rule. ^ Tie trya ef the Work lir", fy comparing the Total Value of th yiwwJ Simples 5 wVi&tfc Valuer/ fie Whole Mixture: Fcir, jf th^t ti&o& accoMj-^tfeO^eratioii ip|«;;^— i-*?-^&'a the precetfing. BuQiels Wheat /. T^e j>5o\ of ^Barley r?^9.*f Oats t a ^' ■ ■ i l l. M 2 9i A LLtG AT 10 N. All ^ic)i amounts to 14/. xvhich is likewife the Value of i%p fujkls at 2ti 4^. the Bufiel^ for that alfo Amounts tp 33601*01 Another Example. ^ Gpldfinith.teift tnehei xilib. of Gold BuIUondf iSCarrafts W^f vnfb 4 lib. ^21 Cairads TiAe; ffm many Carrafts Tine if % lib. of tbit ^As&vfortbr ' .the Anfwer'is i&M, or i%\Gatasfi ne0 -^ SeetheWcNTk. t2Jib* /^Jib. ' uUb. 96 I4 i6Divifor. 21 <;^ . V *■ ^ S. .1. 84 ../'*1<5 140 ^00 Diri4en4 -128 ^'12 11. qf Allig0tm Aittrmte. A3ig4tion Aherndte is, When having the feveral Ji(i*^ of di^rie Jimples given V todifcovcr fuch i2f<«rtif*x of them, as are neceflkry to make a Mixture^ which may bear a certain J^^ propofed. For the Solution bt0ueftms belonging to A^emm Alligation ^ Obfervethefe ■ • ' RULSS. tf , J. numuU Hank tie Terms in fueb.firt, TUf tbe Given Rate ^ tbe Mixture, mayreprefent tbetlooti gnd tbe feverdllK^ies tftit Simples, i^ayfiand as Branches sjfuingfrom tbat Root. II. Having Ranked rie Terms intieir dmO*4ery Link ti« Bran- ch* togetber (Twii 0fii Tvfo} in fucbfin ; tbatOnitbst ^Greater tbdn tbe Root (orHfie) m^ aJmys '^ eouflei Vfitb inotber tb^t u t^ftti^ii»tle/iwrRdotor Rate. * ^ ' 'K - > III. ^Having Anig^ted tt^ BeaiHjies, tni founi tk Difienences between tb:m and tbe Root \ write tik Diferei^ce tf tub Branch, pifi IfgdinH bit correfpondemypakfoBow^ The Nature of this J(Hle will be underftood, in working feme / /■ A' L L I G-^T I M' 9 J . Qfffiiw I* A ^^^orn Mafter would Mix,¥6ur Torts of Grains to- ffstha, vt(* Whc^t « 41. 6d. the Buflicl ; Wtait at 4/. the uihel ; Rie>t 3 x. the Bofliel ; and Barley at 2 f^i 8 ^ the Boihel ■So as to make i{ Quarters in aO, to be fpldat'3f. 6d. the Boihel ^ How much m\A he take.of each (art d'Gmn. I. Heduee the 1 5 (^tfit^x into Bujhekf and they make iToBufitek. II. Turn idl your J^s of C?r40» into Prn^e i So will the firft Wheat be 54^. the fecond 48i« the Rie at 36 i. and the Barley ^t 32 i. the Bufi)eh HL H^uce the ^ of the required $ufi)el^vif. 3 i. 64. into P^e; and they wiltbe 42rf« And that is ^f/^Mgoi^ bemg thus prepared \ Set the Nush ber^own as in the NLargptCf and Xwithem fo, that a (?re4rfr and a ii)0- may ftiQ be to- gether; as54and32,and48^ith36. Then flace the Prife of the Bii/feff/Ilequirea 3 *. 6 i, or42i. by it felf on the tjsft-bMl, and take the Difference between that, and the Frke *"^ ^Sum- of a BuJhelKif every one partictilar erim: !^ the DiCbence between <4 i. and 42 iithat b 1 3, which muft not be(etagainft<4, but againft that number which is Linked with 54, thatis, againit 32. Alfo, thei?^^«9^ between 42 and 36, whid^ is 6, nrnft notbe retag^ft'36, butagainft 48, which is Linkei widi .36 : Andfo muft aO the Differences be Ordere4 { as is e^iie t^ belcennitbelifttr^e. Then proceed by this . RULE. tkat fioB give tbejuii Quantity of that Particular Jdni^ wbofe Prim fiandetb againS the Difference ym wrof^h vmb t So, ' Multiply 120, (the whole Mrfs to be made) by 10, (the Difi feroice fcindiqg againfk 54) the ProduB will be 1200: Which M- \idei by 34 (the^ftm of all the Differmuee) the Qmiem will be 35t| Bttfiets . And fo much muft be taken of that »^4r, whofe Price 1345. 6d. or{4i. And working fo for all the reft, yQufhall Rnd tfaatthere muft betaken of that GnOn wh^Te Price, tp! r. For, ' Aft 34 is to 120 :: So is lo ; to 35 rf .34 : I20 :: 6 : 21 ,4 34 : 120 :: 6 \ %\ ^\ 34 : 120 ;: 12 : 42 ^f Now to ?wt this Right ; the Pro4«^ win be 5040 i the.Pr/Vff. of the whole^ NUTs in Pe»i^f, . Then for the ?r,ice of; tha ii^if of e.Vjery fprt bt Grain^<^ 4&Ot)e Bufhel'Of Qrdn^ ".la tp.the-MVff of One Btt/iijpj^of that Grtf/ir, SoiSi tite Number of Bu^els to bp taken ortftat Qr^in. Xa.th£i?^/<«<tf~tho(eB«/leS. • ' — "■)-'. So the PriiT^ of the Birft (ore of Whm was $44 a94 tt^Q w^ to be taken of that fcrt ' j ^ ^ 'Bufi)^^ i Therefore^ JWWfily '3 5 rf by 54 tiie Prbduft wHl be 1890^4:? and ip »ttcji wiH tiic^ ^a^- tity of that fort of Wbedt amount unto* And working with aH the' reft^ jn the, (kme msmi^Xi you. wiB find the i^ei of tJiefeveral Qu/oitnkst(k te.asfollpfK^thj v(^, Btilhels * Pence ^ ^ . -n, ^21 *i-|kBu(hcls348>comesJicx)8 '# -^''^721 ^^C at ^^6\ to 7 756 'f| N2 rf*^ ^i^r ^1344 ^4f Which added together do make iooS 7Xi ; 4n<l theJVlfiwKr^^or ojf th.e |ff/»r(}i^r f^^ions being dfvided by the peinpmimor^ tl^ CiM^iVfl* WiH b^ 4?> an4 tbt added to 49^?mak«$ it 5040 Pcnct equal to the ^mP i& tl^e jf^&'2Hfl?/i^ at'3i, dJi- theJS;#/. Qtfe^i0» XL .Onf hath $ forts pf :^puts at feveral prices j D^es at 2 5. Almonis at u. 4^. Currants at 10 i. J^/yi»x at ^^d: Friikii at 4i. and P/^j at 4i« the Pound \ and iVoi)^ take of every fort fome, to make a mixei iuaptitj of 30 /* wei^^t, to fell one widi another for 9 ^ the ^ound, how mudi qndUip |^ke of each ? , ' ' ^ Having Having placed the Numbers iwd their differences, and the fam pf thofediffereages diftinfUy as hatji been Ihewed Ijefore, and thay baleen ^ By the Figure in jthc Margine ; the 'Wc^lsever- inore like that in the former (^ueftioife . So 6 5 4 (3^ I 7 15 Sam ii P^, ')tA> i* ^c* >s the tblrd in^theRule, lo'be repeated till' all the difierences have been employ-a i . '-.:<,«»•*.#» • So 3P'"«ultTp{ied bv 6, 'produced ?o, j?hich divided by 38, *e Qubrieftt is 4'f J , of a poSrid wcighp, ^nd fa much rnuft be taken of Dates, ^2^ii 'Seto«aifr5 times! 30 "is 15,0, which divided by 38, the Quo^ tient is 3^4^ for Almonds. And forking after the fame manner ^h 4, I,- 7, 15, their reijieaive Quantities will be foond to be ttcft: founds^ j)aies '4>'28 jCurrams 3> . 6 .-» Trums • 5, 20 . ■ < ' '■ ■ ■ - ^ In alt 16 15:1 ' Ttet is 26 ']| and'the'redu^ron df the' fcaS^ 30* as it ought to be, and by comparing the prices of Biefe-paf- - tfcdlars ,4dbcd,, .A^ith thc^ price.pf iqi(^.:w?is!jt,^.^| ^Al'^St^ ■W^iglit, 'Which! malces^ 470^- This m?y be be proved-Tika*the ;&icttarthe Reader may b^ pcffeS m ir; f wnido-ftHweafo^ 'arfanowcth;- •,:*:. ^ r «M ^ $ay ^6 J L L iO Jt tO^. Szy firfti 24 times 4 1396, and Z4 times ^26 is 672: forthe Bin; Set them thus^ Secondly, j6 times 3 is 48^ .; — . „ and 16 times 35 is 575j Thirdly, ib times 3 is 30^ ) — ii£ . and I o titties tf is (?o, J And < times 30 is 1 50 ...^ Fourthly,. 4 times 5 is^io^"^ - - and 4 times 20 is So. J Laltiy, 3 tunes u is 33/ 8Qd3timtS33i8 96 9f» 48^ 57<f 30, 60 6o< •} — 10, 156 86 i3i 96 In all 227, 1534 flow tiiis i634 being tiie fum of die Nam<^rators of FraEHons, Itbofe common Denominator is 38» mail: be divided by 38, add the Quotient will be 42^ Which added to the whole nnmbor 227 ; the fum is 270. And 10 much Is go multiplied by 91, which Ihew< the urork to be rights ; The Combination Or linking of Nufiib^h may be taried at ptei- fure, as whereas abov^ I finked 24 and gf, alfo 16 with 4, an4 10' with $ ; it might havef been 24 Widh $1 and x6 With 4, and 10 With 9. Ot 24 with 4, and 16 with 9, and 10 with 5, of which diverfity of linking would Mow diverfity of folutions^ but all true, as the Reader may eafily prov^ by himfelf. Likewife, if the Numbers to be linked were 3, $, 79 or any odd Numbers, one of them may be linked to two fevendfy^ to iaake the Work even. Exm{U ni* If the Numbers were iii to; Zi 6ft2Ai^ and the ^ nein or cmmor^ Ttite required were 9, you might firft link them as you fee herei taking i2 twice ^ orelfe you might take any other twice as you fliall think fit, and fodxe Wolrk will . be every way right, though not the fime; if * 1 the diflferences be rightly fet oiF, and ordedf iifed, » is taught befoieinthe firft ^eftiotfif i^tfim ALLIGATION. 97 f^ufim ni. A Gold(inith wonld mix 3 Tcwts of Silver, A^ B. £. A. is 10 i. weight betcpr,,^. ji. weight better; and C 4i« weight better, to make an lA^ot of 50/. weight, which ihould be in fineods Si. weight better: How much maft be taken of each? Set tbem» theh: ^ifoenoes^ ahit the film of their diftrences, as in die Mar- ^.- Then, r to 10 7 4 2 2 50 Mm« rirS^ <o Moldplyed by 4, 19 200, and dhridcd by 9, the Qnodent is- Sceovdlff 5o>lttitiplye3byi, is $0, and by 9, the Quotient Isrr tbirdyt 50 Multiplyed by 2^ is ioo» and dimed by 9, the Quotient i s ? Fatrtblfp The &me.again b-^-— InaH Which is equal to <o, the Qdantity required. Now the firft Traaion multiplied by 10, (omitting the Deilor ninacor) is 2000 The fecond aUb by 10 glfes 5cx> 'Jhc third by 7 makes 700 the laft by 4 makes - 466 ■ i- 1. ' ' Inatt 3600 that is, ^"^^'U which is equal to 400, add If the whole Ingot p, be multiplied by the bettemefi requircid, namdy, by 8, they mSi produce 400 alfo : So this \s proved. In every Alligation, or linkiijg of two Numbers, this is evi- dent, that if the fum of the Nunibe^rs liiiked be greater than tbe mean Number required, tak^n {p ihahy iimes as there slr^ Numbers to be linked, the queftion would be abfurd ; and the telolution thereof inlpoffible* And this fiull'Ierve for the Kule of Miginim THE (98) THE • • RULE r> Of. pJlse to sit ton. ^k 1HIS Rule firrves to rerolve Quefiioni, wliich are not I ^ prefently fit For the Golden tgky zni tberefore inflead JL of the tme Number which i^Cowtit : Sttppofe any Num- ber (jfittt or SmaOf and make trial of it, whether it refelve tke Qu^ftim without any Error; if (b, it i? the Truit Nkmber; If not^ Note' what Error and whether it ht.too Mucb^ or Too Link ; if Too Much mark it thus, -J-, but if Too linkf thus — / . Then fuppofe, again, another Number, (it imports not whe- ther it be nearer or &rther off} and tiy as before, and mark that Error alfo with -}- or — ', according as you find it to be, either More or kfs^ And then Work according to this Ibnow* • ♦• RULE. ' Multiply. the frff Pofition, By thcfeconi Error, ani iheJeconiV(y fition 4^ the prS Error, and (if the Errors be both -{^ or both — ) SubftraA the LeSsr ProduS from tbe Greater, and keep tbc Remain yor a Dtvidjsnd^ and tbe DiSEcrcncc of tbe Errors^ tbe Dlvifor; tbe Quotleot of tba$ Divifion is tbe True Numoer re- quired. But if tbe Errors he one +, tbe other — , jfc Sum of tbe Pro- du£b added togetber tnuSt be tbe Dividend : J»d tbe Sum of tbe ; Errors, tbe Divifor} the reft of tbe Work is the fame as be- ; fore. Queflion I. A Man is to drive 48 young Turkies 40 Miles, Aiia for every Turkey which comes alive' to the end of the Jotfrrt^^ . he is. to receive 3 d. but for every one which dies by the way, '^ he' is 40 pay 6 d. At the end he received 72 d* How many died by the way ? Let The Rftkfif Fdlfi Vppig^, 99 Let the Firft Sujpofitipn be ; That, by the way, there Dyed Twenty: * ' .-,^ ^ Am. rr 120 -— 36 ■ >^ Tkffthcmhe was to pay. Twenty Sir^Penccs, cffwr- And for 28 whidi Lired he was to Receive aS? Tluee-Flences, or - '3 ^ he {aid more than he receivU— r- B8t he ftould have g^tteo; rrr-^ X«fcp- » .) 'Wherefore the ¥irft Error fe- ' Add; X08 T-r 108 Irt the Second SoBpofition be; That^ by the way, ther* Dyed Ten: ' ' * j 4* Rjt them he was to pay Ten Six Fences; Or- And for the 38 which Lived, he wa^ ^Receive 7 Nine ShUltogs Six Pence, O r \ The Difference is — * ^ * y> ---m ' . Batitiho^ld h > ^Q •^H %54 T —■7^ if^mi^m^ Now. ii 90 10 JwrttipUedhy 1 1 j||?rodttCcth| J J^ "^ The Dlffiarcncc-rr-r-' 720 DivldeRd. Alfo « The IHffiaence of Ifhe Errors 1 And 72o» ^vided by 00, ^ve| in Trie l^iunbcr wWclrB^by the W Eigjit that Dyed, he Paid 4x* <X ^rtytbatlived, UbAefidsed i^QI-Tt Thettilftrehce^ 72 02 ^<ry?/ ion 100 The ^Uef fsfje PofittM, QtuflioH II* If it were required to oake tip a pound SmUng of Shillings, and Groats only ; and lb as the rlamber of Groats inay be to tiie Number of SUDingi, as 7t» i : How nuoy Shi^ lings muft there be P . > •v. Tirft, Tuppore %hc Shillings to be 4 SlriB^gw then the Groats mtrft be eqnal to 16 $• vff«, . 48 tirom bat the Shillings taken 7 times are 2$, to which 48 fliould be egoal^ but is -)- 20 Secondly, fuppofe the Shilling$ ' 2 jtben the Groats (making 18 x.) arc $4 which fhould be equal to 7 times 1, butts -f- 4^ Multiply 4 by 40, Produ^ is 1^^ then Multiply 2 by 20^ the Produft is 40, which taken from 160, relfe forth? Dividend • %%o .And tbedifierence of Errors is 3p tafiiy, ' 1 20 (Dv'ided by 20,' the Quotient ii' 6 The Number of Shillings therefore is 6 And tiie Number of Groats is . 42 For as 7 to I9 To is 6 times 7 which is 43, #b 6 times z^ which is d : So the Work is done. « Queflion III. If tbcrebe 4 fcvcraj weights, A.B.C.p. of which D. is 24 Ounces, and C is, double to B. and triple to A. and />. widi twice u^.- is double to C and quadruple tttBm How mach dorfi every one of thefe Weights weigh ? Tirft, fuppofe u^. to be 8 then J>* vvith twice A* is 24, and 16, that is 40,. of which C being the half is- 20, and B. I p.. J / Now thrice ^, is ^u to which C ibouldb^ ecpial, butis*' *'— 4 • • < • — Secondly, let ^. be fuppofed 4 therii7.more,'twice-^. is^i andC. 16 ' \ andB.is8, but thrice ^. is sa, towhidiift . . , ., v ; . ihould be equ^l, but is , ^ -4* 4 Then « • ^ The Ruk of Falfe Pofiioff. - loi . Then ? multiplied by 4, gives ^i, and 4 by 4, prodnceth 16 : BoA thefe Produfts give 4S for the Divideml : And the fuiil of the JErrors C becaufe the firft is — , the other 4- ) giyes 8 for die Divlfor^ and the quotient win be 6, to whkh ^. is e^ qtal, and twice J. more 2). i^ 36, of which C being hidf Is 18, andB. is 9, and thrice^, is equal toC. namely 18, and all right* Whereas the firft Error is equal here to the fecond, it fcBows that the Pofitions were eqnaUy Fal&$ and therefote d^eir dif^ feience which is 4^ being parted iODO two equal Parts, 2 and % if 2 be taken from 8, the remain isthe troe Nomber 6, or if 3 be added to 4, T^hich was the feomd pofition) the fum wiU be aUb5. And farther, wheniberer the JBrrars be ooe •fi,. the other ^, tiioagh they be not Eqaal ; yet dien, if the DiffereMce between tlie Pofitions be partaed into two Pws, which are in Pbo^pcctioa one to anotiE?r, as the two Errors are one to another re{pe£live- ly : TTiehlf the frit fm be taken from the Jirfl T^nm (if diat be die Grtmr^ oradd^ to it (if it be the Ufs) the lame Nomber required isthejreby had. \ - ' - .^ ^ As, let die laft Qtieftion be refomed).. *- .^ . > And let the firft Pofitioo for i<. be x% Then the firft Error will be ~ 18 Then let the Second Pofition be ^ And (b the fecond Error will be J^^ And the difference of Pofition is 1 2 Which divided into two Parts 9 and 3, which havp that Pro-, portioa one to another as have the Errors 18 and 6, then if the firft part 9, be taken from die firft Pofition 15, there retuains the true dumber 6. Or elfe if the fecond part 3, added to the fecond Pofition 3 ; Thereby alfo is made the true Number 6. The way of parting X2 (or any other) into two parts propo^ tioq^ ^jt|fi die Errors, is eafily done by the GoUtn Kgle^ thus : Asdie finn of the Errors 24, is to the d'ifierence of Pofition 12 ; So is die greater Error 18, to the greater Part required, namely 9; Many 10 1 Tbf Rftk (f Fdlfe Fofitm. Many other Qoeftions are in the otlier Books exemplified and mowht by this Jlule ; but feeing I intend not to write a great Book- and alTo becaufe fome of thofe Queftions may be refolved witbont this J(«fe, I Witt a4d no more; Only mention one of thofe Q«eftiQQ8« If tiwe be a Ciikm wkb 4 Cocks, which holds 8 jBarrels of Water, and the firft Ctck win roa it all out in 6 hours, tbc Ibcond in 4, the third 19 3» ^^ tl^e laft in 2 hours: In what time ibaaaiUof tbemrunitoat? . If Che firft in 6 hours runs 8 ' the feoond in the finne titee W0UI4 fon 12 Ae third . »* Chetoft . ^ . • Inaa ' U^ Thenfay, it66tequiretf, whatS? \. . Thea^er^l, that isf of an hour; in which time all tbe 4 Cockstogether wouldruAfWfr^ the &BaicdMf W^ter* TflE S » - r --» ( loj > TUB RULE Of CERES ««rf VIRGINUM. H IS ^Ciie laoft oncertain. and uimeceflary Rule in JriA- metieki being fddom ufed except in sponugQgtfiiimsto pu2zel young beginners, with cafie Problems : Sadi aa Slow* ((fiftknl. A Cateferboxidit 8 Jinliof twoforts^ as Geefe and Hens for aoj. the Geefe cm 4/. a Piece, tbft Hens.;tr»'a pie<9{ How many did he buy of each fort? This may be done by tlK ^ule 4f FMi and alfo thus: Mnl* tipiy the whole number 8, into tne leaft price 3, it prodaceth 16, which taken from die whole price 20, refb 4 fbr a Dividend i which divided by 2, which is the difference of the articular prices the Quotient is 2, for the NthUberof Geefe % and 6 muft be for the Hens: The Proof is eafie. • - 5. 8 U / • **r-vm -J C 2 Geefe at 4 gives : iv^^ ^ Hem at ^ gives i> " ^< >t^ UiillliM 8 20 eueflionU. If 21 Perfons, Mff», >^<?«itf» andc*/7ir^ fp^nd ^ Shillings J fo that every Man pays 2 y. every lyonun i s. every CI4U id. How many is there of each fort? THE RULE, MiUiflji the dumber rf* Perfons hf the leaft Exp^nc*, mi take the ProduS of h from the whole Expence, the rcSt jhaU be the Diridend; vHA divided by the diirerence betwixt t')e grea:eit \ 1 04 TAe KuU of tkres amU Firgham, ini leaft tmkultr Expences : the Quotim it * Number, wKrf tie Number rf Ue» (.«r On ffbicbftend mo6) comet ne/r to\ Im ereateft ad leaft ft^poWA Oe Qimim ii t Numier, dm **»» Oe Nkmier 1/ Mew (or tiofe i^i* ffoU moS) cmm be mud kfs. ' So here at MnWplyea by 6 i that is, by i, the Prpdua b in » which taken £wn a6, jjefts 15 i, tor the Dividettd: And Sn^ttjff f from t. tel6 I i for S* Divifor. and the Quo. ^t is *h *1^ '^ fiunething more Oan 10} die MoQiber of ' Then turn the Dividend, and the Dhrifor both into whole To they reduced will be 31 and j, as before ss to be fcen m the Qpodent* Multiply the Divifor J, by 9>Cwhidi is tte Nnn^ thrProaiik is ay, which taken fiom }i,/whK^is die redi^ rSividSd) thereaainis 4t &r the Nttmbcrcf mmn j and the CUUmranA be 8, 9Meaat2«.each 18*. 4 Women at t f . eadi 4 8 Childicn at 6 it each ^4. Ia'aU2i 26 But the nniAber of Mat «n«Ll^.> jSfjS^SJSf •^ betf. £x*m^ 4 I 7?be RmU of Ceres md Virgimm. 105 Eumj/te It t Men at 2 $• ^cb ti ^ 7 Womea at 1 1. each 7 6 Chil4rai at ^i. each g In all at .. U.ai 16 Or the nimibcr of Men ma/ be 7, i^chiiHiltiplidd by ff prodtt* cerh 21, which takers ictm ji, renWikis id ibr the Woirary «i4 4 Children. MxmfU Jit. 7Menj!t2s. each .* 14#. ^ 10 Wofttett at I s. each 10 4 Children at ^^«eaG&' ^ 2 In all 21 ' In all 26 So there are already (cm ^ vark>cii|qtotKHis of this Qpeftjon^irbjcli make ^is Rule the jefs tfi be regajrded. B»& further,' the numberNrf Men may be 10, and not morc^ for if you put them II, that. Wiltipii<gf*by 3, produceth 33, which is^ greater than 3I9 from "which it ihould be taken^ but I fay it may be 10, and thei) ther^isooty one Woipanri «gsd t^n Children : this confirms the' former part of the Rule. Now for; the latter part, if the Dividend 31 be divided by the fum of the two extream expences Creduced by doub« ling as the Dividend is) 4, the Quotient will be 7 *• And the Men m&y her 7^ as hath bjEren flievyed j hut they may be alfo but 6, afW feweir. they cannot he : As 6 Men, 13 WomeQ, aad 2 Childj^ V SfHT if you mt the?i 5» that jpultipM by 3. produceth' :^5,.whteh Wkcn ftpm 31, there repiaiflf 16 for the Women, ind: fo therms ftouW tp Md Children which is co^ fxary to ^be Suppoficion* . • - Andiurtijer, befcaufe the %ociftnt ]»as 7 |,.th^. Number 9f Men^Tiighc be 9>i if pure Arithmeci^al Diviiioq be only re- . gar^d: And thea tke. Women alfo afe in Number 7-*, and xj^ ChJldren 5. J, «s aiay eafily be .try«d j I -need pot exem- Jifie it. gueflioH TIT. Tf thferef be an fixhibitibn of 900 /. per jinmm Co 30 Perfons: Some C/<rjtx, feme Mejfevgers^ and fome X>oor* keepers f at 60/. ezdi' Clerks 40/. each lAe^efiger^ and 20/. each Doorkwer ; how many muft there be of each fort ? P MuU <j'i6 '■ The Rule of Ceres snd Virginum. Multiply Caccoi#)g to Jhe Tkvlc) ,30 by io, the Produft is 600, which taken from pcx), remains 300 for the Dividend ; and 60 want 20, that fe 40, for theDivifor f*Afad the Quotient is 7{, and more die Clerks cannot be ; Al(b divide by 60 more 20, that is 8ot Quotient is 3 4, and much fewer the Clerks cannot be. Not to ftand upon . the Pradions fin this : csLfe: of dividing Men} the Clerks may be 7, 6, 4, 3^: And the Meffengers i, -S> 5f 79 Of 9» and die Doorkeepers 2a, 21, 20, 19, or 18, that jtbp C)prks\cannot (ia whole Numbers^ be more than 7. or lefs 9, may thus be proved ; Firft, let them be 8, than ^^, times 40 is 320, which is more than 300, out of which it iBould Be taken ; Secondly, let them lie 2, then 2 times 40 is 80, out of ^00 remains 2^o,^whicli divided by 20, gives die Quotient ti» for die Meflengers; fo the Clerks ^d MeUengers being 13^ the Remain thereof to 30, namely 17, inuft be Doorkeepers. » But, 2 Clerks at 60/. each , 120/. II Meffengersat4o/. each 440 17 Doorkeepers at 2qh each -360 In all 30 In all 920 - Which is 20/. too much, therefore the Clerks ,j;anttot be two# * Niffte^"^ It may' be asked , why the Remain 220 ihould be di- < Tided by 20: Whereas the like Remain in the former Example, ^ namely, 16, was taken (widiout any Divifion) abfolutely fbr * the Number o^ Women, or Middle Number ? 1 anfwer, al- 'though the greateft or firft -Number being founds ''as here to be 2) the relidue of 2 to 30, might be rightly Pitted into - two fit Parts in the fame manner as the firft QocftiOL of this •Rule wa* refolved, or dfe by the l^le of Faife: Yet t^ give -further fatisfeaion, the caufe of this i$, the diflference betwixt "the twof lefier Expences, was there J, whioh (before the Dvi* fion was reduced to 1, which neither multiplies nor divid^ ,,anyv Number, but leaves it the. fame, Wfiercas, in this laft. .the' middle Expence (or Exhibition) being 40, and the leaft*!^oi the difference 4of them was 20, by which dividing the Remain of The RttU of Ceres mJL J^irginum. 1 07 •f the laft Subftraftion t The Quotient k ever the Number of the middle Ferfons. Which may ferve as an addition to tho Hule, where the forts of things are but three. ^Jiiott IV. If there be 10 Perfons of four feveral Countries, EngJ^ Frencbf Dutch and Spmifi)^ to pay a Debt of looo/. So that every EngH^lAm jwiys 50 /• every french-bUn 70 /• every Duuh-Nlan iiol. and every SfAnitrl^ i$q A Mow many is therp of each ? ^ ^■ The Dividend (according to the former Rule) is 50CX Now to make the Divifbr, uke his fiim that pays leaft- (namely 50} out of each of the other tliree 150, 130, and 70, and the Remains will be 100, So ^d 20. ' ' ..iff Add the firft and leaft for tliec'1>iviror, it is 120. - « > • « •And the Quotient will be 47 f, and the' JfiwWi igannot be more* \ ' . ' Secondly, add the firft and fecond together for 'the Dlviror, it is i8oj and the Quotient is 24^, an4 the Spmari^ cauQOt be leTs. - ^ '' ' . I mean, they cannot be much more ti^n 4, or left, than 2 : And therefore, feeing any one Solution will ferve, lee' them be* 3, and by that mukiply 100, and take the ProdjaQ outqf KqOf there remains 200 for. a (econd Dividend, '^whith\di vided oy (the fecond Remain) So, the Quotientis 2{ t-THerefore the jDnrrfc*^ »>«i are 2, which multiplied by 80^ m-^^e J^o;. takei, that, out of 200, there remans 40 for a third Divlaend i WKrCh divi- ded by (the Third RemainJ 30, the Quotient is 2 for tfe Irincb^ men alfo ; and confequently the Englijh muft be 3, becaufe all •f them are 10 : But the Sfmatds^'mj be alfo 4 or 2. Example. f'\: •• - "t 1. «. . . 4 Spatfiards at 150/. eic^ \^ 0^600 j i ot I Dutch-man at 136/; * ^ J130 j j .1 French-man' at 70/. . .^ 70 ' I 4Engli«iri56;. iach'^^^ • 200 ' "^ \ 10 In all 1000 P 2 2 Spaniards \ foS '^ The Bjik bf ttres dndVirginutn. 2 Spaniards at 150/. each 3 Dutch at 130/. each 3 French at 70 /• each 2 EnglUh at 50 U each 300 390 210 icx> 10 In an looQ The reafon why thf SpfnUris and Englifi^ as al(b the Dutch and French are equs^ in Number, is becaufe their Payments differ equally from loO) which is die Mean Sum witli which 10 Men Ifaould pay toob/» ^making it fin this Qjseftion, and many other of this Nature, may be anfwered by the Rule of .^l%4tiMF< Thus, SCO if 160 give 10, what 50? Ahtwer J$ 3t^, (diat is in this cafe 3) for the Spdi: niardSf and as many for the Bnglijhy becaulc their refpeflive differences from ico, the one 50 more i the Bnglifi 50 (e&, ajre ^ quaU And alfo, becaufe the other two differteces 30 and 30 ar^ equa)^ the Hmsbej^oS tkeFrtnch is equsl iq tte Number of IhJB fiutch. , V Btit both tho(e Jhfumbers together aie 4, beaiufe 3 Sfmiris^ aitd 3 £»g/i^, taken out of 10, the Remain muft be 4.^ Wherefore the l^omber of the French is 8, and the Dutch Or tkns; {f«? ' — -501 the Mea ia Jtke fame order. If itfo zieiiaire io» what 30 ? »4p . I Anfwer^ Tke Rule of Cms dni Virginum. T09 Anfwcr is ilff^ (tl^ Ji.. i; tWs o^ and confequently % Bngl^v^ii therefdre the FrinA and DmA each 3. But where aiif one of tbe ptsticnlar fiims iseqoal to tfacMtaa Susi) there this caofiot fp well he done Vf ABigttim. ^ txmpk* If ofle ftould buy rz loaves trf Bread for f^ >iird# ib flue ^me might be wthfemy^ fijme fenny ^ fojne b^-f^/t iodfbiM /4rr*fcg Loaves; And it be required to know bow tAt^y te muft buy of eadi^ • Then beaufe of 12 loaves for la pence^ the MeanPrice is i, but one of the particulars being alfo x, there Ihould be no penr jiy loam, bccaufe there is no difereve betwoen the Mean Ffke, and z Pamy^ Bnt it may be found by the Rule of Ceres and Vkgimmt i» Ije cithcTf ^ 4 Two-penny loaves 8 VeACt a Penny Loaves 2 Pence 2 Halfpenny loaves t «««iy 4 Parduns Loaves t Penny m^^ to rtl >« laaves. . la aD »a Pence, Or ^1 3 Two-penny lo^voi 4 Penny Loaves 3 Half-penny Loaves : 2 Farthing l^ves InaU 12 InaDia ill'' » • r« TTT — • . 1 . y' ' ■» flip 3 POSTSCRIPT. To tbU i^fc qf Ceres mi Virginum, (to fuppjy a Vaaney) I have addei hy way of Poftfcript; /in»/<?w Enigmatical Que- fttons wMtbek Anfwers dffixed : Legving the tfurnnerbowtg refitve thcmjtotkfifn^mmjof theUmier. Qfefliou I. There are four fereral Meafiires^ as ^, B, C, 1> ; Of which, D hoick 24 Pints, C holds as much again as B; and 9 times »Bwchsi^iilx And J}^ With twice A^ wiHhold.twice as much asC; and 4 tim^s as much as B. How many Pints doth each of thefe Mcafufiei^hQild fcverally ? * Anfwer, ji holds 6 Pints. /> 24 •■•••' C 1% •Mm 0% ^ 9, So that € holds as much again as 0, and ) times as much as 'A\ and 1> with twic^ jtf, holds as much as C> and fiHiij times as mudiasB^ ..... ' Queflion IL One ;took afum of Money with him, and went to a Gaming-Houfe ; where, at his firft Game he doubled the Money he brought with him;- At the fecond Game he loft 120/. Atthetliird Game he doubled the >loney he hkd Remainmg : at the'fourth Game , he loft 120 /. aiTd bad no Money left. What was the Sum of Money that he took wi^i him ?. ; Anfwor^ 90/. Forafmuch as he loft 120^ at the fourth Game's £nd, and had then no Money left : It is evident that he had but i 20 /• at the toA of the third Gattie : Which was thedoal^of ihe Money which he had at the end of ^fecond Game, namely, 60 /• which with i2o/. that he loft atthe.endofthe fecond Game makes \%oU which was the dpi^ble of the Money thjit he brought with him, name* ly, 90 /.which may be thus Proved.' ' '. '' ,"- h The Money brt>ji^ht with him ^' ' - -^ ' 90 The firft Gamer dofubled it •„ •: .: 180 The fecond he*Ioft.^2o7. ^ - -120 Then he ha<f le^ ' .1 : 60 At the third he doubled it 1 20 At the fourth he loft 120 bV There Remains . 000 Queftim in. Two Perfons ^mes and Ptfu/, had between them 2 •ertain Number of Sheep in two Droves ; fames (aid to ?««/, if you f ut 70 of your Sheep into my Drove, I Ihall have three times as ma- ny F 5 T S € ft IP r. Ill »y Sheep as you have ; But Pml fald to fams^ if you put 7(rtf your Sheep into my Drove, I ihall have 5 times as many ^ you. How many Sheep had each of them ? AnfwcTy famcshsA 1 10, and PaullaaA 130 ■■ ■ For, if y6u take 7ofroa%P«tt/, and add them tojmes^ then^ifiii^jwjll haveiSo, and P4ii/but 60^ which is but one third part of wJat^PdMi has — — Butif you take 70 from ^tfifirf and give them to /«a;: then f<iriwU have 200, and ^imes but 40, which is but one fifth Part of what fml has.' - .'*'• f^^ion IV. A father gave to his £ideft Son 2$2 Crowns, and to his Youngeft he gave but 2'<J Crowns : And to every Son fuccdSvely, from the Youngeft, he gave 28 Crowns more than to the preceding. How inany ^ns had the Padier ? And bow many do the Crowns amount unto? Anfwcr^ He had 9 Sods. And the Number of Crowns 1260. rr^4^imV. A Drover driving of Sbciep. before hiQi:. Qge meets him, and fays, good fpeed Friend with thy 20 Sheep. Nay fays the Drover, 1 have not 20 Sheep; but if I had as many more; and half fomaoymore; and2Sheep; and half a Sheep; thenlihould have 20 Sheep. How many Sheep had he ? Anfvfer^ 7 Sheep ; For 7 and 7 is 1 4, and half 7 is 3 and a half, that is i7andahalf, and 2 anda half is 2o. > Queftim VI. There is 273 ?• to be divided amongft four Peribns: Andrcvfy Bennet, Ciriftophct and DMniei ^ Of w)iicl),;'H^irew is to kive a part unknown ; Tknnct is to liavt twice t^ xduch as JndreWf and 30/. more; Cbriftofber is to have 3 times as much as Andrtvf^ wanting 52>/. And Dmicl]& to have 5 times as much as Andrem^ and 20 /. more . How much of the 273 /• muft each Perfon have. f^Andrevf I. In all 273 Queftim Vn. There was a May-poje, which In a windy Night was broken, fothat'the top thereof lit upon the Ground, at 30 Foot diftance from the bottom thereof; and the Piece broken off was 50 Foot long : I demand how long the May-pole was in all ; and how long was the Handing Part ? Anfwer, ; 112 FO ST 9 C R I PT. Anfwer, The whble tcngth was Tbe Piece ftanding Foot 40 Thele few Qoeftioo^fliidl ftffice in this place ; fiich as are is- lighiediathi$kki40f Diveifiooinay perofem^ . rtHMria *«i*MM ♦ •■ 4 . . Tht End of the FtrB P a r t. :] ^ Decimal Arithmetick. The Second PART. CONTAINING Thd Gtounds and Reafon thereof : And ap- plycd to Pradice ; L la all the Rules of Vulgar Arithmetick. II. Inlnter^, Simple and Compound. III. In difcount and Rebate of Money. IV. In Equation of Payments. With Tables of ^11 theft. ALSO, In the Extraaion of the Square and Cube Roots. In the Menfuration of Superficies and Solids. And in the Works of fewral Artificers : As, Painters^ Plalfierers^] Carfcmirs^ Brkk'Lfycrs^ and Mafons* Whereby, this Decimal Arithmecick, will be as fcrviceable to allof fuch Profeifions; As the foregoing Part was for Merchants and other Tradefmeh. By William Lejibourn^ Philomath. LONDON:^ Printed for Amfljom and John Churchill at the Slack'Swan in Pater'Nofier''Rcfw^ \ 790. r <"0 «■■■■■■«■■* Decimal Arithmetick PART 11. INTRODVCTION. HIS Second Pm^ iwhich Tceateth of DECIMAL A KJTHIA E riCKJ I ftall divide into Four SeaUms. In the FirS^ fhall be Taught how to H^dUcc Vulgar FraBiant into J>echiuU Ftru^ or FrdSions \ and there- by to make Tables (if you pleafe) to^exprefs the feveral Jknminmwu of the Coins^ JVeighs and Metfures of your own oc other Ccumriesj in Decimal Parts or FraSknsi And how &> make qTc of fiidi TMes upon all occafions. The Seconi SeSm^ conuins ffctasim\ and how to Work all the j^/w of Vulgar Arithmetick^ (treated of in the Firfk Part) Decimally : And to ExtraA the Square and Cube Roots* The Third SeBiim treateth of iiintple and Compound ImereS^^ I>ifi(mftt and Ret«te of Mdney-—. ^JEqtmion ot Pajments, 8»^* ^cbafe of Annuities, Valuation of LeafeSf ^ and the like^ with Ta^ l>kt of them ready Computed.. The Fll>urth SeSion, Teacheth how to Meafure Superficies and Sitids \ As Board, Glafs, Land, Pavemcm, &c And Solids, 2sStone^ '^imber. Spheres, Bullets, Columns, &€. And alfo of the Works of the fcvcral Artificers relating to Building j 2& Bricklayers, Carpenters, NU* fm^ ^ojfners. Painters, Glafiers, &c. SECT. I. Of the Nature of Decimal Arithmetick. I Will not in this Place infift upon the Excellencies or Antiquity , of this kmd of-Aritbmetid ; but fall imuiediately upon the PraSice Pt iti. Whichlfl)aUdo by the Solution of icxciSLiProblcmt. 0^2 I'ROBU ii6 Decimal Jriphmetick* ^ P R O BL. I. A Vulgdt Irtilm being given; how to Reduce die fame into t Decimal fan gr Frdtion^ RULE. To the Numerator tf tbe fra^im giy^i. ^^ ^^^t ifumber of C^^hets jfou pleafe I tkn divide the Nunierator by tbe Denomina*- tor, tbe QQotknt Jball be tbe Decimal Fra£tioh requ^d. Example I* Let It he req^ui^ci^ to re4i^ccrT^ intoa Decimal. Firityto the Numerator 4) add fiv^ Cyphers, u>wiU itbe4ooooO| divide this number by the Denominator 17, and the Quotient will be235is9y which is the Decimal required. - •• f And here Note, that Decimal Fr avians are not written in a fmaiter Figure with a Line ^between them, as Vnlgar Fusions are^ but of the (ame Figure, only there muft be a Comma or Point put between the whole Number and the Fra&km^ and that is the diftindion : And (b the former Dedmal muift be Written thils, • 23529. - Xxamfk XL If you would exprefs 235^7 in a Decimal Way> it muft be Writt^ as followeth. By .the laft Example you find that -^^ reduced to a Decimal, was «23529, therefore 235 ^^ nmft be Written thus: 235, 23599* In Decimal FraBims the Numera$or is only exprefs'd, and the De* mmhator only intimated ; for this l^le is general. Of hnv moff Figures fievet tbe Numerator ^ a Decimal FraBim doth ccnfiRy of foma$iy Cyphers rntb a Unite before tbenty datb tbe Denominator of tbe fame FraBim cmfi9. So this Decimal 12, 625, if it were Written ia a Vulgar way, would be 12 v-|.J J. but in a Decimal, on- ly 12, 625, the Comma or Point between 12 and 625 diftinguiih- eth the vmie Number from the FraBion , and the FraBion 625 cbnfiffing of three Figures, intimates that the Denminamr thereof muft conliftof three Cyphers amd an Unite before them ; fo the De- cimal before exprefs'd, 235, 23529, if it wete Written in the Vulgar .way, would be 235 r!-! I-H- But it fufEceth to Exprefs in Decimals, the Numerators only, and omit the Denominators^ the Denomiiiatsrs of all Decimal FraBi- ons being either 10, 100, iooo, loooo, looocx), Vc according to the Number of figures contained in the Numerators, Accor- I Decimal jirithmetick. X17 According to this Rule, ^ou IhaH find that '^l^illbeinDeci-n 5. 80000 cftonly Sthm^j .S 12 I pmalsbyadding > 12.42857* iB^-rf J 5 Cyphers. J i32»5294i. And by this means, all manner of FraSiom of Coins^ Weighs tod Meafures, may be reduced from Vulgar FrgSums^ to DeH* msi FraSions ; as by the aext Problem will appear. P R O B L. n. Sm to €xgfefs Engliih-CoiB in DetkuA Numbers. let it be required Co expcds 9 Shillings fwhichis ^ of apoonl Sterling) in a Dedmal ; To the Nnmeratar 9 add two Cyphers, making it 900, which divide by twenty, the Quotient is 45, for the pecimal of 9i. So;theDecimal of i^s. will be 65, andlbfor any Number of Sbillii\gs« • ^ Here Note, that in the Redufiion of Vulgar Fra£Uons into De- cimals, that many times the iirft, fecond or third Places of the Decinul FraQions are Cyphers, as in the following Tables the Decimal of one Farthing is .001041^7, and the roUbn is, be^ - caufe if you reduce ^4. into aDecimal(for one &rthlng is the 95otJi parr of a pound Sterling) you fhall by adding of fix Cyphers to the Numerator find the Quotient to be 104167, but two Cyphers ffluftbe][^aced before it; becaufe dividing loooooQ by96o, the - f lace of Uftkes lathe Divifor ait the iirft demand extendeth un- to Che third Cypher in the Dividend for in reducing of Vulgar FrafElons to Decifn^s, this is « A general R U L £• Tbit if thepl^e of Unites in tbt Dlviror, at ticfirS Demand^ ex- tend hkt unto the firS of the Cyphers annexed to the Numerator ef $he FraBion^ tb^rc mtiQ be no Cypher />«r bi^'ore in tbe Quotient^ but if tbe place of Unites extend unto tbe feconaCyphct added^ then or^e Cypher nuQ be plated before in tbe quotient^ if unto tbe tbird Cypher, tbetftiM Cyphers mud be placed before in tbe Quotient^ &c. ^ According to which Rule, if you make tryal you Ihall find diat the Decimal of j s, will be 35, the Decimal of 5^* will be ^02083 3 3, the Decimal of two Farthings will be G020S333, as in the Table. By thefe Rules laft delivered are the eniuing Tables of Enslifb money ^ tP'eight and Mcafiire compos'd, and the like maybe done for a foreign Coin^ &c. according as every Mans occafion ihall re- quire. The (ii8) t HE T A B L E O F EngUJh Coins in Decihlals: Ej^lifi f^om. St. 19 18 17 16 15 u 13 12 II 10 7 6 5 4 3 2 t •95 •9 •7$ •7 •6 •55 •5 •45 .4 •35 •3 •25 •2 •^5 a •G>5 < 10 9 5 .0458^333 ^•04 166667 •0375 •03333335 ^2916667 •025 •020S3333 •0166666^ •0125 •0833333 .04166667 •003125 •00208333 .00104167 Tr<y Weigh in Decimdls. .^% 66666 J .83333333 •75 .6666666y •58333333 •5 .41066667 •33333333 •^' , •16666667 •08^33333 •^ •07916667 .075 .07083333 •06666667 .0625 .05833333 TdbUs of Retbt^ioft. T.W. 13 .05415^67 12 •05, 11 •04583333 10 •04166667 9 •0375 S •03333333 ^ ' 7 •02916667 6 .025 5 .02083333 4* .01666667 .3 .0125 2 .00833333 1 .00416667 Cr: 23 .00399395 32 .00381944 31 .00364583 30 .00347223 J9 .00329861 18 .003125 . X7 ^00295139 16 •00277778 15 ^00260417 U .00243056 13 .00225694 . 12 .00208333 11 .00190972 10 .00173611 9 .0015625 8 .00138889 7 •0012x528 6 .00104106 5 •00086805 • 4 .00069444 3 .00052083 3 .00044732 X •000173 IX Averdupcu great Weight MDec :unals. — Oun. Founds 27 •75 .5 •25 •24107x43 36 25 24 33 32 31 30 19 x8 17 x6 15 X4 13 X2 XI xo 9 8 7 6 5 4 3 3 I iJ. 15 14 n X2 XX XO 9 8 7 6 5 4 3 3 X iatf Lotmc€» X ^n. J 119 33314285 22321428 21428571 20535714 19642857 1875 X7857I43 16964286 16071438 t5X7857x 14285714 13392857 125 ii^7U3 10714286 •09821428 •0892857 X •080357x4 •07141857 •0625 •05357U3 •04464286 •03571428 •0267857X •01785714 •00892857 •00837053 .0078125 .00725446 .00669643 •006x3839 .00558035 •00502232 •00446429 •00390635 •00334821 •003790x8 •00223214 .00x674x1 .0011x607 •00055804 •00041853 ;ooo27902 •0001 395 X AvcriufoU jiveriufok li$th Weight i» Decimals. Ounces J>rms. MM haf. 1 JKU 1$ U ts II 9 8 7 6 5 4 3 2 S •937 •875 •8l2$ .6875 .5625 •5 •4375 •375 .312$ .187$ .12< .062$ 15 /)58593V5 14 /)54687$ J3 .05078115 12 .046875 II .04296875 10 .0390625 9 ^.0351526$ 8 .03125 7 -02734375 6 N^34375 $ \oi953ii5 .01^625 .01171875 X)078i25 •00390625 4 3 2 I M^ .60292969 .00195311 •00097656 wmmmm f^m 5 4 3 2 I I |tf. .62$ •5 •37$ •^5 .125 •09375 •062$ .03125 ■w^ Dry Mfdji^cf in DecK inab. Bujbeh. 1 7 6 5 4 I 2 I t87$ '75 .625 •S •375 ■*5 .125 Peeks. I ftf . a- 2 I .09375 •062; .03225 .0234375 .015625 0078125 Pmts. 3 2 1 .0058594 •0039063 .0019531 Long Mtajhres^ the Im^ hittg Tards dMd istn^ UftU Meafufi cuaals, mim Fimt. 7 6 .87i •75 Tables of Redu£Hon. ' tti 1 g». •046875 .03125 U)i5325 Time ih Decimals. Mo^ Da. 9 8 7 6 5 4 3 2 I 30 29 sS 27 26 24 23 22 21 20 .916667 .833333 •75 »66666y •833333 •5 , •416667 •333333 •^5 •1666S7 .083333 •082193 •097454 .076714 •073973 .071233 •068495 .06575$ .063016 .060274 .05753/S - .054795 9 1.052055 18 17 16 ?3 12 .049316 .046577 .043837 .041097 •P38357 .035617 •0328f7 II •030137 10 .027397 9 .024657 8 •02 1 918 7 .02917^ 6 •016438 5 •013698 4 •010959 . 3 .008^192 2 .0054795 .0027,397 Dozens in Decimals^ .9166667 .833333i •75 .666666J •5833333 •5 ^166667 •3333333 •25 .1666667^ .0833333 .076,388 ^ .0694444 •06^55 . , •0555555 .04S611]; .0416667 .0347222 .02083^5 .013888^ •006^444 i 'Xt Th^ tit Decimal Aruhmetkk. The ufe of the foregoing TABLES. THc TaIUs preceding are ia Nrnnbcr nmc; The firft being of EngUJhCoin', The fecond of TrByl^eigh'y The third of Averdufou ffcat Weigh ; The fourth of AuermpoU little Weight \ The fifth df X/juifi Meafures^^ the fixth of Dry Mcdfurts ; The feventh of Long Meafures ; The eighth of Timef and the niilth cf Dozens : Thefe feveral Tables are made by the Rules immedi" atly going before them, and their ufe is to exprefs in Dectnul Numbers either Mot^^ WcigfUt or Mgafure^ as by the following FHfofmous win appear. P R O B L. I. Htm by tU Table to exprefs EngliCh-Coin in Decimals. The firft of the nine Tables is for this purpole ; therefore if you would exprefs either MBingSf Tenee or Wartbings ia Deci- mal Numbers^ yott mufi: repair to the firft Table* which is of JEftgliJb<oik J aad there againft 13 Shillings you mall find .6;, which is the Decimal of 13 (hillings, alfo againft feven pence you (hall find .02916667, which is tlie Decimal reprefenting 7 pence: Alfo againft 2 fiirdiings you ihall find .00208333, which is the Decimal JUifwering 2 Farthings , and the like is to be done for ajiy pdier Number of Shillings, Pence or Farthings. " But if it be required to find the Decimal of divers Denomi* nations of Coin in one^^um, as of Shillings, Pence and Farthings together, you muft add the Decimals of all the particulars toge-* ther, and the fum of them Oiall be the Decimal fought. Examples. If you would know the Decimal of i3j4--p-7i»— 2 4« in one Number : In the Table you (hail find tl^^t, 1 ^ 7 i. 5^ is <{ .02< t 2 J. J L-00208333 The Decimal of ^ 7^. 5^ is ^ .02916667 [. J U The Decimal of 13 x. 7^. 2 j. .68125000 P R O B 1^ r ' Decimal AfithmttUk. 12 j PROBL. n. Hm hy xhz Table to exprefs Troy Wekht i» Decimals. The fecond Table is of Tr(^ Weighty, the feveral Denominations whereof are Ounces^ Pcnny-'wei^Sj and Grains : So that by the Table you (ball find that the Decimal belonging to five Ounces 13.41666667, the Decimal belonging to 17 Penny-\veight is 907083333, and the Decimal belonging to 13 Grains is •o022s6949 and fo of any other number of Ounces, Penny-weights and Grains , feverally. But if it weie required to exprefs thefe for any other! feveral Denominations in one Decimal Fraction, then you muft (as be- fore you did for Mohey; take out of the Table the feveral Be-. cimals belonging to the refpeAive Quantities, and add them to- gether, fo ihall the Sum of that Addition be the Decimal fought* Example. If it were required to find a Decimal which Ihould Tcprcfent 5 Ounces, 17 Penny-weight, 13 Grains. r $^«» T f. 41666667 The Decimal of ^ 17 P. w. yis^ •^57083333 Decimal of 5 t7ir. 17 P. w. 13 Gr. .4^975694 PROBL. in. H(fw by the Table to exprefs Averdupois great Vf eight in Decimals. The third Table is of Averdupois grest Weighs the feveral De- nominations whereof are Quarters oi Hundreds j Pounds^ Ounces and garters of Ounces \ thus you (hall find in the Table, that the De- cimal of 3 Quarters of a Hundred is .75, the Decimal of 22 pounds IS .19642857, the Decimal of 7 Ounces is .00390625, and the Decimaljof 3 Quarters of an Ounce is .00041853, and in this manner you may find the correfpondent Decimal belonging to any number of Quarters, Pounds, Ounces, and parts of Oun- ces feveraHy. But if it be required to find one Decimal Number which {hall reprefent divers Denominations, yon muft firft find the Decimal belonging to the (everal particulars, and add them together, the fum Whereof Ihall bexhe entire Decimal required. R 2 ExmpU^ 724 Decimal ArithmetUk. Sxdmpk. Let it be Required to fiad a Decimal which (hall rc- 'prcfent 3 (garters, 22 Pounds, 7 Ounces | of an Ounce, • • . . * t . TheDcctoalof)"^- i is >;:9«428,7 T ' ' J lOu. r , ^.00390625 C 3J2tt. J .(^••00041853 Pecimal of 3 i?. 22 1. 7 > aw. -JSOTI 335 P R O B L- IV. .' fr<m> by the Table to exprcfs Averdupois little Weight in Decimals.' ^ ^ The fourth Table is of Averdupou Jittk Weighty the Denomina- tions whereof are OunceSy J^rams znd Quarters of Vramsy fo that the Decimal of 1 1 Ounces is .6875, the Decimal of five Drams *is .01959125, and the Decimal of one Quarter of a Dram. is •00097656. ^ . ' ' ' But if it be required to find one Decimal Nun)ber, which (hall reprcfent 1 1 Ounces, 5 Drams and a half. ' Jht Pedmal of<{ . 5,i?r. j> is <i .oi§53i25 L 1 C«* J L»ooo97656 Decimal of 1 1 Cun. 5 j Dr. .70800781 %■ • PROBL. V. J^im' ^ rfe Table ^ ^xprf/> Liquid Meafuresiv Decimals. Becaufe there is fo ,great variety of Liqui4 Meafures • that hardly any two commodities are fold by the fame, the difiFerence of the Gallon continually making alteration, we have therefore Jn this fifth Table made the greateft Denomination to be one 'Gallon, the next lefs Denonainatlon being Pints and quarters of ^ints, io'that in the Table you (hall find the Decimal belonging ito three P^t^ to jbe .375, and the Decimal belonging to two <5:iarters, or half a Pint, to be .0625, and fofbr any other. . ' But for tp exprefs Pints and parts of Pints in one entire De- cimal Number^ you mufl: add the Decimals of the feveral Deno- ininations together, and their Sum iball be the entire Decimal. ' ' . . . . » ■ ■ ' So Decimal Arithmetick. 125 So if you were to exprels 3 Pints and a half. The Decimal of I L!ffpmt.}K:o6V ' The Decimal of 3 i Pints 4375 P R O B L. VI. ffm by the Table to express Dry roeafures m Decimals. The fixth Table is of J?ry MedfureSy the fe^ral Denominations whereof are BujheJs^ Ptcksy qugrterr of Pecks' and Pims^ (b may you find the Decimal of five Bufhels to be .625 the Decimal of two pecks to be .0625, the Decimal pf three quarters of a peck to be .023437, the Decimal of two Pints to be .0099063. Thug are the correfpondent Decimals belotiging to the leveral Denomi- nations found. ' But if you would have one number to exprefi 5 BuQids, .2 ;Pecks, three quarters of a Peck, and' 2 Pints, The Decimal of < ^^> IS ^^2^ J^^^ ^2 Pints, ji C.0039063 Dedmalof 5 Bnjkejs 2\Pech 2 Pints. .714S438 ' PRQBL. VII. fjvwhy the. Table to exprefs Long meafurcs /» Decimals. Thefeventh Table is of Long J^eaJUres, the Integers being Tards ani Ells:^znd the lefFer Denoniinarionsare Quarters t)f Tards or J?&, Ndis, and quarters of Nails. - So you may find in the Ta- ble that the Decimal of three quarters of a Yard, or an Ell, is .7$, the Decimal of two Nails, is .125, and the Decimal of one j^Darter of a Nail is .o 1 5695. But if you would have one number to exprefs 3 quairters of a Yard, or an Ell, two Nails and one quarter of a Nail ; ^ZQuar. 1 'r.7.5' The Decimal of <{ 2 Nails. }> is ^ . 1 2 ^ \^i(l^ofaNa.] Ltcis^s^ . 'Decimal of 3 C^. 2N. iQtu . ^890625 1 R O B L. 126 Decimd Arithme$ick. P R O B L. VIII. //ow by tbt Table to exprefs tht parts of Time in Decimals. Time is ofually divided into rears^ Months and Da^s: So the eighth Table which is of TjW, confifteth of thefe two Denomina- tions, Months and Vays^ you may find that the Decimal of 5 Months is •41667, the Decimal of 26 days is .071233. Thefe are the principal Decimals, but the compound Decimal Number re- prefenting 5 Months, 26 days, is .487900, as youlhall find, if you add .071233, which is the Decimal of 26 days, to .416667, which is the Decimal of five Months. / P R O B L. IK. How hy the Table to exprejfi Dozens in Decimals; The laft Table is of Do^ens^ the Integer being a Groffe^ and thefmall Denominations are Dozens, and parts of Dozens, io may you find the Decimal of feven Dozens tb be .$833333, and the Decimal of five parts of a dozen to be .0347222, andthele two Numbers added together, make .6220555, which is the Number ' which reprefenteth 7 dozen, andT^J parts of a dozen. In the fetting down of Decimal Fraflions, to add dieni together, yott muft always obferve to fet Trimes under Primes^ Seconds un-r der Seconds, &c. which the points before the feveral Frafiions will dired you to do. ' « Hitherto we haVe (hewed the ufe of fthe foregoing Tables in exprdTingrOf Fraftions in decimal Numbers. It refteth now - to (hew the ufe of them in finding what Fraftion either rf Money, Weight or Meafure, any decimal Number given doth reprefent, and that Ihall be made evident by the enfaing Propolition. ' \ PROBL. X. A Decimal Niimbtr being given, haw to find what ffiSitn h do$b reprefent* 1 Let .02916667 be a decimal Number, reprelcnting Tome Fra£li- on-part of EngHjh Coin : Becaufe it is required to find the valae j of this Fraftion in BngUjh Coin, you muft therefore repair to the Table of EngliJhCoin ; in the fecond Column of which Table leek for the Numb;;r given (v/^. .029 16667} which you (hall find to fbnd [ Decimal Jrithmetick. 127 fhnd againft 7 pence, and fo much is the vslue of the decimal Fraft ion .02916667, in jEiig/i/fcCw»* Alfo if the decimal Fraftion .75 were given, you ihall find the Value thereof to be 15 Shillings, and the value of .003125 to be tjirce farthings. ,. , ^ Likewifc in in the Table of trof Weighty if .41666667 were given, it would fignifie five ounces, and .05416667 would cxprcfs 13 Penny-weight and •00x7361 1 will cxprefs ten Grains, ^c. After this manner may you find the value of any decimal Num- ber given, either in Money^ Weight or NUafures^ when the Num- iier given may be exaftly found m the Table : But if the Number given cannot: be found exaftly in the Table unto which it is di- rcftcd at one entrance >Then yon muft find in the famcTaWe, die neareft Number you can, left than the given Number, and take the plumber that anfwers unta it in the firft Column, which win be the greatcft Fraftion of the Number required : Then fubftrafting the Decimal thus found, out of the Decimal given, m Ihall have a Remainder, which Riemainder feek alfo m tho fccond Column of the Table, if it may be found, if not, feek the neareft lefs and the Number anfwering thereto, in the firft Co- lumn, fliall be the next greateft Fraftion ; then Subftrading this Decimal fbund out of the former Remainder, there will be another RcBtoinder, which alfo feek in the Table, and proceed as in the fewer ; An Example or two will make all plain. Exampk 1. Let .68125000 be a Decimal given, reprefenting fome part of EngUfi Coin : If you look in the Table of BffgUlk Coin for .68125000, yqu cannot find it, but the neareft Number in lie Table lefs than it, is .65, againft which I find .13 j. fo that 13 J. is the greateft Fraftion-part of Bxglijh Coin agreeing to this Humber. ^ . This done, Subftraft .65 out of .68125000, and there will Re- main .03125000, which Number alfo you muft feek in the Table of EngUJb Corny but being you cannot find it theFe, you muft take the neareft Number lefs than it, which is .Q.2916667, againft which 1 find 7 pence, which is the next greateft^ Ffafliofl-pait of IngVtfii Coin agreeing to this Number. Again, fubftraft ^02916667, out' of .03125000, and there will Remain .002Q8333, which Number feek in the Table, aiid yoa ihall find it to ftand againft 2 Farthings, and f® much dorh this laft Remainder fignifie 'm£nglijh Coin, and the whrte given Num- ber .68125000 doth reprefent In^nglrjh Coin thirteen Shillings fe- vea Pence two Farthings , as by the Operation following doth appear. V t 128 Decimal Arithmetiok. •681250C0 Numbergive|i. J. ,65 the next Icffer Number ia the Table, rcprefenting 13X. .03 12 $oco firft Remainder, ;629i6667 the next leffer Number in tlie Table, reprefenting 7 d[ .00208333 fecondjlemainder, which tieprefents two Farthings. So doth the whole Number reprefent 1 3 j. 7 i. 2 j. ' Examplelt Let the Decin^l .87426934 reprereatiHgfdniieFraai-. ' on of a pound Sterling, be give;n. If you look intheTabl^of£«g///fe Coin for .87426934 you^annot find it ; but the neareft Kumfc!