THE Merchant's Magazine: OR, Trades-man's Treasury. CONTAINING Vulgar arithmetic in Whole Numbers, with the Reason and Demonstration of each Rule, adorned with curious Copper cuts of the chief Tables and Titles: Also Vulgar and Decimal Fractions, after a New, easy and Practical Method. Merchants accounts, or Rules of practise; showing how to cast up the Value of merchandise, and to make Allowance for Tare and Trett, more compendiously than hath hitherto been made public; with Tables of Foreign Coin in Sterling, and a large Table for reducing the one to the other: Also foreign Weight and Measure compared with the English, and the Weight and Value of the Currant Gold of this Kingdom: Likewise, Rules of Barter, Loss and Gain, Rules of Fellowship, and Equating Time of Payment. Also how to find the simplo or Compound Interest of any sum for any Time, and a Table of simplo Interest( for one Day or upward) at any Rate of Interest; useful for those concerned in the Bank of England. Book-keeping, after a Plain, easy and Natural Method; showing how to Enter, Post, Close, and balance any account, &c. And LASTLY, Maxims to be observed in Drawing, and Accepting Bills of Exchange, Foreign, or domestic, &c. With many other things throughout the Whole, not extant before. Accommodated chiefly to the practise of Merchants and tradesman: But is likewise useful for Schools, Bankers, Diversion of Gentlemen, the Business of mechanics, Land-waiters, and other Officers of Their MAJESTY's Customs and Excise. By Edw. Hatton, Gent. LONDON, Printed for, and Sold by Chr. Coningsby, at the Golden Turk's-Head, against St. Dunstan's Church in Fleet-street, 1695. The Reader is desired to correct the following Mistakes of the Press, before he begins to peruse this Treatise. page. 18, Line. 13, for [ like manner if in] red, like manner in. P. 27 l. 15, for[ Units contained in 42212, but what is contained in the Products 34600, 6920, and 692.] r. Units contained in 34600, 6920 and 692, than in the Product 42212. P. 33 l. 41, for[ remains 15] r. remains 1. P. 52. l. 8, for[ 17) 6412] r. 6412. l. 23, for[ 6 Quarter] r. 6 Farthings. P. 64 l. 4, for [ parts of Unit] r. parts of a Unit. P. 87 l. 27, for[ 1/ 14 of 46723] r. 1/ 12 of 46723, and l. 28, for[ 1/ 12 of 47632] r. 1/ 20 of 47632. P. 92 l. 23, for[ Example 1.] r. Example. P. 107, next after[ 9 ∶ 00 ∶ 23 net] is omitted these words by the Press:[ or take 1/ 7 of the Gross for answer, as by the Table of Aliquot Parts foregoing.] P. 112. l. 11. for[ 22 ∶ 00 ∶ 00 ∶] r. 22 ∶ 01 ∶ 00. P. 125. l. 31. for[ 3000 one years] r. 30.00 one years. P. 126. l. 6. for[ 2.92 l.] r. 2.22. And l. 8. for[ 20.44. l. Answer, or 20 l. 8 s. 9 d. 2 qr.] r. 15.54 l. Answer, or 15 l. 10 s. 9 d. 2 qr.. P. 132. l. 10. for[ 700.7= Add 7.] r. 700.7 Add. Fol. 1. of the Wast-Book, l. 28, 29, 30. for[ ∷] r.[ ∶]. Fol. 2. of Ditto Book, for[ July 9. 1694.] r.[ July 9.] Fol. 3. l. 2. for[ 10. Ditto.] r. July 10. 1694. Fol. 2. of the Journal, l. 4. for[ 307 ∶ 00 ∶ 00.] r. 307 ∶ 10 ∶ 00. And in like manner, red the same sum on Debtor side of Cash and Creditor-side of drugs in the Ledger. In the account of Cash on Creditor side in the Ledger, for[ 2505 ∶ 6 ∶ 00.] r. 2505 ∶ 16 ∶ 00. and on Debtor and Creditor side of Ditto account, for[ 3344 ∶ 10] r. 3345. on the Debtor side of drugs, and Creditor side of Profit and Loss, for[ 284.] r. 284 ∶ 10 ∶ 00. And the Total of Debtor and Creditor sides of drugs, r. 982 ∶ 10 ∶ 00. And the Total of Debtor and Creditor sides of Profit and Loss, r. 1138 ∶ 14 ∶ 6. and the same in the account of Stock. In Debtor side of Balance, for[ To Cash, &c. 2505 ∶ 06.] r. 2505 ∶ 16. And on Creditor side, for 3927 ∶ 14 ∶ 06.] r. 3928 ∶ 4 ∶ 6. and the same in the account of Stock. And on Debtor and Creditor side of the Balance sum, for[ 4067 ∶ 14 ∶ 06.] r. 4068 ∶ 4 ∶ 6. In the Transcript of the Balance, Fol. 170, for[ Imprimis, &c. 2505 ∶ 6.] r. Imprimis, &c. 2505 ∶ 16. TO HENRY SPELMAN OF WICKMERE, IN THE County of NORFOLK, Esq Edward Hatton, As an acknowledgement of Sundry Favours, humbly Dedicateth this Treatise. licenced, Edw. cook. Nov. 30. 1694. TO THE READER. HAving for many Years past spent some leisure hours in the Study of arithmetic, Geometry, &c. I was often solicited to writ something Mathematical. But considering the many ingenious Tracts of that Subject already extant, together with the Censoriousness of the Age, I refused it; knowing that to writ what others had done before, without making some improvements, would look like a Transcript, and not be agreeable to the End, which every Author ought to propose to himself. i. e. To make some new Discoveries, and advance the Art he makes his Subject a Degree nearer Perfection: However being requested by the Bookseller to publish something of the Use of some Arithmetical Copper-Cutts which he had by him, I complied with his Desire, hoping I had acquired such a competent Knowledge in arithmetic, as to offer several things I had not observed in Print before. I resolved also to make such Additions, as might render the Book of general Use to Men of Business, especially to young Merchants, for whom it was chiefly intended. And the practise of merchandise being of so great Consequence to a Nation; I have endeavoured so to handle arithmetic( as a Foundation) and to apply that to Merchants accounts, and both to Book-keeping, as might be most likely to accomplish those concerned in that honourable Employment; for it would signify little for a young Merchant to understand arithmetic without Merchants accounts, and to know the latter without the former is impossible; and though he should assume a good Knowledge of both; yet if he is ignorant of the Art of Book-keeping, that alone will prove a great Deficiency. I have therefore in the following Treatise made all three as plain as can be desired, they being the Essentials on which the whole theoric of a Merchant's Business is composed. As to the particular Parts of the Book: The Rules of arithmetic will be of Universal use; the Rules of practise for the Retailers, as well as Merchants; that part about Tare and Trett for Land-waiters; the Book-keeping part, and that of Bills of Exchange, and Coin, to Bankers; Decimal arithmetic to Gaugers, and others concerned in Measuring; as Joiners, Carpenters, &c. and the whole not onely for Merchants, &c. but for Schools, both for the ease of the Master, and improvement of the Scholar. Notwithstanding all which I doubt not, but the Work will be variously censured, and perhaps the most unkindly by those( so ungrateful is the present Age) who have most reason to be obliging; however I shall not( for my part) pretend to anticipate Answers to all those Cavils, which an envious critic may raise. But to the impartial Reader shall say thus much, That I have taken all imaginable care to prevent Errors, and to explain those things most clearly, which others have either but transiently touched, or wholly omitted; and though it should prove after all, that some few Mistakes should be unluckily crept in; yet my Ignorance of any, will in some measure pled my Excuse, since( Divines say) No Action is farther Criminal than the Will of the Agent is Concomitant; and I am sure the Candid Reader will friendly look over small Faults, for the sake of what is Novel and Genuine. If what is here offered may be advantageous to the public, and thereby answer the End for which it was designed, I shall esteem it a plenary Satisfaction for all the Pains I have taken therein; my design in this Work, being chiefly thy proficiency in these useful, but mysterious Arts, and that it may have that Effect, is the hearty wishes of E. Hatton. To the Ingenious author. BY Numbers powerful, and harmonious Aid, This stately fabric of the World was made. The mighty Fiat was no sooner said, But tuneful Numbers readily obeyed And the rude Chaos, Form, and Beauty had. Since, to Mankind subservient they become, And suffer not that his wild Fancy rome, But when it erring strays, conduct it home. By a long Series found of mighty use, human Affairs, to method to reduce. By these,( after long Hazard, Toil and Pains, Th'adventrous Merchant counts his Loss or Gains, What is his Charge, and what that Charge maintains. By these, each Art, and Science, is made known, And their dark Mysteries, revealed and shown. By these, we Wars and Sieges undertake, Great Conquests gain, and brave Defences make. By these, we sadly count for a past Life, Made up of Labour, Sorrow, Care and Strife. By these, we compass Earth, and Seas about. By these, all's done, and nothing done without. Yet, we were in Traditions dull tract got, And this still copied what a former wrote, And talked thereof, as Parrots do by root. But you, to show your Pity, and your Love, Reason and practise, make together move, And a dull Age, as 'twere by Force improve. Whilst others, poorly cost along the shore, By Reason's Compass, you have ventured o'er, And taught us foreign Truths, unknown before. Go on, but know, great Danger you must run Of Rocks called critics, you may split upon. I'll but this short Description of 'em mention, They all things damn, for want of apprehension. But( for their interest) let the Wise be kind, By this they'll judge what still remains behind, In the Rich Treasury of your wealthy Mind. P. C. Advertisement. ALL Sorts of Mathematical Instruments in Silver, Brass, or Wood, are made and sold by John Worgan under St. Dunstan's Church in Fleetstreet. Where any Nobleman, Gentleman, or Merchant, may hear of fit Persons to collect Rents, or keep Books, or teach the mathematics. Chap: 1. Numeration Teacheth to red or writ any Number set down or name The Table                 1 unites 1                               2 1 Tenns 1 2                           3 2 1 Hundreds 1 2 3                       4 3 2 1 Thousands 1 2 3 4                   5 4 3 2 1 X Thousands 1 2 3 4 5               6 5 4 3 2 1 C Thousands 1 2 3 4 5 6           7 6 5 4 3 2 1 Millions 1 2 3 4 5 6 7       8 7 6 5 4 3 2 1 X Millions 1 2 3 4 5 6 7 8   9 8 7 6 5 4 3 2 1 C Millions 1 2 3 4 5 6 7 8 9 NOTATION OF Whole NUMBERS. IN Order to the Right understanding how any Number is to be red or written, there are these 4 things to be considered: 1. The Characters by which all Numbers are expressed. 2. The Species or Kinds of Numbers. 3. The Order or Place: And, 4. The Multitude or Value signified by any Number. First, The Characters by which all Numbers are expressed in Writing are these Ten; Viz. 1, 2, 3, 4, 5, 6, 7, 8, 9, and( 0) cipher, by which, all Numbers, how great soever, are expressed. Secondly, The Species or Kinds of Numbers are 3; Viz. 1. Digits. 2. Articles. 3. mixed Numbers. 1. A Digit is any of the Nine forementioned Figures singly expressed, Viz. 1, 2, 3, 4, 5, 6, 7, 8, 9; which possess but one Degree or Place. 2. An Article is any of the 9 Digits with a cipher or ciphers, placed to the Right-hand; as 10, 100, 300, 5000, 6000, &c. 3. A mixed Number is composed of Digits, or ciphers and Digits, promiscuously placed together: As 12, 24, 96, 112, 102, 1769, &c. Thirdly, The Order of the Places of Numbers is reckoned from the Right-hand toward the Left, as in the Table foregoing: Toward the Left-hand, 1 is in the First Place, 2 in the Second, 3 in the Third, &c. But the Order of reading Numbers is from the left-hand, toward the Right, as shall be shewed by and by. The Denomination of the Places is reckoned as followeth, and as in the foregoing Table. NUMERATION OF Whole NUMBERS. The Order of Places. 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 The Denomination of the Places. 7 6 5 4 3 2 1 2 3 4 5 6 7 8 9 C Millions of Millions, &c. X Millions of Millions. Millions of Millions. C Thousands of Millions. X of Thousands of Millions. Thousands of Millions. C Millions. X Millions. Millions. C Thousands. X Thousands. Thousands. Hundreds. Tens. Units. Fourthly, Having premised this it will be easy to red any Number, observing onely these two things, Viz. 1. The place any Digit possesseth. 2. The value of that Digit. 1. By the preceding Table it is plain, that the first place toward the Right-hand, is the place of Units, the second the place of Tens, the third the place of Hundreds, &c. 2. Therefore suppose the Digit 9 stands in the Units place, the value of it is 9, that is 9 Units; if it stands in the second place, it is 9 Tens, that is Ninety; if in the third, or hundreds place, 'tis 9 Hundred, &c. So we will suppose that the Digit 7 stands in the fifteenth place which( by the foregoing Table) is Hundreds of Millions of Millions; and the value of that Digit possessing that place being( 7) admitting therefore that all the places toward the Right-hand of the said( 7) were supplied by ciphers, the value of the Number would be Seven Hundred Millions of Millions; and in like manner the 15 Figures in the foregoing Table are thus red. Seven Hundred Sixty Five Millions of Millions, Four Hundred Thirty Two Thousand Millions, One Hundred Twenty Three Millions, Four Hundred Fifty Six Thousand, Seven Hundred Eighty Nine. And by the same Rules are all other Numbers expressed; where Note, that any of the places is Ten times the value of the place next it, toward the Right-hand; as 1111111111, or 1000000000, 100000000, 10000000, 1000000, 100000, 10000, 1000, 100; 10, 1, that is, One Thousand Millions, is Ten times One Hundred Millions, which is Ten times Ten Millions, which is Ten times One Million, which is Ten times One Hundred Thousand, which is Ten times Ten Thousand, which is Ten times One Thousand, which is Ten times One Hundred, which is Ten times Ten, which is Ten times One; and the whole Number being red, is One Thousand One Hundred and Eleven Millions, One Hundred and Eleven Thousand, One Hundred and Eleven. Chap: 2. Addition Teacheth to add sevl sums together to make them one Total.   Pounds Yards Gallons Quarts   1270 2734 3546 2795   1021 3946 5737 1501   2370 6542 7845 7160   8426 5763 6721 5072   5603 9708 9654 8540   7206 1246 4060 3026   4570 2090 7241 1867   3872 3427 5426 9742   2106 7602 3213 6381   7285 9841 9762 7690   1707 1056 2187 3786   2834 5407 8615 4502 Total         Proof         ADDITION OF Whole NUMBERS. ADdition is either simplo or Compound. 1. simplo Addition is when Numbers are to be added that have but one Name or Denomination, as Pounds to Pounds, Feet to Feet, &c. 2. Compound Addition is when Numbers of divers Denominations are added together, as Pounds, Ounces, and Drams, to Pounds, Ounces, &c. in both which Cases these two Rules are to be considered. The First is for the right placing the Numbers to be added. The Second is for the adding together those Numbers after they are stated. The Rule for placing the Numbers that are to be added. Observe to writ the Units place of all your lower Numbers, under the like place of the Number above, Tens place under Tens, Hundreds under Hundreds, &c. as in the Example foregoing, and those that follow: And if the Numbers to be added are of divers Denominations, you are to place all the lower Numbers under those of the same Denomination above; as if you add 17 s. to 2 l. 7 s. you must place the Numbers thus; l. s. 2: 07. 0: 17. 2. The Rule for adding Numbers of one Name together, let the Denomination be what it will, is; sum up every Series or lineal Row of Figures, beginning at the undermost Figure toward the Right-hand, and place the Digit above Ten or Tens in that first Rank under the Line as followeth, and carry the said Ten or Tens to the next Rank toward the Left-hand, calling them so many Units( for they are no more of that next Rank) and add all the Rest of the Ranks as you have done the First; but if there is nothing above even Tens when you have added any Rank together, then place a cipher under that Rank, proceeding to carry the Tens, as is before directed: As in the following Example.   l. Admit I have owing to me for Holland-Cloath 3794 For Thread 896 For cambric 6285 For latin wire 3745 For Sugar 2392 For Nutmegs 3058   Total 20170 To know what sum I am Creditor by, or what is owing me in all, I sum up the Particulars beginning with 8 at the Angle toward the Right-hand, as was before directed, saying, 8 and 2 is 10, and 5 is 15, and 5 is 20, and 6 is 26, and 4 is 30; put a cipher under the Line, and carry 3 to the next Rank toward the Left-hand, saying, 3 and 5 is 8, and 9 is 17, and 4 is 21, and 8 is 29, and 9 is 38, and 9 is 47; put the Seven under the Line, and carry 4 to the next Rank, saying, 4 and 3 is 7, and 7 is 14, and 2 is 16, and 8 is 24, and 7 is 31; put the 1 under the Line, and carry the 3 to the next Rank, saying, 3 and 3 is 6, and 2 is 8, and 3 is a 11, and 6 is 17, and 3 is 20; which put all down because you have no more Ranks, so will you find yourself Creditor by 20170 Pound, and after the like Manner is any other Number of one Denomination added. Secondly, For adding Numbers of divers Denominations together, observe this Rule; Having the Numbers placed as is before directed, and as in the Examples following: Consider how many Units of the least Denomination in the Numbers given to be added, make a Unit of the next superior Denomination; and how many Units soever you find of the next greater Denomination contained in the whole Rank, or Series of the next lesser Denomination, so many must you carry to the said Rank of greater Denomination: And if any thing remain over and above a Unit or Units of the next higher Denomination, such over-plus is to be placed under the Line. Addition of Money d s li s d li s d 12 1 270 16 08 ½ 397 17 10 ¼ 24 2             36 3 954 15 10 254 10 07 48 4 106 01 11 ¼ 632 15 11 60 5 378 10 09 197 16 02 ½ 72 6 452 19 00 ¾ 540 17 06 84 7 670 12 07 325 19 08 96 8 201 00 06 ½ 697 03 00 108 9 815 01 05 402 00 05 ¾ 120 10 720 17 04 ¼ 354 04 03 132 11 147 18 03 976 14 09 144 12 706 14 02 ¾ 214 11 01 ½     908 03 10 107 13 04 Total             Proof             To instance in the foregoing Example of Pounds, Shillings and Pence toward the Left-hand; where note by the way, that ¼ is one Farthing or a Quarter of any thing. ½ is one Half-penny or 2 Quarters of any thing. ¾ is three Farthings or 3 Quarters of any thing. So in the Example aforesaid, 3 Farthings and 1 is 4, and 2 is 6, and 3 is 9, and 1 is 10, and 2 is 12 Farthings, or 3 Pence; which carry to the Pence, saying, 3 and 10 pence is 13 pence, and 2 is 15, and 3 is 18, and 4 is 22, and 5 is 27, and 6 is 33, and 7 is 40, and 9 is 49, and 11 is 60, and 10 is 70, and 8 is 78 pence, that is 6s. 6 d.; put the 6 d under the Line, and carry the 6 s to the shillings, saying, 6 and 3 is 9, and 4( taking but Units place of the shillings) is 13 and 8 is 21, and 7 is 28, and 1 is 29, and 2 is 31, and 9 is 40, and 1 is 41, and 5 is 46, and 6 is 52; put the 2 shillings under the Line, and carry the 5 to the Tens place of shillings, saying, 5 and 8 Ten shillings is 13; put the odd Ten shillings under the Line, and carry 12 Ten shillings to the pound, calling them 6 pounds( by taking half of them) saying, 6 and 8 is 14, and so forward as in the last Example of one Denomination, so you will find the sum to be 6333 l: 12s: 6d: by the same Rule and Method, may you find the Total of any other Number of pounds, shillings and pence: But, Note that because in Quarters of Hundreds, Ounces, &c. the Difficulty of proceeding in that Method would be great; therefore your best way will be when you add Ounces in Averdupoize-weight, &c. to make a point or prick at every 16, so will you avoid charging the Memory, and may with ease carry the said points or pricks to the pounds; each point being 1 pound, a few Examples will make it plain, which take as followeth: Example 1. Example 2. Troy-weight. Averdupoize-weight.   lb. ou. dw. gr. Ton. C. Qr. lb. ou. dr.   416 08 15 20 147 16 3 14 15 15   842 01 10 05 119 03 1 04 01 00   26 11 04 17 7 19 3 27 10 08   22 06 19 23 1 10 2 10 13 12 Total 1308 04 10 17 276 10 3 01 09 03 Example 3. Example 4. Example 5. Of Wine-Measure. Beer-Measure. Ale-Measure.   Ton. butts. H-hds. Gal. Bar. Fir. Gal. Bar. Fir. Gal.   31 01 01 38 31 03 08 71 03 07   10 01 00 10 17 02 01 18 01 01   5 00 01 60 12 01 05 28 00 02   7 01 01 09 72 03 06 13 02 06 Tot. 55 01 00 54 134 03 02 132 00 00 By these few Examples with the help of the following Tables, the Learner will be enabled to add any Numbers proposed, of what Denomination soever. I shall therefore trouble him with no more, but proceed to the Tables which are useful, not only in this Rule of Addition, but likewise in the following parts of arithmetic. TABLES OF English Coin, and Weight. TABLE 1. Of English Coin.     In 1 Pound Sterling are, 960 Farthings. 4 Farthings is 1 Penny. 240 Pence. 12 Pence 1 Shilling. 80.3 Pences. 20 Shillings 1 Pound. 60 Groats. TABLE 2. Of Troy-weight. 32 Natural Grains of Wheat, or 24 Artificial Grains 1 Penny-weight. In a Pound Troy are, 5760 Grains. 240 P-weight 12 Ounces. 20 Penny-weight 1 Ounce.     12 Ounces 1 Pound.     TABLE 3. Of Averdupoize-weight. 4 Qrs of a Dram 1 Dram. In 1 Ton are 573440 Drams. 16 Drams 1 Ounce. ●●840 Ounces. 16 Ounces 1 Pound. l. 2240 Pounds. 28 Pounds 1 Qr of 112 the C. 20 Hundred of 112l. to the Hundred. 20 Hundred 1 Ton.   19½ Hundred 1 Ton of led.   Note that 56 lb, is one Half, and 84 lb, three Quarters of 112 lb; which is the Merchant's Hundred. TABLE 4. Of Apothecary-weight. 20 Grains 1 Scruple, marked ℈. In a Pound Troy are, 5760 Grains. 3 Scruples 1 Dram ʒ. 288 Scruples. 8 Drams 1 Ounce ℥. 96 Drams. 12 Ounces 1 Pound lb. 12 Ounces. TABLES OF Liquid-MEASURE. TABLE 5. Of Wine-Measure. 42 Gallons 1 Tierce. In 1 Ton are, 6 Tierce or 252 Gallons. 1½ Tierce or 63 Gallons. 1 Hogs-head. 4 Hogs-heads. 2 Pipes or Butts. 2 Hogs-heads 1 Pipe or Butt.     2 Pipes 1 Ton.     TABLE 6. Of Beer-Measure. 2 Pints 1 Quart. In a Barrel are, 288 Pints. 2 Quarts 1 Pottle. 144 Quarts. 2 Pottles or 231 solid Inches is, 1 Gallon. 72 Pottles. 36 Gallons or 8316 Solid Inches. 9 Gallons 1 Firkin. 4 Firkins. 2 Firkins ●●●●derkin. 2 Kilderkins. 2 Kilderkins 1 Barre●.     TABLE 7. Of Ale-Measure. 2 Pints 1 Quart. In a Barrel are, 256 Pints. 2 Quarts 1 Pottle. 128 Quarts. 2 Pottles or 282 solid Inches is, 1 Gallon. 64 Pottles. 32 Gallons or 9024 Solid Inches. 8 Gallons 1 Firkin. 4 Firkins. 2 Firkins 1 Kilderkin. 2 Kilderkins. 2 Kilderkins 1 Barrel.     TABLES OF MEASURE. TABLE 8. Of Dry-Measure. 2 Pints 1 Quart. In one Last are, 5120 Pints. 2 Quarts 1 Pottle. 2560 Quarts. 2 Pottles 1 Gallon. 1280 Pottles. 2 Gallons 1 perk. 640 Gallons. 4 Pecks 1 Bushel Corn-Measure. 320 Pecks. 5 Pecks 1 Bushel Water-Measure. 80 Bushels. 8 Bushel 1 Quarter of a Chaldren. 10 Quarters. 4 Quarters 1 Chaldren. 2 Weys. 5 Quarters 1 Wey.     2 Weys 1 Last.     TABLE 9. Of Long-Measure. 3 Barly-Corns 1 Inch. In a Mile are, 190080 Barly-Corns. 12 Inches 1 Foot. 63360 Inches. 3 Foot or 36 Inches 1 Yard. 5280 Feet. 1760 Yards. 45 Inches 1 Ell English. 320 Polls or Perches. 27 Inches 1 Ell Flemish. 2 Yards 1 Fathom. 8 Furlongs. 5 Yards and ½ 1 Poll or Perch. Note, that though 5½ Yards is a Statute-poll; yet in some Countries a Poll is 7½ Yards, some 8 Yards called customary Measure. 40 Perches 1 Furlong. 8 Furlongs 1 English Mile. TABLES OF Superficial-MEASURE. TABLE 10. Of Square or Superficial-Measure. 16 Quarter of an Inch 1 Inch. In a Squ. Mile are, 4014489600 Sq. Inch. 27878400 Feet. 144 Inches 1 Foot. 3097600 Yards. 9 Foot 1 Yard. 102400 Polls. 30 Yards and ½ 1 Poll. 2560 Roods. 40 Poll long, and 1 broad 1 Rod of Land or Qr of an Acre. 640 Acres. 4 Square Rods 1 Acre.     640 Acres 1 Square Mile.     TABLE 11. Of Time. 60″( Second) 1 Minute. In a Year are, 31557600 Secon. 60 Minutes 1 Hour. 525960 Min. 24 Hours 1 Natural Day. 8766 Hours. 7 Days 1 Week. 365 Days& 4 Weeks 1 Month. 6 Hours. 13 Months 1 Day and 6 Hours 1 Solar Year. But note that from the time the Sun leaveth 1 tropic to the time it returns to that tropic is 365 Days, 5 Hours, 49 Minutes, 4 Seconds, and 21 Thirds. The Use of the Foregoing TABLES. YOU have for each of the foregoing Denominations of money, Weight, Measure, &c. 2 Sorts of Tables, that toward the Left-hand, showing how many Units of an inferior Denomination, are contained in a Unit of the next superior Denomination, by which you may know how to add or subtract any Numbers of those Denominations. The other Tables toward the Right hand, show how many Units of any of the lower Denominations is contained in the highest Denomination, which is very useful for the speedy reducing of any thing from one Denomination to another. As to the particular Tables: The First is of English Coin. The Second Table is of Troy-weight; by which Weight is weighed Bread, Corn, Jewels, Gold, Silver, Amber, Electuaries; and all Measures for wet and dry Commodities are taken from this Weight, by a Statute made in the 51 of Henry III. it was provided that 24 Artificial Grains should contain 32 Grains of Wheat, taken out of the Middle of the Ear, and well dried; from which the other Denominations proceed as in the foregoing Table. The Third Table is of Averdupoize weight, 16 Ounces or one Pound of which is equal to 14 ou. 12 p-w. Troy: By this Weight is weighed all manner of things that have waste, as Physical and Grocery Drugs, Rozen, Wax, Pitch, Tarr, Tallow, Butter, Cheese, Soap, Hemp, Flax, Flesh, &c. all base Metals, as Iron, Steel, tin, Copper, led, copperess, alum, &c. The Fourth Table is of Apothecary-weights, which they use in compounding their Medicines, though they buy and sell their Drugs by the Averdupoize-weight. The Fifth is a Table of Wine-Measure. The Sixth of Beer: And The Seventh of Ale-Measure; where you may note, that the Ale-Measure is greater than the Wine, or Beer, for the Ale Gallon containeth 282 Inches; whereas the Beer( or Wine) Gallon, containeth but 231 Inches. The Eighth is a Table of Dry-Measure, by which is measured all kind of dry Substances; as Salt, Sea-Coal, Grain, Meal, &c. The Ninth Table is of Long-Measure, whereby Long-Measure is meant, that wherein only Length is considered; as the measuring of Roads, Cloath, or any other thing, where no notice is taken of Breadth. The Tenth is a Table of Superficial or Square-Measure, which is that wherein Length and Breadth is considered; by which kind of Measure the Content or Area of Board, Glass, Flooring, Tiling, plastering, Land, Painting, and many other things are measured; and note that as the superficial Foot containeth 144 Inches, that is, 12 in length and 12 in breadth; so the solid Foot containeth 1728 Inches, which is 12 long, 12 broad, and 12 thick. The Eleventh Table is of Time, of which I need say nothing, but shall proceed to show, The Reason and Demonstration of Addition. BY Euclid, Lib. 1. Axium 9. the whole is equal to all its Parts taken together. To instance in the right line ( gh) which is equal a b c d e f g h to the 3 lines ab, cd, and of; For the 3 lines ab, cd, and of, being all the parts contained in the whole line gh, and the line ab, added to cd, and that sum to of, making up the line gh; therefore the line gh, is equal to the sum of the parts, viz. to ab, more cd, more of, taken together: Or, in Numbers suppose I say, that 15, 16 and 42, Parts 15 16 42 Total 73 make up 73, it cannot be denied since there are no more Units in 73, but 15, 16 and 42; nor no more Units in 15, 16 and 42; than 73; therefore the Number 73, must be equal to the Numbers 15, 16 and 42. 2. The reasonableness of the Rules given for adding Numbers together will thus appear from what was said in Numeration, of the places of Numbers: For in Addition every degree or place must be added to the like degree or place; So that in the Units place, if there is one or two Tens, it is plain that those one or two Tens must be added to the Numbers in Ten's place, because they are of the same Denomination; so if in a Rank of Figures[ in Ten's place] I find 2 or 3 times Ten, for every 10 in Ten's place, I have one Hundred to carry to the Hundred's place; for 10 times 10 is one Hundred, and as was observed before, Hundreds must be added to the Hundreds, because 'tis of the same degree or place: For EXAMPLE. In adding 149 l. 17 s. to 18 l. 8 s; I say, l. s.   149 17 Parts 18 8 168 5 Total 8 shillings and 7 shillings, is 15 shillings, which is 1 Ten to be carried to the place of Ten shillings; which one Ten being added to the other Ten, makes 2 Tens or 20 shillings: Which being one Unit of the next Denomination, viz. Pounds: I therefore add the 1 pound to the Unit's place of pounds, saying, 1 and 8 is 9, and 9 is 18 pounds; which 8 being Units of Pounds, I place it in the Units of Pound's place of the sum, and carry the 10 pound to the 10 Pound's place, of which Denomination it is one: So 1 Ten and 1 Ten is 2 Tens, and 4 Tens is 6 Tens, which being less than 10 Tens, or 100, I have nothing to carry to the Hundred's place; so I place 6 in Ten's place of the sum, and put the 100 to the Left hand in Hundred's place of the sum; this being observed, any sum may be added( though with much trouble) without carrying any thing from one degree or place to another( which is done purely to save trouble) for instance; Let it be required to add, 2976 4132 8647   15755 The sum of the Figures in Unit's place is 15 Of Ten's place is 140 Of Hundred's place is 1600 And of Thousand's place is 14000 The Total of which is 15755 Because Addition and Substraction do prove each other, I shall show the proof of Addition after Substraction, which followeth. Chap: 3. Substraction Teacheth to take a Lesser Number from a Greater and to know the Remainder   J li s d Rec 7582094 457 ∶ 09 ∶ 07 ¼ Paid 6129762 126 ∶ 16 ∶ 10 ½ Remains 1452332 330 ∶ 12 ∶ 08 ¾ Proof 7582094 457 ∶ 09 ∶ 07 ¼ Borr 5469387 602 ∶ 11 ∶ 08 Paid● 2672052 74 ∶ 19 ∶ 10 ½ Rest         Proof         Dr. 1020060 894 ∶ 00 ∶ 01 ½ Cr. 40709 372 ∶ 14 ∶ 09 ¾ balance         Proofs         SUBSTRACTION OF Whole NUMBERS. AS in Addition, so Substraction is; either, 1. Of one Denomination: Or, 2. Of Numbers of divers Denominations, and is the Converse of Addition. §. 