^ in.the Table lefs than it, is .85, againft which I find 17 Shilling^ fo ■ that 17 Shillings is the greateft Fraftion-part of EngJiJh Conty a- ' gi:e^ing to this Number. . , Then fubftraftkig. .8^ out of .87426934 there wiir Remain '. ',;o242j6934, which number alfo you muft feek in the Table of Euglijh Coiny but feeing you cannot find it there, you muft ta^Q the neareft ^Number lel§ tha;i it, which is .020^3333, agairifi which I find five Pence, Avhich is the n^xt greateft Fraaion-part of. EngJiJhCoin. . , J , Xaftly, fui)ftraa .0208333 ?, out of .02426934, and there will ' Remain .60343601, which Nuniber you muft alfo feek in the Ta- ble of Englifb Coin ; but not finding it exaftly there, you mtift ^ke the neareft Number lefs,. which i^ .003125, againft which you Ihall find 3 Farthings, which is the ne^^t grpateft^FraSion-part of EngUjb Coin, and the Decimal .874269^4, doth in value fignifie 17 Shillings,5 Pence g Farthings, and fbmething more, for .003125 15 the Decimalof 3 Farthings j and the Number you are. to look .for- in the Table is .00343601, greater than the Decimal of 3 Farthings; whcrefi)re, if youfubftraft .Q03i25out of .06343601; there will Remain .31101, which is the Ui^l parjt of a Farthiog' *hich is iflconfiderable. See the following Ojieration. • ^^ ' -.87426934 Decimal given , « .85 Decimal of ■ jjs. ^>242i5934 FirftRemamder. .0^083333 Decimal of 5 dj .00343601 Second Remainder. ^] .003125. . Decimal o f — 3 j, r — ^— — .6003 1 JO I Decimal part of a Farthing. \j> J- f And ♦. f And her^Note, that whatfoever hath been here fzid,£oa* cerning the.ufes of the Table of EngUjhCoiny the fame order is CO be obferved jn th,q ufe.of ,the ^otherT^ibles. of Weighty Mtaffirt^ Time^ &c* as by the followiag Examples (H you rnaJs/t trial} will appear. it i Exsmphi^j. If.thisDecigial .418975604, were glveu to know the value thereof in 7roy Weighty you mall find it to contain x$ Ounces, i.7l?eflny-weightsa.nd i3Graiii§i * . . : IT. ,Al(b .if. f 95053 3 5 were a Decimal given, .and it were required to find. the*)iralue t^rjeof.in Aver^n^U great lyeigbt^ yca (hall find it tqcpntain .3, quarters of an Ounce'. , . . , ...,,, III. Likewife, if .70800781 were i decimal Frjfllc;a given, ypu ftalJ find the value thereof in AverdupoU ' little We'igbi to be li Ounces, 5 Drams, and one quarter of a Dram. , IV. If .457< wecca Decimal^vdiofe value is required in Iij«/i Meafures you miW findr It to contain Bpi^jts ^pd an half. j V. Let .7148438 bQ: a Decimal given, vJIiofe value is required in DrjMeafure^ you. lhall.find.it. to contain 5 Bulhels; i Pecks, 3 Qparters of a Peck, anA 2 Pints. .- •; Thir§ have I (hewed ypii the ufe of the(e decimal Tables in ex* prcfling of 'the Fraaioo-parts of ULmey^ Weighty Meafure^ i&cJ ?at becaufe tjiefe Tables may not be always at hand, when there 15 need of Them, I will here (hew you how the value of any De- cimal given i may be known by Multiplication only ; and this is THE RULE. f multiply the Decimal gfi/e», by th^ Number of Inown parts rf tk ^hferiour Deriomlriatidn,' vobich'are eqid to the Integer, the ftodjifi it the value $f tb^ Decimal propojei bt that inferjonr &en07 jBiiqtiqn ; apd if tlkre happen' to be any Decimal in the Produft, yon ^.iniike manner jini the value thereof intherjexf inferiour Denp-, ?«iMtion, and fo proceed tlllyoii come to the teafl knovm parts of the ^ger'^: , . • . - o\ ; - , ' ' ' ^ Example. Let .6739583^*be a Decimal given, repreferitin| tljci rraQioji of d^ Found Merltngl tirft nyiltiply .^7395X34. by 20 Cthe Number of Shillings in a Pound Sterling) and the ProdUfli^ *ffl be 13-47916620, fropa which cutting oflF the laft ejjht Figures, pith, a point, ordalh.qf tl^e Pen (becaiife..there\ypre ciglit Figures 2 'he given Fraftiori) there will ll[anctbefdre the point (towsffds fclefthand) 13, \vhich are Shillings, andtheR?;n:iain^er,!479i668j binding behind the pointy j.will be thie Frai^ion-partof ^4^e Sh[ilin( •^^liR^> whiqh hfumbef .,.47916680^, YOU mi^fi, muItVply jbi '^ the Miimbsr of' Pence iii 6'r^ fhilUng ) and' thi ftoaiif " ' S \ t |o Decimal Arkhmetkk. wilf be, 575000160, from which Number cut off thelaft elghc •figures as before, and there will be 5 left to the left Hand, which are 5 Pence, and the Figures on the right Hand of the IPoint, v/^. j5oooi6oare the Fraftion-partof one l%iny Sterlings ^ vhich therefore multiply by 4 (the number of Farthings in one I'cnny) and the Produftof that Multiplication will be 300000640, from which cut off the laft eight Figures to the right Hand^ aod there will be left 3 towards the left Hand , which reprefen* teth 3 Farthings, and the remaining Figures towards the rigUfc Hand are but the Fradion-part of a Fardiing, which we ti^re- fbre rejed. And thus you may find by Mult^Jiemim only, t^t this Fradicxi •67395^34 doth reprefent in the known parts of * Engiifi CQtn^ 13 Shillings, 5 Pence, 3 Farthings, as bytbefollow* ing operation appeareth. ( •67395834 20 Shillings . 13, 479x6680 13 95833360 47916680 Ml Pence 5, 7500016a 4 Farthings 3,00000640 In like manner, if this Frafiion .94809028 were Civen, repre* (cotingfome Fradion-part of trof ]V€igk\ you flu^lluxd thev^oe thereof to be 11 Ounces, 7 Penny Weight, 13 Grains, as by the? pperation Mowing appeareth* •94809028 12 189618056 94^09028 Ounces. II, 3770833$ 20 Pennyweights. 7,54166720 24 21^666880 20S333440 Grains. i3|ccooi28o la I ^ De^imdl Jrabmetick. iji la this manoer may any Decimal given be reduced inco ths iQiown parts of the Integer by MulupUca$m only. And f Here no^, that whereas in the preceding Tables the Deci- pial FraOions coaftft cffivcn or eig^ht Figures, we iball ia tl^ profecution oi our Work make uTe only of four or fvf of the firft of them, which will be fufEcien t in ordinary praftic^ and come near enough to the tiurh in any ordinary queftion whatfoever. So if inftead of •02916667, which is the Fraftion-part of 7 Pence, you take out only .02^16. it Will be fufEcient. (-.058333331 r.o5«33l ^n Trojf IVeigtt. Alfo for< .0058594 J> take <^ .0058 t^lti Dry Miafure. i . U5833332 J 1^5833 J ^ar/wi^. Thos much concer(iing the conftriifticm and ufe of the decimal YAkif we ihall now come to the Practice of Duimal Amhrmick* Th End of the Firfi Scaion. SECT. IL (y the Nature ^ Decimals, and how hy them to fefform the Works j(f thefeveral Rules in Arith* tnetickt 1. Of Notation of Decimals* No T A 1 1 N of Decimals is contrary to that of whoto Nombeiis ; For whereas in whole Numbers the value of Figures are increafed ten-fokl by continual addition df Cyphers (iowards the nght Hand : So on the contrary, the valu:;i of d)e places of Decimals do decreafe in the fame proportion. And whereas in whole NuiAbers, Cyphers in the firft place to- wards the left Hand are unneceffarf, yet in D.cimils, tney are abfolutely fteceflary co difcover the true Denominator. Al o Ci- phers at the end (prtowards the right Haid} of d?cimij Nj n'vri ^eof no vAJMit ^^ ^ fiogle Figure in drcImAls ngm^'S ai mnct> » a ' » . ai* J J a Decimd Arithmetiek. aftRefame Figure would do, if there were. Cyphers placed behind it, fo 7 is equivalent unto 70, 700, or 7000, ^e. Por the Di<kf- ininators of decimal Raftions are always Cyphers with an Unite towards the left Hand, as hath been already intimated. So fJv being reduced to its leaft Terms will be ^^ and yT. ; , ; .; ^\\\ be redu- ced to^^alfo, and foof any other»^asby the Tablefollowingdoth evidently appear.' ^ < - j ' 9876^4321 1 W3456789 - •<*;• if'^C^ X 00000000 10000000 .ICOQOOO I 00000 ^ 10000 t thoufand 1000 3 hundred' 100 t> Ten IP I , •00000001 •oooocbi * •OCOOOX' •OCOOl .0061' •cot or " yvj ' •01 or -rs* or •ST ^^aim 11. Of Addition of Dtcitnzls. IN Addition of Decimals, ^e fame order is to be obfervedas ia 'Addition of Numbers of one Denomination before' taugtitfift' tiie iirft Part, in which there is no difficulty : But in Decimal Numbers the chief care to-be takeV i^- in placing your wh(He Numbers and Fraflions in their diie order, which you (hall eafily ftiA certainly do, if you obferve this general ^ule^ vi^. to plate jourvobole Numbers ani Ff anions tmeiindsr anotkr^ fixhtntlk Foinxi ^] ^eperati n whi^h (\a> decimal NuQibers) difiingiiilhibe' 'mhle, Numbers from the Frafffom^ fldnd direBly (me under tli otbeKf ,tben are you to proceed in the addition of them in all re(pe6b, as yoa 4id in whole Numbers* . ' ,4 Kxamie I. Let it be required to add tdgetfacr in .one Tarn tho^ ,fev6ral ftims following, in a decimal Way^ vi^. 36//. «i. id» %^IL px. id. iilU \6s. 9^* and 6JL is. '^'d.'^{ .. }./ !-. Firft, let ao\vn 3(5//; and a Poiat or Coiuma after it, then fit the Praftion-^^rt qf aj:..S.^. look in your Table of jp^^Wi/J Oi;^^^ V-'here you ib^ll;fir\dthe accimal FraiSion ofl:^/ si &pe, .1333 iherefore fprs^/i. !2J. 3^.;i;etdown 36. I333.-. ' .' \ .~ Secondly, for your 29 //V o >• 2d. fetdowas^^ 0083. . ' " ' Thirdly, for your 31 //. 16 s. ^d^ fetdown 31- 837^. Laftly, for yonr 6 //. 2 s. 5 <^. let down 6. 1208 as you Tee xlonc* in the opr/a'tion following. ^ -.•: *• . ... *» ^ T • • - ^% Decimal Jrhhmetick. i j j ' h s. d. 36 02 t'^ ' /'B^j 1333 31 16 pr-^*- "^31, 8375 (5 02 5 J C 6, 1208 :io3 62 o ' 103, 0999 ♦. ' Your decimal ^umbers being thps placed ia due order one nnr der another', ^oceed to the adding of them tc^ether, as if they wer6 whole Numbers, and you (hall lind the fom or total of dieqi tobe 103, 0999. Now die 103 wliidi^ftand: towards the left Hand, are io| pounds; and die ^0999 whicb' ftands' towards die right Hand of Ae Comma, Js the Fraftion-part of one pound Sterling, tfaeTaloe whereof you may find (by the Propoutidn-beforegoing) to be two ihiiling3/<?^e, wbkh IKduU be two ihpng3exaOy bat it want* cth&mewhat,.v/^. the -rrsi part of a Farthing, which is inftnfibic; ibr if by the foore-mentionM Rule you feek the vaioe of the decimal Fradion, V0999, you ihall find it to be 1 Ihillirig, 11 pence, | £urthing^ and the ^^-rrl- part of a Farthing, which yon may call in all 2 (hillings,, for decimal Numbers will feldom hap^ to give tbe exaft value of J^rafilons, but will be either greater or leftr than they ought to Q? ; but in fuch a fum as this is, the flioufioulth ^rt of a ^ardiing » not to be regarded. ; Example IT. Let it be required to add together in a decimal way ffefefums following, vi^. 29T. r8x; jd. 3 j, 63/. ii#. 9ik hf. 129/. 4x. oi. 2 J. and 3/* 7s. lod. i]. Pirft^ for 29 /. 18 J. 7i." 3 f. fet down 29. 731^29. Secondly*) For -6^ /• 1 1 x. 2 i. 3 j. let dowti 63* 5 5937». . ^ Thirdly^ for 294^44;.^ 2^. fet down 129. 20208.. , ^ Laftly, for 3 Zr 7s. iod. i j. fet down 3. 35^271 asyp^j leefcff &t down in the M^igine. - ' Your de(;imalNumbers thus placed In order, add themfiOgether. as if they were whole Numbers, and you fhall 6nd the fum of them to contain 226. 08645. 29. 93229 iSlow the 226, which ftattd towards the left Hand 63. 55937 pf the Commajj are 226 pounds, and the other Fi- 129. 2020S giires towards the rightHand, vi^ 0S645 are the 3* 39271 Fraftion-parts of a Pound Sterling, Which if vou— — — i reduce by the fore-mehtiofted Propofiticfn, you (nail 226. 08645 hi the Value thereof to be 1 Shilling, 8 Pence, j Farthings, fo tb? whole f«m is 226 h 1 x. 8 ^« 3 }• And Ml ??4 Decimal Arithmetiek. And here note, that what hath been faid, as concerning Nimey^ the fame is alio to be uoderftood of Weighs Meafm^^ Time^ &c« as by^he following Examples will appear. Otter MxamtfUsfef PraBice. ExmpUl* Example IL In Money. In Trcf Weighs 135.8835 7- .9741a 135. wv 95. 5583 3. 2875 «* «34« 7^1 S)4^ H^« 7^* ^* ^5330 3. 52187: 18. 249)0 Jbiiii4»lr in. ta Averiufoif IhHk Weif^ 12. 7227 7^. S594 91. 4^8$ 3^ 139S ■^i."av 146. C398 Z\6liL qoOu. gdr. £x4mfle 1% tn Aoirdufoii greg^Wrigh. 37.9442 9- J0<3 33. 6786 to. OCOO 12. 8142 I*. 103. 74U ■» w * in. (y* Suhfirn^ion of Decimals. XHe ^ubftraftion of Decinkab difieredh nothing fittfti tiie Safv ftraiting of one whole Number from another^ «nd the de- timal Numbers to be Sid>ftrafied oneirom another, muft he piked in th/c. fiune order, as in Mditian of decimal Numbers, ^e Pr^ke bf SubJ^f^ion Ihall be feen in the following Examples. Exampk I. tet it be required to Subftra^l 3 1 /. i5 1. 9 i. out of J 36/. 2X. '8^.. Firft^ for your 3d /. 2 J. 8 d. fet down tlje Decimal thereofj which is ^6* 1^33* . Secondly for your 31 /. ^^s, 9^, fet down the Decimal thereof : . ThiJ J r, t)ec$mai Jrkhmetick^ ^25 This done draw a Line under them, andSubftraflUig theteflec from the Greater you (hall find the Remainder to be 4. 2958 the 4 on the left fide of the Comma are ibor 36, i j}] Poaads , and the .29<i which flandeth towards the 3i» 837$. right Hand, is the Fraction-part of a Ponnd, the Value — _—» whereof beiiig foudit, will be found tobei^ir.iii. 4* ^95! So that if you Subftrafl ^ih x6x. 9^. there will Re« main 4/. ^^s. iid. Bat if divers Sums be to be Snbftrafted out of one greater Sbm then you muft firft add all the leveral fmaHer Sums togcttar, anl Subftraft the Sum of them from the Greater given Sum, (b Audi the refidue be the Sum deflred* ExmfUsfar TrdSicc^ Example !• Eximpk 11^ . In Monef. In jtt>efiupou great JFiijjfu Lent 27S4. 837$ BougJit 103. 742$ Sold 37. 944a ^^ 2^ 1222 a^amm^mim^i^mm SJer^f 329- 0083 ' Unfold 65-9442 tS V^^- ^575 ^5^.3 J- 5W-7«^ funes. / ^. j2c,^ , paid in all 103.0999 refbto 2681.7376 pay 268i/i.i4x. oi» " Example III. In 7rer.^<f%*f* JPeliveredtoaGoldfmith of old Plate 7. 97415 Received of new Plate 5. 5.9670 \ Refts ia the Goldfmiths Hands 2. 7743 2 hi. /^oun* iQp.w.i^gfi wmmmmi^ > IV. Of MuhipUcation ^/ Decimals. MVttt?HCAtIOlT of Pethnals diffcreth nothing at all from the mUipUcmoH of whole Numbers, for making the^ ;tcr Number the Multiplicandy andtheleffer Number thcMul- r* the Nujobcriffuing from that Rlultiplicaticnftiallbe called F^Qduft. Now I * > n6 DmmdlJriihmetiek. ^ ^^ Now ia the Multiplication of decimal . Numbers oii<Jj>y aaoth'er, if there be any Fraftion eiliher.la tbe^MMltiplicand or Muitiplier^ ,01; f raftiottin both : So many Figures as the Fraftioqscontain» fomany Figures muft.be cut off from the ProduU towards the rightJHani^ which-lhall be the Fraftion of the ProduB^ arid the Figures tq\iifacd| thejeft Handof ^e Comipa,in the.troduft^ Ihail be thetnt^irs of the Produ&* ' FxMMpleJ. Lti it be irquiredtOmtildply 54 pounds, fire Ml- lings, three pence^. by 16 pounds, fixlhillings, fix pence. • ^ Firft, feek the. Decimal of 34/. 5 Ji 3 df. which you fliall find to be 34. 2625, make this your Multiplicand, thj^n feek the Decim^al of 16/. 6 s. 6d. which you fliall find to be 16. 32$ ^ make this De- cimal Number your Multiplier ; MultipliQ^nd 3 '4.2 6 2 $ then draw a Line, and Multiply MUltipIiet I 6.3 2 5 thefe two Ntimbcrs together, as ^ ,' .« . ■■ V ■ ■ if they were whole Numbers, ' I 7 I 3 I 2 5 a|id you fliall find the Prod udrf 685260 them to' be 5'59.3353i25. Now \ 1027875 becaufe there are four figures in 255750 the Multiplicand which are 5 4.2 625 Fraftions^ namely, thofefour to* ^-^— — ■ ■ wards the right Hand, v/^. 1^625, 37 9,3 953125 and^ there are alfo three figures . ; . Jn the Multiplier, which are Fraftions, namely, thefe three towards the right Hand, vi^. 325* that is in all feven figiires, reprefenting Fraftions, I therefore cut oflFfrom theProduft the feven figures towards therl^t Hand, by making of a Comma thqre, todiftinguilhthe whole Number from the Fraftion: So is559tliemtegerorwholeNumber,and.3353i2f the Fraftion of this Multiplication. ^ Example iV If thcire beFraftlons in tlie Multiplicaicid,"andnone in the Multiplier, yet the work Is ftiH the fame, for you muft cut, off only fp many Figures fropi die Pfoduft, a% 5 7 ^ 7-7 5 ^^rc are Fraftions either in Multiplicand, Mul- 235 tiplier, or both : So if it were required to mul- ■ ' ' ^" " "" tiply 5767 Yards, and 3 quarters of a Yard, bj .^88387$ 235 Yards, you muft firft fet down 5767. 75 m [73032 ^ -. . 1730325 your 5767 Yar4s, . and .threp quartgrs, which Number mi^ft b6 your Imitiplicmdi Ahd alio fet^ « down 235 ya^ds for your AI«/t^/i^r^ then muf I 3 5 5 4 2 1.2 5 tiplying them together, as if tJiej^ Wero"^ Wh(^ Numbers , you Ihall find the ProduB to. |)e M 5 5421' 25, and becaufe there are only two Fi-aftion- figurest. both' which are in the Multiplicand,- caniely,' the two laft thereof Decimal Arithmetick. i^j •7<5, and none in the Multiplier. I therefore cut off only two figures of the Produft, Namely, the twolaft, which are .25, fo is theProdaft of this Multiplication 1955421, 25 which is 1355421 fquare yards, and one quarter of a Yard. And fo if a Garden or other piece of Land, lying fquare, fliould contain in length 5767 Yards and three Quarters, and in breadth 235 Yards, the whole piece would contain 1355421 fquare Yards, and one quarter of a Yard» * Example III. If decimal Fraftions be to be multiplied by decimal Fraftions, you miift th^n (as before) multiply them as whole Numbers, and from the Produft cut off lb .953 many Figures towards the right Hand, as there are .782 . - Figures in the Multiplicand and the Multiplier. So if m it were required to multiply .953 by .782, you fliall 1906 find their Produft tobe .745246, which being butSx 7624 figures in all,I cut them off and that Fraction .745 246 is 667*1 the Produft qf the Multiplication of the two given Fraftions. • 745^4^ Mxample IV. If any two decimal Fraftions being multiplied to- gether, the Produft thereof dothnotconfiftof fo many places as afe required (by the former rules) to be #0752 cutoff, you muft then fupply that defeft by prefixing a .063 Cypher, or Cyphers before the produft towards the ■ ■! leftHand: So if thefe decimal Fraftions .063 and .0752 ' 02256 were to be multiplied, their PrOduft would be 47376. 004512 Now (by the former Rules) you (hould cut off (even '■ Hgures of the Produft towards the right Hand, but .0047376 this produft 47376. confifteth but of five Figures; wherefore to make it fevea Figures, I prefix two Cyphers before the produft on, the left Hand, making it .0047376, and that is die true Produft' produced by this Multiplication. Example y. If you would multiply any Decimal (either Frfiftion only, ,br whole Nupiber and Fraftion together) by 10, 100,1000, C»c. You muft add fo many Cyphers to the Multiplicand, as there are Cyphers in the Multiplier,' and ciit off fo many Figures as there are Fraftions in tl^e Multiplicand, and the Number fhall.be the Produft required: So if "7,856025 were a Decimal given to be multiplied by 100, addtwo Cyphers to the Number given, making it 7.85602 50D, then becaufe there were fix Figures of this Number tlQwards the right Hand, it will' be 78 5.602500, which is the true ?rodiift required. s - ' 'I T Sxamplts ^ 3 ^ Decimal ArithmeticL Examples for Traltice, Example I. Example 11. • 7-4 3 2 2.6 I 2 2.3 5 8 32 7432 44592 14864 44716 67074 7x5-45 6 Example IV. 3 7 5.6 2 1 8 100 ■% 1939752 Example III. •352 .24 I 408 704 37 562.1 80© • - / .8448- N ^ ji ComfeKdicus way for the Multiflicatwn cf De- cimal mixt Numbers. IT hath been much obiefled againft Decimal Muhiplicatkn^ for that, in hen tlie Mixt NuMtrs to be Multiplied together, do conhft of many Decimal Farts j the TroduSt incrcafing to ii, 9, icor I 1 2 Places of Fans ; v hereas 2, 3 or 4 at the moft, w ill be fuificient. J It is true, but that C?^jf5/cK may be removed by thisfoHcwing CiW pevdium: For them^annerof \^orkiEg >* hereof this is the RULE. Set ditcTJilebi^gefi vf rfew^MixtKum.bers ; ard (ahlcvgktlere heftveral Places cj Paits, bcib w tie MultiplitsFid m:d Multiplier) fettbe place af Unity of ike Integer of the Multiplier, fty;i/tr that iiofure of Parts in rif^ Multiplicand, vkcfe Number cf V^xisjcuvctuldhve role ifitle Picdifl: Ihev Jet all tie Figures cj tke Multiplier, tke timri.rj rraj to nlat tkejnaegiitfj : Jr,d bj eaik Figure of lif Mul- tiplier i Decimal Arithmetick. I j^ tiplier, Multiply th: Figure over it in tbi Multiplicand (hiving re gerito the Figure going before) andfet down the FroduSt^ &c. This will bell appear by Examples, Exmplel. Let it be required to Multiply 34.262$, by 16.325, fo that there may be but two Places of Parts in tiie FroduB. I. Setdown the biggfrATtfw^ffr 34.2625 for the Ma/rf/)//Vtf«i 5 aad (becaufe you would have but two Places of Farts in your FroduB) ^tti\izVnites FUce of the Integer ya the Multiplier ^ which is 6, un- der the fecond place of Fjrts in the Multiplicsvdj which is 6 alfo ; and tranfpofe all the other Figures of the Multiplier ^ as you fee tiiem fet down in the Margine : For th? Multiplier given , was 16.325, and here it is made 523.61, j^nd6the 1 hceof 'U«;t;inthe Multipiier^ ftands under 6 (the fecond place of Farts) in the Muh tipliurtd, II, The MuUiplicani and Multiplier being placed in thii order, you are to begin your Work at the Right Hand, asfolloweth ; Saying, Multiplicand 3 4.2 6 2 5 ones 5 is 5, for which (becaufe it is Multiplier 5 2 3.6 i half, keep i in mind) but fet it not ■ ■■- down J but fay again, once 2 is 2 34263 and 1 in mind is 3 ; fet 3 under i, 20557 and go on with that Line, faying 10 2 8 oace 6 is 6, once \t is 2, once 4 is 4, ' 6 8 and once 3 is 3, fetting th*m down ; 17 fothall the firft Line, or Frodu^ be 3^263. Produft 5 59.3 3 IV. Then go to the third Figure of the Multiplier 3 •, and fay, 3 times 6 is 18, for which fit being above 15) bear 2 in mind ; and Tay, 3 times 2 is 6 and 2 in mind is 8, which fernndir 7, and fay, 3 times4is 12, fet down 2 and bear i, then 3 times 3 is 9 and i i^ 10, which being thelaft fet down, and fo the third LLie oir Producl is I02S» 4 J 1 2 V. Then V jio Decimal Arifhtnetick. V. Then go to the fourth Figure 2, faying 2 times 2 is 4, which ' being lefsthan 5 bear none in mind but go to the next Figure, fay- ingi 2 times 4 is 8, fet 8 under 8, and &y 3 times 2 is 6, which fet down alfo ; fo is the fourth Line or Produft 68. .VI. For the laft Figure of the Multiplier^ 5, fay 5 times 4 is 20, for which bear 2 in mind, and fay 5 times 3 is 15 and 2 in mind is 17, which is the laft Line or ProduO, and your Multiplication is ended. VTI.' Draw a Line under the feveral Produilsy and add them ta^ gether, in the fame order they ftand, and the General Produd will be 55933 ; from which feperate two Places to the right Hand^ and ths Froduii will be 559.33. And this may you do, and have any Wumber of Parts in the General Produft. Sxample IL Let it be required to multiply the fame Mixt Num- bers 34.2625, and 16.325, fo that there may be no Places of Parts in the General ProduQ. Multiplicand 3 4.2 6 2 6 in this Cafe, fet the VnHes Place Multiplier 52 3.6 I . of the Multiplier under, the Vnites *—~— •"■"""-"-"" Place in the Multiplicand (and tranf- 3 4 3 pofe all the Figures of the Multi* 205 flier y as before) and as you fee them / ^.o placed in the Alirgi»r, Then, General Produft 5 5 9» ' I. Begin with the firft Figure of the Multiplier ij and fay, once 6 is 6, (which being above 5) bear i in mind, and fay, once 2 is 2 and I in mind is 3 ; fet 3 under i ; then fay, once 4 is 4; and once 3 is 3 J fet them both down j and fo your HriJ Proi«ff wUl be 343« II. Qo to the feeond Figure 6, and fay, 6 times 2 is 1 2 ; . for which bear 1 in mind, and fay, 6 times 4 is 24, and i in mind is 25, fet 5 under^ 3 and bear 2 in mind : Then fay, 6 times 3 is 18, and 2 in mind is 20 J which fet down 5 fo is your feeond ProduS 205. III. For the third Figure of the Multiplier 3, fay, 3 times 4 is 12, for which bear i, and fay, 3 times 3 is 9, and i in mind is jlo, fet down 10, for your third Produft. IV. Say Decimal Arithmetick. 1 41 IV. Say, 2 times 3 is 6, which (being abovt 5) fet down ifor your fourth and laft ProduB. V. Draw a Line, and add all the ProduSs together, their Sum wiH be <'59. Jntegeh, free from all Decimal Pans. 'We may have occafion hereafter to make ufe of this Cmpcndkwa way of Multiplication of Mixt Numbers } .And that the Learner may be perfea therein, take thefe following Examples ready wrought. Example III. Let it be required to Multiply 72.2543, by 5.1642, fo that there may be but two Places of Parts in the Produd. 7 2.2 5 4 3 'Multiplicand 2461.$ Multiplier 36127 Here 8 Places Ihould 723 have been cut ofF but 433 two was fufficient. 29 3 7 3.1 3 Produft ExamplelW. Let it be required to Multiply 259879.890625, by 1.1173698. Multiplicand 2 5 9 8 7 9.8 9 o 6 2 5 Multiplier 8963711.1 2 5 9 8 7 ^.8 9 1 2 5 9 8 7*9 8 9 2598-799 1819.159 77-964 15-59^ 2.3 J 8 .207 ?rodufl 2 9 o 3 8 1;9 3 9 V. 0/ 142 t)ecimtl Arithmetick. V* Of Divifio^ of Decimals* AS Dlvlfion of whole Numbers is the hardelt of the four Sffnhs , of Vulgsr A^itbnt:ticky fo the Divifionof Vicm^UU the moll difticultof the four kinds of Djcimil Antbm nkky but I hope to make itplain to the underftanding of the meaneft capacity. The fevera^ varieties that mvj happen in 'Diviiion, are prihcipally Qf not only thefe) four. Namely, Firft, to diviii whole Numbers gndFraShns. Secondly,' T(? diviii whole r^^mbsrsby Wxt^ or Mixt Numbers by whole* Thirdly, To divide a greater Fr^iofi by a lefs;^ and Laftly, To divide i, lejfcr Fra^ian by a greater. In Diviiionof Decimals this Rule is general, ;/ the Dividend Be gredter than the DWiCoTy the Quotient wiUbeeither a whole Number ^4 mixt, but f/tfc Dividend be lefsthantbs Divifor, ths Quoricnt vfillbe a Decimal. And (for convenience in working, if there be need) any Number of Cyphers may be annexed to the Dividend, that thereby the Quotient may extend to as many places as the tenour of the Quellion (hall require. The mapner of the working of Divifi'on in Decimals,, is the fame with that before delivered in whole Numbers in the firft part of Vulgar Arithmedck, as will appear by the Examples following, iaevery of the four preralfed Varieties. The RULE for the firft variety. ♦ Tt^Dividendtfwirif Divifor, ^«»^^fltfc mixt Numbers, or one of ' them being A^h:AQ'H\XTah^i: and the other A mixt; or th: Dividend being a Decimal, and the Divifor a whole Number or a mixt, the frrdTFigure in the Qiiotient will be of the f>ime Place or Degree, with that Figure or Cypher of tbt Dividend, which at thefirS demand flan- dethy or (at Isafl) ttfufpofed to ftand direSkly over the place of Unite /»r£>e Divifor. E<xm}li I. Wl};re thitzrms giuen are both mixt Numbers. Let It be required to divide 559. 3354i2<; by 15.325. Here the Term^ given are both oimixi STimbsrs^ which being placed according to the Rales ddivered before for the Diviiion of whole Numbers, thv? Figure In the Dividend, which at the firft demand, ftandeth o- ver6, che placer of U aires in the Divifor is 5, andbecaufe this ftan- d:?th in the place of Tenths, therefore the firft Figure in the Qvio- tiant is in the place of Tenths alio, and the whole Number coniilt- ech of two of the focemoft Places, and the reft is a Decimal, thus the* H? DecimAl Jrithmetick." the QpoticHt fought in our prefcnt Example is 34. 2625, of whi'ci 34 the two firft Fig uresis the Integer or whole Number, 262< the Decimal Fraftion. ' Dtvifor Dividend Quotient » 6-3 2 5) 5 5 9'3 3S4.i.2 5 (3 4.26 2 5 48975 69585 65300 42854 32650 I 0204 I 97950 40912 ^m 3^650 82625 81625 1000 Example II. One of the Terms given^ being a whole Number tbt other mixt. ' The mixt Number 1355421.26 being divided by the whole Number 235, the Quotient will be 5767.75 and the firft Figure in tlic place of Thoufands, as by the Operation doth appear. / I or j[A4 PecimlJrithmetick. Div^or Dividend Qumient 235) 1355421.26 (5 7 <^ 7-7 5 1175 1804 1645 1592 1410 1821 1645 1762 1645 1176 1175 1 The R. U L E for the fccond variety. »*f» the Dividend u d whole cr mixt Number, and the Divifor ii Decimal, add at manyCjfpbers to the Dividend as there are places in the Divitbr ; for ihcimegral Pan of the Qmieift mil confiS of as many places as the Divifor, and tt€ Places arifingfrom the integral Pans of the DiYiieni added together. Example I. Let 34&75bethemixt Nambergiven, todivided by the Decimal ,25, to the Number given, I add two Cyphers, the Number of Places ill the Divifor, and then it will be 348,75oo, which being divided oy .2 5, the integral Part of the Quotient will be 1995, becaufe the whole part of the Dividend 348, being divi- ded by .25 giveth two places, and the Number of Places in the Di- vifor being two, giveth two more; and fo the integral part con- fiftetl) of four figures, as by the Operation. Divifor Decimal Arifhmetick^ 145 Pivifor Dividend Quotient .25} 9 4 8*7 5 o o (i 3 9 5«o o • • # 25 98 75 237 32 5 125 The Rule for the third Variety. f. Wben the term given 4reb&th Decifflals» tbe "DbrHtaiiepfg th greater^ tbe integral part of tbe Quotient yoiU confix of m ftymjh?Ucc:> us ihiDWibv dotb. Example. Let the Decimal .73958 be divickd tiy the Decimal .32 the integral part of the Quotient will be 23, becaufe the Di- vifor doth confiftof two Places, as by the operation, doth appear. 3?) .739 5 S (2311 — — — r 99 . . 96 32 6 U Th. 1^6 Decimal Arithwetick. The Rale for the fourth Variety. • Wien the t§rms given ire both Decimals, conjifting of equsl Places^* the Dividend being th leffcrTerm, place the uiviiendas a Nume- rator, and the Divifor as Denominator ; fo itfucb vulgar FraStion the Quotient fought: But if the Terms confiQ net of equal places fttffly the plate or places wanting in either of the Terms^ by annexing a Cypher or Cyphers on the right ffand^ and then proceed aa before* Thus if •27 be given to be divided by I93 . the Quotient will be -*-J. Alfo if .35 be given to be divided by .7S563, the Quoticuf by annexing 3 C^hcrsto.35,* the leffer decimal given, will be jMv^?, which vulgar Fractions may be reduced into Decimals it need be, by the firfliPropofition in this Second part of decimal Aritbmetick. Examples for PraSicem 44) .3<6'7^ (0081, 8cc. .25) 2481.00 (9924 i*MBi* ■*** 352 47 M«i 44 3 225 0231 225 60 50 ^ 109 loo 000 Having given you Bxam^es of th« four foregoing Rules in the fe* Teral csues of Diviiion in Decimals. I will now bring all the forementioned J^ks into One general Jf^ky and give you Exampks •f aU ^ Varioties, that can poffibly arife in Decimal Divifion. / • A C M7 1 " . ' ^ A Supplement to Decimal Divijion Contammg One General Rule, for finding the true Value of 4^;^ Decimal Quotient, 4^i Exam- ples Z;^ all Cafes. "PXtcimals to be Divided may he either, ClVkfe Islfmkfr A U^bole Number by 2,^Mixt Number 4? ' V ' ; CtvioJe Number 5 V*' A HfUsct Number by z^Mixt Number 6^ , -(Decimal frtL^n. 77 C whole Number sS» A Decimal Fr0ionhy a<P Mm Number 93 (J)ecimAl FraBion. Whereas in MuUipUcatiofi^ you always cut ofF fo many Figure^ from the ProduB as tihcre were Decimal Fans both in the MultipH" csnd ^nd Multiplier: It thence neceffarily follows; That the Num- ber oiDecin^al Parti in the Divifor and gy^t/c^t in any Divifion^ iBuft be equal to the Decimal Parts in the Dividend. For which obrcrvc this GENERALRULE, When your Work of Divifion is ended ; Confider hro manf De- cimal Parts are in the Dividend, more than in the Divifor; For that Exccfs U the Number of Decimal Parts mhichmu^ be feparatedin the (Jpotient, for Decimal Parts. But if there be potfomOny figures fa tfe Quotient, aa tbe'fkid Excefs ^, thei, rnuHiefupplied by prefix- ing ef Jo many Cynhers in the Quotient, before the fignificant Figures thereof^ towards the left Hand, vfhb d Point or. Comma beforetbem. So will the Quotient difcover its true Value. Examples in all Cafes ready wrought. ' TO Divide 8 9 76, by 2 3 4 $ 6 Divifor . Dividend * Quotient . 23456) 8976,0000 (ii2 6j^c. U2 11.) Divide 148 . A SuffUmem^ &c. II.; Divide 5 « 6, by 3 6.4 8 <^ ?• Divifor Dividcnid Quotient 3^4860 586.000000 (l6.06y^C. • • • • III.) Divide 7 2, by .0 4 3 2 Divifor Dividend Quotient •0432; 72.000000 Cx666.66,tfr. • • • • • IV.Divide 5 8.2 7 1 8 7 5> by 7 2 5; Divifor Kvidcnd Quotient 725) 5127 I 87 5 (.08037$, (?<^. • • • • • y.) Divide 8 8 $.6 9 8 6 4^ by i 2.2 4* Divifor Dividend Quotient 72.3^1) 885.69864 (12.24, SP'r. . • VI.) Divide 5 5 2 8.6 2 5, by .425. Divifin: Dividend Quotient .425) '5328.625000 (12537.941 VII.) Divide .0 6 8 3, by 2 3. Divifor Dividend Quotient 23) .06283 (.00273, cy^. . . '• VUl.; Divide .8 4 6, by 2.4 3. DivHbr Dividend Quotient 2.4 3) v8 46 o o o (.3 4*8 I, £?r. . • IX.) Diyide.ij56592, by.7358, Divifor Dividend Quotient 47358) .1^6659^ (-24 Of C 149 3 VI. Of the Rule of Three in FraOiions Vulgar and Decimal. W Hat the Rule of Three is, and the piaimer of working, is already ibewed inthefirft part, that which we h(are inten- ded is only to add fome Examples in Fraflions Vulgar, as wen as Decimal *, thai by comparing the work in both, the ExceOent uTe of Decimal Arithmetick mi^t the better appear. And how to Gonyert the known parts of Mtmey, IFeidn^ or Merfures EngUjhy into Decimals bath been ahready mewed, both jlrnbmetiealJj^ and by Tdbiets ; yet to prevent the feveral Admons and SubfiraBms In thofe Tabks^l have here annexed ztLothetDecimdl Tibky for the more fpeedy ReduAion of En^Ujh Money under two Shillings, all Sums of Money above, not havmg Pence or Farthings annexed, being as eafily reduced by Memory as by TaikSf dad this I have the rather done, becaufe the fame Tahle will aUb reduce ibc Coins of Fnmce^ and the parts of trqjf Weighty if an Ounce be made the Integer which in point of FraAice is much more ufeM than the Found. i4-M> The - 1$ o Decimal Arithmttick. ^» The Table of RED UCT ION. I 2 4 5 6 001042 002083 Gr.i 003125 004166 2 - 7 8 9 026014 027083 028125 029166 1 I? 14 16 ■ 005208 P06250 (^0729 1 008333 4 • O302Q8 031250 032291 - • 009375 010410 011458 012500 5 6 oMn5 035406 036458 03 7 500 038541 039583 040625 041666 17 18 ■ 013541 P14583 015625 016666 7 8 9 10 ^ 1 10 II 12 1 20 21 22 24 #• 017708 018750 01 979 1 020833 042708 043756 044791 045833 /- . * 021874 022956 023958 025000 II 12 046875 047916 048958 050000 * 1 Dtiimdl Arithmetiik. Ml The Table of REDUCTION. \ 051082 076014 o52o8jGr. 1 * 077085 13 s. d. 055125 s. d. O78J25 I. I. 054160 2 I 7 079066 14 # 055208 080208 056250 3 • 081250 15 057292 082921 I 2 058333 4 I 8 083333 16 059375 ' 084375 060416 5 > 085406 >7 061458 086458 » ' 3 062500 6 1 I 9 097500 18 . 065542 [088541 064583 7 089583 19 065625 090625 I /4 066666 S « X 10 091666 20 067708 • 092708 • « 068750 9 093750 21 - 069792 1094791 I 5 070853 10 I II 095833 22 071874 * 098675 072910 1 11 099716 25 075958 1 2 0099858 000000 24 1 07500c > 12 9 t / 1^2 Deeimdl Arithmetick. Thcfc things fremifed, we will now (hew the ufc of the T^iWe in fome prattical Quefkions belonging to the Rule of Three Direft. Qiisjtion I. If I of a Yard of Cloth, coft yf of a pound: What fliall 1 7 Yards coft at the fame rate ? If I coft rf > what fhall 17 coft ? Anf. 14 /. ^. Firft, multiply tI hy ^^ the produft is 'H, then divide »f J- by 2, the quotient is '^it- ^S^"^» '^ ^^^ ^^^*^^ "^4- by 84, the quotient is 14^!* ^r i" the leaft terms 14 pound ^ of a Pound. And the value of this Fraftion 4 of a pound, will be found by the third Rule of Reduftion of Fraftions to be 1 1 Shillings 5 Pence and 4 of a Pcny, which is fomewhat above two Farthings, and * of aFarthing. tbefam Quefiim in Decimals. If I of a Yard of Cloth coft rf of a pound, what (hall 17 Yards coft at the fame rate? : To anfwer this (^eftion -J of a yard, and -rrof ^ pound muft firft be reduced into Decimals, either by Divifion, or by the Ta- bles of RcduSion : By both which ways of Redu&ion the Deci- mal of ^ win be .875, and the decimals of ^f will be .75, and then the Terms of the Qpeftion will Stand thus ^ If .S75 parts of a Yard coft .75 parts of a poukd, what Ihall 17 yards coft at the fame rate ? If 087$— ——0.7$-: ^^7* H^rc if you multij^y the fecond term 0,7$ by 17 the third term given, the Produft will be 12.75, and this Produft divided by ^75^ gives in the quotient i4.$7i42, tiat is, 14 pound 57142 parts of a pound,^or i45Decades, that is 14 pounds 10 Shillings, and .7142 parts of a Decade for two Shillings) which by the preceding Tables is i ^ 5^» 2Farthmgs, and .©059 parts of a Farthing. Queflion II. If. a piece of Gold Plate weighing 19 Ounces 5 penny weight and 5 Gcains, be worth 62 Pouad 10 Shillings 6 Pence ; what is one Ounce of the.&me Gold worth ? This Qaeftion in vulgar Fraftions muft be expreffed thus* If I /. -IfH- Troy-weight, coft 62/. f J4, what Ihall -^ J- of a pound Troy coft at the fame rate ? To anfwer this qaeftion, the Fraftions j /. ??iX and 62 1. i*-J-, muft be firft reduced into improper Fraftions and then the Fr;afti- ,on 7-iintx) the leaft known Parts of a pound Troy, and then the Oneftion will ftand thus. If fJfJ give ' 't li> what fliall -5 Mi give ? ' Now / Decimal Arithmetkk. IjfJ Now beaufe it Is neceffary the terms given be reduced intp their Jeaft Denominations, bdForethequeftionberefolved, therefore ±ie anfwer may be found, by ufing the terms given thus redoced aS: whole numbers, not having any regard to the Denominators' of thefeFr^aiohs; Saying thus, If 9197 grains, coft i$oq6 pence, what ihall 480 grains coft ?^ And here if you multiply 15006 by 480, the Prodilft will bS 7202880, which being'divided by 9197, the Quotient Will be 785 pence >f5f parts of a peny, and dividing 783 by i a, it will hp 65 Ihillings 3 pence ^-Bf) or 9 A 5x. 3i^-lf|4« Andaltfaoi^h thii, queftloH is' thus more eafriy anfwered than it would have beeiiy if the terms had been wrought as vaigar FraSions, yet the fame^ terms being reduced to Decimals, the anfwer of the queftion will yet l>e found with moreeafe, as ihall appear by the operation: feUowingk' If a piece of Gold plate weighing 19 ounces 5 pduy weight an<i 5 grains, "be worth 62 /. los. 6 d, what is an ounce of the fam^ Ggld worth? Hie Decimal of 19 ounces 3 peny weight and 5 grains^ making aii Ounce the Integer, is by this Table 19*16041, for that 19 ounces are 19 Integers, 2 peny weight is one tenth of an ounce, and the De- cimal of 1 rf. Wi 5 grains is by tjhis Table .06041 ; and the Decimal of 62^. 10 X. 6d^ by the fame Table is 62;525, and becaufe anll- "iiite or Integer is t)ie third term given, there needs no ifiukiplica^' tion> if therefore you 4iyide^a. 52 5 UiefecondtermV by 16.16041 the firft term propounded^ the Qiiotient will be 5.2632, that is f pounds 5 fhilliogs 3 pencej and fdmewhat more, as by the opera^^ tionfollQwJng it doth appear. - ' Divifcr , Divideni Quotiem. 19.16041) 62/52500000 Cj'2 6 3iAf*' • • • 5 748.1 "2 3 ^ 'Ot, 5049770 if. /. ' a. qi 3832082 12116880 ^^ ■■ I- I..-. L**i^ 11496246 6206340 5*748123 4582170 w-A •• i i . 7 ,, - , . "T , V ; f 3 5 3 tf^f 3832082 75 c® 8'* , .^ .,^ 1 54 Decimd Arithmetic^ m. Quiflim. If { Elk and a qujuter of lionen Qoth coft 2h 16 X. 8 if. iq. what ftall 27S £11$ and a half coft at the fame rate? If you would work this Queftion by whole Numbers, your eaiieft way isfirftto reduce all the terms into their leaft Denominations, diat b to lay, the Ells into quarters, and the pounds, Ihiilings, pence and farthings, all into fiirthin^. To (hall your { Ells and a quarter be 21 quarters, and your 278 Ells ao^ si half will be 1114 CTarters» and your 2/* 16 s. %d. $q. will be 2723 farthings, and uen will your queftion ftand thus in whole Numbers. ~ ?Uinert finhings quarters. 21-— .coft 2723 >*^ what will' ■ iii4r-coft? : Then multiplying the fecond Number by the third, that is, 2723 J by 1114, the Produft will be 9033422, which divided by 12, the ] Quotieat will be 144448 fiurthings, which being again reduced hi- i to pound% (hillings and pence, giveth 1 50 /. 9 s. and 4 pence, as by | the operation Mowing doth appear* j 5723 21) 3033422 (i44448{tj. 1114. .,.•••• 10892 21 Or, 2723 ,93 /• s, d. 2723 — »$o— 09 — 4lJ 2723 84 i , 9 3 3039422 84 94 84 102 182 16% Bu Decimd Arithmetick. i$5 But if you would work the fame Queftion by Decimal Numben, YOU may lave the labour rf reducing the tenns to thefr leaft De- nominations, for 5 EUsanda quarter is in decimal Numbers 5.2$, and 278Ensandanhalf is 278.5, and2l.i6j. Si. 3«- isiapeci- mals 2.^364, and then your Queftion m Decimals wiH ftandthus: £Us h EBs. U 1 Ass«25 : to 2.%i6^ :: So 278.5 ; to i'io.^6^2. If YOU multiply (according to the Rule) thefecond terra by the third that i^ 2.8964 by 278.5,_the Produft of that multiplication will be 789.93400, which divided by the firft term 5.25, the Quo- tient will be 150.4642, which Decimal reprefenteth 150/. 9^1. 4^., and fo much in money wHl 2 78 Ells and a half coft. The 0? E I^At 10 N.. £Us "/• Ells. 5.25 - 2.8364 - 278.5 _ 2 7«>5 >4i 820 2269 I 2 198548 . 5^728 5.25) 789-93740 (150.4642 • • . . . r 5 2 5 Or, 2649 /• S. dm 2625 2437 150 — 9-'4 •■' I 2100 3374 3150 2240 ^ ^2100 I 40 , X 2 I have ^ .15^ Decimsi ArithmetUk. I have l)een the larger in this Rule, apd efpecially In tjiis Exatn? jfle, which is incumbred with Fraftions fufficicnt, becaufe I wouM liave the Reader the better difccrn the difference between the Vulgar and the Decimal waiy, and alfo to fee how expeditious the pne is over the other. Now this example being thus largely ex- plained, I ih^ll with the more brevity pafs over the Rules follow- ing, giving one Example or two at the moft in each Rule. And thus much fliall fuffice for the GMm K^U\ or l(ttU (f 7hree DM in Fraftions; yil. Of the Rule of Three Reverfe. Queflion. I. .'A Lends p* 233 /• 16 s. 2d. for a year without Intereft, upon S\ condition that J. ffioulddo the like courtefie for A when feqdired. A. hath occafion for money 7 months; howmiich money ought B. to lend ji* to requite his courtefie, and fave himfelf )iarmlefs? I will not in this place tell you what the J(ttle of three I^erfe % nor the maimer of working thereof, that being already lufBci- cntlv declared in the firft part,' but give yoti the Example, and the forking thereof which take as folio we th : So will the Queftion be jfehos, ^ted; . . ^ months Decimal Jrithmetkk. i j^ wombs h s, iL numtkm 12 : 233 6 8 :: 7 IVhicb in DecinudsfimdstbuSf numbs /• mmbs. 12 : 2 3 3.3 3 :: 7 12 ■■ ■ t I ^6666 23333 7) 2799.96 f399-99^ • • • • 9 4 2 1 Or, 400 o o <53 69 63 66 3 Here you fee that i2 months and 7 months are whole NumberSy andfo we let them alone without any Reduftion, but tbcDcci* mal of 233/. 6s. id. will be found by dieforc-mention'd Tables andRules to be 233.33, which isthe middle term in the Queftioa and of the fame quality with that, niuft thefourtii termioligjhtbe, therefore if (according to the Rule delivered in the firft part) you i multiply 233.33 by 12, the Produft will 2799.96, which divided by 7, giveth in the Quotient J99.99, which is the Decimal of 400 h j and fo much money ought B. to lend A. for 7 months. Qutftim n. If when tha price of a Quarter of Wheat b V2m [ 5^. 6i. the penny white Loaf (hall weigh X2 0u. leJ^.W. I de- mand what the penny white Loaf (hall We^g^, wken the pcicb of the Quarter ot Wheat is 7 x. 6d. the Quarter? ' ' ' If .58 Decimal Arithmetick* If you place the Numbers according to the ten<^ of the Qpcfti- QBy they will fland as foUoweth. /• X. d. Ou. p.w. s. i. 'X— 5— 6 12 — 16 7—6 In PecimaJSf flm, 1.275 12.8 .375 »-^« -375) KJ.320^ (4 3-SrH 10200 I 25^0 1 $00 12^75 1320 X 6.3200 1 1 2 $ Or, *— 6 Ou. P.tr. <7r. i?7 5 43 10 3 .75 Here if you multiply 1.275, which is the Decimal of i/. 51. 6i. by 12.8. which is the Decimal of 12 ounces 16 penny weight, you (hall find the Produft of that multiplication to be 16,3200, which being divided by .375, which is the Decimal Of 7 ;• 6 i. the Quotient will be 43.5, which is theDecimal;of 43 ounces, to pen- ny we'^ht 3 grains; and fo much ought the penny white loaf to weigh, when the quarter of wheat is fold for 7 j . 6 i. yUL Of the Rale oj Proportion^ confining of jhe Nunwerj. Queflhn I. • IP 100 fc in 12 months yields 6/. intereft, what intereft (ball 264/. 16 X. 5^. yield in 15 months at the fame rate P Set down your numbers in Decimal^ as in the Example follow- ing appeareth, fo ftall you find the Decimal, of 264 /. \6 x. 5 4* to be 264^82081 all the reft being whole niitnbers,* having no Frafiions jpyned with tliem we negleft, and work with them as they are, 10 will the feveral numbers of your queftion (if rightly difpofed) fiand as fblloweth : - If Decimal Arithmetkk. 1 59 /• mo. h U If I 00 in. I 2 gain 6, what 264.8208 g^in in ) 5? 12 6 111 ■ - !■ r iin s 2 o o I 5 8 8*9 2 4 3 79446240 1 5889248 1200) 23855.8720 ri9.86i5^ • • • 4 • • 1200 Or, I 0800 10338 96 00 73»7 19 X7. S^ ? 7200 I 872 1200 6720 5000. 720' Your numbers being thus orderly difpofed, you muft according to the Rule before delivered ip the firft Part, multiply thff firft and fecond terms* together, which in this Example are 100 and 12, whofe Product is 1200, which is your Divifor*^ Then multi- ply the three laft terms ^ne into another, as 264«82o'i fwhich is the Decimal of 264/. 16 s. 5^0 by 6, and the Product thereof will be 1588.9248, which Number again multiplied by 15, (which is the laft term} the Produft, will be 23839.8720 which is your JDMdend^ fiid this number being divided by. your former Pro- duft, givcthin theQuotient 19.86 15, which is the Decimal of 19/. 17 f. 2d. 3 }. fer^y and fp much doth the fimple Intereft of 264^/. 16/; and 5 i. amount unto in 15 months, aner the rate of Cixfcr Centum foi ^Ye^u • n . \ i ;, QuefitonlU i6o ^Pecimd Jrkhfmkk. . Queftim iL If the carriage of 23 hundred and 9 quarters of any thing 127 miles, coft4lL 13 ^< 6^. what (hall the carriage of 47 iiundred and an half of fuch like commodity coft, being carried 381 miles. Place your numbers in cmler as in the following Exaipple doth appear^ then multiply the firft and fecond terms together fer your Dlyifor, and the diree laft one into another for your Dividend, and fo will the Quotient of this Divifif^ anfwer the queftion demanded, 9dA the wwk willfland |ls followeth* Cm ml, U C.,^ ,miU If 23*75 ««7 4*^7< 37'5 381 127* 381 mmf N 1662$ 475 4750 3800 *375 "^425 mitiam^mmmmmmmmm 301 6.2 { 18097.$ . 4.6 7 5 ^904875 1T6 6825 1085850 723900 I I 111 ■ ti 5oi*j25) 84^05.8125 C28,o$ • • • • "603250 ^428081 Or, •' K S4 JU 24 I 3OOO 28 I Q 1508125 »*• 1508125 Here you fee that the firft and (ec(^ Terms muttiDtied togedie^ produced 30i6.25» which mnft be yanr Pivifor. anf the three laft Terms being muteiplied one into anotbar^ produce 846o$.8i250) wfaidi number, divided by 3016*2 5) giveth m the Quotient 28.050^ which Decimal reprefenteth 28 /< one. Ibillin^i and fomuch will the \ Decimal 'Arkhtmkk. %$% tiie xrarriage pf 47 hundred and a Jaif coft being ourried 381 miles* Quiftim. in. If 24 yards of ftuf of throe quarters broad, ooA . 4IL 14 X. what fluiljaS yards of the fame ftalf coft being 5qiiar«» ters broad. If you place your nambers according to the direfiions of thk- Rule, they will Smd thos^ yards partes h s. jw^ir |i . If 24 of } coft 4 14, whatlhaQ 328 coftof 5 ■':• ■] Jn VccimOs ih^ J4rds trMd ^ J^ris ironi 24 3 >7 3^8 S ;-■ 3-' 4.7 . - . 72 . "72) 77oSio (107^0$ I 1 1 J. 1 1 — —i^ Or, 7^~ h s. i. ^ o 8 107 t I :■* ' "^ ii ■ •* ' • 4 ^^ ' - ■ -^ • ■- "a * . wa n I <■. • • 360 4=^ * Firft^ xflttltipIjuhet^ofirftnumb6r>, as 24and 3 togethef, thef make 72 for Divifor, then^^nultiply 4.7, .vrhich is the Decimal of 4 /. 14 X. by 328, and the Produa is 15416, which again multiplied by 5, the laft number giveth 77080 1 unto this Produft, (that there may be a competent number of Sgutes lathe quotient>) I add two Cyphers, making it 7768000, which I divide by 72^ and* the Quotient is 107.055, which is 107/, is. 6d. and. [9 mncSi is. 328 yards of ftuff worth, being 5 quarters broads Y IX- 0/ * • t6t . IkcimdJrithmeUck, IX. Of Barter, ^*|'<WaMcrchant$ having two ftveral Commodities are willing 1 to Barter, or Exchange the one with the other. The one ha^ indigo, which he vnlltm^t 4 s* the pound for ready money, but in Barter he will have 41. 9 d. the pound, and theoth^er Mer« chant hath Kerfies, which for ready money he will fell for 3 ^. 6i. the vardl^ Now the queftioq is, at what price he n^ rate his Kerues bi Barter, te equalize die ^d. adnmce UfOft tte pound of Indigo? The Tenor of Ae edition is this. If 41 in Barter,- require 9 iLwftat Siattl3& &d* require? T<ntri^unihr$ placed vnB ft aitbiASp s» JU s* d» ' '^ 4 d 3 ^ In Decimak thiSf ;» .375 i.ji 3^^ ^ ■ ■ Hi " ■ .•>-«^.2 5 (328 .* * * ^1 I. ■ 4 Or, 16 i. s. 4. f^ 00 ; J itf^ Sa^ then by tfie Hule of Three Dired, if 2 Deades^or 4i» in Barter ]:eqnire..975, which is the Decimal of 9^. wlvtihan i»75 required Which is the Decimal of 3 J. «ii j Firfli I I Decimal j^hbmetick. t6^ Erft, multiply .57$ by 1-75, tbeproduft is .^562?, tot being I it is a Fra&ion, I cut off the two h& Figures, becaufe we require I only three Figures in the Quotient, which divided by 2, giveth in : die Quotient .3 28, which is the Decimal of 7 i. 3 q. this jd. ^q. ' added to this 3 x. 6A maketh 4 milliag^ i penny 3 farthings, and fo much ought, he to rate his Kerfies at by the yard in Barter, to \ lavehimfelf harinlefi*. ^mmm^mm^mi'mmmmmtmmmmm^mf' OpHree Perfons ^i B and C bought 4000 Sheep, which coft X 4S3 1. 6 s. % d. of which money A paid ^03 /• B paid 165 /• I5X. 8^. ^nd Cpaid 114/. \\s» FirBf faj by tbt ^tt of 'Arte DireS. 1. If 483/. 6x. id. biiy 4000 Sheep, how many Sheep ihall 203/. (which is A"s Ihare) buy? Anfwcr^ \^%04 ' " " . T . 2, Say, If 48 J /. 6u %d. buy^ooo Sheep, l^ow many Sheep to 165/. 15/. %d. (whicfrii fi^lhare; buy?-^iwr, 1372. 