1. When you have placed the Numbers in order( the lesser under the greater, as is usual, unless it may, as sometimes it does, save the trouble of removing a Number) this is the Rule. RULE. Having drawn a line under the Numbers given, begin with the Digit standing in Unit's place of the Number to be substracted, and take it from the Figure possessing the like place of the greater Number; placing the excess or difference under the line, doing in like manner with all the Rest. But if the Figure in the lesser Number be greater than the Figure possessing the like place in the greater Number; then you must add 10 to the said lesser Figure, and so proceed to take the said greater Number from the sum, placing the Remainder under the line, and because the 10 borrowed was supposed to be taken from the next Figure toward the Left-hand; therefore add one to that Figure, and so proceed to subtract, as in the former, placing the Excess under the line, as before. EXAMPLE. Admit I have laid out Cash, the sum of 4579 pounds, out of 6947 pounds, which I had in bank; what sum remains yet in my Hands? The Numbers being placed: Take 9 in the Unit's l. 6947 4579 Remains 2368 place of the lower line from 7 in the like place in the upper line, but because you cannot, borrow 10 from the 70, which stands in the Ten's place, and add to the 7 which stands in the upper line, making it 17, so 9 from 17 will leave 8; which put under the line, and say 1( that is 1 Ten) I borrowed and 7 is 8 from 4 and 10 you borrowed as before, that is from 14, leaves 6, which place under the line, saying, 1 you borrowed and 5 is 6 from 9 leaves 3, and 4 from 6 leaves 2, which being put under the line there will appear to remain in my hands 2368 pounds. In like manner, if in the first Example foregoing, if you take 6129762 Hundred from 7582094, there will remain 1452332, for 2 from 4 and there rests 2, 6 from 9 rests 3, 7 from 10( which I borrowed) rests 3; 1 borrowed and 9 is 10 from 12 rests 2, 1 borrowed and 2 is 3 from 8 rests 5; 1 from 5 rests 4, and 6 from 7 and there resteth 1; so the difference between the Numbers given is 1452332; § 2. How to subtract Numbers of divers Denominations. In the first Example foregoing of Pounds, Shillings, Pence and Farthings; you have 126l. 16 s. 10½d. to deduct from 457l. 9 s. 7¼d. To perform which, begin with the farthings, saying, 2 from 1 you cannot, but 2 from 4 farthings( or 1 penny which you borrow from the pence) and the 1 farthing in the upper line, that is, from 5, and the remainder is 3 farthings; which put under the line as you see, saying, 1 I borrowed and 10 pence is 11 pence from 7 pence you cannot, but from 19( borrowing 1 shilling or 12 pence from the shillings and adding to the 7) and the remainder is 8 pence; which put under the line, and say, 1 shilling you borrowed and 16 shillings is 17 from 29 shillings( borrowing 20 shillings or 1 pound from the pounds and adding to the 9 in the upper line) and there resteth 12 shillings, which place under the line. And say, 1 pound you borrowed and 6( in the Pound's place) is 7 from 7 leaveth( 0) put( 0) under the Unit's place of pounds, and say 2 from 5 and there resteth 3, and 1 from 4 and there remains 3; so the Remainder is, 330 l. 12 s. 8¾d. More Examples follow.     C. Qrs. lb. Bought Cotton-Wool   121 ∶ 1 ∶ 20 Sold out   92 ∶ 3 ∶ 27 Remains   28 ∶ 1 ∶ 21     l. ℥. p w. gr. Bought Silver-weight   19 ∶ 00 ∶ 09 ∶ 14   l. ℥. p.w. gr.       Sold out at one time 4 ∶ 10 ∶ 16 ∶ 00 In all 17 ∶ 08 ∶ 00 ∶ 16 At another time 12 ∶ 09 ∶ 04 ∶ 16   Resteth unsold 1 ∶ 04 ∶ 08 ∶ 22 § 3. A Second Way of Substraction. I think it a much better way when any thing is borrowed to add to the Figure in the greater Number, in case 'tis too little: To take what is borrowed from the Figure standing next toward the Right-hand of the Figure that is too little, and suppose the Figure from whence you borrowed any Number to be so much less: So will you never need to pay what was borrowed, as is before taught. EXAMPLE. Here instead of saying 4 from 11 rests 7, and 1 From 37921 Take 29184 Difference 8737 borrowed and 8 is 9 from 12 rests 3: It will be much less trouble to suppose the 10 borrowed to be actually taken from the 2, as it really is, and so the rest of the Figures; so must you say 4 from 11 rests 7, 8 from 11 rests 3, 1 from 8 rests 7, 9 from 17 rests 8, 2 from 2 rest 0. This way of Substraction is much more natural and reasonable than the former, or common way; but the Learner may use which he pleaseth, though I doubt not but were this way as much practised as the former, it would be found much better. § 4. The Reason and Demonstration of Substraction. From the Axium of the whole being equal to all its parts taken together, we may demonstrate( or undeniably prove) the premises. For the Number from whence we make Substraction is the whole, and the Number to be substracted is part of that whole: Now if the part be taken from the whole, what remains will be the true difference between the part and the whole; for the whole containeth no more parts than the sum made of the part taken away, and the part remaining, and the part taken from the whole, is onely so much less than the whole as the part remaining; therefore the part remaining is the true excess or difference between the whole and the part taken from it. As to the reason of the Rule for Substraction, I need say no more than what is above concerning the second way of Substraction, and what was said in the Reason of Addition, and what follows in the proving Substraction. § 5. The Proof of Substraction two Ways. The Demonstration foregoing is sufficient to prove the Truth of Substraction; but because there was no Example take these following. The sum of the Subtrahend and remainder is equal to the Number given, from whence Substraction is to be made, for instance. From 56742   Take 39752 Add. Remains 16990 Proof 56742 the sum, equal to the Number, from whence Substraction is to be made; or, thus by Substraction, From the whole 56742 Take the part 39752 Remains 16990 Which deduct from the whole, And there resteth the part given to be deducted, 39752 Proof. § 6. The Proof of Addition two Ways. After you have taken the sum of the Numbers given to be added you may prove the Truth of that sum by separating the said Numbers into two parts with a line, and the sum of those parts will( if there is no Error) be equal to the Aggregate or sum of all the Numbers given. EXAMPLE.   l. s.   l. s.   5762 4 The sum of these parts       397 12 7182 17   1023 01       8942 13 The sum of these parts 10543 01   1600 08 The total 17725 18             The total of these is equal to the first total, 17725 18 Proof. Or thus by Substraction.   l. s. The total sum of the Numbers given is 17725 18 from which deduct the first part 5762 4 Remains 11963 14 from which deduct the 2d. part of the Tot. 397 12 Remains 11566 02 from which deduct the 3d. part 1023 01 Remains 10543 01 from which deduct the fourth Number 8942 13 And there remains 1600 08 from which deduct the fifth Number 1600 08 And the remainder is 0000 00 Which proves the Work. Chap: 4. Multiplication Table 3 Times 3 ∶ 9   4 ∶ 12   5 ∶ 15   6 ∶ 18   7 ∶ 21   8 ∶ 24   9 ∶ 27 4 Times 4 ∶ 16   5 ∶ 20   6 ∶ 24   7 ∶ 28   8 ∶ 32   9 ∶ 36 5 Times 5 ∶ 25   6 ∶ 30   7 ∶ 35   8 ∶ 40   9 ∶ 45 6 Times 6 ∶ 36   7 ∶ 42 8 ∶ 48   9 ∶ 54 7 Times 7 ∶ 49   8 ∶ 56   9 ∶ 63 8 Times 8 ∶ 64   9 ∶ 72 9 Times 9 ∶ 81   10 ∶ 90   11 ∶ 99 12 Times 2 ∶ 24   3 ∶ 36   4 ∶ 48   5 ∶ 60   6 ∶ 72   7 ∶ 84   8 ∶ 96   9 ∶ 108   10 ∶ 120   11 ∶ 132   12 ∶ 144 MULTIPLICATION OF Whole NUMBERS. MUltiplication is a Rule by which any Number may be so increased by multiplying it by another, as to produce a third Number, which shall bear such reason or proportion to either of the Numbers given, as the other does to a Unit. The two Numbers given to be multiplied, are for shortness termed the Factors: Or, The one( commonly the greater) is called the Multiplicand, and is that Number given to be multiplied. The other is called the Multiplier, and is that Number by which the Multiplicand is multiplied. The third Number, which is that produced by multiplying the two given Numbers together, is called the Product, or in Geometry it is called the Rectangle. By this Rule is compendiously performed many Additions, as 4 times 80 is 320, which would require 3 Additions to know; as you see in the margin. Multiplicand 80 Factors. Multiplier 4 Product 320   80 Add. 80 80 80 320 sum Multiplication is either simplo or Compound. simplo when the Factors are both Digits: And, Compound when the Factors are one or both mixed Numbers or Articles. Before you go any farther you must get the foregoing Tables by heart, which supposing you have done take the following Rules for working any sum propounded. CASE 1. When the Product of each Figure by the Multiplier is less than Ten, how to multiply by a Digit. RULE Having placed the Factor's Units under Units, as in the margin, multiply each Figure in the Multiplicand by the Multiplier, and place the several Products under the line, beginning with the Figure in Unit's place of the Multiplicand. EXAMPLE. What comes 3214 pound of Tea to, at 3214 2 Product l. 6428 Answ. 2 pound per pound? Say, 2 times 4 is 8, which put under a line as in the margin; saying, 2 times 1 is 2, 2 times 2 is 4, and 2 times 3 is 6; so the Answer is, 6428l. CASE 2. When the product of any of the Figures in the Multiplicand is 10 or any Number of Tens. RULE. Put down in the General product the Number of Units, that the product of any Figure in the Multiplicand is above 10, or any Number of Tens, and carry the said 10 or Tens to the product of the next Figure, and so proceed till all the Figures in the Multiplicand are multiplied by the Multiplier. EXAMPLE. What is the price of 3484 Bags of Cotton, 3484 Factors 9 Ans. l. 31356 Product at 9 pounds per Bag? According to the Rule, say, 9 times 4 is 36, put the 6 under the line, and carry 3, saying, 9 times 8 is 72, and 3 I carry is 75, put down the 5 and carry 7, saying, 9 times 4 is 36, and 7 is 43, put down 3 and carry 4, saying, 9 times 3 is 27, and 4 I carry is 31, which put down; so will you find the Answer 31356l. As per margin. CASE 3. When the Factors are each above 10, how to find the Product. RULE. Multiply the Figures in the Multiplicand by that standing in the Unit's place of the Multiplier, as before, and in like manner multiply the Multiplicand by the Figure standing in the Ten's place of the Multiplier; but you must place the Unit's place of the second product under Ten's place of the first, and the other degrees in order, Tens under Hundreds, Hundreds under Thousands of the first product; which done, add the products together, and the Aggregate or sum is the General product required. EXAMPLE. What is the price of 549 Ton of Iron, 594 Factors 18 4752   594   Ans. l. 10692 Product at 18 pounds per Ton? Note that if there had been 3 Figures in the Multiplier, when you were to multiply by that in Hundred's place; the first Figure of the product must have been placed under the 9 in the lower of these products: The following Examples will make it plain. CASE 4. When you have any Number of ciphers toward the Right-hand of the Multiplicand and Multiplier: Multiply by the significant Figures, and put the ciphers toward the Right-hand of the Product. EXAMPLE 1. Admit the Earth's Circumference is 360 Degrees, and that one Degree is 60 Miles, how many Miles is it round the Earth? 360 Degrees Factors. 60 Miles Answer 21600 Product.   EXAMPLE 2. If in a Mile is 1000 Paces, how many Paces in 1000 Miles? 1000 Factors. 1000 Answer 1000000 Product. In this and the like Examples where the Multiplier is onely a Unit with ciphers; place those ciphers to the Right-hand of the Multiplicand, and you have the true Product; for 1 neither augmenteth nor diminisheth any Number by Multiplication nor Division, and is therefore by some said to be no Number; but with what reason I know not. EXAMPLE 3. Multiply 3942300 by ●20020 78846 78846 78924846000 Product. Note that when ciphers stand in the Middle of the Multiplier, place the Unit's place of the Product made of the Figure standing next the ciphers toward the Left-hand, so many places forward as there are ciphers in the said Middle; as you see in the Example. The Demonstration and Reason of Multiplication. If two lines( or Numbers) be given, and one of them be divided into any Number of Parts; the product made of the two whole lines( or Numbers) is equal to the product made of the whole line( or Numbers) and the several parts of that divided. Vid. Euclid's Elem. Prop. 1. Lib. 2. To instance in Numbers( because we are now treating of arithmetic and not of Geometry) if 346 were to be multiplied by 122, which 122 suppose divided into 3 parts, viz. 100, 20 and 2: I say the product made of 346 by 122 is equal to the sum of the products, viz. 346 by 100, 346 by 20, and 346 by 2; as followeth. 346 346 346 100 20 2 1st. Product= 34600 6920= 2d. product 692= 3d. product. 2d. Product 6920     3d. Product 692     sum— 42212 the product equal to the product of 346     by 122     692     692     346     Viz. 42212 For there being no more Units in 42212 than in the products 34600, 6920, and 692, nor any more Units contained in 42212, but what is contained in the products 34600, 6920 and 692; therefore the products 34600, 6920 and 692, are equal to the product 42212. From hence also is the reason of placing the Units of the 324 123 972 6480 32400 The sum 39852 second product under Tens of the first, Units of the third under Tens of the second, &c. as in the Example; where 324 by 3 is 972, 324 by 20( ●2 being in Ten's place) is 6480, and 324 by 100( 1 being in Hundred's place) is 32400. Chap: 5. Division Teacheth, to Divide or separate any Number or Quantity into as many Parts as you please IN this Rule Observe 4 Things Vizt. THE Dividene Divisor Quotient Remain. DIVISION OF Whole NUMBERS. THE Dividend is the Number given to be divided. The Divisor is the Number by which the Dividend is divided. The Quotient is the Number of times that the Divisor is contained in the Dividend. The remainder is the Number that may remain of the Dividend after the Divisor is had, as many times in it as is expressed in the Quotient; from whence it follows that the remainder must be always less than the Divisor, or otherwise the Divisor might be had once more in it. As Multiplication is a compendious way of Addition, so Division is the work of many Substraction; for if 12 be divided by 4, the Quotient would be 3; for 4 may be taken 3 times out of 12. Dividend Divisor 4) 12( 3 Quotient ● Remainder. 1st. from 12 take 4 remains 8 2d. take 4 remains 4 3d. take 4 remains 0 There are several ways that I could easily show for the dividing one Number by another; but I shall onely incert one, which is plainer than- canceling, and shorter than the other ways commonly practised, and is therefore in my opinion the best way. CASE 1. To divide any Number by a Divisor consisting but of one place. Let it be required to divide Divisor] Dividend[ Quotient 7) 37642( 5377 26 54 52 3 Remainder 3̣7̣6̣4̣2 by 7? Having made a crooked( or any other) line at each end of the Dividend to separate it from the Divisor and Quotient, make a point or prick under 7 in the Dividend( not under 3, because you cannot take the Divisor from the 3) and say how often is 7( the Divisor) contained in 37 the first Branch toward the Left-hand of the Dividend, the Answer is 5 times, which( 5) put in the Quotient, and multiply the Divisor thereby saying, 5 times 7 is 35, which deduct from the said 37, and put the remainder( which is 2) under a line, as in the Example: 7) 37642( 5377 26 54 52 3 Remains Then make a prick under the 6( as a distinguishing Mark, that no Figure may be brought down twice) and place it to the Right-hand the remainder( 2) and ask how often 7 is contained in 26, he Answer is 3 times, which put in the Quotient as before multiplying the Divisor thereby; as 3 times 7 is 21 from 26 and there remains( 5) which put under the 6 drawing a line between the 26 and the 5. Then make a prick under the next Figure toward the Right-hand in the Dividend, Viz. under( 4) and place it to the Right-hand, the 5 making it 54, and ask how often the Divisor( 7) can be had in 54, the Answer is 7 times, which put in the Quotient and say, 7 times 7 is 49 from 54 and there rests( 5) which put under a line as before. Lastly make a prick under the 2 in the Dividend, and place it to the Right-hand the remainder 5, which makes 52, and ask how often the Divisor( 7) can be had in 52, the Answer is 7 times; which put in the Quotient, and multiply the Divisor thereby, saying, 7 times 7 is 49, which deduct from the 52, and the remainder is 3, and you have no more Figures in the Dividend: So the Work is finished, and I find that 5377 is one seventh part of 37642. The last Operation is thus contracted. How often 7 in 37 Answer 5 Times 7 is 35 From 37 Rests 2 To which bringing down the next Figure makes   26 3 21 26 5 26 54 7 49 54 5 54 52 7 49 52 3 52 EXAMPLE 2. By the foregoing Method is the Number following divided, viz. 917640 by 9. Dividend Divisor 9) 9̣1̣7̣6̣4̣0̣( 101960 Quotient 17 86 54 ( 00) Remains Note that when the Divisor cannot be had in any part of the Dividend, that is brought down under a line: In such case you are to put a cipher in the Quotient, and bring down the next Figure in the Dividend; as in the Example 9 cannot be had in( 1.) therefore( 0) is put in the Quotient, and 7 brought down, which makes the 1 to be 17, &c. CASE 2. To divide any Number by a Divisor consisting of 2, 3 or 4 places. RULE. It many times happeneth that in dividing a sum by 2, 3, &c. Figures, That though you can have the first Figure of the Divisor in the Dividend, yet you cannot have the rest of the Figures of the Divisor in the like Number of Figures of the Dividend; as if 316 be divided by 182, in this case 1( the first of the Divisor) can be had 3 times in 3( the first Figure in the Dividend, but the rest of the Figures in the Divisor, viz. 82, cannot be had 3 times in 16( the rest of the Figures in the Dividend) therefore you must make trial whether the Divisor can be had one time less in the Dividend; as here, see if 182 ●an be had 2 times in 316, by multiplying( in your mind, or on some piece of wast Paper) 182 by 2, which product if you find yet more than the Dividend 316,( as in this Example you will, and consequently cannot be deducted from it) then take 182 but 1 time in 316, and put 1 in the Quotient. Take good notice of this for it is the onely difficult thing in Division, and that it may appear plain, take the Example following. Let it be required to divide 75231 by 24? 24) 75̣2̣3̣1̣( 3134 32 83 111 15 Remainder To perform this: 1. Make a point under 5, because you can deduct 24 from 75, otherwise the point must have been made under the third place. 2. Ask how often 2 can be had in 7; the Answer is 3 times. 3. Before you put the 3 in the Quotient, make trial in your mind, if the product of 24( the Divisor) by 3 do not exceed 75, which you will find it does not. 4. Therefore put 3 in the Quotient, and say, 3 times 4( the Unit's place of the Divisor) is 12, which deduct from the 5, and 10 that you borrow( for you must always borrow so many Tens, as that the said product of the Figure in the Quotient and Divisor may be deducted) that is, from 15, and the Remainder is 3; which put under a line, and carry the 1 Ten you borrowed in your mind, saying, 3 times 2( in the Divisor) is 6, and 1 you borrowed is 7, from the 7 in the Dividend, and the remainder is( 0). 5. To the remainder( 3) 24) 75̣2̣3̣1̣( 3134 Remains 3̣2 Brought down Remains 8̣3 Brought down Remains 11̣1 Brought down 15 Remainder bring down the next Figure in the Dividend, which is 2. 6. Ask how often 2 can be had in 3, or how often 24( your Divisor) can be had in 32, the Answer is 1. 7. Put 1 in the Quotient. 8. Multiply 24( the Divisor) by 1, saying, 1 time 4 is 4 from 12( borrowing 10) and there rests 8; which put under the line, saying, 1 time 2 is 2, and 1 borrowed is 3 from 3 and the remainder is( 0). 9. To the remainder 8 bring down the next Figure in the Dividend, which is 3,( always making a point under the Figure you bring under the line for the reason aforesaid) so have you 83, inquire therefore, 10. How often 2 the first Figure in the Divisor toward the Left-hand can be had in 8; the Answer is 4 times; but if you make trial you will find the product of 24 by 4 to exceed 83, so that you can but have 24, 3 times in 83. 11. Put 3 therefore in the Quotient, as you see in the Example. 12. Multiply 24 the Divisor by 3, saying, 3 times 4 is 12 from 13( borrowing 10 to add to the 3, last brought down) and there remains 1; which put under the line, as you see, saying, 3 times 2 is 6, and 1 borrowed is 7, from 8 and there remains 1; which being in the Ten's place, makes Eleven. 13. To this 11 bring down the last Figure from the Left-hand in the Dividend, viz.( 1) and you have 111. 14. inquire how often 24( the Divisor) can be had in 111, or how often 2 in 11( because 111 is 1 place more than 24( the Answer is but 4 times,( for if you take it 5 times, you cannot deduct 5 times 24 from 111.) 15. Multiply 24 the Divisor by the Figure you put in the Quotient, which is by 4, 1 saying, 4 times 4 is 16 from 21( borrowing 2 Tens to add to the 1 in Unit's place) and there rests 5, and carry 2, and 4 times 2 in the Divisor is 8, and 2 borrowed is 10, from 11 the last remainder, and there remains 15: So the Work being finished, I find that 24 is contained in 75231, 3134 times, which I have made so plain( proceeding step by step) that any one though of ordinary Capacity may understand it, and by it any other of the like Nature, though the Divisor consists of never so many Figures; take one other Example of this Case. Let it be required to divide 319462 by 548? 548) 3194̣6̣2̣( 582 4546 1622 526 remainder By the Rules foregoing this last Operation, will be performed as followeth. How often 548 in 3194 4546 1622 Answer 5 8 2 5 Times 8 is 40 from 44, rests 4 4 is 20,& 4 is 24, from 29, rests 5 5 is 25,& 2 is 27, from 31, rests 4   Remains 454 To which bring the 6& inquire. 8 Times 8 is 64 from 66, rests 2 4 is 32,& 6 is 38, from 44, rests 6 5 is 40,& 4 is 44, from 45, rests 1   Remains 162 To which bring the 2& inquire. 2 Times 8 is 16 from 22, rests 6 4 is 8,& 2 is 10, from 12, rests 2 5 is 10,& 1 is 11, from 16, rests 5   Rmainer 526 CASE 3. When any Number of ciphers possess the 1st. 2d. 3d, &c. places of the Divisor, how to abbreviate the Work. RULE. As many ciphers as you have in the Divisor toward the Right-hand; so many Figures separate( toward the Right-hand of the Dividend) from the rest by a point or dash with the Pen, and divide the remaining Figures toward the Left-hand in the Dividend, by the significant Figures in the Divisor, leaving out the ciphers: See the Operation following. 15629ˌ00) 137428̣1̣ˌ20( 87   123961   1455820 Remains to be divided into 1562900 parts, which will be less than a Unit. EXAMPLE 2. EXAMPLE 3. 1ˌ00) 365ˌ4( 36 54 remainder to be divided by 100. Note that when the Divisor is a Unit with ciphers, as this last Example; then if you separate so many Figures from the Right-hand of the Dividend, as there are ciphers toward the Right-hand in the Divisor( as was taught before) that part of the Dividend toward the Left-hand of the Dash is the Quotient, and that to the Right-hand is the remainder; as in this Example, you see 36 is the Quotient, and 54 the remainder; because when the ciphers are cut off the Divisor, there remains onely 1 to divide by, and it has been taught before that no Number is made less by dividing by 1. § 2. The Manner of working Division explained, and the Reason of it shewed. The two great Difficulties that are in Division are, 1. That when a Number is to be divided by another, consisting of several Degrees or Places of Figures, it cannot be known without Trial, how often the Divisor can be had in the Dividend. 2. The suctstracting the several Products made of the Quotient, and Divisor from the Left-hand of the Dividend, seems incoherent with the Rules of Substraction, of deducting Unit's place from Units, Ten's place from Tens, &c. To explain an● remove both which Difficulties, take the Example and Rules following, where the whole Work of Division is made plain, and easy to be understood by a mean Capacity. The Example I make use of shall be to divide 19467281 by 426? The Work of Division explained. Products of the Divisor. 1 426) 1946̣7281 ( 40000 First Quotient     17040000   2 852 2427281 5000 Second Quotient 3 1278 2130000   4 1704 297281 600 Third Quotient 5 2130 255600   6 2556 41681 90 Fourth Quotient 7 2982 38340   8 3408 3341 7 Fifth Quotient 9 3834 2982       ( Rem. 359.) 45697 The sum of these Quotients, which is the true General Quotient. In this Example, 1. I have made Products of the Divisor, multiplying it by the several Digits against which the said Products stand. 2. As is usual I prick under the 6 in the Dividend, because I can take the Divisor from the 4 first Figures toward the Left-hand of the Dividend. 3. I consider what place the first Figure in the Quotient toward the Left-hand will possess, which is always the same with the Figure, under which the first Point or Prick is made, and in this Example is Tens of Thousand's place; so that what Figure soever is first put in the Quotient, is so many Tens of Thousands. 4. I look in my 9 Products, which of them is next to, and less than the 4 first Figures to the Left-hand of the Dividend, and find the Product 1704 to be next; right against which in the Series of Digits stands 4, wherefore I put 4 in the Quotient which is 40000, because( as was said in the last step) the Quotient will have 5 Places. 5. I multiply the said 1704 by 10000, because the 4 is in that place, or the Divisor by 40000, and the Product is 17040000, which( according to the true Rules of Substraction) is to be taken from the whole Dividend, and the remainder( as in the Example) is 2427281. 6. I look as before, which of the 9 Products is next to, and less than the 4 first places toward the Left-hand of my new Dividend 2427281( because none of the Products can be had in 3 places) and I find 2130, right against which stands the Digit 5, which must be 5000, because it is to stand in the Thousand's place of the Quotient; where having placed it, multiply( as before) the 2130 by 1000, or the Divisor by 5000, and deduct the Product from the new Dividend 2427281, proceeding with the rest of the Figures till nothing, or a Number less than my Divisor remain; which done, 7. I sum up the 5 Quotients as in the Example, which make the General Quotient 45697, and so the Work is ended. § 3. The Demonstration of Division. The Design of Division is to discover how often one Number is contained in another, and( if nothing remain after Division) the Quote is an even part of the Dividend, and contains a Unit so often as the Dividend containeth the Divisor. The Divisor sheweth how many parts the Dividend is to be divided into, and the Quotient is one of those parts; as if 400 were divided into 8 parts, 8 will be found to be contained in 400, 50 times; so that 50 is one Eighth part of 400, for 400 is 8 times 50, and consequently 50 is one Eighth of 400, and the like may be said of other Numbers. § 4. The Proof of Division. Division may be proved by dividing the Dividend by the Quotient, and the Quotient will be your Divisor: Or, you may prove it( as is more usual) by Multiplication; for if you Multiply the Quotient and the Divisor together, the Product will be equal to your Dividend. To instance, in the Numbers following: If 1728 be divided by 12, the Quotient will be 144; and if for proof, you divide 1728 by 144, the Quotient will be your former Divisor( 12): Or, if you multiply 144, the Quotient, by 12, the Product will be 1728: See the Work. 12) 17̣2̣8̣ ( 144 Quotient Multiply     12 Divisor   52 288       144     48         1728 The Dividend for proof   Rem. 0.     