3* Say again, if 483^ U6$.%L buy 4000 Sheep, how many Sheep Jhall 114/,. |i^ (which is C's Ihare) buy? Anjwer^ 948. Your numb^ reduced to Dechnals and orderly placed will ftand as ia the foUoyf iu|( Q|tQr^ioii« Y 5 JFirB, 1^4 Decimal jrithmetkk. h fieep U V 4S3.3 J33 boy 4000 how many 203 ?. 4ooq 812000 483*1333) 8x20000000 (1680 Sheep • • '• ♦833333 32866670 mm 2899999? ' 38666720 -^ / .«M^ ♦«3-3333' 3866^664 ' 00000560 feemdljfar B* Jhcep •4o^O' I. .U5.783J 40C0, ^ 6631333000 483*3333) ^^31332000 (1372 Sheep V V ♦ •• 4833333 17979990 14499999, " 34806010 •m mm ' 33833331 9666790 9666666 124 Jtirilf Decimal Arithmetick, ^ i6< Itirdlffir C 483.3333—4000 458200.00 483.3 J33 ; 4582000000 948 Jbeif. . . 43499997 23200030 »9333392 38666980 38666664 316 Tie sum^ «/ Vf^orldng. For jrf. Multiply 203/. (which is>fsftare)by4ood f which is the dumber of ffiecp bought) and the Produft is 812000 which number fhoMld be divided by 483.3333, but being his greater than 81200, I therefore* add four Cyphers thereto, that I may have four figures in .the Quotient, aod ifipakes 8120000000, which divided by 483. 3333, giveth in the Quotient 1680, and fo inany iheep belcmg to a. 2. For B, multiply 165.7833 ( which is the Decimal of B'$ fcare ) by 4000, (the number of fheep boughtj and it produceth 6631332000, which divided by 483.3333, giveth in *e Quotient 1 372, and fo many (beep betong to fi. 3. ForC. multiply 114.55, (which is the Decimal of C*sfliare) by 40^, (the number of Iheep bought; it produceth 45820ooo,whM:h number fhould be divided by 483.3333, but being it is not large ^ough to give figures enough in the Quotient,! therefore add two Cyphers, making it 45^2oooooo,%hich divided by 483.3333,giveth w the Quotient 948, and fo many flieep ought C to have. Now for proof, if you add the number of (heep that A. 1680 * -^j B and C. Ihould feverally have, you fliall find them in B. 1 3 70 afl to make 4060, which demodft^tes the Work to be C. 048 *i^. Z^ 4000" - XI 0/ x^ Deeimd Ariihmtkk, Xi. Of Lofs Mti Gai», IF me rtrl tfStuftia 6s.Sd. md IfeUtheftm agtbtfor 8 s; 6 (L- rittfiuS tgti» i» Ufmg out loo li. tipmi fucb tCommodigif Take tbe diflference between the price that your Commoditj eoft, and the price for which you Tell it^ that i^ in this Example^ thediflFerencebetween6;. 8i.and 8j. 6iwliidiisii.ioi.then fiy by the Rule of Three Direft, . » If 6s. id. gain IX. idrf. what win looK. gain? ' If you place your numbers according to the Rule of Three Direfi, as they are here given, they willftand asfoUoweth, $• rf. ' . X. d* & If 6 Sgain 1 ID what ii^ill loo gain? In J>ech(MlSf thus U. U. lu ,^333.. ■■ i ' u ii [ .42$ ■ U. IOO 100 .mi) 42500.00 (127.$ .' . . . ' in 'I' ll 9333 9170 Qt ■ ■> ■■ ■ iJi/ - ■ U* ■ Sm 6666 1^7 %Qi ^5040 TTi \ t r 23331 17090 ^♦7^S 325 Your numbers being plaoed, multiply .425, whicfcis the Ded-- mal of I J, lo d. by 100 //. and the Produft is 42500, to which I add two Cyphers ( that I may have a competent number of figures ia theOuorient) and it makes 42500.00, which divided by .3335, tte Decunaj r ^ Decmdl Jrhhmetkk. tdy Decimal of 6s.%d. giveth in the Quotient it7*<, which is 12T//. 5Livcrsorioxfothereisa7A lox. gained in laying out of I will here prove this qaeftionbytheconrerfe. If by one yard of Stuff which is fold for 8 x. 6 d. there was gained 47 //. 10 i. In laving butof 100 //. I demand what the faid fluff coft a yard at the nrft Hand ? Add 100/. and 27/. 10 s. together, and they make 127^ lOj. then fiy by the Rule of Three Direft, If 197/* 10 i. give u>o/« what fbsSlis. 64* give? InDecimdls^ thu 12 7*5 ■ I o o > 4 2 5 429 127.5; 42.too (.3333- 3 8 ^t 4250 Or, Here if you multiply .4.25, which is^ Decimal of 8 x. 6 i. by 100, you inall have 42.500, to whaeh if you add a Cypher, yoa make it 42500.0, this number bd^ cfivided by 127,5, which is the Decimal of 127/. 10 s. givecfr tn-^ie Quotient .3333, and if yoa had added more Cyphers to 19^ Dividend, you mould have had more Threes in the Qiiotient, and no other Figures, but thdfe four are enough, and are a Decimal Fraftion reprefenting 6s» id. andfvmUchdld the yard of Sthff coflrat the firft Haild. ' * • • • . %IL (^ tofs dndCidin ul>o»Tme^ wrought by the D9tdfk Bcuk of yhree. 1:F one Ett of Lockerim coft me 2x. li. ready mony, aixdiielt ' the (ame again for 2/. loi. th^ Elj,. to be' paid at the expira- tion of three Months; I demand 'whai- 1 it ill gain in* 12 Months, hying out xoo /. upon that Commodity ? T.n$ i68 Deeimd Arithmetick. Tbis and fuch lik» Queftioas, aithough they may be wrought by the Rule of Three Direft, at two Operations, yet they arc heft performed by the Double Rule of three compounded of five Num- bers, wherefore the Queftion may be thus ftated. If 21. ZL in three' Months, gain 2i. what Ihall looh gain ii| 1 1 Months ? If you take your Numbers out of your Scale, and place them according as wasdire£led in the firft part of this Book, you (ball find them to ftand thus, Thefe Numbers reduced to Decimals^ and placed orderly ac« cording to the Tenor of the Rule, will fland a» in the followins Operation. ^ u fHOm i' U tno. If 1.333 in 3 g^ 83, what (hall 100 gain in 12 3 *oo^ 3.999 *?^. 12 16600 8300 5.999) 99600 .(aj/eri • • 7998 19620 ^19995 375 Your Nmnbers being pbced according to the Tenor of the Queftion, if you nroltiply i*333f which is the Decimal of 2s. id. by 3 months, the ProduS will be 3.999, which niuft be yoor Divifor, then multiply 83, which is the Decimal of 2 i. by 100/, and it makes .8300, that again multiplied by 12 months, givetli for the Prpdu£l 99600 for your Dividend, wherefore if you Divide 99600 by 3999, it will give you in the Quotient 25 almoft, which is 25 /• for the Decimal Pradion remaining is fo fmall, that it wanteth not near a&rthing of 25 /. and therefore we call it 25 /• and fo it is exaftly, as you may try, if you reduce all the Number* to their leaft Denoniinations, and woric aiis before taugbt in Vul^ gar jtnbmetick. IwiU ' Decimal Jirithmetick. 1 will prove this Queftion by theconverie. I 169 If one Ell of Lockerham coft^e 2s. 8 i. ready money, for what price (hail I fell the fame again to 6e paid at the end of three months So thati may gain 25/. in loo/l for 12 months? SAy bf the l{ule cf Three DireB, • If 100/. in 12 Months g^ia 25 /. what fhall 2 f. Si. gain in ? Months? If you Reduce your Numbers to Decimals, and place them ac- cording to the Dpuble Rule of Three, they will flsmd asfoUowetl]^ ' 100—12 — 25 — 1*5 33 — 5 12 25 ' ^00 loo 1200 666^1 2666 f""^ 33325 3 1200) 99975 0^3 9600 3600 375* Your Numbers being thus placed, if you muhiiply loo/. by t% months, you ihall find the Produd to be 1 200, whidi is your Di- rifbr. Then multiply 2$ /. by 1.333, which is the Decimal of 2 /• Si. and the Produet thereof will be 33325, which multiply agaia by 3, andtheProduftwillbe 9^75 for your Dividend, this 9997$ divided by 1200, giveth in did Quotient .83, which is the Decimal of 2 d. which 2 i. added to 2 s\id. the price which the £11 of Locker- axQ^ft, giveth 2 x. 10 i. and at that price muft you fell the fanlft at 3 months time, fo that you may gain 25 /• in the ;6o /. in n Months* mm / / E «'7ol] APPENDIX. » XIIL Of ExchAnges. ^nr^O giVe, 6t Exchange one €tmmo4ity for znothcTy or Commo' JL dity fof Money ; or Mptey for Cqmwiodity\ or part Alo»e^ and part CbimMtfrth 's called ^^n^r ; JBut ^xftoijge (according to the ordinary Notion of Nlerchandiiing) is to give Coyn for C^ ; that is, to give z Sum oi Money in one ]?hcey for a 5/i7, ordering the payment of the like Jum f according to thi^Valiu agreed upon^ in another Tldce^ cither at home, which is called Inland^ or in ano- ther Country y which is QZ^A J^ or eign Exchange. And as there is a P^ of Exchange of Money ^ fo there is a Picr or Equation of Weights and McafureSf whereby to value Foreign Goods bought or fold in any two diflferent Countries^ &c Now to perform this Work, there is notMng required more than the Golden ^le^ (or I^k. of Tbree^ if 6rft the J^e, J^io or Troportion betweer^ the Coyns^ Weights and Merfures of any 7w# Countries be firft known, which is bett obtain'd by Experience, rather then taken upon Tj^ft • All that I fliairdo in this Cafe, is tp inftruft the ingenious in the inanner of Work ; and make ufe of fuch J(ates and Prcponions^ as I RnAfetitewn by Mr. lewis l^oberts in his Maf of Commerce. ' , . I fhall illuftrate tht» Jfkh^ Exchange ^ by the Working of feve- ' ral Queftionsthereuato relating. 4 ^^^f^ I.»HoW nwny Rid^^ C^db Bid^r <ont«toiflg li i* 2lf 20 n\\iikl receive for 25^ /. 6s» ^4. 2f. Steriing?^ - .• . ' . • ' h s» i. j. J(«&r /^ X. i, )• i^» ii 5 604, ; V :: 2^1 3 6 87 :i;^^7 .ilcre if you reduce your Number^ to their leaft Denominati* ons, or r^t; them down in Decimals, and multiply and divide ac- cording to the Golden Rule, you (hall find in your Quotient 2^7, and f© mahy Mlders ought to be received for 251/. 6 s. ^d. 2}. Sterling. JSuemonlh How many French Cronns (csLch French Crown bein^ Talued'at €s. Sterling) iballl receive for 492/. i3f. Sterling? A% e AfftuAix, 1^1 i. F.C. I. .s, F.C. ' Multiply and divide according to the Golden Rule, and you fhall have in your Quotient id^R, and fo many French Crowns are to be received for 492 /. i8j. Sterling, Quefiim 111. A Merchant delivered at Paru 1643 Crowns of 6 f. Sterling the Piece, How many Pounds Sterling ought to be received at London ? '^ . F» C9 Sm JF« u • / • s» 6 :: 1643 ^74921! 3 :: 1643 :5 49^ 9 ^g^i : 6 :: 1643 ^^49218 Multiply and Divide, and you (hall have in your Quotient 492 7. 18 s. and (bmuch Sterling Mon^ ought to be delivered at Imiov^ for 1643 French Crowns^ of 6 s. the Crow» Sterling* f^flion IV. Jf 3 yards at Io»rf(?», be 4 Ells at Antwerp^ how »any yards at Loffdin make 84 Ells at Antworp ? . i MAn. r.Lon. £i/An. nLon* As 4 : 3 : ; 84 : 63 And fo nlany yards at London ^ are equal to 84 ^^^^ ^^ Antwerp: Qttefiion V. How many yards of I(?»rf(7» make 2 7 Ells of Antwerp when icx) Ells of Antwerp make 60 Ells of Lions y and 20 Ells of , Lkms make 25 yards of London? I tbefirnmn: J/fr Lions r^r^j London JElx Lions. a» 25 ■ ■ ^o 60- ' i5o|o ^ . .' '75 /^>75. That is 75 YfUrds oiL&ndon is equal to 100 Ells of Antwerp. z 2 r/jf / ly* Jffeniix, Tbe Seemi IFork. EBt Antwerp rWx London Ells, Antwerp, xoo ■ 75 27 ,27 '■ Tards rfhondon 20125 fecit 20.25 f^jHw VI. If ioo/. Sterling be 134/. 6x. 4^. Fteiii/t, what is one pound Sterling worth? Lfl. h s. L Fhft* L s. i» q^ 6 10 : 34365 * 5^^ : 15464 : I :?i6io2 iioo : 134 3666 : I : 3 1 Multiply and Divide, and you fliall have in the Quotient 1.34366, or I /. 6 X. 1*0 d* 2 q.fere and fo much is one pound Ster- ling worth. Quefiion VII. How many Ells of Fntfieiford make 42 * Ells of Vienns in JufirUj when 35 EUs of Vienna make 25 at Uons^ 3 Ells of Lionsy 5 Ells of Jnmerp^ and 100 Ells of Antrserf^ 125 Ells at Frgftckford. £i7.An. r/LFran. ElL An. £]/.Fran. i^ 100 : 125 ::' 3 ; 6.2$ ES.Vio. J!?zr.Fran. EB.U0. J!?i/.Fran. 8) 3 ; 6.25 :: 24 : 50 jEi7.Vi. EH.'Bnn. Ell.Vi. JEi7.Fran. 3) 35 • 50 :: 42.25 : 60.35 Thus hare I given you a few Exchanges, I will hereinfert fomefew Tables derived from Mr. Lewis ^o^rji his Map of Ccnmerce a- fere-faid, of the truth of which I am not a competent Judged but ftall leave that to the fcrutiny of fuch as have occafion to trade ^to Forreign Countries. TABL^E Afpendix. 17? A T AB LE fhe wing what one pound ofjv^rdupois Weight at London^ maketh in I divers Cities, and other remarkable places. One Pound of A Ntwerp XX Amftcrd^m Abevile Ancona Burdeaux Avigon Burgoyne Bolpnia Bridges- Calabria Cakis Corlftan- ? tinople^ ' Deep Dantfick Ferrara Florence Flanders^ 7 don maketb at i Geneva Genoa Hamburg Holland Lisborn Lions Leghorn Milan Mirandola Norimberg Naples Paris Prague Placentia CRochel: . lb. •9615 .9 .91 1.282 ^.12 .91 ,91 I«2< •98 1.3698 1.07 r .8474 \ Loder; .91 u\6 •282 Aver-du-pois- 1 Flanders^ 7 S ^ rjk Jf^eigbt at Lon-^ in general 5 ^^ i.oo •934$ {1.4084 futtle M?85 grofi •92 .88 1 ^•07 common ni^eight •98 filk weight .9 cufl»mers weight 1.4285 '•3333 .88 1.4085 l.8p •83 .13888. Rome t74 Affeniix. One nrml tff Aver-do-pois i Tholonla rtloine Rouan 'might 4J Lon- don iMkts at Turin, Vcnetia ] Vienna ^ I { [ Ih. 1.27 .87$ by Vlcont '9017 common w, .108 *112 .1219$ 1.^625 fiittle 943? grofe •813 The Ufe of the preceding Table. How much weight^atBd/owi^, will 65 5 fi. Averdupois make ? look in the Table for BoJa»w, and right againft it you (hall find 1.25, which (heweth tiiat one pound Averdupois at Io»i(»r isequal to las /. at BoloniA \ Therefore fay by the Rule of Three : . If I U Averdupois give 1.25 /• at BolmU^ what (hall 635 //. Averdupois give ? Anjwr 8i8«75. As by the operation following doth appear, Av. //.Bol. luAv. ii.'OoL As I : 1.25 : : 655 : 818.75, \ ■ ■ A F^ I jlfpetidix. '»7 A T A B L E flie wing what one pound Weight in divqrs Forreign Cities,4nd remvk- ablc Places, maketh at Loadon of Jverdttpoit Weight. . ^ i*K A .Ntwerp I Amfterlam Abeville Ancona Avignon Burdeaux Burgoyne Bolonia Bridges Calabria \^ Calais Deep Dantfick Ferr^ra' . ' ^ Florence \fCl general S Geneva 7fubtlft Genoa •• ' \ >grofe -. Hamburg Holland Lisbon 9 common weight Lions >filk weight 3 cuftom weight Xcgom Milan Mirandola Norimbergi Naples Paris Prague Placentia Rochel Rome 1 > .S3 o •a < c o lb. X.04 i.iiti X.09S9 - -78 8928 A.0989 1.Q980 l«C2Q4 .934$ t.0989 .862 .7$ •9433 J-P.7 ^\ 1.0865 1.0526 1.135 •9345 1*0204 la 1 1 1 •75 •7 •75 1.136 •71. 1.123<, 1.204:3 •72 .8928 .7^:4 .: -W' N*«r— ■»*i f ' »r " ' ' Ro^n ■ / 1^6 appendix. ^ 8 f 7hyyicoiint, 3 common weight SevU Tholoufa V Turin- -^ \ fubtle, Vcnetia 'f^grols. ' g LVicrina' ; : lb. IS St J? •I 5,^<l I ■fr 1*1428 i«io89 •9250 •8928 .82 •64 i.o6 The Ufe of the foregoing Table. In 76^2 //..weight at MirmioU^ how many pound Weight Averdupbis ? . . ' of . Look in the Table for NtirmdoU^ and- r igjht agaijoft it you ihail f nd .75) which Iheweth that one pound AverdupoU is equal X9 l75or-}- of a pound at MrmdoU^ wherefore lay by the Rule of Three, If tU at MirmioU^ gives .7$ or -J- of a'pound^veriiipoK, what lhall76;2/.ofMir^K0/iigive? AnJifKr 5739, as by thp operation ibiiowjng doth appear. U.Mf. .Av. li.M/« As I : .75 : ; 7652 U.Ar. 'itmtmmmlm A *•;, Affendix.. 'Aim UlM^B MBHM mum 177 A T AB L E reducing Englilh Ells to the Meafures of divers Foragn Cities, and remarkable places. I n .o B o < r A MfterAim jrV Antwerp B/idges Arras Norimbcrg Colen LiQe Maftricht Frankfbrd Dantfick Vienna Paris Houan Lions Calais __ . >Linnen Venice -^5^, Lucques Florence Milan Leghorn Madera-* 7 ines 5 Sevil Lisbon Caftiiia Andaluzia Granada Genoa Saragofa Rome Barfeloni Valentia ■r.-.M^ V a>i> Tr II I •Ah Aulil9 \ > firac^ I40328 Catied A afc -^—' • ■ "it I i"i ^■•■MaHMMlllAb liMiMAMiM XJ&: Jffendix. The Ufe of this TaWe. [ In 632 EUs m LondoBy bon many Braces at Florence ? Look in the Table for i7arevre, and right againft it you (hall find 2.04, whichfhewethy that one BU at London maketh at Florence 2.204 Braces ; whereforefay by the Rule of Three. if one Bll at London give 2.04 Braces at Florence, hon ^anj Braces fitaUo'^2 Ells give f AnTwer 1289.28, as by the operation following doth appear. Enio». Bra.FU EWLon* Bra.FL . 2 As I ; 2.04 ; : 632 ; 1289.28 A ^fpenMx. ^9f A T A B L E reducing the Meafures of diyers Foreign Cities^ and remarkable places, to EngUjhWi% . • ^ - / ^ V 7^ i AlVJfterdam Ant\yerp Bridges -Arras , • t Norimberg. Coleu Lifle Maftricl^t, Frankfotd' Dantfick ' ' \^ Vienna Paris Rduan Lions Calais^ r Venice . ? Li innen SHk ^ < 4<4 Lucques Florence Milaft Leghbrn Madera Sevil Lisbon Caftilia Andalufia k Granada One Palmzt Genoa §: /"Saragofa O ^ Rome Barfelona Valentia ; Id •> J « ■ •59 .'<5o97 •6q6 - . • . -5474 .4867 •6024 . -/<^369 . ;'*-4792 ^ ■ • .7228 .9896 ^ ( ■§ s 1.0526 .9708 .9836 .6369 1 (>(}J > t :i\ - '0 \\ J y^< •5555 f>§ .5102 * •5 4901 •4H7 •5 ^ •9681 .7409 I. .7207 •7339 •7339 .2079 1.8181 1.7857 1.4095 .8247 I Aa2 •/■ " r ^ %t^ jifpendix. TheUfcof this Table. fii $727 Braces a Leghorn, bow nmf Ells Englifli. Look in tbe Table for Lsghom^ and right againft ityou (hall find ;$, wl^ch flieweth that ime Brace at Leghorn maketh at London . $ or JaifanEll,whercforefayby the Rule of Three. Jf one Brae dt Leghorn give •$ EBs at London, vhatjodl 5727 praises give f Anfwer 28^3*5 £Us Union. Br. Ir. EB. Lon. Br. Le. EH Dm. fA X ; .$ ; ; $7727 ; 2863.$ trm ^m-mm . I • - • 4 I 1 XIV. <y ' C i8i ] XIV. Of Extra^ion of Roots. I. T70r the Square Root ; That is, a Squire NUmher being given ; }^ to find the l^ot or Side of it in a dumber ; which l(oot or iirdc, being NLuhipUed into ity^^, muft therefore produce that Square Number^m The ^afon of this J^/e is taken from the IVth Fropofition of the Second Book of Euclid^ which faith ; If A Right Line be divided by chance 'y the Squares made ef the Parts, tdgetber mtb theKc&Single madeof tbeV^itstvoice^ /j Equal to the Square of the rohole. JLLVSTJ^ATJOir. Let tbe Line A6, be divided by chanpe in (fae point £, it is mat nifeft that the Square of AB, that is to fay, the Square ABDC, is equal to.the Square of LO, that is of AB, and tx> the Square of £B, and to the two Refiangle Figures of Ad and DO, that is the Re- dangle AO (which is made of the Parts A£^ and BE) twice ; ac- cording to the Tropofition, Now let AB, be fappofed 20, AE i $, and BE 5. Then the Square of AB, (which is made |>y multiplying the Root 20 into it felf ) is equal to 400 And n 182 Of ExtraSiion of Roots. And the Square AE, that is, 1 5 times 1 5, is equal to 225 And the Square of BE is 5 times 5 25 And the Reftanglc AO, 5 times 15 ^ 75 And the Redangle DO, 5 times 15 75 In all 400 Which is equal to the Square AB, as before. The Genefis, for Extrafting of the Square Root. Example I. Let h be required to find tbst Sqxx^e Number, Vfbojh Side J* 57. I. Write down the J^ot 57, as in the I^- Root E^^^9 with the interval of one Figure between the 5 and the 7, and draw a Line under them; and alfo, two down-right Lines^ the one next after the Figure 5, the other after the Figiure — _ r^ — -— - 7 J ^o that the I^Tumbers to be found, may be or- 49 I square ^^j,jy pj^^^^ f^^ Addition : Then let the ^ot given be fuppofed to be divided into thefe two TartSy 50 and 7; Then, II. Multiply 5 into itfelf, the Produa is 25, which fetunderthe Li'fje^ and under 25, Vnite under Vnite. III. Double 5, and it makes, 10, which multiply by 7, makes 70; which kt under 25, but one place forward towards the J(i^bt- Hand. . IV. Multiply 7 into^tfelf, the ProduB \% 49,- which fet under 7, Vnite under Vnite^ and dbw a Line under all. V. Add. the three Numbers between the two Z/w^j^tojgether, in the fame order as they \. ftand ; the Sum of -them Vill be 3249, .which is 3 Square Number : And thei^c^ of it 57, which may be ■ppved by muitjj^yijig 57 by 5 7, for tlie PrcduBWiW he 9249, ~ J 4? 4 . ' * ?^w;?/<?ILSoif the iVwM^<?r 792 aJ^ot, _7 I '.?2 I'Root. were brokea ,intd thefe two Parts 7 and 52 y Z7, • ^h- Sumof thePra^ttflj a riling from- the fi- . v^, ^ ■ veral Branches between the Lifm found as ,po ± t^^- ^"^^ Example y the ^wm of them will be ')5nb2|4^<iaaic 'y^^^^^ a52«^re A^wwkr, the^.o? wh-reof Of Extraction of Roots. 185 • 32 24 • 45 ("57 Root 107; 7 1 7 49 49 Refolvend Produf! o I 00 I Remain. To Extra^ the Sqxizre Koot. When a Number is given to have the Sqtuire I(pot thereof .rx-^ trdtedj as fuppofe this Number 3249: Set the Number down as |n the Margine ; and make a Point or Pr/Vib over 9, xhe place of Vnitjy and another over 2, thefecond J/gwr^ from it towards the left Hand ; obferving the fame order ftill if there were moreSef* condary Figures in the Number given . . Then,draw down*right Lines on the J^gibf Side of each Figure that bath a Prick over it : And put a Quo- tient Une^ (as in Divifion) on the right fide of the given Number^ and foyour Number is prepared for Extra- Bion^nd will ftand as in theMargine ; And to perform the Work of-Bx- tra^ionj fellow thefe DireSiom. I. Seeing 32 (being the Figures of the firft Period) not being a Square Number* find the neareft Square Number among the nine JHgitSj which is lefe than 32, which you will find to be 25 (for 36 is'greaterj whofe J^or is 5, fet 5 in the Quotienty and 25, the Square of it, under 32, and draw a Line under all : And Subftrafting 2 5 . from 32 the remainder 7 fet under the Line. II. To this l(emainder J y hung down 49 (the two Figures 0^ the next Period) fo will the Number be 749, which you may call the ^olvend > Then on the Leftrhmd of 749 make a Crooked Line for a Quotienty in which put the double of the Figure 5 ij[i Che Quo- tient^ which is 10, and ask, how maiiy times 10 can you have in 74j the anfwer will be 7 times; put 7 in the Quotient^ to 5, making it $7, and alfo in the other Quotienty making that 107 ; Thea ^[lultiply 107, by\7 (the Figure h^ put in the Quotients) and the ProduS will be 749, which fet under 749, and fiibftrafting it from 749 above, the J^mainder will be nothing ; which fhews that the Number 3249 is a Square Number^ and that thei^w thereof is 57* As in Example i of tlie Genejis. Bxample IL Let it he required to find the Square Root of 27846729. ' ^ur Number being fet down, wrrii Pcints over the proper Fi- g«rw; and down-right and i?«ot/e»j lines drawn, as was before di- refted, and as you fee here done in the Margins; You may bc^in Mr ExtraSiion in this mannen I .-The 27 3± 84 67 29 102) 2 2 84 04 io47> 80 73 ^7 29 io$47) Relblvcnd Produft_ Refolvend Produft 29 [ Refolvend ^9 I Produft ] I o I oofocTl Remainder. 7 7 38 38 184 Of ExtraSfion ef Roots. 1. The Figures of the f irS period being 27, the nearcft Scpiare ifumber thereto is 25, whofe J^oof is 5 ; fee 5 in the Quotient^ and 25 under 27, and under it draw a Line, and fubftratling 25 from 27, the Remainder will be 2, which fct under 25. II.Tothis/^m4/V (-5277 Root der 2,bring down the two Figures of the next Ferioij viz. 84^ making it 2 84 for the fittt I(efolrveni. III. Double 'the Fipire in the Quo- tient 5, it makes lOy which fet in a Quo« tient, on the Left* hand of 284, and ask, how many times 10 in 28, the Anfwer will be 2 5 fct 2 in the Quotient for the F^cty and alfo in the other Quotient by 10, making it I02. IV. Multiply 102, by 2, 4he laft Fi^e in the I^pet^Quctiem and the Pr(7^«^ will be 204, which fet under 284, and Subftrading 204 from 284, the /(emainder will be 80, which fet under 204^ V. To this J(efnainder 80, bring down the two, Figures of the next Period^ viz, 67, making thefecond J(ejhlvend to be 8067. VI. Double the Oiiotient 5^,- and it makes 104, which fet in a Quotient on the Left-hand or the Reiblvend 8067, and ask, how many times 104, maybe had in 806, the anfwer will be 7 times} put 7 in the Root Quotient, and alfo in the other Quotient on the Left-hand*, making it to be 1047. VII. Multiply this Quotient 1047, by 7, the laft Figure put in- to the Root Quotient and the Produft will be 7329, which fet un- der 8067, and Subftrafting 7329 from 8067, there will remain 738^ which fet under the Line. VIII. To this I(£mdinder 738, bring down the two F/gKr« of the next Period^ viz. 29, making thethird J^/^/v^wi tobe73829, IX. Double the Root Quotient, which is now 527, and it makes 1054, which fet in a Qiiotient on the Left-hand of the Refolvend 73829, and ask how many times 1054 may be had in 7382 ; the Anfwer will be 7 times, put 7 in the Root Quotient, and aMb in the other Quotient^ on the Left-hand of the third Refolvend, making that Quotient to be 10547. • X. Multiply this Quotient 1054^, by 7, the laft Flgdre put in the Root Quotient, and the Produft will be 73B29, which Subftrafted from the Third Refovend 73829, the Remainder will be nothing J which Ihews that the given Number 27846729 is a Sgilir^ Num-* Of ExtroBion of Roots. 1 89 ber^ and the Root of It is $277 ; which may be eafily pcoted by Multiplying 5277 into It feif for the Produft of thatMultiplicatiOB wlllbe2784$729* Other Examples for FraSice^ A) 5 4 03 50 87 21 42^ (22439 K^oot 05 ?4 444J 4483) 44869; »9 171 50 76 ReTolvend Piodtta ReTolvend • Piodaa 74 I 87 40 40 3i 38 1 ReTolvend iPro dttg '21 i ReTolVeni 21 I Produft }. I I 00 I 00 I 00 1 Remainder. 49 (307 Root y I loo Refolvend Prodaft 49lProdtta 00 Refnai amer A Third Exmfknf aNkmb^ notSpiare. cj 06 CS017 {Loot ail.) 64 64 1601) 28 J Refolvend 16027; I 28 03 I 16 I or [ . jiJt^rodaft 12 02 \^ II 21 ( 89 -u. Refolvend ";» Pfoduft 80 I 17 1 Remainder -I This Given Mamber 64280306 is not a Square Knmben foryoB ^i that after the £xtra£lion is ended/ there remains 8017, whicli ^s the Numerator of a Fraftion ; and then, double the Root ^d add a Unite to the double, and and it will be 16035 for the denominator ; and fo the near Root of 64280306 will be So i j^Siii^ Hb ■ ^ . Thii 190 Jftfendix4 This i^ the ufual and common Way ; But to find the broken parts of the Root moreexa^^ly, you muft add a competent Number of Pair of Cyphers, to the given Number j as if yoii would have the Root to the Tenth n Part of a U- T .2 > Hundred C nite,' you muft J '4 >r^„^u.^ erf. Thoufand . C then add to the S 6 C wPners, ^c. Ten ThoufandV Number. C g •^ And in (0 doing, the Broken part of the Root is always aDeci* mal, confiWing of fo many Places, as there are Pair of*^ Cyphers annexed. So, if this Number 43623 were a Number given, to find the Square Root of it, to the Thoufand part of an Unite. Firfl*, fet the whole Number down, and Point it as before; And becaufe you would have the Root to the Thoufand part of an U- nite; addxathe Number Three Pair of Cyphen, and P9int toem alfo, fo will ttie Work Sand as in this Example. (TV.; 4 4 36 23 00 00 00 (2o8«86i Root 4o8> '^6 (.23 32 i 64 4168) ■IT765) 41^7222) 3 3 ■j 59 13- 11 Refolvend Produft 00 44 56 I 00 Refolvend Produft 50 41 04 71 00 21 Refolvend Produft Refolvend Produft , , ~j 1 I I 8 I 26 I 79 I Remainder. And then if you Extraft according to the former direft ions (and as you fee here done) you fhall find the Root to be' 208.861, and yet therc^is a'Remainder of 82679, to which if two Cyphers were added, the next Figure in the Quotient will not amount quite to 2. And if according to this Artifice, you would Extraft the Square Root of 10} you will findittobe 3ai52, (S»c. To fni the Square Upot of aVulgar FraSion robich U Cornmenfur^bU. You muftfirft Reduce the Fraftion into ks^Leaft Terms; for it may fo &U put, that the Fraftion in its given Terms may be Incom* menfun^e^; but being^reduced to its Leaftt it may ))e Commen« furable ; And then, This is the - RULE. Of Extraftion of Roots. 191 RULE. ExtraH the Sauare 1^9% oftke Fra3t(m*s Numerator for Numerator of tbeJ(oot land the Square ^otof the Denominator of tbe Ffrf^ioHj for the Denominator of the I^ot. \ Example. Let it be required to find the Square Root of this Fraaiort ^-Irh ■; . ' ^ This Fraftion reduced to Jt's l^aft terms is '-J, and bo*, the Square Root of 1 6 is 4 for the Denominator ; and the Squari R^ot of 4^1, is 7 fi)r the Denominator ; fo ±e Square Root A-^ or -} J is i* Alfo, the Square Root of f J wHl be found to pe -l-jafld or-j-f, tobej^r|,er. • • Tofnd the Square J^ot of a Ccmmenfurablc Mixt Number, , For the Effefting hereof, this is the R U L £• Reduce the Mixt Number into an Improper Fraftion ; And then, the Square Root of theNuoaeratorandpenominator of tHat Improper Fraftion, (hall . be the Square Root of the given Mixt Number. . • ' • • Example. Let it be required - 16 find the Square Root of this Miict Number 944}:. • ^ ' This Mixt Number reduced into an- Improper Fraftion, will be **'S>- which being in it's Leaft Terms, needs no hiore reducing : So then, the SquareRoot of 2209 is 47, and tfte Square Root of 64 is 8, and the Square Root of the Mixt Number 34I5 (or of Note. When a Mixt Number, or a Proper Fraftion i&Incommcn- foteibte to its Square -Root, prefix tb& tTharafter [f^, before ic,lb ,tte S^rc RooCcDf 7f win be thus ex,preffed, V f 7*— Alfo, ch- Square Root o£vt4 muft be thus expreffed Vj-J^ . For.thefe and fuch • lite cannot be expre€ed by 'any Racicmkl Numbers ihacfoeven H. Ofthf'CiAe Root. A Cube is a Soli^ Figure; comaM under Six equal Squares. CEuclidjLib. ii1Def.2sy/mUfhay beftlyhprefmedby d Dye. Of Cube Numbers there are Three, dlftinft kinds or Species ; Vfr« SmgkfCotmtmd,9iDiiJrraMiitaU*^ , » ' > , • .. ' . Bb2 i.sngjf, T9t ^ ' Jffendix. X. JfNgZf^S^chttecaUed Sh^kCube Skmbersy which are made of anyone Single fignificant Figure Multiplied Twice into it felf : As I Multiplies nothmgiand fo is both fgot and CubeiSi^t 2 times 2 is 4,and 2 times 4 is 8:ro that 2 is the Root^nd 8 the Cube : airo,^ tinsies 9 is 9 ; and 3 times 9 is 27 ; and here 3 is the J^,and 27 the Cube. Andfoof all the Nine J>igh Numbers^ as in i^is Table. C I I 2 3 Multiplied into r n 4 9 4 MUiuipiica inro j^ J J, it felf, produceth ^ ^ ^ 6 7 8 19 the Square Kuwr ber* 36 49 <54, 81J And that Multiplied ^again into the Side produceth the Ca^c ' number. X S 27 64 12$ 226 ?4? 512 L729 2. Comfwndj fuch are called Compouni Cube Numbers^ whofe J^ticoniift of more Figures than One; So, if 12 be the j^^then 12 times i2» v/^. 144 isthe^ffi^re; and 12 times i44jsi72&die Cube\ AlfOjtheCfttfof 22 is 10648, Vc 9* frraionaJ: Thofe are Qlled JrrationdlCube Numbers^ whole ezaft Cube Jfgot cannot be found out by any Ar^fice yet difcovered, either in Wbok Numbers^ FraSiotis or Decimals \ and fach are the Cube JfoctsQi 2,4,7,io,and infinite others. < Tie Genefis for ExtraSing the Cube Root V lEirampIe, Ut it be re<ptirei to find tbat Cube Number ; wbofe (ide,or Root is 57* I. Set down the Upot 57, widi the interval of two Figures, between 5 and 7,asin tbeMarginei ^nd draw a Line under them, and alfo two down* right Linesyonenext after 7s. the other after 5, for the more orderly placing of theNumbersto be added : Then let the J^ot given be fiip- pofed to be divided Jnto tWQ parts,v/^. $00 and 7. 2. Set the Cube of .5, which is 125 under $ (Vnitesunicr Vnites^iic) and the Cube of 7, which is 343, under 7i two Lines, or places, below tlic Cif^ of 25, (Vnites under Vnit^s^ ^ . ^ ^ 3. Triple the Squere of 5, which is 25, and it makes 7$, which Multiply by 7, and it makes 525, which (et under 125, and place forwarder ro the right Hand. 41 Triple^jiiinakesij, which Multiply by 49,, CtbcJ(ir4re of 7) it 5 125 52 7 ••7l ... $.• 35- 34? t i»5 f m I OfExtra^sM of Roots. 19 j itjnakes 73^, which fet under 5 2 $,ene place forwarder towar^flie right Hand. 5. .Draw a Line, and add all the Numbers together, in the fame order as they ftand, and the Sum of them will be 1 85 199, which ii a Ci^ J^umber^ of which 57 is the Hoot : Whidi may beeafily proved, -bj Multiplying 5 7 into it fel^aod that FroiH& again by 57,1b the laft Prodn^ will be iS5i93. aberExmpks for FrgSice: 4|..8| X ' .tf4 - • • • 38 4.. 7 (58. <i? 9 ..9 729 • •• 218 7 . 21 87. 729 10 I 992 970 1 299 To Extras the Cube l(oot. When a Number is given, to have the Cube Hoot thereof found; 'You muft,Pirft, write the Number down; then put a Prick over the firft Figure towards the right hand, which is the Place of Unites; and fo over every Third Figure fit>m diat Place of Unites, towards the left hand : T|ten, by every Pointed Figure, drawa down right Line, for the more orderly fettingof the Figures to be added and Subftrafted; and alfo a crooked line for a Quotient, on the right Hand, and fo is your Number prepared for Extradion ; Tor the performance whereof, obferve tiiefe following DirediOns. Example I. Let tt be rcqukei to find the Cube Hm of this Com- pmiCube Kumber i%$i^l. , ' • 1. Set down the Number given 18:5193 and make a Prick over 3) the Place of Unites, and miifmg two Phces, make another Prick over 5 : Then make a Crooked Line for a Quotient, and by 5 and $ di;aw two down riglit Lines : So will your Number ftand thus, 185 I 193! ( 2. The three firft Figures of the given Number towards the Left Hand, are 185, which is the Firft Period; feek(by the foregoing Table) the ncareft Cube Number to it, being lefsj which you will find to be 12$, whofe Cube Root is 5 in the Quotient, and the Cube thereof 125, whofe Cube Root is 5> place Ae Root $ in theQuo- tieat, and the Cube thereof 12$ under 185, drawing a Line un^ deri25>aRdSubftraaingitfrom 185, there wiU remain 60, which fet 194 Affe^ihL fist tiatotiieLine: AiidroisiiieFirftoperatk»i,{brtfaePirftPe^ ended* 9. Tio the 60 which remainedjbring down the three Figures of tbe neztPeriodyv/^. 199, making the 60 to be6oi9^,which is caDed the; Rjefolvend: under which draw a Line ; and then the work will ftandtbus. 4. Triple die Root in the Quotient 5* i^ makes 1 5, which fet under the Re(bl\rend, in (bcbordo^, the place of Unites in this Triple, ^ may fland under the Place of Tens in the Re- ■ • — ' iblvend ; fo the Triple of the Root 5, being die 1 5 , fet the Unite $ under 9, the Place of Tens in the Refolvend and. then the Work will ftand dius. i85|i9JRefolvend 60I193I 125 60 193 (5 19 J hi Refolvend Hie Triple of the Root 5 {• Triple die Square of the Root 5, and it makes 75 (for $ times /$ is 2<^ and 9 times 25 is 75) which place under the Triple of the Roo t , in (udi order that the pla ce of Unites in this, may fcmd under Che place of Tens in die Triple ^^nd then the Work will ftand thus. 185 12' 60 »P3 i% 193 I Refolvend 15 j Triple of the Root 5 5 I Triple of the Square of the Root 5 g. pra)Bir a Line under the Triple of the Square 6f the Root, and add mat and the Triple of the Root tqgether,inthe rame<)rder'as diey ftand, fo fliall their Sum be 765, for a Divifor 9 under whi<* alfo draw a Line, and the Work will ftand thus. 185 125 \9i / ($ 60 1 193 I Relblf«nd ^ii5 WpVofth^ 65 Divifor uare of theRoot < ' * - mmm ■ I »' ■«-»• 7 Draw Of ExtraStion if Roots. 195 ' 7. Draw a Crooked Line on the Left-hand of the Relblvend, In which place this Divifor laft fiwnd, v/^. 765* to which the wh(de Uefolvend (except th^ place <rf Unity; muft be a Dividend ; that is, ask how maity times 76$ you can have in 6019 1 the anfwer will be 7 times, which put in the Quotient: And this 7 is the Second f ipre of the Root; And then the Work will ftand thus, ♦ •' ' 185 193 125 (57 7^5) I ^ I 193 I Refolvend ■* 15 j Triple of the Root 5 7 15 J Triple of the Square of the Root 5 I 7 1 65 1 Divifor 8, Cube 7, the laft Figure in the Root Quotietft, and It is 343, which place under the Refolvend, in fuch order, that^Unites mky ftand under Unites ; fowill 343, the Cube now fbundlfbnd under 193 of the Refolvend : And the Work will ftand thus^ 185 125 193 C57 wm^mmmm' 755; I 60 \ 193 I Refolvend 15 Triple.of theRpot 5 I Triplp of the Square of tfie Root 5 I 7 I 5 I Divifor wmmi» I I 343 1 TheCubeof7,thelaft.Fig*i|iQuo. . 9- Multiply the Square of 7,,thelaft'.l1gtire oiPtSffeRW/tfallie- iy 495 by the Ttiple Root next under thfe R^fblVetid*, vi^. by 15^ jn<l the Ftoduft Will be 735, which pjace'uhder 343, (tteOito «ftfetdown) in fuch order, that the place of Tens int&rtt Arid' ^ will the Work ftand thus. 18s 196 AffentUx. 18$ 125 193 C57 7^5) I 60 } 193 I Rerolvend '•^ I 15 I Triple of the Root $ 7 I $ I Triple of the Square of the Root $ I 7 1^9 IDivifor II 343 I The Cube of7,the 6ft Figure of the Root 7 I 35 I The Square of7, by the Triple Square of 5 ■ Xo. Multiply the Triple Square of the Root 5» vt^. 7 $» by 7, the ieoood Figure of the Root, the Produft will be $25, which place under 735, iafuchotder, that the place of Unites in this; nuryfiand under the place of Tens in that: And then will the Work fiand dius, 185 125 IW (57 765) L 60 1 193 1 Refolvend ft 1$ t Triple of the Root $ ^ 7 I 5 I Tripleof the Square of the Root $ I 7|65 IDivifor 7 52 343 35 15 The Cube of 7, the laft Figure of the Root The Square oiF 7,by the Triple Square of 5 Triple Square of Root Si in the Root 7 Laftly, draw a Line under the three laft Numbers* and add tbem together in the fiune order as they ftand, and their Sum will be 601939 which may be called the Sabtiahend, becauleit is always to be Subftrafted from the Refolvend ; But in the fixamjple, they are equal one to the other and the Remainder is ocoooywhich ihews, that the given Number 185193 is a Compound Cube Number, and 1 that the Cube Root of it is $7. And fo the i^hole Work will ftand ashereyoufee» / •8s. Of ExtraStioH of Roots^ 197 185 125 I9E (57 The Root 785) I 60 I 193 I Rcfolvcnd 7 I 65 I DivKbr Triple of Root 5 Triple of the Square of Root \ ^m The Cube of 7, the \%tt Figure of the Root 3 $ t Square of 7, by the Triple Square of 5 Triple Square of Root 5, in Root 7 Subtrahend Remainder. "Now, If you compare this Example, with the former Genefif^ you fliail find that the flrft Number there 125, will be e^ual to the leareft Square lefsthan 185 : And the Second Number there^ 525 isthe fame with the Triple Squafe of the Root 5, multipRed in the Root 7 — And again, the third Number there 73 5, is the fame with the Square of the Root 7, multiplied in the Triple of Root 5 x And laftly, the Number 343 there, is the fame with the Cube of the Rooty. ^ ' Another SxamfJefor PraAice^ 492; . 9$ 64 ^ 256 256 15ft 263 (4567 Cube feooc* I I FirftRefolvend iia^Mlrt 12 8 rrjwi \ Triple of the Quotient 4 Triple Square oF the Quotient 4 •.«■* 3 24 <oS85) M7 «5 00 •q I ^?'l^s>l- 1 607 35 ^^i9H^yi I 3 6ch I 85 I 2 id t Divifor ^ Cube of laft Figure iuQuotient $ Sq. of 5., in the Trip. Sq. of 4 Triple Sq. of Root 4 in Root 5 Subtrahend .*— . I ■■ . ■ . - .... ■■■■.■■ ■ I ■ ■ »^i— 4— »fci— I Second Relblvend Tripleof the Quotient 45 TripleSquare ofthe Quotie ht d<' Divifor 1 TTIliTfl 48 <54'> 693 60 o ti6] mitm Cube ot 6, die ialt big. ia Quot4 Squ* of 6> by the Trip* Sq. 0F45 Trip* Sq.pt 4^^ m the lait Fig* 6 I Subtrahend I 437 1 33^ I 263 I Third ICelbl vend .jnyhMfcrMfc r ri-n*«> rM m 643 9.^4.1) ^ 9-8 Appsniix. ^z 39 \\%) I Ur I 3^^ I 26^ I Third Refolygnd. U^6 62 |- I 62 13 380 68 8 Triple ot the Quotient 456 Triple Square ot the Quotient 994 I 48 I Divifor 436 670 66^ 343 32 6 • I I 437 I 33<^ I I oco I 000 263 000 cube of 7, the laft Fig. in Qiiot. Sq. of 7, by^'the Trip. Sq, of 456« Tri.Sq.of 456,in laftFig.inQpcy Subtrahend Remainder. Note, when at any time in your Extraftion, you finda Subtra- hrndTO be greater than die Refoivend next before (from whence it is, always to be Subftrafted) your work is erroneous; addmuft be rectified by putting a lefs Figure in the Quotient. In the two foregoing Examples, both the Numbers propofed proved to be exaft Cubical Numbers: Wherefore, I will here add. another Example (ready wrought) of a Number not Cubical, and ihew how to find the Fraction part of the Root in Decimal Parts, to the 10, 100, 1000, tfr. partsx)f a Unite, or farther if tbeniceity of the work do fo require. Let therefore the Cube Root^ of this Number 23456789, be re- quired to be found. ' V The Number being written down. Pointed, and the Cube Root thereof Extrafted,7as is done in* the following C^i^tion: The Cube Root will be found to be 286, and then there will be a Re- mainder of 63133, now to find the Root nearer; add to the Num- ber given, 3,6,.9or 12 Cyphers^ whereby to attain more Dedmal Parts in the Root : — So in tiiis $^ample I have added twice three Cyphers; with which, I go on with the Extraftion till I have two Decimal Parts in the Root : For the Root is 286 Integers,.and by the adding of the fix Cyphers, it hath two Decimal Parts in the Root, namely .25 which is too little: For that Work being ended, you will find a Remainder of 173^359375, to which, if you add three Cyphers more, there will be 3 Places of" Decimal Parts in the Root5 but this gives the Root to -^ part of a Unite and if more Triples of Cyphers were added, it would give the Root to the ^.•iT, or ,vr?T Part of a Unite, eS'r. -the Of Extraciion of Roots. .lie P E K^A 1 1 N". • • • • ■ • • 23 8 456 789 COO 000 (286.25 126J Vis I J.SO I 1^ I I 2d ( j R_Muivciiu I Divifor 23604) I I I ')04 I 7^$^ 1 I t Ssmrrahepd 84 I I 2^6 I 04 I .j 216 Ktlbivcn4 * - - - ! Divifor ^»" • "• — •*"5"»"f"ii5f""»" I I I 441 I 6<; 6 I [ I" Subtrahen d 2454738) I - I 63 f 1:^3 I oco 1 I Reiblvend T 5^8 I 8 S8 I- 1^24 1. 547 j >» 1 i Divilor r 2457599^^) I Z19 ( 1 1 1 f 9^8 L 1 Subifahen d I4'| 621-) C72 I o03 i. R-^^^lv^;"pd I 1 2 I 457] 399i'Q^ I ^ivi^Pr , 12 1 2 2S6; 146' 566 125 I 1 ■ I I 732] 359,1 375 Tcco, oi. 199 Cca ro 200 AffendiX. To find the Cube ^oot of 4 Vulgar FraSion^ or Mixt iTumber. Let the Fraftion or Mixt Number be firft Reduced into its \ l^eaft Terms; And then, this is the j RULE. The Cuhe Koots of the Numerators and Denominators of Vulgar Fray Bions or Mixt Numbers \ Jhail be the Numerators and I>enominators cf $be Cube J^ots of thefe Fr anions or Mixt Numbers. So, this Proper FraftionyJ^ neither the Numerator nor Deno- minator are Cube Numbers ; but Reduced to its leaft Terms it be* comes -5^, and then, the Cube Root of 8 is 2, and the Cube Root of 27 is 3, fo that the Cube Root of the Proper, Fraftion, is |. Alfo, the Cube Root of this Mixt Number 2o J-l^, will be found to be ' ^- or 2 ^-. For the Mixt Number 20 J J or Improper Fraftion ''ii- being in its Leaft Terms; the Cube Root of 1331, is 11 for the Numerator of the Root, and the Cube Root of 64 is 4, for thepenoipin^torof the Cpbe Root* 1 Note, When a Vulgar Fraftioij or Mixt Number jslncomme9» furableto.its Root; fuchare ufuallydiusexpreffed, vi^. The Cuhe Root of I thus Vc f ; and the Cube Root of 2 1 thus Vc 2 -J-, tff. •i^m,^mmm^<^rmmmmmm^mim^^ SECT. IL / Of JmereFty Simpk atidCompound ; Difcount or jRf- bate of Money ^ and of Mqu^tion of VaymenUy &c. With Tables of d of them. nV >f ^^^y P^^ ^"^ ^^ ^^^> ^ divided into Three Parts, vi^. \\f\ Principal, Tin^c and Intereft ; The Firft fignifies the Sum, X y JL or Value, of the Money, or Goods fo Lent — Time is the torbearaacc of it ; as Years, Months, Weeks or Days — Intereft, is the Profit that arifeth from the other Two, life or Intereft, is either Simple or Compound ^ — Simple Inte- reft is Computed from the Principal and Time only , upon a Certain Rate agreed upoo. But — Compound Intereft (after the Firft Year, or other Time Limited fo? the $:ft Payment) attrafts a Proporti- onal Of Extraction of Roots. 201 onal Ufe Smpofed upon lhe6/, or other Rate due at the firft Years End, f if continued longer) And therefore is cafled Compound orlntereftuponlntereft. ^ ' > ' ■■ I. Of Simple l/fterefff. pOr the Arithmetical Working of Queftions whereof this is the PJ(^0P0J(^TI0N. ^ As the Principal and Time, for which a Loan Is alIow*d, Js to the Intereft thereof ; So will any other Sum of Money to be Borrowed, Be to the Intereft for the fame Time, ^s^mplel. What is the Intereft of 145/. for a Year, zt'6hper CenuibvzYczr. Proport.'] As loo 7. : is to 6 /. : : So is 14$ /. : to 8.70 /. RUL^ Mufmly 14$ h ty6l. tie ProduS wiU be 870, wUeb divided by lobh' (by cutting off two Figures') the Quotient will be 8.70; wUci l(educed is 8 1. 14 s. for the Imereh of 145 1, for a Tear. Example II. What is the Intereft of 250/. for 5 Months, at 8/. PQenhptrAn. Vrrihart S ^^ ^^^ '• ^^ ^^ MOUtilS : Is tO 8 /. irapa/7t2Sois25o/,in5Months;To8/.6x.8i. RULE. , Multiply IQO 1. by 12 The ProduB will be 1200 for a Divifor : Then Multiply 250 1. by 8 1. the ProduS will be 2oco, and that Multiplied by ^ ^ Months :,producetb\oQQOi for a Dividend: Then \oooo divided bf 1200 {adding Cyphers') giveth in the Quotient 8.3 33 1. ; Which 8.333 Reduced, is 8 /. 6 x. SV. for the Intereft of 250/. in 5 Months, at 8 hper Cent. Example III. One lent 650/. which was re-paid again at the esd 'of 6 Months, 3 Weeks and 3 Days ; What came the Intereft thereof to dit 6 per Cent. Proport, \ \ 502 Appendix. ^^^ C As too /• in 365 Days: Isto6/. rropprt.^^'^ 650/. in 192 Days : To 20/. 10 x. j i 3 }. fere. RULE. ^ • Multiply ^6$ky *oo» ^ Prcduit wiU be 36^00 for d Dtvifor ; Ah fif Multiply 6^0 by 6, andtbe ProduB of that by 192, tbelaS Produ3 mU be 74.8800 ; For it Dividend : Wbicb Divided by 36500, mUgivc ^ in tie Quotient 20.5 1 $• j^nd Anfwers the Queflion^ For the Quotient 20.515 Reduced, is 20/. 10 J. 3d. ^q. andfo much dodi thelntereft of 650 /. amount unto in 6 Months 3 Weeks and 3 Days : (or 192 Days) at 6 Lper Cent, per An^ Sxmple IV. What will the Simple Intereft of 26 5 /. 1 3 f . 4 i. i f amount unto in a Year zt &l.pier Cent. per An.- - ^^^ 5 AsAoo /. Is to 6 /. in a Year, -'^^'^^ So 18 265 /. 13 s. 4i I J. To 15 /. 18 1. 9i. 2 J. in a Year. RULE. induce tbe.26^h i^s.^d. i qJnto a Deantalf and itmU be 26^.667^ JUf multiplied by 6 1, (tbs J(ate of Inter eB) producetb i$94, whieb di- vided by 100 (by cutting off two Figures) the Quotient VfiU btf 1 5*94 /• Which Reduced is 15 f. 18 J..9 </. 2 j. for the Intereft of 265 /. i3f.4(f. I j.foraYear. ^ I^OBLEM. How to find the Intereft due upon any Sum of Money, for any Number of Days, and at any Rate of Intereft: For ^le Woricing of Queftions of this Nature, This is a / t GENERAL RULE/ . Multiply tbe Principal Sum by the J{ate of Interest ; and that PrgduS Iff the Number of Days propofed: tfotsla^ ProduB Divide (ah^ys^ by J6500, (the Number of D^ys in one Teofy voitbtvoo Cyphers added) tbe ' Quotient TviU^fiver tbe Queftion demanded. Example I, What will the Intereft of 750/. amount unto, in 232 Days, at 6 /. per Cent, per An. ? . Multiply 752 (the Principal Sum) by 61. tlie Rate of Intereft; tbeTrodod will be 4512; which Miilciplied by 232 (tbe Number, of Day^; produceth 1046784; Which Divided by 36500 (adding Cyphers Of Inter eHy Simple and Cemfound. 20} Cy^er? if need be) the Quotieat will be 28.677, which Decimji FraSion Reduced, is 28 /• i j i. 7 i* And fo much will the Intereft of 752 /.amount unto in 232 Days. . Sample II. What wiQ the Intereft of 8/. 10 s. amount unto, ia 120 Days, at 9 1, j^ Ccm.fcr An. The Principal Sum 1 . ■ 8 a$ The Rate of Intereft 1 9 ^* TheProduft is« 1 " . ? 1 ^ , n 724*$ Which Multiplied by the Number of Days, —.——.» 120 The Product thereof is 1 ■■ ■ 869 40«q Divifir Dividend (hmicm 36;oo} 86940.000 02.382 Which Quotient Reduced, is 2/. js. id. for the Jntfitdft^. to/. 10^. in laoDays. fliMfiftfk Pound s 204 Simfie IntereH dt VI per Ctnu For Pou I « 1 Month • > _ 2 Months. 3 Months • > /. J-. d. i- I. $• d* q. /. J* d. 9- I I D 6 2 2 3 t 2 2 2 043 7 1 3 i- 2 071 lo 3\ 4 4 3 09 2 • I 2 1 ■• 5 6 tf I 00 p I 6 7 1 12 2 I 9 2 7 8 2 » 4 3 2 I I 8 9 2 J I 7 I 2 4 3 9 .10 3 192 2 8 2 lo I 0* ■ 1 ■ * 1 200 3 .11 W' I I 222 3 3 2 12 2 2- 243 3 7 t n 3 2 2 7 X 3 Id 3 M 4 5^ 292 4 2 2 15 Id d 300 4 6 , 7 I 322 , 4 9 2 I? 8 2 3 4 3 5 I I iB 9 2 ^ 7 I b 5 4 3 19 10 3 3 f 2 460 5 8 2 20 30 2 b d 3 600 *9 40 .0 4 800 12 50 5 10 1$ 60 6 12 18 70 80 7 14 p I I 8 16 I 4 90 9 18 I 7 loo 10 fi I 000 I 10 200 I • 2 0,00 3 900 I lo 3 000 4 10 p • 1 400 2. fO 4 000 6 500 2 lo 6 0007 lo doo. r 1 000 9 700 ^ 10 8 000 10 lo fico 4_ 9 000 12 0, Pou I 3 4 5 7 8 9\ lo II 12 I? H I(^ 17 i8 19 2o 30 40 50 60 70 80 loo 200 4 in^aUi!>« /. o ,0 o o o o- o o o o o o o o o o o o o \ o o I I I I I 2 4 300 I 6 400 500 600 700 $ o o I I 2 2 2 3 4 4 4 5 5 6 1 1 8 12 16 o o L j« 4 9 2 7 o 4 9 2 7 o 4 9 2 7 o 4 9 2. 7 o 12 o 16 o o o 4 o 8 o o o o o o 3 2 2 1 o 5 mouchs. /. (L q. 3 2 2 I o o o o o o I I 2 2 6 o 6 o 3 2 2 1 o 3 2 2 I o o o o o o o o o o o o o o o o o o o o o o I I I t I o o o o o 8000 10' o o o 12 Q 00 14 o o o. o .0 .^jg -2 2 2 $ 7 10 12 t5 17 JmL 3 3 4 4 5 o o 6 o 5 1 7. (5 o 6 o 8 8 9 9 o o lo o 10 o 5 10 15 6 o o o o o ■ o 5 o 10 o o o 10' o o o 10 o O 10 o i3 a. o o o. o o o o o o o u o o o o 6 moatbs. Sb L f; 0^ I X 2 3 1 a 4 o o o o o o o o o o o o o o o o o p o o 6 7 7 8 9 7 a 9 4 o o o o o o o o o o o o o o o SL Pd «MMnMMi^ o o o o o 12 O o I X I 2 18 4 10 \6 2 o o a o 2 2 3 6 8. o 14 o o o o o o o 12 18 21 o o o 6 6 a o iSniTiiS X a 1 o 4 12 49a 5 4 3 600 ■■ I ■ I* V X a a Mi o I 9 7 X 10 a 2 10 9 2 11 4 3 o -4 o o o d o o o o o d o c o o Mi I' 2o6 Simfk ^merelf 4^ VLper Ceat,Jpr y i fl J months. «««M ~ 8 mbiitii^. ' "9 months. • POU 1 /. s. d. «• U u d. «• > • -^ • U s. d. ±' <S ^ ^ 2 Q 9 2 3 3. I 4 3 0. 1 7 I I 9 2r 3 2 I I 0, 2 4 3 1.8 2 4 2 9 .2 032 2 037 I 6 .3 ^ 0.40 046 / 4 2 2 4 ,9 2 5 4 3 1 4 to. 3 057 I 063 2 8 5 7- I ^4 3. 0.72 2 \ 9 ^ 3 2 07a 2 081 I • lo 7 ■ < 090 II j 7 8 2 08 9 2 9 lo 3 12 ;o 8 4 3 9 1 I 10 9 2 »3 9 » I 10 4 3 11 8 2 14 9 9 2 11 2 2 «o 12 7 13 6 .1 15 i6 10 ,^ 12 II 2 2 12 9 » 14 4 3 17 II 10 3 13 7 I IS 3 2. i8 12 7 .1 14 4 3 16 2 2 *9 ^3 3 2 15 2 2 0-17 I . I 20 30 14 a Q i5 . , 18 I I 1 40 I 7 40 I 8 I 12 O: I 16 50 I 15 200 2 5 60 2 2 280 2 14 70 80 2 ,. 9 2 16 3 3 2 i5 340 3 . 12 f>o 31 3 3 12 ,0 4 I .0 '0. loo 3 lo 400 4 10 P 2CO 7 8 9 300 400 10 10 12 19 10 '4 16 0. 18 Q V $00 17 lo 20 22 10 ^00 ^i 24 27 700 24 lo 28 31 10 800 28 1^2 ^5 ^0 0, Sifi^k JntertU tt Vl< per Cent.yw' % Pdtt io months, j 1 1 months. A Year. f /. s. i. q. )/• $• ' L q* h ^^. d. q. ■ I I I I I I 22 2 1 O 2 old 2. 2^2. 0243 3 3 0332 *o 3 7 I f 4 4 0443 0492 6 5 560 0,6 6 6 .0 6 7 J P 7 22 7 7 07 « 2 0843 ' 8 8 8 p 2 .0' 9 .7'. r 9 9 9 10) t i> 10 9 2 lo n 1 'P II 12 ; II II 12 I I • 0. 13 2 2 I 12 )2 13 2 2 14 4 3 1 »3 P J* H i : i 0' 15 7 I 1 I H H o' 15 4 3 itf 9 2 ^ •' M 16 18 ^ C 16 17 7 I . 19 2 2 17 17 lo 18 8 2 1043 V '8 18 19 9 2|i i'7 If ip '9 X , 10 3 I 2 9 21 * 20 30 \\ X 2 I 4 1 I Io I 13 X 16 1 40 2 2 4 00 2 8 0. k 50 2 10 Q 9 2 15 p 3000 60 3 e 3 6 o* 3 12 ' 70 3 lo 3 *7 4 40. 8o' 4 1 4800 4 .16 ti ' 90 4 10 4 19 5806 lOO 5 5 Io 6900 200 10 II 12 900 M 16 10 p 18 400 2d 22 24 500 2$ 27 10 : 30 00 600 ?b 33 00 36 • 700 35 38 10 42 ^ 800 40 44 48 Q .0 1 0^ Aff^ndix. \ Is Xi. Of Difcounty or Rekate of M?m^^ TJ'Or t^cDifcoittit, or Hctate of Afoney j Tl^is }s ^ P ^^qFO KTiO N. . ^ , As looA and telateittftata Yeaneo^, Istoxoo/. Soisiod/. Tp the Prdeat Wortli of too h . $9p2^e ^t VI. per Cent* A$to6hUtoiooh :Soiitoo/.tD94^4li Dh^tft phUeni ^gnknt — f. di j. \d6i) ro6.co6oo (94*339$ Or, 18,10 2 So that 100 /• to be paid fl|t a Years eiid^ is wort^ in preljbnt M{>- niy 941.^ ^.9i<.2j. ' (And this is the difierence between Intereft and Rebate : For yon ' are not to Rebate 6 ^ out of 100 7, to be paid at the Years end, but %h\is.