Or thus by Division: 144) 172̣8̣ ( 12 The former Divisor   288     0 Rem. § 5. The Proof of Multiplication. The onely true way to prove Multiplication, is by Division; for if you divide the Product by either the Multiplicand, or Multiplier, the Quotient will be the other. EXAMPLE. In the Example of the second Case of this Chapter, 3484 being multiplied by 9, produceth 31356: 9) 31̣3̣5̣6̣ ( 3484 Quotient   43     75     36     0 Remains And if 31356 be divided by 9 the one Factor, the Quotient is the other Factor, as in the Example. Some Authors have taught to prove Multiplication, by taking the Nines out of the Factors singly, and multiplying the Remainers together, and taking the Nines( if any be) out of the Product, noting that remainder; then take the Nines out of the first Product, and if the remainder be equal to the forementioned, they conclude the Work to be right: but that does not at all follow, for by this Rule you may prove a Thousand false Products as true ones: Example, Admit 3765 were to be multiplied by 58, the true Product is 218370,( but if you suppose the Product 398370, which is 180000 too much,) or 245370, which is 27000 too much, they will both prove right according to this Method; nor is there any other Method to prove Multiplication by, so true and concise as by Division; though 'tis indeed needless to prove every sum you work, by any Method, provided you be careful in the Operation; or it may not be amiss if your Work is great, to run it over twice very carefully, and if you find both times agree, 'tis to be supposed your Work is right. Chap: 6. th Reduction is a Rule consisting of two Parts, vizt. 1st. The Reduceing of a Number, from a greater to a lesser Denomination, as Pounds into Shillings, Hundreds into Pounds, Yards into Feet &c. which is called Reduction Descending, and is performed by Multiplication. 2d. The Reduceing a Number from a lesser to a greater name or Denomination, as Feet into yards, Gallons into barrels, Farthings into Pounds &c. which. is called Reduction Ascending, and is performed by Division. So that all Questions in Reduction are resolved either by Multiplication, or Division, or both, which shall be farther explained by the Questions following. Reduction Descending. § 2. CASE I. WHEN a Number of one Denomination is given to be reduced into a lesser Denomination. RULE. Multiply the given Number by such a Number of Units of the inferior Denomination into which you would have the Number given reduced, as are contained in a Unit of the Denomination which is given, and the Product is the Answer. EXAMPLE 1. In 476 Pounds, how many Farthings? 476 Pounds l. Multiply 960 The Farthings in 1 2856     4284     456960 Farthings for Answer.   EXAMPLE 2. In 87 Hundred Weight, how many Pounds? 87 Hundred Multiply 112 Pounds in one Hundred 174     87     87     9744 Pounds for Answer.   EXAMPLE 3. In 527 Ells Flemish, how many Quarters of a Yard, each Ell being three Quarters of a Yard? 527 Ells Multiply 3 Quarters of a Yard in an Ell 1581 Quarters of a Yard for Answer. EXAMPLE 4. In 328 Bails of Dowlass, how many Pieces? 328 Bails Multiply 3 Pieces in a Bale 984 Pieces for Answer. EXAMPLE 5. In 484 Gross of Tape, each Gross 12 Dozen, each Dozen 2 Pieces, and each Piece 36 Yards, how many Yards? 484 Gross Multiply 12 Dozen in a Gross 968     484     5808 Dozen in 484 Gross Multiply 72 Yards in a Dozen 11616     40656     418176 Yards for Answer.   CASE 2. When it is required to reduce Numbers of divers Denominations, into the lowest Denomination. RULE. Work as in the last Case; but if you have any Number of the next inferior Denomination to that you are reducing, add such Number to the Product. EXAMPLE 1. In 364 l. 05 s. 5 d. How many Pence? l. s. d.     364 ∶ 05 ∶ 5 ∶   Multiply and Add the 5 s. 20 ∶ The shillings in a pound   l. s.   7285 Shillings in 364 ∶ 05 ∶ Multiply and Add the 5 d. 12 Pence in a shilling   14575       7285         l. s. d. 87425 Pence in 364 ∶ 05 ∶ 5 ∶ For Answer. In the last Example in reducing the pounds, say,( o) times 4( in the pounds) is( o), but 5( in the shillings) is 5 shillings; then say, 2 times 4 is 8, &c. And when you come to the shillings, say, 2 times 5 shillings is 10, and 5 in the Pence place is 15 pence, put down 5, and carry 1, &c. Note that if you had any thing in the Ten's place, either in the shillings, pence, &c. you must add them when you multiply by the Figure in the Ten's place of the Multiplier. EXAMPLE 2. In 48 l. 17 s. 11 d. 2 q. How many Farthings? l. s. d. q.     48 ∶ 17 ∶ 11 ∶ 2 ∶ Multiply and Add 17 s. 20 ∶ The shillings in a pound   l. s.   977 Shillings in 48 ∶ 17 ∶ Multiply adding the 11 d. 12 Pence in a shilling   1955       978         l. s. d.   11735 Pence in 48 ∶ 17 ∶ 11 ∶ Multiply and Add 2 q. 4 Farthings in a Penny     l. s. d.   46942 Farthings in 48 ∶ 17 ∶ 11 ½ For Answer. EXAMPLE 3. In 47 C. 2 Qrs. 24 lb. How many Pounds? C. Q. lb.       47 ∶ 2 ∶ 24 ∶     Multiply and Add the 2 Qrs. 4 Quarters in 112 lb.       C. Q.   190 Quarters in 47 ∶ 2 Multiply and Add the 24 lb. 28 Pounds in 1 Q. of C. 1524           382                 C. Q. lb. 5344 Pounds in 47 ∶ 2 ∶ 24 For Answer. This Question is more briefly resolved, as in C. Q. lb. 47 ∶ 2 ∶ 24 47     470     478     5344 lb Answer. the margin, by first putting down your 47 C. 4 times, and the 2 Q. 24 lb, which is 80 lb, in Ten's and Unit's place; so the sum is the Answer. § 3. Reduction Ascending. To reduce Numbers from a lesser to a greater Denomination. CASE 1. When the Number given is to be reduced to the next superior Denomination. RULE. Divide the said given Number by such a Number of Units of the Denomination given, as make a Unit of the next superior Denomination, and the Quotient is the Answer. EXAMPLE 1. In 984 Pieces of Dowlass, how many Bails, each 3 Pieces: See the Operation. 3) 9̣8̣4̣ ( 328 Bails for Answer.   8     24     0   EXAMPLE 2. In 9744 Pounds, how many Hundreds? 11ˌ2) 974̣4̣ ( 87 Hundred for Answer.   784     0   CASE 2. When a Number is to be reduced to a Denomination higher than the next superior Denomination. RULE. Divide the given Number, as before, by such a Number of Units of the Denomination given, as makes a Unit of the next higher Denomination, and note the remainder. Then divide that Quotient by so many Units of that Name or Denomination, which it is of as makes a Unit of the next higher Denomination to the said Quotient, &c. noting the Remainers, as in the Examples following. EXAMPLE 1. In 87425 Pence, how many Shillings and Pounds? EXAMPLE 2. In 5344 lb. How many Quarters, and Hundreds? EXAMPLE 3. In 418176 Yards, how many Gross of Tape? Divide the given Number by 72, and that Quotient by 12, for Answer; because 72 Yards is 1 Dozen, and 12 Dozen 1 Gross. These Questions are the Converse of those in Reduction Descending, and may serve for proof of them, and likewise to show the Learner the Coherence of the Rules. § 4. Reduction Ascending and Descending. Questions performed by Multiplication and Division are these that follow; and such like. EXAMPLE 1. In 874 Ells Flemish, how many Ells English? Multiply the given Number by 3, and divide the Product by 5, and the Quotient is the Answer.   847 Ells Flemish Multiply   3 Quarters of a Yard in 1 Ell 5) 2̣6̣2̣2 ( 524 Ells English for Answer.   12       22       2 Rem.   Note that the remainder is always of the same Denomination with the Dividend. EXAMPLE 2. In 846 Dollars, each 4 s. 6 d. How many pounds Sterling? EXAMPLE 3. In 46 C. of Cotton-wool, how many Pounds, and what the Price, at 15 d, a Pound? Answer 322 l. Chap: 7th. Sect. 1. The Golden Rule Is so called from it's extraordinary usefulness, not only in Arithmetical Questions, but in all parts of the mathematics. It is also called the Rule of Three, because there is always Three Numbers given to find a Fourth, and it is properly called the Rule of Proportion, because the First Number bears such proportion to the second, as the third, does to the Fourth. The design of this Rule is to show how to find a Fourth proportional Number: by having Three given Numbers, which is deducible from the 16th. prop: of the sixth Book of Euclid's Elements. The Rule is Multiply the 2d.& 3d. Numbers together,& divide the Product by the first number,& the Quotient thence arising is the Fourth Number sought or Divide the 2d. Number by the 1st.,& multiply the Quotient by the 3d. number,& the product is the Number required. For the 4th. number contains the 3d. so often as the 2d. contains the first, and this is called direct Proportion. THE SINGLE RULE OF Direct Proportion. ALL the Difficulty in this Rule consisteth in the right stating the three Numbers given; for when you have done that, you have onely Multiplication and Division, and the Work is performed: The Rule therefore for stating any Question in this kind of Proportion is. RULE. Consider that of the three Numbers given you, have always Two of one Denomination: And, That Number which is of another Denomination, must be always placed in the second Place; and to the Left-hand thereof must be placed that Number( of the Two of one Denomination) on which the Second has dependence, and the other of the said Numbers of one Denomination, must be placed next the Right-hand: As supposing it were required to know what the Interest of 75 pound is at the rate 8 Pound per Cent. per Annum, the Numbers will be stated thus: L. prin. L. int. L. prin. 100 ∶ 8 ∶ ∶ 75. In this Example there are two Numbers that are Principal money, and one that is Interest; therefore the Interest( according to the Rule) must stand in the Middle, or second Place; the Principal on which the Interest dependeth, Viz 100( 8 l. being the Interest thereof) must stand in the first Place toward the Left-hand, and the other Principal on which the fourth Number( which is the Number sought for) dependeth, must possess the first Place toward the Right-hand. By these Rules foregoing, you may with Ease and Certainty perform any Operation in Direct Proportion, and for your farther Information take the Examples following. EXAMPLE 1. If the Interest of 100 l. for one Year be 8 l. what is the Interest of 75 Pound for the same Time? EXAMPLE 2. If 32 Rundlets of Brandy cost 96 pounds, what will 4 Rundlets cost at that rate? EXAMPLE 3. If 12 gabs of Cotton-wool cost 184 l, what will 17 gabs cost? Note that( as in the last Example) when any thing remains that is reducible to a lower Denomination; after it is so reduced, it must be divided continually by the first Number. CASE 2. When any of the three Numbers given happen to be of divers Denominations, you may reduce them into the lowest Denomination. And if your first Number require to be reduced; your Third must be reduced likewise into the same Denomination as the first, and the contrary: For the first and third Number, before you begin your Operation, must be always reduced to one Name, as was said before. EXAMPLE 1. If 17 Hogs-heads of Sugar cost 320l. 12 s, what will 5 of those Hogs-heads be worth? Note that when you have multiplied the second and third Numbers together, and divided the Product by the First, the Quotient is of the same Denomination, as the second Number is; after you have reduced it( as in the last Example) into its lowest Denomination given. EXAMPLE 2. If 4 C. 1 Q. 24 lb of Sugar cost 14 l. what will 18 C. cost? Note farther, that what Farthings remains to be divided by the common Divisor( as in the last Example) because you can reduce them into no lower Denomination, you may place them over your Divisor, as Fractions of a Farthing, which shall be explained when we come to vulgar Fractions, &c. CASE 3. When the first Number of the 3 given, is but a Unit, the Operation is performed by Multiplication onely. EXAMPLE 1. If I give 15 Shillings for a Pound of Thread, what will 250 lb cost me at that Rate? lb. s. lb.     1 ∶ 15 ∷ 250         15         125         25           l. s.     3750 Shillings Answer, or 187 ∶ 10 ∶ EXAMPLE 2. At 14l. 10 s. 6 d. per bag of hops, what cost 55 gabs? Bag. l. s. d. Bags       1 ∶ 14 ∶ 10 ∶ 6 ∷ 55.         20               290 Shillings             12             3486 Pence   Multiply     55 the 3d. Numb.     17430             17430                       l. s. d. 191730 Pence Answer, or 798 ∶ 17 ∶ 6 CASE 4. When the third Number of the 3 given( or that toward the Right-hand) is a Unit; such Operation is performed by Division onely; if the Number need no reducing. EXAMPLE 1. If 40 Pieces of broad Cloath cost 590 l. what will one Piece cost? Pieces. l. Pieces. 40 ∶ 590 ∷ 1 4ˌ0) 5̣9̣ˌ0 ( 14¾ l. or 14 l. 15 s. Answer.   19     3 Pounds remains. EXAMPLE 2. If 14 Hogs heads of Tobacco, poise net 9285 lb, cost 619 l. 10 s. what will one Pound cost at that Rate? lb l. s. lb s. d. q. 9285 ∶ 619 ∶ 10 ∷ 1. Answer 1 ∶ 4 ∶ 0 480/ 9285   20           9285) 12390 s. ( 1 Shilling         3105 Shillings remains         12           9285) 37260 d. ( 4 Pence           120 Pence remains         4             480 Farthings remains to divide by 9285. § 2. Whereas in the former Section of Direct Proportion, the fourth Number was always proportionably greater than the Third, as the Second was greater than the First: in this kind of Proportion, on the contrary, the greater the third Number is, the less is the Fourth, and the less the Third is, the greater is the Fourth; and it is therefore called Indirect or Reverse Proportion. And whereas in the last Section the Product of the First and Fourth is equal to that of the Second and Third; in the Proportion I am now treating of, the Product of the Third and Fourth is equal to that of the first and second Numbers; which may serve as a Proof for both. The Method of stating any Question in this Proportion, is the same with Direct; but to find the Number required this is the RULE. Multiply the first and second Numbers toward the Left-hand together, and divide the Product by the Third, and the Quotient arising is the Answer. A Rule to know whether a Question proposed be to be Answered by the Rule of Proportion, Direct or Indirect. Having stated the three Numbers given as is formerly directed, calling the middle Number the mean, and the two outermost Numbers, the extremes: Consider from the Nature of the Question, whether the third Number requires more or less than the second Number; if it requires more, the lesser extreme is to be your Divisor; but if the Third require less, the greater extreme is your Divisor: Now so often as this lesser, and the greater extreme happeneth to be the third Number; so often is your Proportion Indirect, but when they are the first Number, the Proportion is Direct; an Example or two will make it plain. EXAMPLE 1. If a Board is 9 Inches broad, how In. br. long. In. br. 12 ∶ 12 ∶ 9   12   9) 144 ( 16 Inches in length for Answer.   54   0   much in length will make a square Foot, say, if 12 Inches broad require 12 in length, to make a square Foot, what length will 9. Inches broad require: It will require more length, because there is less breadth: See the Work. EXAMPLE 2. How many Yards of Silk 3 Quarters broad, will line 9 Yards of broad Cloath, that is 2 Yards broad? Say, if 6 Quarters wide require 9 Yards in length, what will 3 Quarters wide require in length. Qrs. br. Yar. long. Qrs. br. 6 ∶ 9 ∶ 3   6   3) 54 ( 18 Yards in length for Answer.   24     0   EXAMPLE 3. If when the price of a Bushel of Wheat is 6 s. 3 d. the Penny-loaf weigheth 9 ℥; what must the Penny-loaf weigh, when the price of a Bushel of the same Wheat is 4s. 6 d. the Question is thus stated.   s. d. ℥. s. d.         6 ∶ 3 ∶ 9 ∶ 4 ∶ 6         12     12           75 Pence   54 Pence your Divisor   9               54) 67̣5̣ ( 12 ℥               135                 27 ℥ remains   Multiply         20 Penny-weight     54) 540 ( 10 Penny-weight                 l. ℥. p-w.   0 Remains   Answer 1 ∶ 00 ∶ 10 § 3. The Double Rule of Direct Proportion. In this kind of Proportion there are 5 Numbers given to find a sixth, which sixth will bear such Proportion to the Product made of the fourth and fifth Numbers, as the third Number does to the Product made of the first and second Numbers. The Rule for stating the five Numbers given; is, Make that the third Number from the Left-hand, which is of the same Denomination with the Number sought, then place the two Numbers in the first and second Place to the Left-hand, which are conjunctive in the sense of the Question to the Third, and the other two Numbers in such Order, that the First may be of the same Denomination with the Fourth, and the second of the same with the Fifth; which done, RULE. Divide the Product of the 3 next the Right-hand multiplied one in another, by the Product of the two First to the Left-hand, and the Quotient is the sixth Number sought for. EXAMPLE. If 100 l. in Twelve Months gain 6 l. what will 500 l. gain in Eight Months?   L. prin. Month. L. int. L. prin. Month.   100 ∶ 1̣2 ∶ 6 ∶ 500 ∶ 8   12     6   Divisor= 1200     3000           8         12ˌ00) 2̣4̣0ˌ00 ( 20 Pound Answer.         00 Remains By the Work you may perceive that 500 l. will gain 20 l. in 8 Months, at the Rate of 100 Principle, gaining 6 l. Interest in 12 Months. This Question or any other of this Nature may be resolved at two Single Rules of Proportion, thus: If 100 l. require 6l. what will 500l. require, the Answer is 30 l. Then say, if 12 Months require 30 l. what will 8 Months require? the Answer( as before) is 20 l. § 4. The Double Rule of Indirect Proportion. The Rule for stating your Question. Place the three first Numbers toward the Left-hand in the same Order you did in the last Section, and for the other Two, place that the Fourth, which is of the same Denomination with your second Number, and consequently the other next the Right-hand: So will your first and last, Viz. that required be of one Denomination, your second and fourth of another, and your third and fifth of another. And, The Rule for performing the Operation; is, Divide the Product of the first multiplied in the second, and that Product in the fifth, by the Product made of the third and fourth, and the Quotient is the Answer. EXAMPLE. What Principle will raise 20 l. in Eight Months at 6 per Cent. per Annum. L. prin. Month. L. int. Month. L. int. 100 ∶ 12. 6 ∶ 8 ∶ 20 12   8     1200   48= Your Divisor 20           L. in.       48) 2̣4̣000 ( 500 Quotient for Answer; which proves the last Operation. 0 Remains       § 5. The Reason and Demonstration of the Single Rule of Direct Proportion. At the beginning of this Chapter, it is said, That if 4 Numbers are Geometrically proportional: The Rectangle or Product made of the Means, is equal to that of the two extremes from Euclid. lib. 6. prop. 16. from which I shall prove the Method for finding the fourth Proportional. EXAMPLE. Admit 4 is in proportion to 12, as 18 is to a fourth Number unknown, for which put ( u) they will stand thus: 4 ∶ 12 ∷ 18. u 1. i. e. As 4 is in proportion to 12, so is 18 to the unknown Number; then from the forementioned Proposition. 4u = 216 2. i. e. Four times u( which represents the unknown Number) the Product of the first and fourth, is equal to 12 Times 18, viz. 216, the Product of the two Means; then it necessary follows. u= 216/ 4 3. i. e. That ( u) is equal to 216 divided by 4, for if 4 Times ( u) is equal to 216, then one Time ( u) must be equal to one fourth part of 216: And, 216/ 4= 54 4. Since ( u) or the unknown Number, is equal to one fourth part of 216, and that ¼ part of 216 is equal to 54; therefore u is equal to 54, which is the fourth Number sought; and if you compare the several Steps, you will find the fourth Number to be discovered after the same Method given for finding it, at the beginning of this Chapter; which is by multiplying the second and third Numbers together, and dividing the Product by the First. Or thus, from this Axium. That the fourth Number containeth the Third; so often as the Second does the First. Hence 12/ 4= u/ 18 that is ¼ of 12 is equal to one 18th of ( u) Now 12/ 4= 3 therefore u/ 18= 3 i. e. Twelve divided by 4 is equal to 3, therefore u divided by 18 must be equal to 3. And if u/ 18= 3 then 3 × 18= u i. e. If u divided by 18 is equal to 3, then 3 Times 18 must be equal to u, and consequently ( u) is equal to 54, for 3 Times 18 is 54, as before. Note ( x) signifies multiplied by. § 6. The Demonstration of the Single Rule of Indirect Proportion. By the Definition of this Rule in Section the second foregoing: the Product of the first and second Numbers, is equal to that of the Third and Fourth; from whence this Demonstration; for instance, in finding a Number in a Reverse or Indirect Proportion to 6 ∶ 9 ∶ 3 ∶ u. Therefore by the Definition. 6 × 9= 3 × u, or 54= 3 u i. e. The Rectangle of the two first Numbers 6 by 9, is equal to that of u by 3. Now if 54= 3 u, u= 54/ 3. i. e. If 54 is equal to three Times ( u) then it follows that one Time ( u) is equal to one third part of 54: 54/ 3= 18 Therefore u= 18   i. e. One third of 54 being 18, therefore u is equal to 18, which was required; so the Definition is cleared. By the same Rules may the Double Rules of Proportion be demonstrated; but this Book being chiefly designed for the practise of young Merchants; my designed Brevity requireth, that I pass forward to what is more practical. § 7. The Proof of the Rules of Proportion. Every kind of Proportion I have discoursed of, may have the Operations proved two Ways. CASE 1. Single Direct Proportion. When four Numbers are in Direct Proportion, the Product made of the First and Fourth, is equal to that of the Second and Third; otherwise the Work is not rightly performed. 2dly, The second Way is thus: As the fourth Number is to the Third, so is the Second to the First; otherwise the Work is not right. CASE 2. Single Indirect Proportion. When four Numbers are in an Indirect Proportion; the Rectangle of the First and Second, is equal to that of the Third and Fourth; otherwise there is an error in the Work. 2dly, Thus: As the First to the Third, so is the fourth Number to the Second in a Direct Proportion; otherwise the Operation is not rightly performed. CASE 3. Double Direct Proportion. When a sixth Number is found in a Direct Proportion; the Rectangle of the First, Second and Sixth, is equal to that of the Third, fourth and fifth Numbers, if the Work is not Erroneous. 2dly, Thus: As the Product of the fourth and fifth Numbers is to the Sixth; so is the Product of the First and Second to the Third, in a Single Direct Proportion. CASE 4. Double Indirect Proportion. When five Numbers are given, and a Sixth found in an Indirect or Reverse Proportion; the Rectangle( provided the Work is stated by the Rules foregoing in the fourth Section of this Chapter) of the First, Second and Fifth, is equal to that of the Third, Fourth and sixth Numbers, if the Work is rightly performed. 2dly, Thus: As the fifth Number is to the Product of the Third and Fourth; so is the Sixth to the Product made of the First and Second, by one Single Direct Proportion. CHAP. the 8th. of FRACTIONS. Vulgar FRACTIONS. § 1. Notation and Numeration of Vulgar Fractions. A Fraction is one or more Parts of Unit or Integer, according as the same is divided. Every Fraction consisteth of two Parts, viz. a Numerator and a Denominator. The Denominator is placed( in Writing) below the line you writ in, and sheweth how many Parts the Integer, or Unit is divided into. The Numerator of a Fraction is( in Writing) placed above the line, and sheweth how many of the said Parts, expressed by the Denominator, are contained in the Fraction: For instance, 3 Numerator. 4 Denominator. In reading Fractions the Numerator is first mentioned, then the Denominator; as the Fraction above is red, three fourth Parts of any thing: i.e. The Denominator sheweth that the Integer is divided into four Parts; and the Numerator, that three of those fourth Parts, are contained in the Fraction: So by the same Reason ¼ is one fourth Part, ½ is one half, or two fourth Parts, ⅔ is two third Parts, ⅚ is 5 sixth Parts, &c. As in the following Table. One Half, &c. is ½, l⅓, ¼, ⅕, ⅙, 1/ 7, ⅛, 1/ 9, 1/ 10   Two Thirds, &c. is ⅔, 2/ 4, ⅖, 2/ 6, 2/ 7, 2/ 8, 2/ 9, 2/ 10     Three Fourths, &c. is ¾, ⅗, 3/ 6, 3/ 7, ⅜, 3/ 9, 3/ 10       Four Fifths, &c. is ⅘, 4/ 6, 4/ 7, 4/ 8, 4/ 9, 4/ 10         Five Sixths, &c. is ⅚, 5/ 7, ⅝, 5/ 9 5/ 10         Or thus; the whole line a b, being a Unit, divided into 18 equal Parts; the line a d is 3/ 18, the line a p is 7/ 18, the line a m 15/ 18, &c. Six Sevenths, &c. is 6/ 7, 6/ 8, 6/ 9, 6/ 10           Seven Eighths, &c. is ⅞, 7/ 9, 7/ 10             Eight Ninths, &c. is 8/ 9, 8/ 10               And Nine Tenths, is 9/ 10                 Fractions are either proper or improper. A Fraction properly so called, or a proper Fraction, is when the Numerator is less than the Denominator, as the Fractions foregoing. An improper Fraction is when the Numerator is greater than the Denominator; as 7/ 4, 28/ 20, &c. Again, Fractions are either simplo or Compound. A simplo Fraction is when the Fraction is immediately the Fraction of a Unit or Integer; as those foregoing in the Table, &c. A Compound Fraction is a Fraction of a Fraction, as ½ of ¼ of a Pound Sterling, which is equal to 2 s. 6 d. or it is when a Unit is divided into any Number of Parts, and each of those Parts are again subdivided into Parts; these last Parts are Compound Fractions, being the Fractions of the Fractions of a Unit. So the whole line ( r s) being a Unit, the line r 1, is l⅓, r 2 is ⅖, because the Unit is divided into five Parts; which five Parts being subdivided into four Parts, as under the line: I say, each of these last parts are Fractions of a fifth part; so the line r q is ½ of l⅓ of the line r s; the line r p is ¾ of l⅓ of it, &c. § 2. Reduction of Vulgar Fractions. It may seem strange to some, that Reduction is here taught before Addition, &c. but 'tis necessary it should be so, because Reduction is made use of in all the subsequent Rules, to fit and prepare Fractions for Addition, Substraction, &c. CASE 1. When a mixed Number is given to be reduced to an improper Fraction. RULE. Multiply the Integers by the Denominator of the Fraction, and to the Product add the Numerator, and place the sum over the Denominator for a new Fraction. EXAMPLE. Reduce 12¾ to an improper Fraction; see the Marginal 12¾   51/ 4 Answer Operation. CASE 2. When an improper Fraction is given to be reduced to a whole or mixed Number. RULE. Divide the Numerator of the Fraction by the Denominator, and the Quotient is a whole Number; and if any thing remain it must be placed over the Divisor. EXAMPLE. Reduce 51/ 4 to a whole or mixed Number. 4) 5̣1̣( 12¾ Answer, which proves the last Case. 11 3 Remains CASE 3. When Fractions have different Denominators, and are to be reduced to a common Denominator. RULE. Multiply the Numerator of each Fraction singly, into all the Denominators of the Fractions given, excepting its own, and the Product is a new Numerator; and if you multiply all the Denominators one in another, the Product is a common Denominator. EXAMPLE. Reduce ⅔, ¾ and 5/ 7 to a common Denominator. 2 The first Numerator. 3 Multiply Denominator. 4 The 2d Denominator. 4 Multiply Denominator. 8 Product. Mult. 12 Multiply Denominator. 7 The 3d Denom. Mult. 7 Multiply Denominator. 56 The first new Numerator. 84 The common Denominator. 3 The Numerat. of the 2d. Lastly, 3 The Denom. of the 1st. 5 The Numerat. of the 3d. Mult.   4 The Denom. of the 2d. Mult. 9 The Product.   7 The Denom. of the 3d. 20 The Product. Mult.   3 The Demom. of the 1st. Mult. 63 The 2d new Numerator.     60 The 3d new Numerator. Now if you place each new Numerator over the common Denominator, you will have 56/ 84 Equivalent to ⅔ The first Fraction given. 63/ 84 ¾ The second Fraction. 60/ 84 5/ 7 The third Fraction. CASE 4. To reduce a Fraction into its lowest Term. RULE. Take ½, l⅓, or ¼, &c. of the Numerator and Denominator. EXAMPLE. Reduce 56/ 84 to its lowest Terms: Say half of 56 is 28, and ½ of 84 is 42, then ½, 28 is 14, and ½ 42 is 21; and because you cannot take half 14/ 21, make trial if you can take l⅓, &c. but since you can onely take 1/ 7 of both; say the Sevens in 14 is 2, and the 7 in 21 is 3: So is the given Fraction equivalent to ⅔, and proves the first Fraction in the last Case to be right: See the Work. 56/ 84 or 28/ 42 or 14/ 21 or ⅔ Take   Take   Take   Lowest Term. ½   ½   1/ 7   There are other Rules for the performing this, but none so proper for the young Merchant's practise. CASE 5. To reduce a Compound Fraction to a simplo one, equivalent to the Compound. RULE. Multiply all the Numerators one in another, for the Numerator of the Answer, and the Denominators one in another, for that of the Answer. EXAMPLE. Reduce ¼ of 1/ 12 of 1/ 20 into a simplo Fraction. The Product of the Denominators 4, 12, and 20, is 960, and the Product of 1, 1 and 1 is 1; so the simplo Fraction sought for is 1/ 960. CASE 6. To find the Value of any Fraction, whether the same be of Coin, Measure or Weight, &c. RULE. Multiply the Numerator of the Fraction by such a Number of Units of the next Denomination, inferior to that the Fraction is of, as is equal to a Unit of the Denomination the Fraction is part of, and divide the Product by the Denominator, so the Quotient will answer your Question; but if any thing remain, reduce that to the next lower Denomination, and divide as before. EXAMPLE. What is the Value of 134/ 146 of a Hundred Weight? See the Operation.   