%i^2i. which is lefs than 6l.by6j.9i^2(. Thus fora Xear. But if you would know the Rebate of any other Sum, a|t any 0- ther Rate, and for any other Time, More or L^fs than a Year \ .Thenufetftis; • ' . ' P HO P oi^T JO i^. As xoo h with the Jntereft thereof for any Time required, /. , ^ tstoioo/. ^ IS the Debt due to be Paid at the Expiration o{ that Time ' To the prefent Worth thereof. ' Sxgmpk. .If a liegacy, or other ^m of Money, as ^4$ ;. hecoi^e due to be Paidat fix Months end ; What Sum of Money willdifcbiarie !-• ' profert^'} Asioj/. i5toioo/.Sois}45fcto J,34*95i/« \ JDlv^tfT phfideni Qv^iept - ^ \ •'loj)- 345oo;ooo f 334-95^ )Vhich Quotient 334*9^1 reduced, is 334/. i9x.orf. } f. ^ndtb ninch oog^t to-be^paid tor the ^45/. abating forthefald^MootlfS ib L 6 X. 1 1 i« 2 q. whereas the Intere^l thd^reof ibr the fapf TimPy is to /. 7 X. the Difference being 6 x. 2 f • And according to this Pr^ ^rtion, is the fonol^ifgTa^Ie of pIfcottntorKebate, tSsdculated': lit'. , r jyifhttt^fifr Rddttf if VT. per CenL/dr 209 (t Poa I 9 i 4 5- 6 7 8 9 10' II 12 13 14 «5 16 I month. /• ' s. rf*. f. o I 2 4' 19 10 9 8 7 6 5 6 7 8 9 19 19 19 4 3 2 10 II. 12 18 19 18 18 i!i II 9 8 7 17 I i^ i8^ 17 18 19 19 20 18 iR i^ 18^ 18 i 1 29 17 4?' ' 15 30 40 50 £0 80 ,90 loo 200 jpo 400 [ 398.0:" ?! 600 ;597 o: 3; •8«4:796 ( I '1 7/- 12"^ 89 n 99 'to./ I99<0; i 29810 I 3 3 2 2 I 2 months. /• - X« - ^ ^ft ]• a 2 3' 19 9' 19 -7 o|4 19 19 19 4 2 o 2 1 3 2 o i 3 2 1 o 9 5 7 8 18 18 18 18 18 9 1 5 2 o 3 I o 2 1 o 3 2 I o o i 3 • 2 I o 10 II 12 ' 13 14 17 9 17 7- 17 5- 17 *' 17 o- 3 2 a 3 2 17 18 19 1(5 Id i<5 i5 ko o 3 2 o 2 1 I 1 I I 29 14 f39 li 49 10 -J9'''8*^i ^ J 3 months* i. q. 4- I I 2 3 4 19 I9» 19 18 18 |8. 1 9 6 2 o I 5 5 7 8 9 18 17 17 17 17 2 3 It ^ 7 a 4 ^ o < 2 T 3 I 29 o I 2 3 394 I 492 I? 591:2^ 589i|;,i| r ."•v. *«»^< > .i;^* v'»»\. sto D^ouHty or R^ste, 4f VL per Ccat,fcr r 50 J 49 y I tfo 1 5? 1 ^70 6a. 8 >* 4 i 0,4. 16 < ~ 12, o - 3 48. IS 7 , 3 3 t8o. 7»:.8 7 700 800 2 a 58 10 9 6B $ 10 II I 1 3 I 3 ^3P2i3: 5884' M, 6>moactts. X. i. f. »> 5 18 II 18 i >7 8 17 » 29 2 38 *I^ 48 10 78 , 87 16 ^bm 3 I 91 ,," 2 1 apa^^S 9 I 3 I Q 9 «15«5 7 3 M92;l8 I 1780^ ^^^mmmmmiBfim 10. I 3 6 P 2 I 3 97, 1 »^,3 a5^» 5 4 (^ 9 3 o a o o o I i5 6 o 15 II o 1$ 4 o 14 9 14 2 O 10 13 7. II 13 12 13 5 1 »3 II 10 I »4 u 3 I »$" * 10 8 I itf 10 I I »7 9 6 I 18 : 8 U I 19 8 4 I I 3887 48? 8 582 10 ^7? »2 77^ It 5 3 II <$ 1 % 3 10 3 o a * 3 3 3 o o o 3 3 3 . /. DiJitmMyOrRebate^dpVl.^CtDLfi^ an , 7 n^onths* 8 months. 9 months.. . r > Pou 1 /. *.; <(• f. /• 5. if. ]• I. J. - d. q. 1 19 4 19, 2 '3 17- < 3 2 1 18 7 3 I 18 $ a I 18 " 3 I 1 3 2 17 11 3 2 17 8 r 2 17 5 ■ 4 3 17 3 2 3 16 II 3 Id d 3 5 4 16 $ 2 4 Id I 3 4 1$ 8 I ■ r 6 5 15 " I 5 15 4 .2 5* 14 10 ^ 1 5 15 3 1 d 14 7 2 d 13 JI 1 13 11 H 7 14 7 7 13 10 I 7 9 8 13 11 8 13 I 8 12 3 to 9 13 2 3 9 *2 3 3 9 II 4 3 II 10 12 6 3 10 n 6 2^ 10 10 (S > ' 12 II It 10 2 II 10 9 I II 9 ^ ^0 » ^ 12 11 ^2 12 10 12 8 ^3 H 13 10 6 I .13 9 2 3 I? 7 iJ I M 14 9, lO I 14 8 5 .H 7 •■ il ■' 1 i6 15 9 2 1 If 7 8 I 11 ^ —2; • --} ' 17 16 8 6 16 ' 6 11 Id 541 ] i8 17 7 10 17 d I 3 17 4 6' 19 18 7 I 3 18 '5 4 2 18 ' 3 :^7 3 20 19 .6 ;$ 3 19 4 '7 2 19 2 9 1 ', 50 28 19 8 2 28 Id 11 28- 14- -2, « 401 38 12 11 1 38 9 2 3 38 5 <? 2 1 50 48 621 48 I ^6 2 47 d ;ii 1 6o $7 19 7 57 ij 10 1 57 8 4; 70 8o 67 12 8 67 d I 3 dd 19 ' 8; • 2 77 5 10 3 7d 18 '5 2 76 rf -i: - ^ ^ 86 19 I 2 8d 10 9 t 96 « ''5< -3 ' * 100 9d 12 4 2 96 3 I P ^95" 13' la 2 t 200 1934 ^ 3 192 6 13 191 7 ^9. 30O 289 17 1 1 286' 9 2 3 287 I''>'7. ' 2 400 38^ 9' 5 ^ 384 12 3 3 382 1$ tf, al 500 483 1 10 480 15 4 2 -'47« 9' 4(-2|l 600 579 14 2 2 576 10 5 ' 2 574 J ;3; o| 700 675 6 6 '3 d73 1^2 d($9 ■■i7'--«; ■ all . 800 772 if 11 1 ^169 4 t 2 J£l 'If ^a'-o|] Difcomt, orRithatef ^^fVI.perCent./tfr fl ' 10 mondis. 11 months. > A year. j . I 3 5 4 5 7 8 9 II 13 13 >4 1$ itf »7 l8 i9 40 60 70 90 100 200 300 400 500 ^CO 700 800 1 L s*. d. «• /• f • ~ if. q. !• s, (L f .19 1 18 I 2 17 « 3 i5 2 4 15 2 2 I 3 I 3 Q. 18 II 2 I 17 II 2 Id 10 2 3 15 10 4 14 9 2 18 10 2 » 17 « 3 2 id 7 I 3 15 5 3 4 14 4 5 14 3 5 13 4 7 " 4 8 It 5 9 10 $ 2 2 I 9 5 13 9 6 12 8 2 7 II 8 .8 10 7 2 9970 5 19 2 2 d 12 I 7 10 11 1 8993 .9881 10 9 ^ 11 8 5 12 7 7 13 5 8 14 J 8 I 3 2 2 10 8 5 I ." 7 5 3 12 d 5 I 13 5 4 3 14 4 4 I 10 ^7 d 2 If d 50 U 5 3 2 13 4 I 3 14 3 I J5 4 9 Id 3 9 17 2 10 18 I 10 W II I 3 I »5 3 3 3 Id 2 3 I 17 I 2 3 18 2 I 18 19 I 3 15 I 10 3 id 9 id 19 7 2 17 18 d 18 17 4 I 28 II $ 38^-1 10 47 12 4 57 2 10 6d 19 ' 4 I 3 2 I 28 8 8 3 37 18 3 2 47 7 10 2 55 17 .5 I dd 7 I 28 d 0,2 37 14 8 2 47 3 4 3 50 12^ 1 66 9 " 76 3 . 9 85 14 3 95 4 9 190 9 tf 285 H 3 3 2 I I 2 76 16 7 85 d 20 94 15 8 3 189 11 5 3 2844 2 2 75 9 5 1 84 18 1 I 94 <^ 9 2 1 18813 7 283 P 4 2 380 19 47<^. 3-9! 57* 8 ^ ddd 13 4 751 ir* I 2 3 3 1 .379 2 It I 473 18 8 $58 14 f dd3. 10 t 3. . 758^ t !•- 2- 377 7 2 471 13 II 2 5dd 090 ddo 7 d 2 in. o/, i '■ ■ ■ i ~ jEquatioffs of Paymeffts. 21 J * III. Of jEquAtion of Faj/ments. TT^ Qpations of Payments^ are of Two kinds; vii* 1. Several Equal PSayments, due at Equidiftaot Times. 2. Of fevcral Unequal Payments, at fereral times not Equi* diftant. F6r the Firft of thefe : [[ ^Equation of Equal Payments at Equi- diflant times.; This is tb^ General * R U L E; From the whole Account of the Annuity, or, Monthly Payment, SabftraA tiie Aggregate of th^ feveral Payments ; and the Re- ' maiflderfif yeari[f> multiply by 365. Or(if Monthly) bj 30.416 I>ay8. Tfaeiu Divide that Produd by the Annual or Monthly In- tereft of the mA Aggr^ate, and the 'Quotient will give the Num- ber of Days before the £nd<or Term) of the Amiual ^r Monthly \ ftyment; Example I. IfA,betoPayunto B loohpcrJn. for Five Years; and they agree, that A (hall Pay to B, the whole 500 / at one intire hyment, at 6 per Cem : TheQueftion is.. What time before the Eipirationof die whole Fiveifeari, Amuft Pay B the 500 /i According to the R U L E, »• • Prom the whqleAccount of the AQpaity in the Five Years. $60 Subftraft the Aggregate 500 ; : v. . . • ■ ■ '■ ■■' Tht flemainder is 1 , , j 60.. Which multiplied by ^ : — ^ 36$ The Produft will be > ■ ■ a 1^00. That pi vided by 30 f the Five Years Intereft) the Quot. is— 7 30 So that the 500 /. mtfft be Paid 7^ days before the etpirajti>jj| pf . ^e Five Yeacs ; Divide then- 730 fay 365 days, tlie Quotient will 05^2. Years ; 5a that A muftPay^inco^ the $oo/.atthee^Qi . Three Years, S e iSx- r ^ 1 1 14,. J^^endix, ' Example II. A is to Pay unto B, 62 Lper Jtinum for Four Years, ^ but they mutually agree, that Afhall Pay the Aggregate of the 1 whole Sum, w>,248 atpne\intirePaym^^t,at6/.pfrCCTft, ! How many days-before the expiration of the v^hdle Four Years, mnft tliis Payment be made? . ; Look in tlie following Table of 6 /7frC«n. for the Account of . ' icG /. in 4 Years, which you will fimito be 436 : This multip4icd by 62 (the. Annual .Payment).the'Produft will be '27092 : -And • that Divided bj 190, the Quotient, will be 270,92 /. few: the Aecqunt of 6«/. in 4 Years': —Being thus prepared 5 Proceed according to the R U L E ; as followeth ' "'•1. - -' ' - ' - . ''*.'' ' ' '. I. The Annual l?ayment for 4 Years, is" -'• ■ » -1- ; 6 2.0 o ^ 2. The Account of that in 4 Years, is ^« 2 7 0,3 2 3 . The .Aggregate of 4 Years l^ytficnt, is 2 4 8.0 o ' 4. The Annuallntereft of the Aggregate, is i 4.8 8 0;it^f th^ whole Account -1 — -^ r. ■ ; a 7 0.3 2 Subftr^ the Aj^regatc, -«~ — ». 2 4 8.0 o i I ■ The Remainder is — ■ j \ ■ • - 2 2.3.2 Which multiplied by 365 days ■ — — — - -. 3^5 - The Produa is \ . g i 4 6.8 o Which 'Divided by the Annual Intereft 1 4.8 8 The Qpotient is u ' . .y ■ . Z-^^ ^ 5 4 7. 5 And fo many Dayes before the Expiration of the Four Years,muft tlie Suni of 248 h be Paid, at One intire Payment. For Proof hereof I fayp. If 248 /. be put out at Intereft at 6 /. perCenuiov One Year, and 1 72.5 Days : J fay. If the Intereft thereof in that time do amount urtto 270.32 /• the Work is Proved ' Thus ; The Intereft of 1 /. for i day, is ■ .00 0164383 Which multiplcd by — — — — — 5 47.5 iM-rihi The R:'§dua is ■ ■ ■ ' ■ ■ ,© S 9.9 996925 .Which multiplied by ,. — - 248 ■ ■ ' ' ■ . '■ - j ^ The Produft is -^- — — 22 .31999977680 Towhkhadd— — . 248. ThcSumi s - '■ I 11 fi . 270t3 1999 977680 Proving r- j£quati(ws of payments. 21*5 Proving the Work to be Exaft. ' / ' ' . Hitherto of Yearly Payments r Now for Monthly Payments. Etmph in. ';s^ 19 to Pay unto B^ioS'l. a Month, for Five Moiaths : At what time mulV he pay it ad one Intire Payment, at 6UferCtnt? > ... / /". ' .; [.The Intereft ofri J... foroAe Momh; is; ■ V r -' ' .(^.Q^^ Vhichi 500 /. Vi^ 2. The full Amount of 5 Monthly Payments of 100//. is 500 The Double Of the McftfithlyTntereft' *is-^^— — . 5.0 00 ^hichmUltipUddby.ijoo, is the MOfithly' IntefeA of? . 1 I ■ — ^1— iJm The Whole Amouoti .L.. i ,j !... i , ■■ ."!>' — r 5PjS ' 3. From the Whole Amount 505, Dedua the Agreg. —500 The Remaindcxis . . u * n. . . / , — .0 o < 4.' Multiply gb.4id (the- Days jii'acb mpleat Month );by ^v thi Produft wm be i^a.isSc-pwrhich Divideif by 2,50b (che Monthly Intereft of <oo /L) i^be (jmgtipnt .will be;cc.^"^2 d^ys^; thatjs ^o dj^ys, and about. 20- hQiirsji w-hlcli Ts tw,o oorupJeat MonUis'befqre'the ^ Months: Qt^ j Months after the da/of ' Ag;reemcnt. " ' » • - Of the Mciuation of Vaequal Payments^ at Times THjs manner. oi-Work- will JielLJii:.jaadi?rft<^>y^^^ ' Therefore, * . . Example I. O^e, .Qjve^ $jO0 /* vi[hic]^ he^isto Pay, at three foye- ral Unequal Payments ; F%^.' At the end of four MGitttl^s3.oo /.'at the end ofSix Months 100/. And at thpendof 12 Months 109/, Bat the Dqbter agrees with the Credftor-to Dlfcharge the Dsbt \yih 500 /. ) at one intire Payment. The Qucftion is ; At wliat time 500 /, niay be Pai4, )^ithoi*jJ[>amma[ge or Pcejudicp to either of th^^ two Parties'? " ^ ' ';. ' ...''' £ ea , > 1 II 6 Affendix. • • • k to K^ohiC Queflionspf thk Nature ; Tifs it the General Pirftyfind the true amount; of each of the Sums^ from the¥irft day of the Agreejnentito the Laft day of Payment ; as fpppafing them to be forborii to the Laft. Then but of that, Dedud the Aggr^te of die Refpeflive Payments, and multiply the Remainder (if An- nual) by 365 Days, (if Moiithly)by 30.41^ ; and theP^uft, Divide by the Annual or Monthly I&tereft of chefaid Aggr^afte ; And the Quotient gives the Number of Days, from the £a1fl:.day of Payment} Accounting backwards. 7U mamer.rftbe JV^k if aa faSowffh. ; , Tirft, The Length of Time from the Day of Agreement, to the laftDay-of the Payment, isjuft t2 Months" '" ; . ' ' ij So Then, 1* 300 J.Biyable alter 4 Months, an5T)eTngforb6rn7 i to the end of la Monljbs, hath 8 Months Interefl^ to^ ^,^.o o o Account fof, vif ' . ' . .-.' ^ m >mm • •» 4 t ■'y • •» t 1*^ « J . II. iboJ-Papble aftpr 6 Months, arid being for-' born to the end of 1 2 Mopths,^ bath 6 MOnthsIntereft ^ j.o o b to Account for, vi^. ' ^ - -— — n.'^'"-''/- -- in. Tothefe Sums add the-— r— ~ $ ocd o o iThe Whole amount is ■ — -^ < i < .0 o © IV. From the Whole amount 515/. SubftraJa die? Aggregate of the Sums ^ J 5 o 0.0 o The Remainder is ^-^^ — 7 ' ■■ ^ r/>i\ j 5!oJ) o Being thus far prepared: The Proportion is. As 2.5 A ftiie Intercft of 500 i. for i Month) IstoiMonth; *' ' \ So is 1 5* (the Remainder) ' . To 6 Months. ' As 2.$ /: is to I Mon : : So is i$ /: to 6 Months : And that Refolves the Queftion 5 So that if the whole 500 /.'be Paid at ^ Months end, there will be no Lofi or Dammage to either Debtor or Creditor. ' ^ tj Fof JEjquations of Bsj/ments. 217 FoTiAe Prwf if tUtf I. 300 /. was Due at 4 Months end, and being con-"? /. tinued 2 Months ^nger, the Ip^tereft.for 2 Months isS 3 3:0 } /.and the Whole amount i s ■ ■ j II. ioo}« Paid at 6 Months end, which is thetime7 it was Due, is thereforeju ft . f 1 o o,© o HI. Theother 100/. Paid at 6 Months before the") time th^re muft be an Al»tement.made of 3 /• and isj 9 7»o o The Total 5 o co a « Sothat as thei:^ isin the PirflSuman Increa&qf ^ /. atidin the laft a Decres^fe pf 3/, which ;are to be fetone agalnft theochert and the Whole aj^iou^t is the Aggr^s^tieof the Refjpedive Sums ; and being Paid at the end of 6 Months, makes the ^uation juft $00/. . . Example II, jiOwes tp iB 1 00 /. fisr j^num for Kve Years, andE they agree, that A fhall Pay it oiFit the end of any of the Four Years ; for at the end of Five Years nothing Left than the whole 560/. will Pay the Debt. * ' ;. . Thus the Prefent )Second/Years\474.S7to Worth of the whole ^I^*!i>^'* is-^ $00.0000 Amount 56P /.at the /F?^^\ /S^S-joiS ThefeNumbers are found, by help of the Number in theThifll and Fourth Columnsofa Table in Cvrfm Matbetmicm, Pag. 116.- And the Proportion forfii)4iijgthcmis As 106 /. (the Amduiit 6f the Annual Payment) Is te 94-3^-9^ (the ptefent Worth C)f the Firft year) So is 560 /. (the Whole Amount in Five Years; To 528. joi^^ (the prefent worth at the Fourth Year's end; Jni fi (f the reS* /. 94.33962*^ ^528.301$ .183.92856/^80 is J 500.0000 Tor, As«^3i8^Is to<26949i52^56o/.<474.5762 '351.61290C to JA$u6i2St ,430.7692 3 J 0430.7690 IV. Other IV. Other Ruksfxnr Mquations of Far/menps: 1 Shall here inrert fome other Rules for the Aqiations of Pay- ments;' which (although not foexaa,)are performed with more eafe : And aldioBgh m great Sums 'and long times df Payment, there may be fome fmall Difference : Yet finall Sums and fliort times being of more frequent Ufe, and the Difference buc fmall j Ifliall bere ibewthemanner of Working diem this other way. ^ 1. When the Term &f Pajmem are efuU Example 1. J Owes to B 400 /« tq be Paid at Four Six Months , that is, 100 L at 6 Months, 1 00 ?• at 12 Months; lool. at iZ Months ; and loo /. at 24. Months : But it is agreed to Pay the whole Mony at One- intire Payment. What is the true time. of Pay- aicnt ? ' In this^Queftipn, both the Sums to be Paid, and the Times of Paymeritarci Equal ^ and to Relbive tlie Qj^eftion, This is the •• • • • • . r : • . RULE., Add One.Term of Time of Payment fo the Terms of Time given; And half that Number of ? Terms fhall be the Time of Payment. - •/' - ^ ''/• '.^ '' " " "' In this Queftiou, the Terms of E^ifonent are FoUr Vand the diftanoe of each of diem SMc Montis; in all. 24 Months; to which Add One Term more,and it makes 90 Mopths; the half whereof 15 i^ Months : At the fixpiration of wliich Time, the Payment is to be made. Example II. If ><be to Pay ijnto J?, lop /• a year fer Viyp years ; At what Terfti' of Time muft**-rfPay: B the Whole 5o<?/. at One intire Payment ? . , '. ' " />' ' . , ^ ' Herfcthe Ternis of Time arc fiv-^vtb? Which add bfte,they make 6 j the half whereof is Three years; at the end whereof, Aovtght to Pay unto B the whole ^o h at ©fl^ teHre Payment. And this a- greesexaftty with theFirftExampleof the former way of Working* * ' -* y . ■• . . •. ^/ ^ ..." - Exam^e lit. ' A is to'Payunt6BrS48 /. at Four j^qua^l Payments' vi^. 62/. at 12 Months, ^2 /. at 24.'Mvinch3: 627. at 56 Months: and 62^. at 48 Months : At wliat Tiiirt ought- it to he paid at One intli:e pay-.Ti^nr ? ' • . >• "•■/ Add MquAtions^j^f Foments i * 219 Add 12 Months to the 4S Months, and they make 6oAlonths{ the half whereof is 30 Months; at theen4 ot whidi Time, ^isto pay unto B, the 248 U at One entire paynient : And 30 Mooths contains 547. $ Days ; exaftly agreeing with the foregoing Second 11. When the Sums of Merry f and Times of Payment are Vifereni. Exaiif^le I. Ais to pay unto B, 400 /. in this manner : vi>. ioo> at 3 Months ; 100 /• at 6 Months ; loo /• at 12 Months ; and 100 /• at 24 Months : At what time wilt it become payable at One in£iie payment ? toEifolve thiSf or the like ^ Queflion\ This is the ' ^ ' .\ ^ feu LB. * • ■> . ' ' ' multiply tbefeveral Sums^ bytHe times of their J^effeSive Taymem J mi add tbem altogether I Then Divide that Sum by the Whole Deht^ 'mithe Qumentfiall give you the term rf time j for the Entire Payment a Once. .'.'-..:. - •..' ' Thus, loo /. Multiplied by^ 6 ^Months, is^ 6qo K^PoumL 12C ^'1200 *24«-^ ^2400 The Surp is — 4500 . And this Sum Divided by 400 i ^e Whole Debt, ^yeth in the Quotient 11.25, or 1 1 Monfhs arid "One Quarter of a Month; At the Expiration of which Time (after the Time of Contraft^ muft the Whole 400 /.be paid at Once? '/.,'-: ..' , ^ . • • ■ V txample II. A is to pay unto S, a Legacy of 500 /; in tfils man-, ner; 300/. at 4 Months; 100 /..at 6' Months; and 100 A at 12 Months: At what Time muft it be paid ;at Once? 3oo7 ^ ^? Ci2oo9 looVMultiplicdby^^ ^Vraakes' <^ 6co^ Pounds. 1003 6^23 c^^^o3 500 The Sum 3000 Which Sum Divided by 500, the Qiioticntis 6 Months, and at that 1 220 Afpendix. I that Time oogtic the $00 /• to be paid at Once: And this agrees with the fame Queftion Wrought by the Prececding Exmpk l\ And diis ReTolves the Qpeftion according to the Intereft Tabic ; For, C3007 C 4? C 67 The Int«rcftof-^ioo>For V 6>MoQtfas,is^ i>F(mds. tioo3 ^123 |i 6> Andtfae Inteieftof 5ooiL in 6 Mbntiis, is 15 /• But to Refolve this or the like Queftion morQexaftly, is by the Rebate, and not by the Intereft ot the Afony \ And then, .C3007 C 47 094- 2. 4. I The Rebate of< 100 > For ^ ^>Months,i$^ 97. u 9* o .1003 C«3* C P4» 6* 9« 2 Total —4 85. ic. 10. 3 But the Rebate of 500 /.for 6 Mon. is but — 485. 8. 8. 3 TheDificrence— ooo» 2« 2* o And dius Aucb concerning Simple Intereft and Rebate. IV. OfCmfomdIntereB: With the Confirum^ 0n md Vfe of the Tables thereof. XHe Tables, which I (hall here ihew tbe Conftru£lion and Ufe of are in Number f^ive ; And all of them Calculated ac- 3g to the Prdent Rate of Intereft by Law now eftablUQbt* ed in England \ that is to fay, at Six Pound per Cenufer Jmum for a year, Compound Intereft: And from i year to 21 years ; The two Firft of the Tables (as being of more frequent Ufe than tie o- thertliree, which are deduced from them^ are Caculated for Years, Montiis, Weeks and Days ; And the other three but for Years on- ly 5 And by one or other of thefe Five Tables, all Queftions of Com- pound Intereft Mony may be comprehended and Solved \ The Tables here follow ; And atcerthemtiieir Conftruition and Ufes.' TABLE f. 221 TABLE I. log what One Pound be- )m any number ofyears, weeks and days, C""- years; will amount un- ounting Intercft upon at \l. Per Cent. TAbLB 11. Shewiag the Decreafe of One Pound.Or, what One Pound at the End of any Number of Years, Months, Weeks and Days (under J I yearsj U Worth in ready Mony, Rebating Intereft upon Intereft, at VI . ger Cem. -P. a 5 4 5 6 I 9 11 Y. 3 3 4 S 6 I 9 10 14 15 16 \l 20 31 32 as 24 25 26 19 JO Dec. p. Jooo Months. .859919 .79209J •747958 2 3 t I I 9 10 II Months )103 IBS! 1.004867 t.009760 1.014674 i.oi5(Si3 1.024576 1.02? s«3 1.054574 1.039610 1.04467 1 1.049756 1.054865 .9951 5fi .990335 .976015 .971286 .966581 «6iS59 •957239 .9J2643 .947988 iSsi .363 .584 !947 1084 .704960 .665057 U5274I2 .591898 .558394 |929 310 .526787 .442 SCO .41726s 12>2 IWe.;!-.^ ! Weeks.] •6)5 z S 1.0011 18 1.001237 1.003358 I 2 3 .998883 .997767 .•9S>«!3 BS! .395644 ■3713«4 . •350345 •330512 .311804 4SS 1 Days. 1 1 Days, j :"s 3 ■ 3 4 ■I 1.000 160 1.000319 1.000479 1.000369 1.000798 1.000953 3 3 4 5 6 .999840 .999681 .999521 .999361 .999202 .999042 ■?5« 975 :1,J .2941(5 .277505 .1617J7 .246978 .232998 ■Hi' 1158 L838 1349 .219810 .207367 .195630 .184556 .174110 ^ t 'Ta- 1 2 1 22 " \ Jppendi}c. i /v LLb 111. 1 ABLE IV. TABLE v.. Shewmgwhat i/. Slicwing the pre- Shewing whatAn- will amount un- fcnt worth of i/. nuity { payable to, it being For- annuity,tobepaid yearlyj 1 pound « horn any number yearly to continue will Purchafe, for « of years under 9 1 any Number of any Number of at VI. per Cctn. years under 31, at years under 31. at * Compound Intc- VI per Ce«^ Com- VI /^erC^r. Com- > rtft. pound Intereft. pound Intereft. • Y. 1 DecPar. Y. Deeper. 1 0.94340 Y. 1 DecPar^ 1 1 .oGo'bo , 1 1 .05000 2 2.06000 2 1.83339 2 .54544 / 5 3.18300 3 2.5730! 3 .374.n 4 4.37461 4 3.46^10 4 .28859 1 \ 5 6 5.^3709 5.97^31 5 6 4.21235 5 5 •23740 4,91732 .20335 7 8.39383 7 5.5B238 7 .119}^ 8 9.89749 8 5.20979 8. . .i6i03 ^ 9 n.49131 9 5.80x69 '. 9 .14702 10 • ,1^.18079 10 7.36008 10 .13585 II 14.97154 11 7.88687 II .1257$> 12 15.869^4 12 8.38384 ' 12 •11927 13 18.8821^ n 8.85268 13 ' .11296 1^ *2,i.oi5o5 H 9.29498 14 .10758 15 15 ; 23.2759* 25.57252 15 i5 9.71224 15 i5 •10296 10.10589' .09895^ 17 28.21287 17 10.47725 ?7 ,09544 18 30.90555 18 10.82750 18 •09255 1 ^9 33'759S>9 i 19 iia58ii 1$ •08962 20 3<5.78559 r 20 11.45992 20 .08718 * h2l '39*99^n 21 H.75407 21 .o8>co 22 •43,39?28 > 22 12.04158 22 .08304 .23 46.99582 23 ''2.50337. .23 .081:^7 24 50.81557 ^24 12.55035 24 .07959 • 25 25 - 54.85451 ^9. 1 5538 • 25 26 12.78335 13.C0315 25 25 .07822 .07690 ■ 27 55.70575 27 i;.2io53 27 .07559 2g 58.52810 28 I3.4c5i5 28 .07459 29 73.53979 29 13.59071 ^ - 2JP •^7375 • 30 ] '9.05818 ^0 , 1 2.75.1 ?2 30 .07264 nt N i r C 223 ] ■ The Confirumon (?/ T A B L E I^ T9[e firft Colamn of this Table, havkig [[YeanT at th> head thereof, begins at i, and fo proceeds to ^o; and the Nan* : bers in the next Column ftanding againft any number of Years, are Decimal Numbers, which fhew what one Pound (ox2js) i«; worth (or will amount unto) being forborn any number of Years under 31 : And this Table is mxde according to this Proportion. As 100 /• Is to 1 00 1, and the increafe thereof in one Y ear, vi^ • 1 66 7. Soisi /• CorioxO To I /• and the increafe of it in a Year, vi\* i.o6ooo» WhicKis the Decimal of 1 /.and the Increafe of itin a Year, and hthe firft Number in the fecond Column of the Table againft 1 Year. - , Then for the Second Number, As iOD/.Istoxo6/. So is i.c6/.To i.i2^5ofor theSecond Year. ' Then for the Third Year. As 100/. Is to 106U So is i.i236to 1.19102 For the Third Year , St fie ^ Infimmm : Thus for tie whole Years. But, To find Decimal N ambers, for any pirt of a Year ; a.^ Months, Weeks^ Dxys-j or for Half-years or Quarterly-Payment;. Take the Decimal for one Y:?ars Increafe, v/^. 1.06001 the S^jiuire Root whereof is 1.0290, and is theD;cimilof the Increafe of i f. in 6 Months: AAd the Mean proportional between 1.02956 and i.66opowillbe 1.04.4.67, and is the Decimal for th^ Increafe of li. in9Months: And the mean Proportional b*nveenan Unite ^or i) aad 1.02956, will be 1.014.67 and is the Dicimil of th* Increafe of I /. in 3 MDuths. ''And thusyou may with facility difcover all thi o- Nambers in this firftT'able. TheVfe of r ABLE L Exm;)le I. What will 1 367. 1 «; j . 6 j. amount un^o, it b*in§ for- bocn 20 years; 2t6LpsrCcm.psrJnnuniCo^[yomdl(i:\'c\\? Look in the Table tor 20 years, and right igainlli / vi ihill fiii 3-^^7136, which is the Increafe of i/.in 20 years ^ Multiply chu Ff2 Njn-)-'r S24 Jffendix. Namber by I36w77^ ^e Decimal ofi36/.x$x. 6i« the Prodaflwin be 43 8*65520 which Reduced is 438 /• 13 x. 1 i» 1 f. and unto lb much will 1 36 A 1 5 X. 5 i. be augmented unto hi 20 years. Example IT. What will 200 /. increafe unto, if ibrbom 6 Misnths at 6 Uper Cent, per Annum Intereft upon Intercft ? The Increafe of i /• ifi6 months is 1.029563, which mUltlplyed by 200 /. the ProdufI will 26^9 ti6oo /• which Rjeduced is 205 U 18 $• 3 d. and to fo much will 200 /. be Increafed hi 6 months. « fxdmpU ni. What will the Increafe of 300/. amount unto, if &rborn 3 Weeks, at 6 /• pfr Cent. Compound Intereft. The Increafe of i /. in 3 Weeks is 1.003 3 5S, which multJplyed by 300 /• the Produft is 301,007400, and that Reduced is 301 /• o /• I i. 3 f • from which Subftra£t 300 /• and the remainder i /• o <• i i. 3 f is the Increafe of the 300/.m 3 Weeks. Example IV. If 3600/. be ibrbom 5 days, what fiiall be the In- creafe of it. Sit 6 1, per Cent, per Antfim Compound Intereft ? The Increafe of i /.in 5 days is 1.000798 which multiplyed by a6oo/.the Principal, the Produft^ill be 5602.872800, which n*- duced is 3^2 /.i 7 x. 54-2}. from which Subftfaft 3600, theRe- jnaiuder 2 /. 1 7 1. 5 A 2 f is the Increafe in five Days. The ConfiruBion ^ T ABLE It. the P H^O P Jl^T 10 N'S, AS 100 /.with the Increafe of it due at a years find, irff, 106 1. Is to the Principal, 100 /. So is I /• due at the fame time : To .943 396, the difcoont of i /• for a year, vlf . t8 ^ 10 rf. 1 f • Then, As 10^ /, Is to the Decimal laft found •943396 So is lOQ/. prefentpay To.889996'theI>ecunalforthefecQndyeir, vh. 171. 8i. 2 a. ]Etfic 4d infinitum^ fcr whole years \ But^ To find the Decimal Numbers for parts of a year upon Difcount ; orTor half years or ^arteriy-paymen ts^ Thefe are compofe4 after the fan?e manner as the Table of money forborn, excepting only in the Pointing of the Numbers for Ex- trafting O^cRoots^ the Decimals in the firft Table being all Mixt. Numbers The ConfiruSlion if T. A B L E H. 22 $ numbers; and thefc for Difcount are every' one proper Frafiiims; liaTing a Point prefixed : Wherefore, in theTe proper Fraaions, make the firft toict under the fccond^ Figure on the Left-hand : As for Example .943396 is the Decimal for the years Rebateof i /• put the firft point over the Figure 4, and fo in order to the Right hand; the Root thus £xtra£led will be .971286, for tfaecfiTcount of I /• in 6 months. The Square Root of that ag^in will be .985 5 38, for 3 months ; and thus proceed with mean Proportionals until the Places are all compleat between the Radius and the Decimal laft found : As for halt yearly and quarterly Payments, they are dif covered as were dide before in the Forb^rance of Money ; of which you may fee variety of ways in my Cwrfm Matknmicm. The Vfe (f T ABLE IL • Example I. If 3$6 /• be payable at the end of 7 Years, what is K worth in prefent Money, Difcount or Rebating^ after the R^te of 6 /• per Cent, per Annum Compound Intereft ? Look in tlie Table II. for 7 years, againft which is w66^$7 aild fo much ready Money is 1 /• or 20 x. due at 7 years Bnd wcordi prefimt- ly; multiply .665057 by 256, the whole Sum, the Produft win be 236.761292 which Reduced is 236/. 15 s.^i.l }• and fo much is the 356/. worth in preient Money. ^ V ExMnple II. At the end of 6 months, >f is to pay unto B 500 but they agree dnt it ihall be paid prefently upon Difcount after tl rate of 6 Uper Cent. Intereft upon Intereft. U the Exgmf}e III. jtfhathaLeafe inRtveriion, which at the Expira- tkmof 7 years is valued to be worth 1200/; which Leafed would Purchafe with prefent ready Mone^r ; Rebating aft^ the Rate of 6 f^Ce»r.^i^«iMf Compound Intereft, what ready Money mnftS give A for this Leafe ? Look in Table II. for 7 years, and againft it is .665057 the De- <:imal for the worth of 1 /. or 20 j. due at 7 years end : This Deci- mal multiply by the worth of the Lfcafe after 7 years 1200/. the Prodttft will be 798*068400: which Reduced is 79$ /, i x.4 </. 2 f. aod fo oitich prefent Money nu;iftJB disburfs, to purchafe the Leafe. Example f 226 Affendix. SxAMplclV. AisiXixoY unto B, a Lc/^acie of 1800/. at three feveral Payments, vi^, 600 L at the end of fix months, 6go /.more at la months, and 600^ more at 18 months; B defires the money prefently; and -4 is willing upon Y>VcoMatoi SperCenuperAnnum Compound Intereft; What prefeat money will fatisfie the whole Legacie ? / ' Look in Table^I. for the Decimal of the Difcount for 6 months, where you Ihall find it to be .971286, which multiplyed by ^00/. (the firft Payment to be due at the half years end) tbeProdud will Decimals 582.771660 566.637600 582- 566- .X5- •00- 549-7^4200 J 549 — 15 -9 -8 1 I I 2 3 be 582.771600, whidi Reduced is 582 hi$j* $d. Then is there &y0 1, upon a years Rebate, the Decimal for i year is .943 396, which multiplied by 600/. the Prpduft is 566.037600 \ add that Reduced is 566 L OS. 9 d. due upon the years Rebate, . as appears in the Tabte above Now the laft payment is 600/. upon 18 months Rebite; now to find a Decimal Number for thisj do thus: This Decimal for one years Difcount is .943396, and for 2 years .889996, thefe two multiplied, theproduft will be 8^9618666416 and the Square Root thereof .916307, and this multiplied by 600 /» theiaft pay- ment, the prodnft will be 549.784200, which Reduced is 549/. i$s.Sdy and the Total 1698 hiis. 10 i. which Sum will difcharge all thelliree payments at one Time. -U The Co»firu£tio» <?/ T A B L E III. THis Table js deduced from Table I. For4f you add the prin- cipal 1.006000 /. to tlie firttNumber in thefirttTable i.o6oooo Cwaich is the principal and Intereft of i /.for a year,) the .Sum of them will be 2.060000/. and thatis'the Number ftandi'ng agahift the feconi year In this rhird Table — Again, to this fecond Num- ber ill Table III, add the fecond Number in Table I, v/:^. i«{236oo; the Sum vvlilb* ^,1^3600, for the third Hiiiiiber in Table III. — - And to thh third Namber in Tabielll, add the third Numberin Tablel, theSam will b?4.^746i6, for chetburdi Number in Table III, and fo of j^il :'ve r ^ft. I f The Conpu^ion 0/ TABLE III, VI. 227 The tV^^TABLEIII. / I Bs^dmple T. What vill an Junuhji bf 20/. payable yearly, be augmented unto iDi2y?ars; being all that timeforborn? Ac- counting Tntereft upon Intereft at 6 h per Cevtfer jimunu Look for 12 years in the Table IILagainft which fiands i6.26^g^^ which (hews that an Jnnuhy of 1 /. a year, in 12 years will amount unto. i6,86994 /• wherefore multiply itf.86994 by2o,theproduft will be*^ 3 3 7.9980/. which Reduced is 337 /• js. 11 d. ^q. And lb much will the Atifiui^ of 20 hfer^.bc augmented unto, if forbom 12 years* # • Example II. If an Jfinuity of '60 1. :ayear, be forbom 7 years ; How much yi'dX it amount unto when that term of years is ex- pired ? . ^ ^ . The Number ftanding againft 7 years ih this Table ill. Is ^•393^^9 which multiplied by 60, the produft will 503^62980 /« which Decimal Reduced is 503J. 12 x. jd. And unto fojnuch will tlie Annunj be augmented unto in 7 years. The Conjlrumon of TABLE IV. As theThird'TaWe was Deduced from 7ahle 1.' So this Fourth Table is Deduced ftQm-7iii»/e 11.- Now the Number ftanding againft i year in Table II. is •943996' and this muft alfobe the firft lumber in-TidWe IV. to that Adii889996 (the fecond. Number in labl& II.J theSum 1.833592 is the fecond Number in Title IV.— Again,to this fecond Number in Tahle IV. ^dd tlji?thir4 I^umber in lable II. Fi^... 839619, the Sum will be 2.67 301?, which muft be the third Number jR lii/^ IV. £5 /f,&c^ . , ./ TheVreofTAhtB IV. " ■ Example T. What is the pi'eftirt Wdfth of an Jmuttji or Rent, of 50 1. per jifinumy payable yearly, for 21 years, accounting Com- pound Intere<t,,a Iter the Rate of t h per Cent. The Number flandlhg againft z 1 years, in TahTeTSf. is 11. 76407, the prelent Worthof 1 1. j^nuity fon 21 years : Wherefore multi- ply 1 1.76407, by 50, the prCduit will be 588.i03<ol. wliich Deci- ^ iTial Reduced is 5881.4s. od. 3 q. And fo much is rLe piefent 'i Worth of tli6 50 1. a year Worth for 21 yenrs in pre't nt iV:or! y ? 77x' 22l JffeHdix. ; . rhe ConfiruSionof TABLE Vi FartteCmfftntSimtftUsfMe^ tbk utbe PMJ>fOJHttOhr. AS X.83339* the Decinu^l for 2 years Roit KebatedCasia , I$equaluiValaetoxL jlifmii(^for2yeais, Sois t K of Annual Amtnhj^ for the fame term of time in pro* portion to the Decimal Pur<*afcd by I U Or, As 1.83399: is to iL .*: So is 1.00000 : to .$4544, And this is the fecondNumber in T^i/e V. Now for the third years Decimal. As 2.67aoi, die Decimal for 3 years Rent Rdate Is to an Unite with Cyphers 1x0000000000 : So will I LforaPurduife, be in proportion To .3741 1. The Number ftandingagainft 3 years inMle V. The Vfe ffT ABLE V. Bxmple L What Antmhj to begin prefently and to continue 28 yean, will 640 L Purchafe : Accounting Compound Intereftito the Rate 6 1. /«r Cnir. ^ ifcmf «!• The Number ftanding agamft 28years in /^/l?V.is4>7459 \ which ihews, diat iL for 20 s.) win {Purchafe aii^itofuji^ Worth •07459 Cor I s. 5 d. 3 q. to continue 28 years ;) Wherefore, multiply .07459, by 6401* the produft Will be 47.737601. Whic^ Reduced 1S47KI4S. 9d« And famuchjxer ^IMniiii will 640 L Purchafe for 28 years to Conimenceprelendy. Example II. What ^ImnMr^Rent or Penfion,win 250 L in rea- dy iMony, Purchafei for 7 years : Compound Intereft aUowied ^t 6 1. per Genu per Anmrn. The Decimal ftanding again* 7 yean, is .17913, which multiplied by 250 1. the produft will be 44.78250 L which Reduced is 44 1. 1 5 s. 8 d. 1 q. And fuch Amuitj will 250 U Purchafe for 7 yean. Mx' Minfuf'Ation of Suf erf ^ies Md Solids y &c. 229 j Example lit. A Citizen giving over Trade Refignesorerto a ' Servant of his,his Shop ready Furniihed, the Wares Prized, and the Leafeof.hisHoufe valued^all together being Worth 1 6 «; 81. which . theMafter was willing to Receive in equal and Annual payin^nts I in ) years ; ^the Compound Intcreft agreed upon, at 6 Lper Ctnt.ftr An, What mufttlie Annual payment be? the whole Stock being Valued at \6<^%\. and the term of time for payment 7 years, multiply the Decimal ftanding againft 7 years, I whichis .19613, by 1658 L the produft will be 296.99754* which I Reduced is 296 1. 19 S. iiq. (which you may call ^7910 said fo 1' much paid yearly for 7 years, will Difchsirge the Debt. ^ E C T. HL Of the Menfaration of Superfcies and Solids ; And of the Works of the fever al Artifcers relating to Building. I. Of the Menftiration of Plaln^ or Stiperficial J^gnres.* PLain, or Superficial Figures are fuch as confift of Length and Breaddionly ; not having any Commenfurable Depth or Thicknefs ; As Board, Glafi, Pavements, Land, t^fc. !• HVTotdMeafureaS^uare. Figure L ' lite R U L B« Multiply the length of a>iy of the Sides of the ^qUtxre (the Dim^fion thereof being tiken in any hini of Meajure) asFeet^ Inches^ Tards^ Poks^or Perches, ^c. into H felf : Thi TroduB of that Multiplication fi>aSgive the Superficial Contenty^or Area of that Squarey) infuAMea^ f^e as the Side of the Square was Meafured hy: Whether^ Inches^Feetf Perches J or any other Meafure. let ABC -D,be a Square,whore Side is 24 foot and a half/or 94.5 foot:) multiply 24.5, by 24.5 theprodua willbe6oo.2 5foot,which is 600 foot and a quarter, and fo many Square feet are contained in that Square whofeSide 1524.5 foot. <^ § 2, Iforv 2. Hm to Mtdfkreg T^tMogrm^ ot Long SijfMre. Fig. It^ The RULE. » ■^ f/bilt^fJytbe Length bf the Breadsbf tbe ProduS fiaB give the Artii or SuferficuUComitn (y the Figure. I^ EFGH^ be a Long Square or Parallelogram ; whofe length B Ft is 36.3 5 Pole or Perches \ and the Breadth £F 12.$ Perches. \ilultiply3d.2$ bv 12.5, the produawinbe453.Ji25 Perches^fot ^Arca, or Superficial Content of the Parallelogram. 3. How to Meafure a Triangle. Fig. IIL The R U L E, For the Meafurbig of any triangle^tbere are Three ways ^ dUvbicb are comfrtfei in tbk One GENERAL RULE. From the Angle which it opyfite to the Longefl or fhorted Siit of ahj triangle^ let fall a Perpendicular j 7ie», ( u) Half the Length of the LongeS or Shorted Side^Multiplied into the Length of the Perpendicular f the Produ^ of that Multiplication Jhall give the Area of the Triangle. OTf (2O The Length of the LongeS or Shorten Side of the Triangle, be- ing JAuluplied by half the Length of the Perpendicular \ the ProduS fiaHgive the Area or Superficial Content of the Triangle. Let K^LM. beaTriangle, whofe {sSeft}^*^® is X M , from die Angle j^ oppofite to the ^ shorteftj ^*^^ L JW, let 611 the Perpendicular -^iNT. Then, In the Firft Triangle^the Longeft Side I M, is 22 Inches, and.the jPerpendlcular^iV'9lnches^alFtheLongeft,(ide 1 i,multiplyed b? 9, ttie Perpendicular, produeeth 99 Inches for the Area of the Firft Jriangte; . « Or, 22, the Length of the Longeft Sjde multiplied by 4.$ Inches, half the Length of the Perpendicular, tlie product will alfo be 99.00 Tor the Superficial Contept of the Tj^iangle in Inches. Again, In the Second Triangle,Let the Shorteft Side I. M be 7 Inches and ^(Jiarter Cor7.a5 Inches) and the Perpendicular ^iV24 Inches ; I fay, that 3.625 f Halt the Length of the Shorteft Side,) multiplied by 24, (the Length of the Perpendicular) the prgduft will be 87 Inches for tfie Content of the Triangle.Or, 7.25,(the Length of tlie Shorteft; > , - Side,) Meufitration of Sufirficies and Solids ^ 8rc. 2} i Side,) xnultipliei l)^ 12, (half rfiePerpcBdicular) the prodoft will be 87 inches alfo, for the Content of the Triangle. 4« Htm to Meafure a Figure of Four utircqual SidtSy comtaonlj isSiiAtrAfeitA. Fig. IV^ \jitDEFG be a Trapezia, from i? tp F, thel-ongeft Diagonal pf'the Figure, dniw the Right Line DF\ which fliall Reduce the Trapezia GDEF into two Triangles, EDF, and Gi>F ; this Line or Diagonal being common to both Triangles ; Then from the Angles E and G Let fall two perpendiculars E Hy and G K^ upon the Diagonal Z> F. >}ow, to find the Area of this Trapezia, this ^the RULE, Muftiplf tbe tobole Length of the Diagonal by the half Sum of the two perpendiculars 'y that FroduujbaBbe the Area of the Trape^ta : — &r, Muhiplj half the Length of tbe piagoinaljby the Sum of tie tvfo Ferpendi* hilars^ and that Froml fiall give you the Area^ or Content of the Irape^ia. In the Trapezia II £ F G^, the Diagonal p F, is Cpmmon to both the Triangles^ and is in length 73 Perches ; the Perpendicu- lar GJ(^\^ 28 Perches; and the Perpendicular Elfii Perches: their Sujm is 46, and their half Sum 23 Perches. Now, If you multiply 73 ftl^e whole Diagonal) by 23, (halfthe Sum of the Perpendiculars) the product will be 1679 for the Area,orSttr eicial Content of the Trapezia, in Perches.— Or, 36.5 (halfthe gtli of the EKagonaUmultiplied by 45(the Sum of the Perpen- diculars^ theprodu^ will be 167935 before, for the Area, of die Trapezia. <• Ifow to Meafure an Irregular SuperSHes hmfing many uneauat Sties. Fig.V. '^ . Let ^B Ci? fiF<7 be an Irregular Piece ofLaad<a8a Wood,or the like) to be Meafured. before this Piece can be Meafiired it jnuft be Reduced into Tome of the foregoing Rggilar Figures, as into Trapezia's or Triangles, by drawing of Lines from Angle to Angle within the Plot ; Andbyfuch means, this Irregular Plot is Divided into two Trapezia's, and One Triangle ; For* the Line drawn from Fto C, cuts off the Trapezia -4 B CF5 Alio another Trapezia FCDE is cut off by drawing the Line F£, and then there is left the Triangle EFG. The Figure>eing thus Reduced, let fall the feveral Perpendicu- lars DF^FLy B//, F/, and FO, which Meafured, let eaich Dia- gonal and Perpendicular, be fuch Number of Pcrdie^ as arefetto ttem in the Figure. G g a And 5J2 Appendix. And now are you ready for Caftingup of the Content or Area ^ m Perches. • » .. FirH^ VoT the Triangle F G E. whofeBafe J^(?isa4,and Perpen- dicular FO, lo. Now 34 multiplied by $ (or 17 by 10) the produft will be 170 Tercbes. ■ / ■ Secondly ^ Por the 7Jrf5^i<i -/(BCF, wjierfe the Dl^gpnal or Connmon Bafe to the two Triangies^ \sA C, and whofe J-cngth is 3 ^ Fercbe's'j and the two Perpendiculars and FI 2^Percbesj and BB % Perches^ their Sum is a2 Fercbesj half whereof 16 multiplied by tl}e Common Bafe 55,' t^egroduft will be s$o Perches for thel>^' ;><^/^ JBCF. ' " ' * * • . - , Thirdly t For the Trapexis CDEF^ the Diagonal or Common Bafe, whereof is C £, and contains 32 Perribfi : And the two Perpendicu-^ la^sare JFI. 17, and i> ^,13 Per^ri&M, the Sum of lyjtb beirig.30 Perches^ halfwhereofi5» multiplied by 32, theprb4i;£\ Willbe-Adci rercbes-y for the Trapezia CDEF. Perches The triangle CFGE 7 C170 Thernfpe^/<< -^-^jBCF >Contajns^56o * The Trapezia C'^DEFy ^460 In an 1190 ^ ,6» i^^tttwMf^ffirtf ^jf Jugular Polygon^ or Figure cf any Kum^ her of Equal Sides. Fig. VI. * . , . The R U L E. Multiply the length of one of the Sides of the polygon, imp half the ^ Number of the Sides ; and that produB Multiply by half the length of ^ Perpendtfiular let faki from the Centre of the Polygon upon any of the Sides ; this Second ProduB Jbalt* be the Area ' or Centre cf the Folygon, Let ABCDEbe ii Regular Polygon of Five Si4es,each Side con- taining 2 $ Inches, and ietthe- Length of the Perpendicular F G be 17.^ Inches: fi.) Multiply the length of the Side CJ? 2$; by 5 the produft is xo5, the half whereof is (52,$ and that prod utt - tmjltiplied by 1 7.2, the laft produft will be 107$ Inches for the Area or Content of "the Polygon. « * * •• ^ 7, Hovto Meafure Circles^ and Parts of Circles. Tig. VII. The Proportion that the Diameter of any Circle bears to the Circumference of that Circle, is>^ As 7 to 22 (in tjie leaft terms;) Or, As u 3 to 355 ( in more exaft terms,: ) And by ■ either of thefe ^Proportions, may the following Problems be Refplved. . . . . • •* '^ ' \' * ''' • "■''- ' '■ ' '""■ ■■• ' ■ ■ * ' ■■- » 8 The Menfiiration of Suferfcies 4nd Solids^ &c« 2 j j S. 7£e Diameter of a Circle being given to findtk Grcufrfcf fence. Let the Diameter of a Cii^cle J^^ be 27.6 Inches ; Multiply 27.6 by 2Z, the produft will be 607.2 ; wjiich Divided by 7, the Quotient will be 86.7 Inches for the Circumference. Qr morjpexaft-, Multiply 27.<^, by 355j the produft will be 9798,o, which Di- vided by 1 13, the Quotient will he 86.707 Inches for the Qr^^jm-s firence. " 9. The Cireuntfefence t/ 4 Circk hekig givtu to find tie J>u* meter. ' Multiply the Circumference by (7 or 113,; and Divider the proi duft by 2 2, (or 3 5 5 J the Quotient will give the Diameter. So the Circumterehce beinjg 86.7 (or 86.707) the Diameter will be found to be 2 7.6 Inches. ^'Mnltiply 86.7 by 7 the produft will be 606. 9,which Divide by 22i the Quotient will be 27.6 for thcf Diameter. Or, Multiply 86.7 by 113, tht pitodtift will be 9797.1, whichdivided I by 3^5, the Quotient will bel 27*62 Inches for the Qiametar, a%. before, but nearer. 10. The Diameter of a Circle being given to find tbe Area^ oe Superficial Content thereof. The RULE. Multiply tbe Diameter into ft f elf and that ProduS again^ by 22 (flf )^'y) Divide thif feiofM ProduS by 2% (or ^^2) tbe Quotient mil be tbc Ma^ or Superficial Content of tbe Circle. So, the Diametei' of a Circle being 27.^ Inches; That multiplied into it felfproduceth 761.76, and that multiplied by 22pr6ducetli ' 1^758.7^ Which divided by 98, gives in the. Quotient <98«$2 Inches fBr tfieAVea of the Circle.— Or 761.76 multiplied 0735$ produceth 270424^80, and that divided by 4521 giveth in die Quo« tient 598.284 the Area more'ezaftly^ 1 1. Jbe Circumference of a Circle given to find the Area* The R U L E. Multiply tbe Circumference into itfelfandtbat ProduB Mnlfipfjf by 7 (> 113 :; Ditfde tipffecondPraduiby%% (pr 1420) Tbe Qiotiem tnU give the Area '^ or Content of tbe Circle. So the Circumference of a Circle being 86.7 Inches. That mul ' tifiiedfoto it felfproduceth 75i6.89,whichmult!pljiedby 7thepro- 8^4 • Jff^^f duft is ^^18.23, and that divided by 88,' givetSia tfacQaoticnt 54^.82 Incbeifiktjie Area of the Circle. 12. 7||k Diameter of s Circle being givm^ find th Side rfd Sfyffs Vfkicb JbnB be equal in Area to that Circle^ The It U L £• * lihfltipyi'tbt' Vlanieter of the given Circle {dv?aysybjtlis pednuH •SS6227, the Produ^Jbat be the fide (f aSqme^Eiuai in Jrea to tb$ Circle^ ' . So, the Dhmeter of a Circle being ^7,69 that mtrltiplied intQ ^886227, the produft will be 24.4598^52 ^ That is 2446 Inches fere^ Ahdthat is die ^ideofa Square, Equal in Ai^ea to die Circle whofe Diameter is 27.6 ^ches;For 24.4$ muitiplied into it felf pro? dbcedi 598.2916 Inches. Note. There fnaall diiFepences 'do arife partly, for want of xAof e places in the Decimal Parts \ and pardy into the Diflference pf the Prpportioas between 7 to 2a and 113 and 3^5. 13. lo Meafure aSemicv^k. The R a L & Multiply half tU piameter of the Circle Into it felf and that produB (^always) by 2^^ and thai produS divide (^alwiffs) by i^ The Quotient viU. be the Area,of the Semicircle. ' " Sothat the Diameter being 13.8, that multiplied into it ielf pro? duceth 190.44, which multiplied by 22,produceth4i89.68y which divided by 14, giveth in the Quodent 299.26 Inches, for the Area of 5ie Semicirdc; : ' ■ 14. ToV^ap^eHQuairam^ • The R U L E. HiMkifly half the Diameer into it felf and that produS (ah»ais) bf^ II and that ProdnO^ Divide {always) yi^the Quotient gives the QuO' irams Area. So i3.8|halftheDiameter,mnltiplIed ihto itfelf produceth 19044, which multiplied by 11, prodaceth2094.84,and that divided by 14 ^yediindie Quotient h?*^^ Inches >{or the Area ofthe Quadrant, 15. Hem Menfarition opStforfatf $Hi. Soudf^ Sec. f^ii t5; Ifaw to mafure Ojf ?m^ Pmhn or ^gn^t ^ $ Grcff4 The RULE. i/bdtipjjf bdtf the Dimeter^ byhtifthe JtciyUne\ mi that FfoiuA triObe theJreacftbeScSor'^ orpdrtion^ So, 1 3. 8, half the Dianieter,behig multiplied by 5.52 tie length of half the Arch-line; the prodnft will be 73.41^^)1 the Axtiaef the Seftor. ; ; i5. jfforv to find a line e^udl to half the Arcb-lin^^ F G ^, . , Draw the right Line FC^ and divide it into Four e^al |)fti^ in H^ ^ L^ take one of thofe parts, as I C, knd fet it frdttt Ccd G^ tiien a right Line drawn from Gto Hy (hall be equal to half the Arch-line PGC II. Of the Menfitraiion of SoUds* I . ■ : Solid Figures or Bodies, are fuch as do confift of Th^IKiiien- fions as. Length, Breadth and Depth, or TJtiicIi;ael3 ; A^itpne, Timber, Columns, Globes, Bullets, C^'r. > * _ 17. Ilovf to NLstfure a Cube. Bj-'VJII. RULE. Uuh^lytbe Side tf the Cube m$o Itfilf'^ undsbm Pr^i/k^ 4imhithe Side : The hSt ProduS will be the Solidity or Solid content of we Cnbe. So, let ABC DEFG be a Cube, the SJd? whereof -*4 5, &c. let be 5,2 Inches,Cor any other Meafure) 5.2fliui|fipliefl intoitfelf^ produceth 27.04 and that multiplied again by $^2Vthe laft produ^l will be 140.608 inches 5 which is the SoU4 Content of that Cube, wbofe Side is 5.2 Inches. 18. Jlon to Meafure a Long Cube or PmSelftifeiotfi, Fi2i IX* RULE. ^Multiply the Side offhe Square at the End^into itfeff^ni'tbdt ProduS by tbt length f the lafi ProduBJhdJl'give'tbiSoJidCi^fenf. ' ' So,tet HI^L M N-O P, be a Long Cube,the Side of wbofe Square at the End letbe 5.7 Inches, and the Length thereof 3.5 F,oot,(or .42 Inches) 6.7 multiplied into it felf, produceth 44.89, and that snttltiplied by 43, the Length, makes the Second pro^ufl. to hi S885.3S Inches : Which is the Solid Content of the Lohg Cube. 19* H(m toMerfure im Oblong PMraUeUpidedon. Fig. X. ' RULE. ' Multiply the ttw un-eiiual Sides m the end of tjje Piece , «fte bjibe other ^ andtbe ProduB of tbem Multiply by the Length if the Peice ; the IdS ProduS Jballbe the Solid Content y the Oblong Pa - raUdifipedon. Sb let QJJ tV Z he an bbioiig ?aralleripip<*dori,wtibre Breadth at the End QZ Let be 61 5,andl>spth at the End Vk^ Let be 9.4 Inches^ and let the Length thereof be 16 Foot 3 Inches (or 19 2 Indies* (Multiply 6«5 we Breadth, by 54 Inches the Depth, the I^odu^ thereof is 35*19 wHidi mnltipiied by 192 inches^ the Length } the produft will be6739.2 Inches the Solid Content of the Oblong Fcuallelipipedon in Indies; Whicbif you bivide by 17^8 (the So- lid Inches contained in One Solid Foot of Timber or Stone) the Quotient will be 3rHvl* Eoot. And fo much Solid Stone or Timber is in that Piece. Note^ F6rafinuch as in the following Menfurations of Solids, and in Meafuringofthe Works of Artificers : PimenHons are generally taken in Feetjand Inches, but the Content is, for the more part, required to be given in Feet and Parts of a Foot, and therefore, it will be necefiary here to Ihew, 2o« Himo to Kiduce Inches^ md Pms'of Inches into Decimal Parts of 4 Foot* quarter ') Kotethat ^^ quartersCofanlnch^: "'^ ^ -, quarters C is ^ Inch 1^ Tr Andthefe Fradions Reduced into Decimal Parts, as hath been Taugjhtin the beginning of the Second Part of the Decimal Par^ of ArithmetickjProWew I. And according to the I{uk there giverf, the Decimal Farts AnTwering to, ^ I quartern C*^*^? 1 Inch •^ ^.oBg*^ And fo ofallthereftas in the Table following. Menjuration ofSuf^fcks And Solids ^ &C. 237 i»-^ A Table Ihewjng the Decimal Parts Ap- fwering to every Inch and Quarter of an Inch ^ in One Foot, the Foot being Divided into 1000 Parts Inch. Quart* Inch. o I 2 3 1 o 1 2 3 3 o I 2 3 o I 2 . 3 o i 2 3 1 o i 2 3 •021 .042 •063 •080 •loi •122 .143 .16 .181 .202 • 223 .271 •2p2 •313 -34 .361 .382 .403 .42 .441 • 4^2 •483 Inch* Quart. Inch. 6 o i 2 3 7 o t i m * 3 n ■ " I- S 6 1 2 3 9 ■ . o I 2 3 to n • SSmmmam^ One Foot i.ooo I 2 3 o i 2 3 Dec. Parts. :5 .521 .542 .563 •601 •622 .643 .681 .702 .723 •771 .792 •813 .84 .861 .882 903 ^ « • .92 .941 .962 .983 M««k »H Hi fi , < 2} 8 Affendix. 21. T^Meafure the Solid content of aSquared Stone or Piece of Tin- Ser^ vfbofe End is a J(egttl£t Polygon^ of 5, 6, S, or 9 eiiual fides* Fig. XI. RULE. Find the Area, of Superficial Content, o/ti&e Polygon At the End (.tithe ith before-going) vthichhAdy multiply that hxt^i^ by lie length rf the Vitce, ihdtjMtlgiveyouthe SoMitY. So let ABCDEIG HK^^y be a Square Piece of Stone or Tim- ber of 7 Sides, each Side being 4 Inches and a Quarter (or. 961 Parts) and the Perpendicular from the Centre ©> ^ Inches and a half, for .292 Parts) and let the Length thereof be 8 Foot 3 Inches, (or 8.25 Dec. Parts.) (u) 4 and a Quarter Inches for .36 1 Parts) multiplied by 9 Inches , and a half for 3.$) (half the Number of the Sides,) the produft will be 1.2 03 5 which multiplied by 8 Foot 3 Inches, (ot 8.25) the laft produft will be 104.23875 (or 104 Foot 2 Inches and 3 Quar- ten) for the Solidity. 22. How to Meafure <tPrifme. Fig. XII. RULE. 22. Multiply the AreA of the BAfe (or End) by the Length : the pro- du& is the Solid Content. JjctABC'EHO be a prifme^ the Side of the Triangle at the End BB^ is lainches (or 1.5 Parts) and the Perpendicular CP 13 Inches for 1.021 Parts) and the Length C7C 7 Foot 9 Inches or C7.75 Parts.) (i.) Multiply 1.021 the Perpendicular by half the Side of the Tianglcfil£,vi^..75,thcprH"^ "'•"'*-' -^--— ^^ >\^„w:,.i««x^-f*. 7.75 the Length the Pre multiplication 5.93 5) tbi Inches, and almoit a (Quarter of an Inch, 22* To Meafure a Pyramid. Fig. XIIL RULE. Multiply the AreA of the Bafe^ by One third Part of the Altitudei ^ tbeproduQJhaU be the Solid Content. In the Pyramid ABCD jF,the Bafe at the Bottom being a Square, ' and each fide thereof is 18, and the height thereof 540. (i.) Multiply 18 by 18, thcpro"duais324, which multipliedbf i8o(one third part ot the height)the produft is 58320, for the So- lid content of thePyramidt 24.ft ■ Menfuration of Saperfaes and Solids ^ 8cC. 239 24. te Mcafure a Cone* ¥ig. XIV. RULE. Multiply the Ared, or Content cf theOrcuUr Bafe \ hy metbird pm^of the height of the CopCj the produS jhaUbe the Solid Content fftbe Cone. ' LetO X rz be a Cone,whoreBafe;rr2; is found (by the.torfc. or I u* hereof) to be 346.5 ;and the Height thereof OS/y^o : Mul- tiply 34^. CtheBafe) by 180 (One third Part of the height) the pfo- -duft will be 62370 for tjie Solid Content of tlie Cone. 25. To Meafure a Segment or Fruftum of a ^yrmid or Cone* Fig. xin. . The Fruftum of a Pyramid, or Cone, i^ when the Smaller Hnd thereof is cut off ; and in. this form are moft Timber Trees either Round or Squared ; and fuch is the Fruftum of the Pyramid J{srVj BCD El whofe Solidity let be required. RULE. Multiply the Areas of the two Ends one into the other \ find the Square^otof that produB\ Then add the tvoo Areas^ and this ^oot together 'j md Multiply their Sum by One third part of the length of the Frufiumj and theproduBJhall he theSo\id Content of the Fruftum, So,the Square ofthe greater End BCI>£,(tbe fide being 1 8; is 324f and the Square of the Leffer End ]{STV Cthe fide being 1 2) 15,144 : thefe two S quares multiplied together ,the produ^ is 46656 •, the Sq, Root of which is 216 : Add thefe three together, f/^. 324,144 ana 216; their Sum will be 684 ; which multiplied by 60 (One third Part of 180, the Altitude ofthe truftum3theprodu<l will be 41040 ' which is the Solid Content of the Fruftum : And thefe (if they were Inches wouldmake ^37-flf Foot. 26. JIow to Meafure a Cylinder or Column. Fig. XV. RULE. Multiply the Area or Content of the Circle at the Safe {or End) of the Cylinder \ by the Jtieig^ (or length )f hereof ; tbepraduS mil be the Solid Content ofthe Cylinder or Column. So in the Cylinder ABC i?Xet the Diameter ofthe Circle at the £nd A By be 7.00, then will the Area of the Circle be found to be 38.5 •• This muh:iplie4 by the Length thereof -rf Cor BZ>, ,12.00 produceth 462 , for the Solid Content of the Cylinder. H h 2 27. Hovf 1 27. ffwK to Medfure Spheres^ Globes or Bullets. Fig. JCVJ. The Axis or Diameter of a Globe (or Bullet) given; to find the Superficial Content thereof, - . * ' RULE. MuJtipljf ^ S(fuare of the Dimeter '(always) ^y 3 $ $ ; and divide thdproduB (always') by ii^y the Quotient Jball Jhei9 the Siiperfcij^ tontent of that Glebe. '^ LetjiBCDy be a Sphere or Bullet, wWe. Diameter .r4Bis2i: .this multiplied^nto it felf produce;)! 441 for the Square of jthe Dia- tneter : This multiplied ty 355theprodua will be 156555: And thisproduftdividedoy ii3,givethintheQiioticnt 1385.44 for'thc Superficial Content of the Globe or Bullet. 28. TlfeAxis, or Diameter 0/^ Globe for Bullet^ ^zveir to jiffirfe Solid Content t&sreo^. .• . ...... ,.^- . RULE. Multiply the Cube of the Diameter (^always) by y^'^ a^itheprodud: ther€cf( always) divide by 6j^^d ihe Quotient fhall be tbt Solidity oftbe (plobe cr Bullet : ' ' ^ So the Axis -<^ B, bemg2i, the Cube thereof 159261-, which mul- tiplied by 355,theProduft will be 328765 5, which divided by67^i gi veth in the Quotient 4849. 048 ; which is the Solid Content of th3it Globe or Bullet. According to thefe Rules may any Regular Piecepf Stone or Timber be Meafared : I (hall conclude thi^of M^afuring of Solids with (hewing you how to IMeafure any' Tapering Timber Tre^, which is not exaftly Scpare at both Ends in Foot Meafure. 29. Let there be a Timber Tree, whofeBr^th atone Bnd is 2*3 Eqot and the Depth 1.8 Foot : ^nd the ]^H^r^^( the other- End 3.5 Foot and the Depth 2.1 Fgotj and letT||f Length of the Tree be 42.6 Foot. FirSly 2.3 multiplied by 1.8, gives 4.14 for the Area of the Lejfer End t And 3.5 by2.i gives 7.25, tor the At:9a of the Greater Efed. ' Secondly f*mu\t\^\Y 7.25 by 4.14, the produft will be' 30.0156 : the Square Root whereof is 5.48 : Add thofe thipe Numbers toge- ther,' w^. -4. i'4— 5.48 and 7.25 and their Sum will be 16.87: This multiplied by 14.2 (One third Part of the 42.6, the Length) tie produft. will be 239,55 Foot, and fo many Solid Foot of Timber is there in the Tree, ...,<,, . . >;. ^ i ■- ■ JII. Of ? c J-^. XW. Of Mea/uri»g and Buildwgy &c* 241 III. Of the Miafuring of theWprks of th feverd Artifcers relating to Building : and tvhat Method ^ and Cufioms are Objerved therein^ T He Principal Artificer^ relating to Building (or repairing) of Hpufes and other Edifices, are the Carpent.et, Bricklayer, Plaifterer, Mafonjoyner, Painter and Cjlafier,CJ'r. Of which, foi^ of their Works are Meafiired by the Square of Ten Foot (or 100 Square Foot.) — Some by the Rod,Pole or Perch of i5Foot and ai Half (272 J:, Square Foot making ^ Rod:>7r-Some by the Yard Square of 3 Foot, 9 Square Feet making a Yayd Square : — Some by the Foot Superficial containing 144 Square Inches; Or by the Foot Solid containing 1728 Solid Inches. of tbc Carpenters Jf^orh « The Carpenters Works which arfe MeafureaMe are Roofing, flooring. Partitioning, CS»r. All which ^^ Mcafured by the Square of 100 Superficial Feet. * I. Of l^ofingr\ It is a Rule iregeived ampng Wo^rk^^men ; that the Flat of the Ground upon which any Hpufc is Ere^ed, and fi^lf the iame Flat (the Dimenfions uken y^ithinthe Walls) is equal to the Meafurfe of the Roof that will Cqyer tb^t Houfe, as to the Carpenr ters yVork, So that, If a Houfe within the Walls be 46 Fbpt Long, and |8 Foot Broad, it will contain 1242 Square Feet ; For 46 multiplied by 18 is 828 and H^lf that is 414 their Sum 1242 Superficial Feet; 0iat is, 12 J^quare42 Foot, or i2i'quare i Quarter and 17 Foot j ^d fo many Square ' feet will that Roof Contain, mtCy 2 5 foot is one Quarter? 5ofootisHalf - J^of a Square 75fopt is 3QuaiitQrs j' t . ?I. Of Floorifjg. The Length multiplied by the Breadth gircs the Content. So, If a Floor be 57 foot 3 Inches (or 5 7.2 5) Long,ahd it8 foot 6 Inches Cor 28.5; Broad multiply 57«25 by 285, the pro- 4aftwill be 1531.625 Feet, that is 16 Square i Quarter, 6 foot 7-5. Inches : find fo mu.cl^ doth that Fl9or Contain» ' . 42^ Jfpwdix. Ill, Of Tmhimng. Partitioning is Meafured by multiplying the Length thereof into the Height ; fo that if a Partition be i{2 foot ^ Inches (or 82.5) Long, and 12 foot 3 Incbes»(or 12.2$) High: 82.5, multiplied by 12.25, the Produft will be 1010.625 feet,thatis 10 Square foot and; -J- Inches,for theContent ofthat Partitioning* There are other Works about a Building performed by the Car- penter, which are Meafured by the Foot, Running Meafure,tiiat is, by the Number ot Feet in Length only, Guttering r Skirt-Board ^^ Cornices f Lintale O Rails and Balafters ) Moldings ^Breft-fomers^ Timber-Fronts Shelfing ^DreflingjSV* There are other things alfo, Valued^er Peice, O^r. Doors and Cafes C7 Balcony Doors 9 CStairs and Cafes Window Lights 2 > Cellar Doors > ^ Mantle-trees iantem Lights ^•i ^<>^"^^s, Pilafters 3 ^Pedinaents And diners other things not here mentioned. Note J 1. In Meafuring of Flooring, you muft deduft out thereof for the Well-holeis, for the Stairs, and Chimney-ways : And in Partitioning for Doors, tfc. except (by Contraft) they are to be in- cluded. But in Roofing, feldom any Deductions ace made ibr I^les for the Chimney Shafts, the Vacancies for Lutheren Lightsor Skip Lights, there being more trouble in the ordering of them then the Stuff is Worth to Cover them. II. of the Brick'layers Worh The Principal are Ti7i»^,. WMng and Cbinmies, I. Of tiling. Tiling is Meafured by the Square of to Foot, as Flooring and Roofing were, fo that ioetween the Roo6ng and the Tiling the Difference will not be much, yet the Tiling will always: be the greater. And befides, fometimes the Bricklayers will re* quk^'RvniiiagMeafure for Hips andA^lleys, which in fome Cafes ;iqay be allowed, but in Plain R00& not. An Example of this kind is ne^dlefs^ tl. 0£ WkBing : All Walls in Brickwork whether for Indofing of Gardens or for Houfes, are Meafored by the Rod of 16-; foot, Meafured uoon the Superficies of the Wall, each Rod containing 272I Superficial feet ; but this is not ally For allBr ick-walls of what Thick- Of Meafaring mJL Buildings Src. a^j Thicknefs foever, they muft be Reduced (all the parts them; to a Standard Thicknefs, that is, to One Brick anda ha|f Thick. Example I. If dBnchroAU be 192 fotmLong^ and iz foat£tigbf bov9 many J^d if. cammed therein ? Multiply 192 by 12, theproduftwil! be 2304, which divided' by ^72 (for: die ;- of a foot is always rejt^aed,;] the Quotient will give . 8 Rod, and .128 Remaining, which divide by ^8 (the feet in a- quar- ter of a Rod,) and the QuotientivTll be 1 quarter, and 60 foot over 5 So that the whole Wall contains 8 Rod, i quarter, and^ofdot ; which wants but S foot of 8 Rod and a half, ^ But there is yet fomething elfe to be confidered^ in the Meafiiring of Brick-work; Namely, the Thicknefs of the Walls; for the Thicker the Wall is, the more Rods will be Contained in any Mumbei*of feet Meafurcd upon the Superficies : For, If the Wall be oneBrlok ahdahalf Thick, then 272; foot Meafured upon the Superficies of the Wall, will be a Rod Standard ; But if the Wall be Thicfcter than One Brick and a half, the 2725: foot Meafured upon the Super- ficies of the Wall, will make more than a Standard Rod : And if Thinner than a Brick and half 272 J. feet will be Lefs than a Standard Rod : And therefore. For the ready Reducing of Brick-work of any Thicknefs, to that of a Standard Thicknefs, vi^^ OneBdckanda half: Obferve this » » GENERAL RULE. Muhipiytbe Number of Superficial Feeutbat are found to be contain-- ed upon the Superficies of ^ WaMj by the NMmb& of ht^ Bricks wbicb that U^aU ft in Thicknefs ; One third. Part of tbm FrodM^Jhatt be the Mimheri^ ^ds and Parts of a ^dcoattameiin that WiM^dme^to Standarfl Thicknefs'. Example IT. If a Wall be 72 footLong 19 foot High, and be Fire Bricks and a half Thick; Hpv^ raarty Rod of Bpick4W(w]p(Re- duced to Standard ) is tlier« Containediki that Wall ?. , Multiply 72 by 19, theProduft will be 136*8, andfofiMMf^Strp^:- ncial feet there are upon the Superficies of the Walte^*-*--M«K5ply 1368 by II (the Number of half Bricks^lD^t the Wa»iifr^mtfak:kneGi) theproduawillbei'5o48. One third Part whereof is $016, and fo , many feet doth that Wall contain, it being Reduced to Standard- Thicknefs-^Now if you divide thefe 5016 feet by ^ji the Quot4ent will be.i8 Rod,^a4LMo R^emaining;which diviiied by 68,the Quoti- ent is I quarter, and 52 foot Remaiair^;So,that thewholse Wau Re- duced to Scandard^Thicknefs, Contains 9 18 Rod, i quarter.and 52 feet. .V Other 2/^ Afpendix. bther Brick-layers Work, vi^. Ctimks^ The Chimnies in moft -Buildiags are agreed for by the Hearth in each Room, and fometimes they are included in the Building, and Paid by the Rod} and Meafured in this manner — If the Oiimnies ftand fingle, tlieUfualway is to Gir^ it about; and iF the Jaums are but One Brick thick, and wrought upright over the Mantle^tree to the next Floor ; then Girt it about for a Length, and the Height of the Story for a Breadth, at One Brick thick. .— — But if the Chimny ftand againft a Wall that is before Mea- fured^ then the- Breadth of the Breft and the Breadth of thef Two Jaums is the Length, and the Height of the Story the Breadth at One Brick and half, if the jaums be (b Thick, arid nothing to be Deduded for the Area between the Hearth atid Mantle- tree ; be- caufe of the Wyths, and Thickening of the Hearth above, i^of Ijhe Chimny Shafts, Girt them about in the fmalleft Place, for the Breadth^ and the Height of the ^aft for the Length, at One Brick thick, in confideration gf Wyths, Pargetting and ^cafFolding. Other Brick-layers Works which are Meafured by the Foot Running Mcafure, are Cornices of all forts fStreight Arches 7 Hyps and Valley's. Facices \Skeen Arches 3 Water Courfes, ^c Other Works valued by the Piece, vix* Peers, Pilallers, Plain or Ruftick, Pediments, ^c Paving al^ is Brick-layers Work, with Brick or Tiles; and is Meafured by the Yard Square^or 9 Foot, fo, if a Cellar be Paved with Brick 32 foot Long,and 19 foot Broad ; multiply 32 by i9,the produ£l will be^o^i which divided by 9, the Quotient is 67 Yards, and 5 footi Nate^ That in Meafurlng of the Brick-'work iii Houfes, if yo^ take the Dimenfiods of the fides on the Outfide,yott muft take the Dimenfionsc^ the Ends on the lofide, And Note alfo that Dedufii- ons, arc to be made, for all Doors^ Windows, ^c. according to the feveral thicknefles of the Walls in which fuch Doors or Windows are* IIL of the MafoHs W^* jA^ms do Meafure all their Works by the Fck)t, either Superfi- cial or Solid, and therefore I need give no Examples, for the Rules btfore delivered in this Se£lion for Meafuring of Superficies and 8Uid9, are rufficieot tp perform any Work done by the MaTons^ ■^ ^ * Of Meafurif^ ami BmUingy 8rCr 94 j| IV, of tb€ FldJkrersJ^rvrh Plalfterers Works are principally of Two kinds, MamdrtC'J Work Lath'd and Plaiftered, which they call OrUing— (2.;Work, Rendered^ which is Principally of Two Idndsi vi^. npJMi Bride* walls or between Quarters, both which are Mmnitd by the Yaxd Square* I • Of Ceiling ; If a Ceiling be s^fbot 9 inches, ("or 58«7jS) tpf« and 23 foot 7 inches (or 23.58; firoad}— Multiply 58.^5 h| 23.5S, the produft will be 1 914^7, fi)ot, whichdiyideaby9 givd in Ihe Quotient 213 yards and 6 foot, for the content of that Ceiling. 2. Of Partitioning; If the Partitioning between theRopms oa ' one Floor be i32fbot about, and the Story 12 foot High-^Mnlti^. ply 1 32 by 12, the product will be 1 584, which divided by 9 giVei in the Quotient i;^6 yards, for the Content of that partition. Note uli there be any Doorsi Windows, or tbelike, in your Partitioning. youmUft.nuikeaDeduftion,forthem« Nott 2. When ybu MeafUre tlendring upon Brick- 1nrotk,yotf stte ttf make no deduOions^ but to MeafureaU where your TroiArelgcJ^^-^ But when you MeafureRendring between Charters, you niay very well dedua,ofle $ii&.Part for the Quarters, Braces and Interftices, or^ Note 3i Whiting and Colouring are both Meafurcdby the Yard ^ and in Meafuringthe Whiting of. Cieling Partitioning and Rend- ring upon Walls will be die fame with the PUiftering,yet in' Whiting and enuring between Quarters, you may very well add one Fourth or Fifth part olfore. V.Vfthe ^opitrsWorh '> Joyners do Meafure their Work by the yardS^tt of 9*F6ot,ba(r they have a Cuftam, and &y, we 6uglit tor Meafure where otir Plain touches : And therefore ; In talking the Kmenfions of * RoomE. where there is a Cornice abotit, and fwellihg I^hncls WithMiJld^ ings ; they with a Line, do Girt over every Member of theOnmke and fwelling Panneis and Moldings DowiiWardb, and take thaiT Length for the Height; ^ut for the Girt of the ft<^m sibout, they, take that only as Flat. Example , If a Room Waxitfeotei Were thiifs Girt dbwn*ardsif fliould be found by the ftring to be i $ foot 7 inches, (br i $.58jand, in Compafs about 286 fdot, how man/. yat!ds doth t^t Robm con- tain of Girt Meafure? ~ . Multiply 28i foot, by 15-58^ the produfl: will be 44$$.«8,of' i\i6 foot,' which divided by 9, tbe Quotient will be 49$ yard's^ and OnefoiU, for the content. -^ ^ In Meafuring of Joyners Work, there is another thing Obfcr- rable ) that is, in the Meafuring of Doors, Windovi*lhuttehv Cup- li boardi^ ^risf Doors and Drawers, atid fuch Works a$ are Wrought oti \fi^ fides } for thefe they dt> Account to be paid for Work md y^^^joik^ ftriilcked the Work i$ more; though the Stuff be the : ^fip^^f I^ tho Wlndow-flmttdn abott a Room be 7lfe6t 4 »(*bs for 78.54.; and ^te Height of them 6 fi?et 6 inches, Cor 0'. 5 ) how many yards are contained in thefe Ihutters, at Work and halt? ' ; JiuWply79«34>by6.$,theprodqft will be509.2i,the half where- tf is i$4*i5o and thefe two added make 765,8i,fbot, (or 7i54foot,) wliich fividcd l)y p, the Qjioticnt is 84 yards 8' foot', (or 85 yarfs.) 2>fore, that Deduftions ar^.tobemadeforall WindQw-lights ; Jfat you maft Meafbrc the Window Boards, Safeta and Jaum Board$.by tberafelves, as Work only fingle. VI. Of the taiHsers mrk. ' The taking of tfec; IMttienfiOns for Painters-work is the fame as for tbe Joy ners, JnrCHceiti(fta£jae Moldings, ^c. Th&Dimeofions ia «di^; tBei Cafting up, and Reducing of tb^. Feet into yards, i8^.4ftogptber the ftme as the Plaifterers and Joynefs were: And tji^ ncv^r 6y Work aad bjlf \ but,Once,Twice or Thrice Primed orCokMired* . .Exwples herein are tmneceffary, Only take Notice, that Wii)- dQW-U^ts, Window Barj, QOfemcnts, and fuch like thhigs,they reckon at fomu/chjptfrpieo?. ' ' VII. Cf th GUfiers Work. \ Glafiers do Meafure their Work by the Foot Superficial, fa that the Length and Breadth omltipUed into each ether produceth the ^qantiity of feet in that Pain of Glafs. Soif aPainof Qlafe be4foot 9 inchesCor 4.75)Long, and 3 foot a inch^Qr3.2$)Broa<fcIfyou multiply 4.75 by 3.25,the produSwill «^ X5*4?75»**t is J5 footj 5 inches and a quarter, for the Content. ' Jyr«r, TbatwheQ Windows have half Rounds at the Top, they Meafure them at the fidl Height and Breadth as ifthey were Sqwsre. Air<> Round and Oval Windows are meafurcdat the foil Len^ and Breadth of their DiameterXikewife Crocket-windows in Stone- woi^ are allmcafured at then: foil Square } And there is x^afon for fuch allQwance : For the Trouble in taking the Dimenfibns to Work by, ,Thc VVate of Glafs in Working, and the Timefpentoe traordinary in fetting up of fuch Windows, is for more than the Glafs faved can be valued at. And thus much concerning mea- ftring. But for farther fatisfeftion, you may confult Curfu^ Mofk- rnmcus } and my Book for Building, &c. Tht End of the S<(mA fm. INSTRUMENTAL ' ARITHMETICK. The Third PART. TEACHING, By a new Artifice ^not heretofore PublijDbed, to my Knowledge, in any Language.) , The manner bow to let down any Dtcimal FraElim | required : Or a DecimstFrahion being given to fiod the value thereof in En^Ufii JHory^tight or Meafure-^ by inipedion only. By certain SCALES contrived, fuitable to the Coinsj Weights and Meafures now Ufcd in I I EngUnd. And for Extrading of tbc Square * and Cubcjloots. ALSO, By NEP air's BONES WHEREBY, Multiflication^ Divifwn^ and the ExtroEHm of the Square and Cube Roots are performed with won- derful Speed and Exa&iefs. By William Leybaurn^ Philomath. . LONDON^ Printed for Aamjham aadjohn Churchill at the Blacl^Swan in Pater'No^er^Rm. 1700. I » * I » *• %■ V ♦'^ II ' r N [2493 Yumental Arithmetick, SECT. I. ^ Decimal Scales. He Jrhiemetich of which we now come to treat, and which I 'AJZL call iMfifttmemdl jtrtthnetuht is not any new kind of jthbrnC' /iiiw/iv. ^m; jg indeed the Tame with Dcchml Arithme$ick before taught { go| f, whereas in Decimal Arhhmticl there were certain Tables made mfmw neu Weight and M; ^/tfre^by help ofwhich.the Decimal of any 9)^' i< ftion of mmey^ Weight and Meafure, might be fet down ("as it ;./ fftfc^ in whole numters j.here in this Inftrumemalpan^ we have con- ed certain Scales oi^iArniey^ Weigh wai Meafitre equally divided » the feveral Denominations into which the Weighs doiA Meafures which they are contrived, may be equally divided. Unto all ch Scales there is joyneda Scale of leo, 1000, or iocx)o, e* 1 parts, aco^ng to the length of the Scale, fo that by infpefti* ronly you may readily and exadly without Addition (as in ufing ^ forementioned Tahles you muft neceifarilv do) fet down the cimal FraSim of any part of Monej^ Weigjk or Merfure^ witk ^t celerity and exafinefs, if the Sca^ be any thing well divided^ dbe butotareafonable lengtht Now the Scales which I have chiefly made choice of in this Work ; being of moft Uie vnithEngliJh-xni^i (though other Saks may made for the Coim^ WeigjbtsQx Meafires of any other Country ^ ell, and upon the lame ground) are chiefly thde, vif« 1 Of Money. a Of Troy Weight. 3 Of Averdupois great Weight. 4 Of Averdupois little Weight. 5 Of Liquid Meafures. 6 Of Dry Meafures. 7 Of Xong Meafures. 8 Of Dozens. - ' 1Anto.every thefe ijf Scales, is joyned anotlier Sdk pt locor 1000 equfid iftd$« thefe «rcales aircrmade to face one aifother, & 250 InJirumemAl Jrithmetick. that if yoa look upon any one Divifion in the one, you ihanalfodifJ ccrn plainly what Divifion or part of a Divifion anfwereth' unto in tbeother. . . Tbeic Jcales being thus difpofed, astheymayeaiilybengo&lq ^ukr of Silver^ Br (Us or Wood ; but beft or all upon a f<fuar€ J(i made in form of a pMysBelepipedon^ will by infpeaion only give any Decimal. fradion required without Addition^ (or on the trary) reduce the fraiiion into the known parts of the Ihteger, infpeaion alfo, without SidfiraBioH. Let thus much fuffice for a genbral defcription of what I mean 1 Scales, tiie particular defcription of them will more plainly appear^ when we treat of Niimeratidn upoa the Jules, unto, which w( (hall now proceed. But fir ft take a view of the Scales as they an her^ difpofed, and ^ they may befet upon fujch a Ruler as Ihqc here mentioned. .. ^ NufmeMtitm ufon the Scahs. •1 THe Jtale hereto be defcribedare in nYimber Kght, asi been Already ftiewed, and as by the figure of ^hemappci Kow Numeration upon tf 5cale, is to f rid upon tpbaipm of tbcSX 0tjmwbtrupmthefmeScAtmUhlU We will begin with the firfl:, and fo proceed tffl we have ^\ an Example in every on^^ I. The firift Scal6 is ot Bngliffb Mohey, and is divided into 24c- gual parts, which rcprefent 24 pence or 2 Ihaiings, thefe parts aire irambred with Arithmetical figures, -from the be^nning thetetf, by '1,2,3,4,5, &r. to 24, each divifion reprefentins onepeny, ^d ^the Whole 24 divifioHs reprefent 24 pence, or 2 millings ;\Jt) that where t^e figure 1 ftandeth that part of the .Tcale rq>menteCh onepeiiy, where the figure 2 ftandeth, it reprefentetft 2 A where the figure iS ftandediitreptefenteih 18 pence, or otiis fliilliflg ilx penceand fo of any other ngtkre of die fiime 5^cale. i;iien becaufe there are four fiurthrngs coutdiiied in a pefty» each of thefe pence (or divifions; is fub^Kvided iitto 4 other equal pirts by Siort Lines, every one of thefec^veifentingooe&tdikift, lb istbe whole ScaXt divided in all into 96 equal part9,«bich txt Xm nsambcr of &rthirig$ contained in two killings. Thus if you faiok into, the Jcaleof Monyfo]^8pence 3 farthings, youSiaQ find it at tbe Letter 4t^ which Letter is here put only tor example fake. Alfo if you wpuM find ill the ^cale tbe ^e of 18 i, halttff«T» you ftall find it at«,ahd thusmayyou find the places of aiiymiw^rofjpeiK)ea$d fivtluogs Wider two ihiUings upon the J'cale* Infirumntal Arithrnktick. 5^5 1 Unto this Jcale of Mony (as to alkthe reft of the J'caks) there is 6yned another Jcale of icoO| the ufe of which Jcale is this. Whea ^oohave found any number of pence or farthings upon the i&cale )rMony, you ihall find upon fte Salt of looo, what parts of a . iionfand is the Decimal of dioTe pence and farthing^ ; Thus whea At|]e.fcakofMony,youfind it the Letter ^ 8 i. 3 a.if you caft your ^t dlret)if cxoS^ tx) the Scaled 1000, you fhall find 364 to ftand di* re% againft S^^i^g.which 364is the decimal of 8 i/^ja. Alfo ifyou find oponthe 5cale tif Moiiy i8i.half-peny,which is at tne Letters, you l^n find againft it in the ^cale of iooo,this number.77 i,which is the Decimal of i8rf. half-jpeny. And iri this manner may the Peci<* , mal of any number of pence or ferthings under two Ihillingsbe Aoft ^fily and exaAly obtaihM. Now on the contraryj fiippofe a Deciifial Fraftion were given, reprefentingfome part of JB»^///b Coin; if you look ihthe Jcaleof looo for your number gfven right againft it in the Jcale of Mooy, }oa iholl find what number of pence and fiirthin^s is reprefented thereby. As for Bicample, Xuppofe .364 were a Decimal given,and it were required to find wMt part of Coin It doth reprefent^Look in the Jcaleof looofor the number •364,and right againft it you ihall fiol Spence 3 forthings^ Alfo if .77 1 were a Decimal given» if you look inthe.Ccaleofiood for .7711 you ihall find againft that number iS pence 2 farthings. And thus of any other* Ey what hath been already faid, it may be eafily difcerned of whait exceeding expedition fhefe .Scales thus difpofed are of, for I dace af*; firm, that I will (et down 2 (if not 3)numbersi by the Scale, as foonasoneby theTables^andif the Scale be but of any reafooable. Iftigth, altogether with as much exafltnefs, but if I Ihould vary air unite in thelaft place, in my eftimation in the Scaleof loocsitis Botany thing material. I have been very tedious in ifewing the ufe of tliefe Scales ^ find the fr'adion-^arts of mony ; but the reafon is, becaufe I intend to be the briefer in the reft, for Weight zxA lAufure ; the maimer of Working (when the Divifion of theScak is known) being the fame ifl aU refpefts. witliout the leaft alteration, 2. The fecond Scale is of Trof-Weighyt^o peny weight being the [Integer, which Scale is divided into 48 equal parts or divifions, each "which divifions containsone grain, and are numbered by Arith- etical figures at every three grains by 3,6,9, 1 2 ,Csr^«to 24,and at the ice where 24fliould ftand, there ftandeth P. W. which fignifieth e peny weight, or 24 grains, this P. W. ftandeth in the middle AeLine. Then is the lame Scale continued farther by Arith- petical figures- 3,6,9,12,0^^. as before to 24, and there is writ- ^l^.W. again, reprefenting two peny weight, or 48 grains. .' The Scale being thus divided, it is eafie to find the place where ' y number of Grains under 48 fhall be ^ipon the Scale ; As for ex-* pie, if it be required to find whgre 8 peny weight fhall fall, look upon J i^i InfifumentAt Arithmetich. upon the Scale ^f Troy-weij^ht, from the beginning thereof, ixA \ count the figtfres 3 and 6 then count alfo two of the fmaDer 'Dm* ' (ions, and that makes S^grains, which youlhall find tofiandat the ^ Letter i. ^hich is the place of 8 grains ; Alfo if upon the Scak ^ you find the place of one peny weight 10 grains, you ihall find it at the' Letter e^ and foof any number of grains under 48, or tvo jefly weight. But if you had a Decimal given, and would know what number ; of grains it reprefenteth, if you feek your Decimal given in the the Scile of 1000, right againft it in the Scale of Troy- weight, yatt ihaH find the nufflJbcrofgrainsreprefented thereby. ETUtmpki Let .167 be a Decimal firaftion given. If you look isL the Scale of locx), for 167, right againft h in the Scale of Ttoj" weight, you fhi^ll find 8 peny weight. Alfo if 708 were ^ Decimal fradtioA given. If you feek 708 in tbe Scale of 1600, right againft it you ihall find i peny weight 10 gndns. 3 The third Stale is of Averdupoii great jnfeigkf 28 pounds, or one quarter of an htindered, being the Integer^ thisSuleisnum' bred by x, 2, 3, 4, ^e. to 28, whidi 28 reprefenteth 28 /.or a quar- ter of a hundred, and each of thofe is fub-divided into four finall parts each reprefenting one quarter of a pound. Now if you would kuoW what is the Decimal of any number of 'pounds or quarters under 2 8, if you feek the number of pounds in the. Scale of Averdupois great Weighty riglit againft it in the Scale of Xpoo, you fhall find the Decimal thereof. Thus if it were required to find the f)ecimaiof S pound and ap £a1f, if you look upon the Scale for 8 pound and an half, you ihall find it at Letter g^ and right againft it in t^e Scale of 1000 you ihall find. 3049 which is the Decimal of 8 pounds and an half. 44 The fourth Scale is of Averdupois little Weightf id ounces or cine pound being the Integer; This Scale isfirft divided into i6e* qual parts, and numbred by i,2,3,4,Cs^<;. toid^eachDiviiion re* prdenting one Ounce. Then again, each ^f thefe ounces is fub-di^ vided into 8 other fmaller parts or diviiions, each of which divifi<»s reprefenteth two Drams 5 but if your Scale be large enough, you ihayhaveeachotince divided into 16 equal parts or divifions,each divifion reprefeilting one Dram. Now to find' tile Decimal belonging to any number of Ofluce& and Drams, repair to the Scale of Averdupois littkWeighyZtdon It the quantity of ounces and drains required,and right againft it in the Scale of 1000, you ihall have the Decimal thercQf*^ Thusifit wete required to find the Decimal of 6 Ounces and 6. Drams, if you lo6k this in the Scale of Averdupois littU Weighs Jiou ihall find it at the Letter ^, and right againft it in the Scale of xqoo : you fhall find 398, which is tlic Decimal of ^gunces and 6 drams. . , ' 5; -The Inftrumentat Arithmetick. «5) • 5k The fifth Scale is of J?ry neafuresj one Quarttrr or 8 Bufliels king the Integer j this Scale is firft divided into & equal parts, aa4 numbered by i| 2, 3, &r. to 8, each of which divifions reprefent^th a Buihel, and each o^ thofe parts is again fub-dlvided, firft into 4 equal parts or divifions, each reprefentingoaePeck, and then tho(o ag^in fub-divided into 4 other fmaller parts, reprefenting Quartersi Halves, and Three Quarters of a Peck. Now if you would know the Decimal belonging toany nund>er of Bufhels (under 8 Bufhels one Quarter) Pecks and parts of a Peck» if you feek the number of Bufhels, Pecks, and parts of a Peck in the tole of Dry Meafur^Sy right againft it in the Jcale pf 1000, yott Ihall have the Decimal required. Asfor Example, if it were required to find the Decimal bdopg* iog to 5 Bufhels 2 Pecks, and half a Peck; it yott look into the Scale of X>ry Meafures you &all find $ Bufhels, 2 Pecks^ and an half to •fland at the Letter k^ and right againfl it in the i'cale of 1000, yp« Ihall find .702, which is the Decimal anfwering tQ 5 Bufliclt, 2 Pecks and a half. 6. The fixth J'cale is of Uiuid Merfuresj the Integer being 36 Gallons, 6r one Barrel, this 5cale is divided firft into 36 equal parts • or divilions, and numbered by 5, 10, 15, ^c. to 3^, then every of thefe diviilons, is again fub-diviided into 4 other fmall divifions, each reprefenting a quart, but (if die ^cale be large enough) you may fcb-di vide each Gallon into 8 parts, fo will every part reprefent one pint. Now to find the Decimal belonging to any number of OanoAs(un« der 36 Gallons or one Barrel} quarts or pinjcs,^rcpair to the .fcaleof U!\jiid Meafures^nd feek there upon the Scale, tlie number of gallons, quarts, pints, and againft it in the J'cale of iooO| you ihall find the Decimal thereunto belonging. So if it were required to find a Decimal reprefenting 10 gallons and two quarts, or 4 pints, which is all one; if youfeek in the Scale MdLiquU Meafures for 10 gallons, 2 quarts, you (hall find it at the letter mt againft which^n the Scale of 1000, you Ihall find. 2929 which is the decin^al of 1 gallons 4 pints. 7 The feventh 5cale is of Long Meafwrcsy the Integer being J^ds or EUSf this Jcale is divided into 4 equal parts, and numbred by 1,2,3,4 reprefenting i quarter.2 quarters, 3 quarters or 4 quar* tersofa Yard,or,JEll, thefe are again fub-divided, firft int0 4 other equal parts, reprefenting Nails, and thgfc may be agaip fub-divid- ed at pleafure it need be. Now if you would know what decimal belongeth to any numr ber of Quarters or Nailes of a Yard orBll, if you feek the num- herof quarters and nailes inthe^caleof LongtAgafwtc^ the i'caleof looo will give you the decimal thereof. K k. Thus 1^4 luifiimefftd Arithemetich. Thus if it be required to find the decimal belonging to i ^^^£y Und 3 iiailesj if you feek this in flie Scalcf of Long Medjure, you " find it ftand at the Letter o , againft which in the i'cale of looo yott (hall find .437, which is the Decimal anfwering to 1 quarter and kiails of a Yard or Btt. ^ . < 8 The Eighth and Laft Jcale is ofreprefimative ImhiSy the wbold .ycale being divided into 12 equal parts , arid numbred byj i,2,;,&^. tol2.dndthofe parts are again fub-divided into halvcsl quarters^ and half quarters, as Carpenters-Rules aire ufually di< Vided. - "* Unto thisScale (asutito all the other) there is joyned a Scale 1 coo, this Scale Will readily difcover what is the Decimal belong Ing to a Ay number of Inches, halves or quarters, andtheufeisliif fa^fe with the Scalesljefore mentioned. Thus I have given you a brief Defcription of thcfe Scales, an the ufes of them, aftd db now foppofemy Reader to be perfeflly a< quainted with the way of numbering or accounting upon them wherefore I intend only to give you a Queflrion or two inthemo^ vfuiil Rules of Arithmetick, and fo conclude* ADDITION. 'X)iJ}iitMikionv^ and the manner of Working of it hath been W already taught, bothin thefirftand fecond Parts, wcwil now come to an Example, which let be in Addhim of Bnglifi) Coin* and let the fums to be added be 3^ /• 8f. 81^.29 /• ox. ^^.3 1 /• i6x. 9 i, and 6 /. 2 X, $ i. Firft, fet down 36 U 29 /• 31 /. and 6 /• one under another, i8 (bch order as you fee herein the margine, drawing a Line by the fide of them as70u fee done, and alio a Line under them. , This done, feeing that your firft number to be ftt down to 36 /. is 8 J. 8 d. you muft for the 8 j.(be- (caufe two ifaillings, which we call a Decade^ or the lenth part of a pound* is made the Integer, in the - ^Scale of Mony ) fet down 4, which is done by mc- «iory^ ani| after it make a Comma ; Then your next number to be fet by 29 /. being os.ad. for 36 | 4> the 6 s. fet down a Cypher 5 Thirdly, for your num- 29 0, ijertobefetby3i/.being 16 i. 9 i. for the 16 5. fet 31 8, idown 8 J)ee4des with a comma after it, and Laftly, the 6 i» number to be fet by 6 /. being 2 f. 5 i. for the 2 s. ■— ' I fet down iI>ecade<:ommdL after it, and tten lirill your work ftand, as here you (ee. Tbct 36 29 6 \ tnfirumentd Aritbmrtiek. «55 36 29 6 4,335 0,085 1,208 Then take your Scale in hand, and feeing your firft iiumberof 'pence are 8 d, look in your Scale of money for 8 4- and 9gainft it in your Scale of looo, ,you Iha^l find 333, which fet in the Lipe with [.' 96/. % s. behind the comnaa, then your next number of pence l)C- ing 2 i* look in your fule for a ^« and againft it in the fcale of i c 00, you (ball find 083, which fet to 29 A 01. behind the comma* Then your third number of Pence being 9 d. look in your your fcale for 9 i. and againft it the fcale of iooo,vou mall find 376, \slQiich£et togi/. 16 s. and Lamy, your laft number of pence being 5 d. look in your fcale for 5 ^« and againft it you fhall fi^d 208, which fet to 6 /• 2 s. and then will your whole work fland, ■ ■ . ■ as here you fee* Your fums being thus fet down, which is done with mor^^ci^ijly tlian you can imagine, till you make trial and he£bmething perfis^ therein, you muft then add all the numbers together, as in Addition of Decimals and you Ihall find the fumof them to be io3|4,ooo. Now to know thisinmony, is as eafie as it wast0fet(everal fwns down, for the figures 103, which ftand behind the down right line, are 103 /• and figure 4 which Ifands between the down right line and com- ma are, 4. Decades or 8 r. and being the reft to the right hand are all Cyphers, they lignifie neither pence nor farthings, foi? the total of this Addition 103 /• 8j. o^. That the manner of working^may appear i»9re plain, I will give vou another Ihprt Example as difficult as Iff^sil^- vent, which I performed by a fcale of Wood but of 8 inches long, tec the funjs to be added together be thefe fottowing : 36 4>333 S9 o>o83 3' 8,37<^ 6 1,208 103 ] 49000 u. 3J2 159 2x7 i7 6 5 4 8. I u 3 709 09 4 1 Firft fet 4qmwi your fevfjral $ums of pounjis or>e 332 under another as before,' and draw a line by the. 159 iide of them, ^md another under them. So will they 2 1 7 ftand as here you fee. -« — j vYouf fmns of ppunds being t;hu^.Qi:derly placed aod lines drawn^ repair to your Scale, and fteing your firft num- 'bfer of fhillings, pence and farthings is 1 7 j. 4-rf. \ ^n ibr your 171. fet down ^Decades which is i^s. with a comma after it, then will ttere feftto4>efetdown ix.4^«i4.ori6i.i)^wh)!chifyou fe^k jnyour K k 2 Scale ■ ' 1 ft 5^ I^ft^^^^nfal Afithmetick. Scale of mony, you ihiU find to ftand againil it in the Scaleof looo fhis nnmber 677, which is the Decimal of 4 s.\L i q, a.YoTirfecond aamber of fliillingf, pence and farthings is 6 ^ Si* I f.for your 6 x. fet down g Decsldes which is 6 s. and there wifl tc- mainSi* i^jl which if you feek inyourScaldof mony, you (haU&Ml t% ftand againft it in the Scale of 1000, thisnumber .344, which is the Decimal of 8 i. i|. 3. Your third namberofihitlings,pence,andfarthlags,is5x.;^ 3 J. for your 5 j. fet down 2 Decades, which is 4x. with acomfiu after it, thea wil] therefi: to befet down \s. 3 i. x f. or 15 ^ ?}• /• which if you feek in your Scale of niony, you (hall 332 I 8,677 find to flrand againft it in the Scale of 1000, this 159 3»944' number .656, which is the Decimalof i $i. 3 q. oxn* 317 2,656. 3 i* 3 f • and the three Sums to be added together wiQ r— ftand as here you fee. 709 4,677 Thefe Sums being added together according to the Rule for Addition of Decimals, you (hall find the Sum of them to be 70914,677 now t» know what this is inniony, take notice that the 709 which ftands to the left hand of the down right line are 709 pounds, and the figure 4. which ftands b^ecn the down right line and the comma, are 4 decades or 8 s. but(be- caufe the firft figure ne?t after the comma is above Cs, vi(.6) yoa mufl add i s. to the 4 decades, making them 9 s, then willtbete remain 177. wherefore if you look in theSdIe of 1000 for 177, yoo (ball find ^gainft it in the Scale of mony 4 i. i ^ So is the whole Sam of this Addition 709 /. 9^* 4 ;• as by the precedling work doth appear, f Here note, that when you had fet down your 709 /. 4 ^^' des or 8 x. there remained beyond the comma, 677, which if ba4 fought in your Scale of 1000 you (hould have found againft it)R fhe Scaleof mony 1 $ i. i j. or 1 j, 3 i i f (which is all one^as before > for it appeareth plainly by the Scale, that 590 in tlfc lineot lOOO is equal to one ftilling. \ • iSy BSf R ACTIO N, Vbfiraam (as hath been before fiid) is the ftkjng one or more fmaller Sums put of one greater ; I fhall only give you Example or two, as I have taken the numbers firom a Scale. fix(tntpk. Delivered to a Gol(|t7rmid)»firft of oldi Plate 297 ouQces, jf 3 peny- weight, 1 9 grains. ,^ceiyed of tic 6nje Cold-rmith M i§i ob«w, 1 j P5"T V Infirnmefad Arithmetick. 257 weight aivd 7 grains, and after that received of the lame Gold-fmith 32 ounces 19 peny-weighc and 92 grains, what Plate remainesin (])C CoM-fmiths hands? Take your Numbers out of the Scale of Troy-weight,and fipt them down as you here fee. ^ 1 Delwerti • ounces 29716,895 I(£Cfived firB J^ceived more f 1655^546 329,979 " Kficeiyei In aU 19815,62$ Hefts in the Gold-fmitbt bands Ounces 99H,27£ or 99 penjf^m. p. 2 13 Then add thefeveral weights of Plate received tifgether, and tbey make 19815,625, or 198 ounces 1 1 pen-weight, 6 grains,which jf youfubftrad from 29716896,0c 25 7 ounces, 13 peny-welght, 19 gr. which was the q^uantity of Plate delivered, th^re. will remain . 9911,271, or 99 ounc^, 2 peny-weighc, 13 gr. and fo much Plate is ftill in ^e Gold'finitbs hand. And let thusmueh rufficefbrJtti- jfraBhn, Now we fhould proceed to MuUhticofm and Divifion^ but when die numbers are taken from the ^cale and fet down, the manner of working doth not at all differ frt>m Multiplication and Divifion of JPecimU before taught. But before I teach you how to multiply and Divide Inftcomentally, I Ihall fhew you, ThefATther ufe of tht Decimal Scales ^ and how hy them tofm the Square or Cube Root of any Num^ her. And dljbj any Root hem^ Qtven^ to find the Square of the Cube number of that Root. And both thefe by mfpeftion only, witl^out the help of either Pm, C0j^p4|}ei or any other Motion. T?Or tbc effeSing hereof, there is now inferted, among the fore- r mentioned Decimal Scales of Mony,Weight, Mearure,0^r. name- ly between the Scales of Ayerdupois little l^eighy and tjiat of Ihry Mcafures, Two other Scales, one having written at the boginnlng there- S 5 8 Infirmhenul Jrithmetick. thereof Ae Word Square, and to the other there is added the Word Cube, and between them, there is a third line, which hath writtea upon it;, the word Root. And by thefe three Scales thus united, the Sqoare and Cube Roots of any number may be extrafted by in- {pe^iononly. For, If you find any number whofe Square Root you requh:e,in the Line (NT Scale of Squares, right againft it, in the Line of Roots, you (hall lia^e iht Square Root of that number. Thus, If the number 64 were given, and it were required to find the Square Root thereof. Find the given number 64, ^pon the Line or Scale of Squares (which you may do at thel^tter a) and right againft it, in the Scale of Roots ftands the figure 8, which ihews, that 8 is the Square Root of 64* And in thela^ manner you may find tbp Square Root of any other nmnber. in the Line I ]! 1 in the Scale of Squares , 1 Z I of Roots, Itor, againft ^ |^ >you ihaU y V which is the find ( \ \ Square Root I J I thereof. In like snaaner. If the number 64 were given, and it were required to find the Cube Root thereof. Find the given number 64 in the Scale of Cubes, (which you may 00, by counting the fame number between the fecond and third figuresof lupon the Scale, atnJie letter ^) and right againft it, ia the Line or Scale of Roots, ftands the figure 4, whicfi fliews,that 4 is the Cube Root of 64. And in the 6me manner you may find the Cube Root of any other numben I 343 I intheScate ' ^ * it5| youfhall 64 £nd 27 I L 8J in the Scale of Roots, ^ which is die Cube Root thereof. L »J L2j And by this Artifice, not only the Roots of direCl Square aid Cube numbers may be found, but in numbers that benotdireftl/ Sqnare or Cube, the Fradion pot of the Root is nearly difcover- I V i M oT i l^ 4^ ^1 4 L^ yk ?>x L / ■ • Injlrumenul Jrithmetick.^ «59 1 have hitherto gWen you Examples in fuch Square an4 Cube ntiinbers, as are common and iamiliar, and that any man may com- pute almoft by iriemory \ but by tbcfe the Deraonftration of .the ' Artifice is difcoyered, the Lines of Squares, and Cubes being only Square and Cube numbers transferred to Lines. And now let OS pr<Jceed to greater numbers. And ' I. For the Sqttart Root. In the Extraftion of a Square Root, it is ufual to fet a Prick un- acF the firft figure, the third, the fifth, the feventh, and fo for- wards, beginning fi-om the right hand towards the left. And as many Pricks as M under the Square Number given, of fo many figures will the Root of that Number confift : So that if theNum-' ber given be lefsthan loo^the Root (hail be only of one figure, if lefsthan loooo, itlhall be but two figures, if lefsth^ io6oooo,it fliall be three figures, and fo forward. Henc6, it is. That the Line or Scale of Squaresi is divided firft into loo parts, and if the Number given be greater than ioo,the firft diviiion fWhich is the place where the firft figure of i ftandeth, and which before did fignifie One) muft now fignifie lOO, and the whole line ftall be loooo. If ferther, the nuthbe be greater than 10000, you muft count or efteem the firft Figure of i to be loooo, and then will the whole line be loopooo parti; and if this be too little to exprefs the number given, you may proceed fer- 'tlier, and call the firft i. ioooqoq, and Lo increafe the Line lOO times more; but this is fufficient. Thus when ariy Square Number is given, if you fet it 4own, and prick it, Cor imagine it to befo) if the laft prick to the left hand ihall h\\ under the laft Figure, -(^which will always be when the :T?igtJres in the^ivenNumberbeodd; you miift find all fudi Num- bers upon the! Lme, between the two Figures of i. ' ■■ ■■ ; — ^But if the laft prick ihall fell under the laft Figurebuf ooeof theftven VNumber, twhichit will always do, when the Figures of '.the Num- ber ^iven are even> then you muft find the Number.givenin the Line of Square, between the fccon4 Figure of i audio, at the cad of the Line. Root againft fired. Thus, 56 TheS^are<g^ ■ >will'be W<IJ / .Rootof J?f^ Ctobe Jj? 36000 1 tJ^9* • • And V > 96o Ibtftrttmektd Jrithmetick, y And in this manner inay the nearcft Root of any number not cxadly Square be obtained. For 7.^^. pwilllic C^? The neareftV^'? \Jomd J^S Rootof ^72500. rtobe y69 725000 Nabout ^851 • • • And thus on the contrary, a Number may be Squared, as may partly appear by wliat hath been before delivered ; for if you find the Root in the Scale of Roots, you have itsSquareintheLii^cof Squares and, fo C21 7intheScale^725 7250 Cj^the Square 72400 \ thereof* 851-^ find ^725000" I Thus much for the Square Rout. Now r27 -^m thebcale r Acainfk-^^^ Cof R^ts, J C.8<i-^find ^ IL For the Qtbe Root. In the Extraftion of the Cube Root, itisufualto fet Pricks un* der the Firft Figure, the Fourth, the Seventh, the Tenth, and fo on, pricking always the third Figure from the right hand towards the left. And as many pricks as fall under the Cubick Number given, fo many Figures mall be in the Root. So that if the Num- ber given be lefs than 1000, the Root fhall confift only of one Figure, if lefsthan loooooo, it Ihall be only of two Figures; if it be above 1000000, and lefs than ' loodoooooo, it will be only three Figures. Hence it is, That the Line of Cubes is divided firft into 1000 parts ; And if the number given be greater than 1000, the fiift figure of i (which before did fignifie only Onejmuftnowiignifie 1000, and the fecond Figures of 1 , muft now fignifie loooo, and tbe third 1.9 muft fignifie looooo, and the whole Line niuft be efteem- ed to ^)e 1 00000. Farther, If the number given be greater tbafl loooooo, the firft ly muft fignifie loooooo, the fecond 100000O9 the third looooooo, an^ the whole line 1 000000000 parts. And if thefe be yet too lirae, you may proceed farther \ but let thefe fuffice. Thus when any Cube Number is given, if you fet it down, and prick it; If thelaft prick to the left hand (hall fall under the laft figure, the Number fliall be reckoned between the firft and fc- ... cond Conci Figrnts^ of 1 , and the firft Figure of the Root IbaU be always either i or 2 ^ If the laft pri(£ fail mider tbelaft Figure bat ^ne, the number given inuft be re^JcoAed between thelecondaad third Figure bf 1, and the firft Figores of the Root (hall always be either a, 9, or 4. ' But if the laft prick Ihall &11 under the laft Figure but two, then theNumbe]^ given moft be reckcmed V between the third Figure of i, and 10 at the end of the Lime. ^ . This being confidered, find the the Number given, whofe Cube tloot isdefired, in the proper SeQion upon the Line or Scale ot <3ubes, and right ag^inft it in the Scale of Roots, you (hall bate its Cube Root defir'd. Thus. iRbot ofe^j 849 C ^ <:8490ooooo 3 be about ^ ,47 And the like of any other. ^ On the Contrary, a'nuinber may Be Cubed ; for if you find the, number in the Line of Roots you ihall have the Cube thereof rignc againft it in the Scale of Cubes, giving the true denomination to the Cube, according as the part of the Line againft which the Hj^at ftandeth doth require. Thus have you by this Inftrumental way of working, thbfe things. which in the ordinary courfe are moft hard and intricate, rdndted very ^miliar and eafie : And although' at all times you do not make ufe of them, yet they are ready helps to confirm you in your work* ing Without the tedious way ot proving by Re verfe working. •■■•*• tr ...M, ..I. . . ■ . .^ -^ .,f'. : - SECT. IL By NE pair's 5 6>/£& IN the foregoing Argumeiit I toM you. That the Author and txk* venter of this kiqd of Inftrument, of which I intend to fflew the Ufe^ caOed it HABDOLOGiJ^ and the word he thus defines: ^ ' RABDOLOOIA, eS Jrs Con^andi per Vifguks fUmeri^ t^et. That is, KjiBPOlOGiA is the Art of CountiBg by Numbering Rodsi LI 1.0/ 260 Jf^rtint^mdl ArithmUidc. i # L Of the Fahrick of thefe RadSy dccording to the Inventor^ s Dejcription of them. ^ i ^Hefe Rods may be made either of Silver, Brafs, Box, Ebony, I orlvory, of which laft fabftatice I fuppofe they werf atfim m ade, for that chey are (for the moft part) by all that know or ofe . them, caned l^EPJI^S BONUS. Btiit let th<^ matter of which they are made be what it will, their form (according to this defcription) is exaftly a fquare Parallde* pipedon, the length being abont.tluree Inches, and the breadth of them about One tenth part of ihe length. Bat th^ length of thefe Hoilsare not confined to three Inches, btit let the length be what It wlH, th^ breadtti muft be a tenth part thereof^ hot that may be accounted a competent breadth that is capable of receiving of tvo iHimer teal Figures, for there is never upon one Rod required more to be fet on the breadth thereof. The breadth of thefe Rods being exactly one tenth pact of the length thereof, when loof tfaefeare laid together they do exafiiy make atjeometrial Square^ and if aoof diem be tabulated or bid together , they will make a right-angled Parallelc^ram , whofc length is double to its breadth. If 30 be tabulated, the Figure willbe ftilla Parklfelogram, whofe length will be three thncsthc: ircidthj and fo if 40, four times the length, ^ fic^ ^c. fhe Rods beinjg thus prepared of exaft length and breadth, let eachot them be divided into 10 equal Parts, with this iVov//i, that Nine of the Ten parts ftand ia the middle of each Rod,and the odier tenth part muft be divided into two parts, half whereof muft be fet at the one end, and the other half at the other end of the fame Rod. Then from iide to fide draw right Lines from divifion to divlfion, fo is your Rod divided Into Squares on every fide thereof. Laftiy, from comer to corner of every oiF thefe Squares draw a Diagonal Line, and that will divide every Square into two Triangles. The Rods b^tfig thus prepared and lined, firft into Squares, and then into Triangles, they are then fit to be numbreS. tte Kgure 1. at the beginning of die Book, (hews the Form of ote of thefe Rods lincdas itoughtto be. It B0» Infirumehtal Arithmetick. 