134 Hundred   Multiply       4 Quarters of Hundred     146) 536 ( 3 Quarters of Hundred         98 Quarters remains Multiply       28 Pound in a Quarter       784               196             146) 274̣4̣ ( 18 Pound             1284               116 Pounds remain Multiply       16 Ounces in a Pound       696               116             146) 185̣6̣ ( 12 Ounces             396               104 Ounces remains Multiply       16 Drams in 1 Ounce       624               104             146) 166̣4̣ ( 11 Drams, 58/ 146                 Q. lb. ℥. dr.       Answer 03 ∶ 18 ∶ 12 ∶ 11 58/ 146   204               58             CASE 7. To reduce Fractions of a lower Denomination to a higher. RULE. Consider what Denomination your Fraction is of, and how many of that make a Unit of the next, &c. to the Denomination you would have your Fraction reduced to; then work as in the fifth Case of this Chapter. EXAMPLE. Reduce ⅕ of an Ounce Averdupoize into the Fraction of a Hundred Weight, 16 Ounces being 1 Pound; ⅕ of an Ounce is ⅕ of 1/ 16 of a Pound, then I consider that 28 Pound is a Quarter of a Hundred, and that 4 Quarters is 1 Hundred; therefore ⅕ of an Ounce is, ⅕ of 1/ 16 of 1/ 28 of ¼ of a Hundred; which by the fifth Case foregoing is 1/ 8960 of a Hundred. CASE 8. If you would reduce a Fraction of a higher to a Fraction of a lower Denomination. RULE. Reduce the Numerator of the Fraction into that Denomination you would have your Fraction of, and place it over the Denominator given for a new Fraction. EXAMPLE. Reduce 1/ 8960 of a Hundred into the Fraction of an Ounce.   112 Pound Multiply   16 Ounces   672       112     Product 1792 Ounces in the Numerator; so the Answer is 1792/ 8960, which Fraction in its lowest Term is ⅕, and proves the last Case: See the Work. 1792/ 8960, 896/ 4480, 448/ 2240, 112/ 560, 56/ 280, 8/ 40, ⅕. Proof, or more brief by dividing the first by 1792. Take Take Take Take Take Take   ½ ½ ¼ ½ 1/ 7 ⅛     § 3. Addition. CASE 1. When a simplo Fraction is to be added to a simplo. RULE. If the Fractions are not in a common Denominator, reduce them to one by the third Case of the last Section; then add the Numerators together, and divide the sum by one Denominator, and the Quotient is the sum required, and if any thing remain place it over the Divisor. EXAMPLE. To ⅔ add 5/ 6? The Fractions in a common Denominator are, 12/ 18, 15/ 18. 12 The first Numerator 15 The Second 27 The sum, which divided by 18, is 1 9/ 18, or 1½ For Answer. CASE 2. When a mixed Number is to be added to a mixed. RULE. Work with the Fractional parts as before, and afterward add the sum of the Fractions to the sum of the Integers, and you have your Desire. EXAMPLE. To 1½ add 74⅗. The sum of the Fractions by the last Case is 1 1/ 10, which added to 1 and 74 makes 76 1/ 10. 1 1/ 10 Add. 1 74 76 1/ 10 The sum required. Or you may perform the Work by reducing the given Numbers to improper Fractions, as in Case 1. of the last Section, and so proceeding, as in the first Case of this Section. CASE 3. When a Compound Fraction is to be added to a simplo. RULE. Reduce the Compound Fraction to a simplo by the fifth Case of the last Section; then find the sum by the first Case of this Section. EXAMPLE. To 15/ 42 add ⅜ of ⅔. The Compound Fractions in a simplo are, 6/ 24 or ¼. The common Denominator of ¼ and 15/ 42 is as followeth: 42/ 168 and 60/ 168. The sum of the Numerators is 102, and of the Fractions 102/ 168 for Answer. § 4. Substraction. CASE 1. When a simplo Fraction is to be deducted from a simplo. RULE. Reduce the Fractions to a common Denominator, as before; then take the Numerator of the Subtrahend from the other, and place the remainder over the common Denominator, and you have the Difference sought. EXAMPLE. From ¾ take 5/ 12; See the Work. 36 The first Numerator. 48 The common Denominator. 20 The 2d Numerator.   16 Difference. Answer 16/ 48 or l⅓. CASE 2. When a Compound Fraction is to be deducted from a simplo. RULE. Reduce the Compound Fraction to a simplo, by the fifth Case of Section 2. then work as in the last Section. EXAMPLE. From 13/ 14 take ⅔ of 8/ 9. The Compound Fraction in a simplo is 16/ 27. 13 16 27 27 14 14 91 64 108 26 16 27 351 The 1st. number. 224 The 2d. number. 378 The come. Den. 224 Deduct.     127 Remains. So the Answer is 127/ 378.   CASE 3. When a simplo Fraction is to be deducted from a whole Number. RULE. Deduct the Numerator from the Denominator, and place the remainder over the Denominator; then deduct 1 from the Integer, and place the remainder before the remaining Fraction, and you have the Answer. EXAMPLE. From 12 take ⅝. The Answer is 11●/ ●. Or thus: According to the Rules foregoing, place 1 under the 12, and so proceed as in the first Case of this Section; but the first way is the briefer. Note that the 1 borrowed from the 12( in the first Method) is 8/ ●, so that if from 8/ 8 you take ⅝ there rests ⅜. § 5. Multiplication. CASE 1. When you are to multiply a simplo Fraction by a simplo. RULE. Multiply all the Numerators one in another, for the Numerator of the Product, and likewise the Denominators for the Denominator of the Product. EXAMPLE. Multiply ½ by ⅔: Answer 2/ 6 or l⅓. Note that( contrary to whole Numbers) the Product is less than either of the Factors, and is the same thing as though you divided in whole Numbers; for in the last Example the Product of ½ by ⅔ is but ⅔, the same as though you took the half of ⅔, for half of ⅔ is l⅓. CASE 2. When you multiply a whole Number by a Fraction. RULE. Multiply the Integer by the Numerator of the Fraction, and the Product placed over the Denominator, is the Answer. EXAMPLE. Multiply 126 by ●/ 7. Answer 378/ ●●7 or 54, by Case 2. of§ 2. CASE 3. When you multiply a simplo by a Compound Fraction. RULE. Reduce the Compound Fraction into a simplo, and work as in Case 1. of this Section. EXAMPLE. Multiply 18/ 19 by 5/ 9 of 3/ 7. Answer 270/ 1197, or 30/ 133. § 6. Division. CASE 1. When you would divide a simplo Fraction by a simplo. RULE. Having placed the Dividend and Divisor, as in whole Numbers, multiply the Numerator of the Divisor, in the Denominator of the Dividend, for the Denominator of the Quotient: And the Denominator of the Divisor in the Numerator of the Dividend, for the Numerator of the Quotient. EXAMPLE. Divide 11/ 12 by l⅓. ½) 11/ 12( 2●/ 12 or 1 ⅚. CASE 2. When you divide a whole Number by a Fraction. RULE. Place a Unit under the whole Number, and work as in the last. EXAMPLE. Divide 54 by 3/ 7. See the Operation. 3/ 7) 54/ ●●( 378/ 9, Answer, or 126, which proves the second Case of the last Section. CASE 3. When you divide a simplo Fraction by a Compound. RULE. Reduce the Compound to a simplo Fraction, and work as in Case the first. EXAMPLE. Divide 30/ 133 by 5/ 9 of 3/ 7. The Compound Fraction is 15/ 63. 15/ 63) 30/ 133( 1890/ 1995 or 18/ 19, which proves Case 3. of Sect. 5. Having in the two last Sections shewed the way of multiplying and dividing Fractions, it would be needless to say any thing of the Golden-Rule, since there is nothing in it but what has been already done; observing onely to multiply and divide by the fractional Way instead of whole Numbers. § 7. Reduction of Decimal Fractions. A Decimal Fraction is onely different from a Vulgar in this: That the Denominator of a Decimal Fraction is either 10, or some power of 10, viz. 100, 1000, 10000, &c. so that the Denominator is easily known without expressing it; for in a Decimal Fraction there is a Point or Prick toward the Left-hand of the Numerator, which Point always possesses the like place, as the first Figure toward the Left-hand would, if it were to be wrote down: Thus 1/ 10 is .1 the Prick being in the Ten's place, and therefore denotes the Denominator to be 10; 12/ 100 is .12, 125/ 1000 is .125; 1964/ 10000 is .1964; 17/ 1000 is .017; 24/ 10000 is .0024, &c. The manner to reduce a Vulgar Fraction to a Decimal, is by this Proportion. RULE. As the Denominator of the Vulgar Fraction given, Is in proportion to its Numerator: So is 1000, To the Numerator of the Decimal, whose Denominator is 1000. Or so is 10000 to the Decimal, whose Denominator is 10000, &c. EXAMPLE. What is ⅛ in a Decimal Fraction? See the Operation. 8 ∶ 1 ∷ 1000       1     8) 10̣0̣0̣ ( .125 Answer.     20       40       0   But because it sometimes happens that a cipher or more is to possess the 1, 2, &c. Places of the Decimal toward the Left-hand; therefore take this General RULE. As many ciphers as you have in the third Number of the 3 in proportion as above, so many Places must you prick off in the Quotient toward the Right-hand. EXAMPLE 2. How is 9 d. expressed in the Decimal of a Pound Sterling? RULE. Consider that in a Pound are 240 Pence; therefore 9 d. is 9/ 240 l. in a Vulgar Fraction by the seventh Case of Section the Second foregoing, for 9 d. is 9/ 12 of 1/ 20 of a Pound. Then say as in the last Example. 240 ∶ 9 ∷ 10000       9 l.   240) 90̣0̣0̣0 ( .0375 Answ.     .1800       1200       0 Remains In this Example, because I had 4 ciphers in the third Number: therefore I must prick 4 places off toward the Right-hand the Quotient for Decimals; but because the said Quotient did but consist of 3 places: Therefore I supply the fourth to the Left-hand with a cipher. Note that the greater your third Number is, the nearer do you bring your Decimal to Truth, when any thing happens to remain, as in the Examples following; but in most Cases where the Decimal is not to be multiplied by a great Number, it is sufficient that the fourth Number be 1000. But when you reduce ¼ or ½ or ¾ to Decimals, or any Number of shillings to the Decimal of a Pound, it is sufficient in these Cases if your third Number be 100. EXAMPLE 3. How is 3 Farthings wrote in the Decimal Fraction of a Pound Sterling? Work as you see in the Qrs in a lb. Qrs     960 ∶ 3 ∷ 100000       3 l.   960) 3000̣0̣ ( .00302     2000       80 Remains margin by the Rules given in the last Example, and you will find the Answer to be .00302 or 302 Hundred Thousand Parts of a Pound. EXAMPLE 4. How is 12 Pounds expressed in the Decimal of 112, or One Hundred? The Vulgar Fraction by the last Examples is 1●/ 112 Hundred; therefore the Decimal is .1071, as followeth. 112 ∶ 12 ∷ 10000       12     112) 120̣0̣0̣0̣ ( .1071     800       160       48 Remains, which is inconsiderable being less than 1/ 10000 of a Unit. EXAMPLE 5. How is 13 Shillings in the Decimal of a Pound? In a Vulgar Fraction 13 s. is 13/ 20 l. and in a Decimal .65 l. 20 ∶ 13 ∷ 100       13     20) 130̣0̣ ( .65 Answer.     100       0 Rem. EXAMPLE 6. How is 14 s. 6 d. in the Decimal of a Pound? In 14 s. 6 d. are 174 d. and the Decimal( by the second Example) is .725 l. 240 ∶ 174 ∷ 1000 174 240) 1740̣0̣0̣( .725 l. Answer. 600 1200 0 Remains Note that you may by the Rule following, writ down any Number of Shillings in the Decimal of a Pound, without any Proportion. RULE. If your Shillings are an even Number, half of them is the Decimal of a Pound; but if they are odd put a cipher to the Right-hand, and then the half is the Decimal of a Pound. Thus 14s. is .7l. 16s. is .8 l. &c. Likewise 13 s. or 130 is .65l. 15 s. or 150 is .75l, &c. You may likewise writ down any Number of Pence or Farthings in the Decimal of a Pound, without working by the foregoing Rules For if you reduce the given Pence into Farthings, and place a cipher to the Left hand, you have the Decimal of a Pound required; but if the said Farthings exceed 14: you may add one( for reason given in the next Case) and another for each 39 Farthings. Thus 3 d. is .012 l. 9d. is .037 l. 11 d. is .046l. CASE 2. When it is required to find the Value of any Decimal. RULE. Multiply the Decimal given, by such a Number of Units of the next inferior Denomination as make a Unit of that your Decimal is of, and prick from the Right-hand of the Product so many places as your Decimal consisteth of: So those towards the Left-hand the said Point or Prick are Integers, and those to the Right-hand it, are parts of a Unit of those Integers. EXAMPLE 1. What is the Value of .1071 of a Hundred? See the Operation. .1071 Hundred Multiply 4 Quarters in a Hundred c .4284 Quarters of a Hundred Multiply 28 Pound 34272   8568   l. 11 .9952 Parts of a Pound.   Note that if you suppose the Denominator of your Decimal to be 10000, you will find this way of finding the value of a Decimal Fraction to differ nothing from that of Vulgar: Case 6. Sect. 2. of this Chapter. In the last Example you see that the Value of .1071 Hundred is 11 Pound, and the Parts being another Pound wanting less than a Hundred part of a Unit, you may call the Value 12 Pound; which proves the Work in the fourth Case of the last Section: And Note that as often as the Decimal( as in the Example last preceding) is above .75, in the lowest Denomination: you may call it a Unit. EXAMPLE 2. What is the Value of .747 of a pound Troy? See the Work.   .747 Parts of a Pound Multiply   12 Ounces in a Pound   1494       747     Ounces 8.964 Parts of an Ounce Multiply   20 Penny-weight Penny-weight 19.280 Parts of a Penny-weight Multiply   24 Grains   112       56     Grains 6.720 Parts of a Grain.   So that by the Operation you may perceive that the Value of .747 l. is 8 ℥. 19 d-w. 6 Grains, and about ¾ of a Grain. EXAMPLE 3. What is the Value of .9184 of a Pound Sterling? Answer, 18 s. 4 d. 1 q.   .9184 Parts of a Pound Multiply   20 Shillings in a Pound Shillings 18.3680 Parts of a Shilling Multiply   12 Pence in a Shilling   736       368     Pence 4.4160 Parts of a Penny Multiply   4 Farthings Farthings 1.6640 Parts of a Farthing.   EXAMPLE 4. Note that the Value of a Decimal of a Pound, as in the last Example, may be found by inspection, by this RULE. Double the Figure standing next the Point in the Decimal given, and if the next Figure toward the Right-hand the aforesaid Figure is 5 or more, add 1 to the Product: Then what Figure stands in the second place above or under 5 reckon so many Tens of Farthings, and what stands in the third place from the prick is so many Farthings, which as often as they are above 13 make less by 1, or above 39 less by 2. So .347 l. is 6 s. 11 ¼, &c. The Reason of this Rule. That place in a Decimal Fraction next the prick is called Primes, being so many Tenth parts of a Pound: Now 1/ 10 of a Pound being 2 shillings; therefore whatever Figure possesseth that place must be multiplied by 2. The reason why you add 1 to the Product as often as the second Figure from the prick is 5, or more, is because .05 of a Pound is 1 shilling; for if .1l. be 2 shillings; then half .1 which is .05 must be 1 shillings. Lastly, Your reckoning the second and third Places from the prick so many Farthings, supposes 1000 Farthings in a Pound, and there being but 960, that Rule must be something erroneous, but 'tis near enough the Truth for ordinary practise, especially if for the 40 Farthings which the 1000 exceeds the 960, you make this allowance of deducting 1 at every 25; for if 1000 is 40 too much, 500 is 20 too much, 250 is 10 too much, 50 Farthings is 2 Farthings too much, and 25 is 1 Farthing too much: So that your Computation for 13 Farthings is ½ a Farthing too much, and if you deduct a Farthing at all Decimals between 13, and 38 or 39, it may be near enough; for less than a Farthing is never received or paid in English Coin. Thus I hope the Rule is made plain, and by it you will find .750 is 7 Tenths of a Pound, or 14 s. and .050 l. or 50 Farthings made less by 2 for the Reason aforesaid, is 48 Farthings or 1 Shilling more, which makes 15 s. also .194 l. is 1 Tenth of a Pound or 2 Shillings, .050 l. or 1 Shilling more, which makes 3 s. and 44 Farthings( the 9 being 4 above 5) made less by 2, for the Reason aforesaid, is 10½ d. So the Value is 3 s. 10½d. § 8. Addition There is no difference between Addition of Decimals, and whole Numbers of one Denomination; observing onely to place the Decimals Point under Point, as in the Examples. Example 1. Example 2. Example 3. 46 .9765 l. .39462 .987 360 .146 .0013 .3642 41 .007 .99 .853 72 .9 .176 .9761 521 .0295 Total 1 .56192 Total 3 .1803 Total. § 9. Substraction of Decimals. Place the Numbers as in the last, and proceed as in Substraction of one Denomination. Example 1. Example 2. Example 3. From 39 .0049 From 160 .99 From 389 .0 Take 7 .947 Take 94 .8462 Take 0 .346 Rem. 31 .0579 Rem. 66 .1438 Rem. 388 .654 § 10. Multiplication of Decimals. In this Rule you are to place the Factors, and work as in whole Numbers; but after you have found the Product, observe this General RULE. As many Decimal places as you have in both the Factors, so many places must you prick off toward the Right hand of the Product. And if so many places happen not to be contained in the said Product,( as it will happen when you multiply 2 Fractions together that are of little value) you are to make up the Number by ciphers toward the Left-hand the said Product. Example 1. Multiply 3 .467 By 19 .01 3467 31203 3467 Product 65 .90767 Example 2. Multiply 36492 By .032 72984 109476 Product 1167 .744 Example 3. Multiply .13461 By 42 26922 53844 Product 5 .65362 Example 4. Multiply .1264 By .247 8848 5056 2528 Prod. .0312208 Example 5. Multiply .01832 By .007 Product .00012824 § 11. Division. Division is the same with that of whole Numbers, all the difficulty therefore is to know how many Decimal places to prick off toward the Right-hand the Quotient. for which take this RULE. Take notice how many Decimal places you have in the Dividend, and how many in the Divisor; and how many the Difference is: So many places must you prick off to the Right-hand of the Quotient: But if so many places are not in the Quotient, as the said Difference; make up the Number by prefixing ciphers toward the Left-hand. EXAMPLE 1. Divide 12.43210 by 9465. See the Operation. 9.465) . 12432̣1̣0̣ ( 1.31   29671     12760     Remains 3295   Note that in this and most other Examples in Division of Decimals, it will be necessary to place ciphers toward the Right-hand of the Dividend, and that you may know what Number of ciphers to put to the Right-hand of any Dividend, observe this RULE. Consider how many Decimal places you would have in the Quotient( as 3 is sufficient, if it is not afterward to be multiplied by any thing) and also how many Decimal places you have in your Divisor, and so many as 1.47) 3. 46̣0̣0̣0̣ ( 2.353   520     790     550     109 Remains; which being less than 1 Thousandth part of a Unit is not material; so much for Division. The Golden Rule is the same with that in whole Numbers, observing Multiplication and Division of Decimals, as they are already taught. you have in both, make so many Decimal places in the Dividend, by adding ciphers if need require, as in the Example in the margin, where 3.46 is divided by 1.47, and because I would have 3 Decimals in the Quotient, and there are 2 in the Divisor; I must make 5 Decimal places in the Dividend. Sect. j. practise A short way of Casting up all Sorts of Merchandise The TABLE and even, ts l Sterl. g s d l L q   of a Tun 10 0 ½ 10 0 ½ 6 8 l⅓ 5 0 ¼ 5 0 ¼ 4 0 ⅕ 4 0 ⅕ 2 2 ⅛ 3 4 ⅙ 2 0 1/ 10 2 6 ⅛ l     of a Dwr. 2 0 1/ 10 14   ⅛ 1 8 1/ 12 16   1/ 7 ● even pts a shilling d   s l     of ½ a Dwr. 6   ½ 14   ¼ 4   l⅓ 8   1/ 7 3   ¼ 7 ⅛ 2   ⅙ l     of ¼ a Dwr 1 ½   ⅛ 14   ½ 1   1/ 12 7   ¼       4   1/ 7         3 ½   ⅛ Merchants accounts; OR, RULES of practise. BEfore you enter upon these Rules following, it is necessary you should have the foregoing Tables of the Aliquot parts of Money and Weight well fixed in your mind, and likewise the Table following of the 9 Digits by 12, which will enable you to multiply or divide any Number by 12, as though it were but a Digit. 12 Times 1= 12 12 Times 4= 48 12 Times 7= 84 2= 24 5= 60 8= 96 3= 36 6= 72 9= 108 As a necessary Introduction to practise, you are also to learn to divide a Number by any of the 9 Digits or 12, without putting down more Figures than the Number to be divided and the Quotient: For the Rules of practise being of daily use with the Merchants, ought to be performed with all imaginable Brevity, I shall therefore give the following Examples, to inform the Learner how to take ½, l⅓, ¼, &c. of any Number, and then proceed to what I chiefly design in this Section; namely to show how the Value of any Quantity of merchandise may be found with most Dispatch. Admit then you would take half 3164: Say the Two's in 3 is 1( and the 1 over makes the next 1 Eleven) ½ of 3164 is= 1582 l⅓ of 18765 is= 6255 1/ 12 of 46723 is= 3893 7. Rem. 1/ 20 of 47632 is= 2381 12 Rem. Two's in 11 is 5, and the 1 over makes the Six 16) Two's in 16 is 8, Two's in 4 is 2; so that the half of 3164 is 1582. Also by the same Rule l⅓ of 18765 is 6255; 1/ 14 of 46723 is 3893 and 7 remains, and 1/ 12 of 47632 is 2381, if according to the third Case of the fifth Chapter, you cut off the Figure in Unit's place of the Divided, and take ½ the rest; and in these Cases what remains is always of the same Name with the Dividend. CASE 1. When the Price of a Unit or Integer of a Commodity is one Shilling. RULE. Take 1/ 20 of the given Number for the Answer. EXAMPLE. What is 46743 Pound of Cotton-wool worth at 12d . per Pound? See the Operation. 1/ 20 of 4674ˌ3 is 2337 l. 3s. Ans. CASE 2. When the Price of any Commodity is 2 Shillings. RULE. Take 1/ 10 of the given Number, as in the 1976ˌ4 at 2s. 1976l. 8s . Facit. third Example of the third Case of Chapter the Fifth. EXAMPLE. What the Price of 19764 Yards at 2s. Note that what remains is always( as was said before) of the same Denomination with the Dividend, so that in the last Example, 4 remaining is 4 two Shillings, or 8 s. CASE 3. When the Price of the Unit is any other even Number of Shillings under 20 s. Take this RULE. Take ½ the Price of the Integer, and by that multiply the sum given, and the Product is Pounds; only when you multiply the first Figure toward the Right-hand, double the Excess of the Product above Ten or Tensfor Shillings, and carry the said Tens to the Pounds, as in the Examples following. EXAMPLE 1. What the Price of 4323 Yards at 6s. 4323 Yards at 6 s. l. 1296 ∶ 18 s. Answer per Yard? Work as in the margin. EXAMPLE 2. What the Price of 16947 Yards at 8 s 16947 Yards at 8 s. l. 6778 ∶ 16 s. Facit per Yard? In this Example, say, 4 times 7 is 28, twice 8 is 16 s. and carry 2 Pound, 4 times 4 is 16 and 2 is 18, 8 and carry 1, &c. EXAMPLE 3. What the Price of 7943 Yards of 7943 Yards at 18 s. 7148 l. 14 s. Facit. broad Cloath at 18 s. per Yard? Note that from the Rule in this Case of an even Number of Shillings are excepted 10 s. and 2 s. for when the Price of the Unit is 2 s. work as in the second Case, and if the Price of it is 10 s. take half the Integers given, because there are twice 10 s. in a Pound. EXAMPLE 4. What the Price of 369 Ells of Holland 369 Ells at 10 s. 184 l. 10 s. Facit. at 10 s. per Ell? CASE 4. When the given Price of a Unit of any wears or Commodity, is any odd Number of Shillings under 20. RULE. Work for the next even Number of Shillings, that are less than the said odd Number, by the Rules in the last Case; and for the odd Shilling work as in the first Case, and the sum is the Answer in Pounds. EXAMPLE 1. What is the Price of 859 Yards of Muslin 859 Yards at 17 s. Add 687 l. 4 s. at 16 s. Add 42 l. 19. at 1 s. 730 l. 3 Facit. at 17 s. per Yard? See the margin. From this last Rule is excepted 5 s. for if the Price of the Integer is 5 s. take ¼ of the given Number, because 5 s. is ¼ of a Pound. EXAMPLE 2. What is the Price of 3743 lb of Coffee 3743 l. at 5 s. 935 l. 15 s. Facit. at 5 s. per Pound? See the Work in the margin; where observe that the 3 remaining is 3 five Shillings or 15 s. CASE 5. When the price of the Integer is 1 d. or any other Number of pence, which are the Aliquot or even part of a Shilling. RULE. Divide the Number given by the said part, and those shillings into pounds by the first Case. EXAMPLE 1. At 1 d. per Pound what cost 9764 lb? 1/ 12 of 9764 lb at 1 d. is 813ˌ s. 8 d. Rem. Facit 40 l. 13 s. 8 d. EXAMPLE 2. What cost 13147 lb of damaged raisins ⅙ of 13147 lb at 2 d. is 219ˌ1 s. 2 d. Rem. Facit 109 l. 11 s. 2d. at 2 d. per Pound? Ans. 109 l. 11 s. 2 d. Or take 1/ 120 of the given Number by cutting off the cipher. EXAMPLE 3. What cost 87341 l. of Sugar at 3 d. ¼ of 87341 lb at 3 d. is 2183ˌ5 s. 3d. Rem. Facit 1091 l. 15 s. 3 d. per Pound? Ans. 1091 l. 15 s. 3 d. Or take 1/ 80 of the given Number. EXAMPLE 4. What cost 3097 Pound of raisins at l⅓ of 3097 lb at 4 d. is 103ˌ2 s. 4 d. Rem. Facit 51 l. 12 s. 4 d. 4 d. per Pound? Ans. 51 l. 12 s. 4 d. Or take 1/ 60 of the given Number. See the Operation of each in the margin. EXAMPLE 5. What cost 14032 Pound of Sugar at 1/ 40 of 1403ˌ2 lb at 6 d. Maketh 350 l. 16 s. 6 d. per Pound? In this last Example of 6 d. you need onely to take a fourth part of the given Number, except the Units place which you cut off, and you have the Answer 350 l. and the 32 Six-pences that remains are 16 s. CASE 6. When the Price of a Unit or Integer of any Commodity is any Number of pence under 12, that are not an even part of a Shilling; as 5 d. 7 d. 8 d. 9 d. 10 d. or 11 d. you are to work as in this and the following Cases. EXAMPLE. What cost 34071 Pound of figs at 5 d? RULE. Because 5 d. is a sixth part of half a Crown, take ⅙ and then 1/ ● of the Quotient for pounds; as followeth. ⅙ 34071 lb at 5 d. ⅛ 5678 ∶ 15 d. Remains Facit 709 l. 16 s. 3 d. CASE 7. When the price of the Unit of any thing is 7 d. RULE. Take first 1/ 80 of the given Number, because 80 Three-pences make 1 l. then take 1/ 60 of the given Number, and add to the 1/ 80, and the sum is the Answer in pounds, for 60 Groats is 1 l. See the Work following. EXAMPLE. What cost 321 Pound of damaged Cotton at 7 d. per Pound? 1/ ●0ˌ 32spclowvertline; 1 l. at 7 d. Add is 4 l. 3 d. Remains Add and 1/ 60 is 5 l. 7 s. Remains Facit 9 l. 7 s. 3 d. Note that the 1 remaining above the first Quotient is 1 Three-pence, and the 21 remaining above the second Quotient is 21 Four-pence( by the Rules foregoing) or 7 s. so the Answer is 9 l. 7 s. 3d. CASE 8. When the given price of a Unit or Integer is 8 d. RULE. Take 1/ 60 of the given Number, and put it down twice, and the sum is the Answer in pounds. EXAMPLE 1. What cost 3746 Yards of Ribbon at 8 d. per Yard? 1/ 60ˌ of 374ˌ6 Yards at 8 d. Add is 62 l. 8 s. 8 d. Add 62 l. 8 ∶ 8 Facit 124 l. 17 ∶ 4 CASE 9. When the given Price of the Integer is 9 d. RULE. Take 1/ 40 of the given Number, for 6 d. and 1/ 80 of it for 3 d. and the sum is the Answer in pounds. EXAMPLE. What cost 4052 Bushels of Coals at 9 d. per Bushel, the Operation followeth. 1/ 40ˌ of 405ˌ2 at 9 d. is 101 ∶ 6 s. Add 1/ 80ˌ is 50 ∶ 13 Add Facit 151 l. 19 CASE 10. When the given Price of the Unit or Integer is 10 d. RULE. Take 1/ 40 for 6 d. and 1/ 60 for 4 d. of the Number given, and the sum is the Answer in pounds. EXAMPLE. What cost 3179 Pound of hops at 10 d. per Pound? 1/ 40ˌ of 317ˌ9 at 10 d. is 79 l. 9 s. 6 d. Add 1/ 60ˌ is 52 l. 19 ∶ 8 Add Maketh 132 l. 9 ∶ 2 CASE 11. When the given Price of the Integer is 11 d. RULE. From the given Number( supposing it shillings) take 1/ 12 thereof, and the remainder is the Answer in shillings; which bring into pounds by Case the first. EXAMPLE. What cost 347 Pound of Copper at 11 d. per Pound? 1/ 12 of 3470 lb at 11 d. is 289 s.= 2 d. Deduct Rem. 318ˌ0 s. 10 Facit 159 l. 00 ∶ 10 d. CASE 12. When the given Price of a Unit or Integer is Farthings under 4. RULE. Take the Aliquot parts of 1 d. or 1 s. and work for the shillings as before. EXAMPLE 1. What cost 19746 Yards of Tape at 1 Farthing per Yard? In this Example take ¼ for pence, 1/ 12 for shillings, and 1/ 20 for pounds. ¼ of 19746 Yards at 1 Farthing 1/ 12 4936 d. 2 q. 1/ 20 41ˌ1 s. 4 d. Facit 20 l. 11 ∶ 4½ EXAMPLE 2. What cost 47390 Yards of Tape at 2 Farthings per Yard? In this Example take ½ for pence, and proceed as in the last. ½ 47390 Yards at 2 Farthings 1/ 12 23695 d. 197ˌ4 s. 7 d. Facit 98 l. 14 ∶ 7 EXAMPLE 3. What cost 41038 Yards of Ditto at 3 Farthings per Yard? In this Example, take ½ for 3 Half-pences, ⅛ of the 3 Half-pences for Shillings, &c. ½ 41038 Yards at 3 Farthings ⅛ 20519 Three Half-pences 254ˌ4 s. 10½d. Facit 127 l. 4 ∶ 10½ CASE 13. When the Price of the Integer is shillings and pence. RULE. Work for the shillings as is before directed, and also for the pence as before taught, and the sum is the Answer in pounds. But if the pence given be an Aliquot part of the Shillings given, you may take such part of the Quotient for shillings, and the sum of the Quotient is the Answer. Or if the shillings and pence together be an Aliquot part of a pound; take such part, and you have the Answer at the first Operation in pounds. EXAMPLE 1. What cost 1914 Ells of Lockram at 1 s. 8 d. per Ell? 1/ 12 of 1914 Ells at 1 s. 8 d. Facit 159 l. 10 s. EXAMPLE 2. What cost 2789 Ells of Bagg-holland at 3 s. 4 d. per Ell? ⅙ 2789 Ells at 3 s. 4 d. Facit 464 l. 16 s. 8 d. Note that the 5 remaining is 5 Three shillings 4 pences. EXAMPLE 3. What cost 978 Gross of Buttons at 6 s. 8 d. per Gross? l⅓ of 978 at 6 s. 8d. Facit 326l. EXAMPLE 4. What cost 796 Ells of Dowlass at 3 s. 10 d. per Ell? Take ⅙ as in the second Example for 3 s. 4 d. and 1/ 40 for the 6 d. and the sum of the Quotients is the Answer, as followeth. ⅙ of 79ˌ6 Ells at 3 s. 10 d. is 132l. 13s. 4 at 3 s. 4 d. 1/ 40 is 19 l. 18 ∶ 0 at 6 d. Facit 152 l. 11 ∶ 4d. for Answer. EXAMPLE 5. At 17 s. 4 d. per Yard, what cost 394 Yards of broad Cloath? Take for 17 s. as is before taught, and 1/ 60 of the given Number for the 4 d. 394 at 17 s. 4 d. 315 l. 4 s. for 16 s. Add 19 l. 14 for 1 s. Add 1/ 60 is 6 l. 11 ∶ 4d. for 4d Add Maketh 341 l. 9 ∶ 4 EXAMPLE 6. What cost 1504 Ells of cambric at 19 s. 9 d. per Ell? Take for the 19 s. as is taught in shillings per Unit, and for the 9 d. as is directed in pence per Unit. 1504 Ells at 19 s. 9 d. 1353 l. 12 s. at 18 s. Add 75 l. 4 at 1 s. Add 37 l. 12 at 6d. Add 18 l. 16 at 3 d. Add Facit 1485 l. 4 s. Note that in this last Example, after you have done with the shillings; you may take ½ of 75 l. 4 s. for the 6 d. because 6 d. is ½ a shilling; and ½ of 37 l. 12 s. for the 3 d. because 3 d. is ½ of 6 d. which is somewhat more brief. EXAMPLE 7. What cost 1904 Ends of Dimity at 14 s. 10 d. per End? 1904 at 14 s. 10 d. Add 1332 l. 16 s. at 14 s. Add 47 l. 12 at 6 d. Add 31 l. 14 ∶ 8 at 4 d. Facit 1412 l. 2 s. 8 d. EXAMPLE 8. What cost 1865 Yards of Fustian at 2 s. 4 d. per Yard? 1865 at 2 s. 4 d. 186 l. 10 s. at 2 s. Add 31 l. 1 ∶ 8 d. at 4d. Add Facit 217 l. 11 ∶ 8 CASE 14. When the given Price of the Integer is Pence under 12, and Farthings under 4. RULE. Work for the Pence as is before taught, and if the Farthings are an even part of the Pence, that you worked for next before the Farthings, take such part; otherwise work for the Farthings as is taught before at Farthings per Unit. EXAMPLE 1. What cost 3471 Dozen of Buttons at 3 ½d ? 