261 IL IGvifthife Rods are to be Numbered. IN the two half Squares which are at the ends of each Rod on c- very fide, there are fet pne fin^e Figure, oh each tide pf evexy odone, in the dlviifion at the- end thereof, fo cvttyRod'lwntain- ing tour fides ; Ten Rods will contain 40 fides, and fo conTeqtieady y^ill have 4 fingle Figures at the ends of every of them ; that is, there *will be upon the Ten Rods among them, four Figures of each kind, that is, four Ones, iiii.fourTwoes, 2222. four Threes', 3335. four Fours, 4444^ fopr Fives, §555* four Sixes,- tt^6. four Sevens, 7777* four Eights, 8S88. four Nines,' 9^99. four Cy|ihe0, 0000. V And here it is to be noted, That wliat Figure foever it be that ^ndeth at the ti^pof ^Rod^lqne, th^.Figi}re tl^t ftandeth alqneon the other fide pf the fame SLo4, "^^^^ thait Filure^ . up the nupber 9. As for^xa^Bple, If i ^i^d.onooe fide, ^ . will ftaisd on tl>e tiif^er Cide, fo 2 and 79 ^r» As U) \\(\% t>- ble, where. 1 2 4 ftandsalone at If<! 6 7 8 19 8 7 6 5 ftaft^e^oa \ .the top of any I \ ! the other \ f fide of any of'' ^ ^ fi^c of H^ theIVo4$,then 3 2 1 LoJ f^ujc ^91^ Thisvalfo is to be obfervcdin the $gu):ii]ig of cyeryRod, th^t wl)it Figpre foever ftaM^tli ajpt^at ^I>e top or fuperiour part of the Rod* ^the Figure or Fi§uj-es that iftand in «th e two Triangles nicxt UQd«rncath it, is double to the Fi^jire whidi ftandleth at the to$. And the Figures whicli ftp^jl. i/ithe next qwpTr Ja?i^s below, that that is three ^times^as m^cji ^s the Figure, al)oye.' Apfl that in tliye ,foar;ji Place, or Triangje§, is. -^ur times ?s piuch as ilie Figure fi/- .hQSie|:C^c. tillyo^u cp^p/tc} jhe,l9w#Triar\§fes' in th^t R^, aiid thea the Figure or Fijgupe§ thatftat^ in ^hpfe Triangles arenii^ tiiDflSj8amHcha$.the i;ig]licj>^'lv<:h t^ndeth.at tbetppof.tiieRQd/ 80 if a 'Rpd.haye.4 gtiti^^; tpptjiereof, jtlie t^o^ Triangles which ;u:ejuftafla»cj|:tttBideri^, haye H intj[ie;n, which is double to 4: la ibe next two TrisiagiesbelQw .there is^x^aMi, itatis j2, which^iatrfWctiO 4;.In thciw Triangles jbclpjvcbcm, is i and ^, »; ' • i • • * •. which / sl62 Jnfirummtal Arithmetkk. ^ which together siake i6» which is four times as much as the 4 at thej^ 5 the .next Tr^ngles havp in (hem ^o, ^,f. Thus have you tiie Fftbridc, Inlcriptipii }suad number of tfaefe Rods, according to the Inventors contrivance of them ; He makes mention of Ten of them, and hath in his 609k fet the Figure of the faid Ten, of one of which TenPhave given you a Scheme at Uie beginning of the Book, which is Figure 2.. I will now proce^ to give you me defcription of thefe Rods' in another morecommo* diousfbrm* ^ . - wm \ JU. A DefcriptioH of thefe Rqds Accor^ng tq tffeir beH MaUteFi jContrivMce^ • • • • » » • • » ■ . THe Deroriptioa which I ihall hin^e give of thefe Rods, varies not at an from that before delivered iA the matter of which th^y are made, for thefe may be made either of Silver, Brafs, Wood, iTory, Vt. Neither do they differ in their diiridmg, not yet in their nupil^ering: Only, whereas my Lord iyilp4i> maketh them Square, each Rod tQ coiitaiii four fides, thefe are made flat, cpnfifting each Rod jKit of two (ides, and conuih in length about sincbesti' and in breadth •} of an Inch* iandjftthicknefs-s-l'Of aninch. < One fet of thefe Rods cohfiftedi of five Pii^ces, and tbereforcf hath but ten l^tes ot 4Sides, wh^nesfs thofe of the (.ord Nefmr% confifted of 40 Plains or Sides. Upon one of thefe five Pieces (z Figure whereof is at the begin- ning of the Book, noted with Figure 3} you haVe a Cypher at the head of the firft piece, and 9 at the bottom thereof. Upon* the fecohd of them yo]a have i at the head, and S at the bott«Hn ; \X^ the third you have 2 at the head, and 7 at the bottom: |lp(m tbi^ fourth 3 at top, and 6 at bottom : And upon the fifth you have 4 ;ftt the top, and $ at the bottoin. ' Every of the two Figures at the ^ top and bottom together make 9 ; as o and 9 is 9, i and 8, 2 and * 7, 3 and 6^ 4 ^^d 5. And here obferve, that the Figures 9, S, 7, 6, 5, wliich itand at.the bot'tofn of the Scheme, ftand with their heels upwards in this manner, '60^9$, and fo do all tiie other Figures under them, tiU you f oine to the double Line which is in the middle of the Scheme^ noted with A and ^, at yhich Line, if fhe Scheme were cut into two pieces; and folded or'paftedon^ back-fide of the other half, fo that thH? 9 at the botto;n were placed iipon the Cypher ar the top, and fo 8 upon i, 7 upon 2, 6 upod ^f, tod 5 upon 4^ then the Scheme cut again into fiye tittle flip petsby. Infirumntd Arithmetick. %6'^ the down-right Lines ; thefe five flippets would exaftly rcprdSit pnefetof thefe Rods, for upon one fide of thefe Pieces, youflipnjd 'fcave a Cypher upon one fide, and 9 on the other: npi)n the next /I apd 8, upon another 2 and 7, on another 9 and 6, and on theothec 5 and 4 ; both the Figures on either fide making 9, as before wa$ de* icribed. Thefe $ flippets do now contain the whole MaltipUcation Ta^<^ of Pythagoras before mentioQed^ but fo few are dot of fufBcient ufe, neiSier are the Ten before m^ntionM of th^ Lord Nepair*$Q3[»s. der; for there can be but four Figures of one kind» which in all osmesis not fufficient. ,; Therefore, as thefe Rods are made now a days, they do com* monly make fix Sets of them, that is, 30 pieces, which contain 60 faces, an4 thefe will l;>e of good ufe, and there will feldom be found ^ yfmt^ which in thofe of the Inveijtpr's there will often be, excej^t you have a .great quantity, which will be ftr mprecumberfometiiaa thefe here ({efcribed^ for there is required ?s mi|ch Metal or Wood in one of his, as in four of thefe, ^t^d thei| for l)is four fides^ we have here Ei^t, ., , ) Concerning a Gjififw thefe R^ds^ '• For the orderly keeping and ready finding of thefe Rods, I have often (for my fetf ai^d othsrss) iiad aBox made of Wahmt^treeor Pear-tree, with five Partitions in it, each Partition, to hdd five op fix Sets of thefe Rods, or more if more Rods were required, . Every pf thefe Partitions being Flgtyr'd onthe fide thereof next the Eyev with fuch Figures as the Rods in fuch a Partition had Figures atthe top, fo that the party that was to ufe theixi, could take theoi a^ readily out of his Partition, as a Printer oj^i take his Letters Put tf his refpeflti^e Boles to make any ^yord• In tliis Box there is alfo convenient Rpoib made for one other Rod, dduble in breadth to thefe here defer ibed^. hut of the fan^ length and thicknefs ; upon the one fide whereof there is a Table or Plate ufefiil in the Extrading of the Square Root, and od the ih • therfideanother for the ExtraSing of the Cube Rbot^ the: Figuw wheri^of is at the beginning of the Book, noted with Squ9i^,!Cube? But I ihall forbear to fay ahythiog' of them, till Irq^me to (h^ you how to ExtraQ the ^quar« and Cube Roots by the hdp of thCQH agd the Rods, p/ / / 2(54 Infirumentd Jrithmetici. Of 4 Body J with a Fram^y upon which to lay your Rois^ when any Of€ration it re he wrought by thtm^ htavm iy the name df at ABEL LET. In tbe nfingof thefe Rods, carets to be had firft of the orderly ItTing of them, and then fecoodly, for die keeping of them in that pofitioa till your work be ended. For the effefting whereof, both jieatly and certainly; there is a little Table or Frame contriv'd, containing in breadth ^ of an Inch more than the length of Ae, 3tods, and in length at pleafure, but it may well be about one zsA a half the length of the breadt h. It ought to be made of a thin peice of Pear or Wahiut-ttee^ or of fuch matter as your Box or Cafe is made of, and it may very commodiottfly be contriv'd to be put into the Box as ever Ihad them made to do, for that 1 found ic convenient to carry^oofe. \ Upon the Superficies of this board, clofe to one of the edges thereof, muft be glewed or otherwife feftened with I^ins, a fmall piece of the fame matter, and alfo of the lame length, breadth, and thickdn^fe of one of your Rods, which muft be divided into 9 equal parts, and Lines drawn crofs the piece, fo will, there be 9 Squares, in which you muft Grave or Scamp the nine Digits, be< sinning with i at tlie top, and fo defcending by 1 ^ 4 toi^ at the tettom thereof : And it were neceffary that thefe Fignces (as alfo thofe wtorh areat the head of every of your Rods> were Graven 0r Stampttl of fomething at bigger Figure than the other f^nresof yomrRodsare. Under die end of this ledge beginning at the Figures, and fo con- finuing the whole lengdi of the Board, mnft another ledge of the fame matter andthidmefs, be glewed, or pinned, and Chen isyoar ^tbeUet fini(hed» A Figure whereof you have at the beginning of the Book, noted with figure^ it is called a tabefkty for tiiat, when the Rods are laid tiiereon, for any Operation, to be wrought by them, we ufually &y, the Rods are Tabulated. TtosniuchfortheFabricI^Inrcription, and Numbering of thefe ILods ; I^vs now come to ihew the Ufes of them : Whidi is in Multiplication, Divifion and Extrafting the Square and Oabc Root% IV. Hif9 ' Infirii9»ent4 jArithmtkk. 265 IV. Him to »ffly to lofdewn imy Nttmhm ty the Rods, « PRO ?• I. Jiny Nkmber heinggivtn^ haw to^^^^^ or lay Jbm the fame by the Rods. LEt it be required to Tabulate or lay down this Number 5 4-^ ^ Firft, from among your Sets of Rods (or out of your Cafe/ taKe four of them, of which let one of them have the Figure 3 al! the top thereof, and lay it upon your Tabellet'dofe to the edge thereof, then, - Secondly, Take another Rod frcHn your Cafe, which hath the- Figure 4 at the top of it, and lay that alio upon your Tabelletdofc by the tide of the other. - Thirdly, Take another Rod whichhath the Figurepattheuil^ of it, and lay that upon your Tabellet \clofe by the other two. And laftly, take a fourth Rod, having the Figure 6 attheherf thereof, and lay that alfo upon your Tabellet clofe bfthe reft. Thefe four Rods thus taken, and laid upon the Tabellet, yon Ihall fee in the uppermoft Row (which ftandeth againfl the Figure I on the fide of your Tabellet; thefe four Figures, 3496,. that is 3496, equal to your given Number. In the fecond ,Row jfagainft the Figure 2 ot your Tabellet) y^u Iha^l find the double thereof. In the third (againft the Figure 3) you Ihail find the triple thcreo£ In the fourth the Quadruple thereof. In the fifth the Quintuple} a^d fo on the ninth and laft the Noncuple of theNun^wgiven. P R O P. II. How thefe Rods will afftgrs when ^tdfidattdj tmd '^tifjg Ta^ bi^latedy how to read the MHltiflicatm rf that Number fi . Tabnlat^dy by any of the Nine Digits. ' ^ The Four Rjods being Tabulated rdccordkig to.the Ptkccpt&dsili^ vered in the .j^cffceding Pnifcf$m^ they^Ul -sy^pear exa/^y 41s they are reprefented in tigfire 4, at the begiBinif|g<of tbe^vBook^ ;wU^ Figure lively rc^refents tiie lour -Rods, Ayipg tupon xhip TabeBet»^ which mind wieily for upon the we Tabuhcu^g, wd sight jegdiiig; of the RodsfoTabttkicedt d^ods^herwhole twiork, r The ^ftodsdlvs Tabul^i *a«d "g^^^ice ttem, iii4jbe:|^^e^4, do to tbe£ye» s^pcac ia<d^ latmroMrfiJ^^^ thereof 266 Imfirumentst Jrithmefick. thereof reprefentrng a Rhomboiadesor Diamond-form : la tbe toA* ingof the Figares wliich are in thefe feverai RhombcHades or Dh* mood^form, obrenre there few Dtreftions following; which will fiillf iUaftratie the whole Bnfinefs idtendedi and therefore eipecially to be minded. N Q T B, L TUt tie Figures ufontbe Egds gte u be redd^ beginmngm the right heni gnd reeding tamsrds tbe left ; which is contrary to our coin- 'laon conrfe of reading and writing, which is from the left hand to- vanb the right. II. Tba in every If^emboidies cr Dimml^ there wre either One Ft gjure^ or Tw Figures^ but never more than Two. III. If there be but one Figure in dUbombm, then thdt Figure i$ the Figure tebefet down ulone (be it ehber g Figure or a QpSer) but ^ there be two Figures in u ^jmsboiades (ot for the moS pirt there if) shen udd them two Figures together^ andfet down their fum in one Figure. IV. But if the fum if the two Figures in one If^omboiades or Did- monddoexceedTen^ thenjoumuSfet down the overplus above Ten^ and k^ One in mind t which One jfou muScarrjtothenext J(^omboiades. V. Note that the frfl towards your ri^ handy and the laStowaris your left hand are but ha^ Mftomboiades or Diamonds y and never have inthem more than one Figure only^ tut aU between theni are whole onet^ andfor the mo3 part have mo Figures in them. VI. jfin either Jf^omboiades or half Kfrnaboiades^ youfnd no Fi^ gures but Cyphers^ you muB not negleu butfet them down aa if they »ere Figares. f Thefe Rules being riglitly underftood, all that follows will be fiimiliar and eafie, juid thefe I IbaQ explain by Example fol- lowing. Mxan^le. F6r the inoftradod of the vteceding Rnlesi we will make uie of thofe Rods which were' bdbre Tabubfed, therefore luiye recourfe to ^^gM^o 4 at the beginning of the Bo^ where this Nmnbeir 3495 is Tabulated. The Figures at the top of the Brar Rods are tbefe: 3,41 9|.6« which fignifie the former given Nomber 34961 and this number fiandsag^inft the Figure ion die fide of the TabeUet4 Thenlfay^ that the Figures in the next row Handing agftiaft the>Ff^re2of the Jabelletare doifile thefcoolo^ which I tbos psov^ < . -. * Repair J i \ . Infirumental Arithmetick. 267 Repair to the Rods as they He upon the Tabellet, and in that row which lieth| againft the Figure 2, you ftalt find in the firft half Utiomboiades towards your right hand fvvhere byVmle i7oumuft, beginj the Figure 2, wherefore fet down- with your Pen upon Pa- per the Figure 2. In the next Rhomboiades in the fame row you jhall find 8 and 1 , which added make 9, fet down 9 on the lefchand of 1: In the next Rhombus you (hall find 8 and i again, which is p alfp, fet down -9 on the left hand of the .other, and \a die laft Rhomboiades youlhall find only 6, wherefore fet down 6 on tfe left hand of 9, fohave you in all 6 992, \ which is doublet© 34.96. Again, the Figures in the row which ftands 3gainft the* Figure 3 in the Tabellet, are triple to 34965 for in the firft half Rhom- boiades towards your right hand, you hive8,fet down 8. In the next Rhom. you have 7 and 1, which is 8,fetdowri 8 again,— la the next you have 2 and 2, ^hich is 4? fet 4own 4. — In the next Hhom, you have 9 and i, which makes iq, fet down o and carry i, but it is the laft Rhom. and becaufe there is never another to car- ry the 1 unto, you muft therefore fet it down, fo have you this num- ber 10488, which is triple to 3496. , Again, the Figures tending againft4inthe Tabellet, are Qua- druple to 3496,— for in the half Rhom. you have 4, fet it 3own : in the next 6 and 2, which is 8, fet that down : In the next 6 and j, which is 9, fet that down : In the next 2 and i, which is 3, (ep that down : and in the kft half Rhom, you have i, which ajfo fet , down ; fo have you 13 984,. which is Quadruple to 3496. Alfo, the Figures againft 5 in the Tabellet : the firft is a Cypher, therefore put down o j the next is $and 3, which is 8, fet down 8 ; the next iso and 4, letdown 4*, the next is $ and 2, that is 7, fet down 7 ; and the laft is 1, therefore fet down i, fo have you in all 17480, which is Quintuple to 3496* Againft 6 in the Tabellet, you have in the firft pjace 6, fet it down; then in the next 4 and 3, that is 7, fet that down; in the next 4 and 5, that is 9/et 9 down ;in the next you have 8 and 2,that is Id, fet down o and carry i to the ^ext Rhom. where ytni find only 1, to which add the i, which you carried from flic Rh6m.befor*,atii| irmakes2,fetdown 2:fbhavisyou269J^,Whicofefix trmes349^. Againft 7 in the TabtHet, you have^irft '2-,' fet it down ; then 3 and 4, which is 7, fet 7 down ; in the next you have 8 and 6, Which is i4,wlijch Being 4 above rb, fet down 4, and carry i to the next Rhom. wlicte you hAve Mod t, w!ucj.is '^^laA i which you carried makes 4, fet down 4; then in the laic place you have only 2, wMch* fet xhnhav fo hatseyou in aft 24472, which Is S^dple |p 3496, or ferai timcsas much* ♦ Againft :« in Che Tafediftt, yt»tt hare firft 8 wMchfet down ; then iw* 4, whic* is.6, fet 4^4owa; thrsn 2 and 7, wjiich is 9, fet 9 ioB*h t then 4 iuid 3, which is 7» ^^ 7 d^^^n 5 and laftl jr 2, ^r^tt M m down . N ^ QjSS Jnfirumentd Arithmetick. down; fo have you 27968, which is Oftuple to 3496, or eight times as much. , Laftly, againft 9 in the Tabtllet, you have in the firft place 4, . fet that down*, in the next you have i and 5, which is 6^ fet6 down-, in the next place yop have 6 and 8, v^hichis 14, fetdowji 4, and carry 1 to the next-RJipm. where you find 7 and 3. that is 10, which witH I which you carried makes ij, fet down ij and carry i totijenext Rhom. where you find only 2, and the i carried 'makes 3, therefore fet down 3, and fo you have. 3 14(54 which is J^Ioncu^leto 3496, or nine times as much as the tabulated number. Thus have I given you Examples, in (hewing you bow the Num- bers upon the Rods are to be read and written down ; and in the fjelivery of this Example, I haye made the wbole work which is Vo follow? w plain and eafie, that the meaneft capacity (I think) if he can but tell hi^ Figures, arid add any two Figures together, he may by this here delivered, read or write down aqy Number, that can be tabulated j and that you may throughly underftand this Chapter before you proceed further, I will give you the Pror jluftsof 7009078 multiplied by all the nine Digits, which I would have your ielf to tabulate, and fee if you find your working by your Rods to agree with thofe which are here written, whicli Num- bers if they do, you neednotfcrupleat themoft difficult th^t cai| |>e propofed tp you, therefore ftudy it, and try it, 7009078 7009678 being mul-'* tiplied by an 3 4 5 6 7 8 !> = <! 14018155 21027234 28036312 35045390 42054468 49063546 $6072624 ^63081702 Tim have I fiifficiently defcribed tbefe Rods, and the manner of f^umbringuponthemi and now I think it tin^to apply them tqtha^ufi f&rvfhich they vfere int^nded;-namelyy the more dificuh parts ef AtiHii fnetick, ^ MultipUcation, Diviiio{i, and Bxtradibn of J(o<ns. vv^T*^* * V. MultifHationbytheRqds^ JN Mijltiplying by the jRocJs, you are to confider (as in Vulgar Arittfmetick; three Terms, Things, or Numbers, viz. !• The Multiplicand, which is' the Nqmber to bs multiplied. ^ 2. The MuMplier{Nlmii is the Number by which the Multipli^^ InfirufHenid Arithmitick. . i6<^ 3; The ?todttSty wh?chis thefum produced by tlie multiplying of the two formei" together. ^ And here Note, that the Proiui dofh contain the MuhipUcand^ fo many times as there be Vnites in the^ MultipUer. Thus much for the dlefinition of MuhipUcation-y now for thef Working thereof by tHe Rods, for Whicn this is * ■ Firft^ Set dorbn Upon jour Taper the MultipUcand^ and orderly under h the Multiplier, It matters not greatly of which of the two giveri Numbers be made Multip'licdnd or Multiplier^ but it is ufual and beft t6 make thef greateft Number Multiplicand^ and tlie leffer Multiplier. Then drai» a Ufie mtbyoUr Pen Under them, and having Tabulated your Multiplicand {0^ greaxer-Number) took roiai Numbers in four J^^dsfland again S that fifS tigui^e towards jour right handf^ and that dumber rebichyoujhaUfind upon jour l^odsfimding again^ tbatfril Figure found in jour Tabelletj fet down under jour Line which jou formerijdrew under jour Multiplicand and Multiplier : jfnd hav- ing "fo done with the fir SI Figure of jour Multiplier^ dofo with ibe red i fetting them down one under another^ removing tverj Figure one place more towards the left bandy than, that which went before it^ aaU done in common Muitiplicatiorij and a/s jou fe6 iH the following Ex- ample, , Example, t . Let it required to multiply 3496> by 489* As if it were required to know how much 489 times 3490 would amount unto* Firiiy Set down your given >Juftibers ^3496, ^nd489, one under another^ and draW your Line under the'm^ is here you^ fee done. Secondlj^ 3496 your Multiplicand being Tabulated, and 9 be- 3496 Multiplicand^ ing the firft Figure to the right hand iri 489 Multiplier^ your Multiplierj look upon your Rods^ **. — ^ what Figures there ftand againft 9 in the 314(^4 fide of your Tabellet, and youlhall find . 279^8 (as by the Fifth tlule) 3.1 464, which is the 13984 { * Produft'of 3496 multiplied by 9, where- ■ '■' ■ ' ' " ■ ■ ■ " fore fet down this NiJnlber 31454 under 1709544 Produft your Line, as you fee in th^ Example. Thirdljy Look what Figures upon the ftods ftand againft 8^ which is the fecond Figure of your Multiplier, and yon (hall find 27968 5 fet this Number under the foriiier, moving it orie place forward towards the left hand. Fourthly^ Look what Figure iipdri the Ro<is liandi againtf 4^ which is the third Figure in yotr Multiplier, and you Ihall'fina 15^984, which fet down under the other, one place mbreto thcJ? ieft hand^ Mm ^ 1 270 Inftrmmnul Aritbmetick. Laftly> Under thefe three Sums draw alpine, and add' tfaethrse Sums together, and they make 1709544, which is the Produft of 3496 multiplied by 489 and this 1709544 the ProduS, contains 3496 the Miiltiplicand, 489 times. Fradife vfeU tbisfrU jExample^d compare h with the ^ds as they are fabul^ed in Fig. 4 at the Beginning of the Booi,as alfo witbtbeforcg4fing J(ukSf and you m^ perform any muftiplication* However I will ^ve you one or two more Examples, and fome other ways of Multi- plicstion. Example 2. Let it be required to. multiply the fame^um 349^9 by 261. 3496 I Set the Numbers down as here is done, then look 261 f upon the Rods for theProduft, of ^496 by i, andyoa (hall find it to be the fame, wherefore fet down 9426 under the Line — ; then look upon the Rods for the Produft of 3496 by 6, and you Ihall find it to be 349^ 20976' 6992 912456 209'?6, which fet down under the other Number, one place more towards the left hand* > Again, look in the Rods for the Prorfuftpf 3496 multiplied by 2, and you fhall find it to be 6992 , wHich fet down under the other two • Laftly, Draw a Line under them, and add the three Numbers to- gether in order as they ftand,and the fum of them will be 912456, which is the Produft of 3496 multiplied by 26^. Etamplc 3. Let it be required to miiltijply the fame Number 3496. by $20. Set down your Numbers as here you fee done 3496 520 6992 174H0 1817920 which you Then becaofe the firfk Figure of your Multiplier to- wardis your right hand is a Cypher, wholly omit it, ^and multiply 3496 by 52 only, fo ihall you find the Pro- duft of 3496 by 2, to tie 6992,which fet down : Alfo the Produftby 5 will be 17480, which fet down under the other, one place further; Then draw a Line and add thefe twoTums together, and they make 181792, to the which if you add a Cypher for the Cypher omitted in your multiplier, the Sum will be the 1 8 1 7920^ wfiich is the Produft of 3496 by 520. Example 4« Let it be required to multiply the fame 3496 by 7003 - Set down your Numbers as before, and as you fee here done ; Then having Tabulatstted 3496, fee what theProdu£t thereof is upon the Rod$,being multiplied by 3 the firft Figure in your Multiplier and you (ball . findit to be 10488, whichifet down under the Line- Then the two next places of yourMultiplier being Cy- phers,mdke two Pricks under the former Number/>ne 2^82488 i ' und^ 8, and the other under 4, as you fee ia the £x- , ample 349<5 7003 10483 24,472 • • Infirumefad Arithmetick. aj i ample ; Cor jiftftead of aj Pricks yon may make two CyphcM,) Then look in die Rods for the Produft of ^496 bv 7iaiid yon fliafl finA it to be 24472, which fet down under the other Sum, beginning'yoiir Number at the fourth place, or beyond the two Pricks or Cyphers. I.aftly,draw a Line aAd add thefe two Sums together,and their Sum is 24482488, which i&the Product of 3496 multiplied by 7009* Thus have you four Bxamples in H^Jt^lication^ ia which are Included all the Vaneties that niay at any time happen* in that Rule, vi^. Two where the Multiplier comifted all of Figures, aa In the iirft and fecond Bxamples they did. — Another where the latter place of the MultiplierconfiftedofaCypher—— And this laft Example where Cyphers were intermixed among thePigures. And thus much for this kind of MuhipJic^icnj but before Iieave, I will Ihew you Another Form of Multiplication. T T^JHereasinthe foregoing PormofAftf/ri/7/i<r^io», which Isthe V V beft and moft ufual, (only I infert this following for varie- ty) You began. T your Rods being TabnlatefU with that Figure of your Muftiplier which ftands next your right hand, but there is no necefiity for that, for you may begin with that I^gurew^ich fiandeth next to your left hand, and by fo doing, and ^ placing your feveral Produith one place more to the right hand, as* yoa did before, place them to the left hand, thofo Prodsfi^ addM tO« getherintht^ Form they then ihnd, mall produoo a Produd equal to the Former. Sxmfkj For our Example we will take the firft Example be- forgoing at the'begianinl of this Seftion, where it was required to Multiply 3496 by 4^9. Set the Numbers down as befoceinthat^ firft Example, and as yoniee here done. Then 3496 being Ta|(^lated, look upon your Rod^, 94^9 forthe Produft thereof mirttijiied by4,( which is the firfk 489 Figureof your Multiplier towards your left liand). and' ~— — ^ you (hall findtheProdud thereof to be 13984, which fet 13984 down.-— Secondly^ look the Prbdud of 3496 by S (your 27968 fecond Figure) andyou Ibal! findittobe 2796S, which 31464'' muftnotbefet downasin the tsfther^rft Example but \ as you fee it in this, 8 the firfkFigapi thereof muftbe 1709544 fet one place forwards towards tht; right hknd^ as in the other it was fet a phice b^ck,wilrd, towards .the left*— Lafflyp. ifeek in your Rodil for tbeProdiift of 3496 by 9 yoiir Jaft Figure, and you (hall find it to be 314^4, which fet under the, otter two Num- bers, yet one j^oe more to ttte ri^r hand;**-Si> a line beiag,djawa under, and thefithree Numbers added together produce 1709544- ^ 272 hifiruTmntdl Arithmetiek. c^ualto that in the firft Example : And] that you may the bet- ter fee the diflference of the work, I have fee them one by the other. As in the firft Example. 3496 31984 27668 . 13984 As in this' Example, 3496 48^ 13984 27968 314^4 1709544 ' I7P9544 Oqe Example' more in Multiplication, which (hall be for Aif vertifeiment and direftion, I will give, and fo conclude Multipli' cation; I faid in the general Rule for Working of Multiplication (at the beginning of this Seftion) tliat it mattered not which of yoUr Num^ bers were made the Multiplicand, or which the Multiplier, of which I will here give you a Prefident where the Idler Number ihall be Tabulated, and the greater Number only fetdown ; and I will work it here according to this laft way of MultipUcation, and the Example fliallbeasfolloweth. Example, I>t itbe required to multiply 868437 by 3:49<^,and let 3496 (the lefler Number) be Tabulated. Let the Numbers befetas you here fee, then 3496 being Ta- bulated. Begin with the firft Figure 3496 * towards the left hand of your Multi- 868437 plier, which here is 8, and upon your Rods find the Produft of 3496 mul- H ' — 27968 tiplied by 8, which is 27968, fet that 2^)976 down under the Line— then find the 27968 Produft of 3496 by 6, the fecond Fi- 13984 sure of your Multiplier, and you IC488 ihall find that to be. 20976, fet this 24472 number under the former, one place more towards the right hand 3036055752 Again the third Figureof yourPrO' duft is 8 whofe Produft is 27968, as before, fet that under the other, ftill one place more to tlie right hand. In this manner da wkh the other Figures of the Mul' tiplier, as 4 the next Figure, whole Ptedaftis 13984, which alfo, fet down a place ferwardt - »So alfo the Produft of a which i» Infifumental Arithmetick. 27 j 10488, which fet down, — And laftly, of 7, which is 24472. ■ j .^11 thefe Produfts being fet down in the order as youfeetliem in jheMargent, if you add them together, the Sum of them will be 30360s 5752, whi(;h is the Prpdua of 3496 multiplied by .868 437, the leffer number being Tabulated. Other roAyi of MultipUcation I could have addedj but tbcfe Jefietm fufficknt. VI. Divifion^ the Rods. AS ' in MuhipJication^ fo in Divijion there are diree J^mbersj - Terms <^ or thifi^s required, vix* 1. The J^iiiiieJai, or number to be divided. 2. The Vivifor, or number by which the Dividend is 4iv}ded, and 3. The Quotient^ which is the. Number iffuing fron) (he Divi- dend's being divided by the Divifor ; And this Quotient doth al- ways confift of fo many Vnites zs the Divifor is times contained in the Dividend. . Thus much for the Definition of Divifion^ now let us come to jthe Pra^ice of it by the J(ods, to perform which this ie *■ T H £ 9. U I, E. Tabulate the Divifor ^ Qnbich if alwAjs the leffer Number of the two given) and fet dorm the Dividend^ Md fet the Divifor on the left hani^ dfjd draio a crooked JJnq on the right hand for your Quotient^ at in common 4^ithmetich. Ihen look uponyour Tabulated l(sds(ahoap)for the Niimber lefs than the Number in the fir Si figures of your Dividendy and what figure fiands again^ fhat Number on the edge of your TabeU fet mu3 be the l^igure you muQ put'in your Quotient ^ and that Num- ber you mn^ always fjibftrait from the Fiffires of your Dividend^ and to the remainder add another Figure, fo proceeding from ligure to Figure tiUyour Divifion be wholly ended. Example f Let it be required to divide 1709 544 by 349^. Having Tabulated 349<5, fet down your Dividend, your Divifor on the left ^^4 thereof, and a-croqk?e4 Line for theXJuotient'on the right hand tjjcrepf, ias by the^Ruie prefixing yon' were dh reeled; an jl, as yptj fee done in the ' Exanipjie a.djpyoiQ^*. '. And .bedalafe at your firft fetting down of your Divif<bC3f495, it yrould reacl^T|f it werefetyndie^jouf pjyidend 17095 44) a&' far 1 274 InJtrmmmtMl as the Figute 5, therefore under the Figure 5 make a prick, to in- tiiittCe how hx you are gone on in your work, and nnder tiiis prid^ draw a Line quite under your Dividend ; then is your Sum fet down ready for work^ and will appear as here you fee \ 3496) 1709544 ( Your Sum thus prepared, ask, how often can yon have 349^ in 17095, look in your Tabulated Rods for 17095, which you cannot there find, but the neaioft number tbereunCQ amongft the Rods, which is lels than 1 709 5 (for you muft always take a -Ids number) is 15984, which number ftands againft the Figrtre^ii the Tabel- let, wherefore fet 4 in your Quotient, and 1^984 under the lAe, andrubftraaij984irorti7095,andtherewiil remaift 3iii,WhiCh ret over 170959 andfo isthe flm part of your Divlfioa ended, and jour W(Mrk will ftand thus. 3m 349*^) 1709544 C4 13984 Then make another Prick under 4, the next Figure of your Di- vidend, fowill the remainuignumberbe3iji4,__ThenloDk among your Rods for the number 3iii4forthe nearefk lefs tluw it) and the neareft left you ftallfind to be 2 7958, which ftands a- gmnft 8 in your TabcBet ; put 8 in your (Quotient, and fet 2 7968 imder 31114, and fubftrad 279^8 6om 31114, fo wiU there re- main 3 14^^ whichfet over 31114, fo is thcfecondpartof youf Divifion ended, and your Work will appear thus. 314^ 3111 449^) 1709544 C4^ >3984 ... a79d8 lafiiy^ Make smotter IWck under the next Figure 0? yourDii*» dendy wiiidiis 4^fo, making the remJtming number to be ^14^4, reek among yotr Tabcdated Rods foi*dris number (<* the aeaiS ^; but lookisg you (hafl -find the very number agjiinft whkfa nq^ w j^ur TaMlcc theFigure9^fet9inthe(;^6^t,and Jfijlrmnentd Arithmetick. > , 27$ the Dumber 31464 under 314649 and fubftraO it from 3^464 the number which ftands above the Line, ai\d nothing remains ; and b^- ing ther2 is never another Figure in your Dividend, your Divifion is ended, your work wiU ftand thus, and 3496 is contained in 170^544,389 times. OOQOO ' ' ,. 3146 3III 349<5) 1709544 C489 • • • 13984 27968 1 1464 Another Example^ and bj dttotberway pf piyifion. tet it be required tif divide 9t2^')6 by 3496, fet down your Divi* tend and Divifor, draw a crooked Line fo;^ your Quotient, and air fo make a Prick under the fourth Figure or your Dividend^ and drasvir a Line under your Dividend, fo is your Sum prepared to be divided, and will ftand thus. / , J490 9124$^ C Then yourDivifor 3496 being Tabulated, lookamongft your Rods for the neareft number 109124 which is lefs, and you (hall find it to be 6992, againft which, itands on your Tabellet the Figure 2, fet 2 in the Quotient, and this Number under the Line, andSub- ftraft it from 91^4^ and there will remain 2132, to which number add the next Figure of your Dividend, Namely $,andit makes 21325, under which number draw a Line, then will your Sum ftand Aus, \ 3496)1 912456 (:2 '^ Then among your Rods feek the neareft number to 2 1:525 and you Chall find 20976 to be the neareft number lefs, againft which^ in your Tabellet ftands 6, fet 6 in the Qiiotient, and 20976 under the lAitCf Subftrafting it from 2 132 5, which when you liavedone N n there a^6 I»firumtBHl Aritbmttkk. there win remain 349, to 349 add the next Figure in your Di?i- dend* wbich i$ 5, yotir laft Fignn*> anS it makes 3496, nmier wludi, draw a Lifie, and yoor Wc^kwill Aand a$ heteyotrfee. • • ■ ' '. ' . «• • • m 3rr325 2697^ 3496 This done look amongft your Rods for the neareft namber (o 3496, and youlhall find the eiaS nmnbetatthetupofthe Rods,a- gainft which ftaoids the Figure 1 on ite Tabdiet, fet i in die Quoti- ent, andfnbftraft 34966001 3496, the remainder is nothiiig, and fo is your Divifion aided, the woric fianding thus, and 34^6^1^ Diviidr iscoopined in 912456 the Dhridend,t6i times. . ^ ft 3496) 912456 (261 6992 21325 20967 3496 ?496 0000 I will only fet itdowH ready wroagfit» leaving the ^fliceof it 1S0 your fel^ ».».;... i . • Ja '"■■•■ V. Utb AerejuiredtQ diviie 7^020 jo<5 bf 34^64 249^) 73620^06 i^o9^l^'i • • • vfc I ^99^ 31005 : 27958 '- -50370 27968 24026 * • > 209.76 3050 . I This $um thus itivided>, producet^ i|i thd iQiiotient 20886, • ji^ 2cf>o remaining, fo thatthe Quotient with ^heFra£lion and allis^ 2o8r86|-*j^.Sotbat.3496 the DiFifur,i$(^taine4 in ,73020 506 the dividend, 20886 titnes,. and 305© rQW^iriing.. this Example ipellfraCttf^tagfAp^jffiihfh^fn b^fore^going^ drefiip' fidem infiruilion for any Student taha^T/er^ and be ffm can perform tbefe^ need mtdefpkir of the mo(tdil^cUlt thai VM^^ Aiji (o I <»ondude with Piviliod* L VIL Of the Extrdam of RO OTS* THe Extraftion o( K^s^ which is the diffioilteft part of Mal- tiplication and Divifion, is expeditioufly and certainly per- formed by the Rods^ for the eafie arid ^s^eolte performance of which, there are two Rods on purpofe, one for the Square, the 6- ther for tho <:ube ^Root, of .which I will Ipeik i firftj Oi their Fa- brick: fecondly^ b£. their Ufe* " ' Of the . leAh^JckjifaUtti^sfor:^^^ df^ctsi 6f^fl]^e-6njematftcr,.and.0f ih^ fiortj^i^th and thicfaiefs df lourgtber JUkU, Jiat there bainadc )amtUx S.od> "^t thireef thnes 1 27^ Imfirumentdl Arkhmtitk. j the breadth of the fbnner» the infcriptio9 odthe one fide fervio; ! to extraft fhe Square, and that oa the other fide for the Cube Itoot> I eachof which are dividdd iatO/three Rowsor Colames, ! That which ferveth for the Square Root, hath in the top or nppet- mpft Square bet weeti the Diagonal thereof, thefe figures^o-i, in die { fecond 0-4, in the third 0-9, in the fourth i-5, in the fifth 2- 5, in the ' fixth 3-6, in the feventh 4-9 in the eighth 6*4, and in the ninth or lowermoft d-i , which are the Square Numbers belonging to the mne Digits* In the fecond Co)umn of the fiune Rod, in the firft Square is io- fcribed 2, in the fecond 4, in the third 6, in the fourth 8, in the fifth 10, in the fixth 12, in thefeventh 14, in the eighth i^^andia the ninth i8. In the laft or third Column, there are the nine Digits orderly de^ fcending, namely, i, 2, ^,4, 5^6, 7, S 9. This Rod thus made, is fitted for the Square Root. That which ferveth for the txiht Root, hath in the top or up- permoft Square of the firft Coluam towards the left hand, between the Diagonal thereof, thefe Figures, 0-0 1, in the fecond 0-08, in the third 0-27, in the fourth 0-64, inthefifiJi 1-25, in thefixth 2-16, inthe feventh 9-43, in the eighth 5-12, and in the ninth 7-29. which are Cube numbers orderly defcending.-- — ^The fecond Co- lunm of this Rod contains thefe Square Numbers, U^%9y t6^ 25, 36,49,64,81, orderly defcending. The thbrdancllaft Column of this Rod bath in it the nine Digits, 1,2, 3, 4, 5,6, 7, 8^^, order- ly defcending alfo. This Rod thus prepared and infcribed, is fit for extrafiingdf the Square and Cube Roots, aFigure of either fide whereof you haveatthebegmningof the Book : That which fereeth for the Square Root having the word S(iuare written by the fide, that for the Cube Root, hath Ci^i^ written by the fide. Thus having given you the Fabrick and InfcrlpCion of thefe Rods,, I will now (hew you their Ufe; And firft. »i»"»i*«"-""""»"^"i"— i"i"«iw»««^-i»i— BMBiiiite Vm. The ExtraSlion of the Square Roof, To extrail the Square Root of any number, y^umuft firft pre- pare it, that is, fet down the number on a Paper, then over the firft & loweft figure next the right hand, make a point with your Pen , and over the third figure make another , over the fifth another Point j and fo forth, over every fecund figtfre of the mim*- ' ( Infirumenul Jrithmetkk. ^jg number make a point, always leaving between each Point one figure unpointed, according totbe ordinary Rule by the pen ^ this being done, you ihall fee how many figures will be intfaeRootfinr lb many points as you have, fojnany figures Ihall you brii% into die Quotient for the Square Root, of the number given vn^tdraw a QuQtien^line, as in Divifion, and your number is prepared &r • Operation, and will fiand as in the Example following, where the number given is 1 19025, and the Root Square thereof is required. This number being fet, and Pointed as afore is (hewed, you may perceive that the Root thereof will be of three figures, becaufe there be three Points over the num- • . . ber given, the two figures belongmg to the higheft 1 19025 ( Point next the left hand are 1 1 • the two figures be- longing to the fecond Point are 9o,and the two figures belonging to the fecond Point are 25. and the figures for the Root anfwerable to thofe feveral Points , zrt to be found by the Rods, a^ fbllow- eth. Take the Frame, the Rods, and Lamina, and lay them before you ; and fir ft place the Lamina in the Frame next to the left hand ledge, with that fide upwards, whereupon, are the infcriptionsbe* longing to the Square Root, and marked at the top with the Let- ter Sj then vconfider what is the greateft Square number in ii. the figures belonging to the' firft Point ^ the Lamina prefently (heweth you that the greateft Square number in 1 1 is 9, and his Root Is }» tor 3 times 3 is 9 ; therefore put 3 in the Quotient for the fink figure of the Root, then fet 9 under the 1 1 and Subfbad It there- from, and there, will remain 2, this 2 let under 9 drawing a Line between themi^andto the 2 remain- • • « ing, bring down the 2 figures uiMier the fecond 1190a; (3 Prick, vi^, 90, making it 290. and fo you have 9 gained the firft Figure of the Root, and the Work ' . will ftand as in this Example. ,290 Having the firft Figure of the Root; to get the fecond, and fo all the reft in order, you muft proceed in this manner; double the Root found, which duple Place or Tabulate upon the Rods between the Lamina and the ledge of the Irame; As in this examjde, the duple of 3^ the Root found, is 6, therefore place a Rod that hath in his top or upper Square 6, between the ledge and the Lamina ; Thea look upon your number given, what Figures^ Or number it is that belongeth unto the fecond Point, which you fee will be 290 inthi& our Example ; Then turn your eye to the Lamina and Rod now Tabulated, and fearch thereupon what number will n)eingle(syet> come neareft unto 290, the number out of which the fecond Figure 0f ,the Root is to be found , And there you may Ibon fee iris the number 256, which of any number upon the Rods^ lefi than 290 loniotli iSMtett th c i ^ im tiiy ftr the next greater nnttiber.vpQii ijtt Sods, abofei96,'is ^^^ .whidb is greater than 290, andthcrefive comics te taken out t>t it ; but .256 is the only number te work tWtlni, agafaift ^'^f^ich, Dathe ledge ami Lamina, is this Figcire 4, idbia 4'«auft you put in the Quotient for the feoond Figure of t^ . Rxxkt, andthen Subftrad 2 56 oa^ nf tiiat 2^, and • • •. . Ihere will reft 94,this34fet under 25^ in its due ^390^ (^4 .filaoe, drawing a Line betwe^ a^6. asd 34 ^ itoi ^ w ill tJK Work iftand as in the Example. Ifvyoa ■■ oloaie, you may write yoor numbers to be Sub- 290 . . . ftcaded) nnder the number out of whick Sub* 256 ihnftionisto be made, as I faavedone heoeJa this fixamptey for inftru&toa ioloe, or you. may omit ' 34 t|iat if yon pieaGrbeing you have them heme yoii' xipon the Bbbds. And now for the third figure of the Root, look upon the num- i)er:given, ami there jittu mall fee that the Remainder 34^ with the rtwnilgutes 25 b^onging to tiie tfaicd Point, being ail joynedtn- Hgether, mate 9425, out of which the third Figure of the Root is ^nxbeextraiCM \ To^find out wfaatthis third Fi^fie Oiail be, double ^tte .Root found already, which is thus done very readily ; Take iferChaRod that on his lop. Square hath 8 the duple of 4, thelaft .Figute ia the Quotient, and this Rod put into the Frame between y^st Lamimiand the Rod thatis already Tabulaoed; This bemg done, -your hsve noimote^ to do,but to look over the Rods and Lamina for fuch a number^ as will cometieareft unto the number 3425, that belongeth <o die third Point,and isiefs than it; and the Fig. diat you fee a^inft that numberfofound, put in the Quotient for the third Figure of the Root; Thus looking upon the Rods • • • you ibaU arthe iirft fight find the verynumber 11902$ (34^ itfelf 942$ that beloageth to the diivd Point, in 9 the fifth Line of Squares, againft the figure 5, ^— -* 'therefore,put $ in the Quotient for tiiethird figure *tt9o €f the Root, and if yoa Subfliaft 342 5, the num* wejS ber now foand from 3425, that number belonging ' to the thM Point, there will be no remaifider,fo ii ^425 the Work done, and the number ^iven, 119021 3425 • is aperfeft Squate number, wA the Sguare Root tiieftof is 345/ which vnu chetbin^ ivquired to 060^9 befoittid* Moir 'l&«if3foanii}ldplytkts945 ixit»illiel£|itprod^ ii9025,the ^rft giveiv ac^nbep^ which prov^K the work t&be truiY wp^VftJiti for note tius Vvermore^ tfiat for the pcoof of the ExtraOiiig^ die Square Root, yoiit rauft miikiply the Roo$ found, byitfidf;. (ti^the prodaft addi^ the remaiOi if aay be) and the total will produce the firft mwnber given, if the work be tfuly wrought, otherwife bot. . Vo€ a feeond Example ,. Let 'there be given this numbfr ti 77 1623 769 A, and the Root thereof required : Now to perform the Work^ firft write dowdi the Number^ and then prepare it witk points and a C^otlent-liae, as you were idfaruded before in the ftr* mer Example; This being done, tl^ num- ber will appear as here you fee in the Mar- 1177162^^76^^ (3 gjtne: Now inn, the Figures belonging • • • • • * ^ the higheft point, the greatcft Square . ..,1,1 Nuinberis.$b^hofeRooti$^iput3iath^ 9 Quotieat and (et 9 under i,i , and fub- 277 ftraft it feora 1 1, fo will the remajnier -, — .— be 2, whkh let under 9, and bring d^wi^ 77 the Figures belonging to the iecond frickmakiflg it 277, 99^ der w^b 4raw a Line; Now fiur the fec^pd Fjgttre of tkm Roof* ITabuIate the duple of the Root ^updi vad tjbaf i$ na p^w to 4«i^ but to^^ce a Rod that hath6» the dopleof 3^ dae J(x)ocibua4, j^ hi^ top Square, between tkeX^imae^ and the Ledge, and on the^ left hand c^ the Lamina, .ti» faid Lamioat being TiMatfd with tfiat &ce upward, th^f, i% foe the Square Root ; then.^iiig that liie Number belongiftg to the fecond Point, out i£ which thejfe- cmid Figure is to be eKtr»fte4 is in this Example 277, therefore fearch upto your Tabuialed Rod and Lam'ma for fuch a Ninii^r, ^swiH come neareft to. that 277, which you ihall quicklvJndxoJte a5<5, smd right ^oil: k, oEthe r^t hand Column ot theL^i- na is the Figuire 4, therefore put 4 m in the Quotient for the (econd Fxgureof the Root, ^ivblm^^'ji^Mm 277 and fet-thexemain-. der'21 Uoder it^ and ^alfo can^ tihe 2 $6, and fo have you done %itfa your fecond poin^ Joid now foe the ibWl FigPJfc of the Root, obfenre thattlie 21 remaining with the other i^, twp^Figpires uncancelled bekmging to tbe third Point, being jpyiiql togf(her, make 21 16, out of which ^iame third F^giu^isto^beessctifa^toi^ to perform wHkhWori^* taks^ a lUd fihat^srieth: in Ws mp9^ S^ine^e.FjgB*e.8, the4^f4e of ^ tbr -ViftUK^Jibft :f(^i»ii4f |iiil*f«^t)MR# mo^tbt ftmcks^ »v. sSa Infirmmmd Arithmetick. tween the Lamina and the other Kdd be* 11771^237^4 (343 fore Tabulated, then look for the Kum- ber upon thofe two ilods, and Canuna, ' that will neareft take away 21 16, tte. 9 Number belon^ng to that' third Poine»' ^77 and at firft fight you Ihall find that the ■ ■■ next Number lelfer than 2116 is 2049, and his Figure for the Quotient, 3, there* fore put a in the Quotient, for the third Figure of the Root and fubftrafi 2049, from his refpeAive Number 2116, and theoe remained! 67, which fet undef 2o^, and draw a Line under it, fo you liave done ^ith the third Point, and third Figure, and are to pro* ce^ to the fourth. Where firft you fee that the 67 laft remaining, with the 23, ]nake 6723, 'the Figures belonging to the fourth point, whereoot joa muft Extraft the fourth Figure of the Root. Therefore go <m as before, taking fuch a Rod, as in its top Square hath 6, the' duple of 9, the Figure laft found, and Tabulate it between the La* mina and the other Rods, and dien feek what number there can be found, that will be lefi than 6723 but at the very firft "fight you ihall feetio number upon the Rod fo fmall as that 6723, ioc the very firft number againft 1, is 117716237694 (34309 6861, which is greater than 6723, * • • • • and therefore cannot be taken outof ' it, fo that here you can put no Figure 9 in the Quotient ; but muft fupply the 277 place with a Cypher, thererore put a o ia the Quotient for the fourth 256 Figure of the Root, all the reft of 2 1 16 the number ftanding as it did. Now to the next point, 'which is the fifth. 2049 Y^° ^^^ t^>s Number belonging 672367 , 6723 76 whereout the fifth Figure of ■ the Root is to be found. . Now here, 617481 in regard your laft Figure of the Root . $4^9 $94 found is a o, you mii^ ever in fuch a cafe (in flead of putting a Rod, that hath the duple <^ the Figure found in hk top Square,) you muft tafctt fuch a Rod as upon one of his fiices carrieth only Cy* phers, and Tabulate it between the Lamina and the other Rods al* ready Tabulated^ This being done, queftion the Rods, what Num- ber yon muft take firomthat number 672376, orwhat number it is there, that being lefs than 672376 yet cometh neareft unto it; and alfo what the Figure (hall be tbatjou muft put in the Quodeot for I ■ » Inftrumenul ArithmeHck. 2^j. for the refpeftive Figure of 'the Root 5 The Rods will fudd^nly re*- Iblyie you, that their greateft number lef^ than 672376 is $17481, ^iiA its refpeftive Figure for the Root is 9^ therefore put 9 in the Quotient for the fifth figure of the Root^ then if you wiH wrjte^ 61748^1 under 672376, and make fubftradion, yout'remaindeJCwUl be 548^5, ^hich joyned to the 94, the tWA laftFigujresof the Num- ber given, make ^489594, for the number out of which the iisth a nd laft Figure is to be found. To find this fixth Figure you muft Tabiflatc lipon the Rods the dapltf of the Figure laft foundhetween the Lamina, andtbe.R<Kl^ already Tabulated ; but here, becaufe the dUpie o£ 9 is 18, con* fifting of two Figures, therefore this , -i, ., : r 1J8 cannot be Tabulated upon one. 117716237694 (343Q9S Rod, as before we did ufe to do, when • • r ■• • • . , - - the duple was contained of but one . ■ ) ' ■' ■ ' ' ' ■ ■ ■• : . . Figure, now in this' cafe j (and fo of 9 all the like,) Firft, Tabulate a Rod 277 . bearing % in its tseV^ Square,, hetween ■■* " I the Rods formerly Tabulated and the 256 Lamina, and next to the Lamina, *» 2116 then for the Unite i, being the higheft Figtlre of the 18, you muft Tabulate . 2049. '- thathpon the laft smd low^'Rod,for- f 672376 merly Tabulated, whidi is done by 1 1 \ m .1 ' *. cncreafing the Figure in- its top 617481 Square^ one Unite naore that it was 54^9594 ^ ,;. before; fo here theiaft Rod before , rrr^ — ^-^ y^ Tabulated, carrying only o; either - • ^4^9104 turn it, . or elfe take it away,, and :'.\ ■ > - .9^ , i , place fuch a face of that Rod, or of i: „ . . . ^ . fomeotheit, that hath inhis upper Squaccr^ib^ Vtoite lin rpo^iand ftead wf that face, with Cyphers, fo is yojjt Ntfrnber i8 the (Juple of 9 Tabulated, this done, look over the: Rqds, for a Number; that. will come neareftnncoi ^489594* in tlxejEigWvilae is this Number 5489504, and its refpf^ftive Figure for tlie RoQt, S ; This 8 put in- to the Quotient, and Subftraftion beingmade as isufed to be done, th^te-'wSll remain 90^ aad.fo is.your whole wbrk ended, and the Square^ Root of 11^716237694 is 34?p0Bj if you multiply, this^ Rootbykftlfj and to titat.produaad4'iCh^.9P:t:hatremaineth,>T3U Ih&nprtiduce again thefirftrNumber given, .which argueth that the work iff truly wrought* i ... ' ^ i. / ;Bu6n<M^ whenunyikirig Sremaineth, the Extraaion being en- • deA,> as .here it dothj dBakeii'Fraftlon ()f;.tl^a,t;.remainder asyou do iniDivifion, in this' manner &. Set .the Mumb^r i^ remaining;^ aftjer! this Ei^traftion Is endod^.oYec alane for Numerator, and for the DenogMiiatiorifet the duple t>f the whole Rof>t/pund, withonegd-, i - " * Oo f dcd. $$4 UjhifmptfU MithmetUk, . ^ diertiinto, as therein this ccaiople 9p.remameth ; this 90^ ptrer a Liiie fo the Nmneiator of the fra^ipn^ tlien double 34309^ fbeRopt, and it is 686106^ tp ivhtch addl ooe Unite, and.tlKnil iijll be 6861979 t&is let do^a under the Lipe hi Denomioatpr to the ^ftkm, and then the true Root fqnare of the number giveii vrill be 343098j^7;4? ^^ ^^^ f^^^ ^^ ti'^ Example. This \% iht TJilgar, ana ordinary way to make a fradion of t^t remauh Ijcr. .