1/ 80ˌ of 3ˌ471 at 3 ½. ⅙ of 43 l. 7 s. 9 d. Add is 7 l. 4 ∶ 7 ½ Add Facit 50 l. 12 ∶ 4 ½ EXAMPLE 2. What cost 9761 Pounds of Sugar at 5 ¼d . per Pound? 1/ 80 of 9761 lb at 5 ¼d. is 122 l. 00 s. 3 d. at 3d. Add 2/ 120 is 81 l. 6 s. 10d. at 2d. Add of which ⅛ is 10 l. 3 s. 4 ¼ at 1 q. Add Facit 213 l. 10 s. 5 ¼d d. EXAMPLE 3. What cost 1794 Pounds of Pepper at 3 ¾d ¾d. per Pound? 1/ 80ˌ of 17ˌ94 at 3 ¾d. is 22 l. 8 s. 6 d. at 3 d. Add ¼ of the last Quote is 5 l. 12 ∶ 1 ½ at 3 q. Add Facit 28 l. 00 ∶ 7 ½ CASE 15. When the Price of the Integer or Unit is Pounds, Shillings, Pence and Farthings. RULE. Multiply the given Number by the Pounds, and to the Product add what the same comes to at Shillings, Pence, and Farthings, as is taught before. EXAMPLE. What cost 276 Hundred, 2 Quarters of Steel at 2 l. 3 s. 8 ½ d per Hundred? For Answer, first multiply the 276 by 2 l. then for 3 s. 4 d. take 1/ ● of 276; for the 4 d. take 1/ 60 of it, and for the ½ Penny take ⅛ of the last Quotient, and for the half Hundred take ½ 2 l. 3 s. 8 ½ d; which is 1 l. 1 s. 10 ¼ d. and the sum of these is the Answer. See the Operation. C. Q. 276 ∶ 2 ∶ At 2 l. 3 s. 8 ½ d. Add 552 l. at 2 l. Add 46 l. at 3 s. 4 d. Add 4 l. 12 ∶ at 4 d. Add 0 l. 11 ∶ 6 d at 2 qrs. which is 1/ ● of 4 l. 12 s. Add 1 l. 1 ∶ 10 ¼ for the half Hundred. Facit 604 l. 5 ∶ 4 ¼ § 2. Concerning Tare and Trett. Tare is an Allowance in merchandise made to the Buyer for the Weight of the bag, Cask, Chest, Freal, Hogshead, &c. in which any Merchants Goods is put, and is sometimes called Cloffe. After this Allowance is deducted from the Gross-weight,( which is the Weight of the Commodity and Cask, Hogs-head, &c. together) the remainder is the Weight of the Commodity only, and is called Nett-weight; the Allowance for Tare is various, as you shall see by and by. Trett is an Allowance made for the Waste that may be mixed with the Commodity, as Dust, Moats, &c. which is always 4 l. at 104, but though the Merchant alloweth this to the Retailer, yet himself is only allowed Tare in paying Custom; so that he payeth as well for the Dust as the best of the Commodity. Note that in such Commodities wherein Trett is allowed, the remainder, after the Tare is deducted is called subtle, out of which subtle the Allowance for Trett is made and when it is deducted the remainder is called net; bUt if no Allowance is made for Trett, that Weight is called net that remaineth after the Tare is deducted, as was said before: So that the Tare is always deducted from the Gross-weight, and the Trett from the subtle; and to show the best Method for discovering and deducting these Allowances is the Work of this Section, and shall he explained in the Cases following; wherein I shall be as plain as I can, because I do not know any where the same is done already, with that Perspicuity which is necessary. CASE 1. When the Allowance is 14 l. per Cent.( as of Almonds, figs, Steel or Hemp) how to compute the Nett-weight. This Case, as also the rest may be resolved several ways, which after I have given you an Example of, I shall pitch upon that which in my opinion is the briefest. EXAMPE 1. What is the Nett-weight of 9 C. 2 Qrs. 7 lb. Gross, Tare at 14 l. per Cent. to be deducted? The First Way. RULE. Reduce the given Weight into Pounds, as in the third Example of Case 2.§ 2. Chap. 6. as followeth: Then say, as 112 lb to 14 its Tare, so is the Pounds given to the Answer in Tare, which deduct from the Pounds Gross, and the remainder is Pounds net. C. Qrs. lb 9 ∶ 2 ∶ 7 9     93     96     1071 lb Gross Gross Tare Gross 112 ∶ 14 ∷ 1071 14 4284 1071 112) 149̣9̣4̣( 133 lb Tare 379 434 98 Remains 1071 lb Gross Deduct 133 lb Tare Answer 938 lb net A Second Way of working the last Question. Reduce the 2 Quarters 7 Pound into the Decimal of Hundred, as is taught in Reduction of Decimals; then deduct 14 the Tare from 112, and the remainder is 98: So must you multiply 9.563 C. by 98, and the Product is net pounds required. 9.563 C. Gross Mult. 98 net pounds in 1 C. Mult. 76504 86067 937 174 Pound net as before, wanting only 826 lb. but this last is nearer the Truth. A Third Way of finding the Nett-weight by practise. Because 14 lb is ⅛ part of 112 ∶ take ⅛ of the given Number. ⅛ of 9 C. 2 Qrs. 7 lb Gross is 1 ∶ 0 ∶ 22 lb Tare deduct Remains 8 ∶ 1 ∶ 13 net, or 937 lb Note that since in the first Method, 98 remained, which wanted but 14/ 112 Pound of a Unit, and consequently the Quotient wanted but so much of 134 Pound Tare; and if the Tare is 134 Pound, the net is but 937, as in these two last Examples. There is a fourth Way of computing the Nett-weight, but 'tis neither so true nor brief as those foregoing. The Method is to reduce the Hundreds, &c. into Pounds, and to take 140 for every 1000, but at that rate the Tare in this last Example is 150, which deducted from the Gross-pounds 1071, the net is but 921 lb, which is 16 pounds too little, and what is lost by this Means let those concerned judge. The Method that I shall practise in the following Cases shall be that in the third Example foregoing, it being short, and most Merchant-like; but because some Questions may be performed sooner by the second Method, with the help of a Decimal Table: I shall therefore likewise incert such a Table, and show its Use by several Examples, after I have given Rules for deducting Tare, according to the Method of Rules of practise foregoing by taking the Aliquot parts of a Hundred-weight, which is the best, when you have not the Table at hand. CASE 2. When the Allowance for Tare is 4 l. per Cent. as for Cotton, Wool, hops, Feathers, Lambs-wool, or Polish. RULE. Take 1/ 7 of ¼ of the Gross-weight, and you have the Tare, which deduct from the Gross, and the remainder is the Netr required. EXAMPLE. What is the Nett-weight of four gabs of Cotton, Wool, whose Number and Weight is as followeth. No. C. Q. lb 31 1 ∶ 3 ∶ 19 35 2 ∶ 2 ∶ 07 36 3 ∶ 0 ∶ 14 40 2 ∶ 1 ∶ 12 Total Gross 9 ∶ 3 ∶ 24 at 4 l. per Cent.   2 ∶ 1 ∶ 27 is ¼, of which take 1/ 7. 1/ 7 is 0 ∶ 1 ∶ 11 The Tare, deduct Remains 9 ∶ 2 ∶ 13 net Note that what Hundreds remains in dividing, must be reduced into Quarters of Hundreds, and what Quarters remains must be reduced into Pounds, and then divided; so in taking ¼ of 9 Hundred, 1 Hundred remains or 4 Quarters, which added to the 3, is 7 Quarters, ¼ of which is 1 Quarter, and 3 remains, or 84 Pound, and the 24 is 108 Pound, ¼ of which is 27. CASE 3. When the Tare to be allowed is 6 l. per Cent. RULE. Take ¼ of the given Number, and 1/ 7 of that fourth for 4 Pound Tare; then to the last Quotient add half itself, and the sum is the Tare required. EXAMPLE. What is the Tare to be allowed for 6 Cask of latin or Iron-Wyre, at 6 l. Viz. No. C.         1 2 ∶ 1 ∶ 17     2 3 ∶ 0 ∶ 07     4 2 ∶ 3 ∶ 18       8 ∶ 1 ∶ 14       7 ∶ 3 ∶ 00   Total Gross 16 ∶ 0 ∶ 14 at 6 l. per Cent.     4 ∶ 0 ∶ 03 is ¼     0 ∶ 2 ∶ 08 is 1/ 7 of the fourth Add     0 ∶ 1 ∶ 04 is ½ of the seventh Add sum 0 ∶ 3 ∶ 12 The Tare, deduct     15 ∶ 1 ∶ 2 net remains No. C. Q. lb 7 2 ∶ 1 ∶ 12 8 3 ∶ 1 ∶ 06 12 2 ∶ 0 ∶ 10   7 ∶ 3 ∶ 00 CASE 4. When the Allowance for Tare is 7 l. per Cent. RULE. Take ½ of 1 Eighth of the given Number for Tare. EXAMPLE. What is the Tare of 9 C. 3 Qrs. 16 lb at 7 l. per Cent. See the Work following. C. Qrs. lb 9 ∶ 3 ∶ 16 at 7 l. Tare 1 ∶ 0 ∶ 26 is ⅛ Deduct 0 ∶ 2 ∶ 13 is ½ of ⅛ being the Tare remainder 9 ∶ 1 ∶ 03 net CASE 5. When the Allowance for Tare is 8 l. per Cent. as for Copper and Brimstone. RULE. Take 1/ 7 of a fourth of the given Number, and put it down twice, and the sum is Tare. EXAMPLE. What is the Tare of 3 fats of Copper, viz. No. C. Q. lb 7 3 ∶ 00 ∶ 17 9 2 ∶ 03 ∶ 04 13 3 ∶ 02 ∶ 01 Total Gross 9 ∶ 01 ∶ 22 at 8 lb Tare.   2 ∶ 01 ∶ 12 is ¼.   0 ∶ 01 ∶ 09 is 1/ 7 of the fourth Add   0 ∶ 01 ∶ 09 is Ditto Add   0 ∶ 02 ∶ 18 Tare, deduct   8 ∶ 03 ∶ 04 net CASE 6. When the Allowance for Tare is 10 Pound per Cent. RULE. From ⅛ of the Gross-weight take 1/ 7 of ¼ of the said Weight, and the remainder is Tare. EXAMPLE. What is the Allowance for 5 Casks of copperess at 10 Pound per Cent. Tare? Viz. No. C. Q. lb   4 2 ∶ 1 ∶ 18   5 3 ∶ 0 ∶ 12     5 ∶ 2 ∶ 02     8 ∶ 3 ∶ 08 Total Gross 14 ∶ 1 ∶ 10 at 10 l. per Cent. Tare     1 ∶ 3 ∶ 05 is ⅛ of the Gross for 14 l.     3 ∶ 2 ∶ 09 is ¼ of the Gross for 28 l.     0 ∶ 2· 01 is 1/ 7 of ¼ for 4 lb deduct from 1/ ● Remains 1 ∶ 1 ∶ 04 Tare, deduct from the Gross Remains 13 ∶ 0 ∶ 06 net No. C. Q. lb 7 3 ∶ 1 ∶ 14 9 2 ∶ 2 ∶ 12 10 2 ∶ 3 ∶ 10   8 ∶ 3 ∶ 08 Or this Question may be resolved as well, by taking 2/ 7 from ⅛ of the Gross, and the remainder is Tare. CASE 7. When the Allowance for Tare is 12 l. per Cent. as of alum, Salt-petre and Tallow. RULE. From ⅛ of the Gross-weight take 1/ 7 of the Eighth, and the remainder is Tare. EXAMPLE. What is the Tare of 15 C. 3 Q. 16 lb of Salt-petre at 12 l. per Cent.   C. Q. lb Gross 15 3 ∶ 16 at 12 l. per Cent.   1 ∶ 3 ∶ 26 is ⅛ of the Gross   0 ∶ 1 ∶ 04 is 1/ 7 of the Eighth deduct from the ⅛   1 ∶ 2 ∶ 22 Tare required, deduct from the Gross Remains 14 ∶ 0 ∶ 22 net CASE 8. When the Allowance for Tare is 16 l. per Cent. RULE. To ⅛ of the Gross add 1/ 7 of the Eighth, and the sum is the Tare required. EXAMPLE. What is the Tare of 10 C. 2 Q. 26 lb of Currants at 16 l. per Cent.   C. Q. lb Gross 10 ∶ 2 ∶ 26 at 16 l. Tare.   1 ∶ 1 ∶ 10 is ⅛ of the Gross Add   0 ∶ 0 ∶ 21 is 1/ 7 of the Eighth Add sum 1 ∶ 2 ∶ 03 Tare deduct from the Gross   9 ∶ 0 ∶ 23 net Thus have I given you Rules for deducting the usual Tares in most Commodities, where 112 lb is allowed to the Hundred Weight, which Method I refer to the Learner, as the best, being brief and commendable according to the Rules of practise; but if the Commodity, or merchandise be such as is bought and sold by the Pound, and not the Hundred, the Method following is much shorter, provided you use the following Table of the Decimal parts of 112 lb, by which you will work your Quarters of Hundreds and Pounds, as though they were Hundreds, as shall be shewed by several Examples following the Table. A TABLE for the speedy finding the Tare, showing what Decimal Part of One Hundred any Number of Pounds are. Qr. lb. C. Qr. lb C. Qr. lb C. Qr. lb C. 0 1 .0089 1 0 .25 2 0 .5 3 0 .75   2 .0178   1 .2589   1 .5089   1 .7589   3 .0267   2 .2678   2 .5188   2 .7679   4 .0357   3 .2767   3 .5277   3 .7768   5 .0446   4 .2857   4 .5367   4 .7857   6 .0535   5 .2946   5 .5456   5 .7946   7 .0624   6 .3035   6 .5545   6 .8036   8 .0714   7 .3124   7 .5634   7 .8125   9 .0803   8 .3214   8 .5724   8 .8214   10 .0892   9 .3303   9 .5813   9 .8303   11 .0982   10 .3392   10 .5902   10 .8393   12 .1071   11 .3481   11 .5991   11 .8482   13 .1161   12 .3570   12 .6081   12 .8571   14 .1250   13 .3660   13 .6170   13 .8660   15 .1339   14 .3749   14 .6259   14 .8750   16 .1429   15 .3838   15 .6348   15 .8839   17 .1518   16 .3927   16 .6438   16 .8928   18 .1607   17 .4017   17 .6527   17 .9017   19 .1697   18 .4106   18 .6616   18 .9107   20 .1786   19 .4195   19 .6705   19 .9196   21 .1875   20 .4284   20 .6795   20 .9285   22 .1964   21 .4374   21 .6884   21 .9374   23 .2054   22 .4463   22 .6973   22 .9464   24 .2143   23 .4552   23 .7062   23 .9553   25 .2232   24 .4641   24 .7152   24 .9642   26 .2321   25 .4731   25 .7241   25 .9731   27 .2411   26 .4820   26 .7330   26 .9821         27 .4919   27 .7410   27 .991. The Calculation of this Table. This is no more than what is taught in Example 4. of Section 7. of Chapter 8. of this Book. EXAMPLE. Admit I would know what Decimal part of a Hundred, 2 Quarters 27 Pound is. In 2 Quarters 27 Pound are 83 Pound, or 83/ 112 C. which by the Rule in the said seventh Section of Chap. 8. Example 1. is thus reduced to a Decimal. 112 ∶ 83 ∷ 10000 83 C. 112) 83̣0̣0̣0̣0( .7410 Answer 460 120 80 By the Work you may see that the 2 Quarters 27 Pound is .7410 C. and is the Tabular Number, answering 2 Quarters 27 Pound; the Use of the Table in Allowance for Tare is as followeth. The Use of the Table foregoing. EXAMPLE 1. What is the Nett-weight of the 4 gabs of Cotton mentioned in Case 2. of this Section, the Gross-weight of which is 9 Hundred, 3 Quarters 24 Pound? RULE. Take the Decimal of 3 Quarters, 24 Pound out of the Table, which is .9642: Then deduct 4 Pound Tare from 112, and the remainder is 108 net pounds in 112 Gross, as was shewed before: Therefore multiply 9.9642 C. by 108, and the Product is Nett-pounds, and parts of a Pound. 9.9642 C. 108 7.97136 9.9642 1076.1336 net pounds for Answer, which is the same as in the said second Case of this Section, as you may prove by reducing that Nett-weight into Pounds. EXAMPLE 2. Let it be required to find the Nett-weight of the 10 Hundred, 2 Quarters, 26 Pound of Currants mentioned in Case 8. foregoing. In order to perform which I look in the Decimal Table, what part of 1 Hundred, 2 Quarters and 26 Pound is, and find it .733, and having deducted the 16 Pound Tare from 112, the remainder is 96, wherefore I multiply 10.733 by 96, and the Product is net pounds. 10.733 Hundred Multiply 96 Nett-pound in 1 C. 64398   96597   1030.368 Nett-pounds, which is equal to the Pounds contained in 9 Hundred, 00 Quarters, 23 Pound the Nett-weight of the Currants in the said 8th. Case. CASE 9. When the Hundred Weight is 5 Score, how to deduct the Tare at 5 l. per Cent. RULE. Take 1/ 20 of the given Number, and you have the Tare required. EXAMPLE 3. What is the Tare of 5 gabs of Cotton-yarn from Alleppo-weight, 1099 Pound at 5 l. per Cent. lb 1/ 20 of 109ˌ9 Gross is 54 Tare, deduct 1045 net By the various Examples in the Cases foregoing, you may easily know how to make Allowance for Tare at any rate per Cent. but in many Commodities the Allowance for Tare is not reckoned per Cent. but so much of the Gross, thus; CASE 10. When the Tare of Raw-silk from Smyrna or Cyprus is to be deducted; the RULE. Is to allow 16 Pound Tare for 3 Hundred Weight and upward; from 3 Hundred Weight down to 200 Weight 14 Pound Tare, and from 200 Weight downward is allowed 12 Pound Tare. EXAMPLE 1. What is the Tare of 4 Bails of Raw-silk, Weight 1088 Pound ( Averdupoize)? Answer 58 Pound.   lb lb No. 1. Qt. 346, Tare 16 3. 300, Tare 16 4. 284, Tare 14 8. 158, Tare 12 Total Gross 1088 Tare, 58   58   Remains lb 1030 net   EXAMPLE 2. Likewise in Virginia Tobacco, all Hoggsheads under 3 Hundred Weight allow 70 Pound Tare, from 3 Hundred to 4 Hundred 80 Pound, from 4 Hundred to 5 Hundred 90 Pound, and from 5 Hundred Weight upward 100 Pound Tare. So in the 6 Hoggsheads following, Weight 27 Hundred, 1 Quarter, 00 Pounds. The Tare is 4 Hundred, 3 Quarters, 8 Pound.   C. Q. lb   C. Q. lb No. 5 2 ∶ 3 ∶ 4, Tare 0 ∶ 2 ∶ 14 6 3 ∶ 1 ∶ 12, Tare 0 ∶ 2 ∶ 24 8 4 ∶ 2 ∶ 00, Tare 0 ∶ 3 ∶ 06 9 5 ∶ 1 ∶ 12, Tare 0 ∶ 3 ∶ 16 10 5 ∶ 2 ∶ 08, Tare 0 ∶ 3 ∶ 16 12 5 ∶ 2 ∶ 20, Tare 0 ∶ 3 ∶ 16 Total Gross 27 ∶ 1 ∶ 00, Tare 4 ∶ 3 ∶ 08 Total Tare 4 ∶ 3 ∶ 08, Deduct     Resteth net 22 ∶ 0 ∶ 20         It would be needless to give any more Examples of the deducing Tare, since by knowing the usual Tare for any Commodity according to the Custom of any Port: The Learner may by help of the foregoing Rules be able with Speed and Exactness to make any Allowance desired. I shall therefore conclude this Section with one Example of Allowance for Tare and Trett. CASE 11. When Allowance is required for Tare and Trett. RULE. Find what is to be allowed for Tare according to the Rules foregoing, which having deducted the remainder( as was said at the beginning of this Section) is subtle, which reduce into Pounds and divide by 26( because that is ¼ part of 104) and the Quotient is what is to be allowed for Trett, which deduct from the subtle, and the remainder is net. EXAMPLE. What is the Nett-weight of the 4 Puncheons of Pruons following, Allowance being made for 14 Pound at 112 for Tare, and 4 Pound at 104 for Trett? No. C. Q. lb No. C. Q. lb 4 2 ∶ 1 ∶ 17 9. 2 ∶ 3 ∶ 17 5 1 ∶ 3 ∶ 20 10. 2 ∶ 1 ∶ 12   4 ∶ 1 ∶ 09   5 ∶ 1 ∶ 01           4 ∶ 1 ∶ 09       Total 9 ∶ 2 ∶ 10 Gross at 14 Pound Tare per Cent.         ⅛ is 1 ∶ 0 ∶ 22 Tare, deduct     Remains 8 ∶ 1 ∶ 16 subtle           8             84             84               940 Pound subtle, which divide by 26, and you have         36 Pound Trett, deduct         904 Pound net, for Answer. Note that Trett is usually allowed in the Port of London for cinnamon, Cloves, Mace, Tobacco, Cotton, Yarn, and Cotton-wool. § 3. Concerning Bartering. Merchants are said to barter when they exchange one Commodity for another, and there is much more difficulty in the Name than the Rule; for that is no other than the Rule of Proportion which has been taught already, as will appear by the Example following. CASE 1. When two Merchants barter, and each rateth his Goods sold in Barter, as though they were sold for ready Money. RULE. Let one Merchant consider what the Goods he is minded to Barter amounteth to: Then by the Rule of Proportion see how much of the other Merchant's Commodity the said amount will buy, and so much must be given. EXAMPLE. A Merchant has 18 Hundred 2 Quarters of Coffee-berries at 14 Pound, 10 Shillings per Cent. which he is willing to barter with another Merchant for Lime-juice at 20 Pence the Gallon; how much Lime-juice must the second Merchant give the first for his Coffee-berries? The Price of the first Merchant's Coffee is thus by practise. C. Q. l. s. 18 ∶ 2 at 14 ∶ 10 per Cent. 14       72       18       252 Pound at 18 l. 9 l. at 10 s. 7 l. 5 s. for the half Hundr. The Value of the Coffee 268 l. 5 s.     d. Gal. l. s. 20 ∶ 1 ∷ 268 ∶ 5     20       5365 shall.     12     2ˌ0) 6438ˌ0 d.( 3219 gull. Answer. By the Work I find that if 20 Pence buy one Gallon of Lime-juice 268 l. 5 s. or 64380 d. will buy 3219 Gallons; and so much must the second Merchant give the first for his Coffee-berries. CASE 2. When two Merchants Barter, and the one rateth his Goods above the common Price for ready Money, to know how the other Merchant may advance the Price of his Goods in proportion to the first Merchant, and how to Barter without Loss thereb●▪ RULE. Consider what the first Merchants Goods are worth per Integer in ready Money, and how much he advanceth the Price in Barter; then say, if the Price of a Unit of the first Merchant's Commodity advance so much in Barter( mentioning the Difference between his ready Money and Bartering Prices) how much must the Price of a Unit of the second Merchant's Commodity advance above the ready Money rate in Barter; which having found Work according to the Bartering Prices of each; as in the last Example. EXAMPLE. A Merchant hath fifteen Hundred, one Quarter of Alleppo-Gauls, which he valueth at 5 Pound 6 Shillings 8 Pence per Hundred ready Money, but in Barter he will have 5 Pound 10 Shillings per Hundred; another Merchant hath Jambee-pepper at 14 Pence the Pound ready Money; how much Pepper must the second give the first for his Gauls advancing his Price in Barter proportionably? Answer, 1394 66/ 231 Pound of Pepper. See the Operation. l. s. d. 5 ∶ 6 ∶ 8 ∶ 20     106 shall.   12     1280 d.     s. d. d. 3 ∶ 4 ∷ 14 12   40 d. 40 d. 560     4     Q. 128ˌ0) 224ˌ0 Qrs.( 1 96/ 128 or 1¾ Qrs.     96 Here you see that if 5 l. 6 s. 8 d.( or 1280 d) advance 3 s. 4 d. in Barter; then 14 d. must advance 1¾ Farthing. So that now the Gauls being 5 l. 10 s. or 1320d . per Hundred; the Pepper is to be reckoned 14 d. 1¾ q. per Pound. Then find the Value of the Gauls by adding the Pence in 3 s. 4 d. to the foregoing 1280 d. which maketh 1320 d. by which multiply 15 Hundred 1 Quarter( or 15 25, the Decimal of 1 Quarter being .25) and the Product is the Value of the Gauls in Pence, then find the Answer to the Question by the following Proportion, viz. if 14 d. 1¾ qrs. buy 1 Pound of Pepper; how many will 20130 d. buy? d. q. lb d.   14 ∶ 1¾ ∶ 1 ∷ 20130 ∶ 15.25 C. 4     4 Farthings 1320 d. per C. 57¾ qrs.   80520 forth. 3050         4575 Or 231/ 4 qr. By the First Case of Reduction of Vulgar Fractions.   1525   20130.00 d. Then by Division of Vulgar Fractions 231/ 4) 80520/ 1( 322080/ 231 Answer. Which by the second Case of Reduction of Vulgar Fractions is 1394 66/ 231 Pound of Pepper, and so much must be given for the Gauls. § 4. Exchange of Coin. This is also a kind of Barter; though 'tis not called by that Name, and is a Rule by which Merchants know what sum in English Coin will answer any sum of Foreign Coin, paid by their Factor or Correspondent. The English Exchange with all other Nations[ Pence] for Crowns, Ducatts, Pieces of Eight, &c. except with some part of the Netherlands, they Exchange in Pounds Sterling. Because the Exchange of Coin dependeth on the knowledge of the Value of Coin: I shall therefore first show you the Value of English Coin, and then of Foreign. English Gold is reckoned as fine as any Foreign, being 22/ 24 l. fine; i.e. The Pound Troy being divided into 24 Equal parts called characts; 22 of those parts is fine Gold, and 2 C●racts is Silver or Copper alloy, according to the Standard of England. As to the Value of English Gold, the Pound Troy, or 12 ℥, is divided into 44 ½ parts, each part is in Value 20 Shillings, called a Guinea. A TABLE of the Gold Currant in England, with the Value extrinsic and Currant, as also the Weight, take as followeth. Names of Pieces. Weight. extrinsic Value. Currant Value. Crown-Gold. ℥ dw. gr. l. s. d. l. s. d. THE milled 5 l. Piece. 1 ∶ 06 ∶ 22½ 5 ∶ 00 ∶ 0 5 ∶ 10 ∶ 10 The Double Guinea. 0 ∶ 10 ∶ 18¾ 2 ∶ 00 ∶ 0 2 ∶ 04 ∶ 4 The Guinea. 0 ∶ 05 ∶ 09⅖ 1 ∶ 00 ∶ 0 1 ∶ 02 ∶ 2 The ½ Guinea. 0 ∶ 02 ∶ 16 7/ 10 0 ∶ 10 ∶ 0 0 ∶ 11 ∶ 1 The treble broad Piece. 0 ∶ 17 ∶ 06 3 ∶ 00 ∶ 0 3 ∶ 10 ∶ 6 The Jacobus 22 s. Piece. 0 ∶ 06 ∶ 06 1 ∶ 02 ∶ 0 1 ∶ 05 ∶ 6 The Carolus and Jacobus 20 s. 0 ∶ 05 ∶ 18 1 ∶ 00 ∶ 0 1 ∶ 03 ∶ 6 The half Jacobus 22 s. Piece, and the ½ Carolus or Jacobus 20 s. Pieces; as also the Quarters are in Proportion for Weight and Value to the whole. Angel-Gold. The Rose Nobles are of different Weight from 9 d-w. 18 Grains, to 8 d-w. 6 Grains, and worth 4 l. 10 s. per Ounce. Soveraign-Gold. The 20 s. Pieces of Henry VIII. is worth about 3 l. 10 s. per Ounce, of which there are few Currant now.   ℥. d-w. gr. l. s, d. l. s. d. The French pistol. 00 ∶ 04 ∶ 08 00 ∶ 17 ∶ 6 00 ∶ 17 ∶ 06 The Double French pistol, or ½ the pistol is in proportion to the pistol.         ℥ d-w. gr. l. s. d. l. s. d. The Spanish pistol. 00 ∶ 04 ∶ 08 00 ∶ 17 ∶ 6 00 ∶ 17 ∶ 06 And in Proportion the Double, Quadruple, or ½ pistol.             English Silver called Sterling or Esterling( as some say, from the People that first Coined it in the Time of Richard I. who were sent for from the Easterly part of Germany) is 11 ℥. 2 p-w. fine, and 18 p-w. of alloy, which maketh 12 ℥. of Sterling Silver, and is the Standard for it. A Pound or 12 ℥. of Bullion is worth 62 s. But because Silver is not Currant by weight in England, as in other Countries; I shall onely here insert the extrinsic Value, or Value as the same is by Stamp made Currant. Pieces. Value.   s. d. The Penny New and Old 00 ∶ 01 The 2 d. new& old 00 ∶ 02 The 3 d. new& old 00 ∶ 03 The new& old Groat 00 ∶ 04 q. The old 4 Pence ½d. 00 ∶ 04 ∶ 2 The old& new 6 d. 00 ∶ 06 The old Nine-pence 00 ∶ 09 The old& new shall. 1 or 12 q. The old 13 ½d. Piece 1 ∶ 01 ∶ 2 The half 13 ½ d. 0 ∶ 06 ∶ 3 The quarter 13 ½d 0 ∶ 03 ∶ 1 ½ The old& new ½ Crow. 2 ∶ 06 The old& new Crown 5 ∶ 00 There is likewise Currant a Farthing made( now) of Copper, which is ¼ of a Penny. Note that the Gold or Silver which I call[ milled or New] was all Coined since the Restauration of King Charles II. it being before that Time stamped with Hammers. A TABLE of Foreign Coin, in Sterling Silver. Of the Low-Country Coin. Sterling. l. s. d. 1 Stiver is 00 ∶ 00 ∶ 01⅕ 6 Stivers, or 1 s. Flemish 00 ∶ 00 ∶ 07⅕ 33 s. l⅓ Flemish is 01 ∶ 00 ∶ 00 d. 1 Gilder is 00 ∶ 02 ∶ 00 or 24 1 Emden Dollar 00 ∶ 02 ∶ 03⅗ or 27⅗ 1 Zealand or come. Dollar 00 ∶ 03 ∶ 00 or 36 1 Campen Dollar 00 ∶ 02 ∶ 07⅕ or 31⅕ 1 lion Dollar 00 ∶ 04 ∶ 00 or 48 1 Duccatoon 00 ∶ 06 ∶ 03⅗ or 75⅗ 1 Specie Dollar 00 ∶ 05 ∶ 00 or 60 French Coin. Sterling.   s. d. q. 1 Denier is 00 ∶ 00 ∶ 0 12 Deniers, or 1 Soulz 00 ∶ 00 ∶ 3⅗ 20 Soulz or 1 Lievre 01 ∶ 06 ∶ 0 or 18 d. 1 Crown de Furnios is 3 Lievres, or 04 ∶ 06 ∶ 0 or 54     Sterling.     d. Italy. At Leghorn the Lievre is 9 Florance the Crown Currant 63 Naples the Ducatt is 60 Bergonia the Ducatt is 52 Venice the Ducatt de Banco is 52 The Ducat coranto 40 The St. Mark 34     Sterling.     d. Spain. At Cadiz the Ducatt is 66¼ The Piece of Eight is 54 Valentia the Ducatt is 63 A Testoon of Portugall 15 Saragosa the Ducatt 66 Barcelona the Ducatt is 72     Sterling.     d. Germany A Rix Dollar of the Empire 53¾ A Gilder of Norenburg 85 These Tables are called the Par of Exchange, but the Course of Exchange differeth almost every Day from London to these Places, according as Money is plenty, or according to the Time allowed for payment of the Money in Exchange. And as it is necessary for the young Merchant to understand Foreign Coins: So is it also that he be acquainted with Foreign Weights and Measures, for which purpose I have inserted the Tables following: Viz. 1 English Ell is In the Netherlands.   And their Pound-weight is at London, Averdupoize.   Viz. Ells. lb At Antwerp 1.6667 1.041 Amsterdam 1.695 1.111 Bridges 1.64 1.02 In France, Viz.     At Paris 0.95 1.123   Aulns.   lions 1.016 0.934 calais 1.57 0.934 Roven 1.0 1.109 In Italy, Viz. Braces.   At Venice 1.96 0.666 Leghorn 2.0 0.75 Millan 2.3 0.7 In Spain, Viz. Vares.     At Castile 1.38     Granada 1.36       Canes.     Barcelona .713     Valentia 1.21     CASE 1. When it is required to Exchange English for Foreign Coin. EXAMPLE. Admit I have received an account from Cadiz, that my Factor there hath sold wears, for my account for 1470 Pieces of Eight, the Exchange for each being 54½ d. Sterling, what Sterling Money does the said Pieces of Eight amount to? Work thus: pc. 8. d. Ster. pc. 8. l. s. d.     1 ∶ 54.5 ∷ 1470. 333 ∶ 16 ∶ 3 Sterling     54.5               735               588               735                     l. s. d.     80115.0. d. Ans. or 333 ∶ 16 ∶ 3 Or rather thus by practise. s. d. 1470 Pcs. of 8 at 4 ∶ 6 ½ 1/ ● of 294 l. at 4 s.     1/ 12 of 36 l. 15 s. at 6 d. is= 3 l. 1 s. 3 d. at 0 ½ d. 333 l. 16 s. 3 d. sum But for your more ease and dispatch of casting up Bills of this Nature, I have inserted the following Table, whose use comes after the Table. CASE 2. When you would convert Foreign Weight or Measure into English. RULE. Look in the foregoing Table for the Proportion of the Foreign Weight or Measure to the English, and work by the Rule of Proportion. EXAMPLE. In 11465 Aulns of lions, how many Ells English? By the Table I find 1.016 Aulns is 1 Ell English; therefore say, Aulns. Ell. Aulns. 1.016. 1 ∷ 114̣6̣5( 11284.44 Ells English, Answer.   1.016)       1305     289.0     858.0     452.0     456.0     496.0 A TABLE showing how much Sterling Money is contained in any Number of Crowns, Dollars, Ducatts, Pieces of Eight, Flemish-pounds, &c. from 1 to 2000.   