^ ' But the l^fl; a^4 moft certain way to attain unto the true va|qe pf the ^fiipn remaining, and tliat top by the Rods, very eafie ^4 fpeedy, is tp add tvo, four f <»: fix Cyohersi to the remainder, $m4 <!ontiime the Work of E^traftion^ aad then your frafiion will be in Priipesi Seconds, Thirds, ^c. t^atisiQ 10 parts, loopartsj ^n4 looo parts, in the fame maimer 9s in pivifipn^ for npte« that fofeyefy two Cyphers that he 9d4ed or adjoyned to the number . given^ yo\| (ha)l )kve one ^fiipnal Figure in the Quotient, which ^ill jrepreient the fra^ioii in Decimal part; of aii unite, we wi)) add 6 Ciphers tp thti remiamder in this put laft Example, and jt , W^ tben.be 90000000 , apd then we will continue tpe Work, thm^re 'tabi^bte 16, the duple of 8, the Figure laft found | f]^\<h to dp, put a Rod that carried^ 6, the Ipweft Fisure of i6 \ 'pGtt die Lamina, between it and the Ro<ls afore Tabulated, and ^hen inftead (^that Rod Uft in place next the Xfamina, put anp? tijer Rod that hath in his upper (quare one «nite more than that, as hefe 1 change the Rod &om 8 to 9, and the Rods are Tabu? * lifted, and you are now to look out a number that will neareft take iiWay 9000, tl^ number belonging to the firft fraftionrpoint, but th^ kp4s'kiveyoui)pneib fmall, t^defoire pi|to in the Quqftient finr f:)^e firft Figure pf the firaSfion, and becaufe there is no more to da a^ut this nrft Fi^ure,yo]i are qext tp Tabuhite a Rod wit|i Cyphers i^eic die Laniina, anil ttenibe&r a number that will come neareft < Tintb' 900006, the number belonging to the fecond^ point of the fraOioh, but yet ypii flu^ll have no^ upon the Rods fo finaQ t |:herefore put anotliet o iu the Q^tient for the iecondfrafti? pQ Figure : again^ Tabulate a Q.od with o next the. Lamina, and yo4 than yet again find no num^r pn the Rods fo JTmall as9oobOQ0c^ the ni^mber belpngmg to the tbirdrpoint, therefore gut a third Cypher in the Qubttefit for the third Figure tl^ereof, Thush^^e ypu done with your three points of Cyphers^ whichyou ba^e firft added, but becaufe you ate refolded tQ getat lean one fradioa? figure, therefore add tw6 Cyphers more tp die reimMndef , 10^1119. \t ^00000000, an<) alfo Tab«ilate a Q.od l^itJ| o qea^t to thefaminai' hn4 ^^^ ^^ ^f y^^ can find a number litde enough upon 4^ Rod^ ^4 npv^ here at laft you (hall find, one fijjjitificatiife Figure iu d^i quotient of your ira^ipn, fpr the nmber now Iseloogiag to tki; fourth figQiou point is oooQpooooOiCoqfi^iltgJof ip plac^of lvinire% \ inpf^mi ArHhmtkk. «t$| 'Mn^ the nimiiber Tabplatscd isallb now become to beof x^^oes^ .aand>ithal the higiieft Fjg^ireslefs tharithe higlieft Figures of that tiumber whkli betongeth to that fourth ^int, therefore Tcck upoii . ^he Hods for a number le& than that ' 9QOOOCC006, and that yoti fhall hate j^oodoooooo f ioooi in the fecond line Mpoa the Rods^ and ^i • • i i it is 6861960004, whofe reffJqftiye fi- ■ ' r -^ ( n gtireforthe Rck)tfe2;h6wSubftrafti- 2x38039996 ^on being made, there will remain £861960004 2i?8q399904 Thiisbave yougotteft. ' i one Figure iilto the Quoti^t ot yottr fradionVaind th^t in the fourttt place Defcendidg, and ihaybe thus expreffedfraaioo-^ifeT-o-yvf^ or thus,* vw o -of- hgrtJfyiDg 2 parts of loco Of an Unite',; for note^ that fo many fractional l^mts «s you bring mt6 the Quotient tof Jjroducea new Numerator, tlite Denominator is always anVteitef^ trith as many Cyphets jts yoa have made fr^ftronal F/gwes^ This new i4l^9% ^:w.A found fraftion joyned to tte Whole parts of the BLoot foimd, will ftaijd an here in the Example j dt elfe with^ 343Q98vOOoar out a Denominator, thus, vMch isatt ^ one with that otlier. ■ . , ij k - Thefe Examples might be firfBcfcrit to.fccw tfc ^ixelletf £ itfe of the kods in Extrafting the Sijuare Root of any number; but ^ yet to ftew the more variety of wo^H take oi*e Example more } arid if in any thing I be thought too todkius^ow,thatit is otft of a deiire ofplainnefs, even^of fodt a plaiime& as is asfWi^aUe torttet of the Rods, Let there be giVen .tW$ ImmbleSf, .97419^56 and! thi^ S#frtl , feoot thereof reqnirccf,Wrke.downtl»5«tt«D^<«Mrf:Itfe^ Points under each feeond. Figure, a'Qaafi^ient Line, andthed proceed! as before, and ^rft, took ilpootl;e Laminate What is thef greateft S<juare nttmbiflr iChete, ttfttcanbte h^t^tof 97^ tSe twor f iguresbislongtng to thefrft ^ I^teft Vdtxti the Laifiinsif fbcw^ cth that it is 8ifWboi(Mloi»t'hj9; ^t this 9m tte(^otieiitfortbflf firft Figiire thererf^ and^tein S»bftra.d>aiiTOm.97, ^ tbexe'nxaixi is 16 for the Second Figure,' /Tabulate .i«^ t*ie Duple pi 94 ^ptoi* two llods, between the Laioifta vaA tte ledge, and then i^^dt thofe two ftods andi laMihia feek but the n^t^ that e^ds «estf^ ^ to,l64i,^the»ui8lbertotbe'fbcoiid.pbin!t fe^ feall find to be the nbn^ber 1504, aipd Msr<Cp$^iyeFi^eforth^ : tloot 8, whicfi being .{Art into^ Quo0e«t,i and ^ubftra^tionrm^qr iccording to the liiftrw&tonsafotci-dbliTeim^ oiioid]^ rdsi^in« fag tor thef Third! point Willfcfe 1379*^^ aim.to:ftwIottt h&^ipojjwr figoref for tfae ttocrt. Tabulate 16 the Dupkr of 8 laHl foaod, in ibis Irianae^^ i^t jl^tbat (^rriettt iiihi»«>:{r S^iiMrfteFiga/' •; 286 InfirumeBtdl Arithmetick. t 45 betwixt fhe Lamina andtfae former Rods, and increaft the fonseti • iowennoft Rod ooe Unite, by changing it from 8 to 9, thenihali] you fee the number upon the Rod» 97419256 (987oxJ74f neareftuntt) 13792 is 13769 va^ * * * * feventh Line, and after Subftradioit — — ^— made, there win reft 23 , maidng tbe 81 ' number for the leafl point to1>e 1641 23561 XM>w to find the refpedivc * Figure of that fourth point„« Tabu- 1504 late 14, theDupleof 7as beforeyon 13792 were inflrufted, and then you maB ■ ■ ■ ' fee at the very firft, that no number 137^9 upon tiie Rods is fo fmaO as that 2356 23 $6, {the number belonging to the ' laft Point, therefore put o into the Quotient for the lafi: Figure of the Root, and fo have you ended the .Work, for the whole part the J^ fought for, which in this Example appeareth to be 9870, and the J^emainder 2356. But now to make a Fradion of this i^mainder, as you were before ihew- ed,fet the fame 235^ over a bne, at the end of the whole part of the J^ found 9870/ and then duple the i^oot fbund,aad to that duple add one Unite, and the total will be 19741, which fet under the line for Denomiqatx>r , and then the whole work is finifhed, and the true i^oot found anfwering the demand is 9876^* *.Jf, and fiandeth as in the example it appearth* But if youdefir^to beyetmoreexaft, and would have the tru-i efl value and eftimate of yoqr fratHon ; then tdm it into a Deci- mal, and proo^ as before ; firfl add to the remainder fo manyi times two Cyphers as yon defire to have Figures for your Nume-' rator of your new fraoion, that is to fay, two Cyphers if you would have but only one Figure, four Cyphers for two Figures, fix for three Figures, and fo fordi,forasmanyas yon would have, ' and intil you think your fra£lion is fmall enougli : In this £x- • ample we add to the remainder three .Points of Cyphers, that is,* 'fix Cyphers, becaufe we wouki have three Decimal Figures in our fraftion-root ; then Tabulate the duple of that figure of the i^t laft found, but becaufe that is o, which doth neither increafe nor diminifh , therefore Ts^ulate o, next the Lamina, between itaod ' the other i^s,next fee it^n the i^s what number there w^l come ' neareft to 235600, the number that now belong^ to the higlieft Pointoftheira£tioa^potnits,and that is only the very firf):,'i;i^.i97^ ^ therefore put i in the Quotient, for the firfl: fiaftion-ngnre 1 ; ' that win do it,and then make Subffaa{l]on,and there win remainfbr 1 • tiie iecond fradion-point 38199000* . ' • • 'To' s. - Infirumental Arkhemetick^ 287 To' find'thefceond Fra£iioa Figure, Tabulate 2, the duple of i^ and feek the Number that comethneareftto that remainder 381990O) and that is again only the iirft, diereibre-puM ;in the Quotient, and make Snbftradion, and then to your third point v/ill belong 184^^8900, andto find its refpediye Figures for die Root, Tabulate 2,the duple of i laft found, and inquire what ntimber , upon the Rods will come neareft to that 184588900, and that you ihall find in the ninth line of Squares to be 1 77662061, therdbrepufe 9 in die Quotient, and when Subftraftion is made, you will have remaining 6926839 ; thus have you three Figures in the Quotient, which are enough to give the firaftions Value in any ordinary Quefti- ori ; if you pleafe to continue the Work the fourth Figure willbe 3, and now is your Fraftion turned into a Decimal Fraftion, whole Numerator is 11 93, and his Denominator 10000, and bchig fet iFraftion-ways, will ftand thus^|;-;|: Or it may very well be ex- preffed without the Denominator, only with a line, or point of diftinftion thus, 9870, 11 93 and fo the value of this FradiOn is 1193 parts of 16000. • ; The OPEI^ATION' Of large. 97419256.006600 (9870.1193 81 1641 1504 137^2 . ■ 2 3 5'6o6 1 ./ ■ . > 197401 9 81 9900 ^ 19749 74' I 845 88900 > ♦. : 1 7 7^62-66 1 If you multiply d»s Root found 9870^1 193 by it fetf, the Pro- du£t will be 974192 54.996a9a49vfi^ whdni^e, {>ya Point or Line cutoff 8 places of Fauces, according; tO' the Rules of Mttltl* . plication in DcciAa^ .ArM|bDKttd^7 a^^^ the Numh^ ranaining to- wards .«vatds the left hand, will want Cche Fraftkm confidered,^ bitt: 1 1^- fflicde more tfaaooneUnite of Che Number firftghrea^ but if yotf cake Che paias bacto contini^ the work to one place lower, it wiO HOC muit an Unite, and fo the tower ^ work the Fhtftioa^ the nearerftUyottoome to the exa£i troth. ' i r IX. The Eistr4Sio» of the Cube Root by the Rods, To £xtr^ the Cnbe Root of any Number, yon are firft to writedown the Number given, whofe Root is required, and ^malce apoint under the loweftFigtire next to the right hand, and anoUier jpoint under the &^urth Figure^ and (bunder every third Fi- gure,omitting between ever^ two Points two Figures unppinted^and- uen fomany Points as you have under your Nvmber, io many Fi- gures (hall you have in your Root ) next draw.VQiiotient-liney as in the fixtraftion of the Square Root, this beiiig done^ die Num-^ ber is pr^ared fer Extraction ; then go to the Work : Firft, feek what is the greateft Cube Number in the Number Handing above, or belonging to the higheft poitit next the leii^ lumd, which ths Lamina will (hew upon his left ^e of that Face fi>rtbe Cube Root, and in the Column, upon the right fide of the lame Face is the Root thereof, and when either by yotir memory or Lamina, you have found the greateft Cube Root m that Num^' ber belonging to tli^t firft point, then (as in Extraction of the Square Root) fubftraft the greateft Cube Number, to that firft point beloQgmg, or that in tlie Number, to tbe faid firft point be longing, can be found from the £iid Nuo^ber, ^c* then fubftraft it, and fetthe Remainder under^as before In -the Extraftion of the Square Root, and put the re(pedive Digit Number found in tbcf <^otient for the firft Figure of the Root: Now to find the fecond F^nre of the Root, (and fo all the reft, bow miany & ever they be) you muft always Triple tb^ koot found, and that Triple mol*' tiply again by the Root found,^ tnd that laft produft Tabalate tipotf theRodson theleft)iandof.the Lamina, as before in the Bx:tra3i^ on of the fqoareRoot, yon. did tl)e duple of the Root founds thci^ look npod thofe Tabulated R(yte ;md Lamina togethier, what Num*' ber you cad find upon them wiU come neareft to the Number be-c ^lob$id|gtto the Number ae^tfotibwii^ andfa&^th^n.it/ which JNnmbeiriS'tb&fiiviftr^ ^the t^ignreonthe vigbt hand of thela-^ ^iniiilat9tiieil^gittie for HiefiLooi aafwesrable t^ th^tpoioty and tfaat >¥^;|ii^-piil^lh0 C^MJctt ib^ dte Root,* beifenging fy^rmmntAl Jkrithmtick. %6q WoB^ag and anfwerable to that point; but Aoteaiwayy^thsA yon. Jake the Divifor fo often, indno oftenef, biit tbatypu nttyi.yet aUbi take another l^amberfrom the Number beiongiog to thatpoiau srhich other Nupiber is Square of the Digit new found; iiHii(^lie4 into the former Triple, the produfl add to the Divifor^ .witli tbjb provifo, that yOQ^<:e this new produ^ one plat:e higher toward^! the left hand thftn. h the Divifor, that is to 6y , fet tiie km'eft.place pf that new prodnd under the fecoi^d ^ce of the Divifol?, th^ tptal of this Addition SubftraQ from the Number hdongkig to tfai^ point in a^ion, cancJfrlling the &idnumbte,and the remam.ret;uii<9 der» as you ufeto do in Divifioa gad BxtraOion of the Squane Jbo<>t { and fo prot:eed to the next poin^t^ if you hare any more) j^t toi make ^11 plain, we will iilufttate this by variety of £xamples in aO the kinds and dii&rences of Works. Firft, let 110592 be a number given, and the Cubick Root there- pf required ; this Gpbick Root is thtis found. Firft, prepare your Number, that is tofay, write it dowi^ and intake a point under 99 the loweft Figure tl»sreofnext..the. right liand, andoqepther point under o, tb$: foustb Figure thereof, lea-^ Ting two Figures betweenimpoioted, then draw the Quotient-line, and tbtn the number will fiaini^ ready prepared as in the Example, widitwo points, whereby it appearethtbat the Root will confiit of two Figures, Mliicb are to be fonnd out according to the former 4ire£lions; and firftobferve th^ti Id is the number , . . belonging to the firft pointy and upon the Lamina tio^^t (4 you may alfo obferve tiiat the greateft Cube number ? • in that i|o is<54« and his Cube Root 4, therefore >M*MlNI»ar put 4 in the Quotient, for the iicft Figurp of the 64 Root, and then SubftraQ 64 from ijo^ and tjiere 45)^1 jiritt remain 46, dib 4^ fet under, 64 and the Work of the firft point is done^ and hero you may ^now obferve^, that* the 46 remaining, .with the other Pistes 592, make 4^5^2, whkh is the number befonging to the fecond pomt, at»l where ont the fef jcond Figure i& to be fottnd# ; To obtain thb fecond Figure, proceed in thts manneri triple 4^ the Root fouiid, aHa it is 12^ aii4 diat triple multiply bytiie Root found 4, and the Product is 4fi, this Produft 43 Tabuhrteupodtho Hods on the left hand of the Lamina^ between: it and the Ledgi^i tben view dver thefe Rods and Lamina thus Tabulated,. wIki^ nQpibertherfe willicgme neareftuiito 46592 theditn^r belonging ^ tjie fecond poin^aiid: be ljef$ thaa '% you fhall Tee the^msmbei thgtconiiaihinemnbft lo It, 1$ thuX: in. the nindl Line, 43929^ and ht^ jre^dive Figue for the Ro6t js 9^ tiow Square tJiis p,. anl it is 81;^ this 81. multiply hy;ttefen!ler Triple ift, . and it yield^th 97^ 1**8^ 57^ add unto 4^929^ the nhmber found 09 fte Rods <beii^ fet \n Idditip^ Qf^e^^ I^flr ttAn in>x4iiur]r^ as |f ^ his&re ftew^) 2^ InftrwHenid jirithmeticjt: and tke tbUl will be 53949) which if you compare 'with 4659?, the number belonging to the point in aaion, ycru (hall fee it is too great to betaken out of it: Whereby it appeareth that you muft not take9 for your Root, f(M:. by yuur general Rule you muft take tte Divifor no oftner, but that you may take alfb the Prodnd made by the fquare of that new Digit number multiplied into the iirft tri^e, out of that number belonging to that fep^nd point, thece- fbre tak6 a lefs number upon the Rods, as the number 3891 2 in the ei^th line, which will forely ferve the tum^ wherefore put S in the Quotient for the lecond Figure of the Root tbttght for, now write (if you will} this number 38912, juft under 46592, the number be- longing to the faid fecond point \ then according to the Rule, Square this Digit 8 new found, and it giveth 64, this 64 1 10592 (48 multiplied by the former triple 12, produceth 768, ■ write down under the former number 38912, fet- 64 ing the loweft Figure 8 of th is new prpduft direftly 46592 uilderx, the fecond fignre of 38912, and its fecond figure 6 under the third figure 9, and fo of the 0- 38912 ther in Order, then add thefetwo numbers toge- 768 ther) and die total w|ll be 46592, which is equal to that number above, therefore if Subfba.ftion be 46592 made, there will nothing remain, and fo the Work is ended, whereby you may conclude that 11059a, the number given, is a right Cubick number, and that 48 is the Cube Root thereof, which was the thing required to be found. Now \i you would at any time prove youi^ Work, whether you have- wfw^ht truly or not 5 multiply the Root found Cubelickly, and add the renaainder, when any is to that produA, and if the total be the firft number given, then the work is truly wrought, or clfenot, as here in this Bi^mple; If you multiply the Root found 48 by 48, it is 2 3049 and this 2 304 multiplied again byv48, produceth 1 10592, the number firft given, and therefore conclude that the Work i? truly wrought. ? ' For a recondExample,we will take this number 41 06362 5 and feek the Cube Root thereof,firff,preparethe number yby writkig it down, and making a point under the loweft^ figure 5, aiid another under 3, fhe fotirtii^gure, and another under i, the feuenth figure, an^ draw the (M>tient4k)e2 thefe three pditttg do- declar-e • that the Root wiO . confilttrof three figures; ]Now to^l to Work^ tofmd the firft trf them, confider what thegreateft. Cube number in 41 Is, which ap*. pearethon the Lamina to be 37, and its Root'3^ Ihei^pforeput 3 in. the Quotientfor the f^rft Figure of the Rootv ailddian Tubftraft 27 . from 41 , aiul fet the remainder 14 under 27, ^wl ^work of the. firib Figure is ended, and-the number that bdbngietli to the fecond r Point is 15063, out of which the fecond Figure istobefound, be^ hold th^ Example : To g^ dib fecond ^FigOf e,. trqple 3, the Root found. f^t 410^625 (2 14062 infifumntai Jrk^mtkk. |cni&4i.9#9 99 this triple 9 niuld;^y by' ^ tlie Robtfottiia grveth 27^tbis product 27^ Ta- bnilcft^ upon the kft hand of the Ijaminat and ili^ the number tteit will/come neareft unta ^hat 14063 belonging tothe fecbnd point now :im adion^ remembring the fbrmer Cautioa, 'tiat it be taken no oftner, but thaf withal •,. ; . tketfs may beta)^en from tli^ence alio the PiFodi^ prpdu^d^bf' irmklplying the Square of'therefpeftive Figure into the formfer, TTiipte, as here the diviforyinajr ,be had five times, but by reafon of that ot^ number, that liiuftalib betaken from tlience^ cannot be taken alfi), therefore it can behad but only 4 ^imes^ wherefore put 4.ia the (^otient forthe fecond figure of the koot^-and fe^'the num- ber i.o864ibund in that fourth rank of Squares under the numbei^, 14063, the number belongings to the fecond.poW, then multiply 1 6' the Square of 4, the^ digit now found by 9^ the .Produ£^ is 144, this 144 fet under the for- B^mer iQS64r but according to be former Pro>- I^Vi^l, one place higher toi^rardjs thelef^ hsgid^ [ and fet its lower Figure 4 under ^^ the Second r¥igureof 10864, the next 4un^r S^and the i uppermqft being i under o, aa ypu fee it ftand in the example ; the'n add thefe two numbers together, and they make 1 3304^ tills take from 14063 leaveth 1759^ thus is the work of the lecond . point at an end } l^hoid die Ex* ample* Now for the third figure of theRx>ot^ you .., \ are firft'to obferve, that the number belong- 41063625 (34 J ' ing. to the third and laft point is 1459625, * • * from whence the third and laft Figure " ' .of the Root is to be extrafled, which Figure to find out) triple the Root found 34, 'and ; it is 102, and that multiply again by 34, the ' ^.oot found yieldeth 3468, this produft 3468 is the Divifor, an^d this Tabulate upon the R^eds as before, ^ joyn the Lamina clofeto them, then feeK upon tiiefe Rods and lamina what number will come neareft unto that 1759625, the number belonging to the , third point now in aftioii , (remembring ; the former Caution,but here is no need of that in die work of this point,) you (hall find the : luimbQ: for tiie purpofe, to ^be the nun^ber 173412$ ftandlng in the fiftlt line and itsre^ i fpeSive figwre for the Root 5,therefore put $ i«i the Quotient fof tm P| / third 41063625 (34* 1406$ 10864 14/4' ■ill. ■. M i2304r 1759 27 14063 ^' • - 10S64 144 2«3D4 1759^25 1734125 2550 1759625. 2g2 Infirumentd Arithmetick. third fig.of the Root,now tranfcribe the flnmber i734i2$,froiB mof the RiSbinto the Paper, )uft nnderthe 1759625, then to find d)e other number to bie hereunto added ; fquare 5 the figure hift found, aiid it fiiakes 25, thisSquare 35 multiply by the former triple 102, andtheProduftis 25 soythisfet under the other number 17941259 according to the former Provifo, as yon fee it ftand in the Example, and add thefe two numbers tc^ether , and their total will 4)e i759625^nd is equal to that above,belonging to that third point, fo that if fubftraftion fhould be made^there wouidnothing remain which declareth the number given txi be k perfeft Cubick number,^ and the Cubick Root thereof to be 945 which was the thing required to be done \ if you will multiply this Root 345 Cubiddy, \t pro* dttceth the number firft given 41063625, whtdiprovetfa the work to be truly wrouftht, thelikeistobeoUcrvedinallotherWodcs of this nature whatfoever* For a third Example, we win take at an adventurb this great number 859271650667 , and feek the Cube Root thereof; This number being prepared with points, and a Quotient-line \ (heweth by his four points,that his Cube Root will confift of 4 fig«and by the former Dire£tions,the firft figure will appear to be 9,for the greateft Cube number in 859,the number belonging to the nrit point,b 729 , which taken from that 859^ ^aretii 190, 859271550667 (950 which with the 271 between that 190 and , • • • the next point, make 1 9027 1, out of Which , the Pecond figure of the Root is to be ex- 729 trafted; this fecond figure by the former 130271 Rules, will be found to be 5, after the ^^■. work of this fecond point is ended, there 121625 willbe remaining to the third point 18^6650, 675 and the Root found is 95, this tripled, and ■ I alfo multiplied again by the i(oor, pro- 128375 duceth 27075 for the Divifor, this divifor , Tabulated, there is no number to be found upon the J^sfofmalla^ is 18966 50, the number belongings to that third point, therefore put ao in the Quotient for ^he third figure ,of the J^oot, and fo have you done with that third point. Now to find the fourth figure anfwerable to the fourtji poin^ you need here do no more, but Tabulate two c between the Lamina, and the other J^ds, without any altering ofthe^^ds al- ready Tabulated, and then feek the fourth figure as befoi^^^, the reafon is, becaufeo doth neither multiply nor divide, but only raif- eththe places of figures higher towards the left hand, for here if you triple 950,, the ifoot round, it ^ieldeth 2850, and this multi- plied again by 9 50^ the(aid-^otfound,produceth 2707^00, which is the feme with that former Tabulated number, faving only the two Cyphers, and therefore it is, that when there isaointheQuo' tieat, p F hfirumentdl Jrithmetick. 2^1 ^^tent^ there needs no more to be done, [but tp Tabulate two Cyphers between 8592716506^7 (950 'the lamina and tfaore Rods before Ta- • • < • f^ulated, when you fought for that laft ' ■ i*¥igure, which happ^neth to bp a o« 729 . >4ow upon thofe Rods thus Tabulated, 130271 feek what Number upon thofe Rods and Xamlna will come neareft unto that Num« 1 2 162 5 ber, which belongeth to the fourth point, 675 which is here 1896650667. By viewing ■ the Rods, you Ihall find that Number in 128975 tiie feventh rank of Squares to be the Number that will fer^re the turn, vi^. 1^95250343 1895250343: And its refpeftive Figure 139650 to be put into the Quotient 7, To this Number add 139650, the Number made 1896646843 by multiplying 49 the Square of 7, the Digit now found by 2850, die Triple of the Root afore found and the total is 1896646843. This Subftra^ed from that Number to the laftpoint belonging, leaveth remaining over head 3824. Thus have you iiniihed all your points, andhave found the Cube R.o6t of your Number given to be 9507, And being there is a remainder left, it; appeareth that die Number firft given is not a r»rfe£l Cobe Number ; but the greateft Cube Number therein is 59271646S43, and his Cubick Root is 9507. The truth of this Work yen may examine by multiplying the Root found Cubidcly, which if you do, and add the remainder to the product, you (hall produce the firft Number given. Now for the remainder, to make a fraSion thereof in fuch fort, that it may apdy exprefs the neareft Cube Root, I could fliew Teveral ways delivered by feveral Authors, how to bring it neareft to the truth. . But the moft;^ abfolute and beft Rule to get the Cube Root of any Number not Cubici^ is this : Add to the remainder fo many fe- vexal poiilts, or Ternaries of Cyphers, asyoudefireto come nearer to die true Root, (\n the fame manner as you did in theextraflion of diefquare Root) and then continue on the wdrk forei&C^Oion, 9s you do in. whole Numbers, and die Fraction will' be turned into a liecimalPradion, and then fo many as you add Ternaries of Cy- phers, and the Denominator will have an Unite, with as many Cy* phers. as you added Tcxiiaries, or poinds of Cyphers to the Rc- maiiMer* Pp 2 Some 1 .\ S^4 hjhrftmemd Jridnmticic, Some Vfis of theSfuare and Cube Root. l)fef tf ^ Siaars X^b$. WHat the Jjn-o-e and C^egoatAif, fwd how to evr^ them, hath alrejuiy bcro taught ; and for more cafe and expeditloD, There are T/iilesieadf calculated, boch of -the SMtreapd Cube f^m^ from I tu looo. Wecoaie4io)v.tolhew fomcUfrathecsof^ wJudi in fome meaTufc will appear jn Uiefrt^tiottiMoiiviDg; PROPOSITIONS. I. AMp 1^ kmk jf tht Was ef t fflt or Ca^lt tu f,? [ci\ei, ie jg Ton, tmi thebreadib if ilx trench tbottt rie Fan be ^oFoat; ' dmumiff i*i<0 length dSc4ihie Udier ({mS k, jufllj xores^fron .1... , r ■( J ■ . -•-■—-f tbeW4} Fon, ^itbebre4dtb if the Trench about rie Fon be 4oFo«; / 4iwmt^^h<akngtbAScilttigUdii""' "' '' ^ fbe eigt or l/rew (f. tie Trentb, te the u Pydje47tl)of^firftBoo](pf£w;iJ's£/fn«wt/itisdeni05iftrat£d; that, tbefyu^rreof thf Hnotenufil of aUrigljt foigUifl^tiTriaxgles if. tiUfiu)»e Squtri,s ef the a otter fides : I therefore to refoltc tiiis IPropplitioo, fquire thf hei^ of the WiU, which is ;o, fitffr oooj »Uol fquar? the brcadthof the Trssci whicji is4o/w/f iSoo, niefe p*o added together m^e ?^oo,' the Square Root whereof is so ; tai folong mufta Sqiliqglladder be made jQreac^.fir^ni'The ease <)f thefTrWff tothetopof thelf^a. ' "' \ ;p R p E 9 S I T I Q T^. n. ' there he twotomtf «iChicefter4»JYork, t^hUb lie l^ort^ daj Seutil '' atejrem mother, tm4 tttir dififiife H 2?o rvkf, ^i ExCraef heibiiieayWeafio'm Chichefter, \2otpks\ 1 defretQhiirwthe 4ifiaietf}[wkfim^ceit^x? S;'"';^ ",' Ixfefier • wi Tvl Sxtejttt: amant trom zert. Vff \ 1 Jnfirumenul Artthmeticki 295 Vfe of tbe Cube I(^. ONfi chief life of the Cube Kgot^ is to fjind out a pircmordon be^ tween like Solids; fuch are Spheres, Cubes, and fach like,- SIS in Propofitions following, PROPOSITION. L Ij a, Bullet of Brafs of 4 incbes Diameter ^ weigh 9 pmrnd^ wbatfiiSd BuUet of Brafs weigh, whofe Dianteter if 8 mhe$ ? Cube 4, the Diameter of the leffer Ballet, makes 64, likewife CubetheDiameterofthegreater Bullets, makes46o8, Thisdone, fay by the Rule of Proportion ; If the Cube 64 give 9 U. weighty what (hall the Cube number 4608 give? Multiply and divide, yoa fhall l^^ve 72; and fo many pounds will a BuHetof Brat weight whofe Diameter is 8 Inches. PROPOSITION. II. jf a, Fathom of I^peof 10 inches compafs about, do weigh lypouni^ bow much jhaU a Fathom of J(2pe weigh, which is but 8 inclfes torn* pafs about ? The Square of 10 is 100, the Square of 8 is 64 ; wherefore by the Rule of Proportion, 6y As 100 (the Square of 10) ' Is to 64. (the Square of 8) So ii#7 (the weight of the Fathom of Rope of 10 inches) To to -fii pounds (the weight of the Fathom of Rope of-8 inches about.) PROPOSITION, ta. If a Ship of 100 Tun be 20 faot broad at the Midjhip Beam, of what breadth at the Beam JhaU a Ship (jff the like building) be watfial beiQoTun! The Cube of 20 is 8000, then by the Rule of Proportioafay* As 100 Tun (the burthen of the Ship given) Is to 200 Tun (the burthen of the Ship required) So is 8000 (the Cubeof the givei Ship's Beam) To 16000 (the Cube of the required Ship^s Beam.) Now the Cube Root of 16000 is 25 i almoft, and folongattbjS Mid-{hip Beam muft a Ship of the fame Model be, whofe burdiea is 200 Tun. And let thus much fufEce for the U& of Ntpaif^g Bmes^ an^fot^ Inftmmental Arithmetick I will put AN END. r ( ft • • • I . . ] *> • > 1 O F ALGEBRA Part IV. iWaMBOHM ^ LONDON^ Printed for Jwnfiam and 3^i&/» CharehiB at the BlaekStvan in Pater'No^er^Row, 1700. iMMMMMM^MMMMMMMMM V ' ^ ^ § i ' .t 1 «t J* » 4 r . ^'•x "> ••• sK f* ^."i « / v\ > I t 299 ] ALGEBRA. ■ »■• - " ' ' '■ •> I P RO E ME, N jthis ^temfeoi JiJgehraJhr the SymbdhhcK ufed i and ^Iie Method obferved \ Ic is that of Mr. 7to» £fanio$y in fnch tj^iqudtions as areproj^fed in Numbers: Thn of Defcanes in fuch JKqmtimi as are .fo/2% and not in the iSTion^^x* Not, that tius Book is taken out of themi neither aoth it proceed continQally with them ; but Difjunitly ) as the Author hereof Mr. tho* Gibfmy lon§ fmce, vi^* in Anno i6$$« thought fit to intermix them among other thingiS which are ifoc iH them. Now, fbrafmuch as Aleebn^ cannot be well attained unto witfi^ out a competent knowledge in Geomepry\ It ^ill be neceffary for the Reader to acquaint himfelf with EucHfs Elmenti \ cfpecially thtf fir& Six Books \ and in them principally, with thefe 48 Frofefi* f/oif/ following. Propofiticns of Euclid , fit to be fciiown to the AndlQ. In the Fitn Book thefe Mieveri. Vvop. 5, 13, 14» Mt»^ »9i «*# ^ 32. 4i3, 47i 48* in the Second Bookj thefe Twe/vr* Prop* 1 1 a, ^, 4i $ 9 6; 7; 8, 9^ t o 12, U. tnthe TlitdBook^ tWe Jfevei^. Prdp. x4,ioi ii^ 31,' 31, ^5^ jtfi None in the Fourth Book. In the Fifth Book ^ thefe Six. Prop. 1$, 16^ i^, 19, 24, 2;* In the^mi Book^ thefe Tive/vr. Prop. 2^3,4,69718, 13, 14, i5^ 19*24, 31- . Many morePropofitions ouf of thefe and thef Remaining B^oks^ mi^ht be Ufeful : fiut thefe 48, before reckoned^ aipe fuch as (in my mdsment) ought chiefly to be Rjead and remenolbred, being very Ufenilfor attaining and Refolyixlg uSKquatiimi. i M "li III if r I ^ ■ > fc D E F t N IT JO N^. J}efinition t. THe Unknown Quantity of any Equation h daile^ geherallf Foteftm ; or a Fower^ Quantity^ or tirfii. J?efinitioit lU . A i^AaHgU is in nttmbersthe FtoHlAd two nmnbers fimlttply- ing one another* Q.9 la n joo ALGEBRA. In Geometry it is tbe Area^ fpace, or conteat of a rigbt aisled quadrangiilai' figt)re> marfealfaby mukijjllcationoftwo lines, which arc called, (be fid«s |pf wUch oaeji^ the meafiire of the breadth, the other of the length. A I^angUi ParMBckpifedon is the prodaft of a Reftangle mul- tiplied by aright line dr number :' And if that line or number and the Lengthand breadtfaof the RefiangLe be fi:veraUy equal it is a Cube, orDie. DeftnHUm IV. * A Pnfme is a Solid contained widiin five Superficies of which three ire Qiadrangular, and the other two being oppofite. Triangular; Or it is like the top of an ordinary EnglKh houfe cut off by a Plane paffing through or parallel to the Eaves. The reft of this kind I thall not define here but refer the Reader to Buclid* Thinnmes cf the Poteftatcs or Powers. • ■ -. 1 The firft Fomr is called a Side, or Root : The latter word J(p9t is moftufedhere; and it is fignifiedthns, «• 2* Th^ fecond ?<mtr is called a Smare^ and is thus, written, aa. 3. The third is called a Cube, arid is thtis written, a a a. ox fomc- times for brevity, a' 4* Thefoufm, a B/jiMir^ or Squared Square, anciently a Z^^/ yn^k^ figured thus x ^now thus, 4 tf ^ tf, or for brevity, a*. 5* The fifth Power is called a Jrtfiy(|]/ii, and is written thus,4i44i; or briefly thus it ^ tf.The fixth aiquared Cube^^m^cKfoyWritten fht&jaaaMA or tf ^ ' 7* Thefcventbi a Second SurfilidfSaii is vihttibtkaa,ada 4i,or morelhorttf ^ • 9. The eighth is called a Squared Square Squared, or ^^i ^^ ^ien^ Xcn^icht and is written aaaaaaaa^ov t^us 4 ', &c. Cenf^Ssry I. ' Hence itis ^?"»fi^ft that thefe Powers uninterrupted , are in continual proportion, the proportion of them being as 4, to unity : or the converfe. / > ConfeSay II. , It is alfohere plain, that every ?ower hath fo many dimenfions, as the Letters, with which it is written. ¥or a ^ being written with four Letters, if one Letter ftand for one dimenfion, that is leogthor breadtli, the other three arife by three feveral Multiplications, and .eTf^Multipl^tioaadd^aDimenfion^iotbi^fQ^^ • J LG EB RA, JM o I OS: •^ . r 5» ^ 3 rt^'' o ^ ft pa CO ST. ^•5>4^ 8 3^ Roots of the Powers. >0 00>l Ok n "■" V " ff^SSj^^S^^*- I - Ok M oo -l %l» Pv-f^'.M •-' <xNo o >o •r ^ •-• ex « Ovi^ 0\ wv Os .A via i-« ^ o^§>» >0 00>4 « cx--^ -* %JK ^^ M M ^ ov »-« •& j; 4»* t^ >o •" •»* rK >b ►* Oi ■k ^ ^ 2>.»»^ ^* '*' ki »^i '^»j '^^ fc,% " «*^ J5 ^ oo <x U w <x «;g o o 5 M ^ •o So*** ^ 00 ^ p ^ < • • W 1 F i . ♦ • 1 # « . . -f c H A p: V }02 ALGEBRA, C H A P. I. Ap Ea9U»4iiat cf the OtAtd^ers atid Symbols^ ufediHthuTrAa. f*Irft, one (ingle Letter of the Alphabet is urually Dut for any Qaanticy wliatfoeveri as wdl Linens Number -| whether known, unknown. ' i But for die mofl: part, where any Qaaptity is fought, there a (X Ibm^ other Vowel is put for it; and the other Qnantities known, are fig^ified by Omfmiams*' Thefe letters are nbltiply^.one into another by joynlng them together without any prick or comma between, nor doch it import at an which is firft or laft written ilrox be d^b dc^ and ck t\ are all one* ' . ' • t So Jinultiplyed by 4 produceth tf ^ And ir multiplyed by A, produceth 4 b. And 4 3 multiplyed by c, ppduceth Abe. ThelikeofallotUbs wha^t(beyer» except Fraftional Quantities: gs, I and — -: — • — - 4 bJ^ If the firft of thefe wdre to be multiplied by d, k is done by taking away the d under the libe» dnd the* produ(l is d b-^^fg^ . U the fecond were to be multiplied by ^4*^, it is done ^by making away the Denominator b^^ auid tha ProduA wiU be ]^ ^r^jT^rf''' '• » Fgr all Fra£tions as well ia P)ain as in Figurate Arit]imetick, are nothing elfe but Quotients of one number divided by another; an4 are multiplied agam by taking aw^y their Divi{bran41ineof Sepa< ration. f « Divifion is done in Figurative Arithmetick, moft commonly by applying fome line of Separation betweea the Dividend and the J Abe Divifoc So -<- is tf divided by b^ And ^-^ fignifies diat 4i^ is di- b ' f videdby/, - * Butyerif the Letter/ had been found in the Dividend, the ;jp- plication of this Line had not been neceflary^ for it might have bden petter done by taking away that Letter out of tl^e Dividend* ' J L G E B R J. 3©j So a/c divided by/quotient is ^ r mdf/c c divided by/ r quotient is fc by //"quotient is fc by f c quotient is f/ by//*f quotient IS r by/ cf quotient is/ by/quotientis/^c bye quotient is// I » And the lik? may eafily be under ftood of all the reft, » 4 / . ■ ■ ■ - .... X r- Majority > ^^Minority . < ~. . /Subftrataion . -^ ", LRootof a quantity "V Proportionality continued ' '' '" '''^ . * iProportionalVtydifjljnft;; 9 if f f^ So^'>tf(ignigesi^greafterthanr ' ' • -• -^ -^ ^ <^ ^liefs thane i=:r..,,.,; ^ equal >oir • ■ A-|-e;,..,,.c^ddcdtot ..*.' * , jr ^-^r. ...... f'taktnfromi ' . _ J. V . . .--V.72figiufiestber^aiprootpf 72;cr^^^^^ ,; .,. And A' iff dm fmr figriifiech thatasAisto?, fa^is'^tb *, andfoito/ .:...::/.. Likewifei' cff ff gU fignifiesthatasiisto c, fois/to|:. llicle things before exprefled are almoft generally received* •!/M.Wednotoalyior breyityin wrJtingr butperfpioaity in proving aswillbefeentereafter* - -;• " •'^-r Not^ that Wherefoever — is riot exprelfed, there 4- is under- ftood, thooghTt be not expreffei. - ~ Alfo in Trigonometrie. 1 ufe, s^p^Sy for the fine of an angle pXs^ ands.c» j/)for thcfine of the toatplement of a fide /»^ to 90. AJfo, t.?;andt.t. 1/^ for tangent of rp^xk\ tangent- pf the Com- plementbfjj;f,&c. Alfo for Radius; I ufe r, ...,/ ' If the fign of Additi6tj,^iq!ntdy T^r ft^nd befoft any 'qnanTiCy, it fliewsthatf^ttantityj to bi? mbri thfah nothing •, that iifeniethihg. But if the ligrt of Subftraellon, to Wit — iftand befdter any quan- tity ;-it fhews that quantity to be lefi than nothing; or a want of tolaid quantity. ' ^ So 4- 4, fjgnifies four of anything; But —4, fignifiesawantof four, or ipprlers than nothing. ^ In r I |04 4 l.G BU R J. * -^ /» ADDITION, The additioii of a want of aoy thing, isall one with the fbb- Jbaftioo of the bme diing. So if to-^iayooadd—^ it makes 4-7. Andif to -4-12 yoQadd-^i6it makes — 4 Bot if to 4-1^ yoo add 4-1^ it makes -^28 X Iff SVBSTRJCTION, The fabftratiion of -«- is an one with adding 4- Soif from^-iayoafubftraft — srtQiain is 4^17. . And if from 4-12 yon fubftrafl -—16 remain, is 4-28. Addition of 4* ^ 4" ^^ Subftradion of — from — is all one ith Commoa Additio^^and Subftra&ion. And genexaUy for both. In AidhioHi add the qi^ntities together with.toe lame fign. In SuhfiraSiottf add them aUb, but all the (ignsof that which is lDbeSobftra£iedjBx>m the otfaeti^ muft be changed* M X A U P L M^. If to4-^— 24*3f be added 4"5+^'*^3 *^ ^^^ ^ +^ — ^*+3 Bat if froni \ - 6 ■ 24-3f be Subflraded 4*54ri — 3» ^ ^ sKuo IS 4^^-r-a4-^-*5— i4"3f?4> 'I^^ ^^9 isvgenenil, and gjcnerally known. /i^ MVLTIPLICJTlONi tmnltiplied by 4- ever producel]i 4r Q^nltiplled by — ever produeeth — , -^multiplied by -^ ever prodnceth 4^ .' \ More Vairiefes.thcrc are not.. . ' '• The i^tefflftitks that are- accompanied with theic fignt of 4* ft-r« (ia^both Midtiplicrs temg placed one under, anotibec, at in ^ommomn^ltipliatiQn} mnft be multiplied every oipbdow into f?ery OOP c^ve^ ao4 diea this work 19 0^ • . J . • 1 J Si ,►. i J LG EB HJ. J05 So if, •{■bb-\-b — c, be multiplied bf'\-f—g;p\acetbemtim. Saying, -f/ multiplied into +A Ogives -4r**/ And4-/intx):f*ghcs; V . 4-/'* And -(-/into — c gives — /c And— g into -4-A Ogives , — Bhg And — ^ into -f^ gives ' — ^i^ Laftly, —^ into —f gives -J-r^ Which is the true prodnd. I» D IV I S 10 N. r If the line of reparation do notferve the turn, thiat is^ if anyde* fire, (and it may be done) otherwife, it nuttft then bp by f^ekin^ what quantity nuy be multiplied by the Diviibr to. produce thef Dividends. : , So if b b-^-b c-'-^bf^^bg-^cg-^/gf were to be divided by. b'\^, — /i trial muft be made what mixt quantity miUtiplying bJ^c^^ will produce b b-^^b c — bf — bg — cg-^^fg. In Which there is this of Compendium, that feeing the Dividend confifts of iir Members, and tlie Divifor of three, the quotient iBuft be of two \ that is a Binomial only. And becaufe the quantity ^ is found in tin: Dividend, andndtin the Divifor, it muft be in the <|noti^nt« The faid quotient therefore muft be one of thele, f -|-g» *^^^> ^ g — b» It cannot be the firft, for -fjg, intso r-*/ woiHd have pjDoduced — fg : But in the Dividend it is +fgj therefore it niuft be --g. By the fame reafon jt cannot be th^ laft,. a^ alfo becaufe — i, into -{-b produceth — b i, but it is -j-A ^, in the Dividend* The quotient fought, muft there§)i:e be^ — g# • . Some further Rule for faiifing labour heceia mi^fjcgiveu: But every one likes that beft which he finds out himfelf;' NOris it my purpofo to write ^Bopk of Jlgeb^a^ l^t to^cflUfo; fpinuch^f the Rudiments thereof, as thelleaidermay ftandin i^c4brifl.the peN ufingthefonpwingTregtife. . \:' •■] ' \^ .'-,'. /' Wherein becaufe Divifion is leldom needed ; IF IliaVe i littte €J^ ceeded already, and Ihall a little more in treating (but very briefly) of refolviog fomefew Rooted iE^tions, I mil ask the Readers pardon for both together^ C B iV Br r ^ J LG EB RJ, CHAP. IL i Of Mqiutions. AN iBquatlon is when one or more f^tal quantitiesi are equal to one or more other fpecial quantities! and written with the iign of equaFitj betwixt them \ As a tzsh h. Thi$ is called, a fimple Square Equation. And h h^ being a known (^uare, the fquare root thereof being extra Aed, is equal co J, And that is the diing required. But, dd'\-'bM=cCf andtfif*-^i=ii, andlaftly — dd'\-bds^ff\ are at) of them of that kind, which are caQed mixed Aquations, be« canfe^ (the thing required) is multiplied not only into it felf. But into another known quantity, namely into L And note that this known quantity in all mixed Aquations is caD^ the CoeffUienu Note alfo that the three forts of mixed Aquations above etpref^ CnI are all that can happen in Quadratiques : And by fome one of diefe, ail Problemswhatlbevertranfcending plain Divifion, and not xeadiingSoIkls, are to be refolved by finding the Root 4^ according tQtiiefcOldKifks. In the FirB^ AAA^iitszcc* , Unto the quantity given namely cc^aH tkSfugre of UlftU bh , Coefickm^ b mikes -^c cJ^ *— - Which if it be in linesi may be reda^^ 4 ced into one Square, mifirpm tie fide $f$bst S^ugre^ uh haftbc Cocffi* rienff mi the renumUrfigU be a. Which was the thmg deiired« InibeSeemi^ Air^d=Ud. Unto ifiadd— ^s in the former, and the Sum thejreof being al- ways in numbers a Square, or in lines to be reduced to aSquareas aforefaid ; Vmo the Act or fide of that Square^ ddd bdlfthe Coefficiem^ the Sum thereof fliaU be4, or the Root of the Aquadott fongbt for* J LG E B R J. J 07 In tie IaS^^dar{'ba=iff* . JFrom the Square 0/ tdlftbe Coefficient^ vbich it — ukesbe ftantkj 4 hb given , that is //, there will remain—— — //, which being put in- 4 to one Sduare, and the fide thex^ known Jftb^fide be citber added to half tie Coefficient^ orfubftmSed tberefrom^ eitber tbe SM oft bat dddition^ or ibe remain <f tbefubftroBion^ U equal to^ <• For all Quadratique ^Equations of this kind (where a a the great* eft unknown power is wanting) have two Roots, which being both together ever eqiial to tbeCoeffictebt, if upon the Coefficient, ft) a Oiameter, a Semicircle be defcribed^and the fide of //(the quan^ tity. given) be applied therein , perpendicular to the Diameter ^« two fegmentsof^^ are the tUro Roots (bttghti Fqit in the Equation — aa-^^b d=ffj it is by the i4th4 of the 6tb4 of £uclidf as tolloweth) iW p ff a^f. Wherefore either Segment may be it . aud tbe other will be b-^a^ ^XiAfz mean betwixt theni. t-ikewife in thetwoforrtier ^qua* tions, the work may be effeaed Geometrially and proved alio by this prefent Scheme. In which, as the figure intimates^ the Perpendicular t^ptefents the „ fide of r(r in the iirft .Equation, and ^ the fide of di in the fecond. l!>raw a line bg friim the center i td . the top of the tierpehdif War, the cen- ter b being firft uken in the middle of tbe line h to wit, of the Coefficient^ fot fo it is ufually called. And firft, let the pricked line be put for J, Therefi)re by the before re- cited Propofition^ It is, 4+-*^ cff cf aiKtucViLe.!^ R r And joS 1 t\ A LG^B n J. An4, if Cas tJie Rule prefcribetSiJ to the Sjjuareof half h you add the Square of half c, the total (hall be the Square of the line ^g-; by the 47th. of the firft of Euclid, If therefore from the line ^ g, or for which is all one) b f, you take the line b r, which is half the Coefficient b (for the whole Coeffici- ent^ is the fame with sr) the reft, namely tiie pricked liner/, fliall bcequal to ^. For, r /==» 51^ a. 'tn like fort concerning the fecond Equation, d a*^b a^=:dd *. bb If according to the Rule, you add the Squares rf4,and-—together,it gives thtf Squareof the line ^^,to the Root of «h!cb,tq wit A^,if yoa add half the Cioefficient,towit,^r,qr ^j thefumftiall be/yor»r, either equal to a. And then, dsnr^ that is d^ is to </, fo is dtx^rf, ova — ^, as it ought to be'. I intend anon to write foitoethiHgof J?xir*5iwf of J^j, acc3ording to the general Method of refolving all manner of -^quatioas of Powers y how high or compoftd foever. I do not mean to'exempii- fie tl)em any further then the Cubique order. 'There are Authors cnongh, whom they that defire the full of that Artifice, .may at their own leafure jn Books confulr. v And now becaufe I ftall herein make ifome life of ^^quations, •though not higher then Cubiques,or at the moftthe Biquadratique order: I think fittoadmonilh the Reader, that in putting « always for the thing fought, and working therewith, as if it wer-e known, quite through as the queftwn requires, he (hall at laftcome to an itquation but It may be Tuch a one as wants reducing: -of which a liale. REDVCTION ■■ ' " /. of i^uations is done by adding all that's neceffary, or fifeftraffl:. ing all that's unneceflary on jx)th fides the fign of j^uaiity : Or by Subftrafting Con trad jftories if they fcappen on one and tfie faHic lide, untill the itquation, purg'd of all unHceffary members, remain withalltlm'sabrolutely khownon one fide, equal to (as little as may be>all that* s unknown on the /other fide. One example of this ftiallferye as followedi; ■ In the\^uation ^ ^ df — bd'\'dc^bd'=:zgg*^bd'-^dc^ To reduce this, you muft remember what hath been Taid before;' that til? raking away -^Wdntoi any thing, js^all one with the addi- tion of chat thing, ' There- 4 L G E B R A. . J09 Therefore feeing there is on the firft fide a li^Am of ba expreffed "by — h tf^f you take away that— ^t ^,you thereby add ^ 4 on that fide. Wiierefore that it may ftill be an Equation j you muft add, h a, on the otheV fide alfo. Then it will b<?, aa']^d€'\\baz:zgg'^'2ba — dc Again, fubftraft baon eaCb fide, then it is, ^ aa-^^dczzzgg-^-ha — dc ■ Once niore,fubfl:raa ha on each fide,that you may bring it to tjiat fide where a a ftands. ^ ' Then it is, a a — h a-^ dc-jizgg-^dc* % Laftly, (that the Confonants, or known things may come alloo one fidej fubftraft d ^ on each . Then will it be, a a^^ lia^=gg — 2 dc. Take the Reftangle 2dc out of the Square g^, and-let die i:eft be a Square, namely//! Then it is reduced. Having gone a little about, only for exerrife of them that ar^ qv^ite unskilful herein, now they fti?rtlf?e this Reduction might i^ve been quickly done another way,that is,feeing in the iEquat;ioh. aa — h a'\'dc-\'h a^=zgg-^h d—'dc There are in the firft .parts^ Coiitraditipries, to witj—r^artd ' +> rf; they (deftroying one anothei^; might be^aij;cn awaybptlv^t once, So it will be, • aa-^-dcz::! gg^h a — dc. Then if you fubftraft d c and h d on both fides,it will be reduced to da — b^tzzgg — 2'dcy as it was before. An4 gg — 2 ir, being put into one Square /y, ^e Equation' . ^ ' '^a—la=:^ff, may be refolved as the i^qpation a a — 1 4 s;= rf i was by the fecond Rule for plain Aquatipps^ a lictje before expreffed. " And as here the Reduction was made by Addition and Subftradi- on only,fo fometimes it is made byMnlciplication,fom crimes by Divi- fion; in bQth or either of which, this is general : that Whatfoevijr U done teany one Member ^ tnuSi he done to every Member quite throi(gh tbe^uation. ,''-.• Rr2 CHAP. >' JI9 . J LQ EB R^, C H A P. III. I Cftbi refoluticn of ^qustionSy aecordihg to thegh nerd Method comfojed by Mr. Tha HarrioL ALthough (having bcfote (hewed Rules for all forts of miicd rquares>itinay teem prepofteroufly donehereafteptofpeak oi ^lmple Squares \ yetfurafmuch as I pretend not mucli to Method or Order, aodbecaufe the general Method of Mr. Ifdrriotbegm with Squares, I will do fo, but only with one Example. Thap is, let there be an Aquation of a. a =//. Or let it be exhibited in numbers, a a zz 69169 Firft, take notice that all Squares whether Simple or mixed in ^umbers , ^te to be marked with points, the nrft always over the place of Unity or Unities^ and (9 (ucceffively every BinaiS or fecond figure. Cubes with every ternary figure. Biquadratiques with every quaternary. S^lQUdSf every quinquenary, and fon^rwards. • • » This Square number fi> ppinteijs 69169 111 whiclvt)ecauretiierc are three poii^ts, theix are three figures in the Root* ' So that 4 ^ing a fingle letter canfiot fitly reprefenf that Root, but fome trmomial, as is h'\^C'\'dQio\i\d be put equal to ir, and the Square thereof mould be equal to a a^ or 691 69. But becaufe it may be done as well by adding the Gnomons, that is repetition of theftcond working, (as they are commonly called) fdontn as the pointsaremQretbeatiiro; afimoffiji/ will {erve(with lefi trouble^to dotbe&mef }jet that BinmUl be Ir 4- f • And put l'{^cszai Their Squares ihaU be therefore equaL That is,M + 3 ^^ +^ ^3s: 69169, fhi ALGEBRA, }ii Tkc Refoliuwn. • • • The homogeneal number given 69169 Firit fioglc Root i s= 2 and If =14,0000 .Which 4«oooo being fnbftraAed from? the number given 69 169 ,tben there 3 ' ■ Remains of the Number give^ 29169 Ituot decuplate 6 = 20 I>ivilbr 2, i 40.00 ^he fecond fingle root c ss 6 2* if $40.00 ' 2 7 6.0 o Subftraft 276.00 Remains of the Number given 1569 The Root inoeafed ^ = 26 Root kicrealed ? *— .0/5^ and decuplate 5 ''-^T •• piviforis 2(s= 520 The third (ingle Root Vssj 2le l$6o r r 000^ i \ ■■■MMM Tottl 1569 Subftrad 15*69 'Remains of the Number given * * 0000 The Root inoreared 263, is therefore the true Root, as may be proved by recompofition, or multiplying 253 by 263, for the Pro- 4u£l will be 459169, which was the number given. The Cyphers which are jput after in the Divilbrs and SuhftraAs, ^e only to fin up the number of i9aces, by which the number gi« iren or rather the remaining Points would elTe exceed. R>r the like purpofe is uled the decuplation of the Roots, as on* lyto fupply a place until anot)^ figure fucceed in Place of the Cypher. And \ \ „2 ALGEBRA. An<l in nothing elfe doth this work differ from the ordinary Ex- *J^noX the Square Root, commonly taught and known. '"Ihe «>*« Spends upon ^ ^. im. of the fecond Book of «JjL. wliere it is demonftrated, that jjT j ngh Itnehe dmied fy Shno mo pmt, the Siutrenude of the whole, u equd to the S^eTsif the fins, t»i totheReamgle nude of the pom tmue. So it it km mfoUtmtb. , The Square of the greater part, that is, of 2^o~5 1=67600 TheSauare of the leffer part, thatjs, of 3. -r~r-<r (=00009 ,tS ffi^e of the patR, that is, 260 into 3 twice. -2j^oi56o Equal to the whole Square.. • — — > 69169 Nor do there letters reptefiatfijnatntally the tijingsthemfelves in a divided Superficies only, but as properly and (jlearly tte parts d^Solii Bodies, of which, twaor three Ewmples for fetisfeftjon. to which 1 admonift dje Rej^r, to be intent fo the feveral pointings of the quantities according to their due order, as b be- fore expreffed, ani alfo to the:placing of the Dmfors and Subftraas bvCvphers, as before alfo is intimated : For thisto the Ingenious is Sou^, andalongVerbofity to others. wiUfcarce be fQ. Of Cubkd Mnuiawni - . Let there be a Cube daA^fff Or propofed in Numbers a a 0=^17^ 1 92 3 Put (as before) i4"'=^ ThentheirCubcs alfo (hall be equal, rT}at]sbbb^'}bcC'\'ibcc^cccz=:4i7^i923- The Rffobitiim. • * ^. TheHomogencalNumbtrgiveor 4i78is??i [lie feftfii«te<?aWque Root *=?3 ' V ' Andiifc=:27«ooopoQ: . . Subteaa 27.999990, l^jSMll^ ^U II-* -'• «' • w8^si*?; Vnt A L G E i K A J15 The firft Root?v,^_ dedilpl^ed ^^— ^^ '^.bb 2700.000 3.