48 d. 49 d. 50 d. 51 d. 52 d. 53 d. 54 d. 55 d. 56 d.   57 d. 58 d. 59 d. 60 d. 33 s. ¼ d. ⅜ d. ⅝ d. ⅞ d. 1 000.2 000.204 000.208 000.212 000.217 000.221 000.225 000.229 000.233 1 000.237 000.242 000.246 000.25 000.606 00.001 00.002 00.003 00.004 2 000.4 000.408 000.417 000.425 000.433 000.442 000.45 000.458 000.467 2 000.475 000.483 000.492 000.5 001.212 00.002 00.003 00.005 00.007 3 000.6 000 612 000.612 000.637 000.65 000.667 000.675 000.687 000.7 3 000.712 000.725 000.737 000.75 001.818 00.003 00.005 00.008 00.011 4 000.8 000.816 000.833 000.85 000.867 000.883 000.9 000.917 000.933 4 000.95 000.967 000.983 001.0 002.424 00.004 00.006 00.010 00.014 5 001.0 001.021 001.042 001.062 001.083 001.104 001.125 001.146 001.167 5 001.187 001.208 001.229 001.25 003.03 00.005 00.008 00.013 00.018 6 001.2 001.225 001.25 001.275 001.3 001.325 001.35 001.375 001.4 6 001.425 001.45 001.475 001.5 003.636 00.006 00.009 00.015 00.022 7 001.4 001.429 001.458 001.487 001.517 001.546 001.575 001.604 001.633 7 001.662 001.692 001.721 001.75 004.242 00.007 00.011 00.018 00.025 8 001.6 001.633 001.671 001.7 001.733 001.767 001.8 001.833 001.867 8 001.9 001.933 001.967 002.0 004.848 00.008 00.012 00.021 00.029 9 001.8 001.837 001.875 001.912 001.95 001.937 002.025 002.062 002.1 9 002.137 002.175 002.212 002.25 005.454 00.009 00.014 00.024 00.033 10 002.0 002.042 002.083 002.125 002.167 002.208 002.25 002.292 002.333 10 002.375 002.417 002.458 002.5 006.060 00.010 00.016 00.026 00.035 20 004.0 004.083 004.167 004.25 004.333 004.417 004.5 004.583 004.667 20 004.75 004.833 004.917 005.0 012.121 00.021 00.031 00.052 00.073 30 006.0 006.125 006.25 006.375 006.5 006.612 006.75 006.875 007.0 30 007.125 007.25 007.375 007.5 018.181 00.032 00.047 00.078 00.109 40 008.0 008.167 008.333 008.5 008.667 008.833 009.0 009.167 009.333 40 009.5 009.667 009.833 010.0 024.242 00.042 00.062 00.104 00.146 50 010.0 010.208 010.417 010.625 010.833 011.042 011.25 011.458 011.666 50 011.875 012.083 012.292 012.5 030.303 00.052 00.078 00.130 00.182 60 012.0 012.25 012.5 012.75 013. 0● 013.25 013.5 013.75 014.0 60 014.25 014.5 014.75 015.0 036.364 00.062 00.094 00.156 00.219 70 014.0 014.292 014.583 014.875 015.167 015.458 015.75 016.042 016.333 70 016.625 016.917 017.208 017.5 042.424 00.073 00.109 00.182 00.251 80 016.0 016.333 016.667 017.000 017.333 017.666 018.0 018.333 018.667 80 019.0 019.333 019.667 020.0 048.485 00.083 00.125 00.208 00.292 90 018.0 017.375 018.75 019.125 019.5 019.875 020.25 020.625 021.0 90 021.375 021.75 022.125 022.5 054.545 00.094 00.141 00.234 00.328 100 020.0 020.417 020.833 021.25 021.667 022.083 022.5 022.917 023.333 100 023.75 024.167 024.583 025.0 060.606 00.104 00.156 00.26 00.365 200 040.0 040.834 041.667 042.5 043.333 044.167 045.0 045.833 046.667 200 047.5 048.333 049.167 050.0 121.212 00.208 00.312 00.521 00.729 300 060.0 061.25 062.5 063.75 065.0 066.25 067.5 068.75 070.0 300 071.25 072.5 073.75 075.0 181.818 00.312 00.469 00.781 01.094 400 080.0 081.667 083.333 085.000 086.666 088.333 090.0 191.667 093.333 400 095.0 096.667 098.333 100.0 242.424 00.417 00.625 01.041 01.458 500 100.0 102.083 104.167 106.25 108.333 110.417 112.5 114.583 116.667 500 118.75 120.833 122.917 125.0 303.03 00.521 00.781 01.301 01.823 600 120.0 122.5 125.000 127.5 130.0 132.5 135.0 137.5 140.0 600 142.5 145.0 147.5 150.0 363.636 00.625 00.937 01.561 02.190 700 140.0 142.917 145.833 148.75 151.666 154.583 157.5 160.416 163.333 700 166.25 169.167 172.083 175.0 424.242 00.729 01.093 01.822 02.552 800 160.0 163.333 166.667 170.0 173.333 176.667 180.0 183.333 186.667 800 190.0 193.333 196.667 200.0 484.848 00.833 01.249 02.083 02.917 900 180.0 183.75 187.5 191.25 195.0 198.75 202.5 206.25 210.0 900 213.75 217.5 221.25 225.0 545.455 00.937 01.406 02.344 03.281 1000 200.0 204.167 208.333 212.5 216.666 220.833 225.0 229.167 233.333 1000 237.5 241.667 245.833 250.0 606.060 01.042 01.562 02.604 03.646 2000 400.0 408.334 416.667 425.0 433.333 441.667 450.0 458.333 466.667 2000 457.0 483.333 491.667 500.0 1212.121 02.083 03.125 05.209 07.291 The Construction of the Table foregoing. The first Column toward the Left-hand is any Number of Crowns, Dollars, Ducats, Pieces of Eight, Flemish-pounds, or any other Denomination of Foreign Coin, whose Value is as expressed at the Head of each Column; the 13 Columns next, show the Sterling Money of any sum of Foreign Coin is equal to, at any Rate, from 48 d. to 60 d. Sterling, for each Piece of Foreign Coin. The 5th. Column from the Right-hand, sheweth how many Pounds Sterling are contained in any Number of Flemish-pounds from 1 to 2000 at the Rate of 33 s. Flemish for 20 s. Sterling. The 4 Columns next the Right-hand, show, the Amount of any Piece of Coin at ¼, ⅜, ⅝ or ⅞ of a Penny per Piece, whose Use shall be shewed by and by. The Calculation of the Table. Multiply any of the Numbers at the Head of a Column by any of those in the Column next the Left hand, and the Product is the Tabular Number answering the said two Numbers in Sterling money, except the Column of Flemish-pounds, which is thus calculated: As 33 Shillings Flemish Is in proportion to 1 Pound Sterling, So is 20 Shillings Flemish To .6 06060 l. Sterling, which Number being multiplied by any of the Numbers in the Column next the Left-hand, produceth the respective Number in Sterling Money, answerable to the aforesaid Number in the Left-hand Column, supposing them Flemish-pounds. The Use of the Table in casting up Bills of Exchange. Admit you would know how much Sterling Money is contained in 1000 Dollars, each 60½d. Sterling? Look for 1000 in the Column next the Left-hand, and in a right Line toward the Right-hand under 60d. is 250.0 l. Then look under ¼ d. and you find 1.042, which double( for the ½d.) is 2.084 l. The sum of which is the Answer, which makes Sterling. 252.084 l. Note that if the sum given of Foreign Coin is 1500, 1820, 2500, &c. you may take the Answer out at twice or thrice. § 5. Concerning Interest of Coin, and equating the Time of Payments. Interest is either simplo or Compound. simplo Interest is when the Interest onely of the Principal( or sum put out to Interest) is considered. Compound Interest is the Interest of the Principal, and Interest due upon that Principal put together; as if I forbear in my Friend's Hand 100 l. two Years, and am to receive Interest at 6 l. per Cent. per Annum, my hundred Pound at one years End is 106 l. and at the second years End, I have the Interest of the Principal, which is 12 l. and of the first Years Interest, viz. of 6 l. which is 7 s. 2½d. so my 100 l. being forborn two Years, brings me in 12 l. 7 s. 2 ½d. Interest. The Method of finding the simplo or Compound Interest of any sum followeth. CASE 1. When you would find the simplo Interest of any sum for 1, 2, 3, 4, &c. Years. RULE. Make 100 l. the first Number in the Golden-Rule, the Rate of Interest, or Interest of 100 l. the second Number, and the sum given the third Number, and work as before taught in the Rule of Direct Proportion; and having found the Interest for one Year, multiply it by the Number of Years, and you have the Answer. EXAMPLE. What is the simplo Interest of 500 l. for three Years, at 6 l. per Cent. 100 ∶ 6 ∷ 500     6     3000 One years Interest Multiply     3 Years Multiply     90 Pounds, Answer. EXAMPLE 2. What is the Interest of 74 l. for seven Years simplo Interest, being computed at 3 l. per Cent. per Annum? l. l. l.           100 ∶ 3 ∷ 74               2.92l. 1 Years Interest for 74 l. Multiply     7 Years               l. s. d.       20.44 l. Answer, or 20 ∶ 8 ∶ 9½   CASE 2. When the Interest of any sum is required for any Time less than a Year. RULE. Find the Interest of the sum given for a Year; then say, if 365 Days require the Interest found, what Interest will the Days, for which the Interest is to be computed require? multiply and divide, and the Quotient is the Answer. EXAMPLE. What is the Interest of 750 l. from the Fifth of June, to the First of December following, at 8 l. per Cent. per Annum? By your almanac, or otherwise, you will find between the said Times, 178 Days. And that the Interest of 750 l. for 1 Year is 60 l. therefore say, if 365 Days require 60 l. what will 178 Days require? multiply and divide for the Answer: See the Work. 100 ∶ 8 ∷ 750     8   l. 60.00 The Interest for 365 Days. Days. l. Days.       365 ∶ 60 ∷ 178           60             l. s. d. 365) 10680.000( 29.260 l. or 29 ∶ 5 ∶ 2 ½ Answer. A TABLE of simplo Interest at any usual Rate, Viz. At 3, 5, 6 or 8 l. per Cent. per Annum, from 1 l. to 1000 l. for one Day, and may likewise serve for any greater sum, or Number of Days, or Rate of Interest.   3 l. per C. 5 l. per C. 6 l. per C. 8 l. per C. l. 1 Day. 1 Day. 1 Day. 1 Day. 1 .0000821 .00013698 .00016438 .00021917 2 .0001642 .00027397 .00032876 .00043835 3 .0002463 .00041095 .00049315 .00065753 4 .0003284 .00054794 .00065753 .00087671 5 .0004105 .00068493 .00082191 .00109589 6 .0004926 .00082191 .00098629 .00131506 7 .0005747 .00095890 .00115068 .00153424 8 .0006568 .00109568 .00131506 .00175342 9 .0007389 .00123287 .00147944 .00197260 10 .0008219 .00136986 .00164383 .00219178 20 .0016438 .00273972 .00328766 .00438356 30 .0024657 .00410959 .00493149 .00657534 40 .0032876 .00547942 .00657532 .00876712 50 .0041095 .00685930 .00821915 .01095890 60 .0049314 .00821916 .00986298 .01315068 70 .0057533 .00958902 .01150681 .01334246 80 .0065752 .01095888 .01315064 .01753424 90 .0073971 .01232874 .01479447 .01972602 100 .0082191 .01369863 .01643835 .0219178 200 .0164382 .02739726 .03287670 .0438356 300 .0246573 .04109599 .04931505 .0657534 400 .0328764 .05479462 .06575340 .0876712 500 .0410955 .06849315 .08219175 .1095890 600 .0493146 .08219278 .09863010 .1315068 700 .0575337 .09589141 .11506845 .1534246 800 .0657528 .10959004 .13150680 .1753424 900 .0739719 .12328867 .14794515 .1972602 1000 .082191 .13698630 .16438356 .219178 The Construction and Use of the foregoing Table. The first Column toward the Left-hand, is the sum of which you would know the Interest, the second is the Interest of any of those sums for one Day, at 3 l. per Cent. per Annum; the third is the Interest of any of those sums for one Day, at 5 l per Cent. per Annum, the fourth the Interest for the same Time, at 6 l. per Cent. per Annum, and the fifth at 8 l. per Cent. per Annum, for one Day. The use of which is thus: Suppose I would know( as in the Example of the second Case foregoing) the Interest of 750 l. for 178 Days, at 8 l. per Cent. Look in the Table against 700 l. and you will find under 8 l. per Cent. .1534246, which is the Interest of 700 l. for one Day, at 8 l. per Cent. and against 50 l. under 8 l. per Cent. you'll find .0109589, which is the Interest of 50 l. for one Day, at 8 l. per Cent. the sum of which is .1643835, which multiply by the Number of Days your sum is forborn( which in this Example is 178 Days) and the Product is the Answer; which in the Example aforesaid is 29.260 l. or 29 l. 5 s. 2½d. EXAMPLE 2. Suppos● I have bought Goods to the Value of 1000 l. for which I am to pay at the End of six Months by Contract; but a Week afterward I agree to pay the said Money presently, for which I am to have rebate at 8 l. per Cent. how much Money must I pay? In six Months wanting one Week, are 175 Days; therefore multiply the Number in the Table, against 1000, under 8 l. per Cent. viz. .219178 by 175 Days, and the Product is 38.356 l. which I am to be abated of my 1000 l. and am therefore to pay but 961 l. 12 s. 10½ d. .219178 175 1095890 1534246 219178 38.356150 CASE 3. When you would find the Compound Interest of any sum. RULE. Make 100 l. the first Number in the Rule of Proportion, 100 l. and its Interest for a Year the second Number, and the sum you would know the Interest of the third Number; then multiply and divide, and the Quotient is the Principal given, and Interest required for one Year; which make the third Number in the Rule of Proportion, continuing the first and second Numbers as before, &c. So that for every Year your Money is forborn, you have one Operation in the Rule of Proportion. EXAMPLE. What is the Amount of 550 l. 10 s. for three Years Compound Interest, computed at 6 l. per Cent. per Annum? See the Operation. l. l. l. 100 ∶ 106 ∷ 550.5 ∶     106     33030     5505 100 ∶ 106 ∷ 583.530 The first Years Amount.     106     350118     58353 100 ∶ 106 ∷ 618.5418 The second Years Amount.     106     37112508     6185418     655.654308 The Amount for three Years; or 655 l. 13 s. 1 d. which is the Answer. There is a much briefer way of finding the Compound Interest, which is done for any Number of Years, at one Operation by Artificial Numbers, called Logarithms; but since that kind of arithmetic falls not within the Subject of this Book, which tends chiefly to accomplish the Young Merchant; and since Compound Interest is seldom either taken or given by great Traders, I shall therefore omit the former, and say no more of the latter. CASE 4. When several sums of Money are due at several Times, and the debtor and Creditor agree to make but one Payment of the Whole, it may be done without Loss to either by this RULE. Multiply every sum of Money by the Time it becometh due, and divide the sum of the Products by the total Debt, and the Quotient is the true Time, at which the Money ought all to be paid. EXAMPLE. Admit I have 1200 l. owing me, to be paid at 4 several Payments, Viz. 500 l. at two Months End, 300 l. at six Months, 200 l. at ten Months, and 200 l. at twelve Months; the Question is, at what Time the whole may be paid at one Payment, without wrong on either Side?   l.   Mo. Products.   Divisor 1200 500 x 2= 1000 Dividend 7200   300 x 6= 1800 7200 Month. 200 x 10= 2000   ( 6= Quotient, or Answer. 200 x 12= 2400   By the Work you see the whole Debt ought to be paid at the End of six Months, which is the true Equated Time. Note that ( x) signifieth[ multiplied by] and(=)[ equal to.] § 6. Concerning Gain and Loss in the practise of merchandise. CASE 1. When Goods are bought at any Rate, and you desire to know how to retail the same, so as to gain a certain sum by the Sail. RULE. As the whole Quantity of the Goods bought Is in proportion to the Total of the sum given for the Goods, and the sum proposed to be gained; So is any part of the Commodity, To a fourth Number, for which if you sell the said Part, you will gain the sum desired by Sail of the Whole. EXAMPLE. Admit a Druggist buyeth 158 Ounces of black-Amber-greece for 230 l. I demand how he may fell the same( by the Ounce Troy) to gain 50 l. by the bargain. Say, ℥. l. ℥. l. l. s. d. 158 ∶) 280 ∷ 1 ∶ ( 1.772 Answer, or 1 ∶ 15 ∶ 5 ¼ per Ounce.   28̣0. 0̣0̣0̣             1220             1140             340             24           CASE 2. When you would gain a certain sum per Cent. by the Sail of any Commodity, to know how to sell the same. RULE. Consider what the whole Value of your Goods will gain, at your proposed Rate per Cent. per Annum; then work as in the last Case. EXAMPLE. A Furrier buyeth 2100 lb of New-England beaver for 700 l. 14 s. how may he sell the same per Pound, to gain at the Rate of 20 l. per Cent. per Annum? The Operation is thus performed: CASE 3. When Goods are bought at a certain Price, and afterwards sustain Damage, and must therefore be sold at an under Rate, to know how to sell the same to lose a certain sum. RULE. First, find the Value of the Goods at the Rate you gave for them, from which deduct what you are willing to lose, and work the remainder in Proportion, as in the two last Cases. EXAMPLE. An Oylman buyeth of a Merchant 2100 lb of Westphalia Ham, for which he gives 9 d. per Pound; but having sustained Damage, he is willing to lose 8 l. 15 s. by the Sail, at what Price must he sell the same to lose just that sum? The Value of the Hams by practise is 78 l. 15 s. 1/ 40 of 21ˌ00 at 9 d. is= 52 l. 10 s. 1/ 80 is= 26 l. 5 Facit 78 l. 15 s. By the Work I find he must sell for 8 d. per lb The last Question is more briefly resolved thus: If 2100 l. lose 8.75 l. what will 1 lb lose? 21ˌ00) 8. 7ˌ50 ( .004 l. or 1 d. per Pound loss.     So must he sell it for 8 d per lb as was taught before.   3 CASE 4. When Goods are bought at one Price, and sold for a greater to be paid at Time; to know what you gain by 100 l. in a Year at that Rate. RULE. Say by the Double Rule of Direct Proportion foregoing. As the Price your Goods( or any part) cost you, is to the gain by such Goods, or part in the Time you trust the Buyer: So is 100 l. in 12 Months to the sum gained thereby for Answer. EXAMPLE. A Merchant bought 17 C. 2 Qrs. of Logwood at 20 s. 6 d. per Hundred ready Money, and sold the same to a Dyer for 25 s. per Hundred, to be paid at the End of 6 Months; the Question is what he gained at that Rate by 100 l. in a Year. See the Work as by the Rule above.   s. 1 Hundred cost the Merchant 20.5 He sold the same for 25 His gain by 1 Hundred= 4.5 Then say, § 7. Concerning Fellowship, or Trading in Company. CASE 1. When two or more Merchants make a common Stock, and by Trade gain or lose a certain sum; to know what each gaineth or loseth in proportion to his share of the common Stock. RULE. Divide the whole Loss or Gain by the whole Stock, and multiply the Quotient by each Man's share of the Stock, and the several Products are the respective Gain or Loss of each particular Merchant. EXAMPLE. Three Merchants make a common Stock of 16000 l. of which 7000 l. was put in by the first Merchant, 5000 l. by the second; and 4000 l. by the third, and by one Voyage they gain 24000 l. what must each have in proportion to his share? CASE 2. When several Merchants make a common Stock for a certain Time, and at the End thereof make a Dividend, to find each Man's share of the Gain or Loss according to his stock and time. RULE. Multiply each Man's share in the Common-stock, by the Time it continued therein, and proceed with the Products, as with the shares in the last Case. EXAMPLE. Two Merchants make a Common stock for 12 Months; the first put in 2500 l. for 8 Months, the second put in 3000 l. for 12 Months, at the End whereof they make a Dividend of 1680 Pounds gain; how much of that Gain shall each have in proportion to his stock and time of Continuance. l.   Mo. Products.     2500 × 8= 20000     3000 × 12= 36000. Gain divide sum= 56ˌ000) 168̣0̣. ˌ000( .030         00 Remains       l. .03 multiplied by 20000 l. produceth 600.00 The 1st. gained. 36000 l.       1080.00 The 2d. gained   The total Gain= 1680 for proof. I might have inserted more of these and such like Examples, but one being sufficient to explain a General Rule, I shall proceed to the next Chapter. I intended here to incert the custom payable by Merchants for Goods inward and outward; but the Act of Tonage and Poundage being about expiring, and because some Alterations may possibly be made in a New one, I have at present omitted the same, referring the Merchant to the Book of Rates. Note that it's usual to allow 2 lb at every 300 for draft in weighing Cinnamon, Cloves, Mace, Nutmegs, Mollosses, Smyrna Gauls, Tobacco, and Cotton-wool. CHAP. X. Treateth of Book-keeping by debtor and Creditor. HAving in the foregoing Chapters given the young Merchant the Grounds and Reasons of arithmetic, and Rules for casting up any thing that may occur in his daily business; I come in this Chapter to show him how to place the same to account; and that I may do it with all the plainness I can, and in as few words, I shall proceed to show, § 1. The Explanation of this kind of Book-keeping, and the Books requisite to be kept, and their Use. The Method of keeping Books by way of debtor and Creditor or( as some call it) after the Italian manner, Is so regular and precise, that at any Time, the Merchant can be resolved what he gaineth or loseth by every particular Person he dealeth with, or merchandise he dealeth in, and consequently what he is worth to a Farthing. And for your information how these Books are kept, take this General Rule. Any thing whatsoever is received either by the Merchant, or any way for his account by his Servants, whether the same be Money of wears: I say the thing so received for, or upon his account, is in the Ledger( which shall be spoken to by and by) made debtor to the Person received from, or thing for which it is received. Also every thing whatsoever is delivered from the Merchant upon any account, whether Money or wears, the thing so delivered by the Merchant, or any way for his Use or account, is in the Ledger made Creditor By the Person to, or thing for which the same is delivered. My meaning in this Rule shall be fully made appear in all the usual Cases of a Merchant's Dealing, after I have shewed the Books necessary for keeping accounts after this Method, which is done as followeth. 1. The Waste-Book is that wherein every thing is entered, whether received or paid, together with the Time when, by the Day of the Month inserted in the Middle of the page., with the Year of our Lord; and is of no farther use but onely to remind the Book-keeper, that such and such business is to be posted into the Journal, the Cash being never summed up in this Book, it being several Men's accounts of Receipts, and Payments placed together promiscuously. 2. The Iournal is a Book into which every thing is posted out of the Waste-book, which is here to be made debtor or Creditor, and ought to be expressed in a better style or Phrase of Speaking more Merchant-like, it being as it were a Preparatory to the Ledger, whereby is shewed what accounts are to be entred debtor in the Ledger to, or Creditor by other accounts. In this Book the Day of the Month is also placed in the Middle of the page., which is never summed up; unless it contain onely one Man's accounts, for the reason aforesaid. 3. The Ledger is the chief-Book of accounts, and that in which all accounts meet, and are placed debtor on the Left-hand page., and Creditor on the Right; so that the folios on the Right and Left-hand in this Book are numbered alike; because one and the same account is placed on both sides. In this Book the Day of the Month is placed in a narrow Column toward the Left-hand of the page., and the Name of the Month to the Left-hand the Day. At the Head of each Folio in this Book, is written the Name of the City or place where the Books are kept, with the Year; all which you will see in the Example of these three Books after the several Cases; the Denomination of most of your accounts to be entered in this Book, are thus ranked and explained. First place your account of Stock at the beginning of your Ledger, viz. Make Stock-Debter to what you owe, when you begin to keep your Books, let the Debt be upon what account soever, in these words, on the Left-hand Folio as it lieth before you. Stock debtor. To sundry accounts as per Inventory, so much as the same is; or if you owe onely one sum, say Stock is Dr. as per Inventory to that sum; and first of all having taken an Inventory of all you are worth in Cash, wears, or Debts( as you see in the Inventory following) writ on the Right-hand Folio the sum of what you are worth, as appeareth by the particulars in the Inventory, making Stock Creditor in these words. Per Contra Creditor. By sundry accounts as per Inventory, mentioning the Value of all the Cash, wears and Debts you have. The next thing( on the same Folio) is the account of Cash; where note, that before you enter any thing debtor or Creditor in your Ledger you are to look whether you have any thing of the same Denomination in your Inventory, which if you have you must the first thing in the account, make it Debtor to Stock for so much as is in the Inventory of that account, as suppose you have in ready Cash at the Time of taking your Inventory 2000 l. you must make, first Cash Creditor. To Stock l. 2000 ∶ 00 ∶ 00 And afterward make the same account debtor To all Persons from whom you receive any Money, whether the same is in part or in full for wears sold, &c. but if you sell for ready Money you must make Cash debtor to the wears, And the said Persons of whom you receive, or thing for which you receive Money must be made in their own account Creditor by Cash, according to the General Rule foregoing, and as shall be shown in the Cases following. Next to the account of Cash in your Ledger, you may put what account occurs in practise; as the account of Men, wears, Voyages, &c. If a Person buyeth wears of you, and payeth not ready Money, you are to make such Person debtor to such wears, and the wears Creditor by so much sold such a Person. When you ship off Goods to your Factor to be sold for your account, you are in this Book to keep an account of the Voyage in a place by itself, as you do the rest, making Voyage ta such a place( mentioning the Port or Place your Factor resideth at) consigned to such a Person( mentioning your Factor's Name) debtor To the Goods shipped. To custom, Insurance, and all other Charges of the same, and the contrary accounts Creditor by Voyage. When you have advice that the Goods shipped are sold, then in some one place make Factor at such a place, my account Currant( which is the account running between your Factor and you concerning the Goods sent him) debtor to Voyage; And the Voyage Creditor By the account Currant, &c. In this Book is also kept the account of Profit and Loss, by itself thus: Profit and Loss debtor. To what Money you pay and have nothing for it; as to Rebate of Money paid you before due; To Abatement by Composition, when a Person is insolvent, To household expenses, Servants Wages, &c. And, Per Contra Creditor. By all the Cash you receive, and deliver nothing for the same; as By Money received with an Apprentice; By Rebate for paying a sum before due; By Legacy left you by a Friend, and By the sum you gain by every particular Commodity you deal in, or Person you deal with, By Ships in Company, By Voyages, &c. At the beginning of this Book you are to have an Alphabetical Table of all the Persons Names you deal with, and Commodities you deal in, with the account of Profit and Loss, Voyages, accounts Currant, or in Company, &c. referring to the Folio in the Ledger, where such account standeth. 4. The Cash-Book is that wherein you enter all the Money you receive upon any account, on the Left-hand Folio, making Cash-Debter to the thing you receive it for, &c. as was said before, and on the Right-hand Folio enter all the Cash you pay, Creditor by the Person you pay it to( mentioning whether it is in full or in part) or thing you pay it for, and place the Day when you receive or pay it, as in the Ledger, and when you see convenient; as once in a Month or oftener, sum up your account of Cash received and paid, carrying the sum to the account of Cash in the Ledger, which account, without this Book would swell too bog, provided you should enter the particulars there. 5. It is necessary you should keep a Book to enter all the Cash in, which you expend in House-keeping, and once in a Month transfer the same to the Debter-side in the Ledger, thus: household expenses, debtor, To Cash, so much as you bring from your Book of Houshold-Expences; and Cash-Creditor, by Houshold-Expences, in your Cash-book. In this Book is likewise proper to enter the Charge of Apparel, Rent of your Dwelling-house, Pocket-Expences, Servants Wages, &c. 6. A Book of Charges of merchandise, Wherein you must enter the Charge of Custom, Ware-house Room, Postage of Letters, Porterage, Cartage, Wharfage, &c. and once in about a Month make a sum, and transfer it into Creditor-side of your Cash-book, making a refer to the Folio of the Book of Charges of merchandise. 7. A Book of Factories or Invoyces, Which is an account of Goods shipped or sent by you to your Factor, or received from him, &c. In this Book, enter the Goods sent or shipped to be sold for your account, with the Value and Time when sent on the Left-hand Folio; and as you receive Advice of their Sale, enter the same on the Right-hand Folio; so may you readily see how the account stands in that particular. 8. Besides these Books, the Merchant ought to have a Book wherein to enter a Copy of all Letters he sendeth or receiveth upon account of Trade also. 9. A Pocket-book to take the Minits of what business you do abroad, for the ease of your Memory, and to avoid Error. 10. A small Book wherein to Enter all Bills of Exchange, the Merchant accepteth with the sum and Time when payable, and to whom; or if Foreign Bills, the Foreign Coin, Exchange, and what the same is in Sterling-money; and as you pay the same, writ [ Paid] in the margin against the Bill paid. 11. Lastly, A Book of Receipts, wherein to take all Receipts for Money you pay: Expressing first the Day of the Month, then the sum received, and for what, or whether in full or in part, and for whose use, which must be Signed by the person receiving. Thus have I given you an account of all the Books necessary for a Merchant to keep, especially if he is a great Dealer; also the Nature of the account to be inserted in each Book, and the Use thereof. I shall next proceed to give such particular Directions as will enable the Book-keeper to find proper Debtors and Creditors, for most, if not all the Cases he will meet with in the practise of merchandise. § 2. Sheweth how to Enter in your Ledger proper accounts in domestic Trade. Definition. PRoper accounts in domestic business is, when the same is wholly managed by the Merchant or his domestic Servants; as in the Cases following. Case 1. When Money is received for a Debt. Rule.[ Cash Debtor] To him for whose account it was paid. The paying Man's account, Creditor, by Cash. Case 2. When present Money is received for wears. Rule.[ Cash Debtor] To the wears sold, the sum received. Goods sold Creditor by Cash for the same sum. Case 3. When Money is paid for wears, presently, as soon as bought. Rule.[ wears bought, Debtor] To Cash for what paid. Cash Creditor. By wears bought, in the same sum. Case 4. When I pay Money that was due formerly. Rule.[ The Receiver's account] Debtor, To Cash for what paid. Cash account, Creditor, By the person receiving. Case 5. When Money is taken at Interest. Rule.[ Cash Debtor] To the Lending-man for the Principal I receive. [ Profit and Loss] Debtor to the Lending-man for the Interest coming due to him. The Lending-man Creditor By sundry accounts, referring to the folios of Cash, and Profit, and Loss. Case 6. When Money is lent at Interest. Rule.[ The Borrowing-man] Debtor To sundry accounts,( referring to the folios of Cash, and Profit, and Loss.) Cash-Creditor by the Borrowing-man for the Principal lent. Profit and Loss Creditor By the Borrowing-man for the Interest. Case 7. When Interest-money is paid by me, and the Principal continued. Rule.[ Profit and Loss] Debtor to Cash for the sum paid. Cash Creditor by Profit and Loss for the same sum. Case 8. When Money is received by me for Interest, and the Principal continued. Rule.[ Cash Debtor] To Profit and Loss for the Interest received. Profit and Loss, Creditor, by the paying Man for the same. Case 9. When I receive Money by Assignation. Rule.[ Cash Debtor] To the Assignor for the sum received. The Assignor Creditor By Cash for the same sum. Case 10. When I satisfy a Debt by Assignment of another due to me. Rule.[ The Receiver Debtor] To him on whom the Assignment is charged. He on whom the Assignment is charged Creditor By the Accepter. Case 11. When I pay Money to any, by my Creditor's Assignation. Rule.[ The Assignor] Debtor To Cash for the sum paid( mentioning to whom) Cash Creditor By the Assignor for the same sum, mentioning to whom paid, and by whose Assignation. Case 12. When I receive part of a Debt, and( by Composition) give a Discharge in full. Rule.[ Cash Debtor] To the Payer for the sum received. Profit and Loss, Debtor, To him for the sum I abate by Composition. The paying Man Creditor By sundry accounts, referring to the folios of Cash, and Profit, and Loss. Case 13. When wears are bought upon Time. Rule. The[ wears bought Debtor] To the Seller for the value of them. The Seller Creditor By the wears bought, for the like sum. Case 14. When wears are sold upon Time. Rule. The[ Buyer Debtor] To the wears sold for their value. The wears Creditor By the Buying man for the same sum. Case 15. When wears bought are to be paid for at several Payments. Rule. The[ wears Debtor] To the Seller for their value, mentioning the several days of Payment in the Journal. The Seller Creditor By the wears for the like sum. Case 16. When wears are bought part for Ready-money, and part at time. Rule. The[ wears bought Debtor] To sundry accounts,( referring to the account of Cash, and the Seller's account by their folios.) The Selling-man Creditor, By wears bought so much as is left unpaid; And Cash Creditor By wears for so much pa●d Ready-money. Case 17. When sundry Parcels of wears are bought for Ready-money. Rule. The several and[ respective wears] must be made Debtor To Cash for the value it stands me in; and Cash Creditor by sundry accounts for the Total value, referring to the several folios where the several wears stand Booked Case. 18. When several Parcels of whores are sold for Ready-money. Rule.[ Cash Debtor] To sundry accounts( referring to the folios where the several wears sold are Entred in the Ledger) for their whole value. The respective wears Creditor By Cash. Case 19. When wears are sold part for Ready-money, and part at time. Rule. The[ Buyer's account Debtor] To wears sold for the sum unpaid. [ Cash Dr.] To the wears sold for the sum received in part. wears Creditor By sundry accounts( referring to the accounts of the Buyer and Cash) for the whole sum for which the wears are sold. Case 20. When wears are bought at time, and booked, and afterward Ready-money agreed to be paid upon Rebate. Rule. The[ Seller Debtor] To Cash for the sum paid him( deducting the Re-bate.) Cash Creditor by the Seller for the same sum. The[ Seller Debtor] To Profit and Loss for the Rebate. Profit and Loss Creditor By the Seller for the same sum. Case 21 When wears are sold at time and booked, but Money received presently after, for the same, allowing Rebate. Rule.[ Cash Debtor] To the Buyer for the sum received upon Rebate. The Buyer Creditor By Cash for the same sum. [ Profit and Loss] Debtor to the Buyer for the sum Rebated. The Buyer Creditor by Profit and Loss for the same sum. Case 22. When wears are bought and part paid Ready-money, part at Time, and the rest by Assignation. Rule.[ wears Dr.] To sundry accounts, the whole Value referring to the folios of the Seller's account, Cash, and his whose Bill is Assigned. The Seller Cr. By the wears for so much as is yet unpaid. Cash Cr. By the wears for so much as is paid in Cash. [ Him whose Bill you have Assigned] Cr. By the whores, so much as the sum Assigned is. Case 23. When whores are sold for part ready Money, part hy Assignation, and the rest at Time. Rule.[ Cash Dr.] To wears the sum received in part of the Buyer. [ The Person on whom the Assignment is made] Dr. To whores, so much as is Assigned. [ The Buyer Dr.] To wears for the sum he left unpaid. wears Cr. By sundry accounts referring to the folios of Cash, the Person to pay the Money Assigned; and the Buyer. Case 24. When in Lieu of a Debt you receive Goods, whose Value is more than your Debt, which Surplus is returned in Cash immediately. Rule. Make[ wears bought] Dr. To sundry accounts, referring to the Folio of the Payer's account, and that of Cash, in your Ledger. The Payer's account Cr. By whores, the sum paid him by Agreement. Cash Cr. By wears for the Surplus paid back. Case 25. When in Payment of a Debt you sell Goods to your Creditor, whose Value exceeding his Debt, he returneth you the Over-plus. Rule. The[ Buyer Dr.] To wears for so much as his Debt was. [ Cash Dr.] To wears for the Over-plus returned. wears Cr. By sundry accounts referring to the folios of Cash and the Buyer's account. Case 26. When wears are bought in Barter for other wears. Rule. The[ wears bought Dr.] To the wears sold for the Value of the wears sold. wears sold Cr. By wears bought for the Value of those bought. Case 27. When wears are bought part for wears and part for ready Money. Rule.[ wears bought] Dr. To sundry accounts, referring to the folios of wears sold, and Cash. wears sold Cr. By the wears bought for the Value of those sold. Cash Cr. By wears bought for the sum paid in Money. Case 28. When wears are sold for part wears, and part ready Money. Rule. Make[ wears bought] Dr. To wears sold for what they Cost. Cash Dr. To wears sold for the sum received. wears sold Cr. By sundry accounts referring to the folios of the wears bought, and Cash. Case 29. When you pay Money for part of a Ship. Rule.[ Ship]( naming her) Dr. To Cash for the sum paid, naming the Master, and what part you have bought. Cash Cr. By the Ship, the sum paid, mentioning to whom. § 3. accounts proper in Foreign Trade. definite. PRoper accounts in Foreign Trade is, when the Merchant sendeth Goods beyond the Sea to some, Correspondent to be sold for his account. Case 1. When you ship off wears. Rule. Make[ Voyage to such a place( mentioning the place whither you sand them) consigned to your Factor, or Correspondent( mentioning his Name) Dr.] To the wears shipped for their Value, naming the Ship and Master's Name, whores Cr. By Voyage To, &c. Case 2. When you would enter Charges of Goods shipped off. Rule. Make[ Voyage to the place whither your Ship is bound, Dr.] To Charges of merchandise for the sum paid on—( naming the Commodity.) Charges of merchandise Cr. By the Voyage for the same sum. Case 3. When Money is received upon Insurance. Rule. Make[ Cash Dr.] To Insurance— Reckoning, Expressing what sum you Insure, to whom, and on what account. Insurance account Cr. By Cash, &c. Case 4. If the Goods you Insure are lost. Rule. Make[ Insurance— Reckoning] Dr. To the Person to whom you Insured the sum lost. The Person to whom you Insured Cr. By Insurance— Reckoning. Case 5. When you pay Money for Insurance. Rule. Make[ Insurance— Reckoning Dr.] To Cash for the sum paid( mentioning the sum Insured to you, by whom, and on what account) and Cash Cr. By Insurance account, &c. Case 6. If the Goods that are Insured to you are lost. Rule. Make[ The Person that Insured Dr.] To Insurance— Reckoning the sum Insured, and Insurance— Reckoning Cr. By the Person that Insured to you. Case 7. When you Even the account of Insurance. Rule. If the sum on the Cr, Side exceed that of the Dr. Make[ Insurance— Reckoning Dr.] To Profit and Loss for that Excess. But if the Dr. Side exceed the Cr. Make[ Profit and Loss Dr.] To Insurance— Reckoning for that Excess, and in both Cases per contra Creditor— Case 8. When you Receive Advice from your Factor, that Goods formerly consigned to him are sold— Rule. Make( in some place in your Ledger)[ Factor at—( mentioning the Place he liveth at) my account Currant Dr.] To Voyage to such place, for the known Sail in Sterling-Money, being the net proceed of whores, as by his account on the File sold for so much foreign Coin( mentioning the Exchange) Then make Voyage to such place, consigned to such Person( mentioning your Factor's Name) Cr. By my Factor at such place my account Currant so much Sterling Money as you know by his account, the Foreign Coin of the net proceed Amounts to. Note, That the net proceed is when the Charges of custom, bringing the Goods from on Board into the Warehouse Provision, &c. is deducted from the Value the Goods are sold for by a Factor. Case 9. When whores are bought upon Time, and shipped off before Entry in your Books. Rule. Make[ Voyage to such a place Consigned to such a Person, Dr.] To the Selling-man, naming the Quantity, Price, and other Conditions of buying and shipping off. The Sellers Cr. By Voyage, &c. Case 10. When Abatement is made by my Factor for Defect in Goods he formerly sold. Rule. Make[ Profit and Loss Dr.] To Factor at— my account Currant( mentioning for what, and the sum.) Then make Factor at such a place my account Currant Cr. By such sum, &c. Case 11. When whores are bought for Ready-money, and immediately shipped off before Entry. Rule. Make[ Voyage To— Dr.] To Cash for the Value of the Goods shipped, mentioning the Names of the wears, Quantity and Charges, till on Board, &c. Then Cash Cr. By Voyage, &c. Case 12. When you receive the unhappy News of your Goods being cast away. Rule. Make[ Profit and Loss Dr.] To Voyage To such a place, Consigned to such a Person, &c. Then make Voyage Cr. By Profit and Loss for the same sum, &c. Case 13. When I order my Factor beyond the Sea, to ship off Goods to another Factor in another place. Rule.[ Voyage to such a place Consigned to my receiving Factor] Dr. To my sending Factor( mentioning their Names) my account Currant so much for such a thing. Then make my sending Factor( mentioning his Name) at such a place my account Currant Cr. By Voyage to such a place Consigned to my receiving Factor( Naming his Name and Place) for the same sum. Case 14. When I receive the Content of a Bill here, and thereupon draw the same on my Factor, to pay to the Order of him that paid me. Rule. Make[ Cash Dr.] to my Factor at such a place my account Currant for so much Sterling received of such a Person for my Bill drawn on Ditto Factor, payable by him to such a Person, at such a Time, so much Foreign Coin, which at so much Exchange makes Sterling so much. Then make Factor at such a place, my account Currant Cr. By Cash, &c. Case 15. When I receive whores in Return from my Factor or Correspondent. Rule. Make[ wears received Dr.] To Factor at such a place,( who sent the wears) my account Currant so much as the wears cost, mentioning what they are, &c. Then Factor at such a place, my account Currant, Cr. By wears received for their Value, &c. Case 16. When I deliver a Bill here drawn upon my Factor beyond Sea, and receive not the Content till some time after. Rule. The[ Person to whom I deliver my Bill,] Dr. To Factor at such a place, my account Currant in so much Sterling for my Bill of so much Foreign Coin drawn upon such Factor, payable at such a Time, to such a Person, or Order, the Exchange at so much Sterling, for so much Foreign Coin makes Sterling— Factor at such a place, my account Currant Cr. By the Person to whom I deliver my Bill. Case 17. When I receive Money presently, which is the Content of a Bill drawn on some Person here by my Factor. Rule.[ Cash Dr.] To Factor at—( my account Currant) for so much received of such a Person by Bill of Exchange payable at sight, for the Value paid there( i. e. beyond Sea by my Factor) to such a Person. Then Factor at— my account Currant Cr. By Cash, &c. Case 18. When I receive Advice that my Factor at one place has drawn a Bill on my Factor at another place. Rule. Make[ The Drawing Factor my account Currant] Dr. To the paying Factor, my account Currant for so much Foreign Coin drawn by— payable at such a time to such a Person, so much Foreign Coin, which at such Exchange makes Sterling— Then make the accepting Factor my account Currant Cr. By the drawing Factor, my account Currant, mentioning both their Names; sum, &c. § 3. Factorage accounts in domestic Trade. definite. THese accounts are when a Trade is managed by the Factor, or his Servants for the Employer, whom the Factor serveth in Commission. Case 1. When a Factor receives wears from his Employer. Rule. In some one place in your Ledger, make[ The account of Goods for your Employer,] Dr. to Cash for so much paid custom, fraught, &c. Then make Cash Cr. By account of Goods, &c. so much as paid. Case 2. When wears received in Commission by a Factor, are sold for Ready-money. Rule.[ Cash Dr.] To account of Goods for the Employer, the sum received. Then make account of Goods for the Employer Cr.] By Cash the same sum. Case 3. When Commission wears are sold by the Factor in Barter. Rule.[ Goods bought in Barter] Dr. To account of Goods, for account proper of the Employer for their Value. [ account of Goods for account, &c.] Cr. By wears received the same sum. Case 4. When wears in Commission are sold part for Ready-money and part at time. Rule. The[ Buyer Dr.] To account of Goods for account proper of the Employer, the sum left unpaid, Cash Dr. To account of Goods, &c. for the sum received. [ account of Goods for account proper of the Employer] Cr. By sundry accounts for the Total value of the Goods sold, referring to the folios of the accounts of the Buyer, and Cash. Case 5. When wears are sent to an Employer in Return with Charges in shipping off. Rule. Make[ account of wears for account proper of the Owner]( or your Employer naming his Name) his account Currant] Dr. To the Goods shipped, naming the Value and Goods, with the ships, and Masters Names, &c. Also the same account Dr. To Cash paid for custom, and other Charges. Then make wears shipped Cr. By the Employer, his account Currant for the Value. And Cash Cr. By the same account Currant, for the Charges of shipping off. Note, That if these Goods shipped were bought by Order, and on the account of the Employer with Ready-money, and not entered before in your Ledger. Make[ your Employer( naming his Name) his account Currant Dr.] To Cash for the Value of the Goods, and Charges of shipping off. And Cash Cr. per contra. Case 6. When a Bill of Exchange is drawn on a Factor by his Employer, payable at time. Rule. Make[ Employer at such a place( as before) his account Currant] Dr. To him to whom the Bill is payable for the Content thereof. Then make[ him Cr. To whom the Bill is payable]( naming his Name) By your Employer his account Currant for the same sum. Note that if this Bill had been paid to Order of the Employer by the Factor presently; The Employer's account Currant must be made Dr. To Cash for the sum paid( naming to whom) And Cash Cr. By the Contrary for Ditto sum. And the Entry is the same with this last, when the Factor remits Ready-money to his Employer. § 4. Factorage accounts in Foreign Trade. definite. THese accounts are when a Factor cannot carry on the Business of those whom he serves in Commission, without Assistance of Foreign Correspondence, for whose Returns he is accountable to his Employer. Case 1. When Goods sent to Sea are Insured by me. Rule. Make[ Voyage to such a place, for such a ones account( the Employer) Consigned to such a Factor Dr.] To Cash( If you paid the Insurance-money presently) And Cash Cr. By Voyage, &c. But if the Insurance-money was not to be paid presently: Then[ Voyage To, &c.] Dr. To the Insurer. And the Insurer Cr. By Voyage. Case 2. When Goods are shipped by a Factor by Order of his Employer to his Factor in another Country. Rule. Make[ Voyage to such a place for account of your Employer Consigned to your Factor( naming his Name) Dr.] To[ my Employer his account of wears] for Charges at the Receipt of the Goods. And To Cash for Charges of shipping. Then make[ per contra Cr. the account of wears, And Cash. Case 3. When you receive Advice that the wears are sold, which were formerly sent to your Factor. Rule. Make[ Factor at such a place for account of my Employer] Dr. To Voyage to such place, for Ditto account for the net proceed as by Advice. Then make[ Voyage to the same place for account of my Employer] Cr By Factor at— for account of my Employer. Case 4. When you are to enter your Provision for wears sold on a Foreign account. Rule. Make[ Voyage to such a place( where your Factor resideth) for account of my Employer] Dr. To Profit and Loss, for so much as your Provision( or money for your Employment) amounteth to, as by your Agreement. Then make[ Profit and Loss Cr.] By Voyage to— for account of my Employer for the same sum. Case 5. When you receive Advice that your Factor hath made Abatement for Defects in Goods that he formerly sold. Rule. Make[ Voyage to such a place for account of my Employer] Dr. To Factor at such a place for account of your Employer, so much as abated. Then make[ Factor( at such a place) for account of my Employer( at such a place Cr.] By Voyage( To the place your Factor liveth at) for account of my Employer for the same sum. Note that when you close the account of wears sold by your Factor with his Returns, &c. for account of your Employer, you must make Voyage to your Factor; for account of your Employer Dr[ To your Employer's account Currant] for the Balance thereof. And the Contrary Cr. By Voyage to such a place for account of your Employer, for the same sum. § 5. Company accounts. definite. COmpany accounts is, when a Stock is employed in Common between several Merchants in the Way of Trade, and each Partner is to have a Share of the Gain, or bear a Share in the Loss, in Proportion to his Share in the Stock, as is taught in the Rules of Fellowship in the last Chapter. Case 1. When Goods are bought and paid for by myself, for Company accounts. Rule. Make[ wears in Company between my Partner and Me( naming our several Shares of the Stock) debtor] To Cash for the Value of the Goods, &c. [ Cash Cr.] By wears in Company between Partner and Me for the same sum. Then make[ my Partner( naming his Name) his account Currant] Dr. To Ditto Partner's account by me in Company for his Share of the Stock. And his account by me in Company Cr. By his account Currant for the same sum. Note, that if the Goods were bought upon trust, the Entry is the same; if instead of[ Cash] you make the Goods Dr. To, the[ Seller] and him Cr. by the same. Case 2. When I receive my Partner's Share of Cash for the Goods bought in Company. Rule.[ Cash Dr.] To my Partner his account Currant for the sum he paid me. [ His account Currant] Cr. By Cash for the same sum. Case 3. When you( having the Management of the account in Company) give an Assignment To a Cr. upon your Partner, for his Share of Goods bought in Company. Rule. Make[ the Receiver Dr.] To[ your Partner his account Currant] for the sum in the Assignment. And Partner's account Currant Cr. By the Demander for the same sum— Case 4. When I receive Ready-money for Goods sold in Company. Rule. Make[ Cash Dr.] To wears sold in Company( always naming the wears) between my Partner and Me( naming his Name, and each of our Shares) for such Goods sold such a Person, so much as their Value. And[ wears in Company between such a one, and me Cr. By Cash for the same sum. Then make[ Partner's account by me in Company] Dr. To his account Currant for his Share of the Cash received. And[ Partner's account Currant Cr. By his account by me in Company for the same sum. Note, if these wears had been sold at time, the Entry is the same, if instead of making Cash Dr. To wears in Company; you make the Buyer Dr. To the same wears: And[ wears in Company Cr. By the Buyers, &c. Case 5. When Goods are sold in Company, part for Ready-money and part at time. Rule. Make Cash Dr. To wears in Company, between my Partner and Me, for the Money received in part. And the Buy. Dr To the same Ac. for the Money left unpaid: Then make[ wears in Company between my Partner and Me Cr. By sundry accounts( referring To the folios of Cash) and the Buyer's account for the whole Value of the Goods sold. 2dly. Make[ my Partner's account by me in Company] Dr. To his account Currant, for his Share of the whole Value of the wears sold. And my Partner's account Currant Cr. By his account by me in Company for the same sum. Case 6. When I bring into Company wears of my own, that are entred in my Ledger. Rule. Make[ wears in Company( naming their Names) between my Partner and Me] Dr. To wears( naming their name again) in the sum you bring them into Company for, naming for what Quantity. Then make[ wears( as before, entred in your Ledger) Cr.] By the same wears in Company between my Partner and Me, for the Quantity brought into Company at such a price. 2dly. Make[ Partner his account by me in Company Dr.] To my Partner's account Currant for so much Goods brought into Company by me, of which his share of the price is so much. Then make[ Partner his account Currant Cr.] By his account by me in Company for his said Share— Case 7. When wears bought for Company account and Booked, are shipped off To be sold for the same Company account. Rule. Make[ Voyage( to the place whither the ship is bound, and Factor the wears are consigned to) in Company between my Partner and Me] Dr. To Goods shipped for their Value, To Cash for Charge of Shipping, so much as paid for that. Then make[ wears in Company between my Partner and Me] Cr. By[ Voyage in Company between us] for their Value. And[ Cash Cr.] By Charges of Shipping. 2dly. Make[ my Partner his account Currant Dr.] To his account by me in Company for his Share of the Charge in shipping off. And Ditto[ Partner his account by me in Company] Cr. By his account Currant for the same sum— Case 8. When wears are bought on Company account to be paid for at Time, And are shipped off( and Charges paid) before Entry. Rule. Make[ Voyage( to such a place) in Company between my Partner and Me, Consigned to our Factor] Dr. To the Selling-man for the Value of the wears, and To Cash for the Charges of shipping, &c. Then make[ the Seller Cr.] By Voyage to such a place, in Company between my Partner and Me, Consigned to our Factor, for the Value of the Goods shipped. And [ Cash Cr.] By Voyage in Company between my Partner and Me, Consigned to our Factor at such a place, for the Charge till on Board. 2dly, Make[ Partner his account Currant] Dr. To Ditto Partner's account by me in Company, for his Share of the Value of the wears, and Charges till on Board. And[ his account by me in Company] Cr. By his account Currant for the same Share of the Value and Charges of shipping. Note, That if the wears bought in this Case had been paid for in Ready-money, the Entry would be the same, with this Difference onely; That whereas first, Voyage in Company, &c. is made Dr. To the Seller, and he Cr. By Voyage, &c. you must make[ Voyage in Company, &c.] Dr. To Cash for the Value and Charges, And Cash Cr. By Voyage, &c. for the same sum. Case 9. When I receive Advice that wears for Company account are sold by our Factor. Rule.[ Factor at such a place, for Company account between my Partner( so much of the Stock) and( so much) me, our account Currant Dr.] To Voyge to such a place in Company between my Partner and Me( naming our Shares) Consigned to Ditto Factor for the net proceed as by Advice. And [ Voyage to such a place in Company between my Partner and Me( naming our Shares of the Stock) Consigned to such a Factor] Cr. By Factor at such a place for Company account between my Partner and Me, our account Currant for the said net proceed. Case 10. When I receive Advice that our Factor hath made Abatement for Defect in Goods sold( between my Partner and Me) in Company. Rule. Make[ Voyage to— in Company between my Partner and Me( naming our Shares always after the Name) Consigned to—( Factor) Dr.] To Ditto Factor for Company account between my Partner and Me, for the Abatement for the Defect. Then[ Factor at such a place for Company account between my Partner and Me, our account Currant] Cr. By Voyage to— in Company between my Partner and Me, for the same sum abated. Case 11. When Money is remitted to me by our Factor, for wears sold, for account of Company, and by me received. Rule. Make[ Cash Dr.] To Factor at— for Company account between my Partner and Me, our account Currant, for the Money received by Bill, and Factor at— for Company account between my Partner and Me, our account Currant] Cr. By Cash for the same sum— Then make[ my Partner's account by me in Company Dr.] To Ditto Partner's account Currant, for his Share in the Money received. And [ My Partner's account Currant Cr.] By Partner's account by me in Company for the same sum. Note, That if this Money had been payable by Bill, at single or double Usance, &c. the Entry would differ little, onely instead of making[ Cash Dr. To Factor, &c.] make his account that Accepteth the Bill, Dr. To Factor at— &c. and per contra Cr. Case 12. When I receive wears from our Factor, in return for wears sold, by him for Company account, and pay Charges for fraught, Custom, &c. at the Receipt thereof. Rule. Make[ wears received] Dr. To Factor at— for Company account between my Partner and Me, our account Currant for the Value of the Goods and Charges till on Board, as per Advice— so much And[ Factor at— for Company account between, &c. our account Currant] Cr. By wears received for the same sum.— Then make[ wears received] Dr. To Cash for the sum paid at the Receipt for custom, &c. And Cash Cr. By wears received for the same sum. Then to place the account between my Partner and Me. Make[ My Partner's account by me in Company Dr. To his account Currant for his Share, as per Invoice of the Return. And[ my Partner's account Currant] Cr. By his account by me in Company for the same sum. Case 13. When I receive Advice That my Factor has shipped off, and Consigned wears to our Factor in another Country, for Company account. Rule. Make[ Voyage to— Consigned to our Factor for Company account between my Partner and Me] Dr. To Factor at—( Viz. my Factor that shipped the Goods) my account Currant for their Value and Charges, as per Advice of my Factor. And my Factor( that shipped the Goods) at— my account Currant Cr. By Voyage to—( Viz. the place our Factor resideth at) Consigned to our Factor in Company between my Partner and Me, for the same sum. Then make[ my Partner's account Currant] Dr. To his account by me in Company, for his Share of the Value and Charges. And [ My Partner's account by me in Company] Cr. By his account Currant for the same sum— Case 14. When wears are returned by our Factor, to my Factor in another Country, for wears sold for Company account by our said Factor. Rule.