^ 0090.000 mm I III X>iviror ,• 2790,000 Second fiogle Root f=:i4 • '^.bb c io-3co.c»oo '^,b c c 01440.0C0 c<; c coo64.iGyco - / 12304.000 . -Subftraft 12304.060 Remains of the Number given 0247792-3 The Root increafed i^=H ^ Root increafed 7. y / depuplate J . ^ 3.^ b 346800 3.^001020 » » • I *'.' ^ • Divifor 3478^0 ' ' The third fingte Root ^=7 <^,bbc 2427600 3.^ c c CO49980 CC C COOO343 * . : . -Subfttaft . '21477928 ■ • ,1,1! • i/ ' u Remainslaftiy of the Number;gii^cn 000 . : . v\. •• o; ... The Rooc iacr^afijd.A-f'^fc^J^? - . - v.: :.■^^i > 4 M « 1 J «» Which is the true'Root -of the Gwbe 417:819113, ?a?/«iiay;bajpio?Bd by recompofitipn, that is, by MukiplyiDg.347 by^47,v6*d btePro- duft Again by 947, the laft Produft Ihall be equal to the Cul«<HrJi|ch was gi^entoberefolved, - . „ . And as above in die Square, the. Canon of .Ihe'^Re&hJtioas kasihe letters b b^-z b C'\'C c, . being the me Square 'i>fb,-^c^ '. 'Aiil thofe letters did anfwer exactly to^-the, parts of the-Square^'dividef^like in both'the Dlmenfiojas ; ' So hejre aliaJthe .Caaoa:6f rBiPi^^ or / J14 A L Q EB R A. ^^VMeRhhh^lhic-^-^hccJ^cce^ doexadlyanfwerto the Paraor Members (^a Cube, divided into two partS) alike in all tbe tbreeDbneiifionSt as any one may prove upon a Cube made of (bide flendec matter, aiid cut through all three ways, for be (hatl find the wboleCube(fuppo(ed equal to 417^1923 as before) juftly made up of the two Cubes of the two Segments, that is, i^^ and cec^ and three PiarallelepipedoHS , whofe length and breadth are equal to i, and their tliicknef s to r, thofe three are the 3 i i c. And lamy, three other Parallekirpipedons, whofe length and breadth ar^ equal, tor^ and their thidmels to ^, fuch are the^ bcc. See tbe fotming Schemmifme. TheCubeof thegreater Segmen which is 340, bb b r 393040C0 1 'Jbt threp greater ParaDelepipe-'J dons, %bbx r .i^^jCoo The three leflerPttallelepiped ons^^^ ot^bce V •4.4698a The Cube of the lefier Segment, 1 . which is 7^^rr ^ .....343 »■«■ Thewbole Cube given 41781923 Ntttf That the greater Segment is the aggregate of all the fingte Hoots except the laft, being duly valued by a Cypher, as here ic is }4o»btttthe lefler Segment is the laft iingle Root only, ashefe?, I have done this to let the Reader fee,that he may be (ure,let the quantity to be refolved be great or little whatfoever, if he be careful to make his Qmon rijBht , the letters themfelves will direft him vhow to frame his DivUors and SubflnlOs in order to the final refolu- tkm, efpecially in thefe unmixed Quantities, where the points fimit how&rthe SubflraA (hall advance at every operation, beginning firftatdie point next the left hand, not farther, and to the Second point only at the fecond Work , and not otherwiie in all that Ibilow. . And inMix*dJEquations,if they be madeup of Cube withaddlt^ on of certain Squares,or certain Roots,or both Squares and Roots, or by SuMhraftion of the fame Canon of the Refolntion muft ever be made by Multiplying the affumed Root 6 -f"^^^^^^ P^^ ^^ ^ Qpefititious Root If > quite through the Equation in aU the degrees there- A L G E B R J, J15 thereof, for fo fhall arifeaH the feveral parcels of which the fcrtrit Subftrafts are orderly to be made. In a Cubick -Equation , if all the quantities be prefent, there is no need to point any but theCubicksand Roots: yet I have here- diftinguilhed the places of the Squares alfo with little Croffes Ob- * liquely; which labour, when the Workman is intent upon his bufi- nels, may well enough be fpared. ' * Of the Rejblution of Mixed Cubicks. Let the -Equation Aaa-\'d44 — ff^^^^g^g he .propo(ed in I^nmbers. Aslet itbe aaaa-^^2da — 75 0^^:29282970 Therefore i= 32 and //zr.75 And g^g = 292^9^70 Put ^4-i:=0 . T And make the Canon of Refolutionbyfut^ftitutingt^cinthe place of 41 quite through the federal q!imititu^j^ad'\'daar^ffd^ The Ganoa rightly made will be +iii /. Thefe feveral parcels of the Canon, being rightly fubftrafted from the homogeneal Number 2928:^70} the Number flball be thereby refolved, and the Rooctffqond.. . : , \ f^^ Notefirft, ThatalUhe-par^el^n the Canon, which have not the Secondary Root c in tJhcm, as -f- A * f -|- i J i and -//.f| are to be fubftratm at the iirft Operation, the other reinaiiittfg: parcels to be all fubftraQed as often as there (hall be points leftibove* . The Rcfilmoff.^ ^^.,;: v^^^ ;^.,.. The homogeneal Number given. ; !;_:a9?^939 . . r.i roo;i • •• The firft fiaglc: iRoot J^a- • ,/[:, — ; j ;'^ :,/; ■ , -jj --. r :f'bbb o.oooooo C ■ Ar'bbb 128.0000 ^^f fb. . 150.00 r^ 92650.00 Subftrad Remains of the Nomber giTea . \ jj S f Remains Ii6 ALGEBRA. ttemaias of the Number given "^The fivft Hoot decupbte £=20 ^hb I200.000 ^b ' •f6o*ooo aii .'1280.00 d . f*» 32«co •« T3912.'0O mrm Diylfor 1390450 ;rhe recoaa jBnaeRoot ^=9 \ ^ilc ^oScx>.coo ' TCt. •" .729,000 Si5^ i^ti$2o.bo ifrr • ;2592«x>o - f .f. I. ];.78oo2.oo •— //c.»«.d75.o J I I I I <. ^ttiii^ I77934SO «v.... Remains of the Number j^vim » • « The Root incttafed £ s 29 Root increafia decuflate ^ s 290 Remains of the Number given ^{6017970 i:; T. • • i779J4?o 02224520 ■■y- <.. . 3&& 252300 ib •••87^ 2^6 .18560 . - .'. i 1 . , '<>^2f(»> ». 7»^5S ♦ ^ ■» . Tiut4 J L G EB R J^^ J17 TSbirdfm&leRoot c==8 ' * ^^ '^bbc .2018400 ^b c c • •<«5$68o ' • c et ^i^ 2dbc ••148480 if r . • • • .^A^ 2^25.tao ••*-//r. ..•600 f 2224520 ^ / Subftraaiaftiy \2»*4^ Remains of the Number given . > , 090 Whereby it appears tha£ the -whole Root 2^8 is die true Root whereby this -Equation is explicable, as may be proved alfo by re- compolition. fori ^5 = 24389000 3 55ir=. 2018400 ' ' d kb -=: .269iW» 2i5r = ,. l4*^Sb dc czn^ 9 9 '.2048 ' Inall = 2930^'isib^ From wh!Qhfubfti'aa/y5+//r=- . • • 24350 • ■ Remains - 29282970 Which was the whole HomogenealNumb€r.§i;fen. , , ^j>r.at B. • . ' .... Whereas in compofmg the IfJiviisial^ thie.gradual quantities arc tifed, as in thle former exampl^i^atid ^^i well as 3 5 5 and 2 db^ it is to be noted that in praajjce^ .tfiofe Jina^Jg particles j 5, OTr^ May be comitted 5 the other. witlioiir%mVnvjiuftliii^li^l^^ for choofing the Secondary Roots. Having nowc ipftw^ in an Examfdr-.W^l^ar^all A6 pow^ wre prefent, in thcfeoneor two that tblloW, to make tfit W6fk1ftorter, I fliall leave out one or other q£ tl^m.^ , • ' '^ [•- ' lnthe-fiquatidrf4i-i>+//»^ggg* Propounded in Numbers if4'4-*32tJ4c6^i^8348i32, It romc- times happens that the Coeffiefetit%bun(feSiith more binancfigu- , res than theHomogeneal doth winrterilaries,in fuch a cafe t^iat there - may be room njad^ to ,begin the fetriaion . The Coefficient muft be devolved to the ne^t point further to the right hantt,; nWsilhc '^ > I I 31$ ' JZG EB R J. recondytliiidtfirarth, or further, if ni^require^nd there the WoHc is to begin. The Coefficient isalw^ys the known quantity which Multiplies any of the unknown infi^ripf quantities. Example of Devolution* Fut&4''=' ■ I Refohttion. The Homogeneal number given . . 18348192 The firft fingle Root £=2 J^bbb' 0008.000 4-//^ 6408 1 2.0 . • * • » • • 7^6416^12.0 Subftraft . :.. 641512.0 Remains of the Number given 1932012 Jhe firft Root decuphte ^ = 20 3 1 i 0000120Q // oo32p4oe^, \ Divifor 321606 The fecond fingle Root, ^=6 Repains of the number giveo 1 93201^ f %btf ••«.9i6,o ffi .1922436 1932012 Subftraft 19320 i*^ > * - |lemaiasof the number givc» 000 Wliere* J LG E S k A. 519 V^herefore the whole Root is equal to the Root increafed, 25, as may be proved in manner as beforefiid. It fometimes happens alfo In the -Equation A A a — ffa:==!:ggg Patinto Numbers* As.aa4T^io^ooo*A:=::2o^i2^* Tliat the Coefficient abounds jwith more binarie figureffthan thtf . Homojgeneal with ternaries : Wherefore that there may be plac6 for the Rjerolution,put before thie IJomog^oeal J3>Waikl ttelefthaiia. To many Cyphers as will aiFord that to receive as many Cubical I>oints, as the Coefficient doth Quadratical : I^xkbat ttic feft empty point, as it were by anticipafiojiy' begin the Rjefqlutjon*) In which rherc is this of Compendium, tSijij.tEe firftSquafe'Rootextraaed out of the Coefficient, is either equ^l tfl the firft fingle Root of the Homogeneal fought, orjefsthaa it by Unity. But U the Aquation halibut two Dimenfions, As tf 4—254 <t =65024, then tHe^lfirft figure of the tocfficicot. Hamelya, is the firft Root* Examfk $f^ij^icifMwn The Homogeneal Number given . + 020312$ • • • b^-c-znA The Canonist ljffyj^ff\c %: The firfl fingle Root *»* :. . Subftraatheiifferen^^^^ ? - • • Remains of the Number glren ^-. 4753,^1 25 r u • ^ • • The firft Root decuplate I ^. 30 '^^bb 270CC060 '^ff 105000.0 iV - •' ' Diviibt \ 165000*0 iii^ J20 J LG E B R J. ,$bb,c, 54PO.OQO ^bcff .jdo.ooo ctic ...S.ooO ^^ffcs 2t000.0Q I III! II IIM... _ !■ ♦ •|- 35d8oo.o ^^ r \ JUaetaSa^ci tlie Mftunber giireii -|- 103^12; Tbe RoDfr Iircreaftd £3=39 Hoot incret&d decnphte i ±s p6 %lb 307200 •— /jf 105000 ■ ■I ■ Diviror 262200 The tkird fihgle Root fi:$' V. ^bbc 153^000 "^ bee •• ^4000 52^000 1035125 • ' ': -i Subftraft^ ' • • . 103512$ Rctnains of theMumbcr giveri, , ooo^ . :- * •**■■..— _\ Which fliewedi that the Root increafcd i 4- c = 32 5, is the true Root of the Aquation, And it loaybeijproved by recompofitionai formerly. In the -Equation — aaa -^ffd ±^SS^r Whifch. 15 ijQsrplikable by two Roots, as (hall be Ihewed in the4iext€h^ter,.fe^/^ 5,to find them both* Put the ^uatlon intaNumbers. As-— 11444- 5241^^:^ 1244150 , Therefore//= 5241^ a^id" ^244166 =g|g^, ' ?VLtb'\^c:=za Therefore —i * A +// 47 — 3AA1: >sx244t^ EiftrsSh 2 A LG E B R J, J31 ExtraSiion of tkgreattr Root. ' • • • The Homogjjneal Number given 12441^0 The firftfingle Root h^2 Jb 104' _ b b 8.000000 ffb 104832.00 '^ 24832.00 Subftraft +24832.00 Remains of the Number given -^ 1239040 The firft Root decuplate fc= ff' $2^16.0 "^^bb 1200.000 Divifor — 67584.0 The fecond fingle Root czz 1 Remains of the number £iven — 1239940 . ffc .-52415.0 '^^'^bbc i2oo.6(BO ^•^^bcc • .60.000 ''^ccc .;.1.660 • # Subftraft V i— 73^840 ■ • Remains of the number given ' v ^ ' ' . 4-^502200 ■■.••■•'.••-# Tbc Root increafed and decupl^jRfcsiira )24ii5 32300 // 5a4ii5: ■ ' • *--^%ib 1^2: Dit^r-r^j884 The 322 ji L G B B RJ. Use third fiflgle Root c=6 . ^f^^bbc 793800 "-^^Bc c • 22680 "^^ccc • • .216 — S02200 Subftrad —$02200 i*-^ Remains of the number given oCo Rootmcrea(e(i6«f^=2i^, >which ist}ie true Root (bughcw IJ, EduSiion vf the leffer Root by Devolution^ The Homogeneal Number given ,-.... 12441^ • « *=2 ffb 104832.0 ^^Jibb •—•••8.000 -1-1040320 Suhftraft _ -1*1040320 Itetnainsof the Number given ' +203840 The Root increafed and decupled £=20 ff .$24.16. ■^p Divifof. _5iai6 The (econd fingje Root ri=:4 , '^^gJAr . ,4800 -4-203840 SobftraA . "+'2o2^4<^. I •w ReiAalns of the Number given ooe TheRoot increafed 6-|-c=:24 Where 4 i-G E B R 4^ J2J Wherefore 24 is the true Ro'Cf^ fotight,% may be proved by re* ^oitipofitiony as hath bee^ mewed before* . . So this ^Equation is explicable by two Roots, that k^2i6^ aod 24 VIEfA^ Lib: de l(£c^gnhiime ^fuUonfm^ (Aaf. \Z: Vrof. 2. faitfa^ *niat in the Aquation ^'-'AAti^fftescgig^ tiie Coefficient //is ^ompofed of three proportional Square^ and the HoTn<»peneal^2 is madeby Mdltiplicationof the aggregate of the two nr£^ or the twcF lafty (for a,ll is one) into tiie fide ofeach odier. and the Root i inay be the fide ekher of the firft or t&ird. This, (or the fame HX fubftartce^ laith that Noble Author, Anditiseviclent, for ina:W <:r/4.rfi//-f A *///==// Andputr=ui Therefore ccc'^ddc'\^bbc'^cetsuldc^bc OrpUtfcs:^ it is bbb^ddb^cb—bbb=zddii:ccb Both^hieh are inanifeff.. COM p B i^ j> ivii u i^cnce it may be (hefred; thateith'ef of the cpySfititftfs iLosS^i jis it, being found and called r, the other Root e may be found by ar ^[^sdratique iEquationonly. For&ppbiin§ • / ee+rft=f/-^f; Thcfti^- It is tfe4-^c-j-rr==f/. AndV^' ceff ecfff ' EucVtde6.ii. BlitbycoiyftruQionry iii'f bbffU Micf-^^d-^bbs^f. Sti then ibibree and fe±^e« But it was (hewed before that b might be a Robt of tt^is ^italtioir jgg And therefore e a^o is aRoOt of tht Qtniki and the Compendium i5 proyfd., . 'ida^ft^^ggg 1 , I t J34 J^GEBRJ. ExMifU ntfo in NimAers. In tiie bft iEqiiiCtoB 4 4=:^ ^==4^6 56 And: -/^524i6 Pr0m wfakh take ^ ^5=4^656 057^0 [ ' ReAains' . //— r«:=57^o But f tf=: 57^ And Cf=5i84 ■fc"*— ■ >i I 5760 TlteSniftis re4;^fts=S76o Therefore cf-|-cr=//-^r, Which was, C^f, In the Equation "^Aaa^faMszggg^ the G)efficient/is coni^ pofed of three proportional Lines, andg^g is equal to a Solid made by a Square fwhofe fide is equal to the two firft, or the two hh) multiplied into the Remaining Line : And the aggregate of the ficft and (econd msy be if, and the aggregate of the fecond and third ftaUber, . Put i' Vf i^fff Andfuppofe — ^4Li^jgtsszi6: Then 4 may be 3, and^^ 'nif.ViitAf 4c f£cogniu Cap. 1 8« prt^. 6. A n(i ^fefore the ftoot it found, ani eallerf r, the Root e may be foppd by aplain^%quati6n ; for fuppofc^ the nMMe proportiooai^r^ itiifrr^ — ^' j'"^ c^f* * ' ' * '' Arid /r — ty-^^ c^jy Ot ^ yy^cfs^fi? — cc. And making fc^-c fzrx X, it i$yy^ey=:x x. Aitd tfte K&0t y being- iboiid by the firft Rule of Cbi^. 2, It is laftly (making cf yff dff)y^ dzrzc. I will here add a few Rules ^grounded upon Mr. Harriots 6 SeSI- ons) by which the Reader may eafily perceive the Fabrique of Equations, their Roots, increment and decrement, Mult^licati- oa and Dximipji of them* and their- Number in any Equation as followeth. — • CHAP. A L G p B R J, 72$ * • m C H A R IV. R U L B t. \. Every ^pdtioh being compcfed of frmh^fiWi ^ndfcffientiknovm quantities bath its Original by Jioats eojnfofei of a qtiamity known andofrmetnber quantity unhndnon^ and. theje J(fots Multiplied together produce eettjin particular Mjembers tDkb-\* oni-^tAfp^iv^ly^ figged Q'or in every ^^quathn boththefefi^gneranjirefentyv^i^rder* ly placed maie up tbe. JSHquation. As the Equation a a — r A a*^f4 -J- 6 r = o. is made by multiplying a-^b-zsiobja — c s^^r ^id becaufe it was at firft a — A =0 therefore ^i ax /» and thelfte.pf IT. Andsftom hehcc it follows that where tlieiirft term (^orhi^hcft power) itt aqu^dratique ytquation is.figned-^ there 0ie-$ji«atiqp hatth two Roots, as here by Subftra^tlng QnbotlLparts4*44r--— 64 **^/:€t the Aquation will be Air=-^rf£^-^ii,+"r4, ao4 vmft have 2 Roots, . v I .Thefe compound quantities b muitipiyinjg I ihaii call Siff^miak^ .'Whether4-+-/'or4 — *b. not having any need in this Trcatife^ diftinguifh betwixt B/»(?OTii// and ^j^ifttj/x. . ^. The Aquation <i 4— ir if 4" ^*=5=*^ If it bi, i<fr put c — i:^d* tlien the Aquation wili be, 4* ^ ^^ ^ ^ ^ ^ ^* ^^i ^\9^ tlie firft kind mentioned in Chap. 2. bat if it ix; ^ > e^futb — 4^^f and the Aquation will be ?[- a a — fa =r b r^and is lite' iCbe fecor^d fort in the fame Oapter. Jhe Original of the Aquation itf— itf-J-f^-^fr^Psp hcx^ propofed , is -{-<« — ^ir o multiplied. by -^4 t+-^;?c.o, t})at ^ az::zb\xyaz:z^Yr'C»' This xquatian hstii bat am true Root, wLich js by and one folfe, which is f. - |. By this whicli hath been (aid icis plain th^t (ome ^qut(i<M^ have as many Roots as Dimedions^Jome not (b maj^y, but pc^i^ can have more ; for the number of dimenfions being th^ fame with tbe number /of Multipliers (if all diverfe) can be butallRppt9.Npr can tbe aequation be divided by any other thingtban oat! of thof^ Bmmiah by whofe. Multiplication it wasn(iade« But if the Multipliers how many foeyer be ftill the lamf , there can be but one Root; For let i^* 4 -»- i sr o be Multiplied ^f quadra^ cal}y , the produft is -J- tf 4 44 — i^b aaa-^rSbb^^h^^'bbba-J^ t il ft. where it is plain there can be no nchtt l^oot but b. I meai^ none: greater or lels than it: becaufe in trti^Ui bcrcace 4RoottS| bstevery QQefmgulaf )y equal to b. % if there may, let it be 1*, and iet d he gre^^r or lefs tbani, it tmportsnot wbichi And feeing d:sxay Suwitate iln thePiace of «, ^te t)iroa^ the Aquation » it will ba Tt2 dddi J26 ALGEBRA. <fiii~4*i"4-<^5iirf—4*Hi+lW«=s:o. Which if a > J, pr clfc i <; *, i$ at thefrft fight impoffiljle : For the difFercna between the 4- and— is always equal to the power of the difference between 6 and i, which power is here a Biquadrat, therefore i= 5, And againfeeing phis ^Equation may be derived bv patting Regnal to i, fir fiibftituting b in the place of i quite tlirough, It wiif be ' T\'2btbb'\-6Blfpb^^bbbh'[T4bpbi Which is manifeft, thcrefdre again ir? i, which is contrary to the (hftpofition, therefore ^ is the only Root of this /Equation, for in.* deed, the /Equation propofed being made only of multiplications of i — A = o cannot be divided, that is refolved^ by any other Bioo- Ibualthentf — *, of which it was made. ' ' 4, Hence it is that the laft term in every /Equation may be called the H6m6geiieal, becaufe it is naturally made by multiplication of the Koots of the /Equation, though the Coefficients in Tome or- ^inarv /Equations are di^uiied with other Charaders, which hap- j^ns by Addition or Subftraftion of them, to reduce the can(»ical Ifequation to fewer Numbers, whereby the redundancy of the Signs 4* and — is to be t^ken away, this is to be feen above in this Rule, where the iEquation * . 4rtf^ — bA-\'ca'^bcz^o is reduced to -^aa^da-^be^zzc^ by making i=c—^ and +-ittf4-*^+^*"^*^=o, reduced to -{-aa -^fa-^ b c =?o, by making b.-^c T=zf Where the CoefEcientf d or f, is not a part of the Homogene^l h Cy but a difference by which b is greater or left than c : By help bf which difference, the Equation which confifted canonically of four Members, hadinow but three* ' 5* And &is Redu^ion is ufeful, for a$ M. J>gi Cmcs faitb, and ^ivhichmay be feen true by the way of Multiplication above (hewed, every /Equation hath fo many true Roots as the Signes 4* and^- therdn are changed, which in the canonical ^nation 4- A 4— bA-^ca^-^bc^TLOyZtt changed three times^ whereas the itquation hath not three true Roots, hut one true and one falle that is ^ sLndr, and the common Equation reduced changed! the Signs but once, that is ftom -^-iato — A r in the former^ or from 4-^tf to — /i in the latter: And from thence it may be known that thq /Equation hath but One true Root, The likeconfideratioa ipught to be in others. ' And whereas the (akl Dcs Cmes doth bften mention ial(e Roots, it is ti> be noted that fuch ar^ lefs than nothing, z%'^a^b=:oz Or 4-^=^ — *» ahdif any true Root, as -f-i-r^tfssjobemulti^ plied by this -i^a-^-bzrzo, there will arifc an -Equation -^ad J^ I, a',^ c 4 -r- he =?: o Where the Sign 4- follows twice, the Sign •^— twice, and they are Once changed, which ihould intjmate. (accoi^ ^f^is^ I>^Cffu()i two {alfe]^ts,^d<me true: For he (kith. So i ^ 4 LG EtiR A ..pl «iai3T timas as 4- or — r eonje ewke together, To niatff &I^ jLoot? there are, this Squation therefore muft be rcduceii, By ' liiaking b — e = iif~i>f,oreireif i<cdien make ■:—>=;: /;,'fiji,'it-wi5. be either -f-rfi 4- (i*—A* = o, Ck+ia— /a— AJ^pp'wJijcb cxmfirms that w^ich DesCurns (aith (rf twice -|- or -r-iNamely, that there .are as maj)y talfe Roots, in tjie jEqamJoq, .as ^ioij — ^oioe twico tpgether, a«d fo many tri^ ^oopas'.Tfr.and .t. iji fiianged. - ' '] And where the Roots arc all 61fe, the -tqiiatio/iis impoflHjle, ^s sJrb—s> multiplied hy <-4,f^o, ptodufeth <(|J-i'rf'-p;tf« -trAc=:o which caqnot be. And diere&ire,iivhep(l(efe &^n ^- quation pretcii4o4 like -x i -f- 6 < 4" "* = T * '^' Pf f^i^^^ jo^ement in^ybemade, v ' . 6. The lame i>ff Cmet (a|th alfo that ^H thq [aire R quatioi), may be turned to trueones, andthetriiep fhangeingtbe Signs of thefecbud, fourth, and eye And Ibis is evident, for pf the JEquation j-t — 24' a — 875=0 by fuch change is ma^c -t-« M-^-"'-' a — 87 — where the firft had t!)ree true Roots,, am the latter hath three felfe and bat one trite,. This Wkeij atalladvMitures, to ferve for an Example onlj otherwhatfoevetwiUjdothelike. ..,: : , -t-, r M- . jLU l¥ n, 7Je uvlmapn Spm ^ m JSipi^M tuafh Intrufid or Aea-e^elt bjfumfing anatber «4'"^" vtrnttty ^t?— theiicimtmmiiliro- iaent, auttj tltt Bitigmi^ t^edpqfi"^ tHi JSqItatiin ti it watiefore^ the ha unkaemi Huimti^ : And if tbitmcTiMent te faryUttofu* apanef tie CoeffiHcnt y tht ficond Term, at Vni^ H ^tiedinun-; tbePS etifeiejti Term be one +■ theothet — ). xbm by fiukinaSifi'it ' 'dtcreaji of' the J(f« the fecmi tcr/n if t&eiiAipaibti^kbOett a- war, imd mnulJed. _ '^:.vA\\^': ,; • intheaqiiation +:«*« + t'*i<^>fi*==;;D, the Rootiraafbc Increafed by nlaltihg f— 9 ==«, ao4 Subftitufing e - j in the platit .of t quite throng the arquation, <»s^: thereby fliail arifca neif — bbt^-=z—hhe .Which ,j8 7 h(lEB RJ. Which is equal to the former as you fee agreeing in the parties- cnlars, a.ndthe1U»ot ebeing'fbund, a mayWhadbycafting away q from (• ^"J JL jVe wiii'deftroy — 3 J' f. ^'^ ^° the fecond Term ee will beQuit««kenoutof the aequatibnas ismanifeft, &r the aequatioa To purged will be +«?«r 3H<+-\««r**'=° ^"f ^ Snfttcaaineoo each part^-2flji--6*f having firft made * * <: -^ katasziii, itwiUbethen + eee— 3H«='<''''« Themafl- ni of fijchteduaioAOf Solids, ftall follow in thenext Chapter. In like fort the Root t, mi£ht have been dccwafed by any quan- titv< as «, which if it be proportioned to b as afbcriaid; woaid take away the fecond Term of an atqnation, where the Signs, of the firft and fecond Terms are not like ; as in the aquation + 4 ayt ^bit—bbe=i9t byphtting3X=fr, ande4-x=4.- ThePro- Hem wiU be 'firilv perftpncd by making e^xthe Root of die net* aqi^tion, as before was «- f, olf iving the fame or^ ,n com- pding the particulars, due refpea had to the Si^ti^s -^ and — , mtere they ought to be altered. .. .^j . ^. ^ ^ fomirrtduced equation +e»-3f?*«=iiim.ght be fnrdiec leduoed (if need require) to-f e»— i>^»s=:<i<<i. NO T E. . Thi9liu«ment«tiW apdiimtoutiW! of th^ Roots in fuch manner as toSfSway the fecond T«;rm of any donation, > b of exceBent rfc fai ftdb «q«Mtioos.as hayfc three or W.mcnfions, and camiet 2» a«J dS> with any Bino^ia^ made of 4 + or -r fome other SLfJ^ -Mot^v as 6.f, or the like, be reduced td fewer dimeo- STwSyhUamin tJ^tfuch'an^uation is ^id. and car.. Srawrartifice already, or lOcely to beittvrtttfcd, berefolredby ?iScSSf"^t by W of ^J»^ Conique &aionsit may; in SbSfelt iSSer' ncceffary or estnamly ftcHitatirtg, to tate a- ^Se f^d Term { if tl»eie be anyj frqm thearquatm, as IhaH be fecn hereafter in its place. , , . RULE III. n,, uMluotBn Riot of any jeqmion my be mulitPliei (or iividfi Je term in it, h*mg firfl ^umi tnotber unknown «MW^»/» «ul. if * ' A LG EBk J.' ^29 required to iiioltipiy the Root <t by 4, Affumc f =4 ^ and write Which i$an 3cquation, and the- ».oot^ ^is.^^airupk to 4, ;a£ma]r be proved thus. . ^, •.*•/. i?at it=^4i = ?;=== i and |4^2t l » Theft tf*4=64' r r A= i6>lfut V iaKial^ < • V ■•MM In all 128 I , \ ■'■ '■•- .' • S'lH 'It . Therefore #* 4" ^tf tf-h ^^<^---:*c^^^^' • ' Again^ Pute:t=i6 Allelfelthi^ififneRiff^' Then fetf=;:4o96" 4^eff::t::3072i l6irt:ffS=:u>24>But(54^fi^=r:8l^2 . In aft 8192 .therefbri^'e^^4:4^^^ + ^^^^^-^^^^ ^ Aadr;='»6x=:4*wIiichwa&tQlie;prftV:e^^^ ' V\ - ^ ^ The tttiHtyidf'tHi^' ItMle wjft k>pwtt' in-Wifeciflf .S^^Stei'S^^ feacd with PtafKdljs,^ w^hoIe;NtrirftdfS W m^W^ Roots by the denomHiatoFbr drti«TtirtStc*s bf-^tfee ftaWohV^fer^ftt foc^^f xneaiis tlie Coefficceurof > ihefttoab^^^Bey^i }s- tRuAti^lied by the fame as before, multiplying 4 by 4, multiplying alfo^ by the fame Num- ber 4. And many times b^ this ILiSe^^Uations may be freed from Surd Numbers alfo ; efpecially if fuch befound in the fecond Term, . Put cfNfrV frasc*' • . . -.r ...;:r; . Bnt tf yet it be required to avoid the Frattion 9| f, then jaaKe J A it. Art multiplying 8.by3, j^fby?, and 128 by 27, there viU be a new third xquatiim. I I jjQ A LQ £ B R Ji Which cbitfifts of entire Numbers, laving one true Root which ^ 9, and the Root of the middle aquation w^ 3, which is the tbom €hereof,'and the iCoot of the firft equation was 3^ 8 And &6wl hope this Rule and the ufe of it is plain enough. NOTE L It maV bfe flitcd,"that if tlie Sirds in the fecond and lafl Terms of thefirftatquation,towit:^tf4tf4-y8tf44--H*— 4^*'»*'^^'*"^^' terly incommenfurable, the reduAion had dot been fo feafible. For althou^ 4 V 2 multiplied by the Cube 6f V 8 that is by 8 V 8 pro-' 4uced 9a y 16. which is equal tQthefntire number laiS, yet if it had been 2 V a 6t 2 ^-5, or any flidi printes to be.multiplied by 8 y 8 the i^odua would have been 16 y 24 or i6 V 40 dioctgh thi^ laftmay (by the note after the Confeflary in' €6^. 6.) be reduced by multi^ying it again by V 40 unto the intire number 640 • Ne-' trerthelofi this fecond multiplication by a Surd, renders^ the z^a- tiott inexplicable, at ieaik by the precedent Rule. ^OTM II. It may be {urther noted, that if Tnftead of e = 4 4 one would put itszf shuts not being fo liquid, as Numbers, the x^tion would then be eee — fbee-\-ffece^fffbcd:=zo increafihg the di- inenficms of the lefler Terms^ for remedy whereof thtee tines are to be found in proportion 6ne to another as afe th* magmtudes fh.ffc.fffb. of which let thfe firft linls bfc ftipj^ed to contain Vtim z» omtias the fuporficies/idoth (for which. purppfeX|»rry muftbealittefet, and agreed on before, j The Names of thefe lines' whenfoundmay be called;, £,i, and the equation may be writteir « * * « • Not E ifi^ , . ':Butitisdg8(ia tobe noted, that w^rlc the Uo^ A iy and «, are dbmmenfurable in length the t^ee lines i, Kg^ may be very eafily found, for then th^ may Wfi^med by Numbers and if /be put for i;iijr^thene==tf anddfieworl^fireftrate, but where the laid line^ are incommenfurable in lengdi tlUs Reduoion is^dy^ys Jiard if not imipoflible: For thefe incommcnfiirable lines do moft commonly xeprelent Tu^ Surd iKumbenifRS cannot by anjr Rcduftioabecotft* .! ...,.....:■ J L Q EB R A, 1^1 r R 11 L E IV. lie JEpaxton aAA—ibba=2ccc^ prany other like it, by pn cc'\- bb ' I i iing = tf may if c>b be brought to eee=x:ccc-\-ddi or if c = bto^ee = ccc,cr hfiJ^.if c<:bthen w eee=rrc^ V — d did id: Which IdH mdy be catted an impojftble J¥:qmtiGn. bbfff Put ef bfl< And becatife d fe c^ual to the fum 6l e 6b ^ the Extreams, wfiicb ari e -f- ---^ therefore,- . • e From thencie it wfll bb ^^e"" '\^^bbe'^'\'b^ee.'{-b^ ') •i——-— .i^ — — =4-^^^ I eee \ And — 2b be^ — 2bbbbee , \ , ■' ,z=. — '^bba fee » Therefore rejefling'the contradiaories, and mulflpiying all by eec,^ it is, -f-e*^ -f- ^'^ 1= 2 c' ^*. Therdorc-^e'^ -^2 a; cc c^ zzz-^bbbbbb. And'\-e^—2c^€^'{-c^z^-\'C'^ — ^^.. Therefore , (fot e^' — c^ zzzV e"" -:^^2 cU^' ^ c^) -{^e^zrzcce-^ If now in the fifft eafe c be greiater than f, then'pu t c'^ — b^zr. d^\ Then it will ht eee z^icce^ i dddddd Tb'*t ii, eee = ccc ■^ddd. Which' is the J&quation promifed in the firft cafe. Secondly, If b be equal to^, then^*^ — i>^==o And it will ea-^ filyfollow, feeingfasis fliewedabove) that c<^ — ac^'e* -^C" ±=.o^ therefore the Root of it e ' -^ <: ^ ==; o; that is & e ff --- c c c the fecond , ^equation prefcrib^d- Laftlyj by the third cafe, feeing c is lefs than ^^ Vnlcccccc-^bbbbbbt^-^dddddd: ', Then it will be eee^zccc-^-V-^ dddddd the aeqtratioh pire- , , fcribed* in the third cafe, a;id (becaufe of tlib inex^^lieability of i"^ dddddd} impoflible. tttt C Nf I / ??2 J LG En RJ. COMPBN^njVM. Whereas Mr. Harriot faith Propter V — d*^ inexplicabilitatent^ tSc* The faid quantity V — i* is not explicable becauie — d'^ arifeth by multiplying -f-i* by — <f»betwitt Which two there is no mean } fyc no one thing can produce d^ but d^ only, and — i* is noc pro- duced by ^- ^* or — d^ becaufe by both, this therefore may fervc for a Compendium to fave labour which might elfe be'loft, in feek- iDg that which is impoffibl^ to be found* NOTX * i ureA'for**^*^^, andi^for^iii, and^^r^ for bBhci^i^^ and the like, (as i>ej Gir^s hath dyne) only for abridgment, as in t he Defin ition of the Powers is already (hewed. And rr<r-J- VT^nrp with that line over to diftingui(h betwixt i(^<^<^ — ^^ as one quantity, and V c^ taken by it felf and — b^ taken apart aUor, for by fnch miftakes may great errors fucceed. 1 will add no more Rules, thefe 4 may be multiplied by anyone that doth not find thefe fufEcicnt for his purpofe , at his own pleafure. ^ CHAP- V. 0/ ReduSfionof Solids. HAving; fpoken in Chap. 4 l^le 2. of making ^^r — 2qif =zdddj and in J^le 4. of C" — b^z=:dddddd, 1 think it not amifs here to Ihewhow fuch Addition and Subftraftion of Solids jnay be performed. ^ , And it may be noted that ii^ is for brevity fake there ufurped fof np c. or fome' other folinomial reftangle Paratlilpipeddjjy equal to the Btnondal reftangle Solid bbc^2qq(ii for if this Binomial could Cby plain Geometry), be given in a CiAe, as is ddd^ fomethingelfe rfiight be done which here I will not fpeak of. Now therefore feeing ^ = 3 g as there it is, the equation may be written v ^ jjf-— 2gjj=<^^^j orrathet Mak«- ALGEBRA. 33 j Make =/, therefore 2 ^ j ^ == 2 j/r , ' ' ' 't ' Secondly, make 9 j j — 2 if^jggt from thence it is plain that h b c^^2qqq=zggcy which wasnrffto bedone.^ Thirdly, to reduce c*^ — b^ into one intire Solid, though not into a Squared Cube as i^, as is ufurped by Mr. Harriot for brevity in ivriting^ or facility in reafoning, f^. 100^ fuppofkig that done which caonot be done by ftreight liaes and Circles hitherto. Now therefore feeing r^ — A^is produced by multiplicatio4 of cce'\'bbb\x\X.o ccc^^bbb. cc bb Make — /, and- =^, and /+^2=J and f-^g^p^ b c therefore bcfz=.ccc^ and bcg:=zbbb^ and bcqz=iccc-\'bbbm Secondly alfo bcp-rzccc'-^bbb: And therefore bbccpq-zz£ccccc^ *-^— bbbbbb^ which was fecondly to be done. Example in Numbers. Put ^=::2 and ^^=3; Then/rr» = 729, and bbbbbbz=z64f andtlietbenf^r<rrf — bbbbbb^ that is 729 — 64=665, whidi is produced by multiplying 274-8 by 27 — 8, that is, 3 s by 19. Now make/=:^., and^=f, then/+g= ^'^ = ^ and/— gz:; 3-5-==]^. And^r ^ = 35, and bsp^zi^. And laftly ^or^^j=: Moreover, if you niake/>j=xx, the Solid is further reduced to b be ex x^ which although if be not a Squared Cube, yet it hath afquare Root, -namely ^rx, which maybe of good ufi in many cafes to refolve Equations into AnalogXmes, ^f which kind of Deraonftration, by help of EucHde 6. 14, fome notice is. taken, be- fore in Ci^^. 2. N 07 E. I The three Cafes of the Aquation a^ — ^bbaTn 20^ 9 mentioned in the beginning of the fourth Kuk of the l»lt Chap, are called by Mr. .Harriot^ the firft Bjferbolicalj the fecond Parabolical^ the third- Ellipticalf becaufe of lome fimilitude between them and thofe fefti- ©ns, of which three Cafe^, the firft isrefoluble by a Conique Sedli- on, the fecond by a Circle, and the third not at all. MultipUcation and Divifion of Solids is altogetheraseafieas Ad- dition or Subftraftion, for if one would divide ccch^ ^^, maker cc ex -— ^=:x, and again mak^ ^—^ = ^, and then ^ is the Quotient h - I , re ^cdf U u 2 Example JJ4 ALGEBRA. JBxample in J^umhers* ^ ccc Put ^;=2 and ^51^3, then =6L to fiiul iriROt^nake H C€ CX C£C * . . ' b ' kb asitfhouidbe. Again, if c^ flbould be divided by *, it is now r- — = r, and ccc pultiplying by i it is — — :=^b^f b -' Again, multiplying Vjccxt is — : zzibcc^^ and bcc^ is the i , Quotient requtrpd, * BHt if it be required to bring the Quotient to a Biquadrat, make h\z=.ii^ thenr^ii=Arff^and msSiQciznff^ then the Quoti- ent will be/f/jf. Multiplication h naturally foeafie that there needs no more be faid of it, than what hath been fkidalteady in Cf>4p. i. Now, of -Equations confifting of '^ terms in continuabproporti- pn as tf '^ -}• ^ ^ <* ^ = '"*» ^^ fecondly a^-r-bbbi^^zc'^^ or laftly let }t be—- tf'4-**^^*'*=^^*> let them firft be propofed in Nura- ^bers as 4^^4*2 44 = 24, if' by Rule 1. of Chop. 2. it be wrought, Itwillbefbundy 25-^i=:4tf, and 44=4 or 4=:= 2, Otherwife if -the Square of half the Coefficient be added on both parts, then And their fquare Roots alfo are equal ; that is 44-}- 1 = 5 and 44=:4ortf=2as before, and the latter may prove the former-. 2. In the fecond, let it be 4^ — 10 4 44:;=: 459 add 25 to eadi part, thelitis 4 44444— 104^ ^+2$ =484. Now each part of the Equation is a Square and their Roots alfo \areequal, that is 444 — $=^2, that is 4445=27, and 4=3. 3. Laftly/ If — 4* + 7004^ =5:^,46875 from the Sauateof 7«t^ that is, froms i?2^oq take the Homogeneil 4^^75, there remains 7«5625, whrfe fquare Root is 275: And either 350 + 275: Or 350 — 275, that is either $2$ or 751$ equal to 4444, and 4= 5; Or V ^ ^. 75 = 4, whict Cbarafler V j 5. fignifiei- the Biquadratl- \ I \ A LQ E BR J. IT r E. 3? 5 ' The feft and laft of thdethree Equations, may be done as well in iJnes^Nwnbers (by thefaid three Rules of Chap. 2.) and fp -miy iBquation of 4, ?, i^, or 32 dimenfions, but -Equations of 6, v2,()r24Dimenrions, cannot be effeded fo, becaufe there is ever one or more Cubique Roots to be extrafted, which without two means cannot be done. ;, Tor if it may, then I fay, that two means between any twolijes may thereby be found, for in the fecond iSquacion ^* — bbbana cc =2 c^ by Rule 2 Cbdf. 2 r^ 4- -J: ^* is a fquare, make — = ^, tten bb ^^ -^ ^^^ and Ai' 4rf= c't, and 35 (rrii=::;crrrrr, then make;- — ' ■ t- bb s=/, therefore/<:= i- b 6, and/V b^ = ; h"^. Now becaufe — = ^f^ C makej&=4/, then/ctff**=v^^ Make /<:=//, t\^^n ccbhll:=:^\h^. Again, makeii-|-fc* r=m»l, and i4-l-//=:w« And then it will ht ..c&miiinnz=i hbccd4-i'C<rhhlif that is, ^'^rj-vi*', to the fquare Root hereof .' ^ ' bb ftnn^ add ybbb:T\m% make.J — =/;; then mp=z[.bb^ an4 wi ^/^ bntp^ibbb^ Make— =:^, thsn »i;^jr={J'^'^'. LafHf, w make r 4- j = X, thetiith cmn-^lbbhzrimnx^zaaa^ byCi^, 2. J^/e 2. Now if wi, », and x be. proportional, then the nfiddle- moft is equal to 4, but tnat is uncertain, and cannot; be mad^ t)therwife : But by niakitig rr^z-mnk *iU be r r xi^ «» and x Will then be the leffer of two means between r and x if r <xor the greater mean, if r>x. And fo if r and x had been given; and required to find 2 means between them hYKnr^rniAuon orderly^ one might come to the faid Equation a"^ — bbbAAaz=:c^ oi yfihxch if the Root a be found, two means are alfo found between r and x which «^as to be proved* CHAR fe y 336 ALGEBRA. C H A P. VI. (y Surd Numbtrs, R u L E J, ^T^ffefhupre J(m •f my Kwrnber being multiplied by that NUmber, I froiucetb tffefyuare Jfgot of the Ct3>e cf the Nuniher. Tor V tf multiplied by a produced! dV a^ but aVa=:Vaaa for liking E(pumuhfpUces they will be equal, as if the firft, namely i / tf be multiplied ftill by V a^ the Produft IsaVaa^ that is a a. And if ^AAdht multiplied by i' tf it produceth i/aaaa that v&aa ^Ifo, wherefore tfVtf=Vtf4tf; And therefore 3 V' 3 =5/27 either of which is the Cube of V 3 , and the like of all others. R UI, E II, Sari Numbers are multhlied and. divided like v^hole Numkers^ the JProduS retaining flill the claraQerof the J(opt. That is, V z multiplied by V 3, producetb V 6, a^id fg of all •thers* R U ^ E III. A l^efor Squaring Binomial Surds* Multiply the Quantity to which the Sign V belongs, into the Square of theCocHicient; and the Produti is the Square required. , Examfk in Nun^ers* If the Square of 3 y 7 be required : Multiply 7 into 95 the Pro- duft is ^3 ; the Square required. Or, it theSquareof 4V9bedemanded; 9 into 16, the Produft is 144; which is the Square demanded : The like of all others^ And itat ihews. That all fuch Surds are commenfurable in Powers* K T E. Where I Ihall have occafioA (if any be) to fpeak cif a' Cubique Root, I (hall fign it thus, V c. and the ^iquadratique Root thus V j (• R U L E IV. 70 muftipljf divide^ M w SuifirAS tb€ J^ots of Surd Numbers* of MV I- J L Q E B R J. 537 A! V ir I p L J c^ r 1 6 If. Befides that which hath been faid in the bft Rule abovc^ thefe Roots of Sards may be multiplied and divided, and known by odier names, foas ifometimes the Produfts, or Quotient (hall be rational. Firft therefore any fquare Root doubled is the f^uare Root of the quadruple, as 2^5 = ^26 and 2 V 2o = y 8o. 3 ^5 = 45,4^ 5 = V8o. 5V5 = Vi25. 2 / 10 = V 40. 3 y 10= y 90. 4^ 10= y 160, and 5y io=y 250, C?r. in^nitely ftill multiplying the Numerator, 2,3,4,5,ef<:4 into it lelf, andthe produft into the Surd Number, asif 3 y lossiypo, it arifeth from 3 times 9 into the Surd Number y lo: And the like of all others whatfoevcr. For put y i« = y 10, to be multiplied by another Number, as by a = 10, the produft is a V tf = 10 y lo, which by the flrft Rule is y aaa:=:V 1000, that is, the Numerator i o into it felf making loo, which multiplied again ^y the Surd V 10, gives V icoo. And if it had been at firft y.4r=: y 10, multiplied by any other Number, as e=: 3, the produft muft by the fame method be ^y it z:=:eV 10 that is (by the fame reafon as the former^ VeeazziV And it is plain, that if any Root be multiplied by I -^Ouadruplerf The Produft (hall/Nontuplo.- be the Root ofySddecuple.- the Q Vigintiquintuple.' jTrigintifextuple. And fo forward infinitely, according to the proportion of tW Squares of the Multi^rers, AlfobyDecuplation, asif $y $ = y 12^, then:5 y 5os=:y i2$o: Or if 41/ 4 = v 64 |:hen 4y4o:±:y ^40. And (as above; if 4y 10 = y 160, theft 4y 100 =i: y 1600. Alfo by Subdecuplation, if 2y io=iy,4o, then aY i==:y4; Or if 5 y 20 = y^ 500, then 5y2::=ry5oj And (according to that atorefaid) 3 y 37 = y 333, and jy 3dis y 324, that is, the fquare Root of 3 times 3 times 36. And this may often be of ufe,- not oniy in Numbers but Species, and is therefore to be had in memory by him that would be reac^ in Multiplication of SurdNumbersy or Surd Quantities. Furthermore it may be ufeful to remember that in K^ciprocd^ Surds ;is 4 y 5 and 5 y 4 thefe tyro have that proportioa ohe to another a^ 4 hath to a mean betwixt 4 zni 5» ^ As for Example 4 y 9 hath that proportion to 9 y 4 as hath 4 1» ii which is a mean betwixt 4^ and 9^ for 4y 9s= la^and 9y4- J?8 J L G E B R J. s=i8, but4' 6'^ 12' i8'' or more generally . aVe^ eVdff a^ Vae^f for multiply the Means, itisaeVa dnd multiplvthe HxCreams It is 4. V if ^^, and divide each of ^em by d the firl^ is e V ^ the other is Va e e^ out by the former part of this Role ei A:siLi gee wherefore this is proved. C N S E Cr A RT. Hence it isevident that Roots of themM ves inexplicable may be> fo multiplied as the Produft may bfe rational: For if f 20, bemul-' tiplied by 4 / 5 the Produft will be 4 y 100 = 40. For 2y5 = V2oand 2^20=^80, therefore 4 V 5 =i: ^ 86, but V 80 multiplied by V 20 gives V i4oo=4o, I need fay nothing orJDivifiony for tha£ is no more but by the fame fteps to go back again, as V 1600 divided by V SoQiiotientis / 20. And fo of the reft which hath been faid in Multiplication. NOTE Thefe things being fo, it ^ ill not be hard to find fome Number to compare with any J'»ri Number fo as to make that work ratio- nal and exprimible which fecmed not fo : For there is not any Ser* Number can be given which may not by fome multiplication be made a rational Number : For l^t it be V 5, V 7, i 8, or any of thefe as V 7 multiply it firft by V 7 thatproduceth'7, but multiplyy 7 by any fquare Number: whatfoever, as by 4 omitting the Sign V, it gives 28^ than again multiply V 7 by y 2 8 it produceth V 1 96 = 1 4,, ^ For this is ail one as toinultiply one Squlare Number by another which muft needs produte a Square Number. So here the Square Number 4 was multiplied by 7 and after by 7,- tiiat is by 49, which multipliers cannot produce any other than a Square Number, to wit i p5 EucliL 9. i. And whatfoever hath hitherto been faid of Quadratlques, may ferve for cubiques alfo ; due refpeft always had to the degree or the quantity and ^oot, for any V c. multiplied by 2 gives 8 y r. by 5 It gives 2 7 y r. by 4 it gives 64 y c. that is 2 y r. 8 = y r. 54 and 3y(r. 8=ty(r.2id, and ^ y r. 27 = y <r. 729 the proportion ftiil increaling as the Cube of tteir Multipliers. And the like confideration had, this may be applicable to Blyisr lriit/j««, or any higher order. And ftill whatfoever hath been (aid of Multiplicatiou, ferves in a retrograde way for divifion alfo. ' RULE \ ALGEBRA. 139 \ \ /■ f(. U L B IV. f(fr A Dp ir 10 it. Surd ms are ufually Added ^nd Subftniacd by the Signs 4- ancf — as the Square Hoot of a added to the Square Root Qt«f 2>ttmis V a-4-V8orSubftraaed reftisVS— Va. ^ ^, . , . Butthefe may be added into one Sum, fi>i? feeing 8 is quadruple to 2 therefore, a V 2= V 8. And thcSum is 3 t(iandthc remam is y 2; Likewifc the J(fdird«i iM^ii 8 V a = 2 >^, *rc <?1?^ of Addition, Subftraaion, Multiplication or Divifionj mfjiitj are being added 3 V 52 that is V 288; Subftraacd* V 32> /*^«}?* pU€dV4096; divided V4vl»tt foch as ate neither commenfurable nor reciprocal cannot be amaffed into one Sum. . , '. ^ j^ . AndtheSum of the former Aidition of V S + V abemg already reduced to 3 V 2 may beyet further reduced to V 1 8,for 3 V 2 is equal to the fquare Root of three times two as hath been more thaa oncelhewed. . ju V? Apd generaUy-whenthe Surds given are denominated by Num- bers in quadruple proportion, as t? 2 to V 8, and V 3 toV 12, oc* the leffer and the greater twice being added tt^ether, as JitQ i6, or 3 to 24, the fquare Root of the Sum is equal to the Smni ot ti^ two fquare Roots given to be added, that is, V 2 + V 8 =s y 1 8, ana xi reafon is,^ V i -f / 4= V 9, which ' 9 is^compofed of the leffer once and the greawr twice, that is, as often as the r lis contained in the • 4. . , .au^^a But jf the NumbeTsJje prime one to anoAcr, ^Iftey^^ ded or fubftraaed bx^ljg gig^s.-H- and—, for thefe Rules reacH not to primes. ' ■ ^ , .^ Ui*^- And having faid this Uttleto atquamt fuch asare wont to be a- fraid of operations where Swdsare prefeat, with this which w^ render fome thUigs eafie which, perhaps lecjned bard, aad others whidi were hard, lefs difficult. I wiU now leave this ruggedSul>- Tea, andrecreat a little with A few eafie Praj^a/trowr th« pertorm^ ing of which mayferveto recalinto life and Vt^dtiCC that whiA hath been fpoken of Solids in the former Chapter* C H A P. yiL PROB* h AJt^yrigh tine Mngpven^ to divide it tmmopms^ramtbe 'ks&MgUof the^i^eMdmeofAefms^ tnaj be to tie Square cf'ibe (fiber part^ m[Hchtro]jmlon M u betPixtjnj ma right lines grven. Xx . Let / ^4o J L G E B R !A. Let the right lin^ given be ^. . > . j The fegment to be ^a^ed a^ ^ \ Then the other Segment IS 'i — a. \ And let tiieti^o.liaesghnenb^ r add/* . . JVnd r4az:^sbb — sb^. per i6. 6« tvuliie* iLXxatviyra^A^jia^^sbb.' Vl^ .^zsid, aad y t^ide adl by r. :. ' Thenitis, i»j+i<iCa=rfA. l/lz!kcib:s::ff. ^ . ^'Thcn laft!yit»4«-^i4=://. ^ Andfis^afilyvfound by J^kuoi ' And ^ it haa been requiredto have had the R.edangle^« or — ^ fcrnie other plain to have had any limited proportion to the Sqnare a a^ t^work had heen aknoft the fame, wichfome fmall addition. P R. O B. n. . ] Tcmaicd Salcnon friangk^.^ phickibe Bafe^ Perpendicular^ ati frcfgrtian of m^her Sides JhiU( be given* (I. account that the JBafe Kirhich.rubtends the divided angle.J , Let the bafc given.be ^, the perpendicular ^ , and the prdpordon of the other fides, as r to /• ' Of whidi kt r be the leflfer. An<| for the lelTer Segment of the Safe put 4 : Therefore by fuppofition, .,„ So that the Squares of them are alftf ptbpBitional-, ' ^ * •This is,-., • ;r -;: JR Adby inultiplyittg the means and extreaims Itfs, ssaa^^rraa4^2rrba:=!:.rrcc'{-'rrbb^-^ssct ss Make — f =5^. ' r And divide all the /Equation by r. Then it is. Secondly,* mate ic-—r3=:/, an4jf «2**:^<'r.- Thenitis, f44+2rirfs;rfi£— x^i^< '• ] ^ '^ Agaul . ^ LG gB E^X, l^ - Again make Ii=*. and, i- = t. and Me f.^^^trb.^ ■zrb . ' ■ \ r ■ I^aftly, make — = ?, and *g —'** ~ »» "•• The iEWiorf finally reduced w»be<ih«i*i^e4^««. and « maybe found by tbc^ftRtttefiwf*«art,*qaMions.£^ a. p R o B. in. ' t ><»j< Mopier being' ghien, to fifL tm mbtr Numbers, fotfitHifx Untothe'^oareof^elfwnbergiveaaddUnirsr, tl»e half of die , Sum (hall be^ Hypothenufp, o^ from the faid S^uare.take Unity, , the half of the remain fliall Jw the middle fide. . • ^ , j j ;„„ . ^r let the ^luraber giyen be ., the Square is^^^,t<rw^'ch add n| Uaiti', f.hi Sum is u <t+ 1, the^ half «here6f »s -• #«-^ {. tor^W Snd"?rfrom M taRe Onity* the reft is 44-^ i, the half whereof is, -J at-^l for th6midd!e fide. Butthelefferfide (byliippofition) isj. .The Square of tlie leffer fide is at. , . , , TbeSquareof fliemiddlenioftis ;:rf<ti4->-i'» *-r ♦ .^thtiQfe Squares are •^44^44-' I '^^-tV- But the Smiare of th^ Sypoteiiufc, v^'^ of^ l^^T f ' is equd to thefe, that js, , ^+ 4'*' +• vf ^ + i-, ^ ^r,^. ^ -^ therS^re by the ^i of the firft of BucUde the Propofitioir « _ proved* v . « ^ CO j^o 1 1 ^ iin , ^ . . ' ' Hence it is plain, that the.twogrej^ter fides of any ^^f/^^S^J^^^^^^^ angle difFer by Unity, for if two Squares differ by 2, their hahes differ by I. If it be required to h^ve all the three fides in whole Number?, - then the lefSbr fide m^iff 6e it odd Numbed p a o B. IV. m differemcf the fides of fi I{e^avgU, rohh the Ammd DUgo- ml in oneS'mi. bnvggmn in I^mb6's, to fnd QUt tb^Mes. J4a ALGEBRA. Let th^ diftrencc of the fides be i And the Area and Diagonal together 73 • And put the leflfar fide equal to ^ Then the greater is 4 -)" 7* TheTe two multiplied produce iitf-j-yi, which ,is equal to the Area. And therefore 7} — €a — ia isvtHe Diagonal TheSquareof whichis -4- 5329 — i\6dA} -|-49i«tf — '1022^3 Which reduced and rightly ordered, Is Which by the 47. of the firft of EiuUdcj is equal to the two SquaTe$ of the other fides^, andii4"7> whofe fquares aie^iC, aad4«-^ 14^+49- , ThatiS, '\'A€iAJc\i^iLA€^-^9T€€-^^Ol2€'\-%l7^Z^^a£ + 14*4-49. ' ^ ' That IS, 4-<itf444-t44tf4-^99*rf— io3tf<i4-5j8o:520* Thatis, — 44tf* — i44J*-f-99iii4-xo3^*=5 528o. In which ^nation, becaufe a Add hath four Dimenfions, arid the Homogeneal 5280, butfourPlaces, the Root « cannot coniiftb| more than one Place, or Figure, which muft be found oat by trying every one of the nine Pigits, if need be, and will be found at laft tobe $, therefore the otjfer ndeis5-|-7= 12, the Area 60, and the Diagonal I ^. But if tf had been more or lefsthan $ yet (except fomethiqg elS^ lead a readier way) it is good to try 5 at firft, if it be too little then 7^ if that alfo too little, then 9, fo there will be no need to try the- even Numbers, 6, 8, ^e. for if $ be too little and 7 too great, it ipuft be6r the like reafpii will ferve for 8,4, 2, fo that he which guefleth moft unfortunately, needs not try above four or five Digits, wUdi is no great matter, the lilce happening fometimes iniceking . the Quotient in plain Divifion,for no man is fure f guefs right at fir& But that we may exempliiie this in bigger Numbers^ where 4 9iay conlift of two or more places.. Letthe difference of fides be 71 The Area and diagonal together 1177. Working a$ in the former example, there will ar ife an Equation which being reduced'ahd ordered as before, willl)e ' \ ^^$AAA'^1^7AdA^^26%%AA']ri^l^1^^^=^ti^2^^* And.putting^ b'\-'c:zzAi Then the Canonof the reh^ution will be ~ '-^bibb''^4bbbc'-^6bb^cc'^4bccc'^ccc{: '^I^bbif^/vz6bb€*^d26bcc^ld2ctc . A tG EB R A ^4j 4- 1 57276 3 +16727^ A be orderly Tobftraaed from the ifomogcneal Ntunber g(v$iti X 380288, as fi>Ilowcth.\ ^ • The number given ^ +1580288 The firftfingle Root 4=i. . * — bbbb l.oooo •— 142^^^ 142WOO0 — 2685 J B 2685.00 ' ^ I Inall — 4ao$«oo -1-1572765 167276.^ Subftraft (the difference of + and •^) + 1 2 52260 ■^■* Remains oTtheNnmber given . +0128028 The firft Root decaplate icsio; . •— 4>i> 4o®o * — 6ti 060© . I : — aB 004.0 ^ --«426tB 42600 ■ V , _ — 426i 4260 ;. r-5?7oA 53700 Then —105200 is all the— And .. 4- 16727615111 the 4- Diyifor ,4-62076 is thekdifercttce. v -^ The fe'cond fingle Root iT = 3 Remains of the nnmber given -4-128028 ■*•* try''' •' • 1 " — 4rrrr 12000 — 6tt rr .^406 ' — ^l ecc , 108^ -^C C€C •.•8i ' ; J — 426(5 (T 127800 i * -. —4255 re .38346 " . — ^I42^rr ••3834 — 537aftf 161100 — ^2685^^ 2416$ In an — 373806 4" X67276 c = 501828 Allthe— being 373800 11ie4ifRrencci$4"i2862l Which j^ A L OtE B R A Which being taken fr«» «hp xtamm of the Hm&ber givcti ^ 138028, ^ere remainsfinally nothiiie, To that the giVen Equation itJB«yjrtWw*hythe Root^-f-^qp «'• The leffer Me a is therefore t ?j, to which if the difference given namely, 71 ». lift addfd, the middle fide 84 Is thereby compof^d. Ag^tn, if to that iniddle fide 84 be added Unityy the Hypote- im&of a right angled triangle is ciinpoM, whpfe^defides Jh 13, 84, 8$. The Superficies of this Triangle n half the Parallelogram or Refiangle required. For 84 multiplied by 1 3, gives lo^^fep the Ar^ of the Redangle, to which adding 8t;the Diagonal, ctvpoftth theNomber 11 77, as was required in the Propofition. €0 M^ M/f 3 rv M. it6ng the two jgreater fides of any Reftangle Triangle, exceed one another by Unity fas by die htmet CoroHary) the difference betwixt the two lefier fidea being gives » Ae d^rence betwixt every two fides is alfo given. So that putting d for the leffer fide of the Reftangle, the greater fide is * -I- 7^» ^** ^^^ diagonal a + 72, whofe Square is -\^ a a 4-^ 1444-4- 5184, to which the two Squares of the fides, beings 4 -f «44- 14244* 504^) areequalr Thatis,,' 2 4 4+ 142 4-1- 5041 =4 44* r44 4 — 51^4 And Subftrafting from each part 444-1444^5041 There will iM»i&4"^ ^-~ * '«= 14^. And 4 win be found 1 3, by Che (ecotid Rule of .C%. ^ . liE Si) M TT. In the fecond Probtcm of this Chatter it hath been (hewed ho^ ^pon a Bafe and Perpendicular and Proportion rf the renuining fides given, to defcribe a Triangle^ It is there to be underfl:ood of an Acute- Angled Tri^gle, in . which the Perpendicular fidls within.the Triangle. Illow therefore let it be otherwiib. A9 *