[ Voyage to such a place Consigned to my Factor] Dr. To( our) Factor at— for account of Company between my Partner and Me, our account Currant, for the Value of the Goods shipped. And ( our)[ Factor at— for account of Company, between my Partner and Me our account Currant] Cr. By Voyage( to such a place) Consigned to[ my] Factor for the said Value and Charges. Then make[ my Partner's account by me in Company] Dr. To his account Currant for his Proportion, as per Advice received of the account. His account Currant Cr. By his account by me in Company, for the same sum. Case 15. When my Partner draws a Bill upon me payable at sight. Rule. Make[ Partner( naming his Name) his account Currant Dr. To Cash for the Content of the Bill paid. And Cash Cr. By[ my Partner's account Currant] for the same sum. Case 16. When you close an account in Company, observe this Rule. Make[ wears ( &c.) in Company between my Partner( naming his Share of the Stock, and so much me) Dr. To sundry accounts, for closing the account, viz. [ To Profit and Loss] for my Share of the Gain by Trading. [ To Ditto] for my Provision( or Employment) at so much per Cent. as by Agreement. And [ Profit and Loss] Cr. By the sum, your Provision and Share of the Gain amounts to. Then wears, &c. in Company( as before) Dr. To my Partner's account Currant for his Share of the Gain. And[ His account Currant Cr.] By wears in Company, &c. for the same sum. Company accounts are generally esteemed very difficult: But if a Person has a good Understanding in proper accounts, and Factorage, he will find this very easy, there being little Difference more than this 1. In the Title of an account in Company, To take in his Partner's Name in Company, mentioning his, and your Shares of the Stock, &c. 2. After any thing is bought, sold, shipped off, received, &c. and Booked as in a proper or Factorage account( having respect to the Title of Company account as aforesaid) you must take Care to make your Partner or Partner's account Currant Dr. To or Cr. By his account by you in Company, which you will easily know how to do by the 16 Cases foregoing. § 6. The Method of keeping the Waste-Book, journal, and Ledger. THE Waste-Book of me C. D. of London, Merchant: Containing all my Dealings from the First day of July 1694. In the Name of God. Amen. An Inventory taken July the first, Containing all my Estate in Cash, wears, and Debts, which I have at this Day: And also what Debts are owing by me to others, &c.         li. s. d. My whole Estate this day in Money, wears and Debts is— 3159 li. 10 s.       ( Viz.) li. s. d.       Imprimis. I have in ready Cash 1540 ∶ 00 ∶ 00       Item. I have Drugs, viz.   li. s. d. 340 l. of Scammony at 10 s. per l. 170 ∶ 00 ∶ 0 565 l. Opium at 6 s. per l. 169 ∶ 10 ∶ 0 105 C. galangal at 40 s. per C. 210 ∶ 00 ∶ 0               549 ∶ 10 ∶ 00       Item. I have Raw Silk, viz. 440 l. of tripoli Belladine at 16 s. per li. 352 ∶ 0 ∶ 0 650 l. Legee of Smirna at 12 s. per lb 390 ∶ 0 ∶ 0               742 ∶ 00 ∶ 00       Item. I have at Aleppo, consigned to Gilbert Gainwell my Factor there, these Norwich wears remaining unsold, viz. 18 Serge Denims that cost 6 l. each, 108 ∷ 0 ∷ 0 30 Grograms at 3 l. per piece 90 ∷ 0 ∷ 0 40 Barateens at 3 l. 5 s. each 130 ∷ 0 ∷ 0             88 pieces in all, which cost       328 ∶ 00 ∶ 00               3173 04 00 Item. I am Indebted to several persons, viz.             To William Richardson due the 3d. instant, 150 ∶ 00 ∶ 00       To Richard Nicholson to balance his account in my old Ledger 80 ∶ 00 ∶ 00       To Charles Rolling due the 16th instant, 140 ∶ 00 ∶ 00               170     The Method of the Entries in the Waste-book. July 2. 1694. li. s. d. Sold George Higgs 300 l. of Scammony for ready Money at 20 s. 6 d. per lb 307 10 00 3. Ditto.       Paid William Richardson in full 150     4. Ditto.       Bought of Richard Nicholson the Norwich wears following, viz.   li. s. d. 10 Grograms at 3 l. per Piece 30 ∶ 00 ∶ 00 24 Barateens at 3 l. 6 s. per Piece 79 ∶ 04 ∶ 00       34 Pieces in all, amounting to 109 04 00 Of which I have paid 80 l. ready Money,       And the rest which is 29 l. 4 s. to be paid in 8 days.       5. Ditto.       Received Advice from Gilbert Gainwell, my Factor at Aleppo, that he hath sold to sundry persons for my account 60 Pieces of Norwich wears, the net proceed of which, as by the particulars in his account on the File is 1500 Dollars, the Exchange at 4 s. 6 d. per Dollar, makes Sterling 337 10 00 Lent George Higgs the sum of 500 l. for 3 Months, for which he is to pay me Interest at the rate of 8 l. per Cent. per Annum.       So that the Money lent is 500     And the Interest thereof comes to 10     July 9. 1694.       Sold William Short the following drugs, viz.   li. s. d. 40 l. of Scammony at 21 s. per l. 42 ∶ 00 ∶ 00 350 l. of Opium at 12 s. per l. 210 ∶ 00 ∶ 00       390 l. in all, for 252     of which I have received 160 l. and the rest, which is 92 l. to be paid in 3 months.       10. Ditto.       Received from my Factor Gilbert Gainwell at Aleppo by my Order and on my account 8 Chests of Myrrh, containing 30 C. net, which at 22 Dollars per C. comes to 660 Dollars, the Exchange at 4 s. 6 d. per Dollar makes Sterling 148 10 00 Richard Nicholson hath assigned the 80 l. due to him from me, for the Balance of his Account in my old Ledger to James Silver, which I have paid to Ditto Silver on demand. 80 00 00 12. Ditto.       Gilbert Gainwell Factor at Aleppo, hath remitted to me 600 Dollars, payable here at triple Usance, by Matthew Clessold, for the value delivered there to Mahoat Janezwar the first of April last, the Exchange at 4 s. 8 d. per Dollar, makes Sterling 140 00 00 Which Bill is accepted.       Paid Richard Nicholson in full 29 04 00 13. Ditto.       Sold Alderman Ryley Mercer, the following Norwich wears, viz.   li. s. d. 10 Grograms at 3 l. 10 s. per Piece 35 ∶ 00 ∶ 00 24 Barateens at 4 l. 4 s. per Piece 100 ∶ 16 ∶ 00         135 16 00 For which he hath given me an Assignment on Peter Paygood, to be paid me in 8 days, which I have accepted.       Sold William Short the following Raw Silk for ready Money, viz.   li. s. d. 350 l. of Tripoly-Belladine at 30 s. per lb 525 ∶ 00 ∶ 00 650 l. Legee at 25 s. per lb 812 ∶ 10 ∶ 00         1337 10   ( The Entry of the Inventory in the Journal.) THE JOURNAL of me C. D. of London, Merchant: Containing all my Dealings from the First day of July 1694. In the Name of God. Amen. An Inventory taken the first of July, 1694. of my present Estate, in Money, wears, and Debts, this day owing to me, and what Debts are owing by me, &c.         li. s. d Sundry accounts are Debtor to Stock in the sum of 3159 l. 10 s. for so much Cash, wears and Debts, owing to me this day, viz. li. s. d.       Cash for so much in Chest 1540 ∶ 00 ∶ 00       Drugs, viz.   li. s. d. 340 l. of Scammony at 10. s. per l. 170 ∶ 00 ∶ 0 565 l. Opium at 6 s. per l. 169 ∶ 10 ∶ 0 105 C. galangal at 40 s. per C. 210 ∶ 00 ∶ 0               549 ∶ 10 ∶ 00       Raw Silk for 1090 lb, viz. 440 l. of Tripoly-Belladine at 16 s. per li. 352 ∶ 0 ∶ 0 650 l. Legee of Smirna at 12 s. per lb 390 ∶ 0 ∶ 0               742 ∶ 00 ∶ 00       Voyage to Aleppo, consigned to Gilbert Gainwell my Factor there, for Norwich wears remaining unsold, viz. 18 Serge Denims that cost 6l. each, 108 ∶ 0 ∶ 0 30 Grograms at 3 l. per piece 90 ∶ 0 ∶ 0 40 Barateens at 3 l. 5 s. each 130 ∶ 0 ∶ 0             in all 88 pieces, which amounts to 328 ∶ 00 ∶ 00               3159 10   Stock is Debtor To Sundry accounts 370 l.             Due to Sundry Persons. viz.             To William Richardson due the 3d ●nstant, 150 ∶ 00 ∶ 00       To Richard Nicholson for the Fo●t of his old account 80 ∶ 00 ∶ 00       To Charles Rolling due the 16th instant, 140 ∶ 00 ∶ 00               370     The End of the Inventory. ( The Method of Journal Entries.) July 2. 1694. l. s. d. Cash debtor To drugs for 300 l. of Scammony sold George Higgs for ready Money at 20 s. 6 d. per l. 307 00 00 Ditto 3.       William Richardson Debtor to Cash paid him in full 150 00 00 Ditto 4.       Norwich wears debtor To sundry accounts 109 l. 4 s. for 34 Pieces bought of Richard Nicholson, viz.   li. s. d. 10 Grograms at 3 l. per Piece 30 ∶ 00 ∶ 00 24 Barateens at 3 l. 6 s. each 79 ∶ 04 ∶ 00         109 04 00 To Cash paid Ditto Nicholson in part 80 ∶ 00 ∶ 00 To Ditto Nicholson, to pay him the 12th Instant 29 ∶ 04 ∶ 00       Ditto 5.       Gil. Gainwell at Aleppo my account Currant, Debtor to Voyage to Aleppo, consigned to Ditto Gainwell the sum of 337 l. 10 s. for the net proceed of wears sold, as per his account for 1500 Dollars, the exchange at 4 s. 6 d. Sterling per Dollar, makes English Coin 337 10 00 George Higgs debtor to sundry accounts the sum of 510 l. for 500 l. lent him at Interest for 3 Months, at 8 per cent. per ann. viz.   li. s. d. To Cash for the Principal lent 500 ∶ 00 ∶ 00 To Profit and Loss for the Interest 10 ∶ 00 ∶ 00         510 00 00 Ditto 9.       Sundry accounts Debtor to drugs the sum of 252 l. for 390 lb sold William Short as followeth,   li. s. d. 40 lb of Scammony at 21 s. per lb 42 ∶ 00 ∶ 00 350 of Opium at 12 s. per lb 210 ∶ 00 ∶ 00       ( viz.) 252 00 00 Cash for 160 l. received, in part of Ditto Short, and Ditto Short Debtor for 92 l. he is to pay me in 3 Months.       ( The Method of Journal Entries.) July 10. 1694. l. s. d. drugs Debtor to Gilbert Gainwell at Aleppo my account Currant, 148 l. 10 s. 00 d. for 8 Chests of Myrrh, poise net 30 C. at 22 Dollars per C. makes 660 Dollars, the exchange at 4 s. 6 d. per Dollar is Sterling 148 10 00 Richard Nicholson Debtor to Cash the sum of 80 l. being the Balance of an account due to him, which I have paid James Silver by Assignation of Ditto Nicholson. 80 00 00 12. Ditto.       Matthew Clessold Dr. to Gilbert Gainwell at Aleppo my account Currant 140 l. by Bill remitted to me by Ditto Gainwell payable at triple Usance for the Value delivered there, to Mahoat Janezwar 140 00 00 Richard Nicholson Debtor to Cash paid him in full 29 04 00 13. Ditto.       Peter Paygood Debtor to Norwich wears the sum of 135 l. 16 s. for 34 pieces sold Alderman Ryley, viz.   l. s. d. 10 Grograms at 3 l. 10 s. per piece 35 00 ∶ 00 24 Barateens at 4 l. 4 s. per piece 100 ∶ 16 ∶ 00         135 16 00 For which sum Ditto Paygood hath given me his Bill to pay in 8 days by Assignation of Ditto Ryley       Cash Debtor to Raw Silk 1337 l. 10 s. for 1000 pounds, sold to Simon Strutt for Ready Money, viz.   l. s. d. 350 lb of Tripoly-Belladine at 30 s. per lb 525 ∶ 00 ∶ 00 650 Legee at 25 s. per lb 812 ∶ 10.00         1337 10 00 THE LEDGER. London, Anno Domini, 1694. 1694   Stock debtor. l. s. d. July 1 To sundry accounts as per Inventory 370         To Balance 3927 14 6       4297 14 6 1694   Cash Debtor.       July 1 To Stock 1540       2 To drugs in part, of George Higgs 307       9 To Ditto, received of W. Short in part 160       13 To Raw Silk for 1000 lb 1337 10         3344 10       Norwich wears Debtor.       July 4 To sundry accounts for 34 Pieces 109 04       To Profit and Loss gained by this account 26 12         135 16   1694   Voyage to Aleppo consigned to Gilb. Gainwell Dr.       July 1 To Stock for wears unsold 328         To Profit and Loss gained by this account 150 2 6       478 2 6 1694   Gilb. Gainwell at Aleppo my account Cur. Dr.       July 5 To Voyage to Aleppo consigned to Ditto Gainwell 337 10   1694   William Short Debtor.       July 9 To drugs, due October the 9th. next 92     1694   Per Contra Creditor. l. s. d. July 1 By sundry accounts as per Inventory 3159 10       By Profit and Loss gained by 2 Weeks Trade 1138 4 6       4297 14 6 1694   Per Contra Creditor.       July 3 By W. Richardson, in full 150       4 By Norwich wears, in part 80       5 By Geo. Higgs, lent at Interest 500       10 By Richard Nicholson paid James Silver 80       12 By Ditto Nicholson, in full 29 4       By Balance, remains in Cash 2505 6         3344 10   1694   Per Contra Creditor.       July 13 By Peter Paygood, for 34 Pieces 135 16   1694   Per Contra Creditor.       July 5 By Gilb. Gainwell at Aleppo my account Cur. 337 10       By Balance for wears unsold 140 12 6       478 2 6 1694   Per Contra Creditor.       July 10 By drugs for 8 Chests 148 10     12 By Matthew Clessold 140         By Balance due to me 49           337 10       Per Contra Creditor.           By Balance 92     1694   drugs Debtor. li. s. d. July 1 To Stock 549 10     10 To Gilb. Gainwell at Aleppo, my account Cur. 148 10       To Profit and Loss, gained by this account 284           982     1694   Richard Nicholson Debtor.       July 10 To Cash paid James Silver, by Assignation 80       12 To Cash paid Ditto Nicholson in full 29 4         109 4   1694   Matthew Clessold Debtor.       July 12 To Gilb. Gainwell at Aleppo, my account Cur. 140     1694   Peter Paygood Debtor.       July 13 To Norwich wears 135 16   1694   George Higgs Debtor.       July 5 To sundry accounts for Principal and Interest 510         Profit and Loss Debtor.           To Stock gained by 12 Days Trade 1138 4 6     Charles Rolling Debtor.       July 1 To Balance due to him 140     1694   Per Contra Creditor. li. s. d. July 2 By Cash for 300 lb of Scammony 307       9 By sundry accounts 252         By balance rests unsold, viz. 215 lb of Opium at 6 s. 105 C. of galangal at 40 s. and 30 C. of Myrrh, at 4 l. 19 s. per C. which cost 423           982     1694   Per Contra Creditor.       July 1 By Stock 80       4 By Norwich wears 29 4         109 4       Per Contra Creditor.           By balance due to me 140         Per Contra Creditor.           By balance due to me 135 16       Per Contra Creditor.           By balance 510     1694   Per Contra Creditor.       July 5 By George Higgs for Int. Money due Oct. 5. next 10         By Norwich wears 26 12       By Voyage to Aleppo 150 2 6     By drugs 284         By Raw Silk 667 10         1138 4 6 1694   Per Contra Creditor.       July 1 By Stock 140     1694   William Richardson Debtor. l. s. d. July 3 To Cash paid him in full 150     1694   Raw Silk Debtor.       July 1 To Stock 742         To Profit and Loss gained by this account 667 10         1409 10       balance Debtor.           To Cash rest in Chest 2505 6       To Voyage to Aleppo consigned to G. Gainwell 140 12 6     To G. Gainwell my account Currant 49         To William Short 92         To drugs unsold, viz. 215 lb of Opium at 6 s. per lb 105 C. of galangal at 40 s. and 30 C. of Myrrh at 4 l. 19 s. per C. which in all cost 423         To Matthew Clessold 140         To Peter Paygood by Assignation 135 16       To George Higgs 510         To Raw Silk, viz. 90 lb of Tripoly-Belladine remaining unsold, which cost 72           4067 14 6     Per Contra Creditor.           By Stock 150     1694   Per Contra Creditor.       July 13 By Cash 1337 10       By balance rests unsold 90 lb of tripoli Belladine at 16 s. per lb 72           1409 10       Per Contra Creditor.           By Charles Rolling due to him 140         By Stock 3927 14 6       4067 14 6 Note, That the Transcript of the Debter-side of the foregoing Balance will be an Inventory of what you are worth in Cash, wears, and Debts; and that of the Creditor-side( leaving out Stock) will be what you owe, and must when you begin new Books( or a new account) be entred as an Inventory, as followeth. An Inventory of me C. D. of London, Merchant, containing my whole Estate this fourteenth Day of July 1694. In Cash, wears and Debts: And also what Debts are owing by me to others, &c. ( Viz.) l. s. d. Imprimis. I have in ready Cash 2505 ∶ 06 ∶ 00 Item. I have at Aleppo, consigned to G. Gainwell my Factor there, Norwich wears unsold, which cost 140 ∶ 12 ∶ 06 Item. Ditto Gainwell oweth me for Norwich wears sold by him, and their Value not returned to me 49 ∶ 00 ∶ 00 Item. William Short oweth me for Drugs, due Octob. 9. next 92 ∶ 00 ∶ 00 Item. I have Drugs by me, unsold, viz.   l. s. d. 215 lb of Opium at 6s . per lb. 64 ∶ 10 ∶ 0 105 C. of galangal at 40s . per Hundred cost 210 ∶ 00 ∶ 0 30 C. of Myrrh which cost 4 l. 19 s. per Hundred 148 ∶ 10 ∶ 0         423 ∶ 00 ∶ 00 Item. Matthew Clessold oweth me upon Bill due the first of September 140 ∶ 00 ∶ 00 Item. Peter Paygood oweth me by Assignation of Alderman Ryley for Norwich wears 135 ∶ 16 ∶ 00 carried over 3485 ∶ 14 ∶ 06   l. s. d. li. s. d. Brought over 3485 ∶ 14 ∶ 06       Item. George Higgs oweth me upon Bond 510 ∶ 00 ∶ 00       Item. I have Raw Silk unsold, viz.             90 l. of Tripoly-Belladine, which at 16 s. per lb cost 72 ∶ 00 ∶ 00   4067 14 06 Item. I am indebted as followeth,       ( Viz.)       To Charles Rollings due the 16th. Instant 140 00 00 To your Ledger you ought to have an Index or Alphabet thus: C. Cash Fol. 1 Clessold Matthew Fol. 2. D. Drugs Fol. 2 G. Gainwell Gil. my account Currant Fol. 1 H. Higgs George Fol. 2. N. Norwich wears Fol. 1 Nicholson Richard Fol. 2 P. Paygood Peter Fol. 2 Profit and Loss 2 R. Rolling Charles Fol. 2 Richardson Will. 3 Raw Silk 3 S. Stock Fol. 1 Short Wm. 1 U. Voyage to Aleppo, consigned to Gil. Gainwell 2 This Alphabet referreth to the foregoing Ledger, and is always to be affixed to the Beginning thereof, that so any account whether of Men, wears, Cash, Voyages, or any thing the Merchant dealeth in may be found with Ease. And the surnames are always put first, because there are not so many of one surname, as of one Christened, and consequently are the easier found by the surname. Note, That if you had kept a Ca-shbook for the account of Cash foregoing, The three lines on the Debtor-side, and the five on the Creditor-side, would have been contained in two lines: Thus. Cash Debtor.   l. s. d. To sundry accounts, Fol. 2, 1, 3. 1804 ∶ 10 ∶ 00 on which 2d. 1st. and 3d. folios stand the accounts of Geo. Higgs, W. Short, and Raw. Silk: and on the Creditor-side thus. Cash Creditor.   l. s. d. By sundry accounts, Fol. 3, 1, 2, 2, 2 839 ∶ 04 ∶ 00 on which Pages stand the accounts of W. Richardson, Norwich wears, Geo. Higgs, and Richard Nicholson, and thus may you bring to account all the Cash received and paid in a Month, &c. or any other accounts may refer to the Folio that it is Debtor to, or Creditor by. § 7. Directions for Posting. When you would Post any accounts( which is the entering any thing in its proper place in the Ledger,) as for instance, in the Norwich wears bought of Richard Nicholson the 4th. of July, By the 16. Case of the 2d. Section of this Chapter, the wears are made Debtor to Cash for the sum paid in part, and To R. Nicholson for what resteth due to him. So that( if the account of Norwich wears was not before entred in the Ledger;) I turn to ( N) in the Alphabet or Index; and because I find the 2d. Folio a proper place( there being room) to enter the same: I writ in the said Alphabet[ Norwich wears Fol. 2.] Then on that Folio on the left-hand page., I writ[ Norwich wears Debtor] in a fair Italian or set Roman hand, To sundry accounts Fol. 1, 2, for the whole Value of the wears— 109 l. 4s. i. e. To Cash on Fol. 1, for 80 l. paid in Ready-money, for which sum you must likewise turn to the account of Cash, and make[ Cash Creditor] thereby; and to R. Nicholson, on Fol. 2. for the 29 l. 4 s. due to him, for which sum he must have Credit given him: Therefore turn to R. Nicholson's account by the Alphabet as before, or if the same is not entred, you may do it in the Alphabet, and in some convenient place of the Ledger, as is taught of the Norwich wears, making[ Richard Nicholson Creditor] By Norwich wears 29 l. 4 s. referring if you please to Fol. 1. where Norwich wears stand: But if you think referring to the folios from one account to another is too difficult, you may omit it, making Cash Debtor in the Ledger To, and Cr. By sundry accounts as per Cash Book, and you may easily refer from one account to another by the Day of the Month, &c. as if Norwich wears the 4th. of July is Debtor To Richard Nicholson, Richard Nicholson the same Day of the Month will stand Creditor, By Norwich wears, and the contrary, and by the Day of the Month you may likewise find any account in the Waste-book, Journal, &c. for the particulars of any account in the Ledger, which is there but Entered in short. § 8. Directions for Closing an account. The Closing an account is always in Order to the Balance of it, and is done either by Profit and Loss, or Balance, being thus performed. I shall instance in two of the Examples foregoing, by which you will easily have a right Notion of Closing an account. EXAMPLE 1. In the account of Cash foregoing when I come to Close, Even, or End the account, in Order to Balance I find( by summing up the Debtor and Creditor Sides) that I have received more Cash than I have paid by 2505 l. 06 s. therefore I close the account of Cash, by making Cash Creditor by Balance for 2505 l. 06 s. remaining in Chest. EXAMPLE 2. In the account of Norwich wears: I find the Debtor-side to be 109 l. 04 s. and the Creditor-side 135 l. 16 s. i. e. That I have sold the Goods for more than they cost me by 26 l. 12 s. which is my Gain, Therefore Profit and Loss must be made Creditor by Norwich wears 26 l. 12 s. and consequently Norwich wears must be made Debtor to Profit and Loss 26 l. 12 s. which closes the account. But if the Excess had been on the other Side, that is to say, That I had not sold the Goods for so much as I gave for them: Then Profit and Loss will be Debtor to Norwich wears, and Norwich wears Creditor by Profit and Loss, so much as Lost by the account: So that all accounts of wears are closed by Profit and Loss provided the wears are all sold; but if they are not the account of wears must be always made Creditor by Balance for the wears, remaining unsold, and then closed by the account of Profit and Loss, as the account of drugs foregoing. Note, that in an account of Men: If the Creditor-side exceed the debtor: Then am I indebted to that Man, and the account must be closed by making him Debtor to Balance for so much as is due to him, which is so much as the Creditor side exceeds the Debtor. § 9. Directions for Balancing your accounts. In closing an account you Balance that particular account, but when you would balance all your accounts to see what you are worth, or what you have, and what you owe, do thus: Having closed your particular accounts, except Stock, and Profit and Loss; Take a clean Sheet of Paper, and on the Left-hand Folio make[ Balance Debtor] and on the other Side [ per contra Creditor] Then begin at the Beginning with Cash, as in the foregoing Balance, making Balance Debtor to Cash for so much remaining in Chest. 2dly. I come to the Norwich wears, and find the account closed by the foregoing Rule, with Debtor to Profit and Loss: Therefore I enter on the Creditor-side of that account[ Profit sand Loss Creditor by Norwich wears 26 l. 12 s. 3dly. In the Voyage to Aleppo, I enter Baiance Debtor to Voyage To Aleppo for the Goods unsold at Aleppo, or in any account of wears, the Balance must be always made Debtor to the wears unsold. 4thly. In the account of G. Gainwell my account Currant. Because I find my Factor has not returned the Money for wears that he has sold for my account by 49 l. I make Balance Dr. To G. Gainwell my account Currant for that sum. And in short, Balance is made Debtor to all accounts for the sum that such account is made Creditor by Balance; and Balance is made Creditor by all accounts for the sum that such Account is made Debtor to Balance: And Profit and Loss is made Debtor and Creditor in like Manner; To and by the accounts closed with Profit and Loss. And having closed these accounts, and entered the same in the account of Balance, as taught before: Close the account of Profit and Loss, by making the same Dr. To Stock, for so much as the Creditor side exceeds the Debtor, and the Contrary, which Contrary seldom happeneth, for few that are careful in their Business, Trade, and gain nothing. Then carry the Foot of the account of Profit and Loss( if Gain) To the Creditor-side of Stock, if Loss, to the Debtor-side. Then close the account of Stock, as before taught for other accounts, and make Balance Dr. To or Cr. By the Excess of the Dr. or Cr. Side of Stock, as taught above, and in the Example foregoing of Stock, and Balance; and last of all, sum up the Dr. and Cr. Sides of Balance, and if the sums are equal, your Books have been rightly kept, otherwise not. Note, That in the account of Stock, the sum you owed when you begun Trade, and your present Stock, will always Balance your former Stock, and what you have gained by Trading, if your accounts have been well kept. CHAP. XI. Maxims and Rules to be observed in Drawing and Accepting Bills of Exchange, Foreign, and domestic. 1. BIlls are either Foreign or domestic. 2. Foreign Bills are usually payable in London, and other parts of England, at Single, Double, or triple Usance. 3. domestic Bills are usually payable, either at Sight, or some Number of Days after. 4. A Foreign Bill payable at Usance here in London, is payable a Month and three Days( according to the custom of London) after the Date of the Bill, allowing for the 10 Days of the Month, which Foreigners usually reckon before us; as if a Bill at Amsterdam, Rotterdam, &c. is drawn upon any Person in London, payable to me at Double Usance, which Bill is dated the 12th. of August 1694. This Bill is payably to me two Months and three Days after the Date of the Bill: i.e. The Bill being dated August the 12th. one Usance( or Month) after is September the 12th. and two Usance is October the 12th. from which deducting 10 Days( which they reckon before us in new style) and the remainder is the 2d. of October, to which add three Days( allowed according to the custom of London, over and above the Usance) and the sum is October the 5th. before the Sun going down of which Day the Bill is to be paid. And, 5. If a Foreign Bill is not paid when due, it must be protested in the Office of a public Notary, who protesteth against the Drawer, he on whom it is drawn, &c. for all Charges, Re-charges, and Interest to be paid by them. 6. After the Bill is Protested, the Protest and Bill is Registered, and then the Protest is return'd; but 'tis usual in kindness to him on whom it is drawn, to keep the Bill 3 or 4 Days longer. 7. If the Bill is not yet paid, it is usual to go upon the Exchange to see if any Body will pay the said Bill, for the Honour of the Drawer. 8. If any one is found that will pay it, he must likewise pay you the Charge of the Protest( which is about 2 s. 6 d.) and also the Interest, and other Charges, which he afterwards Charges on the Drawer. 9. But if no one be found that will pay it, then the Bill must be returned with the Charges, Interest, &c. to the Drawer. 10. The Allowance for Payment over and above Usance, is different, according to the Country. As Days. At London 3 Is allowed after the Single, Double, &c. Usance. Roterdam 6 rouen 5 Paris 10 Hamburgh 12 Antwerp 14 11. Though Usance generally signifieth a Month in Bills, drawn To and from London, yet from Venice to London Single Usance is three Months. 12. When you have Money to receive from a Foreign Correspondent, you are to make your Case known to an Exchange-Broker, who will procure Persons that will pay you your Money here; you giving them your Bill for the like sum payable to their Order by your Correspondent, and in this Case you are to inquire how the Exchange goes to such a place where the Money is payable, and make your Bargain as to the Exchange as well as you can; which having done, draw your Bill, mentioning the Sterling Coin, at so much Foreign Coin, for so much Sterling, as by the Tables of Exchange in Chap. 9. 13. A domestic Bill that is payable at sight, is not payable till three Days after the Person on whom it is drawn, seeth it. 14. If a Bill is accepted, the Accepter is become Debtor to him to whom the Bill is payable. And 15. If a Bill is accepted, and not paid in time, he to whom it is payable, may, by the Law of Merchants, seize the Goods of the Accepter. 16. When a Bill or Note for Money is made payable to another or Order; if the Person to whom it is payable goes not in Person to receive the Money, he must writ his Order on the Back-side of the Bill or Note, thus: I Order the Bearer A. B. to receive the Content of this Note, or Bill, And afterwards subscribe your Name. 17. When any one draws a Bill payable to another, the Drawer ought at the same opportunity, to give advice to him by whom it is payable, that he has drawn a Bill on him payable to such a Person, at such a time, for such a sum, for the avoiding all suspicion of Deceit in counterfeiting the Drawer's Hand, &c. 18. When part of the Content of a Note, &c. is onely required to be paid, the sum paid in part must be endorsed on the Back-side of that part most wrote on, as across the Middle, &c. that so the Endorsement cannot be cut off without defacing the Bill. 19. If you draw a Bill on any one that is indebted to you, and it be not paid in that time, which you think it might reasonably be: you must draw a second Bill on him, mentioning it in the Bill to be your second, third, &c. Bill payable to such a Person, &c. The Form of an Inland Bill. Norwich, July the 14th, 1694. AT four Days sight pay Mr. Henry Molyneux, or his Order, Three hundred Pounds, for the Value received here of Ralph Rich, and place it to account, as per Advice from Your Humble Servant, Matthew Mount. To Mr. Tho. Telfast Merchant in London. If this Bill is not paid, draw a Second, thus. Norwich, July the 14th, 1694. AT four Days sight pay this my second Bill of Exchange,( my first not being paid) to Mr. H. M, &c. A Foreign Bill. London, July the 14th, 1694. for 601 l. 4s. 3d. Sterling, 2 Usance at 33 s. Flemish, for 20 s. Sterling. AT Double Usance pay this my first Bill of Exchange unto John Vandersteagen, or his Order, Six hundred and one Pounds four Shillings, Three-pence Sterling, at Thirty three Shillings Flemish for one Pound Sterling, for the Value received here of James Langrique, and pass it to account, as per Advice from Your Friend and Servant, Timothy Trustnone. To Mr. Daniel Denderdorp, Merchant in Antwerp. FINIS.